CODATA recommended values of the fundamental … · II. Special Quantities ... REVIEWS OF MODERN...

98
CODATA recommended values of the fundamental physical constants: 2006 * Peter J. Mohr, Barry N. Taylor, and David B. Newell § National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8420, USA Published 6 June 2008 This paper gives the 2006 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology CODATA for international use. Further, it describes in detail the adjustment of the values of the constants, including the selection of the final set of input data based on the results of least-squares analyses. The 2006 adjustment takes into account the data considered in the 2002 adjustment as well as the data that became available between 31 December 2002, the closing date of that adjustment, and 31 December 2006, the closing date of the new adjustment. The new data have led to a significant reduction in the uncertainties of many recommended values. The 2006 set replaces the previously recommended 2002 CODATA set and may also be found on the World Wide Web at physics.nist.gov/constants. DOI: 10.1103/RevModPhys.80.633 PACS numbers: 06.20.Jr, 12.20.m CONTENTS I. Introduction 635 A. Background 635 B. Time variation of the constants 635 C. Outline of paper 636 II. Special Quantities and Units 636 Relative Atomic Masses 637 A. Relative atomic masses of atoms 637 B. Relative atomic masses of ions and nuclei 638 C. Cyclotron resonance measurement of the electron relative atomic mass A r e 639 V. Atomic Transition Frequencies 639 A. Hydrogen and deuterium transition frequencies, the Rydberg constant R , and the proton and deuteron charge radii R p , R d 639 1. Theory relevant to the Rydberg constant 639 a. Dirac eigenvalue 639 b. Relativistic recoil 640 c. Nuclear polarization 640 d. Self energy 640 e. Vacuum polarization 641 f. Two-photon corrections 642 g. Three-photon corrections 644 h. Finite nuclear size 644 i. Nuclear-size correction to self energy and vacuum polarization 645 j. Radiative-recoil corrections 645 k. Nucleus self energy 645 l. Total energy and uncertainty 645 m. Transition frequencies between levels with n =2 646 2. Experiments on hydrogen and deuterium 646 3. Nuclear radii 647 B. Antiprotonic helium transition frequencies and A r e 647 1. Theory relevant to antiprotonic helium 648 2. Experiments on antiprotonic helium 649 3. Values of A r e inferred from antiprotonic helium 650 C. Hyperfine structure and fine structure 650 1. Hyperfine structure 650 2. Fine structure 651 V. Magnetic Moment Anomalies and g-Factors 651 III. I * This report was prepared by the authors under the auspices of the CODATA Task Group on Fundamental Constants. The members of the task group are: F. Cabiati, Istituto Nazionale di Ricerca Metrologica, Italy K. Fujii, National Metrology Institute of Japan, Japan S. G. Karshenboim, D. I. Mendeleyev All-Russian Research Institute for Metrology, Russian Federation I. Lindgren, Chalmers University of Technology and Göteborg University, Sweden B. A. Mamyrin deceased, A. F. Ioffe Physical-Technical In- stitute, Russian Federation W. Martienssen, Johann Wolfgang Goethe-Universität, Ger- many P. J. Mohr, National Institute of Standards and Technology, United States of America D. B. Newell, National Institute of Standards and Technology, United States of America F. Nez, Laboratoire Kastler-Brossel, France B. W. Petley, National Physical Laboratory, United Kingdom T. J. Quinn, Bureau international des poids et mesures B. N. Taylor, National Institute of Standards and Technology, United States of America W. Wöger, Physikalisch-Technische Bundesanstalt, Germany B. M. Wood, National Research Council, Canada Z. Zhang, National Institute of Metrology, China People’s Re- public of This review is being published simultaneously by the Journal of Physical and Chemical Reference Data. [email protected] [email protected] § [email protected] REVIEWS OF MODERN PHYSICS, VOLUME 80, APRIL–JUNE 2008 0034-6861/2008/802/63398 ©2008 The American Physical Society 633

Transcript of CODATA recommended values of the fundamental … · II. Special Quantities ... REVIEWS OF MODERN...

CODATA recommended values of the fundamental physical constants:2006*

Peter J. Mohr,† Barry N. Taylor,‡ and David B. Newell§

National Institute of Standards and Technology, Gaithersburg,Maryland 20899-8420, USA

Published 6 June 2008

This paper gives the 2006 self-consistent set of values of the basic constants and conversion factors ofphysics and chemistry recommended by the Committee on Data for Science and TechnologyCODATA for international use. Further, it describes in detail the adjustment of the values of theconstants, including the selection of the final set of input data based on the results of least-squaresanalyses. The 2006 adjustment takes into account the data considered in the 2002 adjustment as wellas the data that became available between 31 December 2002, the closing date of that adjustment,and 31 December 2006, the closing date of the new adjustment. The new data have led to a significantreduction in the uncertainties of many recommended values. The 2006 set replaces the previouslyrecommended 2002 CODATA set and may also be found on the World Wide Web atphysics.nist.gov/constants.

DOI: 10.1103/RevModPhys.80.633 PACS numbers: 06.20.Jr, 12.20.m

CONTENTS

I. Introduction 635

A. Background 635

B. Time variation of the constants 635

C. Outline of paper 636

II. Special Quantities and Units 636

Relative Atomic Masses 637

A. Relative atomic masses of atoms 637

B. Relative atomic masses of ions and nuclei 638

C. Cyclotron resonance measurement of the electron

relative atomic mass Are 639

V. Atomic Transition Frequencies 639

A. Hydrogen and deuterium transition frequencies, the

Rydberg constant R, and the proton and deuteron

charge radii Rp, Rd 639

1. Theory relevant to the Rydberg constant 639

a. Dirac eigenvalue 639

b. Relativistic recoil 640

c. Nuclear polarization 640

d. Self energy 640

e. Vacuum polarization 641

f. Two-photon corrections 642

g. Three-photon corrections 644

h. Finite nuclear size 644

i. Nuclear-size correction to self energy and

vacuum polarization 645

j. Radiative-recoil corrections 645

k. Nucleus self energy 645

l. Total energy and uncertainty 645

m. Transition frequencies between levels with

n=2 646

2. Experiments on hydrogen and deuterium 646

3. Nuclear radii 647

B. Antiprotonic helium transition frequencies and Are 647

1. Theory relevant to antiprotonic helium 648

2. Experiments on antiprotonic helium 649

3. Values of Are inferred from antiprotonic

helium 650

C. Hyperfine structure and fine structure 650

1. Hyperfine structure 650

2. Fine structure 651

V. Magnetic Moment Anomalies and g-Factors 651

III.

I

*This report was prepared by the authors under the auspicesof the CODATA Task Group on Fundamental Constants. Themembers of the task group are:F. Cabiati, Istituto Nazionale di Ricerca Metrologica, ItalyK. Fujii, National Metrology Institute of Japan, JapanS. G. Karshenboim, D. I. Mendeleyev All-Russian ResearchInstitute for Metrology, Russian FederationI. Lindgren, Chalmers University of Technology and GöteborgUniversity, SwedenB. A. Mamyrin deceased, A. F. Ioffe Physical-Technical In-stitute, Russian FederationW. Martienssen, Johann Wolfgang Goethe-Universität, Ger-manyP. J. Mohr, National Institute of Standards and Technology,United States of AmericaD. B. Newell, National Institute of Standards and Technology,United States of AmericaF. Nez, Laboratoire Kastler-Brossel, FranceB. W. Petley, National Physical Laboratory, United KingdomT. J. Quinn, Bureau international des poids et mesuresB. N. Taylor, National Institute of Standards and Technology,United States of AmericaW. Wöger, Physikalisch-Technische Bundesanstalt, GermanyB. M. Wood, National Research Council, CanadaZ. Zhang, National Institute of Metrology, China People’s Re-public ofThis review is being published simultaneously by the Journalof Physical and Chemical Reference Data.

[email protected][email protected]§[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 80, APRIL–JUNE 2008

0034-6861/2008/802/63398 ©2008 The American Physical Society633

A. Electron magnetic moment anomaly ae and thefine-structure constant 652

1. Theory of ae 6522. Measurements of ae 654

a. University of Washington 654b. Harvard University 654

3. Values of inferred from ae 654B. Muon magnetic moment anomaly a 655

1. Theory of a 6552. Measurement of a: Brookhaven 656

a. Theoretical value of a and inferred valueof 657

C. Bound electron g-factor in 12C5+ and in 16O7+ andAre 657

1. Theory of the bound electron g-factor 6582. Measurements of ge12C5+ and ge16O7+ 661

a. Experiment on g 12e C5+ and inferred value

of Are 661b. Experiment on ge16O7+ and inferred value

of Are 661c. Relations between ge12C5+ and ge16O7+ 662

VI. Magnetic Moment Ratios and the Muon-Electron MassRatio 662A. Magnetic moment ratios 662

1. Theoretical ratios of atomic bound-particleto free-particle g-factors 662

2. Ratio measurements 663a. Electron to proton magnetic moment ratio

e /p 663b. Deuteron to electron magnetic moment

ratio d /e 664c. Proton to deuteron and triton to proton

magnetic moment ratios p /d and t /p 664d. Electron to shielded proton magnetic

moment ratio e /p 665e. Shielded helion to shielded proton

magnetic moment ratio h /p 666f. Neutron to shielded proton magnetic

moment ratio n /p 666B. Muonium transition frequencies, the muon-proton

magnetic moment ratio /p, and muon-electronmass ratio m /me 6661. Theory of the muonium ground-state

hyperfine splitting 6662. Measurements of muonium transition

frequencies and values of /p and m /me 668a. LAMPF 1982 668b. LAMPF 1999 668c. Combined LAMPF results 669

Electrical Measurements 669. Shielded gyromagnetic ratios , the fine-structure

constant , and the Planck constant h 6691. Low-field measurements 670

a. NIST: Low field 670b. NIM: Low field 670c. KRISS/VNIIM: Low field 670

2. High-field measurements 671a. NIM: High field 671b. NPL: High field 671

B. von Klitzing constant RK and 671

VII.A

1. NIST: Calculable capacitor 6712. NMI: Calculable capacitor 6723. NPL: Calculable capacitor 6724. NIM: Calculable capacitor 6725. LNE: Calculable capacitor 672

C. Josephson constant KJ and h 6721. NMI: Hg electrometer 6722. PTB: Capacitor voltage balance 673

D. Product K2JRK and h 673

1. NPL: Watt balance 6732. NIST: Watt balance 674

a. 1998 measurement 674b. 2007 measurement 674

3. Other values 6754. Inferred value of KJ 675

E. Faraday constant F and h 6751. NIST: Ag coulometer 676

VIII. Measurements Involving Silicon Crystals 676A. 220 lattice spacing of silicon d220 676

1. X-ray optical interferometer measurementsof d220X 677

a. PTB measurement of d220W4.2a 678b. NMIJ measurement of d220NR3 678c. INRIM measurement of d220W4.2a and

d220MO* 6782. d220 difference measurements 679

a. NIST difference measurements 679b. PTB difference measurements 680

B. Molar volume of silicon VmSi and the Avogadroconstant NA 680

C. Gamma-ray determination of the neutron relativeatomic mass Arn 681

D. Quotient of Planck constant and particle massh /mX and 6821. Quotient h /mn 6822. Quotient h /m133Cs 6823. Quotient h /m87Rb 683

IX. Thermal Physical Quantities 684A. Molar gas constant R 684

1. NIST: Speed of sound in argon 6852. NPL: Speed of sound in argon 6853. Other values 685

B. Boltzmann constant k 685C. Stefan-Boltzmann constant 686

X. Newtonian Constant of Gravitation G 686A. Updated values 686

1. Huazhong University of Science andTechnology 686

2. University of Zurich 687B. Determination of 2006 recommended value of G 688C. Prospective values 689

XI. X-ray and Electroweak Quantities 689A. X-ray units 689B. Particle Data Group input 689

XII. Analysis of Data 690A. Comparison of data 690B. Multivariate analysis of data 699

1. Summary of adjustments 6992. Test of the Josephson and quantum Hall

effect relations 704

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XIII. The 2006 CODATA Recommended Values 706A. Calculational details 706B. Tables of values 707

XIV. Summary and Conclusion 716A. Comparison of 2006 and 2002 CODATA

recommended values 716B. Some implications of the 2006 CODATA

recommended values and adjustment for physics andmetrology 719

C. Outlook and suggestions for future work 721Acknowledgments 722Nomenclature 722References 724

I. INTRODUCTION

A. Background

This paper gives the complete 2006 CODATA self-consistent set of recommended values of the fundamen-tal physical constants and describes in detail the 2006least-squares adjustment, including the selection of thefinal set of input data based on the results of least-squares analyses. Prepared under the auspices of theCODATA Task Group on Fundamental Constants, thisis the fifth such report of the Task Group since its estab-lishment in 19691 and the third in the four-year cycle ofreports begun in 1998. The 2006 set of recommendedvalues replaces its immediate predecessor, the 2002 set.The closing date for the availability of the data consid-ered for inclusion in this adjustment was 31 December2006. As a consequence of the new data that becameavailable in the intervening four years, there has been asignificant reduction of the uncertainty of many con-stants. The 2006 set of recommended values first becameavailable on 29 March 2007 at http://physics.nist.gov/constants, a web site of the NIST Fundamental Con-stants Data Center FCDC.

The 1998 and 2002 reports describing the 1998 and2002 adjustments Mohr and Taylor, 2000, 2005, re-ferred to as CODATA-98 and CODATA-02 throughoutthis article, describe in detail much of the currentlyavailable data, its analysis, and the techniques used toobtain a set of best values of the constants using thestandard method of least squares for correlated inputdata. This paper focuses mainly on the new informationthat has become available since 31 December 2002 andreferences the discussions in CODATA-98 andCODATA-02 for earlier work in the interest of brevity.More specifically, if a potential input datum is not dis-cussed in detail, the reader can assume that it or aclosely related datum has been reviewed in eitherCODATA-98 or CODATA-02.

The reader is also referred to these papers for a dis-cussion of the motivation for and the philosophy behind

the periodic adjustment of the values of the constantsand for descriptions of how units, quantity symbols, nu-merical values, numerical calculations, and uncertaintiesare treated, in addition to how the data are character-ized, selected, and evaluated. Since the calculations arecarried out with more significant figures than are dis-played in the text to avoid rounding errors, data withmore digits are available on the FCDC web site for pos-sible independent analysis.

However, because of their importance, we recall indetail the following two points also discussed in thesereferences. First, although it is generally agreed that thecorrectness and overall consistency of the basic theoriesand experimental methods of physics can be tested bycomparing values of particular fundamental constantsobtained from widely differing experiments, throughoutthis adjustment, as a working principle, we assume thevalidity of the underlying physical theory. This includesspecial relativity, quantum mechanics, quantum electro-dynamics QED, the standard model of particle physics,including combined charge conjugation, parity inversion,and time reversal CPT invariance, and the theory ofthe Josephson and quantum Hall effects, especially theexactness of the relationships between the Josephsonand von Klitzing constants KJ and RK and the elemen-tary charge e and Planck constant h. In fact, tests ofthese relations K 2

J=2e /h and RK=h /e using the inputdata of the 2006 adjustment are discussed in Sec.XII.B.2.

The second point has to do with the 31 December2006 closing date for data to be considered for inclusionin the 2006 adjustment. A datum was considered to havemet this date, even though not yet reported in an archi-val journal, as long as a description of the work wasavailable that allowed the Task Group to assign a validstandard uncertainty uxi to the datum. Thus, any inputdatum labeled with an 07 identifier because it was pub-lished in 2007 was, in fact, available by the cutoff date.Also, some references to results that became availableafter the deadline are included, even though the resultswere not used in the adjustment.

B. Time variation of the constants

This subject, which was briefly touched upon inCODATA-02, continues to be an active field of experi-mental and theoretical research, because of its impor-tance to our understanding of physics at the most funda-mental level. Indeed, a large number of papers relevantto the topic have appeared in the last four years; seethe FCDC bibliographic database on the fundamen-tal constants using the keyword time variationat http://physics.nist.gov/constantsbib. For example, seeFortier et al. 2007 and Lea 2007. However, there hasbeen no laboratory observation of the time dependenceof any constant that might be relevant to the recom-mended values.

1The Committee on Data for Science and Technology wasestablished in 1966 as an interdisciplinary committee of theInternational Council for Science.

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C. Outline of paper

Section II touches on special quantities and units, thatis, those that have exact values by definition.

Sections III–XI review all available experimental andtheoretical data that might be relevant to the 2006 ad-justment of the values of the constants. As discussed inAppendix E of CODATA-98, in a least-squares analysisof the fundamental constants, the numerical data, bothexperimental and theoretical, also called observationaldata or input data, are expressed as functions of a set ofindependent variables called adjusted constants. Thefunctions that relate the input data to the adjusted con-stants are called observational equations, and the least-squares procedure provides best estimated values, in theleast-squares sense, of the adjusted constants. The focusof the review-of-data sections is thus the identificationand discussion of the input data and observational equa-tions of interest for the 2006 adjustment. Althoughnot all observational equations that we use are expli-citly given in the text, all are summarized in TablesXXXVIII, XL, and XLII of Sec. XII.B.

As part of our discussion of a particular datum, weoften deduce from it an inferred value of a constant,such as the fine-structure constant or Planck constanth. It should be understood, however, that these inferredvalues are for comparison purposes only; the datumfrom which it is obtained, not the inferred value, is theinput datum in the adjustment.

Although just four years separate the 31 Decemberclosing dates of the 2002 and 2006 adjustments, there area number of important new results to consider. Experi-mental advances include the 2003 Atomic Mass Evalua-tion from the Atomic Mass Data Center AMDC,which provides new values for the relative atomicmasses ArX of a number of relevant atoms; a newvalue of the electron magnetic moment anomaly ae frommeasurements on a single electron in a cylindrical Pen-ning trap, which provides a value of the fine-structureconstant ; better measurements of the relative atomicmasses of 2H, 3H, and 4He; new measurements of tran-sition frequencies in antiprotonic helium pAHe+ atomthat provide a competitive value of the relative atomicmass of the electron Are; improved measurements ofthe nuclear magnetic resonance NMR frequencies ofthe proton and deuteron in the HD molecule and of theproton and triton in the HT molecule; a highly accuratevalue of the Planck constant obtained from an improvedmeasurement of the product K2

JRK using a moving-coilwatt balance; new results using combined x-ray and op-tical interferometers for the 220 lattice spacing ofsingle crystals of naturally occurring silicon; and an ac-curate value of the quotient h /m87Rb obtained bymeasuring the recoil velocity of rubidium-87 atoms uponabsorption or emission of photons—a result that pro-vides an accurate value of that is virtually independentof the electron magnetic moment anomaly.

Theoretical advances include improvements in certainaspects of the theory of the energy levels of hydrogen

and deuterium; improvements in the theory of antipro-tonic helium transition frequencies that, together withthe new transition frequency measurements, have led tothe aforementioned competitive value of Are; a newtheoretical expression for ae that, together with the newexperimental value of ae, has led to the aforementionedvalue of ; improvements in the theory for the g-factorof the bound electron in hydrogenic ions with nuclearspin quantum number i=0 relevant to the determinationof Are; and improved theory of the ground-state hy-perfine splitting of muonium Mu the +e− atom.

Section XII describes the analysis of the data, with theexception of the Newtonian constant of gravitation,which is analyzed in Sec. X. The consistency of the dataand potential impact on the determination of the 2006recommended values were appraised by comparingmeasured values of the same quantity, comparing mea-sured values of different quantities through inferred val-ues of a third quantity such as or h, and finally byusing the method of least squares. Based on these inves-tigations, the final set of input data used in the 2006adjustment was selected.

Section XIII provides, in several tables, the 2006CODATA recommended values of the basic constantsand conversion factors of physics and chemistry, includ-ing the covariance matrix of a selected group of con-stants.

Section XIV concludes the paper with a comparisonof the 2006 and 2002 recommended values of the con-stants, a survey of implications for physics and metrol-ogy of the 2006 values and adjustment, and suggestionsfor future work that can advance our knowledge of thevalues of the constants.

II. SPECIAL QUANTITIES AND UNITS

Table I lists those quantities whose numerical valuesare exactly defined. In the International System of UnitsSI BIPM, 2006, used throughout this paper, the defi-nition of the meter fixes the speed of light in vacuum c,the definition of the ampere fixes the magnetic constantalso called the permeability of vacuum 0, and thedefinition of the mole fixes the molar mass of the carbon12 atom M12C to have the exact values given in thetable. Since the electric constant also called the permit-tivity of vacuum is related to 0 by 0=1/0c2, it too isknown exactly.

The relative atomic mass ArX of an entity X is de-fined by ArX=mX /mu, where mX is the mass of Xand mu is the atomic mass constant defined by

mu = 1 m12C = 1 u 1.66 −2712 10 kg, 1

where m12C is the mass of the carbon 12 atom and u isthe unified atomic mass unit also called the dalton, Da.Clearly, ArX is a dimensionless quantity and Ar

12C=12 exactly. The molar mass MX of entity X, which isthe mass of one mole of X with SI unit kg/mol, is givenby

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MX = NAmX = ArXMu, 2

where NA6.021023/mol is the Avogadro constantand Mu=10−3 kg/mol is the molar mass constant. Thenumerical value of NA is the number of entities in onemole, and since the definition of the mole states that onemole contains the same number of entities as there arein 0.012 kg of carbon 12, M12C=0.012 kg/mol exactly.

The Josephson and quantum Hall effects have playedand continue to play important roles in adjustments ofthe values of the constants, because the Josephson andvon Klitzing constants KJ and RK, which underlie thesetwo effects, are related to e and h by

2e h cKJ = , R 0

K =e2 = . 3

h 2

Although we assume these relations are exact, and noevidence—either theoretical or experimental—has beenput forward that challenges this assumption, the conse-quences of relaxing it are explored in Sec. XII.B.2. Somereferences to recent work related to the Josephson andquantum Hall effects may be found in the FCDC biblio-graphic database see Sec. I.B.

The next-to-last two entries in Table I are the conven-tional values of the Josephson and von Klitzing con-stants adopted by the International Committee forWeights and Measures CIPM and introduced on Janu-ary 1, 1990 to establish worldwide uniformity in themeasurement of electrical quantities. In this paper,all electrical quantities are expressed in SI units. How-ever, those measured in terms of the Josephson andquantum Hall effects with the assumption that KJ andRK have these conventional values are labeled with asubscript 90.

For high-accuracy experiments involving the force ofgravity, such as the watt balance, an accurate measure-ment of the local acceleration of free fall at the site ofthe experiment is required. Fortunately, portable andeasy-to-use commercial absolute gravimeters are avail-able that can provide a local value of g with a relativestandard uncertainty of a few parts in 109. That theseinstruments can achieve such a small uncertainty if prop-erly used is demonstrated by a periodic internationalcomparison of absolute gravimeters ICAG carried outat the International Bureau of Weights and Measures

BIPM, Sèvres, France; the seventh and most recent,denoted ICAG-2005, was completed in September 2005Vitushkin, 2007; the next is scheduled for 2009. In thefuture, atom interferometry or Bloch oscillations usingultracold atoms could provide a competitive or possiblymore accurate method for determining a local value of gPeters et al., 2001;2005.

McGuirk et al., 2002; Cladé et al.,

III. RELATIVE ATOMIC MASSES

Included in the set of adjusted constants are the rela-tive atomic masses ArX of a number of particles, at-oms, and ions. Tables II–VI and the following sectionssummarize the relevant data.

A. Relative atomic masses of atoms

Most values of the relative atomic masses of neutralatoms used in this adjustment are taken from the 2003atomic mass evaluation AME2003 of the Atomic MassData Center, Centre de Spectrométrie Nucléaire et deSpectrométrie de Masse CSNSM, Orsay, France Audiet al., 2003; Wapstra et al., 2003; AMDC, 2006. The re-sults of AME2003 supersede those of both the 1993atomic mass evaluation and the 1995 update. Table IIlists the values from AME2003 of interest here, whileTable III gives the covariance for hydrogen and deute-rium AMDC, 2003. Other non-negligible covariancesof these values are discussed in the appropriate sections.

Table IV gives six values of ArX relevant to the 2006adjustment reported since the completion and publica-tion of AME2003 in late 2003 that we use in place of thecorresponding values in Table II.

The 3H and 3He values are those reported by theSMILETRAP group at the Manne Siegbahn LaboratoryMSL, Stockholm, Sweden Nagy et al., 2006 using aPenning trap and a time-of-flight technique to detect cy-clotron resonances. This new 3He result is in goodagreement with a more accurate, but still preliminary,result from the University of Washington group in Se-attle, WA, USA Van Dyck, 2006. The AME2003 valuesfor 3H and 3He were influenced by an earlier result for

Value

2 458 m s−1

0−7 N A−2=12.566 370 614. . . −710 N A−2

−1=8.854 187 817. . . −110 2 F m−1

g mol−1

0−3 kg mol−1

conventional value of Josephson constant K −1J−90 483 597.9 GHz V

conventional value of von Klitzing constant RK−90 25 812.807

TABLE I. Some exact quantities relevant to the 2006 adjustment.

Quantity Symbol

speed of light in vacuum c, c0 299 79magnetic constant 0 41electric constant c20 0 relative atomic mass of 12C Ar

12C 12

molar mass constant M 10−3u k

molar mass of 12C A 12CMu M12r C 121

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TABLE II. Values of the relative atomic masses of the neutronand various atoms as given in the 2003 atomic mass evaluationtogether with the defined value for 12C.

Relative atomic Relative standacertainty ur

610−10

010−10

rdAtom mass ArX un

n 1.008 664 915 7456 5.1H 1.007 825 032 07 10 1. 2H 2.014 101 777 8536 1.810−10

3H 3.016 049 277725 8.210−10

3He 3.016 029 319126 8.610−10

4He 4.002 603 254 15363 1.610−11

12C 12 Exact16O 15.994 914 619 5616 1.010−11

28Si 27.976 926 532519 6.910−11

29Si 28.976 494 70022 −107.61030Si 29.973 770 17132 1.110−9

36Ar 35.967 545 10528 7.810−10

38Ar 37.962 732 3936 9.510−9

40Ar 39.962 383 122529 7.210−11

87Rb 86.909 180 52612 1.410−10

107Ag 106.905 096846 4.310−8

109Ag 108.904 752331 −82.910133 −10

3He from the University of Washington group, which isin disagreement with their new result.

The values for 4He and 16O are those reported by theUniversity of Washington group Van Dyck et al., 2006using their improved mass spectrometer; they are basedon a thorough reanalysis of data that yielded prelimi-nary results for these atoms that were used inAME2003. They include an experimentally determinedimage-charge correction with a relative standard uncer-tainty ur=7.910−12 in the case of 4He and ur=4.010−12 in the case of 16O. The value of Ar

2H is alsofrom this group and is a near-final result based on theanalysis of ten runs carried out over a 4 year periodVan Dyck, 2006. Because the result is not yet final, thetotal uncertainty is conservatively assigned; ur=9.910−12 for the image-charge correction. This value ofAr

2H is consistent with the preliminary value reported

by Van Dyck et al. 2006 based on the analysis of onlythree runs.

The covariance and correlation coefficient of Ar3H

and Ar3He given in Table V are due to the common

component of uncertainty ur=1.410−10 of the relativeatomic mass of the H +

2 reference ion used in theSMILETRAP measurements; the covariances and cor-relation coefficients of the University of Washington val-ues of A 2H, A 4He, and A 16

r r r O given in Table VIare due to the uncertainties of the image-charge correc-tions, which are based on the same experimentally de-termined relation.

The 29Si value is that implied by the ratioAr

29Si+ /Ar28Si H+=0.999 715 124 181265 obtained

at the Massachusetts Institute of Technology MIT,Cambridge, MA, USA, using a recently developed tech-nique of determining mass ratios by directly comparingthe cyclotron frequencies of two different ions simulta-neously confined in a Penning trap Rainville et al.,2005. The relative atomic mass work of the MIT grouphas now been transferred to Florida State University,Tallahassee, FL, USA. This approach eliminates manycomponents of uncertainty arising from systematic ef-fects. The value for Ar

29Si is given in the supplemen-tary information to Rainville et al. 2005 and has a sig-nificantly smaller uncertainty than the correspondingAME2003 value.

B. Relative atomic masses of ions and nuclei

The relative atomic mass ArX of a neutral atom X isgiven in terms of the relative atomic mass of an ion ofthe atom formed by the removal of n electrons by

TABLE V. The variances, covariance, and correlation coeffi-cient of the values of the SMILETRAP relative atomic massesof tritium and helium 3. The number in bold above the maindiagonal is 1018 times the numerical value of the covariance,the numbers in bold on the main diagonal are 1018 times thenumerical values of the variances, and the number in italicsbelow the main diagonal is the correlation coefficient.

TABLE III. The variances, covariance, and correlation coeffi-cient of the AME2003 values of the relative atomic masses ofhydrogen and deuterium. The number in bold above the maindiagonal is 1018 times the numerical value of the covariance,the numbers in bold on the main diagonal are 1018 times thenumerical values of the variances, and the number in italicsbelow the main diagonal is the correlation coefficient.

A 1 2r H Ar H

Ar1H 0.0107 0.0027

Ar2H 0.0735 0.1272

Cs 132.905 451 93224 1.810

TABLE IV. Values of the relative atomic masses of variousatoms that have become available since the 2003 atomic massevaluation.

AtomRelative atomicmass ArX

Relative standarduncertainty ur

2H 2.014 101 778 04080 4.010−11

3H 3.016 049 278725 8.310−10

3He 3.016 029 321726 8.610−10

4He 4.002 603 254 13162 1.510−11

16O 15.994 914 619 5718 1.110−11

29Si 28.976 494 662520 6.910−11

Ar3H Ar

3He

Ar3H 6.2500 0.1783

Ar3He 0.0274 6.7600

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E X − E Xn+A n

rX = A X +r + nAre − b b

2 . 4muc

Here EbX /muc2 is the relative-atomic-mass equivalentof the total binding energy of the Z electrons of theatom, where Z is the atomic number proton number,and EbXn+ /muc2 is the relative-atomic-mass equivalentof the binding energy of the Z−n electrons of the Xn+

ion. For a fully stripped atom, that is, for n=Z, XZ+ is N,where N represents the nucleus of the atom, andEbXZ+ /muc2=0, which yields the first few equations ofTable XL in Sec. XII.B.

The binding energies Eb used in this work are thesame as those used in the 2002 adjustment; see Table IVof CODATA-02. For tritium, which is not includedthere, we use the value 1.097 185 439107 m−1 Ko-tochigova, 2006. The uncertainties of the binding ener-gies are negligible for our application.

C. Cyclotron resonance measurement of the electron relativeatomic mass Ar(e)

A value of Are is available from a Penning-trapmeasurement carried out by the University of Washing-ton group Farnham et al., 1995; it is used as an inputdatum in the 2006 adjustment, as it was in the 2002 ad-justment:

Are = 0.000 548 579 911112 2.1 10−9 . 5

IV. ATOMIC TRANSITION FREQUENCIES

Atomic transition frequencies in hydrogen, deute-rium, and antiprotonic helium yield information on theRydberg constant, the proton and deuteron charge radii,and the relative atomic mass of the electron. The hyper-fine splitting in hydrogen and fine-structure splitting inhelium do not yield a competitive value of any constantat the current level of accuracy of the relevant experi-ment and/or theory. All of these topics are discussed inthis section.

A. Hydrogen and deuterium transition frequencies, theRydberg constant R, and the proton and deuteron chargeradii Rp, Rd

The Rydberg constant is related to other constants bythe definition

2mecR = . 6

2h

It can be accurately determined by comparing measuredresonant frequencies of transitions in hydrogen H anddeuterium D to the theoretical expressions for the en-ergy level differences in which it is a multiplicative fac-tor.

1. Theory relevant to the Rydberg constant

The theory of the energy levels of hydrogen and deu-terium atoms relevant to the determination of the Ryd-berg constant R, based on measurements of transitionfrequencies, is summarized in this section. Complete in-formation necessary to determine the theoretical valuesof the relevant energy levels is provided, with an empha-sis on results that have become available since the pre-vious adjustment described in CODATA-02. For brevity,references to earlier work, which can be found in Eideset al. 2001b for example, are not included here.

An important consideration is that the theoretical val-ues of the energy levels of different states are highlycorrelated. For example, for S states, the uncalculatedterms are primarily of the form of an unknown commonconstant divided by n3. This fact is taken into account bycalculating covariances between energy levels in addi-tion to the uncertainties of the individual levels as dis-cussed in detail in Sec. IV.A.1.l. In order to take thesecorrelations into account, we distinguish between com-ponents of uncertainty that are proportional to 1/n3, de-noted by u0, and components of uncertainty that are es-sentially random functions of n, denoted by un.

The energy levels of hydrogenlike atoms are deter-mined mainly by the Dirac eigenvalue, QED effectssuch as self energy and vacuum polarization, and nuclearsize and motion effects, all of which are summarized inthe following sections.

a. Dirac eigenvalue

The binding energy of an electron in a static Coulombfield the external electric field of a point nucleus ofcharge Ze with infinite mass is determined predomi-nantly by the Dirac eigenvalue

E = fn,jm c2D e , 7

where

fn,j = Z2 −1/2

1 +n − 2 ,

8

n and j are the principal quantum number and total an-gular momentum of the state, respectively, and

TABLE VI. The variances, covariances, and correlation coef-ficients of the University of Washington values of the relativeatomic masses of deuterium, helium 4, and oxygen 16. Thenumbers in bold above the main diagonal are 1020 times thenumerical values of the covariances, the numbers in bold onthe main diagonal are 1020 times the numerical values of thevariances, and the numbers in italics below the main diagonalare the correlation coefficients.

Ar2H Ar

4He 16OAr

Ar2H 0.6400 0.0631 0.1276

Ar4He 0.1271 0.3844 0.2023

16OAr 0.0886 0.1813 3.2400

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1 = j + −

2 1 2 1/2

j + − Z2 . 92

Although we are interested only in the case in which thenuclear charge is e, we retain the atomic number Z inorder to indicate the nature of various terms.

Corrections to the Dirac eigenvalue that approxi-mately take into account the finite mass of the nucleusmN are included in the more general expression foratomic energy levels, which replaces Eq. 7 Barker andGlover, 1955; Sapirstein and Yennie, 1990,

m2c2

EM = Mc2 + fn,j − 1m 2 − fn,j − 12 rrc 2M

1 − 2l0 Z4m3

+ rc2 + ¯ , 10

2l + 1 2n3mN

where l is the nonrelativistic orbital angular momentumquantum number, is the angular-momentum-parityquantum number 1 = −1j−l+1/2j+ 2 , M=me+mN, andmr=memN/ me+mN is the reduced mass.

b. Relativistic recoil

Relativistic corrections to Eq. 10 associated withmotion of the nucleus are considered relativistic-recoilcorrections. The leading term, to lowest order in Z andall orders in me /mN, is Erickson, 1977; Sapirstein andYennie, 1990

m3 Z5 1 8ES = r m 2

2 l0 ln 3 ec Z −2 − ln kmemN n 3 3 0n,l

1 7 2− l0 − a −

9 3 n m2 − m2 l0

N e

m2 me 2 mNN ln −

r me ln

m

mr , 11

where

an = − 2ln 2

n n

+ 1 1+ 1 −

i=1 i 2n 1 − l0 + l0 .

ll + 12l + 1

12

To lowest order in the mass ratio, higher-order correc-tions in Z have been extensively investigated; the con-tribution of the next two orders in Z is

m 6

E = e ZR m

mN n3 ec2

D60 + D 2 −272Z ln Z + ¯ , 13

where for nS1/2 states Pachucki and Grotch, 1995; Eidesand Grotch, 1997c

D = 4 ln 2 − 760 2 14

and Melnikov and Yelkhovsky, 1999; Pachucki andKarshenboim, 1999

11D72 = − , 15

60

and for states with l1 Golosov et al., 1995; Elkhovski,1996; Jentschura

D60 = and Pachucki, 1996

ll + 13 −

n2 2

4l2 . 16− 12l + 3

In Eq. 16 and subsequent discussion, the first subscripton the coefficient of a term refers to the power of Zand the second subscript to the power of lnZ−2. Therelativistic recoil correction used in the 2006 adjustmentis based on Eqs. 11–16. The estimated uncertainty forS states is taken to be 10% of Eq. 13, and for stateswith l1 it is taken to be 1% of that equation.

Numerical values for the complete contribution of Eq.13 to all orders in Z have been obtained by Shabaevet al. 1998. Although the difference between the all-orders calculation and the truncated power series for Sstates is about three times their quoted uncertainty, thetwo results are consistent within the uncertainty as-signed here. The covariances of the theoretical valuesare calculated by assuming that the uncertainties arepredominately due to uncalculated terms proportionalto me /mN /n3.

c. Nuclear polarization

Interactions between the atomic electron and thenucleus which involve excited states of the nucleus giverise to nuclear polarization corrections. For hydrogen,we use the result Khriplovich and Sen’kov, 2000

EPH = − 0.07013h l0

n3 kHz. 17

For deuterium, the sum of the proton polarizability, theneutron polarizability Khriplovich and Sen’kov, 1998,and the dominant nuclear structure polarizability Friarand Payne, 1997a gives

EPD = − 21.378h l0

n3 kHz. 18

We assume that this effect is negligible in states ofhigher l.

d. Self energy

The one-photon electron self energy is given by

E2 Z4

SE = 3 FZ 2m n ec , 19

where

FZ = A −241 lnZ + A40 + A50Z

+ A Z 2 ln2Z −2 + A Z 2 −262 61 lnZ

+ GSEZZ2. 20

From Erickson and Yennie 1965 and earlier paperscited therein,

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TABLE VII. Bethe logarithms ln k0n , l relevant to the deter-mination of R.

n S P

1 2.984 128 5562 2.811 769 893 −0.030 016 7093 2.767 663 6124 2.749 811 840 −0.041 954 895 −0.006 740 9396 2.735 664 207 −0.008 147 2048 2.730 267 261 −0.008 785 04312 −0.009 342 954

TABLE VIII. Values of the function GSE.

4A41 =

3 l0,

4 10 1A40 = − ln k0n,l +

3 9 l0 − 1 − 2 2l + 1 l0 ,

139A50 = − 2 ln 2

32l0,

A62 = −

l0,

A61 = 4 11 + +

1 28 601+ + ln 2 − 4 ln n −

n

3 180

77 1−

45n2 l0 + 1 −n2 2 1

+ 15 3 j1/2l1

96n2 − 32ll + 1+

3n2 1 −2l − 12l2l + 12l + 22l + 3 l0 .

21

The Bethe logarithms ln k0n , l in Eq. 21 are given inTable VII Drake and Swainson, 1990.

The function GSEZ in Eq. 20 is the higher-ordercontribution in Z to the self energy, and the valuesfor GSE that we use here are listed in Table VIII. ForS and P states with n4, the values in the table arebased on direct numerical evaluations by Jentschura andMohr 2004, 2005 and Jentschura et al. 1999, 2001. Thevalues of GSE for the 6S and 8S states are based onthe low-Z limit of this function GSE0=A60 Jentschura,

Czarnecki, and Pachucki, 2005 together with extrapola-tions of complete numerical calculation results of FZsee Eq. 20 at higher Z Kotochigova and Mohr, 2006.The values of GSE for D states are from Jentschura,Kotochigova, Le Bigot, et al. 2005.

The dominant effect of the finite mass of the nucleuson the self energy correction is taken into account bymultiplying each term of FZ by the reduced-mass fac-tor mr /me3, except that the magnetic moment term−1/ 2 2l+1 in A40 is multiplied instead by the factormr /me2. In addition, the argument Z−2 of the loga-rithms is replaced by me /mrZ−2 Sapirstein andYennie, 1990.

The uncertainty of the self energy contribution to agiven level arises entirely from the uncertainty of GSElisted in Table VIII and is taken to be entirely of type un.

e. Vacuum polarization

The second-order vacuum-polarization level shift is

E2 Z4

VP = m 23 HZ

n ec , 22

where the function HZ is divided into the part corre-sponding to the Uehling potential, denoted here byH1Z, and the higher-order remainder HRZ,where

H1Z = V40 + V50Z + V61Z2 lnZ−2

+ G1VPZZ2, 23

HRZ = GRVPZZ 2 , 24

with

V40 = − 4 15 l0,

V50 = 548l0,

V61 = − 2 15 l0. 25

The part G1VPZ arises from the Uehling potential

with values given in Table IX Mohr, 1982; Kotochigovaet al., 2002. The higher-order remainder GR

VPZ hasbeen considered by Wichmann and Kroll, and the lead-ing terms in powers of Z are Wichmann and Kroll,1956; Mohr, 1975, 1983

D

n S1/2 P1/2 P3/2 D3/2 D5/2

1 −30.290 240202 −31.185 15090 −0.973 5020 −0.486 50203 −31.047 70904 −30.912040 −1.164020 −0.609020 0.031 63226 −30.71147 0.034 17268 −30.60647 0.007 94090 0.034 842212 0.008020 0.035030

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TABLE IX. Values of the function G1VP.

n S1/2 P1/2 P3/2 D3/2 D5/2

1 −0.618 7242 −0.808 872 −0.064 006 −0.014 1323 −0.814 5304 −0.806 579 −0.080 007 −0.017 666 −0.000 0006 −0.791 450 −0.000 0008 −0.781 197 −0.000 000 −0.000 00012 −0.000 000 −0.000 000

2

G 31R 19 2 1 VPZ = − l0 + −

45 27

16 2880Zl0

+ ¯ . 26

Higher-order terms omitted from Eq. 26 are negligible.In a manner similar to that for the self energy, the

leading effect of the finite mass of the nucleus is takeninto account by multiplying Eq. 22 by the factormr /me3 and including a multiplicative factor ofme /mr in the argument of the logarithm in Eq. 23.

There is also a second-order vacuum polarizationlevel shift due to the creation of virtual particle pairsother than the e−e+ pair. The predominant contributionfor nS states arises from +−, with the leading termbeing Eides and Shelyuto, 1995; Karshenboim, 1995

Z2 4 4 mEVP =

n3 −15 e

m2m 3

r m c2. 27 m e

e

The next-order term in the contribution of muonvacuum polarization to nS states is of relative orderZme /m and is therefore negligible. The analogouscontribution E2

VP from +− −18 Hz for the 1S state isalso negligible at the level of uncertainty of current in-terest.

For the hadronic vacuum polarization contribution,we use the result given by Friar et al. 1999 that utilizesall available e+e− scattering data,

E2VP = 0.67115E2

had VP, 28

where the uncertainty is of type u0.The muonic and hadronic vacuum polarization contri-

butions are negligible for P and D states.

f. Two-photon corrections

Corrections from two virtual photons have been par-tially calculated as a power series in Z,

2

E4 Z 4

=n3 mec2F4Z , 29

where

F4Z = B40 + B50Z + B63Z2 ln3Z−2

+ B62Z2 ln2Z−2 + B61Z2 lnZ−2

+ B60Z2 + ¯ . 30

The leading term B40 is well known,

32 102 2179 9B40 = ln 2 − − − 3

2 27 648 4

l0

2 ln 2 2 197 33 1 − + − − − l0 . 31

2 12 144 4 2l + 1

The second term is Pachucki, 1993a, 1994; Eides andShelyuto, 1995; Eides et al., 1997

B50 = − 21.556131l0, 32

and the next coefficient is Karshenboim, 1993; Manoharand Stewart, 2000; Yerokhin, 2000; Pachucki, 2001

B63 = − 8 27 l0. 33

For S states,

the coefficient B62 is given by

16 71 1 1B62 = − ln 2 + + n − ln n − + ,

9 60 n 4n234

where =0.577. . . is Euler’s constant and is the psifunction Abramowitz and Stegun, 1965. The differenceB621−B62n was calculated by Karshenboim 1996and confirmed by Pachucki 2001, who also calculatedthe n-independent additive constant. For P states, thecalculated value is Karshenboim, 1996

4 n2 − 1B62 = . 35

27 n2

This result has been confirmed by Jentschura and Nán-dori 2002, who also showed that for D and higher an-gular momentum states B62=0.

Recent work has led to new results for B61 and higher-order coefficients. In the paper of Jentschura, Czarnecki,and Pachucki 2005, an additional state-independentcontribution to the coefficient B61 for S states is given,which differs slightly 2% from the earlier result of Pa-chucki 2001 quoted in CODATA 2002. The revised co-efficient for S states is

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TABLE X. Values of N used in the 2006 adjustment.

n NnS NnP

1 17.855 672 0312 12.032 141 581 0.003 300 63513 10.449 80914 9.722 4131 −0.000 394 33216 9.031 83218 8.697 6391

413 581 4NnS 20272 616 ln 2B61 = + + −

64 800 3 864 135

22 ln 2 40 ln2 2− + + 3 +

3 9304 32 ln 2

−135 9

3 1 1

+ + n − ln n − + ,4 n 4n2 36

where is the Riemann zeta function Abramowitz andStegun, 1965. The coefficients NnS are listed in TableX. The state-dependent part B61nS−B611S was con-firmed by Jentschura, Czarnecki, and Pachucki 2005 intheir Eqs. 4.26 and 6.3. For higher-l states, B61 hasbeen calculated by Jentschura, Czarnecki, and Pachucki2005; for P states,

4 n2 − 1 166 8B61nP1/2 = NnP +

3 n2 − ln 2405 27

, 37

4 n2 − 1 31 8B61nP3/2 = NnP +

3 n2 − ln 2 38405

; 27

and for D states,

B61nD = 0. 39

The coefficient B61 also vanishes for states with l2.The necessary values of NnP are given in Eq. 17 ofJentschura 2003 and are listed in Table X.

The next term is B60, and recent work has also beendone for this contribution. For S states, the state depen-dence is considered first, and is given by Czarnecki et al.2005 and Jentschura, Czarnecki, and Pachucki 2005,

B60nS − B601S = bLnS − bL1S + An , 40

where

An = 38 4 337 043− ln 2NnS − N1S −

45 3 129 600

94 261 902 609 4 16 4− + + − + ln2 2

21 600n 129 600n2

3 9

n 9n2

76 304 76 53 35+ − + −

45 135n 135n2 ln 2 + − +

15 2n

419 28 003 11−

30n2 2ln 2 + −10 800 2n

31 397 53 419+ + 2 2 + 35

− 2 310 800n 60 8n 120n

+ 37 793 16 304+ ln2 2 − ln 2 + 82ln 2

10 800 9 135

13− 2 − 23

3 + n − ln n . 41

The term An makes a small contribution in the range0.3 to 0.4 for the states under consideration.

The two-loop Bethe logarithms bL in Eq. 40 arelisted in Table XI. The values for n=1 to 6 are fromJentschura 2004 and Pachucki and Jentschura 2003,and the value at n=8 is obtained by extrapolationof the calculated values from n=4 to 6 bL5S=−60.68 with a function of the form

b cbLnS = a + + , 42

n nn + 1

which yields

24bLnS = − 55.8 − . 43

n

It happens that the fit gives c=0. An estimate for B60given by

B60nS = bLnS + 109 NnS + ¯ 44

was derived by Pachucki 2001. The dots represent un-calculated contributions at the relative level of 15% Pa-chucki and Jentschura, 2003. Equation 44 givesB601S=−61.69.2. However, more recently Yerokhin etal. 2003, 2005a, 2005b, 2007 have calculated the1S-state two-loop self energy correction for Z10. Thisis expected to give the main contribution to the higher-order two-loop correction. Their results extrapolated toZ=1 yield a value for the contribution of all terms oforder B60 or higher of −127 1±0.3, which corre-sponds to a value of roughly B60=−12939, assuming alinear extrapolation from Z=1 to 0. This differs byabout a factor of 2 from the result given by Eq. 44. Inview of this difference between the two calculations, forthe 2006 adjustment, we use the average of the two val-ues with an uncertainty that is half the difference, whichgives

TABLE XI. Values of bL, B60, and B71 used in the 2006adjustment. See the text for an explanation of the uncertainty33.7.

n bLnS B60nS B71nS

1 −81.40.3 −95.30.333.72 −66.60.3 −80.20.333.7 1683 −63.50.6 −77.00.633.7 22114 −61.80.8 −75.30.833.7 25126 −59.80.8 −73.30.833.7 28148 −58.82.0 −72.32.033.7 2915

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B601S = − 95.30.333.7 . 45

In Eq. 45, the first number in parentheses is the state-dependent uncertainty unB60 associated with the two-loop Bethe logarithm, and the second number in paren-theses is the state-independent uncertainty u0B60 thatis common to all S-state values of B60. Values of B60 forall relevant S states are given in Table XI. For higher-lstates, B60 has not been calculated, so we take itto be zero, with uncertainties unB60nP=5.0 andunB60nD=1.0. We assume that these uncertainties ac-count for higher-order P- and D-state uncertainties aswell. For S states, higher-order terms have been esti-mated by Jentschura, Czarnecki, and Pachucki 2005with an effective potential model. They find that thenext term has a coefficient of B72 and is state indepen-dent. We thus assume that the uncertainty u0B60nS issufficient to account for the uncertainty due to omittingsuch a term and higher-order state-independent terms.In addition, they find an estimate for the state depen-dence of the next term, given by

B71nS = B

71nS − B711S

427 16= − ln 2

36 3

3 1 1

− + + 2 + n − ln n4 n 4n

46

with a relative uncertainty of 50%. We include this ad-ditional term, which is listed in Table XI, along with theestimated uncertainty unB71=B71/2.

The disagreement of the analytic and numerical calcu-lations results in an uncertainty of the two-photon con-tribution that is larger than the estimated uncertaintyused in the 2002 adjustment. As a result, the uncertain-ties of the recommended values of the Rydberg constantand proton and deuteron radii are slightly larger in the2006 adjustment, although the 2002 and 2006 recom-mended values are consistent with each other. On theother hand, the uncertainty of the 2P state fine structureis reduced as a result of the new analytic calculations.

As in the case of the order self energy and vacuum-polarization contributions, the dominant effect of the fi-nite mass of the nucleus is taken into account by multi-plying each term of the two-photon contribution by thereduced-mass factor mr /me3, except that the magneticmoment term, the second line of Eq. 31, is instead mul-tiplied by the factor mr /me2. In addition, the argumentZ−2 of the logarithms is replaced by me /mrZ−2.

g. Three-photon corrections

The leading contribution from three virtual photons isexpected to

have

E6 =

the form3 Z4

3 mec2C40 + C50Z + ¯ , 47 n

in analogy with Eq. 29 for two photons. The leadingterm C40 is Baikov and Broadhurst, 1995; Eides and

Grotch, 1995a; Laporta and Remiddi, 1996; Melnikovand van Ritbergen,

2000

568a 5 24 85 121 3 84 0713

C40 = − + − −9 24 72 2304

71 ln4 2 2392 ln2 2 47872 ln 2− − +

27 135 108

15914 252 2512 679 441+ − +

3240 9720 93 312

l0

100a 24 2155 83 3 1393

+ − + − −3 24 72 18

25 ln4 2 252 ln2 2 2982 ln 2 2394

− + + +18 18

9 2160

17 1012 28 259 1 − − − l0 , 48

810 5184 2l + 1

where a = n4 n=11 / 2 n4=0.517 479 061. . . . Higher-order

terms have not been calculated, although partial resultshave been obtained Eides and Shelyuto, 2007. An un-certainty is assigned by taking u0C50=30l0 andunC63=1, where C63 is defined by the usual convention.The dominant effect of the finite mass of the nucleus istaken into account by multiplying the term proportionalto l0 by the reduced-mass factor mr /me3 and the termproportional to 1/ 2l+1, the magnetic moment term,by the factor mr /me2.

The contribution from four photons is expected to beof order

44 Z n3 mec2, 49

which is about 10 Hz for the 1S state and is negligible atthe level of uncertainty of current interest.

h. Finite nuclear size

At low Z, the leading contribution due to the finitesize of the nucleus is

E0NS = ENSl0, 50

with

2 3 2

E = m 2r Z

NS m c2 ZRN , 513 m

3e n e

C

where RN is the bound-state root-mean-square rmscharge radius of the nucleus and C is the Comptonwavelength of the electron divided by 2. The leadinghigher-order contributions have been examined by Friar1979b, Friar and Payne 1997b, and Karshenboim1997 see also Borisoglebsky and Trofimenko 1979and Mohr 1983. The expressions that we employ toevaluate the nuclear size correction are the same asthose discussed in more detail in CODATA-98.

For S states, the leading and next-order correctionsare given by

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R ZE m

NS = ENS 1 − C r N m R Z − r

Cln

N

me me C n

5n + 9n − 1+ n + − 2 − CZ2

4n ,

52

where C and C are constants that depend on the de-tails of the assumed charge distribution in the nucleus.The values used here are C=1.71 and C=0.474 forhydrogen or C=2.01 and C=0.384 for deuterium.

For the P1/2 states in hydrogen, the leading term is

Z2n2 − 1ENS = ENS .

4n2 53

For P3/2 states and D states, the nuclear-size contribu-tion is negligible.

i. Nuclear-size correction to self energy and vacuum polarization

For the self energy, the additional contribution due tothe finite size of the nucleus is Pachucki, 1993b; Eidesand Grotch, 1997b; Milstein et al., 2002, 2003a

E 23NSE = 4 ln 2 − 4 ZENSl0, 54

and for the vacuum polarization it is Friar, 1979a, 1981;Hylton, 1985; Eides and Grotch, 1997b

ENVP = 3 ZENS4 l0. 55

For the self energy term, higher-order size correctionsfor S states Milstein et al., 2002 and size corrections forP states have been calculated Jentschura, 2003; Milsteinet al., 2003b, but these corrections are negligible for thecurrent work, and are not included. The D-state correc-tions are assumed to be negligible.

j. Radiative-recoil corrections

The dominant effect of nuclear motion on the self en-ergy and vacuum polarization has been taken into ac-count by including appropriate reduced-mass factors.The additional contributions beyond this prescriptionare termed radiative-recoil effects with leading termsgiven by

m3r Z5 352

ERR = m2

emN 2 m c2 6 3 − 22 ln 2 +n3 e l0 36

448 2− + Zln2Z−2 +

27 3¯ . 56

The constant term in Eq. 56 is the sum of the analyticresult for the electron-line contribution Czarnecki andMelnikov, 2001; Eides et al., 2001a and the vacuum-polarization contribution Eides and Grotch, 1995b; Pa-chucki, 1995. This term agrees with the numerical valuePachucki, 1995 used in CODATA-98. The log-squaredterm has been calculated by Pachucki and Karshenboim1999 and by Melnikov and Yelkhovsky 1999.

For the uncertainty, we take a term of orderZlnZ−2 relative to the square brackets in Eq. 56

with numerical coefficients 10 for u0 and 1 for un. Thesecoefficients are roughly what one would expect for thehigher-order uncalculated terms. For higher-l states inthe present evaluation, we assume that the uncertaintiesof the two- and three-photon corrections are muchlarger than the uncertainty of the radiative-recoil correc-tion. Thus, we assign no uncertainty for the radiative-recoil correction for P and D states.

k. Nucleus self energy

An additional contribution due to the self energy ofthe nucleus has been given by Pachucki 1995,

4Z2Z4 m3

ESEN = r

3n3 m2 c2

Nln mN

mrZ2l0

− ln k0n,l . 57

This correction has also been examined by Eides et al.2001b, who consider how it is modified by the effect ofstructure of the proton. The structure effect would leadto an additional model-dependent constant in the squarebrackets in Eq. 57.

To evaluate the nucleus self energy correction, we useEq. 57 and assign an uncertainty u0 that corresponds toan additive constant of 0.5 in the square brackets for Sstates. For P and D states, the correction is small and itsuncertainty, compared to other uncertainties, is negli-gible.

l. Total energy and uncertainty

The total energy EXnLj of a particular level where L

=S,P, . . . and X=H,D is the sum of the various contri-butions listed above plus an additive correction XnLj thataccounts for the uncertainty in the theoretical expres-sion for EX

nLj. Our theoretical estimate for the value ofXnLj for a particular level is zero with a standard uncer-

tainty of u XnLj equal to the square root of the sum ofthe squares of the individual uncertainties of the contri-butions; as they are defined above, the contributions tothe energy of a given level are independent. Compo-nents of uncertainty associated with the fundamentalconstants are not included here, because they are deter-mined by the least-squares adjustment itself. Thus, wehave for the square of the uncertainty, or variance, of aparticular level

u2 2

u2 X = 0iXLj + uniXLjnLj , 58

i n6

where the individual values u0iXLj /n3 and uniXLj /n3

are the components of uncertainty from each of the con-tributions, labeled by i, discussed above. The factors of1/n3 are isolated so that u0iXLj is explicitly indepen-dent of n.

The covariance of any two ’s follows from Eq. F7 ofAppendix F of CODATA-98. For a given isotope X, wehave

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2X j

un Lj,Xn Lj = u0iXL

3 , 591 2

i n1n2

which follows from the fact that uu0i ,uni=0 anduun i ,un i=0 for n1n2. We also set

1 2

u X Xn L j ,n L j = 0 601 1 1 2 2 2

if L1L2 or j1 j2.For covariances between ’s for hydrogen and deute-

rium, we have for states of the same n

H D u0iHLju DLj + u HLju DLjunLj,nLj = 0i ni ni

i=i n6 ,c

61

and for n1n2

HLu H j

n Lj,D u0i u0iDLj

n Lj =1 2

i=ic

n1n23 , 62

where the summation is over the uncertainties commonto hydrogen and deuterium. In most cases, the uncer-tainties can in fact be viewed as common except for aknown multiplicative factor that contains all of the massdependence. We assume

u Hn L j , Dn L j = 0 631 1 1 2 2 2

if L1L2 or j1 j2.The values of u XnLj of interest for the 2006 adjust-

ment are given in Table XXVIII of Sec. XII, and thenon-negligible covariances of the ’s are given in theform of correlation coefficients in Table XXIX of thatsection. These coefficients are as large as 0.9999.

Since the transitions between levels are measured infrequency units Hz, in order to apply the above equa-tions for the energy level contributions we divide thetheoretical expression for the energy difference E ofthe transition by the Planck constant h to convert it to afrequency. Further, since we take the Rydberg constantR=2mec /2h expressed in m−1 rather than the elec-tron mass me to be an adjusted constant, we replace thegroup of constants 2m 2

ec /2h in E /h by cR.

m. Transition frequencies between levels with n=2

As an indication of the consistency of the theory sum-marized above and the experimental data, we list belowvalues of the transition frequencies between levels withn=2 in hydrogen. These results are based on values ofthe constants obtained in a variation of the 2006 least-squares adjustment in which the measurements of thedirectly related transitions items A38, A39.1, and A39.2in Table XXVIII are not included, and the weakly de-pendent constants Are, Arp, Ard, and are as-signed their 2006 adjusted values. The results are

H2P1/2 − 2S1/2 = 1 057 843.92.5 kHz

2.3 10−6 ,

H2S1/2 − 2P3/2 = 9 911 197.62.5 kHz

2.5 10−7 ,

H2P1/2 − 2P3/2 = 10 969 041.47599 kHz

9.0 10−9 , 64

which agree well with the relevant experimental resultsof Table XXVIII. Although the first two values in Eq.64 have changed only slightly from the results of the2002 adjustment, the third value, the fine-structure split-ting, has an uncertainty that is almost an order of mag-nitude smaller than the 2002 value, due mainly to im-provements in the theory of the two-photon correction.

A value of the fine-structure constant can be ob-tained from data on the hydrogen and deuterium transi-tions. This is done by running a variation of the 2006least-squares adjustment that includes all the transitionfrequency data in Table XXVIII and the 2006 adjustedvalues of Are, Arp, and Ard. The resulting value is

−1 = 137.036 00248 3.5 10−7 , 65

which is consistent with the 2006 recommended value,although substantially less accurate. This result is in-cluded in Table XXXIV.

2. Experiments on hydrogen and deuterium

Table XII summarizes the transition frequency datarelevant to the determination of R. With the exceptionof the first entry, which is the most recent result for the1S1/2–2S1/2 transition frequency in hydrogen from thegroup at the Max-Planck-Institute für QuantenoptikMPQ, Garching, Germany, all these data are the sameas those used in the 2002 adjustment. Since these dataare reviewed in CODATA-98 or CODATA-02, they arenot discussed here. For a brief discussion of data notincluded in Table XII, see Sec. II.B.3 of CODATA-02.

The new MPQ result,

H1S1/2-2S1/2 = 2 466 061 413 187.07434 kHz

1.4 10−14 , 66

was obtained in the course of an experiment tosearch for a temporal variation of the fine-structureconstant Fischer et al., 2004a, 2004b; Hänschet al., 2005; Udem, 2006. It is consistent with, but has asomewhat smaller uncertainty than, the previousresult from the MPQ group, H1S1/2−2S1/2=2 466 061 413 187.10346 kHz 1.910−14 Niering etal., 2000, which was the value used in the 2002 adjust-ment. The improvements that led to the reduction inuncertainty include a more stable external referencecavity for locking the 486 nm cw dye laser, thereby re-ducing its linewidth; an upgraded vacuum system thatlowered the background gas pressure in the interactionregion, thereby reducing the background gas pressureshift and its associated uncertainty; and a significantlyreduced within-day Type A i.e., statistical uncertaintydue to the narrower laser linewidth and better signal-to-noise ratio.

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TABLE XII. Summary of measured transition frequencies considered in the present work for the determination of the Rydbergconstant R H is hydrogen and D is deuterium.

Reported value Rel. stand.Authors Laboratory Frequency intervals /kHz uncert. ur

Fischer et al., 2004a, 2004b MPQ H1S1/2–2S1/2 2 466 061 413 187.07434 1.410−14

Weitz et al., 1995 MPQ H2S1/2–4S1/2− 14H1S1/2–2S1/2 4 797 33810 2.110−6

H2S1/2–4D5/2− 14H1S1/2–2S1/2 6 490 14424 3.710−6

D2S1/2–4S1/2− 14D1S1/2–2S1/2 4 801 69320 4.210−6

D2S1/2–4D5/2− 14D1S1/2–2S1/2 6 494 84141 6.310−6

Huber et al., 1998 MPQ D1S1/2–2S1/2−H1S1/2–2S1/2 670 994 334.6415 2.210−10

de Beauvoir et al., 1997 LKB/SYRTE H2S1/2–8S1/2 770 649 350 012.08.6 1.110−11

H2S1/2–8D3/2 770 649 504 450.08.3 1.110−11

H2S1/2–8D5/2 770 649 561 584.26.4 8.310−12

D2S1/2–8S1/2 770 859 041 245.76.9 8.910−12

D2S1/2–8D3/2 770 859 195 701.86.3 8.210−12

D2S1/2–8D5/2 770 859 252 849.55.9 7.710−12

Schwob et al., 1999, 2001 LKB/SYRTE H2S1/2–12D3/2 799 191 710 472.79.4 1.210−11

H2S1/2–12D5/2 799 191 727 403.77.0 8.710−12

D2S1/2–12D3/2 799 409 168 038.08.6 1.110−11

D2S1/2–12D5/2 799 409 184 966.86.8 8.510−12

Bourzeix et al., 1996 LKB H2S1/2–6S1/2− 14H1S1/2–3S1/2 4 197 60421 4.910−6

H2S1/2–6D5/2− 14H1S1/2–3S1/2 4 699 09910 2.210−6

Berkeland et al., 1995 Yale H2S1/2–4P1/2− 14H1S1/2–2S1/2 4 664 26915 3.210−6

H2S1/2–4P3/2− 14H1S1/2–2S1/2 6 035 37310 1.710−6

Hagley and Pipkin, 1994 Harvard H2S1/2–2P3/2 9 911 20012 1.210−6

Lundeen and Pipkin, 1986 Harvard H2P1/2–2S1/2 1 057 845.09.0 8.510−6

Newton et al., 1979 U. Sussex H2P1/2–2S1/2 1 057 86220 1.910−5

The MPQ result in Eq. 66 and Table XII forH1S1/2–2S1/2 was provided by Udem 2006 ofthe MPQ group. It follows from the measuredvalue H1S1/2–2S1/2=2 466 061 102 474.85134 kHz1.410−14 obtained for the 1S,F=1,mF= ±1→ 2S,F=1,mF = ±1 transition frequency Fischer etal., 2004a, 2004b; Hänsch et al., 2005 by using the wellknown 1S and 2S hyperfine splittings Ramsey, 1990;Kolachevsky et al., 2004 to convert it to the frequencycorresponding to the hyperfine centroid.

3. Nuclear radii

The theoretical expressions for the finite nuclear sizecorrection to the energy levels of hydrogen H and deu-terium D see Sec. IV.A.1.h are functions of the bound-state nuclear rms charge radius of the proton Rp and ofthe deuteron Rd. These values are treated as variables inthe adjustment, so the transition frequency data, to-gether with theory, determine values for the radii. Theradii are also determined by elastic electron-proton scat-tering data in the case of Rp and from elastic electron-deuteron scattering data in the case of Rd. These inde-pendently determined values are used as additionalinformation on the radii. There have been no new re-sults during the last four years and thus we take as input

data for these two radii the values used in the 2002 ad-justment,

Rp = 0.89518 fm, 67

Rd = 2.13010 fm. 68

The result for Rp is due to Sick 2003 see also Sick2007. The result for Rd is that given in Sec. III.B.7 ofCODATA-98 based on the analysis of Sick and Traut-mann 1998.

An experiment currently underway to measure theLamb shift in muonic hydrogen may eventually providea significantly improved value of Rp and hence an im-proved value of R Nebel et al., 2007.

B. Antiprotonic helium transition frequencies and Ar(e)

The antiprotonic helium atom is a three-body systemconsisting of a 4He or 3He nucleus, an antiproton, andan electron, denoted by pHe+. Even though the Bohrradius for the antiproton in the field of the nucleus isabout 1836 times smaller than the electron Bohr radius,in the highly excited states studied experimentally, theaverage orbital radius of the antiproton is comparable tothe electron Bohr radius, giving rise to relatively long-

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TABLE XIII. Summary of data related to the determination of Are from measurements on anti-protonic helium.

Transition Experimental Calculated a bn , l→ n , l value MHz value MHz 2cR 2cR

p 4He+: 32,31→ 31,30 1 132 609 20915 1 132 609 223.5082 0.2179 0.0437

p 4He+: 35,33→ 34,32 804 633 059.08.2 804 633 058.01.0 0.1792 0.0360

p 4He+: 36,34→ 35,33 717 474 00410 717 474 001.11.2 0.1691 0.0340

p 4He+: 37,34→ 36,33 636 878 139.47.7 636 878 151.71.1 0.1581 0.0317

p 4He+: 39,35→ 38,34 501 948 751.64.4 501 948 755.41.2 0.1376 0.0276

p 4He+: 40,35→ 39,34 445 608 557.66.3 445 608 569.31.3 0.1261 0.0253

p 4He+: 37,35→ 38,34 412 885 132.23.9 412 885 132.81.8 −0.1640 −0.0329

p 3He+: 32,31→ 31,30 1 043 128 60813 1 043 128 579.7091 0.2098 0.0524

p 3He+: 34,32→ 33,31 822 809 19012 822 809 170.91.1 0.1841 0.0460

p 3He+: 36,33→ 35,32 646 180 43412 646 180 408.21.2 0.1618 0.0405

p 3He+: 38,34→ 37,33 505 222 295.78.2 505 222 280.91.1 0.1398 0.0350

p 3He+: 36,34 37,33 414 147 507.84.0 414 147 509.31.8 −0.1664 −0.0416

lived states. Also, for the high-l states studied, becauseof the vanishingly small overlap of the antiproton wavefunction with the helium nucleus, strong interactions be-tween the antiproton and the nucleus are negligible.

One of the goals of antiprotonic helium experimentsis to measure the antiproton-electron mass ratio. How-ever, since we assume that CPT is a valid symmetry, forthe purpose of the least-squares adjustment we take themasses of the antiproton and proton to be equal and usethe data to determine the proton-electron mass ratio.Since the proton relative atomic mass is known moreaccurately than the electron relative atomic mass fromother experiments, the mass ratio yields information pri-marily on the electron relative atomic mass. Other ex-periments have demonstrated the equality of the charge-to-mass ratio of p and p to within 9 parts in 1011; seeGabrielse 2006.

1. Theory relevant to antiprotonic helium

Calculations of transition frequencies of antiprotonichelium have been done by Kino et al. 2003 and byKorobov 2003, 2005. The uncertainties of calculationsby Korobov 2005 are of the order of 1 MHz to 2 MHz,while the uncertainties and scatter relative to the experi-mental values of the results of Kino et al. 2003 aresubstantially larger, so we use the results of Korobov2005 in the 2006 adjustment. See also the remarks inHayano 2007 concerning the theory.

The dominant contribution to the energy levels is thenonrelativistic solution of the Schrödinger equation forthe three-body system together with relativistic and ra-diative corrections treated as perturbations. The nonrel-ativistic levels are resonances, because the states can de-cay by the Auger effect in which the electron is ejected.Korobov 2005 calculated the nonrelativistic energy byusing one of two formalisms, depending on whether theAuger rate is small or large. In the case in which the rateis small, the Feshbach formalism is used with an optical

potential. The optical potential is omitted in the calcula-tion of higher-order relativistic and radiative corrections.For broad resonances with a higher Auger rate, the non-relativistic energies are calculated with the complex co-ordinate rotation method. In checking the convergenceof the nonrelativistic levels, attention was paid to theconvergence of the expectation value of the the deltafunction operators used in the evaluation of the relativ-istic and radiative corrections.

Korobov 2005 evaluated the relativistic and radia-tive corrections as perturbations to the nonrelativisticlevels, including relativistic corrections of order 2R,anomalous magnetic moment corrections of order3R and higher, one-loop self energy and vacuum-polarization corrections of order 3R, and higher-orderone-loop and leading two-loop corrections of order4R. Higher-order relativistic corrections of order 4R

and radiative corrections of order 5R were estimatedwith effective operators. The uncertainty estimates ac-count for uncalculated terms of order 5 ln R.

Transition frequencies obtained by Korobov 2005,2006 using the CODATA-02 values of the relevant con-stants are listed in Table XIII under the column header“calculated value.” We denote these values of the fre-quencies by 0pHen , l :n 3 , l, where He is either He+ or4He+. Also calculated are the leading-order changes inthe theoretical values of the transition frequencies as afunction of the relative changes in the mass ratiosArp /Are and ArN /Arp, where N is either 3He2+ or4He2+. If we denote the transition frequencies as func-tions of these mass ratios by pHen , l :n , l, then thechanges can be written as

A p 0pHen,l:n,la n, r

pHen,l: l = , 69Are

Arp

Are

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A He0r pHe

n,l:n,l

bpHen,l:n,l = . 70Arp ArN

Arp

Values of these derivatives, in units of 2cR, are listed inTable XIII in the columns with the headers “a” and “b,”respectively. The zero-order frequencies and the deriva-tives are used in the expression

pHen,l:n,l = 0p

He

n,l:n,l + apHen,l:n,l

A re 0 Arp

− 1Arp

Are

A p 0

+ bpHen,l:n r,l

ArNArN − 1 + ¯ , 71

Arp

which provides a first-order approximation to the tran-sition frequencies as a function of changes to the massratios. This expression is used to incorporate the experi-mental data and calculations for the antiprotonic systemas a function of the mass ratios into the least-squaresadjustment. It should be noted that even though themass ratios are the independent variables in Eq. 71 andthe relative atomic masses Are, Arp, and ArN arethe adjusted constants in the 2006 least-squares adjust-ment, the primary effect of including these data in theadjustment is on the electron relative atomic mass, be-cause independent data in the adjustment provide valuesof the proton and helium nuclei relative atomic masseswith significantly smaller uncertainties.

The uncertainties in the theoretical expressions forthe transition frequencies are included in the adjustmentas additive constants pHen , l :n , l. Values for the the-oretical uncertainties and covariances used in the adjust-ment are given in Sec. XII, Tables XXXII and XXXIII,respectively Korobov, 2006.

2. Experiments on antiprotonic helium

Experimental work on antiprotonic helium began inthe early 1990s and it continues to be an active field ofresearch; a comprehensive review through 2000 hasbeen given by Yamazaki et al. 2002 and a concise re-view through 2006 by Hayano 2007. The first measure-ments of pHe+ transition frequencies at CERN with ur10−6 were reported in 2001 Hori et al., 2001, im-proved results were reported in 2003 Hori et al., 2003,and transition frequencies with uncertainties sufficientlysmall that they can, together with the theory of the tran-sitions, provide a competitive value of Are, were re-ported in 2006 Hori et al., 2006.

The 12 transition frequencies—seven for 4He and fivefor 3He given by Hori et al. 2006—which we take asinput data in the 2006 adjustment, are listed in column 2of Table XIII with the corresponding transitions indi-cated in column 1. To reduce rounding errors, an addi-tional digit for both the frequencies and their uncertain-

ties as provided by Hori 2006 have been included. All12 frequencies are correlated; their correlation coeffi-cients, based on detailed uncertainty budgets for each,also provided by Hori 2006, are given in Table XXXIIIin Sec XII.

In the current version of the experiment, 5.3 MeV an-tiprotons from the CERN Antiproton Decelerator ADare decelerated using a radio-frequency quadrupole de-celerator RFQD to energies in the range 10 keV to 120keV controlled by a dc potential bias on the RFQD’selectrodes. The decelerated antiprotons, about 30% ofthe antiprotons entering the RFQD, are then diverted toa low-pressure cryogenic helium gas target at 10 K by anachromatic momentum analyzer, the purpose of which isto eliminate the large background that the remaining70% of undecelerated antiprotons would have pro-duced.

About 3% of the p stopped in the target form pHe+,in which a p with large principle quantum number n38 and angular momentum quantum number lncirculates in a localized, nearly circular orbit around theHe2+ nucleus while the electron occupies the distributed1S state. These p energy levels are metastable with life-times of several microseconds and de-excite radiatively.There are also short-lived p states with similar values ofn and l but with lifetimes on the order of 10 ns andwhich de-excite by Auger transitions to form pHe2+ hy-drogenlike ions. These undergo Stark collisions, whichcause the rapid annihilation of the p in the heliumnucleus. The annihilation rate versus time elapsed sincepHe+ formation, or delayed annihilation time spectrumDATS, is measured using Cherenkov counters.

With the exception of the 36,34→ 35,33 transitionfrequency, all frequencies given in Table XIII were ob-tained by stimulating transitions from the pHe+ meta-stable states with values of n and l indicated in columnone on the left-hand side of the arrow to the short-lived,Auger-decaying states with values of n and l indicatedon the right-hand side of the arrow.

The megawatt-scale light intensities needed to inducethe pHe+ transitions, which cover the wavelength range265 nm to 726 nm, can only be provided by a pulsedlaser. Frequency and linewidth fluctuations and fre-quency calibration problems associated with such laserswere overcome by starting with a cw “seed” laser beamof frequency , known with u 410−10

cw r through itsstabilization by an optical frequency comb, and then am-plifying the intensity of the laser beam by a factor of 106

in a cw pulse amplifier consisting of three dye cellspumped by a pulsed Nd:YAG laser. The 1 W seed laserbeam with wavelength in the range 574 nm to 673 nmwas obtained from a pumped cw dye laser, and the 1 Wseed laser beam with wavelength in the range 723 nm to941 nm was obtained from a pumped cw Ti:sapphire la-ser. The shorter wavelengths 265 nm to 471 nm forinducing transitions were obtained by frequency dou-bling the amplifier output at 575 nm and 729 nm to 941nm or by frequency tripling its 794 nm output. The fre-quency of the seed laser beam cw, and thus the fre-

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quency pl of the pulse amplified beam, was scannedover a range of ±4 GHz around the pHe+ transition fre-quency by changing the repetition frequency frep of thefrequency comb.

The resonance curve for a transition was obtained byplotting the area under the resulting DATS peak versuspl. Because of the approximate 400 MHz Dopplerbroadening of the resonance due to the 10 K thermalmotion of the pHe+ atoms, a rather sophisticated theo-retical line shape that takes into account many factorsmust be used to obtain the desired transition frequency.

Two other effects of major importance are the so-called chirp effect and linear shifts in the transition fre-quencies due to collisions between the pHe+ and back-ground helium atoms. The frequency pl can deviatefrom cw due to sudden changes in the index of refrac-tion of the dye in the cells of the amplifier. This chirp,which can be expressed as ct=plt−cw, can shiftthe measured pHe+ frequencies from their actual values.Hori et al. 2006 eliminated this effect by measuringct in real time and applying a frequency shift to theseed laser, thereby canceling the dye-cell chirp. This ef-fect is the predominant contributor to the correlationsamong the 12 transitions Hori, 2006. The collisionalshift was eliminated by measuring the frequencies of tentransitions in helium gas targets with helium atom den-sities in the range 21018/cm3 to 31021/cm3 to de-termine d /d. The in vacuo =0 values were ob-tained by applying a suitable correction in the range −14MHz to 1 MHz to the initially measured frequenciesobtained at 21018/cm3.

In contrast to the other 11 transition frequencies inTable XIII, which were obtained by inducing a transitionfrom a long-lived, metastable state to a short-lived,Auger-decaying state, the 36,34→ 35,33 transitionfrequency was obtained by inducing a transition fromthe 36,34 metastable state to the 35,33 metastablestate using three different lasers. This was done by firstdepopulating at time t1 the 35, 33 metastable state byinducing the 35,33→ 34,32 metastable state to short-lived-state transition, then at time t2 inducing the36,34→ 35,33 transition using the cw pulse-amplifiedlaser, and then at time t3 again inducing the 35,33→ 34,32 transition. The resonance curve for the36,34→ 35,33 transition was obtained from theDATS peak resulting from this last induced transition.

The 4 MHz to 15 MHz standard uncertainties of thetransition frequencies in Table XIII arise from the reso-nance line-shape fit 3 MHz to 13 MHz, statistical orType A, not completely eliminating the chirp effect 2MHz to 4 MHz, nonstatistical or Type B, collisionalshifts 0.1 MHz to 2 MHz, Type B, and frequency dou-bling or tripling 1 MHz to 2 MHz, Type B.

3. Values of Ar(e) inferred from antiprotonic helium

From the theory of the 12 antiprotonic transition fre-quencies discussed in Sec IV.B.1, the 2006 recommendedvalues of the relative atomic masses of the proton, alpha

particle , nucleus of the 4He atom, and the helion h,nucleus of the 3He atom, Arp, Ar, and Arh, re-spectively, together with the 12 experimental values forthese frequencies given in Table XIII, we find the fol-lowing three values for Are from the seven p4He+ fre-quencies alone, from the five p3He+ frequencies alone,and from the 12 frequencies together:

Are = 0.000 548 579 910312 2.1 −9 10 , 72

Are = 0.000 548 579 905315 2.7 10−9 , 73

Are = 0.000 548 579 908 8191 1.7 10−9 . 74

The separate inferred values from the p4He+ and p3He+

frequencies differ somewhat, but the value from all 12frequencies not only agrees with the three other avail-able results for Are see Table XXXVI, Sec. XII.A,but has a competitive level of uncertainty as well.

C. Hyperfine structure and fine structure

1. Hyperfine structure

Because the ground-state hyperfine transition fre-quencies H, Mu, and Ps of the comparativelysimple atoms hydrogen, muonium, and positronium, re-spectively, are proportional to 2Rc, in principle avalue of can be obtained by equating an experimentalvalue of one of these transition frequencies to its pre-sumed readily calculable theoretical expression. How-ever, currently only measurements of Mu and thetheory of the muonium hyperfine structure have suffi-ciently small uncertainties to provide a useful result forthe 2006 adjustment, and even in this case the result isnot a competitive value of , but rather the most accu-rate value of the electron-muon mass ratio me /m. In-deed, we discuss the relevant experiments and theory inSec. VI.B.

Although the ground-state hyperfine transition fre-quency of hydrogen has long been of interest as a poten-tial source of an accurate value of because it is experi-mentally known with ur10−12 Ramsey, 1990, therelative uncertainty of the theory is still of the order of10−6. Thus, H cannot yet provide a competitive valueof the fine-structure constant. At present, the mainsources of uncertainty in the theory arise from the inter-nal structure of the proton, namely i the electric chargeand magnetization densities of the proton, which aretaken into account by calculating the proton’s so-calledZemach radius; and ii the polarizability of the protonthat is, protonic excited states. For details of theprogress made over the last four years in reducing theuncertainties from both sources, see Carlson 2007, Pa-chucki 2007, Sick 2007, and references therein. Be-cause the muon is a structureless pointlike particle, thetheory of Mu is free from such uncertainties.

It is also not yet possible to obtain a useful value of from Ps since the most accurate experimental resulthas ur=3.610−6 Ritter et al., 1984. The theoretical un-

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certainty of Ps is not significantly smaller and may infact be larger Adkins et al., 2002; Penin, 2004.

2. Fine structure

As in the case of hyperfine splittings, fine-structuretransition frequencies are proportional to 2Rc andcould be used to deduce a value of . Some data relatedto the fine structure of hydrogen and deuterium are dis-cussed in Sec. IV.A.2 in connection with the Rydbergconstant. They are included in the adjustment becauseof their influence on the adjusted value of R. However,the value of that can be derived from these data is notcompetitive; see Eq. 65. See also Sec. III.B.3 ofCODATA-02 for a discussion of why earlier finestructure-related results in H and D are not considered.

Because the transition frequencies corresponding tothe differences in energy of the three 2 3P levels of4He can be both measured and calculated with reason-able accuracy, the fine structure of 4He has long beenviewed as a potential source of a reliable value of .The three frequencies of interest are 0129.6 GHz,122.29 GHz, and 0231.9 GHz, which correspond tothe intervals 2 3P1–2 3P 3 3 3 3

0, 2 P2–2 P1, and 2 P2–2 P0, re-spectively. The value with the smallest uncertainty forany of these frequencies was obtained at HarvardZelevinsky et al., 2005,

01 = 29 616 951.6670 kHz 2.4 10−8 . 75

It is consistent with the value of 01 reported by Georgeet al. 2001 with ur=3.010−8, and that reported byGiusfredi et al. 2005 with ur=3.410−8. If the theoret-ical expression for 01 were exactly known, the weightedmean of the three results would yield a value of withur810−9.

However, as discussed in CODATA-02, the theory ofthe 2 3PJ transition frequencies is far from satisfactory.First, different calculations disagree, and because of theconsiderable complexity of the calculations and the his-tory of their evolution, there is general agreement thatresults that have not been confirmed by independentevaluation should be taken as tentative. Second, thereare significant disagreements between theory and ex-periment. Recently, Pachucki 2006 has advanced thetheory by calculating the complete contribution to the2 3PJ fine-structure levels of order m7 or 5 Ry, withthe final theoretical result for 01 being

01 = 29 616 943.0117 kHz 5.7 10−9 . 76

This value disagrees with the experimental value givenin Eq. 75 as well as with the theoretical value 01=29 616 946.4218 kHz 6.110−9 given by Drake2002, which also disagrees with the experimental value.These disagreements suggest that there is a problemwith theory and/or experiment that must be resolved be-fore a meaningful value of can be obtained from thehelium fine structure Pachucki, 2006. Therefore, as inthe 2002 adjustment, we do not include 4He fine-structure data in the 2006 adjustment.

V. MAGNETIC MOMENT ANOMALIES AND g-FACTORS

In this section, theory and experiment for the mag-netic moment anomalies of the free electron and muonand the bound-state g-factor of the electron in hydro-genic carbon 12C5+ and in hydrogenic oxygen 16O7+are reviewed.

The magnetic moment of any of the three chargedleptons =e, , is written as

e = g s , 77

2m

where g is the g-factor of the particle, m is its mass,and s is its spin. In Eq. 77, e is the elementary chargeand is positive. For the negatively charged leptons −, g

is negative, and for the corresponding antiparticles +, g

is positive. CPT invariance implies that the masses andabsolute values of the g-factors are the same for eachparticle-antiparticle pair. These leptons have eigenvaluesof spin projection sz= ± /2, and it is conventional towrite, based on Eq. 77,

g e = , 78

2 2m

where in the case of the electron, B=e /2me is theBohr magneton.

The free lepton magnetic moment anomaly a is de-fined as

g = 21 + a , 79

where gD=−2 is the value predicted by the free-electronDirac equation. The theoretical expression for a may bewritten as

ath = aQED + aweak + ahad , 80

where the terms denoted by QED, weak, and had ac-count for the purely quantum electrodynamic, predomi-nantly electroweak, and predominantly hadronic that is,strong interaction contributions to a, respectively.

The QED contribution may be written as Kinoshitaet al., 1990

aQED = A1 + A2m/m + A2m/m

+ A3m/m ,m/m , 81

where for the electron, , ,= e, , and for themuon, , ,= ,e ,. The anomaly for the , whichis poorly known experimentally Yao et al., 2006, is notconsidered here. For recent work on the theory of a,see Eidelman and Passera 2007. In Eq. 81, the termA1 is mass independent, and the mass dependence of A2and A3 arises from vacuum polarization loops with lep-ton , , or both. Each of the four terms on the right-hand side of Eq. 81 can be expressed as a power seriesin the fine-structure constant ,

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2 3 4

i = A2 i + A4

A i + A6 + A8

i i

5+ A10

i + ¯ , 82

where A22 =A2

3 =A43 =0. Coefficients proportional to

/n are of order e2n and are referred to as 2nth-ordercoefficients.

For the mass-independent term A1, the second-ordercoefficient is known exactly, and the fourth- and sixth-order coefficients are known analytically in terms ofreadily evaluated functions,

A21 = 1

2 , 83

A41 = − 0.328 478 965 579 . . . , 84

A61 = 1.181 241 456 . . . . 85

A total of 891 Feynman diagrams give rise to themass-independent eighth-order coefficient A8

1 , and onlya few of these are known analytically. However, in aneffort that has its origins in the 1960s, Kinoshita andcollaborators have calculated A8

1 numerically; the resultof this ongoing project that is used in the 2006 adjust-ment is Kinoshita and Nio, 2006a; Gabrielse et al., 2006,2007

A81 = − 1.728335 . 86

Work was done in the evaluation and checking of thiscoefficient in an effort to obtain a reliable quantitativeresult. A subset of 373 diagrams containing closed elec-tron loops was verified by more than one independentformulation. The remaining 518 diagrams with no closedelectron loops were formulated in only one way. As acheck on this set, extensive cross checking was per-formed on the renormalization terms both among them-selves and with lower-order diagrams that are knownexactly Kinoshita and Nio, 2006a see also Gabrielse etal. 2006, 2007. For the final numerical integrations, anadaptive-iterative Monte Carlo routine was used. Atime-consuming part of the work was checking forround-off error in the integration.

The 0.0035 standard uncertainty of A81 contributes a

standard uncertainty to aeth of 0.8810−10ae, which issmaller than the uncertainty due to uncalculated higher-order contributions. Independent work is in progress onanalytic calculations of eighth-order integrals. See, forexample, Laporta 2001, Laporta et al. 2004, and Mas-trolia and Remiddi 2001.

Little is known about the tenth-order coefficient A101

and higher-order coefficients, although Kinoshita et al.2006 are starting the numerical evaluation of the12 672 Feynman diagrams for this coefficient. To evalu-ate the contribution to the uncertainty of aeth due tolack of knowledge of A10

1 , we follow CODATA-98 toobtain A10

1 =0.03.7. The 3.7 standard uncertainty ofA10

1 contributes a standard uncertainty component toa −1

eth of 2.210 0ae; the uncertainty contributions to

aeth from all other higher-order coefficients, whichshould be significantly smaller, are assumed to be negli-gible.

The 2006 least-squares adjustment was carried out us-ing the theoretical results given above, including thevalue of A8

1 given in Eq. 86. Well after the deadlinefor new data and the recommended values from the ad-justment were made public Mohr et al., 2007, it wasdiscovered by Aoyama et al. 2007 that 2 of the 47 in-tegrals representing 518 QED diagrams that had notpreviously been confirmed independently required acorrected treatment of infrared divergences. The revisedvalue they give is

A81 = − 1.914435 , 87

although the new calculation is still tentative Aoyama etal., 2007. This result would lead to the value

−1 = 137.035 999 07098 7.1 10−10 88

for the inverse fine-structure constant derived from theelectron anomaly using the Harvard measurement resultfor ae Gabrielse et al., 2006, 2007. This number isshifted down from the previous result by 64110−9 andits uncertainty is increased from 96 to 98 see Sec.V.A.3, but it is still consistent with the values obtainedfrom recoil experiments see Table XXVI. If this resultfor A8

1 had been used in the 2006 adjustment, the rec-ommended value of the inverse fine-structure constantwould differ by a similar, although slightly smaller,amount. The effect on the muon anomaly theory is com-pletely negligible.

The mass independent term A1 contributes equally tothe free electron and muon anomalies and the bound-electron g-factors. The mass-dependent terms are differ-ent for the electron and muon and are considered sepa-rately in the following. For the bound-electron g-factor,there are bound-state corrections in addition to the free-electron value of the g-factor, as discussed below.

A. Electron magnetic moment anomaly ae and thefine-structure constant

The combination of theory and experiment for theelectron magnetic moment anomaly yields the value forthe fine-structure constant with the smallest estimateduncertainty see Table XIV.

1. Theory of ae

The mass-dependent coefficients of interest and corre-sponding contributions to the theoretical value of theanomaly aeth, based on the 2006 recommended valuesof the mass ratios, are

A42 me/m = 5.197 386 7826 10−7

→ 24.182 10−10ae, 89

A42 me/m = 1.837 6360 10−9

→ 0.085 10−10ae, 90

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TABLE XIV. Summary of data related to magnetic moments of the electron and muon and inferredvalues of the fine-structure constant. The source data and not the inferred values given here are usedin the adjustment.

Quantity ValueRelative standard

uncertainty ur Identification Sec. and Eq.

ae

−1ae1.159 652 18834210−3

137.035 998 83503.710−9

3.710−9

UWash-87 V.A.2.a 102V.A.3 104

ae

−1ae1.159 652 180 857610−3

137.035 999 711966.610−10

7.010−10

HarvU-06 V.A.2.b 103V.A.3 105

R

a−1

0.003 707 2064201.165 920 936310−3

5.410−7

5.410−7

−6

BNL-06 V.B.2 128V.B.2 129

A62 me/m = − 7.373 941 7227 10−6

→ − 0.797 −1 10 0ae, 91

A62 me/m = − 6.581919 10−8

→ − 0.007 10−10ae, 92

where the standard uncertainties of the coefficients aredue to the uncertainties of the mass ratios but are neg-ligible for a th. The contributions from A6

e 3 me /m ,me /m and all higher-order mass-dependent terms arenegligible as well.

The value for A62 me /m in Eq. 91 has been up-

dated from the value in CODATA-02 and is in agree-ment with the result of Passera 2007 based on a calcu-lation to all orders in the mass ratio. The change is givenby the term

17x63 4381x6 ln2 x 24 761x6 ln x− +

36 30 240 158 760

132x6 1 840 256 147x6

− − , 931344 3 556 224 000

where x=me /m, which was not included in CODATA-02. The earlier result was based on Eq. 4 of Laportaand Remiddi 1993, which only included terms to orderx4. The additional term was kindly provided by Laportaand Remiddi 2006.

For the electroweak contribution, we have

aeweak = 0.029 7352 −1 10 2

= 0.256445 10−10ae, 94

as calculated in CODATA-98 but with the current val-ues of GF and sin2W see Sec. XI.B.

The hadronic contribution is

aehad = 1.68220 10−12 = 1.45017 10−9ae.

95

It is the sum of the following three contributions:a4

e had=1.8751810−12 obtained by Davier and

Höcker 1998, a6ae had=−0.225510−12 given by

Krause 1997, and ae had=0.03185810−12 calcu-

lated by multiplying the corresponding result for themuon given in Sec. V.B.1 by the factor me /m2, sincea

e had is assumed to vary approximately as thesquare of the mass.

Because the dependence on of any contributionother than aeQED is negligible, the anomaly as a func-tion of is given by combining terms that have likepowers of / to yield

aeth = aeQED + aeweak + aehad , 96

where

aeQED = C2e

2

+ C4e

+ C6

e 3

4 58

+ C + C10 e e + ¯ , 97

with

C2e = 0.5,

C4e = − 0.328 478 444 00,

C6e = 1.181 234 017,

C8e = − 1.728335 ,

C10e = 0.03.7 , 98

and aeweak and aehad are as given in Eqs. 94 and95.

The standard uncertainty of aeth from the uncertain-ties of the terms listed above, other than that due to , is

ua th = 0.27 10−12e = 2.4 10−10ae, 99

and is dominated by the uncertainty of the coefficientC10

e .For the purpose of the least-squares calculations car-

ried out in Sec. XII.B, we define an additive correction

R 137.035 6726 1.9 10 V.B.2.a 132

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e to aeth to account for the lack of exact knowledge ofaeth, and hence the complete theoretical expression forthe electron anomaly is

ae,e = aeth + e. 100

Our theoretical estimate of e is zero and its standarduncertainty is uaeth,

e = 0.0027 10−12. 101

2. Measurements of ae

a. University of Washington

The classic series of measurements of the electron andpositron anomalies carried out at the University ofWashington by Van Dyck et al. 1987 yields the value

ae = 1.159 652 188342 −3 −9 10 3.7 10 , 102

as discussed in CODATA-98. This result assumes thatCPT invariance holds for the electron-positron system.

b. Harvard University

A new determination of the electron anomaly using acylindrical Penning trap has been carried out by Odomet al. 2006 at Harvard University, yielding the value

ae = 1.159 652 180 8576 10−3 6.6 10−10 ,

103

which has an uncertainty that is nearly six times smallerthan that of the University of Washington result.

As in the University of Washington experiment, theanomaly is obtained in essence from the relation ae= fa / fc by determining, in the same magnetic flux densityB about 5 T, the anomaly difference frequency fa= fs− fc and cyclotron frequency fc=eB /2me, where fs=geBB /h is the electron spin-flip often called preces-sion frequency. The marked improvement achieved bythe Harvard group, the culmination of a 20 year effort,is due in large part to the use of a cylindrical Penningtrap with a resonant cavity that interacts with thetrapped electron in a readily calculable way, and throughits high Q resonances, significantly increases the lifetimeof the electron in its lowest few energy states by inhib-iting the decay of these states through spontaneousemission. Further, cooling the trap and its vacuum enclo-sure to 100 mK by means of a dilution refrigerator elimi-nates blackbody radiation that could excite the electronfrom these states.

The frequencies fa and fc are determined by applyingquantum-jump spectroscopy QJS to transitions be-tween the lowest spin ms= ±1/2 and cyclotron n=0,1 ,2 quantum states of the electron in the trap. InQJS, the quantum jumps per attempt to drive them aremeasured as a function of drive frequency. The transi-tions are induced by applying a signal of frequency fato trap electrodes or by transmitting microwaves of fre-quency fc into the trap cavity. A change in the cyclo-tron or spin state of the electron is reflected in a shift inz, the self-excited axial oscillation of the electron. The

trap axis and B are in the z direction. This oscillationinduces a signal in a resonant circuit that is amplifiedand fed back to the trap to drive the oscillation. Satu-rated nickel rings surrounding the trap produce a smallmagnetic bottle that provides quantum nondemolitioncouplings of the spin and cyclotron energies to z. Fail-ure to resolve the cyclotron energy levels would result inan increase of uncertainty due to the leading relativisticcorrection / fchfc /mc210−9.

Another unique feature of the Harvard experiment isthat the effect of the trap cavity modes on fc, and henceon the measured value of ae, are directly observed forthe first time. The modes are quantitatively identified asthe familiar transverse electric TE and transverse mag-netic TM modes by observing the response of a cloudof electrons to an axial parametric drive, and, based onthe work of Brown and Gabrielse 1986, the range ofpossible shifts of fc for a cylindrical cavity with a Q500 as used in the Harvard experiment can be readilycalculated. Two measurements of ae were made: one,which resulted in the value of ae given in Eq. 103, wasat a value of B for which fc=149 GHz, far from modesthat couple to the cyclotron motion; the other was at146.8 GHz, close to mode TE127. Within the calibrationand identification uncertainties for the mode frequen-cies, good agreement was found between the measuredand predicted difference in the two values. Indeed, theirweighted mean gives a value of ae that is larger than thevalue in Eq. 103 by only the fractional amount 0.510−10, with u −1

r slightly reduced to 6.510 0.The largest component of uncertainty, 5.210−10, in

the 6.610−10 ur of the Harvard result for ae arises fromfitting the resonance line shapes for fa and fc obtainedfrom the quantum jump spectroscopy data. It is basedon the consistency of three different methods of extract-ing these frequencies from the line shapes. The methodthat yielded the best fits and which was used to obtainthe reported value of ae weights each drive frequency,spin flip or cyclotron, by the number of quantum jumpsit produces, and then uses the weighted average of theresulting spin flip and cyclotron frequencies in the finalcalculation of ae. Although the cavity shifts are wellcharacterized, they account for the second largest frac-tional uncertainty component, 3.410−10. The statisticalType A component, which is the next largest, is only1.510−10.

3. Values of inferred from ae

Equating the theoretical expression with the two ex-perimental values of ae given in Eqs. 102 and 103yields

−1 ae = 137.035 998 8350 3.7 −9 10 104

from the University of Washington result and−1 a −10

e = 137.035 999 71196 7.0 10 105

from the Harvard University result. The contribution ofthe uncertainty in aeth to the relative uncertainty of

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either of these results is 2.410−10. The value in Eq.105 has the smallest uncertainty of any value of alphacurrently available. Both values are included in TableXIV.

B. Muon magnetic moment anomaly a

Comparison of theory and experiment for the muonmagnetic moment anomaly gives a test of the theory ofthe hadronic contributions, with the possibility of reveal-ing physics beyond the standard model.

1. Theory of a

The current theory of a has been throughly reviewedin a number of recent publications; see, for example,Davier et al. 2006, Jegerlehner 2007, Melnikov andVainshtein 2006, Miller et al. 2007, and Passera2005.

The relevant mass-dependent terms and correspond-ing contributions to ath, based on the 2006 recom-mended values of the mass ratios, are

A42 m/me = 1.094 258 308882

→ 506 386.456138 −8 10 a, 106

A42 m/m = 0.000 078 06425

→ 36.12612 10−8a, 107

A62 m/me = 22.868 379 9719

→ 24 581.766 1620 10−8a, 108

A62 m/m = 0.000 360 5121

→ 0.387 5222 −8 10 a, 109

A82 m −8

/me = 132.682372 → 331.28818 10 a,

110

A102 m/me = 66320 111

→3.8512 10−8 a, 112

A63 m/me,m/m = 0.000 527 6617

→ 0.567 2018 10−8a, 113

A83 m/me,m/m = 0.037 59483

→ 0.093 8721 10−8a. 114

These contributions and their uncertainties, as well asthe values including their uncertainties of aweakand ahad given below, should be compared with the5410−8a standard uncertainty of the experimentalvalue of a from Brookhaven National LaboratoryBNL see next section.

Some of the above terms reflect the results of recentcalculations. The value of A6

2 m /m in Eq. 109includes an additional contribution as discussed in

connection with Eq. 91. The terms A82 m /me and

A83 m /me ,m /m have been updated by Kinoshita

and Nio 2004, with the resulting value forA8

2 m /me in Eq. 110 differing from the previousvalue of 127.5041 due to the elimination of variousproblems with earlier calculations, and the resultingvalue for A8

3 m /me ,m /m in Eq. 114 differing fromthe previous value of 0.0793, because diagrams thatwere thought to be negligible do in fact contribute to theresult. Further, the value for A10

2 m /me in Eq. 111from Kinoshita and Nio 2006b replaces the previousvalue, 930170. These authors believed that their result,obtained from the numerical evaluation of all integralsfrom 17 key subsets of Feynman diagrams, accountsfor the leading contributions to A10

2 m /me, andthe work of Kataev 2006, based on the so-calledrenormalization-group-inspired scheme-invariant ap-proach, supports this view.

The electroweak contribution to ath is taken to be

aweak = 1542 10−11, 115

as given by Czarnecki et al. 2003, 2006. This value wasused in the 2002 adjustment and is discussed inCODATA-02.

The hadronic contribution to ath may be written as

a 4 6ahad = a had + a had + a

had + ¯ ,

116

where a4 had and a 6a

had arise from hadronicvacuum polarization and are of order /2 and /3,respectively; and a

had, which arises from hadroniclight-by-light vacuum polarization, is also of order /3.

Values of a4 had are obtained from calculations that

evaluate dispersion integrals over measured cross sec-tions for the scattering of e+e− into hadrons. In addition,in some such calculations, data on decays of the intohadrons are used to replace the e+e− data in certainparts of the calculation. In the 2002 adjustment, resultsfrom both types of calculation were averaged to obtain avalue that would be representative of both approaches.

There have been improvements in the calculationsthat use only e+e− data with the addition of new datafrom the detectors CMD-2 at Novosibirsk, KLOE atFrascati, BaBar at the Stanford Linear Accelerator Cen-ter, and corrected data from the detector SND at No-vosibirsk Davier, 2007; Hagiwara et al., 2007; Jegerleh-ner, 2007. However, there is a persistent disagreementbetween the results that include the decay data andthose that use only e+e− data. In view of the improve-ments in the results based solely on e+e− data and theunresolved questions concerning the assumptions re-quired to incorporate the data into the analysis Melni-kov and Vainshtein, 2006; Davier, 2006; Davier et al.,2006, we use in the 2006 adjustment results based solelyon e+e− data. The value employed is

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a4 had = 69021 10−10, 117

which is the unweighted mean of the values a4 had

=689.44.610−10 Hagiwara et al., 2007 anda4

had=690.94.410−10 Davier, 2007. The uncer-tainty assigned to the value of a4

had, as expressed inEq. 117, is essentially the difference between the val-ues that include data and those that do not. In particu-lar, the result that includes data that we use to estimatethe uncertainty is 711.05.810−11 from Davier et al.2003; the value of a4

had used in the 2002 adjustmentwas based in part on this result. Although there is thesmaller value 701.85.810−11 from Trocóniz and Yn-duráin 2005, we use only the larger value in order toobtain an uncertainty that covers the possibility of phys-ics beyond the standard model not included in the cal-culation of ath. Other, mostly older results fora4

had, but which in general agree with the two valueswe have averaged, are summarized in Table III of Jeger-lehner 2007.

For the second term in Eq. 116, we employ the value

a6a had = − 97.9095 10−11 118

calculated by Hagiwara et al. 2004, which was also usedin the 2002 adjustment.

The light-by-light contribution in Eq. 116 has beencalculated by Melnikov and Vainshtein 2004, 2006, whoobtain the value

a had = 13625 10−11. 119

It is somewhat larger than earlier results, because it in-cludes short distance constraints imposed by quantumchromodynamics QCD that were not included in theprevious calculations. It is consistent with the 95% con-fidence limit upper bound of 15910−11 for a

hadobtained by Erler and Sánchez 2006, the value1104010−11 proposed by Bijnens and Prades 2007,and the value 1253510−11 suggested by Davier andMarciano 2004.

The total hadronic contribution is

ahad = 69421 10−10 = 59518 10−7a. 120

Combining terms in aQED that have like powers of /, we summarize the theory of a as follows:

ath = aQED + aweak + ahad , 121

where

2 3 a QED = C2

+ C4

+ C6

4

+ C8

+ C10

5+

¯ , 122

with

C2 = 0.5,

C4 = 0.765 857 40827 ,

C6 = 24.050 509 5942 ,

C8 = 130.991680 ,

C10 = 66320 , 123

and aweak and ahad are as given in Eqs. 115 and120. The standard uncertainty of ath from the uncer-tainties of the terms listed above, other than that due to, is

uath = 2.1 10−9 = 1.8 10−6a, 124

and is primarily due to the uncertainty of ahad.For the purpose of the least-squares calculations car-

ried out in Sec. XII.B, we define an additive correction to ath to account for the lack of exact knowledgeof ath, and hence the complete theoretical expressionfor the muon anomaly is

a, = ath + . 125

Our theoretical estimate of is zero and its standarduncertainty is uath,

= 0.02.1 10−9. 126

Although ath and aeth have some common compo-nents of uncertainty, the covariance of and e is neg-ligible.

2. Measurement of a: Brookhaven

Experiment E821 at Brookhaven National LaboratoryBNL, Upton, New York, was initiated by the Muong−2 Collaboration in the early 1980s with the goal ofmeasuring a with a significantly smaller uncertaintythan ur=7.210−6. This is the uncertainty achieved inthe third g−2 experiment carried out at the EuropeanOrganization for Nuclear Research CERN, Geneva,Switzerland, in the mid-1970s using both positive andnegative muons and which was the culmination of nearly20 years of effort Bailey et al., 1979.

The basic principle of the experimental determinationof a is similar to that used to determine ae and involvesmeasuring the anomaly difference frequency fa= fs− fc,where fs= ge /2mB /h is the muon spin-flip oftencalled precession frequency in the applied magnetic fluxdensity B and fc=eB /2m is the corresponding muoncyclotron frequency. However, instead of eliminating Bby measuring fc as is done for the electron, B is deter-mined from proton nuclear magnetic resonance NMRmeasurements. As a consequence, the value of /p isrequired to deduce the value of a from the data. Therelevant equation is

Ra = , 127

/p − R

¯ ¯where R= fa / fp and fp is the free proton NMR frequencycorresponding to the average flux density seen by themuons in their orbits in the muon storage ring used inthe experiment. Of course, in the corresponding experi-ment for the electron, a Penning trap is employed ratherthan a storage ring.

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The BNL a experiment was discussed in bothCODATA-98 and CODATA-02. In the 1998 adjust-ment, the CERN final result for R with ur=7.210−6,and the first BNL result for R, obtained from the 1997engineering run using positive muons and with ur=1310−6, were taken as input data. By the time of the 2002adjustment, the BNL experiment had progressed to thepoint where the CERN result was no longer competi-tive, and the input datum used was the BNL mean valueof R with ur=6.710−7 obtained from the 1998, 1999,and 2000 runs using +. The final run of the BNL E821experiment was carried out in 2001 with − and achievedan uncertainty for R of ur=7.010−7, but the result onlybecame available in early 2004, well after the closingdate of the 2002 adjustment.

Based on the data obtained in all five runs and assum-ing CPT invariance, an assumption justified by the con-sistency of the values of R obtained from either + or−, the final report on the E821 experiment gives as thefinal value of R Bennett et al., 2006 see also Miller etal., 2007:

R = 0.003 707 206420 5.4 10−7 , 128

which we take as an input datum in the 2006 adjustment.A new BNL experiment to obtain a value of R with asmaller uncertainty is under discussion Hertzog 2007.

The experimental value of a implied by this value ofR is, from Eq. 127 and the 2006 recommended value of /p the uncertainty of which is inconsequential inthis application,

aexp = 1.165 920 9363 10−3 5.4 10−7 .

129

Further, with the aid of Eq. 217 in Sec. VI.B, Eq. 127can be written as

a, m R = − e e−

, 1301 + ae,e m p

where we have used ge=−21+ae and g=−21+a andreplaced ae and a with their complete theoretical ex-pressions ae ,e and a ,, discussed in Secs. V.A.1and V.B.1, respectively. Equation 130 is, in fact, theobservational equation for the input datum R.

a. Theoretical value of a and inferred value of

Evaluation of the theoretical expression for a in Eq.121 with the 2006 recommended value of , the uncer-tainty of which is negligible in this context, yields

ath = 1.165 918121 10−3 1.8 10−6 , 131

which may be compared to the value in Eq. 129 de-duced from the BNL result for R given in Eq. 128. Theexperimental value exceeds the theoretical value by1.3udiff, where udiff is the standard uncertainty of the dif-ference. It should be recognized, however, that thisagreement is a consequence of the comparatively largeuncertainty we have assigned to a4

had see Eq. 120.

If the result for a4 had that includes tau data were

ignored and the uncertainty of a4 had were based on

the estimated uncertainties of the calculated values us-ing only e+e− data, then the experimental value wouldexceed the theoretical value by 3.5udiff. This inconsis-tency is well known to the high-energy physics commu-nity and is of considerable interest because it may be anindication of “new physics” beyond the Standard Model,such as supersymmetry Stöckinger, 2007.

One might ask, why include the theoretical value fora in the 2006 adjustment given its current problems? Byretaining the theoretical expression with an increaseduncertainty, we ensure that the 2006 recommendedvalue of a reflects, albeit with a comparatively smallweight, the existence of the theoretical value.

The consistency between theory and experiment mayalso be examined by considering the value of obtainedby equating the theoretical expression for a with theBNL experimental value, as done for ae in Sec. V.A.3.The result is

−1 = 137.035 6726 1.9 −6 10 , 132

which is the value included in Table XIV.

C. Bound electron g-factor in 12C5+ and in 16O7+ and Ar(e)

Precise measurements and theoretical calculations forthe g-factor of the electron in hydrogenic 12C and inhydrogenic 16O lead to values of Are that contribute tothe determination of the 2006 recommended value ofthis important constant.

For a ground-state hydrogenic ion AXZ−1+ with massnumber A, atomic number proton number Z, nuclearspin quantum number i=0, and g-factor ge−AXZ−1+ inan applied magnetic flux density B, the ratio of the elec-tron’s spin-flip often called precession frequency fs

= g AXZ−1+e− e /2meB /h to the cyclotron frequency

of the ion f Ac= Z−1eB /2m XZ−1+ in the same mag-

netic flux density is

fsAXZ−1+ g +

e−AXZ−1 A +r

AXZ−1 f AXZ−1+ = − , 133

c 2Z − 1 Are

where ArX is the relative atomic mass of particle X. Ifthe frequency ratio fs / fc is determined experimentallywith high accuracy, and ArAXZ−1+ of the ion is alsoaccurately known, then this expression can be used todetermine an accurate value of Are, assuming thebound-state electron g-factor can be calculated fromQED theory with sufficient accuracy; or the g-factor canbe determined if Are is accurately known from anotherexperiment. In fact, a broad program involving a num-ber of European laboratories has been underway sincethe mid-1990s to measure the frequency ratio and calcu-late the g-factor for different ions, most notably to date12C5+ and 16O7+. The measurements themselves are be-ing performed at the Gesellschaft für Schwerionen-forschung, Darmstadt, Germany GSI by GSI andUniversity of Mainz researchers, and we discuss the ex-

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TABLE XV. Theoretical contributions and total for theg-factor of the electron in hydrogenic carbon 12 based on the2006 recommended values of the constants.

Contribution Value Source

Dirac gD −1.998 721 354 4022 Eq. 1382gSE −0.002 323 672 4264 Eq. 1462gVP 0.000 000 008 5121 Eq. 1524g 0.000 003 545 67725 Eq. 1566g −0.000 000 029 618 Eq. 1588g 0.000 000 000 101 Eq. 15910g 0.000 000 000 0001 Eq. 160

grec −0.000 000 087 63910 Eqs. 161–163gns −0.000 000 000 4081 Eq. 165ge−12C5+ −2.001 041 590 20328 Eq. 166

perimental determinations of fs / fc for 12C5+ and 16O7+ atGSI in Secs. V.C.2.a and V.C.2.b. The theoretical expres-sions for the bound-electron g-factors of these two ionsare reviewed in the next section.

1. Theory of the bound electron g-factor

In this section, we consider an electron in the 1S stateof hydrogenlike carbon 12 or oxygen 16 within theframework of bound-state QED. The measured quantityis the transition frequency between the two Zeeman lev-els of the atom in an externally applied magnetic field.

The energy of a free electron with spin projection sz ina magnetic flux density B in the z direction is

eE = − · B = − ge− szB , 134

2me

and hence the spin-flip energy difference is

E = − ge−BB . 135

In keeping with the definition of the g-factor in Sec. V,the quantity ge− is negative. The analogous expressionfor ions with no nuclear spin is

EbX = − ge−XBB , 136

which defines the bound-state electron g-factor, andwhere X is either 12C5+ or 16O7+.

The theoretical expression for ge−X is written as

ge−X = gD + grad + grec + gns + ¯ , 137

where the individual terms are the Dirac value, the ra-diative corrections, the recoil corrections, and thenuclear size corrections, respectively. These theoreticalcontributions are discussed in the following paragraphs;numerical results based on the 2006 recommended val-ues are summarized in Tables XV and XVI. In the 2006adjustment, in the expression for gD is treated as avariable, but the constants in the rest of the calculationof the g-factors are taken as fixed quantities.

Breit 1928 obtained the exact value

TABLE XVI. Theoretical contributions and total for theg-factor of the electron in hydrogenic oxygen 16 based on the2006 recommended values of the constants.

Contribution Value Source

Dirac gD −1.997 726 003 08 Eq. 138gSE

2 −0.002 324 442 121 Eq. 146gVP

2 0.000 000 026 38 Eq. 152g4 0.000 003 546 5411 Eq. 156g6 −0.000 000 029 63 Eq. 158g8 0.000 000 000 10 Eq. 159g10 0.000 000 000 00 Eq. 160grec −0.000 000 117 021 Eqs. 161–163gns −0.000 000 001 561 Eq. 165ge−16O7+ −2.000 047 020 3811 Eq. 166

gD = − 23 1 + 21 − Z2

= − 21 − 1 Z 2 1 4 1 6 − 3 12Z − 24Z + ¯ 138

from the Dirac equation for an electron in the field of afixed-point charge of magnitude Ze, where the only un-certainty is that due to the uncertainty in .

The radiative

corrections

may be written

as2

C 2 grad = − 2 Z + C4

e e Z + ¯ ,

139

where the coefficients C2ne Z, corresponding to n vir-

tual photons, are slowly varying functions of Z. Thesecoefficients are defined in direct analogy with the corre-sponding coefficients for the free electron C2n

e given inEq. 98 so that

lim C2ne Z = C 2n

e . 140Z→0

The first two terms of the coefficient C2e Z have

been known for some time Faustov, 1970; Grotch, 1970;Close and Osborn, 1971. Recently, Pachucki et al.2004, Pachucki, Czarnecki, Jentschura, et al. 2005,and Pachucki, Jentschura, and Yerokhin 2005 calcu-lated additional terms with the result

C2 1 Z2

e,SEZ 4 32 −2 = 1 + + Z6

lnZ2 9

247 8 8+ − ln k

216 9 0 − ln k3 3

+ Z5RSEZ , 141

where

ln k0 = 2.984 128 556, 142

ln k3 = 3.272 806 545, 143

RSE6 = 22.16010 , 144

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RSE8 = 21.8594 . 145

The quantity ln k0 is the Bethe logarithm for the 1S statesee Table VII and ln k3 is a generalization of the Bethelogarithm relevant to the g-factor calculation. The re-mainder function RSEZ was obtained by Pachucki etal. 2004 and Pachucki, Jentschura, and Yerokhin,2005 by extrapolation of numerical calculations of theself energy for Z8 by Yerokhin et al. 2002 using Eq.141 to remove the lower-order terms. For Z=6 and 8,this yields

C2e,SE6 = 0.500 183 606 6580 ,

C2e,SE8 = 0.500 349 288714 . 146

The lowest-order vacuum-polarization correction con-sists of a wave-function correction and a potential cor-rection. The wave-function correction has been calcu-lated numerically by Beier et al. 2000 with the result inour notation

C2e,VPwf6 = − 0.000 001 840 343143 ,

C2e,VPwf8 = − 0.000 005 712 02826 . 147

Each of these values is the sum of the Uehling potentialcontribution and the higher-order Wichmann-Kroll con-tribution, which were calculated separately with the un-certainties added linearly, as done by Beier et al. 2000.The values in Eq. 147 are consistent with an evaluationof the correction in powers of Z Karshenboim, 2000;Karshenboim et al., 2001a, 2001b. For the potential cor-rection, Beier et al. 2000 found that the Uehling poten-tial contribution is zero and calculated the Wichmann-Kroll contribution numerically over a wide range of ZBeier, 2000. An extrapolation of the numerical valuesfrom higher Z, taken together with the analytic result ofKarshenboim and Milstein 2002,

7C2

e,VPpZ = Z5 +432

¯ , 148

for the lowest-order Wichmann-Kroll contribution,yields

C2e,VPp6 = 0.000 000 007 959569 ,

C2e,VPp8 = 0.000 000 033 23529 . 149

More recently, Lee et al. 2005 obtained the result

C2e,VPp6 = 0.000 000 008 20111 ,

C2e,VPp8 = 0.000 000 034 2311 . 150

The values in Eqs. 149 and 150 disagree somewhat, soin the present analysis we use a value that is an un-weighted average of the two, with half the difference forthe uncertainty. These average values are

C2e,VPp6 = 0.000 000 008 0812 ,

C2e,VPp8 = 0.000 000 033 7350 . 151

The total one-photon vacuum polarization coefficientsare given by the sum of Eqs. 147 and 151,

C2 6 = C2e,VP e,VPwf6 + C2

e,VPp6

= − 0.000 001 832 2612 ,

C2 8 = C2e,VP e,VPwf8 + C2

e,VPp8

= − 0.000 005 678 3050 . 152

The total for the one-photon coefficient C2e Z,

given by the sum of Eqs. 146 and 152, is

C2e 6 = C2 2 e,SE6 + Ce,VP6

= 0.500 181 774 3881 ,

C2e 8 = C2

e,SE8 + C2e,VP8

= 0.500 343 610414 , 153

and the total one-photon contribution g2 to theg-factor is

g2 = − 2C2e Z

= − 0.002 323 663 9144 for Z = 6

= − 0.002 324 415 7467 for Z = 8. 154

The separate one-photon self energy and vacuum polar-ization contributions to the g-factor are given in TablesXV and XVI.

Calculations by Eides and Grotch 1997a using theBargmann-Michel-Telegdi equation and by Czarnecki etal. 2001 using an effective potential approach yield

2n 2n Z 2Ce Z = Ce 1 + + ¯ 155

6

as the leading binding correction to the free-electron co-efficients C2n

e for any order n. For C2e Z, this correc-

tion was known for some time. For higher-order terms, itprovides the leading binding effect.

The two-loop contribution of relative order Z4 wasrecently calculated by Jentschura, et al. 2006 and Pa-chucki, Czarnecki, Jentschura, et al. 2005 for any Sstate. Their result for the ground-state correction is

2C4

Z = C4e Z

e 1 +6

+ Z414lnZ−2

9

991 343 2 4 6792

+ − ln k − ln k +155 520 9 0 3 3 12 960

14412 1441− ln 2 + 3 + OZ5

720 480

= − 0.328 577823 for Z = 6

= − 0.328 657897 for Z = 8, 156

where ln k0 and ln k3 are given in Eqs. 142 and 143.The uncertainty due to uncalculated terms is estimatedby assuming that unknown higher-order terms, of orderZ5 or higher for two loops, are comparable to the

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higher-order one-loop terms scaled by the free-electroncoefficients in each case, with an extra factor of 2 in-cluded Pachucki, Czarnecki, Jentschura, et al., 2005,

uC4e Z = 2Z5C4

e RSEZ . 157

The three- and four-photon terms are calculated withthe leading binding correction included,

2C6Z = C6

e Z e 1 + +

= 1.181 611 . . . for Z = 6

= 1.181 905 . . . for Z = 8, 158

where C6e =1.181 234.

. ., and

Z 2C8

e Z = C8e 1 + +

= − 1.728935 . . . for Z = 6

= − 1.729335 . . . for Z = 8, 159

where C8e =−1.728335 Kinoshita and Nio, 2006a.

This value would shift somewhat if the more recent ten-tative value C8

e =−1.914435 Aoyama et al., 2007 wereused see Sec. V. An uncertainty estimate

C10 10e Z Ce = 0.03.7 160

is included for the five-loop correction.The recoil correction to the bound-state g-factor asso-

ciated with the finite mass of the nucleus is denoted bygrec, which we write here as the sum g0 2

rec+grec cor-responding to terms that are zero and first order in /,respectively. For g0

rec, we have

0 2 Z4

grec = − Z +31 + 1 − Z22

2

− Z 5 m m PZ e +

mNO e

mN

= − 0.000 000 087 711 . . . for Z = 6

= − 0.000 000 117 111 . . . for Z = 8, 161

where mN is the mass of the nucleus. The mass ratios,obtained from the 2006 adjustment, are me /m12C6+=0.000 045 727 5. . . and me /m16O8+=0.000 034 306 5. . ..The recoil terms are the same as in CODATA-02, andreferences to the original calculations are given there.An additional term of the order of the mass ratiosquared is included as

SZZ2 me 2

, 162mN

where SZ is taken to be the average of the disagreeingvalues 1+Z, obtained by Eides 2002 and Eides andGrotch 1997a, and Z /3 obtained by Martynenko andFaustov 2001, 2002 for this term. The uncertainty in SZis taken to be half the difference of the two values.

For g2rec, we have

2 g 2 Z m

rec = e + 3 m

¯

N

= 0.000 000 000 06 . . . for Z = 6

= 0.000 000 000 09 . . . for Z = 8. 163

There is a small correction to the bound-state g-factordue to the finite size of the nucleus, of order

2

ns = − 8 Z 4 Rg N

3 C

+ ¯ , 164

where RN is the bound-state nuclear rms charge radiusand C is the Compton wavelength of the electron di-vided by 2. This term is calculated as in CODATA-02Glazov and Shabaev, 2002 with updated values for thenuclear radii, RN=2.470322 fm and RN=2.701355 fm,from the compilation of Angeli 2004 for 12C and 16O,respectively. This yields the correction

gns = − 0.000 000 000 4081 for 12C,

gns = − 0.000 000 001 561 for 16O. 165

The theoretical value for the g-factor of the electronin hydrogenic carbon 12 or oxygen 16 is the sum of theindividual contributions discussed above and summa-rized in Tables XV and XVI,

g 12 5+e− C = − 2.001 041 590 20328 ,

g 16 7+e− O = − 2.000 047 020 3811 . 166

For the purpose of the least-squares calculations car-ried out in Sec. XII.B, we define gCth to be the sum ofgD as given in Eq. 138, the term −2 /C2

e , and thenumerical values of the remaining terms in Eq. 137 asgiven in Table XV, where the standard uncertainty ofthese latter terms is

ugCth = 0.3 10−10 = 1.4 −1 10 1gCth . 167

The uncertainty in gCth due to the uncertainty in enters the adjustment primarily through the functionaldependence of g

D and the term −2 /C 2e on .

Therefore, this particular component of uncertainty isnot explicitly included in ugCth. To take the uncer-tainty ugCth into account, we employ as the theoret-ical expression for the g-factor

gC,C = gCth + C, 168

where the input value of the additive correction C istaken to be zero and its standard uncertainty isugCth,

0 = −1C 0.0027 10 . 169

Analogous considerations apply for the g-factor in oxy-gen,

ug −10 −11Oth = 1.1 10 = 5.3 10 gOth , 170

gO,O = gOth + O, 171

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TABLE XVII. Summary of experimental data on the electron bound-state g-factor in hydrogeniccarbon and oxygen and inferred values of the relative atomic mass of the electron.

Relative standardInput datum Value uncertainty ur Identification Sec. and Eq.

12C5+ / fc12C5+fs 4376.210 498923 5.210−10 GSI-02 V.C.2.a 175

Are 0.000 548 579 909 3229 5.210−10 V.C.2.a 17716O7+ / fc

16O7+fs 4164.376 183732 7.610−10 GSI-02 V.C.2.b 178Are 0.000 548 579 909 5842 7.610−10 V.C.2.b 181

−1O = 0.01.1 10 0. 172

Since the uncertainties of the theoretical values of thecarbon and oxygen g-factors arise primarily from thesame sources, the quantities C and O are highly corre-lated. Their covariance is

uC, −22O = 27 10 , 173

which corresponds to a correlation coefficient ofrC,O=0.92.

The theoretical value of the ratio of the two g-factors,which is relevant to the comparison to experiment inSec. V.C.2.c, is

g 12 5+e− C

=ge 16O7+ 1.000 497 273 21841 , 174

where the covariance, including the contribution fromthe uncertainty in for this case, is taken into account.

2. Measurements of ge(12C5+) and ge(

16O7+)

The experimental data on the electron bound-stateg-factor in hydrogenic carbon and oxygen and the in-ferred values of Are are summarized in Table XVII.

a. Experiment on ge12C5+ and inferred value of Are

The accurate determination of the frequency ratiofs

12C5+ / fc12C5+ at GSI based on the double Penning-

trap technique was discussed in CODATA-02. See alsothe recent concise review by Werth et al. 2006. Sincethe result used as an input datum in the 2002 adjustmentis unchanged, we take it as an input datum in the 2006adjustment as well Beier et al., 2002; Häffner et al.,2003; Werth, 2003,

f 12C5+s

fc12C5+ = 4376.210 498923 . 175

From Eqs. 133 and 4, we have

fs12C5+ ge−12C5+12 5+ = − 12 − 5Are

fc C 10Are

E 5+b12C − Eb12C

+muc2 , 176

which is the basis for the observational equation for the12C5+ frequency-ratio input datum.

Evaluation of this expression using the result forfs

12C5+ / f 12c C5+ in Eq. 175, the theoretical result for

g 12e− C5+ in Table XV, and the relevant binding ener-

gies in Table IV of CODATA-02, yields

Are = 0.000 548 579 909 3229 5.2 10−10 .

177

This value is consistent with that from antiprotonic he-lium given in Eq. 74 and that from the University ofWashington given in Eq. 5, but has about a factor of 3to 4 smaller uncertainty.

b. Experiment on g 16e O7+ and inferred value of Are

The double Penning-trap determination of the fre-quency ratio fs

16C7+ / fc16C7+ at GSI was also dis-

cussed in CODATA-02, but the value used as an inputdatum was not quite final Verdú et al., 2002, 2003;Werth, 2003. A slightly different value for the ratio wasgiven in the final report of the measurement Verdú etal., 2004, which is the value we take as the input datumin the 2006 adjustment but modified slightly as followsbased on information provided by Verdú 2006: i anunrounded instead of a rounded value for the correctiondue to extrapolating the axial temperature Tz to 0 K wasadded to the uncorrected ratio −0.000 004 7 in place of−0.000 005; and ii a more detailed uncertainty budgetwas employed to evaluate the uncertainty of the ratio.The resulting value is

f 16s O7+

f 16

c O7+ = 4164.376 1837 32 . 178

In analogy with what was done above with the ratiofs

12C5+ / fc12C5+, from Eqs. 133 and 4 we have

f 16O7+ 16−s g 7+

e O 7+

16 7+ = − Ar16O 179

fc O 14Are

with

Ar16O = Ar

16O7+ + 7Are

Eb16O − Eb16O7+−

muc2 , 180

which are the basis for the observational equations forthe oxygen frequency ratio and Ar

16O, respectively.The first expression, evaluated using the result forfs

16C7+ / fc16C7+ in Eq. 178 and the theoretical result

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for ge−16C7+ in Table XVI, in combination with the sec-ond expression, evaluated using the value of Ar

16O inTable IV and the relevant binding energies in Table IVof CODATA-02, yields

Are = 0.000 548 579 909 5842 7.6 10−10 .

181

It is consistent with both the University of Washingtonvalue in Eq. 5 and the value in Eq. 177 obtained fromf 12C5+ 12

s / fc C5+.

c. Relations between g 12 5+e C and ge

16O7+

It should be noted that the GSI frequency ratios for12C5+ and 16O7+ are correlated. Based on the detaileduncertainty budgets of the two results Werth, 2003;Verdú,

2006, we find the correlation coefficient to be

f 12C5+ f 16O7+r s , s

fc12C5+

f 16O7+ = 0.082. 182c

Finally, as a consistency test, it is of interest to com-pare the experimental and theoretical values of the ratioof ge−12C5+ to g 16

e− C7+ Karshenboim and Ivanov,2002. The main reason is that the experimental value ofthe ratio is only weakly dependent on the value of Are.The theoretical value of the ratio is given in Eq. 174and takes into account the covariance of the two theo-retical values. The experimental value of the ratio canbe obtained by combining Eqs. 175, 176, 178–180,and 182, and using the 2006 recommended value forAre. Because of the weak dependence of the experi-mental ratio on Are, the value used is not at all critical.The result is

ge−12C5+16 7+ = 1.000 497 273 6889 8.9 10−10 ,

ge− O

183

in agreement with the theoretical value.

VI. MAGNETIC MOMENT RATIOS AND THE MUON-ELECTRON MASS RATIO

Magnetic moment ratios and the muon-electron massratio are determined by experiments on bound states ofrelevant particles. The free electron and muon magneticmoments are discussed in Sec. V and the theory of theg-factor of an electron bound in an atom with no nuclearspin is considered in Sec. V.C.1.

For nucleons or nuclei with spin I, the magnetic mo-ment can be written as

e = g I 184

2mp

or

= gNi . 185

In Eq. 185, N=e /2mp is the nuclear magneton, de-fined in analogy with the Bohr magneton, and i is the

spin quantum number of the nucleus defined by I2= ii+12 and Iz=−i , . . . , i−1, i, where Iz is the spinprojection. However, in some publications, moments ofnucleons are expressed in terms of the Bohr magnetonwith a corresponding change in the definition of theg-factor.

For atoms with a nonzero nuclear spin, bound-stateg-factors are defined by considering the contribution tothe Hamiltonian from the interaction of the atom withan applied magnetic flux density B. For example, forhydrogen, in the framework of the Pauli approximation,we have

H = He− · p − e−H · B − pH · B

2 = N

Hs · I − ge−H B s · B − gpH I · B ,

186

where H characterizes the strength of the hyperfineinteraction, H is the ground-state hyperfine frequency,s is the spin of the electron, and I is the spin of thenucleus, that is, the proton. Equation 186, or its analogfor other combinations of particles, serves to define thecorresponding bound-state g-factors, which are ge−Hand gpH in this case.

A. Magnetic moment ratios

A number of magnetic moment ratios are of interestfor the 2006 adjustment. The results of measurementsand the inferred values of various quantities are summa-rized in Sec. VI.A.2, and the measurement results them-selves are also summarized in Table XIX.

The inferred moment ratios depend on the relevanttheoretical binding corrections that relate the g-factormeasured in the bound state to the corresponding free-particle g-factor. To use the results of these experimentsin the 2006 adjustment, we employ theoretical expres-sions that give predictions for the moments and g-factorsof the bound particles in terms of free-particle momentsand g-factors as well as adjusted constants; this is dis-cussed in the following section. However, in a number ofcases, the differences between the bound-state and free-state values are sufficiently small that the adjusted con-stants can be taken as exactly known.

1. Theoretical ratios of atomic bound-particle to free-particleg-factors

Theoretical g-factor-related quantities used in the2006 adjustment are the ratio of the g-factor of the elec-tron in the ground state of hydrogen to that of the freeelectron ge−H /ge−; the ratio of the g-factor of the pro-ton in hydrogen to that of the free proton gpH /gp; theanalogous ratios for the electron and deuteron in deute-rium, ge−D /ge− and gdD /gd, respectively; and theanalogous ratios for the electron and positive muon inmuonium, ge−Mu /ge− and g + Mu /g + , respectively.

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These ratios and the references for the relevant calcu-lations are discussed in CODATA-98 and CODATA-02;only a summary of the results is included here.

For the electron in hydrogen, we have

ge−H 1 1 1 = 1 − Z2 − Z4 + Z 2

ge− 3 12 4

1 2 m 1 2

+ Z e + A4 1 − Z2

2 mp 2 1 4

5 m− Z e

2

12 +

m¯ , 187

p

where A41 is given in Eq. 84. For the proton in hydro-

gen, we have

gpH 1 97= 1 − Z − Z3

gp 3 108

1 m 3 + 4+ Z e ap

+6 mp 1 + a

¯ , 188p

where the proton magnetic moment anomaly ap is de-fined by

ap = p − 1 1.793. 189

e/2mp

For deuterium, similar expressions apply for the elec-tron,

ge−D 1 1= 1 − Z 2 − Z 4 1 2 + Z

ge− 3 12 4

1 m 1 1 2+ Z e +

2 mdA42

2 1 −4Z2

5 − Z2 me +

12 ¯ 190

md

and deuteron,

gdD 1 97= 1 − Z − Z3

gd 3 108

1 m+ Z e 3 + 4ad +

6 md 1 + a¯ , 191

d

where the deuteron magnetic moment anomaly ad is de-fined by

ad = d − 1 − 0.143. 192

e/md

In the case of Mu, some additional higher-order termsare included because of the larger mass ratio. For theelectron in muonium, we have

TABLE XVIII. Theoretical values for various bound-particleto free-particle g-factor ratios relevant to the 2006 adjustmentbased on the 2006 recommended values of the constants.

Ratio Value

ge−H /ge− 1−17.705410−6

gpH /gp 1−17.735410−6

ge−D /ge− 1−17.712610−6

gdD /gd 1−17.746110−6

ge−Mu /ge− 1−17.592610−6

g +Mu /g + 1−17.625410−6

ge−Mu 1 1= 1 − Z 22 1

− Z4 + Zge− 3 12 4

1 2+ Z 2 me 1 + A 4 1

− Z2

2 m 1 2

4

5 1− Z 2

12 me − 1 + Z

m 2

Z 2 m 2e

m

+ ¯ , 193

and for the muon in muonium, the ratio is

g + Mu 1 97 1 m= 1 − Z − Z3 + Z e

g + 3 108 2 m

1 m 1+ Z

12 e − 1 + ZZ

m 2

m 2e

+ ¯ . 194m

The numerical values of the corrections in Eqs.187–194, based on the 2006 adjusted values of therelevant constants, are listed in Table XVIII. Uncertain-ties are negligible at the level of uncertainty of the rel-evant experiments.

2. Ratio measurements

a. Electron to proton magnetic moment ratio e /p

The ratio e /p is obtained from measurements of theratio of the magnetic moment of the electron to themagnetic moment of the proton in the 1S state of hydro-gen e−H /pH. We use the value obtained by Win-kler et al. 1972 at MIT,

e−H= − 658.210 705866 1.0 10−8 , 195

pH

where a minor typographical error in the original publi-cation has been corrected Kleppner, 1997. The free-particle ratio e /p follows from the bound-particle ra-tio and the relation

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e− g ge−pH H −1e−H=

p gp ge− pH

= − 658.210 686066 1.0 10−8 , 196

where the bound-state g-factor ratios are given in TableXVIII.

b. Deuteron to electron magnetic moment ratio d /e

From measurements of the ratio dD /e−D in the1S state of deuterium, Phillips et al. 1984 at MIT ob-tained

dD= − 4.664 345 39250 −4 −8 10 1.1 10 .

e−D

197

Although this result has not been published, as in the1998 and 2002 adjustments, we include it as an inputdatum, because the method has been described in detailby Winkler et al. 1972 in connection with their mea-surement of e−H /pH. The free-particle ratio isgiven by

gd e−Dg −1dD

=e− ge− gd

dDe−D

= − 4.664 345 54850 10−4 1.1 10−8 ,

198

with the bound-state g-factor ratios given in TableXVIII.

c. Proton to deuteron and triton to proton magnetic momentratios p /d and t /p

The ratios p /d and t /p can be determined byNMR measurements on the HD molecule bound stateof hydrogen and deuterium and the HT moleculebound state of hydrogen and tritium, 3H, respectively.The relevant expressions are see CODATA-98

pHD = 1 + p

−dHD dHD pHD , 199

d

tHT = 1 − tHT + HT

HT p t , 200p p

where pHD and dHD are the proton and deuteronmagnetic moments in HD, respectively, and pHD anddHD are the corresponding nuclear magnetic shield-ing corrections. Similarly, tHT and pHT are thetriton nucleus of tritium and proton magnetic momentsin HT, respectively, and tHT and pHT are the cor-responding nuclear magnetic shielding corrections.Note that bound= 1−free and the nuclearmagnetic shielding corrections are small.

The determination of d /p from NMR measure-ments on HD by Wimett 1953 and by a Russian groupin St. Petersburg Neronov et al., 1975; Gorshkov et al.,1989 was discussed in CODATA-98. However, for rea-

sons given there, mainly the lack of sufficient informa-tion to assign a reliable uncertainty to the reported val-ues of dHD /pHD and also to the nuclear magneticshielding correction difference dHD−pHD, the re-sults were not used in the 1998 or 2002 adjustments.Further, since neither of these adjustments addressedquantities related to the triton, the determination oft /p from measurements on HT by the Russian groupNeronov and Barzakh, 1977 was not considered in ei-ther of these adjustments. It may be recalled that a sys-tematic error related to the use of separate inductancecoils for the proton and deuteron NMR resonances inthe measurements of Neronov et al. 1975 was elimi-nated in the HT measurements of Neronov and Barzakh1977 as well as in the HD measurements of Gorshkovet al. 1989.

Recently, a member of the earlier St. Petersburggroup together with one or more Russian colleaguespublished the following results based in part on newmeasurements and reexamination of relevant theoryNeronov and Karshenboim, 2003; Karshenboim et al.,2005,

pHD= 3.257 199 53129 8.9 10−9 , 201

dHD

tHT= 1.066 639 88710 9.4 10−9 , 202

pHT

dp dHD − pHD = 152 10−9, 203

tp tHT − pHT = 203 10−9, 204

which together with Eqs. 199 and 200 yield

p = 3.257 199 48230 9.1 10−9 , 205d

t = 1.066 639 90810 9.8 10−9 . 206p

The purpose of the new work Neronov and Karshen-boim, 2003; Karshenboim et al., 2005 was i to checkwhether rotating the NMR sample and using a high-pressure gas as the sample 60 to 130 atmospheres,which was the case in most of the older Russian experi-ments, influenced the results and to report a value ofpHD /dHD with a reliable uncertainty; and ii toreexamine the theoretical values of the nuclear magneticshielding correction differences dp and tp and their un-certainties. It was also anticipated that based on thisnew work, a value of tHT /pHT with a reliableuncertainty could be obtained from the highly precisemeasurements of Neronov and Barzakh 1977.However, Gorshkov et al. 1989, as part of their ex-periment to determine d /p, compared the resultfrom a 100 atmosphere HD rotating sample with a100 atmosphere HD nonrotating sample and found nostatistically significant difference.

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TABLE XIX. Summary of data for magnetic moment ratios of various bound particles.

Quantity Value uncertainty ur Identification Sec. and Eq.

e−H /pH −658.210 705866 1.010−8 MIT-72 VI.A.2.a 195dD /e−D −4.664 345 3925010−4 1.110−8 MIT-84 VI.A.2.b 197pHD /dHD 3.257 199 53129 8.910−9 StPtrsb-03 VI.A.2.c 201dp 15210−9 StPtrsb-03 VI.A.2.c 203tHT /pHT 1.066 639 88710 9.410−9 StPtrsb-03 VI.A.2.c 202tp 20310−9 StPtrsb-03 VI.A.2.c 204e−H /p −658.215 943072 1.110−8 MIT-77 VI.A.2.d 209h /p −0.761 786 131333 4.310−9 NPL-93 VI.A.2.e 211n /p −0.684 996 9416 2.410−7 ILL-79 VI.A.2.f 212

To test the effect of sample rotation and sample pres-sure, Neronov and Karshenboim 2003 performed mea-surements using a commercial NMR spectrometer oper-ating at a magnetic flux density of about 7 T and anonrotating 10 atmosphere HD gas sample. Because ofthe relatively low pressure, the NMR signals were com-paratively weak and a measurement time of 1 h was re-quired. To simplify the measurements, the frequency ofthe proton NMR signal from HD was determined rela-tive to the frequency of the more easily measured pro-ton NMR signal from acetone, CH32CO. Similarly, thefrequency of the deuteron NMR signal from HD wasdetermined relative to the frequency of the more easilymeasured deuteron NMR signal from deuterated ac-etone, CD32CO. A number of tests involving the mea-surement of the hyperfine interaction constant in thecase of the proton triplet NMR spectrum, and the isoto-pic shift in the case of the deuteron, where the deuteronHD doublet NMR spectrum was compared with the sin-glet spectrum of D2, were carried out to investigate thereliability of the new data. The results of the tests werein good agreement with the older results obtained withsample rotation and high gas pressure.

The more recent result for pHD /dHD reportedby Karshenboim et al. 2005, which was obtained withthe same NMR spectrometer employed by Neronov andKarshenboim 2003 but with a 20 atmosphere nonrotat-ing gas sample, agrees with the 10 atmosphere nonrotat-ing sample result of the latter researchers and is inter-preted by Karshenboim et al. 2005 as confirming the2003 result. Although the values of pHD /dHD re-ported by the Russian researchers in 2005, 2003, and1989 agree, the 2003 result as given in Eq. 201 andTable XIX, the uncertainty of which is dominated by theproton NMR line fitting procedure, is taken as the inputdatum in the 2006 adjustment because of the attentionpaid to possible systematic effects, both experimentaland theoretical.

Based on their HD measurements and related analy-sis, especially the fact that sample pressure and rotationdo not appear to be a problem at the current level ofuncertainty, Neronov and Karshenboim 2003 con-cluded that the result for tHT /pHT reported by

Neronov and Barzakh 1977 is reliable but that itshould be assigned about the same relative uncertaintyas their result for pHD /dHD. We therefore in-clude as an input datum in the 2006 adjustment the re-sult for tHT /pHT given in Eq. 202 and TableXIX.

Without reliable theoretically calculated values for theshielding correction differences dp and tp, reliable ex-perimental values for the ratios pHD /dHD andtHT /pHT are of little use. Although Neronov andBarzakh 1977 give theoretical estimates of these quan-tities based on their own calculations, they do not dis-cuss the uncertainties of their estimates. To address thisissue, Neronov and Karshenboim 2003 examined thecalculations and concluded that a reasonable estimate ofthe relative uncertainty is 15%. This leads to the valuesfor dp and tp in Eqs. 203 and 204 and Table XIX,which we also take as input data for the 2006 adjust-ment. For simplicity, we use StPtrsb-03 as the identifierin Table XIX for pHD /dHD, tHT /pHT, dp,and tp, because they are directly or indirectly a conse-quence of the work of Neronov and Karshenboim2003.

The equations for the measured moment ratiospHD /dHD and tHT /pHT in terms of theadjusted constants e− /p, d /e−, t /p, dp, and tpare, from Eqs. 199 and 200,

−1 HD −1

e−p = 1 + d , 207

dHD dp p

e−

tHT

= 1 − tp t . 208pHT p

d. Electron to shielded proton magnetic moment ratio e /p

Based on the measurement of the ratio of the electronmoment in the 1S state of hydrogen to the shielded pro-ton moment at 34.7 °C by Phillips et al. 1977 at MIT,and temperature-dependence measurements of theshielded proton moment by Petley and Donaldson

Relative standard

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1984 at the National Physical Laboratory NPL, Ted-dington, UK, we have

e−H= − 658.215 943072 1.1 10−8 , 209

p

where the prime indicates that the protons are in aspherical sample of pure H2O at 25 °C surrounded byvacuum. Hence

e−=

pge−H

ge−−1e−H

p

= − 658.227 597172 1.1 10−8 , 210

where the bound-state g-factor ratio is given in TableXVIII. Support for the MIT result in Eq. 210 frommeasurements at NPL on the helion see the followingsection is discussed in CODATA-02.

e. Shielded helion to shielded proton magnetic moment ratioh /p

The ratio of the magnetic moment of the helion h, thenucleus of the 3He atom, to the magnetic moment of theproton in H2O was determined in a high-accuracy ex-periment at NPL Flowers et al., 1993 with the result

h = − 0.761 786 131333 4.3 10−9 . 211p

The prime on the symbol for the helion moment indi-cates that the helion is not free, but is bound in a heliumatom. Although the exact shape and temperature of thegaseous 3He sample are unimportant, we assume that itis spherical, at 25 °C, and surrounded by vacuum.

f. Neutron to shielded proton magnetic moment ratio n /p

Based on a measurement carried out at the InstitutMax von Laue–Paul Langevin ILL in Grenoble,France Greene et al., 1977 1979, we have

n = − 0.684 996 9416 2.4 10−7 . 212p

The observational equations for the measured valuesof h /p and n /p are simply

h/p = h/p , 213

n/p = n/p , 214

while the observational equations for the measured val-ues of e−H /pH, dD /e−D, and e−H /p fol-low directly from Eqs. 196, 198, and 210, respec-tively.

B. Muonium transition frequencies, the muon-proton magneticmoment ratio Õp, and muon-electron mass ratiom Õme

Measurements of transition frequencies between Zee-man energy levels in muonium the +e− atom yieldvalues of /p and the muonium ground-state hyper-

fine splitting Mu that depend only weakly on theory.The relevant expression for the magnetic moment ratiois

2 −1+ 2

− f + 2s f f g + Mu= Mu p e p p

4se f 2 , 215p p − 2fpfp g +

where Mu and fp are the sum and difference of twomeasured transition frequencies, fp is the free protonNMR reference frequency corresponding to the mag-netic flux density used in the experiment, g + Mu /g + isthe bound-state correction for the muon in muoniumgiven in Table XVIII, and

e− ge−Muse = , 216

p ge−

where ge−Mu /ge− is the bound-state correction for theelectron in muonium given in the same table.

The muon to electron mass ratio m /me and the muonto proton

m = magnetic moment ratio /p are related by

−1e g . 217me p

p

ge

The theoretical expression for the hyperfine splittingMuth is discussed in the following section and maybe written as

16 m m −3

th = 2Mu cR e 1 + e

3 m m

F,me/m

= FF,me/m , 218

where the function F depends weakly on and me /m.By equating this expression to an experimental value ofMu, one can calculate a value of from a given valueof m /me or one can calculate a value of m /me from agiven value of .

1. Theory of the muonium ground-state hyperfine splitting

This section gives a brief summary of the presenttheory of Mu, the ground-state hyperfine splitting ofmuonium +e− atom. There has been essentially nochange in the theory since the 2002 adjustment. Al-though complete results of the relevant calculations aregiven here, references to the original literature includedin CODATA-98 or CODATA-02 are generally not re-peated.

The hyperfine splitting is given by the Fermi formula,

16 −3

= cR Z3 2 me F 3

m m

1 + e

m . 219

Some of the following theoretical expressions corre-spond to a muon with charge Ze rather than e in orderto identify the source of the terms. The theoretical valueof the hyperfine splitting is given by

Muth = D + rad + rec + r-r + weak

+ had, 220

where the terms labeled D, rad, rec, r-r, weak, and had

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account for the Dirac relativistic, radiative, recoil,radiative-recoil, electroweak, and hadronic strong inter-action contributions to the hyperfine splitting, respec-tively.

The contribution

D, given by the Dirac equation, is

3 17D = F1 + a 1 + Z2 + Z4 + ¯ ,

2 8

221

where a is the muon magnetic moment anomaly.The radiative corrections

are written as

2 = 1 + a D2 rad F Z + D 4 Z

3

+ D6 Z + ¯

, 222

where the functions D2nZ are contributions associ-ated with n virtual photons. The leading term is

D2Z 2 5 2 2 −2 = A1 + ln 2 −2Z + − ln Z

3

+ 281 8− ln 2 −2

3lnZ + 16.9037 . . .

360

5 547Z 2 + ln 2 − lnZ−2

2 96

Z3 + GZZ3, 223

where A21 = 1

2 , as in Eq. 83. The function GZ ac-counts for all higher-order contributions in powers ofZ, and can be divided into parts that correspond to theself energy or vacuum polarization, GZ=GSEZ+GVPZ. We adopt the value

GSE = − 142 , 224

which is the simple mean and standard deviation of thefollowing three values: GSE=−12.02.0 from Blundellet al. 1997, GSE0=−15.91.6 from Nio 2001, 2002,and GSE=−14.31.1 from Yerokhin and Shabaev2001. The vacuum polarization part GVPZ has beencalculated to several orders of Z by Karshenboim et al.1999, 2000. Their expression yields

GVP = 7.2279 . 225

For D4Z, as in CODATA-02, we have

D4Z 4 1 2 −2 = A1 + 0.77174Z + − ln Z3

− 0.6390 . . . lnZ−2 + 102.5Z2

+ ¯ , 226

where A41 is given in Eq. 84.

Finally,

D6Z = A61 + ¯ , 227

where only the leading contribution A61 as given in Eq.

85 is known. Higher-order functions D2nZ with n3 are expected to be negligible.

The recoil contribution is given by

me 3 m Zrec = F −

m 1 −

m /m 2 ln

e

me

1+ Z −2 65

2 ln − 8 ln 2 +

1 + me/m 18

9 27+ ln2

2m m

2 +m 2 − 1 ln 93

+ 2e 2

333 13 m

+ e

2 − − 12 ln 212 m

me 4

Z2

+ 3 m− ln 1

lnZ−2 − ln2Z −2

2 me

6

101+ − 10 ln 2

18lnZ−2 + 4010 Z3

+ ¯ , 228

as discussed in CODATA-02The radiative-recoil

contribution is

2 me 2m 13 mr-r = F − 2 ln + ln

m me 12 me

21 2 35 4+ 3 + + 2 16

+ ln −2 + ln 22

6 9

3

3

341 4− ln −2 − 4010 m

+ − ln3

180

3

me

4 2 m

+ 2 m 2

ln − e 13

3 m F 6 ln 2 +e m 6

+ ¯ , 229

where, for simplicity, the explicit dependence on Z is notshown.

The electroweak contribution due to the exchange ofa Z0 boson is Eides, 1996

weak = − 65 Hz. 230

For the hadronic vacuum polarization contribution,we use the result of Eidelman et al. 2002,

had = 2364 Hz, 231

which takes into account experimental data on the crosssection for e−e+→+− and on the meson leptonicwidth. The leading hadronic contribution is 231.22.9Hz and the next-order term is 52 Hz, giving a total of2364 Hz. The pion and kaon contributions to the had-ronic correction have been considered within a chiralunitary approach and found to be in general agreementwith but have a three times larger uncertainty than thecorresponding contributions given in earlier studies us-ing data from e+-e− scattering Palomar, 2003.

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The standard uncertainty of Muth was discussed inAppendix E of CODATA-02. The four principal sourcesof uncertainty are the terms rad, rec, r-r, and hadin Eq. 220. Included in the 67 Hz uncertainty of r-r isa 41 Hz component, based on the partial calculations ofEides et al. 2002, 2003 and Li et al. 1993, to accountfor a possible uncalculated radiative-recoil contributionof order Fme /m /3 lnm /me and nonlogarith-mic terms. Since the completion of the 2002 adjustment,results of additional partial calculations have been pub-lished that would lead to a small reduction in the 41 Hzestimate Eides and Shelyuto, 2003, 2004, 2007; Eides etal., 2004, 2005. However, because the calculations arenot yet complete and the decrease of the 101 Hz totaluncertainty assigned to Muth for the 2002 adjustmentwould only be a few percent, the Task Group decided toretain the 101 Hz uncertainty for the 2006 adjustment.

We thus have for the standard uncertainty of the the-oretical expression for the muonium hyperfine splittingMuth

u −8Muth = 101 Hz 2.3 10 . 232

For the least-squares calculations, we use as the theoret-ical expression

for the hyperfine splitting,

mMu R,, e ,

m ,Mu = Muth + Mu, 233

where Mu is assigned a priori the value

Mu = 0101 Hz 234

in order to account for the uncertainty of the theoreticalexpression.

The theory summarized above predicts

Mu = 4 463 302 881272 Hz 6.1 10−8 , 235

based on values of the constants obtained from a varia-tion of the 2006 least-squares adjustment that omits asinput data the two LAMPF measured values of Mudiscussed in the following section.

The main source of uncertainty in this value is themass ratio me /m that appears in the theoretical expres-sion as an overall factor. See the text following Eq.

D14 of Appendix D of CODATA-98 for an explana-tion of why the relative uncertainty of the predictedvalue of Mu in Eq. 235 is smaller than the relativeuncertainty of the electron-muon mass ratio as given inEq. 243 of Sec. VI.B.2.c.

2. Measurements of muonium transition frequencies andvalues of Õp and m Õme

The two most precise determinations of muoniumZeeman transition frequencies were carried out at theClinton P. Anderson Meson Physics Facility at Los Ala-mos LAMPF, USA, and reviewed in CODATA-98.The following three sections and Table XX give the keyresults.

a. LAMPF 1982

The results obtained by Mariam 1981 and Mariam etal. 1982, which we take as input data in the currentadjustment as in the two previous adjustments, may besummarized as follows:

Mu = 4 463 302.8816 kHz 3.6 10−8 , 236

f = 627 994.7714 kHz 2.2 10−7 p , 237

rMu,fp = 0.23, 238

where fp is very nearly 57.972 993 MHz, correspondingto the flux density of about 1.3616 T used in the experi-ment, and rMu,fp is the correlation coefficient ofMu and fp.

b. LAMPF 1999

The results obtained by Liu et al. 1999, which wealso take as input data in the current adjustment as inthe 1998 and 2002 adjustments, may be summarized asfollows:

−8Mu = 4 463 302 76553 Hz 1.2 10 , 239

f −8p = 668 223 16657 Hz 8.6 10 , 240

TABLE XX. Summary of data related to the hyperfine splitting in muonium and inferred values of /p, m /me, and from the combined 1982 and 1999 LAMPF data.

Quantity ValueRelative standard

uncertainty ur Identification Sec. and Eq.

Mu

fp4 463 302.8816 kHz627 994.7714 kHz

3.610−8

2.210−7

LAMPF-82LAMPF-82

VI.B.2.a 236VI.B.2.a 237

Mu

fp4 463 302 76553 Hz668 223 16657 Hz

1.210−8

8.610−8

LAMPF-99LAMPF-99

VI.B.2.b 239VI.B.2.b 240

/p

m /me

−1

3.183 345 2437206.768 27624137.036 001780

1.210−7

1.210−7

5.810−8

LAMPFLAMPFLAMPF

VI.B.2.c 242VI.B.2.c 243VI.B.2.c 244

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rMu,fp = 0.19, 241

where fp is exactly 72.320 000 MHz, corresponding tothe flux density of approximately 1.7 T used in the ex-periment, and rMu,fp is the correlation coefficientof Mu and fp.

c. Combined LAMPF results

By carrying out a least-squares adjustment using onlythe LAMPF 1982 and LAMPF 1999 data, the 2006 rec-ommended values of the quantities R, e /p, ge, andg, together with Eqs. 215–218, we obtain

+= 3.183 345 2437 1.2 10−7 242

p

m = 206.768 27624 1.2 10−7 , 243me

−1 = 137.036 001780 5.8 10−8 , 244

where this value of may be called the muonium valueof the fine-structure constant and denoted as −1Mu.

It is noteworthy that the uncertainty of the value ofthe mass ratio m /me given in Eq. 243 is about fourtimes the uncertainty of the 2006 recommended value.The reason is that taken together, the experimentalvalue of and theoretical expression for the hyperfinesplitting essentially determine only the value of theproduct 2m /me, as is evident from Eq. 218. In thefull adjustment, the value of is determined by otherdata with an uncertainty significantly smaller than thatof the value in Eq. 244, which in turn determines thevalue of m /me with a smaller uncertainty than that ofEq. 243.

VII. ELECTRICAL MEASUREMENTS

This section is devoted to a discussion of quantitiesthat require electrical measurements of the most basickind for their determination: the gyromagnetic ratios ofthe shielded proton and helion, the von Klitzing con-stant RK, the Josephson constant KJ, the product K2

JRK,and the Faraday constant. However, some results we dis-cuss were taken as input data for the 2002 adjustmentbut were not included in the final least-squares adjust-ment from which the 2002 recommended values wereobtained, mainly because of their comparatively largeuncertainties and hence low weight. Nevertheless, wetake them as potential input data in the 2006 adjustmentbecause they provide information on the overall consis-tency of the available data and tests of the exactness ofthe relations KJ=2e /h and RK=h /e2. The lone exceptionis the low-field measurement of the gyromagnetic ratioof the helion reported by Tarbeev et al. 1989. Becauseof its large uncertainty and strong disagreement withmany other data, we no longer consider it—seeCODATA-02.

A. Shielded gyromagnetic ratios , the fine-structure constant, and the Planck constant h

The gyromagnetic ratio of a bound particle of spinquantum number i and magnetic moment is given by

2f = = = , 245

B B i

where f is the precession that is, spin-flip frequencyand is the angular precession frequency of the particlein the magnetic flux density B. The SI unit of is s−1

T−1=C kg−1=A s kg−1. In this section, we summarizemeasurements of the gyromagnetic ratio of the shieldedproton

2pp = , 246

and of the shielded helion

2hh = , 247

where, as in previous sections that dealt with magnetic-moment ratios involving these particles, protons arethose in a spherical sample of pure H2O at 25 °C sur-rounded by vacuum; and helions are those in a sphericalsample of low-pressure, pure 3He gas at 25 °C sur-rounded by vacuum.

As discussed in CODATA-98, two methods are usedto determine the shielded gyromagnetic ratio of aparticle: the low- and high-field methods. In either case,the measured current I in the experiment can be ex-pressed in terms of the product KJRK, but B depends onI differently in the two cases. In essence, the low-fieldexperiments determine /KJRK and the high-field ex-periments determine KJRK. This leads to

K R = J K

Γ90 lo , 248KJ−90RK−90

K = J−90RK−90

Γ90 hi , 249KJRK

where Γ90 lo and Γ90 hi are the experimental values of in SI units that would result from the low- and high-field experiments if KJ and RK had the exactly knownconventional values of KJ−90 and RK−90, respectively. Thequantities Γ90 lo and Γ90 hi are the input data used inthe adjustment, but the observational equations takeinto account the fact that KJ−90KJ and RK−90RK.

Accurate values of Γ90 lo and Γ90 hi for the protonand helion are of potential importance because theyprovide information on the values of and h. Assumingthe validity of the relations KJ=2e /h and RK=h /e2, thefollowing expressions apply to the four available protonresults and one available helion result:

KJ−90RK−90 ge− Γp−9 0lo = p 3 , 250

40R e−

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TABLE XXI. Summary of data related to shielded gyromagnetic ratios of the proton and helion, andinferred values of and h.

Quantity ValueRelative standard

uncertainty ur Identification Sec. and Eq.

loΓp−90

−1

−1 T−12.675 154 0530108 s

137.035 9879511.110−7

3.710−8

NIST-89 VII.A.1.a 253VII.A.1.a 254

loΓp−90

−1

−1 T−12.675 153018108 s

137.036 006306.610−7

2.210−7

NIM-95 VII.A.1.b 255VII.A.1.b 256

loΓh−90

−1

−1 T−12.037 895 3737108 s

137.035 9852821.810−7

6.010−8

KR/VN-98 VII.A.1.c 257VII.A.1.c 258

hiΓp−90

h

−1 T−12.675 152543108 s

6.626 0711110−34 J s

1.610−6

1.610−6

NIM-95 VII.A.2.a 259VII.A.2.a 261

hiΓp−90

h

−1 T−12.675 151827108 s

6.626 07296710−34 J s

1.010−6

1.010−6

NPL-79 VII.A.2.b 262VII.A.2.b 263

KJ−90RK−90 ge− Γh−9 0lo = − h 3 , 251

40R e−

c 2 ge− Γp−9 0hi = p 1

. 2522KJ−90RK−90R − e h

Since the five experiments, including necessary correc-tions, were discussed fully in CODATA-98, only a briefsummary is given in the following sections. The five re-sults, together with the value of inferred from eachlow-field measurement and the value of h inferred fromeach high-field measurement, are collected in TableXXI.

1. Low-field measurements

A number of national metrology institutes have longhistories of measuring the gyromagnetic ratio of theshielded proton, motivated, in part, by their need tomonitor the stability of their practical unit of currentbased on groups of standard cells and standard resistors.This was prior to the development of the Josephson andquantum Hall effects for the realization of practical elec-tric units.

a. NIST: Low field

The most recent National Institute of Standards andTechnology NIST, Gaithersburg, USA, low-field mea-surement was reported by Williams et al. 1989. Theirresult is

lo = 2.675 154 0530 108 −1Γp−90 s T−1

1.1 10−7 , 253

where Γp−9 0lo is related to p by Eq. 248.

The value of that may be inferred from this resultfollows from Eq. 250. Using the 2006 recommendedvalues for other relevant quantities, the uncertainties ofwhich are significantly smaller than the uncertainty ofthe NIST result statements that also apply to the fol-lowing four similar calculations, we obtain

−1 = 137.035 987951 3.7 10−8 , 254

where the relative uncertainty is about one-third therelative uncertainty of the NIST value of Γp−9 0lo be-cause of the cube-root dependence of on Γp−9 0lo.

b. NIM: Low field

The latest low-field proton gyromagnetic ratio experi-ment carried out by researchers at the National Instituteof Metrology NIM, Beijing, PRC, yielded Liu et al.,1995

Γp−9 80lo = 2.675 153018 10 s−1 T−1

6.6 −7 10 . 255

Based on Eq. 250, the inferred value of from theNIM result is

−1 = 137.036 00630 2.2 10−7 . 256

c. KRISS/VNIIM: Low field

The determination of h at the Korea Research Insti-tute of Standards and Science KRISS, Taedok ScienceTown, Republic of Korea, was carried out in a collabo-rative effort with researchers from the Mendeleyev All-Russian Research Institute for Metrology VNIIM, St.Petersburg, Russian Federation Kim et al., 1995; Shifrinet al., 1998a, 1998b, 1999; Park et al., 1999. The result ofthis work can be expressed as

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TABLE XXII. Summary of data related to the von Klitzing constant RK and inferred values of .

Quantity Value uncertainty ur Identification Sec. and Eq.

RK

−1

25 812.808 3162

137.036 0037332.410−8

2.410−8

NIST-97 VII.B.1 265VII.B.1 266

RK

−1

25 812.807111

137.035 9973614.410−8

4.410−8

NMI-97 VII.B.2 267VII.B.2 268

RK

−1

25 812.809214

137.036 0083735.410−8

5.410−8

NPL-88 VII.B.3 269VII.B.3 270

RK

−1

25 812.808434

137.036 004181.310−7

1.310−7

NIM-95 VII.B.4 271VII.B.4 272

RK

−1

25 812.808114

137.036 0023735.310−8

5.310−8

LNE-01 VII.B.5 273VII.B.5 274

8Γh−9 0lo = 2.037 895 3737 10 s−1 T−1

1.8 10−7 , 257

and the value of that may be inferred from it throughEq. 251 is

−1 = 137.035 985282 6.0 10−8 . 258

2. High-field measurements

a. NIM: High field

The latest high-field proton gyromagnetic ratio ex-periment at NIM yielded Liu et al., 1995

Γp−9 0hi = 2.675 152543 8 −1 −1 10 s T

1.6 10−6 , 259

where Γp−9 0hi is related to p by Eq. 249. Its correla-tion coefficient with the NIM low-field result in Eq.255 is

rlo,hi = − 0.014. 260

Based on Eq. 252, the value of h that may be inferredfrom the NIM high-field result is

h = 6.626 07111 10−34 J s 1.6 10−6 . 261

b. NPL: High field

The most accurate high-field p experiment was car-ried out at NPL by Kibble and Hunt 1979, with theresult

Γp−9 8 −1 −10hi = 2.675 151827 10 s T

1.0 10−6 . 262

This leads to the inferred value

h = 6.626 072967 10−34 J s 1.0 10−6 , 263

based on Eq. 252.

B. von Klitzing constant RK and

Since the quantum Hall effect, the von Klitzing con-stant RK associated with it, and available determinationsof RK are fully discussed in CODATA-98 andCODATA-02, we only outline the main points here.

The quantity RK is measured by comparing a quan-tized Hall resistance RHi=RK/ i, where i is an integer,to a resistance R whose value is known in terms of the SIunit of resistance . In practice, the latter quantity, theratio R /, is determined by means of a calculable crosscapacitor, a device based on a theorem in electrostaticsfrom the 1950s Thompson and Lampard, 1956; Lam-pard, 1957. The theorem allows one to construct a cy-lindrical capacitor, generally called a Thompson-Lampard calculable capacitor Thompson, 1959, whosecapacitance, to high accuracy, depends only on its length.

As indicated in Sec. II, if one assumes the validity ofthe relation RK=h /e2, then RK and the fine-structureconstant are related by

= 0c/2RK. 264

Hence, the relative uncertainty of the value of thatmay be inferred from a particular experimental value ofRK is the same as the relative uncertainty of that value.

The values of RK we take as input data in the 2006adjustment and the corresponding inferred values of are given in the following sections and summarized inTable XXII.

1. NIST: Calculable capacitor

The result obtained at NIST is Jeffery et al., 1997see also Jeffery et al. 1998

Relative standard

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RK = 25 812.8 1 + 0.32224 10−6

= 25 812.808 3162 2.4 10−8 , 265

and is viewed as superseding the NIST result reportedby Cage et al. 1989. Work by Jeffery et al. 1999 pro-vides additional support for the uncertainty budget ofthe NIST calculable capacitor.

The value of that may be inferred from the NISTvalue of RK is, from Eq. 264,

−1 = 137.036 003733 2.4 10−8 . 266

2. NMI: Calculable capacitor

Based on measurements carried out at the NationalMetrology Institute NMI, Lindfield, Australia, fromDecember 1994 to April 1995, and a complete reassess-ment of uncertainties associated with their calculable ca-pacitor and associated apparatus, Small et al. 1997 re-ported the result

RK = RK−90 1 + 0.44.4 10−8

= 25 812.807111 4.4 10−8 . 267

The value of it implies is

−1 = 137.035 997361 4.4 10−8 . 268

Because of problems associated with the 1989 NMIvalue of RK, only the result reported in 1997 is used inthe 2006 adjustment, as was the case in the 1998 and2002 adjustments.

3. NPL: Calculable capacitor

The NPL calculable capacitor is similar in design tothose of NIST and NMI. The result for RK reported byHartland et al. 1988 is

RK = 25 812.8 1 + 0.35654 10−6

= 25 812.809214 5.4 10−8 , 269

and the value of that one may infer from it is

−1 = 137.036 008373 5.4 10−8 . 270

4. NIM: Calculable capacitor

The NIM calculable cross capacitor differs markedlyfrom the version used at NIST, NMI, and NPL. The fourbars electrodes that comprise the capacitor are hori-zontal rather than vertical and the length that deter-mines its known capacitance is fixed rather than vari-able. The NIM result for RK, as reported by Zhang et al.1995 is

RK = 25 812.808434 1.3 −7 10 , 271

which implies

−1 = 137.036 00418 1.3 10−7 . 272

5. LNE: Calculable capacitor

The value of RK obtained at the Laboratoire Nationald’Essais LNE, Trappes, France, is Trapon et al., 2001,2003

RK = 25 812.808114 5.3 10−8 , 273

which implies−1 = 137.036 002373 5.3 10−8 . 274

The LNE Thompson-Lampard calculable capacitor isunique among all calculable capacitors in that it consistsof five horizontal bars arranged at the corners of a regu-lar pentagon.

C. Josephson constant KJ and h

Again, since the Josephson effect, the Josephson con-stant KJ associated with it, and the available determina-tions of KJ are fully discussed in CODATA-98 andCODATA-02, we only outline the main points here.

The quantity KJ is measured by comparing a Joseph-son voltage UJn=nf /KJ to a high voltage U whosevalue is known in terms of the SI unit of voltage V. Heren is an integer and f is the frequency of the microwaveradiation applied to the Josephson device. In practice,the latter quantity, the ratio U /V, is determined bycounterbalancing an electrostatic force arising from thevoltage U with a known gravitational force.

A measurement of KJ can also provide a value of h.If, as discussed in Sec. II, we assume the validityof the relation K 2

J=2e /h and recall that =e /40c= 2

0ce /2h, we have

8h = . 275

0cK2J

Since ur of the fine-structure constant is significantlysmaller than ur of the measured values of KJ, ur of hderived from Eq. 275 will be essentially twice the ur ofKJ.

The values of KJ we take as input data in the 2006adjustment, and the corresponding inferred values of h,are given in the following two sections and summarizedin Table XXIII. Also summarized in that table are themeasured values of the product K2

JRK and the quantityF90 related to the Faraday constant F, together withtheir corresponding inferred values of h. These resultsare discussed in Secs. VII.D and VII.E.

1. NMI: Hg electrometer

The determination of KJ at NMI, carried out using anapparatus called a liquid-mercury electrometer, yieldedthe result Clothier et al., 1989

K −6J = 483 5941 + 8.087269 10 GHz/V

= 483 597.9113 GHz/V 2.7 10−7 . 276

Equation 275, the NMI value of KJ, and the 2006 rec-ommended value of , which has a much smaller ur,

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TABLE XXIII. Summary of data related to the Josephson constant KJ, the product K2JRK, the

Faraday constant F, and inferred values of h.

Quantity ValueRelative standard

uncertainty ur Identification Sec. and Eq.

KJ

h

483 597.9113 GHz V−1

6.626 06843610−34 J s2.710−7

5.410−7

NMI-89 VII.C.1 276VII.C.1 277

KJ

h

483 597.9615 GHz V−1

6.626 06704210−34 J s3.110−7

6.310−7

PTB-91 VII.C.2 278VII.C.2 279

KJ2RK

h

−16.036 7625121033 J−1 s

6.626 06821310−34 J s

2.010−7

2.010−7

NPL-90 VII.D.1 281VII.D.1 282

KJ2RK

h

−16.036 761 85531033 J−1 s

6.626 068 915810−34 J s

8.710−8

8.710−8

NIST-98 VII.D.2.a 283VII.D.2.a 284

KJ2RK

h

−16.036 761 85221033 J−1 s

6.626 068 912410−34 J s

3.610−8

3.610−8

NIST-07 VII.D.2.b 287VII.D.2.b 288

F90

h

96 485.3913 C mol−1

6.626 06578810−34 J s1.310−6

1.310−6

NIST-80 VII.E.1 295VII.E.1 296

yields an inferred value for the Planck constant of

h = 6.626 068436 10−34 J s 5.4 10−7 . 277

2. PTB: Capacitor voltage balance

The determination of KJ at PTB was carried out usinga voltage balance consisting of two coaxial cylindricalelectrodes Sienknecht and Funck, 1985, 1986; Funckand Sienknecht, 1991. Taking into account the correc-tion associated with the reference capacitor used in thePTB experiment as described in CODATA-98, the resultof the PTB determination is

KJ = 483 597.9615 GHz/V 3.1 10−7 , 278

from which we infer, using Eq. 275,

h = 6.626 067042 10−34 J s 6.3 −7 10 . 279

D. Product K2JRK and h

A value for the product K2JRK is important to deter-

mine the Planck constant h, because if one assumes thatthe relations KJ=2e /h and RK=h /e2 are valid, then

4h = 2 . 280

KJRK

The product K2JRK is determined by comparing electrical

power known in terms of a Josephson voltage and quan-tized Hall resistance to the equivalent mechanical powerknown in the SI unit W=m2 kg s−3. Comparison is car-ried out using an apparatus known as a moving-coil wattbalance first proposed by Kibble 1975 at NPL. To date,

two laboratories, NPL and NIST, have determined K2JRK

using this method.

1. NPL: Watt balance

Shortly after Kibble’s original proposal in 1975,Kibble and Robinson 1977 carried out a feasibilitystudy of the idea based on experience with the NPLapparatus that was used to determine p by the high-field method Kibble and Hunt, 1979. The work contin-ued and led to a result with an uncertainty of about 2parts in 107 Kibble et al., 1990. This result, discussed inCODATA-98 and which was taken as an input datum inthe 1998 and 2002 adjustments, and which we also takeas an input datum in the 2006 adjustment, may be ex-pressed as

K2R = K2 R 1 + 16.1420 10−6J K J-NPL K-NPL

= 6.036 762512 1033 J−1 −1 s 2.0 10−7 ,

281

where KJ−NPL=483 594 GHz/V and RK−NPL=25 812.8092 . The value of h that may be inferredfrom the NPL result is, according to Eq. 280,

h = 6.626 068213 10−34 J s 2.0 10−7 . 282

Based on the experience gained in this experiment,NPL researchers designed and constructed what is es-sentially a completely new apparatus, called the NPLMark II watt balance, that could possibly achieve a re-sult for K2R with an uncertainty of a few parts in 108

J KRobinson and Kibble, 1997; Kibble and Robinson,2003. Although the balance itself employs the same bal-

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ance beam as the previous NPL watt balance, little elsefrom that experiment is retained in the new experiment.

Over 1000 measurements in vacuum were carried outwith the MK II between January 2000 and November2001. Many were made in an effort to identify the causeof an observed fractional change in the value of K2

JRK ofabout 310−7 that occurred in mid-April 2000 Robin-son and Kibble, 2002. A change in the alignment of theapparatus was suspected of contributing to the shift.

Significant improvements were subsequently made inthe experiment, and recently, based on measurementscarried out from October 2006 to March 2007, the initialresult from MK II, h=6.626 070 9544 J s 6.610−8,was reported by Robinson and Kibble 2007 assumingthe validity of Eq. 280. Although this result becameavailable much too late to be considered for the 2006adjustment, we do note that it lies between the value ofh inferred from the 2007 NIST result for K2

JRK discussedin Sec. VII.D.2.b, and that inferred from the measure-ment of the molar volume of silicon VmSi discussed inSec. VIII.B. The NPL work is continuing and a resultwith a smaller uncertainty is anticipated Robinson andKibble, 2007.

2. NIST: Watt balance

a. 1998 measurement

Work on a moving-coil watt balance at NIST beganshortly after Kibble made his 1975 proposal. A first re-sult with ur=1.310−6 was reported by NIST research-ers in 1989 Cage et al., 1989. Significant improvementswere then made to the apparatus and the final resultfrom this phase of the NIST effort was reported in 1998by Williams et al. 1998,

K2JRK = K2

J−90RK−901 − 887 10−9

= 6.036 761 8553 33 −1 −1 −8 10 J s 8.7 10 .

283

A lengthy paper giving the details of the NIST 1998 wattbalance experiment was published in 2005 by Steiner,Newell, and Williams 2005. This was the NIST resulttaken as an input datum in the 1998 and 2002 adjust-ments; although the 1989 result was consistent with thatof 1998, its uncertainty was about 15 times larger. Thevalue of h implied by the 1998 NIST result for K2

JRK is

h = 6.626 068 9158 10−34 −8 J s 8.7 10 . 284

b. 2007 measurement

Based on the lessons learned in the decade-long effortwith a watt balance operating in air that led to their 1998result for K2

JRK, the NIST watt-balance researchers ini-tiated a new program with the goal of measuring K2

JRKwith ur10−8. The experiment was completely disas-sembled, and renovations to the research facility weremade to improve vibration isolation, reduce electromag-netic interference, and incorporate a multilayer tem-perature control system. A new watt balance with major

changes and improvements was constructed with littleremaining of the earlier apparatus except the supercon-ducting magnet used to generate the required radialmagnetic flux density and the wheel used as the balance.

The most notable change in the experiment is that inthe new apparatus the entire balance mechanism andmoving coil are in vacuum, which eliminates the uncer-tainties of the corrections in the previous experiment forthe index of refraction of air in the laser position mea-surements ur=4310−9 and for the buoyancy force ex-erted on the mass standard ur=2310−9. Alignmentuncertainties were reduced by over a factor of 4 by iincorporating a more comprehensive understanding ofall degrees of freedom involving the moving coil; and iithe application of precise alignment techniques for alldegrees of freedom involving the moving coil, the super-conducting magnet, and the velocity measuring interfer-ometers. Hysteresis effects were reduced by a factor of 4by using a diamondlike carbon-coated knife edge andflat Schwarz et al., 2001, employing a hysteresis erasureprocedure, and reducing the balance deflections duringmass exchanges with improved control systems. A pro-grammable Josephson array voltage standard Benz etal., 1997 was connected directly to the experiment,eliminating two voltage transfers required in the old ex-periment and reducing the voltage traceability uncer-tainty by a factor of 15.

A total of 6023 individual values of W90/W wereobtained over the 2-year period from March 2003 toFebruary 2005 as part of the effort to develop and im-prove the new experiment. The results are converted tothe notation used here by the relation W90/W=K2 2

J−90RK−90/KJRK discussed in CODATA-98. The ini-tial result from that work was reported by Steiner, Wil-liams, Newell, et al. 2005,

K2JRK = K2 −9

J−90RK−901 − 2452 10

= 6.036 761 7531 1033 J−1 s−1 5.2 10−8 ,

285

which yields a value for the Planck constant of

h = 6.626 069 0134 10−34 J s 5.2 10−8 . 286

This result for K2JRK was obtained from data spanning

the final 7 months of the 2-year period. It is based on theweighted mean of 48 W90/W measurement sets using aAu mass standard and 174 sets using a PtIr mass stan-dard, where a typical measurement set consists of 12 to15 individual values of W90/W. The 2005 NIST result isconsistent with the 1998 NIST result, but its uncertaintyhas been reduced by a factor of 1.7.

Following this initial effort with the new apparatus,further improvements were made to it in order to reducethe uncertainties from various systematic effects, themost notable reductions being in the determination ofthe local acceleration due to gravity g a factor of 2.5,the effect of balance wheel surface roughness a factorof 10, and the effect of the magnetic susceptibility of themass standard a factor of 1.6. An improved result was

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then obtained based on 2183 values of W90/W recordedin 134 measurement sets from January 2006 to June2006. Due to a wear problem with the gold mass stan-dard, only a PtIr mass standard was used in these mea-surements. The result, first reported in 2006 and subse-quently published in 2007 by Steiner et al. 2007, is

K2JRK = K2

J−90RK−901 − 836 10−9

= 6.036 761 8522 1033 J−1 s−1 3.6 10−8 .

287

The value of h that may be inferred from this value ofK2

JRK is

h = 6.626 068 9124 10−34 J s 3.6 10−8 . 288

The 2007 NIST result for K2JRK is consistent with and

has an uncertainty smaller by a factor of 1.4 than theuncertainty of the 2005 NIST result. However, becausethe two results are from essentially the same experimentand hence are highly correlated, we take only the 2007result as an input datum in the 2006 adjustment.

On the other hand, the experiment on which the NIST2007 result is based is only slightly dependent on theexperiment on which the NIST 1998 result is based, ascan be seen from the above discussions. Thus, in keepingwith our practice in similar cases, most notably the 1982and 1999 LAMPF measurements of muonium Zeemantransition frequencies see Sec. VI.B.2, we also take theNIST 1998 result in Eq. 283 as an input datum in the2006 adjustment. But to ensure that we do not give un-due weight to the NIST work, an analysis of the uncer-tainty budgets of the 1998 and 2007 NIST results wasperformed to determine the level of correlation. Of therelative uncertainty components listed in Table II ofWilliams et al. 1998 and in Table 2 of Steiner, Williams,Newell, et al. 2005 but as updated in Table 1 of Steineret al. 2007, the largest common relative uncertaintycomponents were from the magnetic flux profile fit dueto the use of the same analysis routine 1610−9; leak-age resistance and electrical grounding since the samecurrent supply was used in both experiments 1010−9; and the determination of the local gravitationalacceleration g due to the use of the same absolutegravimeter 710−9. The correlation coefficient wasthus determined to be

rK2JRK-98, K2

JRK-07 = 0.14, 289

which we take into account in our calculations as appro-priate.

3. Other values

Although there is no competitive published value ofK2

JRK other than those from NPL and NIST discussedabove, it is worth noting that at least three additionallaboratories have watt-balance experiments in progress:the Swiss Federal Office of Metrology and AccreditationMETAS, Bern-Wabern, Switzerland, the LNE, and the

BIPM. Descriptions of these efforts may be found inBeer et al. 2003, Genevès et al. 2005, and Picard et al.2007, respectively.

4. Inferred value of KJ

It is of interest to note that a value of KJ with anuncertainty significantly smaller than those of the di-rectly measured values discussed in Sec. VII.C can beobtained from directly measured watt-balance values ofK2

JRK, together with directly measured calculable-capacitor values of RK, without assuming the validity ofthe relations KJ=2e /h and R 2

K=h /e . The relevantexpression is simply KJ= K2

JR 1/2KW/ RKC , where

K2JRKW is from the watt balance and RKC is from the

calculable capacitor.Using the weighted mean of the three watt-balance

results for K2JRK discussed in this section and the

weighted mean of the five calculable-capacitor resultsfor RK discussed in Sec. VII.B, we have

KJ = KJ−901 − 2.81.9 10−8

= 483 597.886594 GHz/V 1.9 10−8 , 290

which is consistent with the two directly measured val-ues but has an uncertainty that is smaller by more thanan order of magnitude. This result is implicitly includedin the least-squares adjustment, even though the explicitvalue for KJ obtained here is not used as an input da-tum.

E. Faraday constant F and h

The Faraday constant F is equal to the Avogadro con-stant NA times the elementary charge e, F=NAe; its SIunit is coulomb per mol, C mol−1=A s mol−1. It deter-mines the amount of substance nX of an entity X thatis deposited or dissolved during electrolysis by the pas-sage of a quantity of electricity, or charge, Q=It, due tothe flow of a current I in a time t. In particular, theFaraday constant F is related to the molar mass MXand valence z of entity X by

ItMXF = , 291

zmdX

where mdX is the mass of entity X dissolved as theresult of transfer of charge Q=It during the electrolysis.It follows from the relations F=NAe, e2=2h /0c,me=2Rh /c2, and NA=AreMu /me, where Mu=10−3 kg mol−1, that

AF = reM 5 1/2

u c

R

20 h . 292

Since, according to Eq. 291, F is proportional to thecurrent I, and I is inversely proportional to the productKJRK if the current is determined in terms of the Jo-sephson and quantum Hall effects, we may write

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K e90 = JRK A c 1/2

r M 5F u , 293

KJ−90RK−90 R 20 h

where F90 is the experimental value of F in SI units thatwould result from the Faraday experiment if KJ=KJ−90and RK=RK−90. The quantity F90 is the input datum usedin the adjustment, but the observational equation ac-counts for the fact that KJ−90KJ and RK−90RK. If oneassumes the validity of KJ=2e /h and RK=h /e2, then interms of adjusted constants, Eq. 293 can be written as

cM A e 2F u

90 = r . 294KJ−90RK−90 Rh

1. NIST: Ag coulometer

There is one high-accuracy experimental value of F90available, that from NIST Bower and Davis, 1980. TheNIST experiment used a silver dissolution coulometerbased on the anodic dissolution by electrolysis of silver,which is monovalent, into a solution of perchloric acidcontaining a small amount of silver perchlorate. The ba-sic chemical reaction is Ag→Ag++e− and occurs at theanode, which in the NIST work was a highly purifiedsilver bar.

As discussed in detail in CODATA-98, the NIST ex-periment leads to

F90 = 96 485.3913 C mol−1 1.3 −6 10 . 295

Note that the new AME2003 values of A 107r Ag and

Ar109Ag in Table II do not change this result.

The value of h that may be inferred from the NISTresult, Eq. 294, and the 2006 recommended values forthe other quantities is

h = 6.626 065788 10−34 J s 1.3 10−6 , 296

where the uncertainties of the other quantities are neg-ligible compared to the uncertainty of F90.

VIII. MEASUREMENTS INVOLVING SILICON CRYSTALS

Here we discuss experiments relevant to the 2006 ad-justment that use highly pure, nearly crystallographicallyperfect, single crystals of silicon. However, because onesuch experiment determines the quotient h /mn, wheremn is the mass of the neutron, for convenience and be-cause any experiment that determines the ratio of thePlanck constant to the mass of a fundamental particle oratom provides a value of the fine-structure constant ,we also discuss in this section two silicon-independentexperiments: the 2002 Stanford University, Stanford,California, USA, measurement of h /m133Cs and the2006 Laboratoire Kastler-Brossel or LKB measurementof h /m87Rb.

In this section, W4.2a, NR3, W04, and NR4 are short-ened forms of the full crystal designations WASO 4.2a,NRLM3, WASO 04, and NRLM4, respectively, for usein quantity symbols. No distinction is made between dif-ferent crystals taken from the same ingot. As we use the

current laboratory name to identify a result rather thanthe laboratory name at the time the measurement wascarried out, we have replaced IMGC and NRLM withINRIM and NMIJ.

A. 220 lattice spacing of silicon d220

A value of the 220 lattice spacing of a silicon crystalin meters is relevant to the 2006 adjustment not onlybecause of its role in determining from h /mn see Sec.VIII.D.1, but also because of its role in determining therelative atomic mass of the neutron Arn see Sec.VIII.C. Further, together with the measured value ofthe molar volume of silicon VmSi, it can provide a com-petitive value of h see Sec. VIII.B.

Various aspects of silicon and its crystal plane spacingsof interest here are reviewed in CODATA-98 andCODATA-02. See also the reviews of Becker 2003,Mana 2001, and Becker 2001. Some points worthnoting are that silicon is a cubic crystal with n=8 atomsper face-centered-cubic unit cell of edge length or lat-tice parameter a=543 pm with d220=a /8. The threenaturally occurring isotopes of Si are 28Si, 29Si, and 30Si,and the amount-of-substance fractions x28Si, x29Si,and x30Si of natural silicon are approximately 0.92,0.05, and 0.03, respectively.

Although the 220 lattice spacing of Si is not a funda-mental constant in the usual sense, for practical pur-poses one can consider a, and hence d220, of an impurity-free, crystallographically perfect or “ideal” siliconcrystal under specified conditions, principally of tem-perature, pressure, and isotopic composition, to be aninvariant of nature. The reference temperature andpressure currently adopted are t90=22.5 °C and p=0that is, vacuum, where t90 is Celsius temperature on theInternational Temperature Scale of 1990 ITS-90Preston-Thomas, 1990a, 1990b. However, no referencevalues for xASi have yet been adopted, because thevariation of a due to the variation of the isotopic com-position of the crystals used in high-accuracy experi-ments is taken to be negligible at the current level ofexperimental uncertainty in a. A much larger effect on ais the impurities that the silicon crystal contains—mainlycarbon C, oxygen O, and nitrogen N—and correc-tions must be applied to convert the 220 lattice spacingd220X of a real crystal X to the 220 lattice spacing d220of an ideal crystal.

Nevertheless, we account for the possible variation inthe lattice spacing of different samples taken from thesame ingot by including an additional component orcomponents of relative standard uncertainty in the un-certainty of any measurement result involving a siliconlattice spacing or spacings. This additional componentis typically 210−8 for each crystal, but it can be larger,for example, 3/2210−8 in the case of crystal MO*

discussed below, because it is known to contain a com-paratively large amount of carbon; see Secs. III.A.c andIII.I of CODATA-98 for details. For simplicity, we do

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TABLE XXIV. Summary of measurements of the absolute 220 lattice spacing of various siliconcrystals and inferred values of d220. The latter follows from the data discussed in Sec. VIII.A.2.

Quantity Value uncertainty ur Identification Sec. and Eq.

d220W4.2a 192 015.56312 fm 6.210−8 PTB-81 VIII.A.1.a 297d220 192 015.56513 fm 6.510−8

d220NR3 192 015.591976 fm 4.010−8 NMIJ-04 VIII.A.1.b 298d220 192 015.597384 fm 4.410−8

d220W4.2a 192 015.571533 fm 1.710−8 INRIM-07 VIII.A.1.c 299d220 192 015.573253 fm 2.810−8

d220MO* 192 015.549851 fm 2.610−8 INRIM-07 VIII.A.1.c 300d220 192 015.568567 fm 3.510−8

h /mnd220W04 −12060.267 00484 m s 4.110−8 PTB-99 VIII.D.1 322d220 192 015.598279 fm 4.110−8 VIII.D.1 325

Relative standard

not explicitly mention our inclusion of such componentsin the following discussion.

Further, because of this component and the use of thesame samples in different experiments, and because ofthe existence of other common components of uncer-tainty in the uncertainty budgets of different experimen-tal results involving silicon crystals, many of the inputdata discussed in the following sections are correlated.In most cases, we do not explicity give the relevant cor-relation coefficients in the text; instead Table XXXI inSec. XII provides all non-negligible correlation coeffi-cients of the input data listed in Table XXX.

1. X-ray optical interferometer measurements of d220(X)

High accuracy measurements of d220X, where X de-notes any one of various crystals, are carried out using acombined x-ray and optical interferometer XROI fab-ricated from a single-crystal of silicon taken from one ofseveral well-characterized single crystal ingots or boules.As discussed in CODATA-98, an XROI is a device thatenables x-ray fringes of unknown period d220X to becompared with optical fringes of known period by mov-ing one of the crystal plates of the XROI, called theanalyzer. Also discussed there are the XROI measure-ments of d220W4.2a, d220MO*, and d220NR3, whichwere carried out at the PTB in Germany Becker et al.,1981, the Istituto Nazionale di Ricerca Metrologica,Torino, Italy INRIM Basile et al., 1994, and the Na-tional Metrology Institute of Japan NMIJ, Tsukuba,Japan Nakayama and Fujimoto, 1997, respectively.

For reasons discussed in CODATA-02 and by Cavag-nero et al. 2004a, 2004b, only the NMIJ 1997 result wastaken as an input datum in the 2002 adjustment. How-ever, further work, published in an Erratum to that pa-per, showed that the results obtained at INRIM given inthe paper were in error. After the error was discovered,additional work was carried out at INRIM to fully un-derstand and correct it. New results were then reportedin 2006 Becker et al., 2007. Thus, as summarized inTable XXIV and compared in Fig. 1, we take as input

data the four absolute 220 lattice spacing values deter-mined in three different laboratories, as discussed in thefollowing three sections. The last value in the table,which is not an XROI result, is discussed in Sec.VIII.D.1.

We point out that not only do we take the 220 latticespacings of the crystals WASO 4.2a, NRLM3, and MO*

as adjusted constants, but also the 220 lattice spacingsof the crystals N, WASO 17, ILL, WASO 04, andNRLM4, because they too were involved in various ex-periments, including the d220 lattice spacing fractionaldifference measurements discussed in Sec. VIII.A.2.The inferred values of d220 in Table XXIV are based onthe data in that section.

(d220/fm − 192 015) × 103

540 550 560 570 580 590 600 610

540 550 560 570 580 590 600 610

710− d220

d220(W4.2a) PTB-81

d220 PTB-81

d220(NR3) NMIJ-04

d220 NMIJ-04

d220 h/mnd220(W04) PTB-99

d220(MO∗) INRIM-07

d220 INRIM-07

d220(W4.2a) INRIM-07

d220 INRIM-07

d220 CODATA-02

d220 CODATA-06

(d220/fm − 192 015) × 103

540 550 560 570 580 590 600 610

540 550 560 570 580 590 600 610

710− d220

d220(W4.2a) PTB-81

d220 PTB-81

d220(NR3) NMIJ-04

d220 NMIJ-04

d220 h/mnd220(W04) PTB-99

d220(MO∗) INRIM-07

d220 INRIM-07

d220(W4.2a) INRIM-07

d220 INRIM-07

d220 CODATA-02

d220 CODATA-06

FIG. 1. Inferred values open circles of d220 from various mea-surements solid circles of d220X. For comparison, the 2002and 2006 CODATA recommended values of d220 are alsoshown.

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a. PTB measurement of d220W4.2a

The following value, identified as PTB-81 in TableXXIV and Fig. 1, is the original result obtained at PTBas reported by Becker et al. 1981 and discussed inCODATA-98:

d220W4.2a = 192 015.56312 fm 6.2 10−8 .

297

b. NMIJ measurement of d220NR3

The following value, identified as NMIJ-04 in TableXXIV and Fig. 1, reflects the NMIJ efforts in the earlyand mid-1990s as well as the work carried out in theearly 2000s:

d220NR3 = 192 015.591976 fm 4.0 −8 10 .

298

This value, reported by Cavagnero et al. 2004a, 2004b,is the weighted mean of the 1997 NIMJ result of Na-kayama and Fujimoto 1997 discussed in CODATA-98and CODATA-02, and the result from a new series ofmeasurements performed at NMIJ from December 2002to February 2003 with nearly the same apparatus. Oneof the principal differences from the earlier experimentwas the much improved temperature control system ofthe room in which the NMIJ XROI was located; the newsystem provided a temperature stability of about1 mK/d and allowed the temperature of the XROI to beset to within 20 mK of 22.5 °C.

The result for d220NR3 from the 2002–2003 measure-ments is based on 61 raw data. In each measurement,the phases of the x-ray and optical fringes optical or-ders were compared at the 0th, 100th, and 201st opticalorders, and then with the analyzer moving in the reversedirection, at the 201st, 100th, and 0th orders. The n /mratio was calculated from the phase of the x-ray fringe atthe 0th and 201st orders, where n is the number of x-rayfringes in m optical fringes optical orders of periodλ /2, where λ is the wavelength of the laser beam used inthe optical interferometer and d220NR3= λ /2 / n /m.

In the new work, the fractional corrections tod220NR3, in parts in 109, total 18135, the largest by farbeing the correction 17333 for laser beam diffraction.The next largest is 5.07.1 for laser beam alignment.The statistical uncertainty is 33 Type A.

Before calculating the weighted mean of the new and1997 results for d220NR3, Cavagnero et al. 2004a,2004b revised the 1997 value based on a reanalysis ofthe old experiment, taking into account what waslearned in the new experiment. The reanalysis resultednot only in a reduction of the statistical uncertainty fromagain, in parts in 109 50 to 1.8 due to a better under-standing of the undulation of n /m values as a function oftime, but also in more reliable estimates of the correc-tions for laser beam diffraction and laser beam align-ment. Indeed, the fractional corrections for the revised1997 NMIJ value of d220NR3 total 19038 compared to

the original total of 17314, and the final uncertainty ofthe revised 1997 value is u −8

r=3.810 compared to ur=4.810−8 of the new value.

For completeness, we note that two possible correc-tions to the NMIJ result have been discussed in the lit-erature. In the Erratum to Cavagnero et al. 2004a,2004b, it is estimated that a fractional correction to thevalue of d220NR3 in Eq. 298 of −1.310−8 may berequired to account for the contamination of the NMIJlaser by a parasitic component of laser radiation as inthe case of the INRIM laser discussed in the next sec-tion. However, it is not applied, because of its compara-tively small size and the fact that no measurements ofd220NR3 have yet been made at NMIJ or INRIMwith a problem-free laser that confirm the correction, ashas been done at INRIM for the crystals WASO 4.2aand MO*.

Fujimoto et al. 2007 estimated, based on a MonteCarlo simulation, that the fractional correction tod220NR3 labeled Fresnel diffraction in Table I ofNakayama and Fujimoto 1997 and equal to 16.0810−8 should be 10310−8. The change arises fromtaking into account the misalignment of the interferingbeams in the laser interferometer. Because this addi-tional diffraction effect was present in both the 1997 and2002–2003 measurements but was not considered in thereanalysis of the 1997 result nor in the analysis of the2002–2003 data, it implies that the weighted mean valuefor d220NR3 in Eq. 298 should be reduced by thisamount and its ur increased from 4.010−8 to 5.010−8. However, because the data required for the cal-culation were not precisely known they were not loggedin the laboratory notebooks because the experimenterswere unaware of their importance, the correction isviewed as somewhat conjectural and thus applying itwould not be justified Mana and Massa, 2006.

c. INRIM measurement of d220W4.2a and d220MO*

The following two new INRIM values, with identifierINRIM-07, were reported by Becker et al. 2007:

d220W4.2a = 192 015.571533 fm 1.7 10−8 ,

299

d220MO* = 192 015.549851 fm 2.6 10−8 .

300

The correlation coefficient of these values is 0.057, basedon the detailed uncertainty budget for d220MO* in Cav-agnero et al. 2004a, 2004b and the similar uncertaintybudget for d220W4.2a provided by Fujimoto et al.2006. Although the 2007 result for d220MO* ofBecker et al. 2007 in Eq. 300 agrees with the 1994INRIM result of Basile et al. 1994, which was used asan input datum in the 1998 adjustment, because of themany advances incorporated in the new work, we nolonger consider the old result.

In addition to the determination described in the pre-vious section of the 220 lattice spacing of crystal

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NRLM3 carried out at NMIJ in 2002–2003 using theNMIJ NRLM3 x-ray interferometer and associatedNMIJ apparatus, Cavagnero et al. 2004a, 2004b re-ported the results of measurements carried out at IN-RIM of the 220 lattice spacings of crystals MO* andNRLM3, where in the latter case it was an INRIM-NMIJ joint effort that used the NIMJ NRLM3 x-ray in-terferometer but the INRIM associated apparatus. Butas indicated above, both results were subsequentlyfound to be in error: the optical laser beam used to mea-sure the displacement of the x-ray interferometer’s ana-lyzer crystal was contaminated by a parasitic componentwith a frequency that differed by about 1.1 GHz fromthe frequency assigned the laser beam.

After eliminating the error by replacing the problemlaser with a 633 nm He-Ne external-cavity diode laserlocked to a 127I2 stabilized laser, the INRIM researchersrepeated the measurements they had previously carriedout with the INRIM MO* x-ray interferometer and withthe refurbished PTB WASO 4.2a x-ray interferometeroriginally used in the PTB experiment that led to the1981 value of d220W4.2a in Eq. 297. The PTB WASO4.2a x-ray interferometer was refurbished at PTBthrough remachining, but the result for d220W4.2a ob-tained at INRIM with the contaminated laser was notincluded in Cavagnero et al. 2004a, 2004b. The valuesof d220W4.2a and d220MO* in Eqs. 299 and 300resulted from the repeated measurements Becker et al.,2007.

In principle, based on the experimentally observedshifts in the measured values of d220W4.2a andd220MO* obtained with the malfunctioning laser andthe properly functioning laser, the value of d220NR3obtained in the INRIM-NMIJ joint effort using the mal-functioning laser mentioned above, and the value ofd220WS5C also obtained with this laser, could be cor-rected and taken as input data. WS5C is an XROImanufactured by INRIM from a WASO 04 sample, butthe value of d220WS5C obtained using the contami-nated laser was also not included in Cavagnero et al.2004a, 2004b. However, because of the somewhat er-ratic history of silicon lattice spacing measurements, theTask Group decided to use only data obtained with alaser known to be functioning properly.

Improvements in the INRIM XROI apparatus sincethe 1994 d *220MO measurement of Basile et al. 1994include i a new two-axis “tip-tilt” platform for theXROI that is electronically controlled to compensate forparasitic rotations and straightness error of the guidingsystem that moves the platform; ii imaging the x-rayinterference pattern formed by the x-ray beam transmit-ted through the moving analyzer in such a way that de-tailed information concerning lattice distortion and ana-lyzer pitch can be extracted on line from the analysis ofthe phases of the x-ray fringes; and iii an upgradedcomputer-aided system for combined interferometer dis-placement and control, x-ray and optical fringe scanning,signal digitization and sampling, environmental monitor-ing, and data analysis.

The values of d220W4.2a and d220MO* in Eqs. 299and 300 are the means of tens of individual values, witheach value being the average of about ten data pointscollected in 1 h measurement cycles during which theanalyzer was translated back and forth by 300 opticalorders. For the two crystals, respectively, the statisticaluncertainties in parts in 109 are 3.5 and 11.6, and thevarious corrections and their uncertainties are laserbeam wavelength, −0.84, −0.84; laser beam diffrac-tion, 12.02.2, 12.02.2; laser beam alignment, 2.53.5,2.53.5; Abbe error, 0.02.8, 0.03.7; trajectory error,0.01.4, 0.03.6; analyzer temperature, 1.05.2, 1.07.9;and abberations, 0.05.0, 0.02.0. The total uncertain-ties are 9.6 and 15.7.

2. d220 difference measurements

To relate the lattice spacings of crystals usedin various experiments, highly accurate measure-ments are made of the fractional difference d220X−d220ref /d220ref of the 220 lattice spacing of asample of a single-crystal ingot X and that of a referencecrystal “ref.” Both NIST and PTB have carried out suchmeasurements, and the fractional differences from thesetwo laboratories that we take as input data in the 2006adjustment are given in the following two sections andare summarized in Table XXV. For details concerningthese measurements, see CODATA-98 and CODATA-02.

a. NIST difference measurements

The following fractional difference involving a crystaldenoted simply as N was obtained as part of the NISTeffort to measure the wavelengths in meters of the K1x-ray lines of Cu, Mo, and W; see Sec. XI.A:

d220W17 − d220N= 722 10−9. 301

d220W17

The following three fractional differences involvingcrystals from the four crystals denoted ILL, WASO 17,MO*, and NRLM3 were obtained as part of the NISTeffort, discussed in Sec. VIII.C, to determine the relativeatomic mass of the neutron Arn Kessler et al., 1999:

d220ILL − d220W17= − 822 10−9 , 302

d220ILL

d220ILL − d220MO*= 8627 10−9, 303

d220ILL

d220ILL − d220NR3= 3422 10−9. 304

d220ILL

The following more recent NIST difference measure-ments, which we also take as input data in the 2006 ad-justment, were provided by Kessler 2006 of NIST andare updates of the results reported by Hanke andKessler 2005:

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TABLE XXV. Summary of measurements of the relative 220 lattice spacings of silicon crystals.

Quantity Value Identification Sec. and Eq.

1−d220W17 /d220ILL −82210−9 NIST-99 VIII.A.2.a 3021−d220MO* /d220ILL 862710−9 NIST-99 VIII.A.2.a 3031−d220NR3 /d220ILL 342210−9 NIST-99 VIII.A.2.a 3041−d220N /d220W17 72210−9 NIST-97 VIII.A.2.a 301d220NR3 /d220W04−1 −112110−9 NIST-06 VIII.A.2.a 305d220NR4 /d220W04−1 252110−9 NIST-06 VIII.A.2.a 306d220W17 /d220W04−1 112110−9 NIST-06 VIII.A.2.a 307d220W4.2a /d220W04−1 −12110−9 PTB-98 VIII.A.2.b 308d220W17 /d220W04−1 222210−9 PTB-98 VIII.A.2.b 309d220MO* /d220W04−1 −1032810−9 PTB-98 VIII.A.2.b 310d220NR3 /d220W04−1 −232110−9 PTB-98 VIII.A.2.b 311

−9d220/d220W04−1 101110 PTB-03 VIII.A.2.b 312

d220NR3 − d220W04= − 1121 10−9, 305

d220W04

d220NR4 − d220W04= 2521 10−9, 306

d220W04

d220W17 − d220W04= 1121 10−9. 307

d220W04

The full designations of the two new crystals involved inthese comparisons are WASO 04 and NRLM4. Themeasurements benefited significantly from the reloca-tion of the NIST lattice comparator to a new laboratorywhere the temperature varied by only about 5 mK inseveral weeks compared to the previous laboratorywhere the temperature varied by about 40 mK in oneday Hanke and Kessler, 2005.

b. PTB difference measurements

Results for the 220 lattice-spacing fractional differ-ences of various crystals that we also take as input datain the 2006 adjustment have been obtained at the PTBMartin et al., 1998,

d220W4.2a − d220W04= − 121 10−9, 308

d220W04

d220W17 − d220W04= 2222 10−9, 309

d220W04

d *220MO − d220W04= − 10328 10−9 , 310

d220W04

d220NR3 − d220W04= − 2321 10−9. 311

d220W04

To relate d220W04 to the 220 lattice spacing d220 ofan ideal silicon crystal, we take as an input datum

d220 − d220W04= 1011 −9 10 312

d220W04

given by Becker et al. 2003, who obtained it by takinginto account the known carbon, oxygen, and nitrogenimpurities in WASO 04. However, following what wasdone in the 1998 and 2002 adjustments, we have in-cluded an additional component of uncertainty of 110−8 to account for the possibility that, even after cor-rection for C, O, and N impurities, the crystal WASO 04,although well characterized as to its purity and crystal-lographic perfection, does not meet all criteria for anideal crystal. Indeed, in general, we prefer to use experi-mentally measured fractional lattice spacing differencesrather than differences implied by the C, O, and N im-purity content of the crystals in order to avoid the needto assume that all crystals of interest meet these criteria.

In order to include this fractional difference in the2002 adjustment, the quantity d220 is also taken as anadjusted constant.

B. Molar volume of silicon Vm(Si) and the Avogadro constantNA

The definition of the molar volume of silicon VmSiand its relationship to the Avogadro constant NA andPlanck constant h as well as other constants is discussedin CODATA-98 and summarized in CODATA-02. Inbrief we have

a3

mSi = Si , 313n

MSi AVmSi = = rSiMu , 314

Si Si

VmSi ArSiMNA =

a3 = u

/n , 8 3 315

d220Si

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2cM A e 2d3V u

mSi = r 220 , 316Rh

which are to be understood in the context of animpurity-free, crystallographically perfect, ideal siliconcrystal at the reference conditions t90=22.5 °C and p=0, and of isotopic composition in the range normallyobserved for crystals used in high-accuracy experiments.Thus mSi, VmSi, MSi, and ArSi are the meanmass, mean molar volume, mean molar mass, and meanrelative atomic mass of the silicon atoms in such a crys-tal, respectively, and Si is the crystal’s macroscopicmass density. Equation 316 is the observational equa-tion for a measured value of VmSi.

It follows from Eq. 314 that the experimentaldetermination of VmSi requires i measurementof the amount-of-substance ratios n29Si /n28Si andn30Si /n28Si of a nearly perfect silicon crystal—andhence amount-of-substance fractions xASi—and thencalculation of ArSi from the well-known values ofArASi; and ii measurement of the macroscopic massdensity Si of the crystal. Determining NA from Eq.315 by measuring VmSi in this way and d220 using xrays is called the x-ray-crystal-density XRCD method.

An extensive international effort has been underwaysince the early 1990s to determine NA using this tech-nique with the smallest possible uncertainty. The effortis being coordinated by the Working Group on theAvogadro Constant WGAC of the Consultative Com-mittee for Mass and Related Quantities CCM of theCIPM. The WGAC, which has representatives from allmajor research groups working in areas relevant to thedetermination of NA, is currently chaired by P. Becker ofPTB.

As discussed in CODATA-02, the value of VmSiused as an input datum in the 2002 adjustment was pro-vided to the CODATA Task Group by the WGAC andwas a consensus value based on independent measure-ments of Si at NMIJ and PTB using a number ofdifferent silicon crystals, and measurements of their mo-lar masses MSi using isotopic mass spectrometry at theInstitute for Reference Materials and MeasurementsIRMM, European Commission, Geel, Belgium. Thisvalue, identified as N/P/I-03 in recognition of the workdone by researchers at NMIJ, PTB, and IRMM,is VmSi=12.058 82573610−6 m3 mol−1 3.010−7.Since then, the data used to obtain it were reanalyzed bythe WGAC, resulting in the slightly revised value Fujiiet al., 2005

V Si = 12.058 825434 10−6 m3 −1m mol

2.8 10−7 , 317

which we take as an input datum in the 2006 adjustmentand identify as N/P/I-05. The slight shift in value andreduction in uncertainty are due to the fact that the ef-fect of nitrogen impurities in the silicon crystals used inthe NMIJ measurements was taken into account in thereanalysis Fujii et al., 2005. Note that the new value of

Ar29Si in Table IV does not change this result.

Based on Eq. 316 and the 2006 recommended valuesof Are, , d220, and R, the value of h implied by thisresult is

h = 6.626 074519 10−34 J s 2.9 10−7 . 318

A comparison of this value of h with those in TablesXXI and XXIII shows that it is generally not in goodagreement with the most accurate of the other values.

In this regard, two relatively recent publications, thefirst describing work performed in China Ding et al.,2005 and the second describing work performed inSwitzerland Reynolds et al., 2006, reported results that,if taken at face value, seem to call into question theuncertainty with which the molar mass of naturallyoccurring silicon is currently known. See also Valkierset al. 2005. These results highlight the importance ofthe current WGAC project to measure VmSi usinghighly enriched silicon crystals with x28Si0.99985Becker et al., 2006, which should simplify the determi-nation of the molar mass.

C. Gamma-ray determination of the neutron relative atomicmass Ar(n)

Although the value of Arn listed in Table II is aresult of AME2003, it is not used in the 2006 adjust-ment. Instead, Arn is obtained as discussed in this sec-tion in order to ensure that its recommended value isconsistent with the best current information on the 220lattice spacing of silicon.

The value of Arn can be obtained by measuring thewavelength of the 2.2 MeV ray in the reaction n+p→d+ in terms of the d220 lattice spacing of a particularsilicon crystal corrected to the commonly used referenceconditions t90=22.5 °C and p=0. The result for thewavelength-to-lattice spacing ratio, obtained fromBragg-angle measurements carried out in 1995 and 1998using a flat crystal spectrometer of the GAMS4 diffrac-tion facility at the high-flux reactor of the Institut Maxvon Laue–Paul Langevin ILL, Grenoble, France, in aNIST and ILL collaboration, is Kessler et al., 1999

λmeas = 0.002 904 302 4650 1.7 10−7 ,d220ILL

319

where d220ILL is the 220 lattice spacing of the siliconcrystals of the ILL GAMS4 spectrometer at t90=22.5 °C and p=0. Relativistic kinematics of the reac-tion yields

2λmeas Are A += rn Arp

,d220ILL R d 2

220ILL Arn + Arp − A2rd

320

where all seven quantities on the right-hand side areadjusted constants.

Dewey et al. 2006 and Rainville et al. 2005 reporteddeterminations of the wavelengths of the gamma rays

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TABLE XXVI. Summary of data related to the quotients h /mnd220W04, h /mCs, and h /mRb,together with inferred values of .

Relative standardQuantity Value uncertainty ur Identification Sec. and Eq.

h /mnd220W04−1

−12060.267 00484 m s137.036 007728

4.110−8

2.110−8

PTB-99 VIII.D.1 322VIII.D.1 324

h /mCs−1

−13.002 369 4324610−9 m2 s137.036 000011

1.510−8

7.710−9

StanfU-02 VIII.D.2 329VIII.D.2 331

h /mRb−1

−14.591 359 2876110−9 m2 s137.035 998 8391

1.310−8

6.710−9

LKB-06 VIII.D.2 332VIII.D.2 334

emitted in the cascade from the neutron capture state tothe ground state in the reactions n+ 28Si 29Si+2,+ 32S→ 33 35

→Si+3, and n+ Cl→ 36Cl+2. The gamma-ray

energies are 3.5 MeV and 4.9 MeV for the Si reaction,5.4 MeV, 2.4 MeV, and 0.8 MeV for the S reaction, and6.1 MeV, 0.5 MeV, and 2.0 MeV for the Cl reaction.While these data together with the relevant relativeatomic masses are potentially an additional source ofinformation on the neutron relative atomic mass, the un-certainties are too large for this purpose; the inferredvalue of Arn has an uncertainty nearly an order ofmagnitude larger than that obtained from Eq. 320. In-stead, this work is viewed as the most accurate test ofE=mc2 to date Rainville et al., 2005.

D. Quotient of Planck constant and particle mass h Õm(X) and

The relation

= R=2mec /2h leads to

2R ArX h 1/2

, 321c Are mX

where ArX is the relative atomic mass of particle Xwith mass mX and Are is the relative atomic mass ofthe electron. Because c is exactly known, ur of R andAre are less than 710−12 and 510−10, respectively,and ur of ArX for many particles and atoms is less thanthat of Are, Eq. 321 can provide a value of with acompetitive uncertainty if h /mX is determined with asufficiently small uncertainty. Here we discuss the deter-mination of h /mX for the neutron n, the 133Cs atom,and the 87Rb atom. The results, including the inferredvalues of , are summarized in Table XXVI.

1. Quotient h Õmn

The PTB determination of h /mn was carried out atthe ILL high-flux reactor. The de Broglie relation p=mnv=h /λ was used to determine h /mn=λv for theneutron by measuring both its de Broglie wavelength λand corresponding velocity v. More specifically, the deBroglie wavelength, λ0.25 nm, of slow neutrons wasdetermined using back reflection from a silicon crystal,and the velocity, v1600 m/s, of the neutrons was de-

ntermined by a special time-of-flight method. The finalresult of the experiment is Krüger et al., 1999

h= 2060.267 00484 m s−1

mnd220W04

4.1 10−8 , 322

where d220W04 is the 220 lattice spacing of the crystalWASO 04 at t90=22.5 °C in vacuum. This result is cor-related with the PTB fractional lattice-spacing differ-ences given in Eqs. 308–311—the correlation coeffi-cients are about 0.2.

The equation for the PTB result, which follows fromEq. 321, is

h A c= re 2

. 323mnd220W04 Arn 2Rd220W04

The value of that can be inferred from this relationand the PTB value of h /mnd220W04, the 2006 recom-mended values of R, Are, and Arn, the NIST andPTB fractional lattice-spacing-differences in Table XXV,and the four XROI values of d220X in Table XXIV forcrystals WASO 4.2a, NRLM3, and MO*, is

−1 = 137.036 007728 2.1 10−8 . 324

This value is included in Table XXVI as the first entry; itdisagrees with the values from the two other h /m re-sults.

It is also of interest to calculate the value of d220 im-plied by the PTB result for h /mnd220W04. Based onEq. 323, the 2006 recommended values of R, Are,Arp, Ard, , the NIST and PTB fractional lattice-spacing-differences in Table XXV, and the value ofλmeas/d220ILL given in Eq. 319, we find

d220 = 192 015.598279 fm 4.1 10−8 . 325

This result is included in Table XXIV as the last entry; itagrees with the NMIJ value, but disagrees with the PTBand INRIM values.

2. Quotient h Õm(133Cs)

The Stanford University atom interferometry experi-ment to measure the atomic recoil frequency shift of

133photons absorbed and emitted by Cs atoms, Cs, in

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order to determine the quotient h /m133Cs is describedin CODATA-02. As discussed there, the expression ap-plicable to the Stanford experiment is

h c2

m 133 = Cs

Cs 2 2 , 326eff

where the frequency eff corresponds to the sum of theenergy difference between the ground-state hyperfinelevel with F=3 and the 6P1/2 state F=3 hyperfine leveland the energy difference between the ground-state hy-perfine level with F=4 and the same 6P1/2 hyperfinelevel. The result for Cs/2 obtained at Stanford isWicht et al., 2002

Cs/2 = 15 006.276 8823 Hz 1.5 10−8 . 327

The Stanford effort included an extensive study ofcorrections due to possible systematic effects. The larg-est component of uncertainty by far contributing to theuncertainty of the final result for Cs, ur=1410−9

Type B, arises from the possible deviation from 1 ofthe index of refraction of the dilute background gas ofcold cesium atoms that move with the signal atoms. Thiscomponent, estimated experimentally, places a lowerlimit on the relative uncertainty of the inferred value of from Eq. 321 of u −9

r=710 . Without it, ur of would be about 3 to 4 parts in 109.

In the 2002 adjustment, the value eff=670 231 933 04481 kHz 1.210−10, based on themeasured frequencies of 133Cs D1-line transitions re-ported by Udem et al. 1999, was used to obtain theratio h /m133Cs from the Stanford value of Cs/2. Re-cently, using a femtosecond laser frequency comb and anarrow-linewidth diode laser, and eliminating Dopplershift by orienting the laser beam perpendicular to the133Cs atomic beam to within 5 rad, Gerginov et al.2006 remeasured the frequencies of the required tran-sitions and obtained a value of eff that agrees with thevalue used in 2002 but which has a ur 15 times smaller,

−12eff = 670 231 932 889.94.8 kHz 7.2 10 .

328

Evaluation of Eq. 326 with this result for eff and thevalue of Cs/2 in Eq. 327 yields

h133 = 3.002 369 43246 −9 10 m2 s−1

m Cs

1.5 10−8 , 329

which we take as an input datum in the 2006 adjustment.The observational equation for this datum is, from Eq.321,

h A= re c 2

m133Cs A 133 . 330r Cs 2R

The value of that may be inferred from this expres-sion, the Stanford result for h /m133Cs in Eq. 329, the2006 recommended values of R and Are, and theASME2003 value of Ar

133Cs in Table II, the uncertain-

ties of which are inconsequential in this application, is−1 = 137.036 000011 7.7 −9 10 , 331

where the dominant component of uncertainty arisesfrom the measured value of the recoil frequency shift, inparticular, the component of uncertainty due to a pos-sible index of refraction effect.

In this regard, we note that Campbell et al. 2005have experimentally demonstrated the reality of one as-pect of such an effect with a two-pulse light grating in-terferometer and shown that it can have a significantimpact on precision measurements with atom interfer-ometers. However, theoretical calculations based onsimulations of the Stanford interferometer by Sarajlic etal. 2006, although incomplete, suggest that the experi-mentally based uncertainty component ur=1410−9 as-signed by Wicht et al. 2002 to account for this effect isreasonable. We also note that Wicht et al. 2005 havedeveloped an improved theory of momentum transferwhen localized atoms and localized optical fields inter-act. The details of such interactions are relevant to pre-cision atom interferometry. When Wicht et al. 2005 ap-plied the theory to the Stanford experiment to evaluatepossible systematic errors arising from wave-front curva-ture and distortion, as well as the Gouy phase shift ofGaussian beams, they found that such errors do not limitthe uncertainty of the value of that can be obtainedfrom the experiment at the level of a few parts in 109,but will play an important role in future precision atom-interferometer photon-recoil experiments to measurewith ur510−10, such as is currently underway at Stan-ford Müller et al., 2006.

3. Quotient h Õm(87Rb)

In the LKB experiment Cladé et al., 2006a, 2006b,the quotient h /m87Rb, and hence , is determined byaccurately measuring the rubidium recoil velocity vr

=k /m87Rb when a rubidium atom absorbs or emits aphoton of wave vector k=2 /λ, where λ is the wave-length of the photon and =c /λ is its frequency. Themeasurements are based on Bloch oscillations in a ver-tical accelerated optical lattice.

The basic principle of the experiment is to preciselymeasure the variation of the atomic velocity induced byan accelerated standing wave using velocity selectiveRaman transitions between two ground-state hyperfinelevels. A Raman pulse of two counterpropagating la-ser beams selects an initial narrow atomic velocity class.After the acceleration process, the final atomic velocitydistribution is probed using a second Raman pulse oftwo counterpropagating laser beams.

The coherent acceleration of the rubidium atomsarises from a succession of stimulated two-photon tran-sitions also using two counterpropagating laser beams.Each transition modifies the atomic velocity by 2vr leav-ing the internal state unchanged. The Doppler shift iscompensated by linearly sweeping the frequency differ-ence of the two lasers. This acceleration can conve-niently be interpreted in terms of Bloch oscillations in

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the fundamental energy band of an optical lattice cre-ated by the standing wave, because the interference ofthe two laser beams leads to a periodic light shift of theatomic energy levels and hence to the atoms experienc-ing a periodic potential Ben Dahan et al., 1996; Peiket al., 1997.

An atom’s momentum evolves by steps of 2k, eachone corresponding to a Bloch oscillation. After N oscil-lations, the optical lattice is adiabatically released andthe final velocity distribution, which is the initial distri-bution shifted by 2Nvr, is measured. Due to the highefficiency of Bloch oscillations, for an acceleration of2000 m s−2, 900 recoil momenta can be transferred to arubidium atom in 3 ms with an efficiency of 99.97% perrecoil.

The atoms are alternately accelerated upwards anddownwards by reversing the direction of the Bloch ac-celeration laser beams, keeping the same delay betweenthe selection and measurement Raman pulses. Theresulting differential measurement is independent ofgravity. In addition, the contribution of some systematiceffects changes sign when the direction of the selectionand measuring Raman beams is exchanged. Hence, foreach up and down trajectory, the selection and measur-ing Raman beams are reversed and two velocity spectraare taken. The mean value of these two measurements isfree from systematic errors to first order. Thus each de-termination of h /m87Rb is obtained from four velocityspectra, each requiring 5 min of integration time, twofrom reversing the Raman beams when the accelerationis in the up direction and two when in the down direc-tion. The Raman and Bloch lasers are stabilized bymeans of an ultrastable Fabry-Pérot cavity and the fre-quency of the cavity is checked several times during the20 min measurement against a well-known two-photontransition in 85Rb.

Taking into account a −9.2±410−10 correction toh /m87Rb not included in the value reported by Cladéet al. 2006a due to a nonzero force gradient arisingfrom a difference in the radius of curvature of the upand down accelerating beams, the result derived from 72measurements of h /m87Rb acquired over 4 days,which we take as an input datum in the 2006 adjustment,is Cladé et al., 2006b

h

m87 = 4.591 359 28761 10−9 m2 s−1

Rb

1.3 −8 10 , 332

where the quoted ur contains a statistical componentfrom the 72 measurements of 8.810−9.

Cladé et al. 2006b examined many possible sourcesof systematic error, both theoretically and experimen-tally, in this rather complex, sophisticated experiment inorder to ensure that their result was correct. These in-clude light shifts, index of refraction effects, and the ef-fect of a gravity gradient, for which the corrections andtheir uncertainties are in fact comparatively small. Moresignificant are the fractional corrections of 16.8±8

10−9 for wave front curvature and Guoy phase,−13.2±410−9 for second-order Zeeman effect, and4410−9 for the alignment of the Raman and Blochbeams. The total of all corrections is given as10.9810.010−9.

From Eq. 321, the observational equation for theLKB value of h /m87Rb in Eq. 332 is

h A c87 = re 2

87 . 333m Rb Ar Rb 2R

Evaluation of this expression with the LKB result andthe 2006 recommended values of R and Are, and thevalue of Ar

87Rb resulting from the final least-squaresadjustment on which the 2006 recommended values arebased, all of whose uncertainties are negligible in thiscontext, yields

−1 = 137.035 998 8391 6.7 10−9 , 334

which is included in Table XXVI. The uncertainty of thisvalue of −1 is smaller than the uncertainty of any othervalue except those in Table XIV deduced from the mea-surement of ae, exceeding the smallest uncertainty of thetwo values of −1ae in that table by a factor of 10.

IX. THERMAL PHYSICAL QUANTITIES

The following sections discuss the molar gas constant,Boltzmann constant, and Stefan-Boltzmann constant—constants associated with phenomena in the fields ofthermodynamics and/or statistical mechanics.

A. Molar gas constant R

The square of the speed of sound c2ap ,T of a real gas

at pressure p and thermodynamic temperature T can bewritten as Colclough, 1973

c2ap,T = A0T + A1Tp + A2Tp2 + A 3

3Tp + ¯ ,

335

where A1T is the first acoustic virial coefficient, A2Tis the second, etc. In the limit p→0, Eq. 335 yields

c2 a0,T = A 0RT

0T = , 336ArXMu

where the expression on the right-hand side is thesquare of the speed of sound for an unbounded idealgas, and 0=cp /cV is the ratio of the specific-heat capac-ity of the gas at constant pressure to that at constantvolume, ArX is the relative atomic mass of the atomsor molecules of the gas, and Mu=10−3 kg mol−1. For amonatomic ideal gas, 0=5/3.

The 2006 recommended value of R, like the 2002 and1998 values, is based on measurements of the speed ofsound in argon carried out in two independent experi-ments, one done in the 1970s at NPL and the other donein the 1980s at NIST. Values of c2

ap ,TTPW, whereTTPW=273.16 K is the triple point of water, were ob-

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tained at various pressures and extrapolated to p=0 inorder to determine A0T 2

TPW=ca0,TTPW, and hence R,from

c2

R = a0,TTPWArArMu , 3370TTPW

which follows from Eq. 336.Because the work of both NIST and NPL is reviewed

in CODATA-98 and CODATA-02 and nothing has oc-curred in the past 4 years that would change the valuesof R implied by their reported values of c2

a0,TTPW, wegive only a summary here. Changes in these values dueto the new values of ArAAr resulting from the 2003atomic mass evaluation as given in Table II, or the newIUPAC compilation of atomic weights of the elementsgiven by Wieser 2006, are negligible.

Since R cannot be expressed as a function of anyother of the 2006 adjusted constants, R itself is taken asan adjusted constant for the NIST and NPL measure-ments.

1. NIST: Speed of sound in argon

In the NIST experiment of Moldover et al. 1988, aspherical acoustic resonator at a temperature T=TTPW

filled with argon was used to determine c2ap ,TTPW. The

final NIST result for the molar gas constant is

R = 8.314 47115 J mol−1 K−1 1.8 10−6 . 338

The mercury employed to determine the volume ofthe spherical resonator was traceable to the mercurywhose density was measured by Cook 1961 see alsoCook and Stone 1957. The mercury employed in theNMI Hg electrometer determination of KJ see VII.C.1was also traceable to the same mercury. Consequently,the NIST value of R and the NMI value of KJ are cor-related with the non-negligible correlation coefficient0.068.

2. NPL: Speed of sound in argon

In contrast to the dimensionally fixed resonator usedin the NIST experiment, the NPL experiment employeda variable path length fixed-frequency cylindrical acous-tic interferometer to measure c2

ap ,TTPW. The final NPLresult for the molar gas constant is Colclough et al.,1979

R = 8.314 50470 J mol−1 K−1 8.4 10−6 . 339

Although both the NIST and NPL values of R arebased on the same values of Ar

40Ar, Ar38Ar, and

Ar36Ar, the uncertainties of these relative atomic

masses are sufficiently small that the covariance of thetwo values of R is negligible.

3. Other values

The most important historical values of R have beenreviewed by Colclough 1984 see also Quinn et al.1976 and CODATA-98. However, because of the large

uncertainties of these early values, they were not consid-ered for use in the 1986, 1998, or 2002 CODATA adjust-ments, and we do not consider them for the 2006 adjust-ment as well.

Also because of its noncompetitive uncertainty ur=3610−6, we exclude from consideration in the 2006adjustment, as in the 2002 adjustment, the value of Robtained from measurements of the speed of sound inargon reported by He and Liu 2002 at the Xián Jiao-tong University, Xián, PRC.

B. Boltzmann constant k

The Boltzmann constant is related to the molar gasconstant R and other adjusted constants by

2Rk = h R

.cAre 2R = 340

Mu NA

No competitive directly measured value of k was avail-able for the 1998 or 2002 adjustments, and the situationremains unchanged for the present adjustment. Thus,the 2006 recommended value with ur=1.710−6 is ob-tained from this relation, as were the 1998 and 2002 rec-ommended values. However, a number of experimentsare currently underway that might lead to competitivevalues of k or R in the future; see Fellmuth et al. 2006for a recent review.

Indeed, one such experiment underway at the PTBbased on dielectric constant gas thermometry DCGTwas discussed in both CODATA-98 and CODATA-02,but no experimental result for A /R, where A is themolar polarizability of the 4He atom, other than thatconsidered in these two reports, has been published bythe PTB group see also Fellmuth et al. 2006 andLuther et al. 1996. However, the relative uncertaintyin the theoretical value of the static electric dipole po-larizability of the ground state of the 4He atom, which isrequired to calculate k from A /R, has been lowered bymore than a factor of 10 to below 210−7 Łach et al.,2004. Nevertheless, the change in its value is negligibleat the level of uncertainty of the PTB result for A /R;hence, the value k=1.380 65410−23 J K−1 3010−6from the PTB experiment given in CODATA-02 is un-changed.

In addition, preliminary results from two otherongoing experiments, the first at NIST by Schmidt et al.2007 and the second at the University of Paris byDaussy et al. 2007, have recently been published.

Schmidt et al. 2007 report R=8.314 48776 J mol K−1 9.110−6, obtained frommeasurements of the index of refraction np ,T of 4Hegas as a function of p and T by measuring the differencein the resonant frequencies of a quasispherical micro-wave resonator when filled with 4He at a given pressureand when evacuated that is, at p=0. This experimenthas some similarities to the PTB DCGT experiment inthat it determines the quantity A /R and hence k. How-ever, in DCGT one measures the difference in capaci-

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tance of a capacitor when filled with 4He at a given pres-sure and at p=0, and hence one determines thedielectric constant of the 4He gas rather than its index ofrefraction. Because 4He is slightly diamagnetic, thismeans that to obtain A /R in the NIST experiment, avalue for A /R is required, where A=40 /3 and 0 isthe diamagnetic susceptibility of a 4He atom.

Daussy et al. 2007 report k=1.380 652610−23 J K−1 19010−6, obtained from measurementsas a function of pressure of the Doppler profile at T=273.15 K the ice point of a well-isolated rovibrationalline in the 2 band of the ammonium molecule, 14NH3,and extrapolation to p=0. The experiment actually mea-sures R=kNA, because the mass of the ammonium mol-ecule in kilograms is required but can only be obtainedwith the requisite accuracy from the molar masses of 14Nand 1H, thereby introducing NA.

It is encouraging that the preliminary values of k andR resulting from these three experiments are consistentwith the 2006 recommended values.

C. Stefan-Boltzmann constant

The Stefan-Boltzmann constant is related to c, h, andthe Boltzmann constant k by

25k4

= , 34115h3c2

which, with the aid of Eq. 340, can be expressed interms of the molar gas constant and other adjusted con-stants as

325h =

15c6 RR 4

. 342AreM 2

u

No competitive directly measured value of was avail-able for the 1998 or 2002 adjustment, and the situationremains unchanged for the 2006 adjustment. Thus, the2006 recommended value with ur=7.010−6 is obtainedfrom this relation, as were the 1998 and 2002 recom-mended values. For a concise summary of experimentsthat might provide a competiive value of , see the re-view by Fellmuth et al. 2006.

X. NEWTONIAN CONSTANT OF GRAVITATION G

Because there is no known quantitative theoretical re-lationship between the Newtonian constant of gravita-tion G and other fundamental constants, and becausecurrently available experimental values of G are inde-pendent of all other data relevant to the 2006 adjust-ment, these experimental values contribute only to thedetermination of the 2006 recommended value of G andcan be considered independently from the other data.

The historic difficulty of determining G, as demon-strated by the inconsistencies among different measure-ments, is described in CODATA-86, CODATA-98, andCODATA-02. Although no new competitive indepen-dent result for G has become available in the last

4 years, adjustments to two existing results considered in2002 have been made by researchers involved in theoriginal work. One of the two results that has changed isfrom the Huazhong University of Science and Technol-ogy HUST and is now identified as HUST-05; the otheris from the University of Zurich UZur and is now iden-tified as UZur-06. These revised results are discussedbelow.

Table XXVII summarizes the various values of G con-sidered here, which are the same as in 2002 with theexception of these two revised results, and Fig. 2 com-pares them graphically. For reference purposes, both thetable and figure include the 2002 and 2006 CODATArecommended values. The result now identified asTR&D-96 was previously identified as TR&D-98. Thechange is because a 1996 reference Karagioz and Iz-mailov, 1996 was found that reports the same result asdoes the 1998 reference Karagioz et al., 1998.

For simplicity, we write G as a numerical factor mul-tiplying G0, where

G = 10−11 30 m kg−1 s−2. 343

A. Updated values

1. Huazhong University of Science and Technology

The HUST group, which determines G by the time-of-swing method using a high-Q torsion pendulum withtwo horizontal, 6.25 kg stainless steel cylindrical sourcemasses labeled A and B positioned on either side of thetest mass, has reported a fractional correction of +36010−6 to their original result given by Luo et al. 1999.It arises in part from recently discovered density inho-mogeneities in the source masses, the result of which is adisplacement of the center of mass of each source massfrom its geometrical center Hu et al., 2005. Using a“weighbridge” with a commercial electronic balance—amethod developed by Davis 1995 to locate the centerof mass of a test object with micrometers precision—Huet al. 2005 found that the axial eccentricities eA and eBof the two source masses were 10.3±2.6 m and6.3±3.7 m, with the result that the equivalent dis-placements between the test and source masses arelarger than the values used by Luo et al. 1999. Assum-ing a linear axial density distribution, the calculated frac-tional correction to the previous result is +21010−6

with an additional component of relative standard un-certainty of 7810−6 due to the uncertainties of the ec-centricities.

The remaining 15010−6 portion of the 36010−6

fractional correction has also been discussed by Hu et al.2005 and arises as follows. In the HUST experiment, Gis determined by comparing the period of the torsionpendulum with and without the source masses. Whenthe source masses are removed, they are replaced by air.Since the masses of the source masses used by Luo et al.1999 are the vacuum masses, a correction for the air,first suggested by R. S. Davis and T. J. Quinn of theBIPM, is required. This correction was privately com-

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TABLE XXVII. Summary of the measurements of the Newtonian constant of gravitation relevant to the 2006 adjustment togetherwith the 2002 and 2006 CODATA recommended values.

Item Source Identificationa Method10 G

−2m3 kg−1 sRel. stand.uncert. ur

2002 CODATA adjustment CODATA-02 6.674210 1.510−4

a. Karagioz and Izmailov, 1996 TR&D-96 Fiber torsion balance, 6.67295 7.510−5

dynamic modeb. Bagley and Luther, 1997 LANL-97 Fiber torsion balance, 6.67407 1.010−4

dynamic modec. Gundlach and Merkowitz, UWash-00 Fiber torsion balance, 6.674 25592 1.410−5

2000, 2002, dynamic compensationd. Quinn et al., 2001 BIPM-01 Strip torsion balance, 6.675 5927 4.010−5

compensation mode,static deflection

e. Kleinvoß, 2002, Kleinvoß UWup-02 Suspended body, 6.674 2298 1.510−4

et al., 2002 displacementf. Armstrong and Fitzgerald, MSL-03 Strip torsion balance, 6.673 8727 4.010−5

2003 compensation modeg. Hu et al., 2005 HUST-05 Fiber torsion balance, 6.67239 1.310−4

dynamic modeh. Schlamminger et al., 2006 UZur-06 Stationary body, 6.674 2512 1.910−5

weight change2006 CODATA adjustment CODATA-06 6.674 28 67 1.010−4

municated to the Task Group by the HUST researchersin 2003 and included in the HUST value of G used in the2002 adjustment.

The HUST revised value of G, including the addi-tional component of uncertainty due to the measure-

G/(10−11 m3 kg−1 s−2)

6.670 6.672 6.674 6.676 6.678

6.670 6.672 6.674 6.676 6.678

−410 G

UWup-02

HUST-05

LANL-97

TR&D-96

BIPM-01

MSL-03

UZur-06

UWash-00

CODATA-02

CODATA-06

FIG. 2. Values of the Newtonian constant of gravitation G.

ment of the eccentricities eA and eB, is item g in TableXXVII.

2. University of Zurich

The University of Zurich result for G discussedin CODATA-02 and used in the 2002 adjustment,G=6.674 0722 G0 3.310−5, was reported by Schlam-minger et al. 2002. It was based on the weighted meanof three highly consistent values obtained from threeseries of measurements carried out at the Paul ScherrerInstitute PSI, Villigen, Switzerland, in 2001 and 2002and denoted Cu, Ta I, and Ta II. The designation Cumeans that the test masses were gold plated copper, andthe designation Ta means that they were tantalum. Fol-lowing the publication of Schlamminger et al. 2002, anextensive reanalysis of the original data was carried outby these authors together with other University of Zur-ich researchers, the result being the value of G in TableXXVII, item h, as given in the detailed final report onthe experiment Schlamminger et al., 2006.

In the University of Zurich approach to determiningG, a modified commercial single-pan balance is used tomeasure the change in the difference in weight of twocylindrical test masses when the relative position of twosource masses is changed. The quantity measured is the

aTR&D: Tribotech Research and Development Company, Moscow, Russian Federation; LANL: Los Alamos National Labora-tory, Los Alamos, New Mexico, USA; UWash: University of Washington, Seattle, Washington, USA; BIPM: International Bureauof Weights and Measures, Sèvres, France; UWup: University of Wuppertal, Wuppertal, Germany; MSL: Measurement StandardsLaboratory, Lower Hutt, New Zealand; HUST: Huazhong University of Science and Technology, Wuhan, PRC; UZur: Universityof Zurich, Zurich, Switzerland.

11

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800 g difference signal obtained at many differentworking points in the balance calibration range usingtwo sets of 16 individual wire weights, allowing an in situmeasurement of the balance nonlinearity over the entire0.2 g balance calibration interval. A more rigorousanalysis using a fitting method with Legendre polynomi-als has now allowed the relative standard uncertaintycontribution to G from balance nonlinearity to be re-duced from 1810−6 to 6.110−6 based on the Cu test-mass data. Various problems with the mass handler forthe wire weights that did not allow application of theLegendre polynomial fitting procedure occurred duringthe Ta test-mass measurements, resulting in large sys-tematic errors. Therefore, the researchers decided to in-clude only the Cu data in their final analysis Schlam-minger et al., 2006.

Each source mass consisted of a cylindrical tank filledwith 7.5103 kg of mercury. Since the mercury repre-sented approximately 94% of the total mass, special carewas taken in determining its mass and density. Thesemeasurements were used further to obtain more accu-rate values for the key tank dimensions and Hg mass.This was done by minimizing a 2 function that de-pended on the tank dimensions and the Hg mass anddensity, and using the dependence of the density onthese dimensions and the Hg mass as a constraint. Cal-culation of the mass integration constant with these im-proved values reduced the ur of this critical quantityfrom 20.610−6 to 6.710−6.

Although the analysis of Schlamminger et al. 2002assumed a linear temporal drift of the balance zeropoint, a careful examination by Schlamminger et al.2006 found that the drift was significantly nonlinearand was influenced by the previous load history of thebalance. A series of Legendre polynomials and a saw-tooth function, respectively, were therefore used to de-scribe the slow and rapid variations of the observed bal-ance zero point with time.

The 2002 value of G obtained from the Cu data was6.674 03 G0, consistent with the Ta I and Ta II values of6.674 09 G0 and 6.674 10 G0 Schlamminger et al., 2002,whereas the value from the present Cu data analysis is6.674 2512 G0, with the 3.310−5 fractional increasedue primarily to the application of the nonlinear zero-point drift correction. A minor contributor to the differ-ence is the inclusion of the very first Cu data set that wasomitted in the 2002 analysis due to a large start-up zero-point drift that is now correctable with the new Leg-endre polynomial-sawtooth function analysis technique,and the exclusion of a data set that had a temperaturestabilization system failure that went undetected by theold data analysis method Schlamminger, 2007.

B. Determination of 2006 recommended value of G

The overall agreement of the eight values of G inTable XXVII items a to h has improved somewhatsince the 2002 adjustment, but the situation is stillfar from satisfactory. Their weighted mean is G

=6.674 27568 G0 with 2=38.6 for degrees of freedom=N−M=8−1=7, Birge ratio RB=2 /=2.35, and nor-malized residuals ri of −2.75, −0.39, −0.22, 4.87, −0.56,−1.50, −2.19, and −0.19, respectively see Appendix E ofCODATA-98. The BIPM-01 value with ri=4.87 isclearly the most problematic. For comparison, the 2002weighted mean was G=6.674 23275 G0 with 2=57.7for =7 and RB=2.87.

If the BIPM value is deleted, the weighted mean isreduced by 1.3 standard uncertainties to G=6.674 18770 G0, and 2=13.3, =6, and RB=1.49. Inthis case, the two remaining data with significant nor-malized residuals are the TR&D-96 and HUST-05 re-sults with ri=−2.57 and −2.10, respectively. If these twodata, which agree with each other, are deleted, theweighted mean is G=6.674 22571 G 2

0 with =2.0, =4, RB=0.70, and with all normalized residuals less than1 except ri=−1.31 for datum MSL-03. Finally, if theUWash-00 and UZur-06 data, which have the smallestassigned uncertainties of the initial eight values andwhich are in excellent agreement with each other, aredeleted from the initial group of eight data, the weightedmean of the remaining six data is G=6.674 384167 G0with 2=38.1, =6, and RB=2.76. The normalized re-siduals for these six data, TR&D-96, LANL-97, BIPM-01, UWup-02, MSL-03, and HUST-05, are −2.97, −0.55,4.46, −0.17, −1.91, and −2.32, respectively.

Finally, if the uncertainties of each of the eight valuesof G are multiplied by the Birge ratio associated withtheir weighted mean, RB=2.35, so that 2 of theirweighted mean becomes equal to its expected value of=7 and RB=1, the normalized residual of the datumBIPM-01 would still be larger than 2.

Based on the results of the above calculations, thehistorical difficulty of determining G, the fact that alleight values of G in Table XXVII are credible, and thatthe two results with the smallest uncertainties,UWash-00 and UZur-06, are highly consistent with oneanother, the Task Group decided to take as the 2006CODATA recommended value of G the weighted meanof all data, but with an uncertainty of 0.000 67 G0, cor-responding to ur=1.010−4,

G = 6.674 2867 10−11 m3 kg−1 s−2 1.0 10−4 .

344

This value exceeds the 2002 recommended value by thefractional amount 1.210−5, which is less than one-tenthof the uncertainty ur=1.510−4 of the 2002 value. Fur-ther, the uncertainty of the 2006 value, ur=1.010−4, istwo-thirds that of the 2002 value.

In assigning this uncertainty to the 2006 recom-mended value of G, the Task Group recognized that ifthe uncertainty was smaller than really justified by thedata, taking into account the history of measurements ofG, it might discourage the initiation of new research ef-forts to determine G, if not the continuation of some ofthe research efforts already underway. Such efforts needto be encouraged in order to provide a more solid and

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redundant data set upon which to base future recom-mended values. On the other hand, if the uncertaintywere too large, for example, if the uncertainty of the2002 recommended value had been retained for the 2006value, then the recommended value would not have re-flected the fact that we now have two data that are inexcellent agreement, have ur less than 210−5, and arethe two most accurate values available.

C. Prospective values

New techniques to measure G using atom interferom-etry are currently under development in at least twolaboratories—the Università de Firenze in Italy andStanford University in the United States. This comes asno surprise since atom interferometry is also being de-veloped to measure the local acceleration due to gravityg see Sec. II. Recent proof-of-principle experimentscombine two vertically separated atomic clouds formingan atom-interferometer-gravity gradiometer that mea-sures the change in the gravity gradient when a wellcharacterized source mass is displaced. Measuring thechange in the gravity gradient allows the rejection ofmany possible systematic errors. Bertoldi et al. 2006 atthe Università de Firenze used a Rb fountain and a fastlaunch juggling sequence of two atomic clouds to mea-sure G to 1%, obtaining the value 6.646 G0; they hopeto reach a final uncertainty of 1 part in 104. Fixler et al.2007 at Stanford used two separate Cs atom interfer-ometer gravimeters to measure G and obtained thevalue 6.69334 G0. The two largest uncertainties fromsystematic effects were the determination of the initialatom velocity and position. The Stanford researchersalso hope to achieve a final uncertainty of 1 part in 104.Although neither of these results is significant for thecurrent analysis of G, future results could be of consid-erable interest.

XI. X-RAY AND ELECTROWEAK QUANTITIES

A. X-ray units

Historically, units that have been used to express thewavelengths of x-ray lines are the copper K1 x unit,symbol xuCuK1, the molybdenum K1 x unit, symbolxuMoK1, and the ångström star, symbol Å*. They aredefined by assigning an exact, conventional value to thewavelength of the CuK1, MoK1, and WK1 x-raylines when each is expressed in its corresponding unit,

λCuK1 = 1537.400 xuCuK1 , 345

λMoK1 = 707.831 xuMoK1 , 346

λWK1 = 0.209 010 0 Å*. 347

The experimental work that determines the best val-ues of these three units was reviewed in CODATA-98,and the relevant data may be summarized as follows:

λCuK1= 0.802 327 1124 3.0 10−7 , 348

d220W4.2a

λWK1= 0.108 852 17598 9.0 10−7 , 349

d220N

λMoK1= 0.369 406 0419 5.3 −7 10 , 350

d220N

λCuK1= 0.802 328 0477 9.6 1

d220N

where d220W4.2a and d220N denote thespacings, at the standard reference conditit90=22.5 °C, of particular silicon crystalsmeasurements. The result in Eq. 348 is frration between researchers from Friedrichversity FSU, Jena, Germany and the PTal., 1991. The lattice spacing d220N is retals of known lattice spacing through Eq.

In order to obtain best values in thesense for xuCuK , xuCuK , and Å*1 1 ,units to be adjusted constants. Thus, theequations for the data of Eqs. 348–351

λCuK1 1537.400 xuCuK1= ,

d220W4.2a d220W4.2a

λWK Å*1 0.209 010 0= ,

d220N d220N

λMoK1 707.831 xuMoK1= ,

d220N d220N

λCuK1 1537.400 xuCuK1= ,

d220N d220N

where d220N is taken to be an adjustedd220W17 and d220W4.2a are adjustedwell.

B. Particle Data Group input

There are a few cases in the 2006 adjustan inexact constant that is used in the anadata is not treated as an adjusted quantityadjustment has a negligible effect on itssuch constants, used in the calculation of texpressions for the electron and muonment anomalies ae and a, are the mass of tm, the Fermi coupling constant GF, and sithe weak mixing angle sin2W, and are obtamost recent report of the Particle Dataet al., 2006,

mc2 = 1776.9929 MeV 1.6 10−4 ,

0−7 , 351

220 latticeons p=0 andused in the

om a collabo--Schiller Uni-B Härtwig etlated to crys-301.least-squares

we take theseobservationalare

352

353

354

355

constant andconstants as

ment in whichlysis of input, because thevalue. Three

he theoreticalmagnetic mo-he tau leptonne squared ofined from theGroup Yao

356

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Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

GF 10−53 = 1.166 371 GeV−2 8.6 10−6 ,

c357

sin2 W = 0.222 5556 2.5 10−3 . 358

To facilitate the calculations, the uncertainty of mc2 is

symmetrized and taken to be 0.29 MeV rather than+0.29 MeV, −0.26 MeV. We use the definition sin2W=1− mW/mZ2, where mW and mZ are, respectively, themasses of the W± and Z0 bosons, because it is employedin the calculation of the electroweak contributions to aeand a Czarnecki et al., 1996. The Particle DataGroup’s recommended value for the mass ratio of thesebosons is mW/mZ=0.881 7332, which leads to the valueof sin2W given above.

XII. ANALYSIS OF DATA

The previously discussed input data are examined inthis section for their mutual compatibility and their po-tential role in determining the 2006 recommended val-ues of the constants. Based on this analysis, the dataare selected for the final least-squares adjustment fromwhich the recommended values are obtained. Becausethe data on the Newtonian constant of gravitation G areindependent of the other data and are analyzed in Sec.X, they are not examined further. The consistency of theinput data is evaluated by directly comparing differentmeasurements of the same quantity, and by directlycomparing the values of a single fundamental constantinferred from measurements of different quantities. Asnoted in the outline section of this paper, the inferredvalue is for comparison purposes only; the datum fromwhich it is obtained, not the inferred value, is the inputdatum in the adjustment. The potential role of a particu-lar input datum is gauged by carrying out a least-squaresadjustment using all initially considered data. A particu-lar measurement of a quantity is included in the finaladjustment if its uncertainty is not more than about tentimes the uncertainty of the value of that quantity pro-vided by other data in the adjustment. The measure weuse is the “self-sensitivity coefficient” of an input datumSc see CODATA-98, which must be greater than 0.01in order for the datum to be included.

The input data are given in Tables XXVIII, XXX, andXXXII and their covariances are given as correlationcoefficients in Tables XXIX, XXXI, and XXXIII. The’s given in Tables XXVIII, XXX, and XXXII are quan-tities added to corresponding theoretical expressions toaccount for the uncertainties of those expressions, aspreviously discussed see, for example, Sec. IV.A.1.l.Note that the value of the Rydberg constant R dependsonly weakly on changes, at the level of the uncertainties,of the data in Tables XXX and XXXII.

A. Comparison of data

The classic Lamb shift is the only quantity among theRydberg constant data with more than one measuredvalue, but there are ten different quantities with morethan one measured value among the other data. Theitem numbers given in Tables XXVIII and XXX for themembers of such groups of data A39, B2, B11, B28,B31, B33–B36, B38, and B58 have a decimal point withan additional digit to label each member.

In fact, all data for which there is more than one mea-surement were directly compared in either the 1998 or2002 adjustments except the following new data: theUniversity of Washington result for Ar

2H, item B2.2,the Harvard University result for ae, item B11.2, theNIST watt-balance result for K2

JRK item B36.3, and theINRIM result for d220W4.2a, item B38.2. The two val-ues of Ar

2H agree well—they differ by only 0.5udiff; thetwo values of ae are in acceptable agreement—they dif-fer by 1.7udiff; the two values of d220W4.2a also agreewell—they differ by 0.7udiff; and the three values ofK2

JRK are highly consistent—their mean and impliedvalue of h are

K2JRK = 6.036 761 8721 1033 J−1 s−1, 359

h = 6.626 068 8923 10−34 J s 360

with 2=0.27 for =N−M=2 degrees of freedom, whereN is the number of measurements and M is the numberof unknowns, and with Birge ratio RB=2 /=0.37 seeAppendix E of CODATA-98. The normalized residualsfor the three values are 0.52, −0.04, and −0.09, and theirweights in the calculation of the weighted mean are 0.03,0.10, and 0.87.

Data for quantities with more than one directly mea-sured value used in earlier adjustments are consistent,with the exception of the VNIIM 1989 result forΓh−9 0lo, which is not included in the present adjustmentsee Sec. VII. We also note that none of these data hasa weight of less than 0.02 in the weighted mean of mea-surements of the same quantity.

The consistency of measurements of various quanti-ties of different types is shown mainly by comparing thevalues of the fine-structure constant or the Planck con-stant h inferred from the measured values of the quan-tities. Such inferred values of and h are given through-out the data review sections, and the results aresummarized and discussed further here.

The consistency of a significant fraction of the data ofTables XXVIII and XXX is indicated in Table XXXIVand Figs. 3–5, which give and graphically compare thevalues of inferred from that data. Figures 3 and 4compare the data that yield values of with ur10−7

and ur10−8, respectively; Fig. 5 also compares thedata that yield values of with ur10−7, but does so

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TABLE XXVIII. Summary of principal input data for the determination of the 2006 recommended value of the Rydberg constantR. The notation for the additive corrections XnLj has the same meaning as the notation XnLj in Sec. IV.A.1.l.

Item standardnumber Input datum Value uncertaintya ur Identification Sec.

A1 H1S1/2 0.03.7 kHz 1.110−12 Theory IV.A.1.lA2 H2S1/2 0.0046 kHz 5.610−13 Theory IV.A.1.lA3 H3S1/2 0.0014 kHz 3.710−13 Theory IV.A.1.lA4 H4S1/2 0.00058 kHz 2.810−13 Theory IV.A.1.lA5 H6S1/2 0.00020 kHz 2.110−13 Theory IV.A.1.lA6 H8S1/2 0.000082 kHz 1.610−13 Theory IV.A.1.lA7 H2P1/2 0.00069 kHz 8.410−14 Theory IV.A.1.lA8 H4P1/2 0.000087 kHz 4.210−14 Theory IV.A.1.lA9 H2P3/2 0.00069 kHz 8.410−14 Theory IV.A.1.lA10 H4P3/2 0.000087 kHz 4.210−14 Theory IV.A.1.lA11 H8D3/2 0.000 0048 kHz 9.310−15 Theory IV.A.1.lA12 H12D3/2 0.000 0015 kHz 6.610−15 Theory IV.A.1.lA13 H4D5/2 0.000038 kHz 1.910−14 Theory IV.A.1.lA14 H6D5/2 0.000011 kHz 1.210−14 Theory IV.A.1.lA15 H8D5/2 0.000 0048 kHz 9.310−15 Theory IV.A.1.lA16 H12D5/2 0.000 0016 kHz 7.010−15 Theory IV.A.1.lA17 D1S1/2 0.03.6 kHz 1.110−12 Theory IV.A.1.lA18 D2S1/2 0.0045 kHz 5.410−13 Theory IV.A.1.lA19 D4S1/2 0.00056 kHz 2.710−13 Theory IV.A.1.lA20 D8S1/2 0.000080 kHz 1.610−13 Theory IV.A.1.lA21 D8D3/2 0.000 0048 kHz 9.310−15 Theory IV.A.1.lA22 D12D3/2 0.000 0015 kHz 6.610−15 Theory IV.A.1.lA23 D4D5/2 0.000038 kHz 1.910−14 Theory IV.A.1.lA24 D8D5/2 0.000 0048 kHz 9.310−15 Theory IV.A.1.lA25 D12D5/2 0.000 0016 kHz 7.010−15 Theory IV.A.1.lA26 H1S1/2−2S1/2 2 466 061 413 187.07434 kHz 1.410−14 MPQ-04 IV.A.2A27 H2S1/2−8S1/2 770 649 350 012.08.6 kHz 1.110−11 LK/SY-97 IV.A.2A28 H2S1/2−8D3/2 770 649 504 450.08.3 kHz 1.110−11 LK/SY-97 IV.A.2A29 H2S1/2−8D5/2 770 649 561 584.26.4 kHz 8.310−12 LK/SY-97 IV.A.2A30 H2S1/2−12D3/2 799 191 710 472.79.4 kHz 1.210−11 LK/SY-98 IV.A.2A31 H2S1/2−12D5/2 799 191 727 403.77.0 kHz 8.710−12 LK/SY-98 IV.A.2A32 H2S1/2−4S1/2− 1

4H1S1/2−2S1/2 4 797 33810 kHz 2.110−6 MPQ-95 IV.A.2

A33 H2S1/2−4D5/2− 14H1S1/2−2S1/2 6 490 14424 kHz 3.710−6 MPQ-95 IV.A.2

A34 H2S1/2−6S1/2− 14H1S1/2−3S1/2 4 197 60421 kHz 4.910−6 LKB-96 IV.A.2

A35 H2S1/2−6D5/2− 14H1S1/2−3S1/2 4 699 09910 kHz 2.210−6 LKB-96 IV.A.2

A36 H2S1/2−4P1/2− 14H1S1/2−2S1/2 4 664 26915 kHz 3.210−6 YaleU-95 IV.A.2

A37 H2S1/2−4P3/2− 14H1S1/2−2S1/2 6 035 37310 kHz 1.710−6 YaleU-95 IV.A.2

A38 H2S1/2−2P3/2 9 911 20012 kHz 1.210−6 HarvU-94 IV.A.2A39.1 H2P1/2−2S1/2 1 057 845.09.0 kHz 8.510−6 HarvU-86 IV.A.2A39.2 H2P1/2−2S1/2 1 057 86220 kHz 1.910−5 USus-79 IV.A.2A40 D2S1/2−8S1/2 770 859 041 245.76.9 kHz 8.910−12 LK/SY-97 IV.A.2A41 D2S1/2−8D3/2 770 859 195 701.86.3 kHz 8.210−12 LK/SY-97 IV.A.2A42 D2S1/2−8D5/2 770 859 252 849.55.9 kHz 7.710−12 LK/SY-97 IV.A.2A43 D2S1/2−12D3/2 799 409 168 038.08.6 kHz 1.110−11 LK/SY-98 IV.A.2A44 D2S1/2−12D5/2 799 409 184 966.86.8 kHz 8.510−12 LK/SY-98 IV.A.2A45 D2S1/2−4S1/2− 1

4D1S1/2−2S1/2 4 801 69320 kHz 4.210−6 MPQ-95 IV.A.2

A46 D2S1/2−4D5/2− 14D1S1/2−2S1/2 6 494 84141 kHz 6.310−6 MPQ-95 IV.A.2

Relative

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TABLE XXVIII. Continued.

Itemnumber Input datum Value

Relativestandarduncertaintya ur Identification Sec.

A47 D1S1/2−2S1/2−H1S1/2−2S1/2 670 994 334.6415 kHz 2.210−10 MPQ-98 IV.A.2A48 Rp 0.89518 fm 2.010−2 Rp-03 IV.A.3A49 Rd 2.13010 fm 4.710−3 Rd-98 IV.A.3

aThe values in brackets are relative to the frequency equivalent of the binding energy of the indicated level.

TABLE XXIX. Correlation coefficients rxi ,xj0.0001 of the input data related to R in TableXXVIII. For simplicity, the two items of data to which a particular correlation coefficient correspondsare identified by their item numbers in Table XXVIII.

rA1,A2=0.9958 rA6,A19=0.8599 rA27,A28=0.3478 rA30,A44=0.1136rA1,A3=0.9955 rA6,A20=0.9913 rA27,A29=0.4532 rA31,A34=0.0278rA1,A4=0.9943 rA7,A8=0.0043 rA27,A30=0.0899 rA31,A35=0.0553rA1,A5=0.8720 rA9,A10=0.0043 rA27,A31=0.1206 rA31,A40=0.1512rA1,A6=0.8711 rA11,A12=0.0005 rA27,A34=0.0225 rA31,A41=0.1647

rA1,A17=0.9887 rA11,A21=0.9999 rA27,A35=0.0448 rA31,A42=0.1750rA1,A18=0.9846 rA11,A22=0.0003 rA27,A40=0.1225 rA31,A43=0.1209rA1,A19=0.9830 rA12,A21=0.0003 rA27,A41=0.1335 rA31,A44=0.1524rA1,A20=0.8544 rA12,A22=0.9999 rA27,A42=0.1419 rA32,A33=0.1049rA2,A3=0.9954 rA13,A14=0.0005 rA27,A43=0.0980 rA32,A45=0.2095rA2,A4=0.9942 rA13,A15=0.0005 rA27,A44=0.1235 rA32,A46=0.0404rA2,A5=0.8719 rA13,A16=0.0004 rA28,A29=0.4696 rA33,A45=0.0271rA2,A6=0.8710 rA13,A23=0.9999 rA28,A30=0.0934 rA33,A46=0.0467

rA2,A17=0.9846 rA13,A24=0.0002 rA28,A31=0.1253 rA34,A35=0.1412rA2,A18=0.9887 rA13,A25=0.0002 rA28,A34=0.0234 rA34,A40=0.0282rA2,A19=0.9829 rA14,A15=0.0005 rA28,A35=0.0466 rA34,A41=0.0307rA2,A20=0.8543 rA14,A16=0.0005 rA28,A40=0.1273 rA34,A42=0.0327rA3,A4=0.9939 rA14,A23=0.0002 rA28,A41=0.1387 rA34,A43=0.0226rA3,A5=0.8717 rA14,A24=0.0003 rA28,A42=0.1475 rA34,A44=0.0284rA3,A6=0.8708 rA14,A25=0.0002 rA28,A43=0.1019 rA35,A40=0.0561

rA3,A17=0.9843 rA15,A16=0.0005 rA28,A44=0.1284 rA35,A41=0.0612rA3,A18=0.9842 rA15,A23=0.0002 rA29,A30=0.1209 rA35,A42=0.0650rA3,A19=0.9827 rA15,A24=0.9999 rA29,A31=0.1622 rA35,A43=0.0449rA3,A20=0.8541 rA15,A25=0.0002 rA29,A34=0.0303 rA35,A44=0.0566rA4,A5=0.8706 rA16,A23=0.0002 rA29,A35=0.0602 rA36,A37=0.0834rA4,A6=0.8698 rA16,A24=0.0002 rA29,A40=0.1648 rA40,A41=0.5699

rA4,A17=0.9831 rA16,A25=0.9999 rA29,A41=0.1795 rA40,A42=0.6117rA4,A18=0.9830 rA17,A18=0.9958 rA29,A42=0.1908 rA40,A43=0.1229rA4,A19=0.9888 rA17,A19=0.9942 rA29,A43=0.1319 rA40,A44=0.1548rA4,A20=0.8530 rA17,A20=0.8641 rA29,A44=0.1662 rA41,A42=0.6667rA5,A6=0.7628 rA18,A19=0.9941 rA30,A31=0.4750 rA41,A43=0.1339

rA5,A17=0.8622 rA18,A20=0.8640 rA30,A34=0.0207 rA41,A44=0.1687rA5,A18=0.8621 rA19,A20=0.8627 rA30,A35=0.0412 rA42,A43=0.1423rA5,A19=0.8607 rA21,A22=0.0001 rA30,A40=0.1127 rA42,A44=0.1793rA5,A20=0.7481 rA23,A24=0.0001 rA30,A41=0.1228 rA43,A44=0.5224rA6,A17=0.8613 rA23,A25=0.0001 rA30,A42=0.1305 rA45,A46=0.0110rA6,A18=0.8612 rA24,A25=0.0001 rA30,A43=0.0901

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TABLE XXX. Summary of principal input data for the determination of the 2006 recommended values of the fundamentalconstants R and G excepted.

Itemnumber Input datum Value

Relativestandarduncertaintya

ur Identification Sec. and Eq.

B1 Ar1H 1.007 825 032 0710 1.010−10 AMDC-03 III.A

B2.1 Ar2H 2.014 101 777 8536 1.810−10 AMDC-03 III.A

B2.2 Ar2H 2.014 101 778 04080 4.010−11 UWash-06 III.A

B3 Ar3H 3.016 049 278725 8.310−10 MSL-06 III.A

B4 Ar3He 3.016 029 321726 8.610−10 MSL-06 III.A

B5 Ar4He 4.002 603 254 13162 1.510−11 UWash-06 III.A

B6 Ar16O 15.994 914 619 5718 1.110−11 UWash-06 III.A

B7 Ar87Rb 86.909 180 52612 1.410−10 AMDC-03 III.A

B8b Ar133Cs 132.905 451 93224 1.810−10 AMDC-03 III.A

B9 Are 0.000 548 579 911112 2.110−9 UWash-95 III.C 5B10 e 0.002710−12 2.410−10 Theory V.A.1 101B11.1 ae 1.159 652 18834210−3 3.710−9 UWash-87 V.A.2.a 102B11.2 ae 1.159 652 180 857610−3 6.610−10 HarvU-06 V.A.2.b 103B12 0.02.110−9 1.810−6 Theory V.B.1 126B13 R 0.003 707 206420 5.410−7 BNL-06 V.B.2 128B14 C 0.002710−10 1.410−11 Theory V.C.1 169B15 O 0.01.110−10 5.310−11 Theory V.C.1 172B16 fs

12C5+ / fc12C5+ 4376.210 498923 5.210−10 GSI-02 V.C.2.a 175

B17 fs16O7+ / fc

16O7+ 4164.376 183732 7.610−10 GSI-02 V.C.2.b 178B18 e−H /pH −658.210 705866 1.010−8 MIT-72 VI.A.2.a 195B19 dD /e−D −4.664 345 3925010−4 1.110−8 MIT-84 VI.A.2.b 197B20 pHD /dHD 3.257 199 53129 8.910−9 StPtrsb-03 VI.A.2.c 201B21 dp 15210−9 StPtrsb-03 VI.A.2.c 203B22 tHT /pHT 1.066 639 88710 9.410−9 StPtrsb-03 VI.A.2.c 202B23 tp 20310−9 StPtrsb-03 VI.A.2.c 204B24 e−H /p −658.215 943072 1.110−8 MIT-77 VI.A.2.d 209B25 h /p −0.761 786 131333 4.310−9 NPL-93 VI.A.2.e 211B26 n /p −0.684 996 9416 2.410−7 ILL-79 VI.A.2.f 212B27 Mu 0101 Hz 2.310−8 Theory VI.B.1 234B28.1 Mu 4 463 302.8816 kHz 3.610−8 LAMPF-82 VI.B.2.a 236B28.2 Mu 4 463 302 76553 Hz 1.210−8 LAMPF-99 VI.B.2.b 239B29 58 MHz 627 994.7714 kHz 2.210−7 LAMPF-82 VI.B.2.a 237B30 72 MHz 668 223 16657 Hz 8.610−8 LAMPF-99 VI.B.2.b 240B31.1b Γp−90 lo 2.675 154 0530108 s−1 T−1 1.110−7 NIST-89 VII.A.1.a 253B31.2b Γp−90 lo 2.675 153018108 s−1 T−1 6.610−7 NIM-95 VII.A.1.b 255B32b Γh−90 lo 2.037 895 3737108 s−1 T−1 1.810−7 KR/VN-98 VII.A.1.c 257B33.1b Γp−90 hi 2.675 152543108 s−1 T−1 1.610−6 NIM-95 VII.A.2.a 259B33.2b Γp−90 hi 2.675 151827108 s−1 T−1 1.010−6 NPL-79 VII.A.2.b 262B34.1b RK 25 812.808 3162 2.410−8 NIST-97 VII.B.1 265B34.2b RK 25 812.807111 4.410−8 NMI-97 VII.B.2 267B34.3b RK 25 812.809214 5.410−8 NPL-88 VII.B.3 269B34.4b RK 25 812.808434 1.310−7 NIM-95 VII.B.4 271B34.5b RK 25 812.808114 5.310−8 LNE-01 VII.B.5 273B35.1b KJ 483 597.9113 GHz V−1 2.710−7 NMI-89 VII.C.1 276B35.2b KJ 483 597.9615 GHz V−1 3.110−7 PTB-91 VII.C.2 278B36.1c KJ

2RK 6.036 7625121033 J−1 s−1 2.010−7 NPL-90 VII.D.1 281

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standardItem uncertaintya

number Input datum Value ur Identification Sec. and Eq.

B36.2c KJ2RK

−16.036 761 85531033 J−1 s 8.710−8 NIST-98 VII.D.2.a 283B36.3c KJ

2RK−16.036 761 85221033 J−1 s 3.610−8 NIST-07 VII.D.2.b 287

B37b F90 96 485.3913 C mol−1 1.310−6 NIST-80 VII.E.1 295B38.1c d220W4.2a 192 015.56312 fm 6.210−8 PTB-81 VIII.A.1.a 297B38.2c d220W4.2a 192 015.571533 fm 1.710−8 INRIM-07 VIII.A.1.c 299B39c d220NR3 192 015.591976 fm 4.010−8 NMIJ-04 VIII.A.1.b 298B40c d220MO* 192 015.549851 fm 2.610−8 INRIM-07 VIII.A.1.c 300B41 1−d220N /d220W17 72210−9 NIST-97 VIII.A.2.a 301B42 1−d220W17 /d220ILL −82210−9 NIST-99 VIII.A.2.a 302B43 1−d220MO* /d220ILL 862710−9 NIST-99 VIII.A.2.a 303B44 1−d220NR3 /d220ILL 342210−9 NIST-99 VIII.A.2.a 304B45 d220NR3 /d220W04−1 −112110−9 NIST-06 VIII.A.2.a 305B46 d220NR4 /d220W04−1 252110−9 NIST-06 VIII.A.2.a 306B47 d220W17 /d220W04−1 112110−9 NIST-06 VIII.A.2.a 307B48 d220W4.2a /d220W04−1 −12110−9 PTB-98 VIII.A.2.b 308B49 d220W17 /d220W04−1 222210−9 PTB-98 VIII.A.2.b 309B50 d220MO* /d220W04−1 −1032810−9 PTB-98 VIII.A.2.b 310B51 d220NR3 /d220W04−1 −232110−9 PTB-98 VIII.A.2.b 311B52 d220/d220W04−1 101110−9 PTB-03 VIII.A.2.b 312B53c VmSi 12.058 82543410−6 m3 mol−1 2.810−7 N/P/I-05 VIII.B 317B54 λmeas/d220ILL −10.002 904 302 4650 m s 1.710−7 NIST-99 VIII.C 319B55c h /mnd220W04 −12060.267 00484 m s 4.110−8 PTB-99 VIII.D.1 322B56b h /m133Cs −13.002 369 4324610−9 m2 s 1.510−8 StanfU-02 VIII.D.2 329B57 h /m87Rb −14.591 359 2876110−9 m2 s 1.310−8 LKB-06 VIII.D.3 332B58.1 R 8.314 47115 J mol−1 K−1 1.810−6 NIST-88 IX.A.1 338B58.2 R 8.314 50470 J mol−1 K−1 8.410−6 NPL-79 IX.A.2 339B59 λCuK1 /d220W4.2a 0.802 327 1124 3.010−7 FSU/PTB-91 XI.A 348B60 λWK1 /d220N 0.108 852 17598 9.010−7 NIST-79 XI.A 349B61 λMoK1 /d220N 0.369 406 0419 5.310−7 NIST-73 XI.A 350B62 λCuK1 /d220N 0.802 328 0477 9.610−7 NIST-73 XI.A 351

aThe values in brackets are relative to the quantities ae, a, g 12 5+ 16 7+e− C , ge− O , or Mu as appropriate.

bDatum not included in the final least-squares adjustment that provides the recommended values of the constants.cDatum included in the final least-squares adjustment with an expanded uncertainty.

through combined values of obtained from similar ex-periments. Most of the values of are in reasonableagreement, implying that most data from which they areobtained are reasonably consistent. There are, however,two important exceptions.

The value of inferred from the PTB measurement ofh /mnd220W04, item B55, is based on the mean valued220 of d220W04 implied by the four direct 220 XROIlattice spacing measurements, items B38.1–B40. Itdisagrees by about 2.8udiff with the value of with thesmallest uncertainty, that inferred from the HarvardUniversity measurement of ae. Also, the value of in-

ferred from the NIST measurement of Γp−9 0lodisagrees with the latter by about 2.3udiff. But it isalso worth noting that the value −1=137.036 0000382.810−8 implied by h /mnd220W04 together withitem B39 alone, the NMIJ XROI measurement ofd220NR3, agrees well with the Harvard ae value of .If instead one uses the three other direct XROI lat-tice spacing measurements, items B38.1, B38.2, andB40, which agree among themselves, one finds −1

=137.036 009228 2.110−8. This value disagrees with from the Harvard ae by 3.3udiff.

TABLE XXX. Continued.

Relative

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TABLE XXXI. Non-negligible correlation coefficients rxi ,xj of the input data in Table XXX. Forsimplicity, the two items of data to which a particular correlation coefficient corresponds are identi-fied by their item numbers in Table XXX.

rB1,B2.1=0.073 rB38.1,B38.2=0.191 rB42,B46=0.065 rB46,B47=0.509

rB2.2,B5=0.127 rB38.2,B40=0.057 rB42,B47=−0.367 rB48,B49=0.469

rB2.2,B6=0.089 rB41,B42=−0.288 rB43,B44=0.421 rB48,B50=0.372

rB5,B6=0.181 rB41,B43=0.096 rB43,B45=0.053 rB48,B51=0.502

rB14,B15=0.919 rB41,B44=0.117 rB43,B46=0.053 rB48,B55=0.258

rB16,B17=0.082 rB41,B45=0.066 rB43,B47=0.053 rB49,B50=0.347

rB28.1,B29=0.227 rB41,B46=0.066 rB44,B45=−0.367 rB49,B51=0.469

rB28.2,B30=0.195 rB41,B47=0.504 rB44,B46=0.065 rB49,B55=0.241

rB31.2,B33.1=−0.014 rB42,B43=0.421 rB44,B47=0.065 rB50,B51=0.372

rB35.1,B58.1=0.068 rB42,B44=0.516 rB45,B46=0.509 rB50,B55=0.192

rB36.2,B36.3=0.140 rB42,B45=0.065 rB45,B47=0.509 rB51,B55=0.258

The values of compared in Fig. 5 follow from Table −1 69 5.0 −9 h/m = 137.035 999 35 10 , 362XXXIV and are, again in order of increasing uncer-tainty,

−1 ae = 137.035 999 68394 6.9 10−10 , 361 −1 R −8K = 137.036 003025 1.8 10 , 363

TABLE XXXII. Summary of principal input data for the determination of the relative atomic massof the electron from antiprotonic helium transitions. The numbers in parentheses n , l :n , l denotethe transition n , l n , l.→

Itemnumber Input datum Value

Relative standarduncertaintya ur Identification Sec.

C1 p 4He+32,31:31,30 0.0082 MHz 7.310−10 JINR-06 IV.B

C2 p 4He+35,33:34,32 0.01.0 MHz 1.310−9 JINR-06 IV.B

C3 p 4He+36,34:35,33 0.01.2 MHz 1.610−9 JINR-06 IV.B

C4 p 4He+39,35:38,34 0.01.1 MHz 1.810−9 JINR-06 IV.B

C5 p 4He+40,35:39,34 0.01.2 MHz 2.410−9 JINR-06 IV.B

C6 p 4He+32,31:31,30 0.01.3 MHz 2.910−9 JINR-06 IV.B

C7 p 4He+37,35:38,34 0.01.8 MHz 4.410−9 JINR-06 IV.B

C8 p 3He+32,31:31,30 0.0091 MHz 8.710−10 JINR-06 IV.B

C9 p 3He+34,32:33,31 0.01.1 MHz 1.410−9 JINR-06 IV.B

C10 p 3He+36,33:35,32 0.01.2 MHz 1.810−9 JINR-06 IV.B

C11 p 3He+38,34:37,33 0.01.1 MHz 2.310−9 JINR-06 IV.B

C12 p 3He+36,34:37,33 0.01.8 MHz 4.410−9 JINR-06 IV.B

C13 p 4He+32,31:31,30 1 132 609 20915 MHz 1.410−8 CERN-06 IV.B

C14 p 4He+35,33:34,32 804 633 059.08.2 MHz 1.010−8 CERN-06 IV.B

C15 p 4He+36,34:35,33 717 474 00410 MHz 1.410−8 CERN-06 IV.B

C16 p 4He+39,35:38,34 636 878 139.47.7 MHz 1.210−8 CERN-06 IV.B

C17 p 4He+40,35:39,34 501 948 751.64.4 MHz 8.810−9 CERN-06 IV.B

C18 p 4He+32,31:31,30 445 608 557.66.3 MHz 1.410−8 CERN-06 IV.B

C19 p 4He+37,35:38,34 412 885 132.23.9 MHz 9.410−9 CERN-06 IV.B

C20 p 3He+32,31:31,30 1 043 128 60813 MHz 1.310−8 CERN-06 IV.B

C21 p 3He+34,32:33,31 822 809 19012 MHz 1.510−8 CERN-06 IV.B

C22 p 3He+36,33:34,32 646 180 43412 MHz 1.910−8 CERN-06 IV.B

C23 p 3He+38,34:37,33 505 222 295.78.2 MHz 1.610−8 CERN-06 IV.B

C24 p 3He+36,34:37,33 414 147 507.84.0 MHz 9.710−9 CERN-06 IV.B

aThe values in brackets are relative to the corresponding transition frequency.

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TABLE XXXIII. Non-negligible correlation coefficients rxi ,xj of the input data in Table XXXII.For simplicity, the two items of data to which a particular correlation coefficient corresponds areidentified by their item numbers in Table XXXII.

rC1,C2=0.929 rC9,C10=0.925 rC14,C23=0.132 rC17,C24=0.287rC1,C3=0.912 rC9,C11=0.949 rC14,C24=0.271 rC18,C19=0.235rC1,C4=0.936 rC9,C12=0.978 rC15,C16=0.223 rC18,C20=0.107rC1,C5=0.883 rC10,C11=0.907 rC15,C17=0.198 rC18,C21=0.118rC1,C6=0.758 rC10,C12=0.934 rC15,C18=0.140 rC18,C22=0.122rC1,C7=0.957 rC11,C12=0.959 rC15,C19=0.223 rC18,C23=0.112rC2,C3=0.900 rC13,C14=0.210 rC15,C20=0.128 rC18,C24=0.229rC2,C4=0.924 rC13,C15=0.167 rC15,C21=0.142 rC19,C20=0.170rC2,C5=0.872 rC13,C16=0.224 rC15,C22=0.141 rC19,C21=0.188rC2,C6=0.748 rC13,C17=0.197 rC15,C23=0.106 rC19,C22=0.191rC2,C7=0.945 rC13,C18=0.138 rC15,C24=0.217 rC19,C23=0.158rC3,C4=0.907 rC13,C19=0.222 rC16,C17=0.268 rC19,C24=0.324rC3,C5=0.856 rC13,C20=0.129 rC16,C18=0.193 rC20,C21=0.109rC3,C6=0.734 rC13,C21=0.142 rC16,C19=0.302 rC20,C22=0.108rC3,C7=0.927 rC13,C22=0.141 rC16,C20=0.172 rC20,C23=0.081rC4,C5=0.878 rC13,C23=0.106 rC16,C21=0.190 rC20,C24=0.166rC4,C6=0.753 rC13,C24=0.216 rC16,C22=0.189 rC21,C22=0.120rC4,C7=0.952 rC14,C15=0.209 rC16,C23=0.144 rC21,C23=0.090rC5,C6=0.711 rC14,C16=0.280 rC16,C24=0.294 rC21,C24=0.184rC5,C7=0.898 rC14,C17=0.247 rC17,C18=0.210 rC22,C23=0.091rC6,C7=0.770 rC14,C18=0.174 rC17,C19=0.295 rC22,C24=0.186rC8,C9=0.978 rC14,C19=0.278 rC17,C20=0.152 rC23,C24=0.154rC8,C10=0.934 rC14,C20=0.161 rC17,C21=0.167rC8,C11=0.959 rC14,C21=0.178 rC17,C22=0.169rC8,C12=0.988 rC14,C22=0.177 rC17,C23=0.141

−1 h/mnd220 = 137.036 007728 2.1 10−8 ,

364

−1 Γp,h−9 0lo = 137.035 987543 3.1 10−8 ,

365

−1 Mu = 137.036 001780 5.8 10−8 . 366

Here −1ae is the weighted mean of the two ae valuesof ; −1h /m is the weighted mean of the h /m87Rband h /m133Cs values; −1RK is the weighted mean ofthe five quantum Hall effect–calculable capacitor values;−1h /mnd220 is the value given in Table XXXIV and isbased on the measurement of h /mnd220W04 and thevalue of d220W04 inferred from the four XROI deter-minations of the 220 lattice spacing of three differentsilicon crystals; −1Γp,h−9 0lo is the weighted mean ofthe two values of −1Γp−9 0lo and one value of−1Γh−9 −1

0lo; and Mu is the value given in TableXXXIV and is based on the 1982 and 1999 measure-ments at LAMPF on muonium.

Figures 3–5 show that even if all data of Table XXXwere retained, the 2006 recommended value of wouldbe determined to a great extent by ae, and in particular,the Harvard University determination of ae.

The consistency of a significant fraction of the data ofTable XXX is indicated in Table XXXV and Figs. 6 and7, which give and graphically compare the values of hinferred from those data. Figure 6 compares the data byshowing each inferred value of h in the table, while Fig.7 compares the data through combined values of h fromsimilar experiments. The values of h are in good agree-ment, implying that the data from which they are ob-tained are consistent, with one important exception. Thevalue of h inferred from VmSi, item B53, disagrees by2.9udiff with the value of h from the weighted mean ofthe three watt-balance values of K2

JRK uncertainty ur=3.410−8—see Eq. 360.

In this regard, it is worth noting that a value of d220 ofan ideal silicon crystal is required to obtain a value of hfrom VmSi see Eq. 316, and the value used to obtainthe inferred value of h given in Eq. 318 and TableXXXV is based on all four XROI lattice spacing mea-surements, items B38.1–B40, plus the indirect valuefrom h /mnd220W04 see Table XXIV and Fig. 1. How-ever, the NMIJ measurement of d220NR3, item B39,and the indirect value of d220 from h /mnd220W04 yieldvalues of h from VmSi that are less consistent with thewatt-balance mean value than the three other directXROI lattice spacing measurements, items B38.1, B38.2,

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TABLE XXXIV. Comparison of the input data in Table XXX through inferred values of the fine-structure constant in order of increasing standard uncertainty.

Primary Item Relative standardsource number Identification Sec. and Eq. −1 uncertainty ur

ae B11.2 HarvU-06 V.A.3 105 137.035 999 71196 7.010−10

ae B11.1 UWash-87 V.A.3 104 137.035 998 8350 3.710−9

h /mRb B57 LKB-06 VIII.D.3 334 137.035 998 8391 6.710−9

h /mCs B56 StanfU-02 VIII.D.2 331 137.036 000011 7.710−9

h /mnd220W04 B55 PTB-99d220 B38.1–B40 Mean VIII.D.1 324 137.036 007728 2.110−8

RK B34.1 NIST-97 VII.B.1 266 137.036 003733 2.410−8

loΓp−90 B31.1 NIST-89 VII.A.1.a 254 137.035 987951 3.710−8

RK B34.2 NMI-97 VII.B.2 268 137.035 997361 4.410−8

RK B34.5 LNE-01 VII.B.5 274 137.036 002373 5.310−8

RK B34.3 NPL-88 VII.B.3 270 137.036 008373 5.410−8

Mu B28.1,B28.2 LAMPF VI.B.2.c 244 137.036 001780 5.810−8

loΓh−90 B32 KR/VN-98 VII.A.1.c 258 137.035 985282 6.010−8

RK B34.4 NIM-95 VII.B.4 272 137.036 00418 1.310−7

loΓp−90 B31.2 NIM-95 VII.A.1.b 256 137.036 00630 2.210−7

H,D A26−A47 Various IV.A.1.m 65 137.036 00248 3.510−7

R B13 BNL-02 V.B.2.a 132 137.035 6726 1.910−6

and B40, which agree among themselves a disagree-ment of about 3.8udiff compared to 2.5udiff. In contrast,the NMIJ measurement of d220NR3 yields a value of from h /mnd220W04 that is in excellent agreement withthe Harvard University value from ae, while the threeother lattice spacing measurements yield a value of in

(α−1 − 137.03) × 105

597 598 599 600 601 602 603 604

597 598 599 600 601 602 603 604

810− α

Γ h−90(lo) KR/VN-98

∆νMu LAMPF

RK NPL-88

RK LNE-01

RK NMI-97

Γp−90(lo) NIST-89

RK NIST-97

h/mn

ae Harvard-06

CODATA-02

d220 PTB-mean

h/m(Cs) Stanford-02

h/m(Rb) LKB-06

ae U Washington-87

CODATA-06

FIG. 3. Values of the fine-structure constant with ur10−7

implied by the input data in Table XXX, in order of decreasinguncertainty from top to bottom, and the 2002 and 2006 CO-DATA recommended values of . See Table XXXIV. Here“mean” indicates the PTB-99 result for h /mnd220W04 usingthe value of d220W04 implied by the four XROI lattice-spacing measurements.

poor agreement with alpha from ae 3.3udiff.The values of h compared in Fig. 7 follow from Table

XXXV and are, again in order of increasing uncertainty,

hK2JRK = 6.626 068 8923 10−34 J s

3.4 10−8 , 367

hVmSi = 6.626 074519 10−34 J s

2.9 10−7 , 368

(α−1 − 137.03) × 105

599.8 600.0 600.2 600.4

599.8 600.0 600.2 600.4

810− α

h/m(Cs) Stanford-02

h/m(Rb) LKB-06

ae U Washington-87

ae Harvard-06

CODATA-02

CODATA-06

FIG. 4. Values of the fine-structure constant with ur10−8

implied by the input data in Table XXX, in order of decreasinguncertainty from top to bottom. See Table XXXIV.

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Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

(α−1 − 137.03) × 105

598 599 600 601 602

598 599 600 601 602

−810 α

∆νMu

Γ p,h−90(lo)

h/mnd220

RK

h/m

ae

CODATA-02

CODATA-06

FIG. 5. Values of the fine-structure constant −7 with ur10implied by the input data in Table XXX, taken as a weightedmean when more than one measurement of a given type isconsidered see Eqs. 361–366, in order of decreasing uncer-tainty from top to bottom.

hKJ = 6.626 067827 10−34 J s

4.1 10−7 , 369

hΓp−9 0hi = 6.626 072457 10−34 J s

8.6 10−7 , 370

hF = 6.626 065788 10−3490 J s

1.3 10−6 . 371

Here hK2JRK is the weighted mean of the three values

of h from the three watt-balance measurements ofK2

JRK; hVmSi is the value given in Table XXXV andbased on all four XROI d220 lattice spacing measure-ments plus the indirect lattice spacing value fromh /mnd220W04; hKJ is the weighted mean of the twodirect Josephson effect measurements of KJ; hΓp−90hi

Primary Item Relative standardsource number Identification Sec. and Eq. h / J s uncertainty ur

KJ2RK B36.3 NIST-07 VII.D.2.b 288 6.626 068 912410−34 3.610−8

KJ2RK B36.2 NIST-98 VII.D.2.a 284 6.626 068 915810−34 8.710−8

KJ2RK B36.1 NPL-90 VII.D.1 282 6.626 06821310−34 2.010−7

VmSi B53 N/P/I-05 VIII.B 318 6.626 07451910−34 2.910−7

KJ B35.1 NMI-89 VII.C.1 277 6.626 06843610−34 5.410−7

KJ B35.2 PTB-91 VII.C.2 279 6.626 06704210−34 6.310−7

hiΓp−90 B33.2 NPL-79 VII.A.2.b 263 6.626 07296710−34 1.010−6

F90 B37 NIST-80 VII.E.1 296 6.626 06578810−34 1.310−6

hiΓp−90 B33.1 NIM-95 VII.A.2.a 261 6.626 0711110−34 1.610−6

is the weighted mean of the two values of h from the twomeasurements of Γp−90hi; and hF90 is the value givenin Table XXXV and comes from the silver coulometermeasurement of F90. Figures 6 and 7 show that even ifall data of Table XXX were retained, the 2006 recom-mended value of h would be determined to a large ex-tent by K2

JRK, and, in particular, the NIST 2007 determi-nation of this quantity.

We conclude our data comparisons by listing in TableXXXVI the four available values of Are. The reason-able agreement of these values shows that the corre-sponding input data are consistent. The most importantof these data are the University of Washington value ofA e, , , f 12C5+ / f 12C5+, f 12 7+

r O 12 7+C O s c s / fc O , andthe antiprotonic helium data, items B9, B14–B17, andC1–C24.

In summary, the data comparisons here have identi-fied the following potential problems: i the measure-

TABLE XXXV. Comparison of the input data in Table XXX through inferred values of the Plancconstant h in order of increasing standard uncertainty.

k

[h/(10−34 J s) − 6.6260] × 105

5 6 7 8 9

5 6 7 8 9

10−6 h

Γ p−90(hi) NIM-95

F90 NIST-80

Γp−90(hi) NPL-79

KJ PTB-91

KJ NMI-89

Vm(Si) N/P/I-05

2KJRK NPL-90

2KJRK NIST-98

2KJRK NIST-07

CODATA-02

CODATA-06

FIG. 6. Values of the Planck constant h implied by the inputdata in Table XXX, in order of decreasing uncertainty fromtop to bottom. See Table XXXV.

10

10

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TABLE XXXVI. Values of Are implied by the input data in Table XXX in order of increasingstandard uncertainty.

Primary Item Relative standardsource number Identification Sec. and Eq. Are uncertainty ur

fsC / fcC B16 GSI-02 V.C.2.a 177 0.000 548 579 909 3229 5.210−10

fsO / fcO B17 GSI-02 V.C.2.b 181 0.000 548 579 909 5842 7.610−10

p He+ C1–C24 JINR/CERN-06 IV.B.3 74 0.000 548 579 908 8191 1.710−9

Are B9 UWash-95 III.C 5 0.000 548 579 911112 2.110−9

ment of VmSi, item B53, is inconsistent with the watt-balance measurements of K2

JRK, items B36.1–B36.3, andsomewhat inconsistent with the mercury-electrometerand voltage-balance measurements of KJ; ii the XROI220 lattice spacing values d220W4.2a, d220W4.2a,and d220MO*, items B38.1, B38.2, and B40, are incon-sistent with the value of d220NR3, item B39, and themeasurement of h /mnd220W04, item B55; iii theNIST-89 measurement of Γp−9 0lo, item B33.1, is incon-sistent with the most accurate data that also determinethe value of the fine-structure constant; iv although nota problem in the sense of i–iii, there are a number ofinput data with uncertainties so large that they are un-likely to make a contribution to the determination of the2006 CODATA recommended values.

Furthermore, we note that some inferred values of in Table XXXIV and most inferred values of h in TableXXXV depend on either one or both of the relationsKJ=2e /h and RK=h /e2. The question of whether relax-ing the assumption that these relations are exact wouldreduce or possibly even eliminate some of the observedinconsistencies, considered in Appendix F of CODATA-02, is addressed below. This study indeed confirms theJosephson and quantum Hall effect relations.

B. Multivariate analysis of data

The multivariate analysis of the data is based on thefact that measured quantities can be expressed as theo-retical functions of fundamental constants. These ex-pressions, or observational equations, are written interms of a particular independent subset of the constantswhose members are called adjusted constants. The goalof the analysis is to find the values of the adjusted con-stants that predict values for the measured data that bestagree with the data themselves in the least-squares sensesee Appendix E of CODATA-98.

The symbol is used to indicate that an observedvalue of an input datum of the particular type shown onthe left-hand side is ideally given by the function of theadjusted constants on the right-hand side; however, thetwo sides are not necessarily equal, because the equationis one of an overdetermined set relating the data to theadjusted constants. The best estimate of a quantity isgiven by its observational equation evaluated with theleast-squares estimated values of the adjusted constantson which it depends.

In essence, we follow the least-squares approach ofAitken 1934 see also Sheppard 1912, who treatedthe case in which the input data are correlated. The 150input data of Tables XXVIII, XXX, and XXXII are of135 distinct types and are expressed as functions of the79 adjusted constants listed in Tables XXXVII, XXXIX,and XLI. The observational equations that relate theinput data to the adjusted constants are given in TablesXXXVIII, XL, and XLII.

Note that the various binding energies E 2bX /muc in

Table XL, such as in the equation for item B1, aretreated as fixed quantities with negligible uncertainties.Similarly, the bound-state g-factor ratios in this table,such as in the equation for item B18, are treated in thesame way. Further, the frequency fp is not an adjustedconstant but is included in the equation for items B29and B30 to indicate that they are functions of fp. Finally,the observational equation for items B29 and B30, basedon Eqs. 215–217 of Sec. VI.B, includes the functionsae ,e and a , as well as the theoretical expres-sion for input data of Type B28, Mu. The latter expres-sion is discussed in Sec. VI.B.1 and is a function of R, ,me /m, a ,, and Mu.

1. Summary of adjustments

A number of adjustments were carried out to gaugethe compatibility of the input data in Tables XXVIII,XXX, and XXXII together with their covariances inTables XXIX, XXXI, and XXXIII and to assess theirinfluence on the values of the adjusted constants. Theresults of 11 of these are given in Tables XLIII and XLVand are discussed here. Because the adjusted value ofthe Rydberg constant R is essentially the same for allsix adjustments summarized in Table XLIII and equal tothat of adjustment 4 of Table XLV, the value of R is notlisted in Table XLIII. It should also be noted that adjust-ment 4 of all three tables is the same adjustment.

Adjustment 1. This initial adjustment includes all inputdata, four of which have normalized residuals ri withabsolute magnitudes significantly greater than 2; the val-ues of ri for these four data resulting from adjustments1–6 are given in Table XLIV. Consistent with the previ-ous discussion, the four most inconsistent items arethe molar volume of silicon VmSi, the quotienth /mnd220W04, the XROI measurement of the 220lattice spacing d220NR3, and the NIST-89 value of

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TABLE XXXVII. The 28 adjusted constants variables usedin the least-squares multivariate analysis of the Rydberg-constant data given in Table XXVIII. These adjusted constantsappear as arguments of the functions on the right-hand side ofthe observational equations of Table XXXVIII. The notationfor hydrogenic energy levels EXnLj and for additive correc-

Xtions XnLj have the same meaning as the notations E andnLjX in Sec. IV.A.1.l.nLj

Adjusted constant Symbol

Rydberg constantbound-state proton rms charge radiusbound-state deuteron rms charge radiusadditive correction to EH1S1/2 /h

additive correction to EH2S1/2 /h

additive correction to EH3S1/2 /h

additive correction to EH4S1/2 /h

additive correction to EH6S1/2 /h

additive correction to EH8S1/2 /h

additive correction to EH2P1/2 /h

additive correction to EH4P1/2 /h

additive correction to EH2P3/2 /h

additive correction to EH4P3/2 /h

additive correction to EH8D3/2 /h

additive correction to EH12D3/2 /h

additive correction to EH4D5/2 /h

additive correction to EH6D5/2 /h

additive correction to EH8D5/2 /h

additive correction to EH12D5/2 /h

additive correction to ED1S1/2 /h

additive correction to ED2S1/2 /h

additive correction to ED4S1/2 /h

additive correction to ED8S1/2 /h

additive correction to ED8D3/2 /h

additive correction to ED12D3/2 /h

additive correction to ED4D5/2 /h

additive correction to ED8D5/2 /h

additive correction to ED12D5/2 /h

R

Rp

Rd

H1S1/2H2S1/2H3S1/2H4S1/2H6S1/2H8S1/2H2P1/2H4P1/2H2P3/2H4P3/2H8D3/2H12D3/2H4D5/2H6D5/2H8D5/2H12D5/2D1S1/2D2S1/2D4S1/2D8S1/2D8D3/2D12D3/2D4D5/2D8D5/2D12D5/2

Γp−9 0lo. All other input data have values of ri consider-ably less than 2, except those for p3He32,31:31,30 andp3He36,33:34,32, items C20 and C22, for which r20

=2.09 and r22=2.06. However, the self-sensitivity coeffi-cients Sc for these input data are considerably less than0.01; hence, because their contribution to the adjustmentis small, their marginally large normalized residuals areof little concern. In this regard, we see from Table XLIVthat three of the four inconsistent data have values of Sc

considerably larger than 0.01; the exception is Γp−9 0lowith Sc=0.0099, which is rounded to 0.010 in the table.

Adjustment 2. Since the four direct lattice spacingmeasurements, items B38.1–B40, are credible, as is themeasurement of h /mnd220W04, item B55, after dueconsideration the CODATA Task Group on Fundamen-tal Constants decided that all five of these input data

should be considered for retention, but that each of theira priori assigned uncertainties should be weighted by themultiplicative factor 1.5 to reduce ri of h /mnd220W04and of d220NR3 to a more acceptable level, that is, toabout 2, while maintaining their relative weights. Thishas been done in adjustment 2. As can be seen fromTable XLIII, this increase of uncertainties has an incon-sequential impact on the value of , and no impact onthe value of h. It does reduce RB, as would be expected.

Adjustment 3. Again, since the measurement ofVmSi, item B53, as well as the three measurements ofK2

JRK, items B36.1–B36.3, and the two measurements ofKJ, items B35.1 and B35.2, are credible, the Task Groupdecided that all six should be considered for retention,but that each of their a priori assigned uncertaintiesshould be weighted by the multiplicative factor 1.5 toreduce ri of VmSi to about 2, while maintaining theirrelative weights. This has been done in adjustment 3.Note that this also reduces ri of h /mnd220W04 from2.03 in adjustment 2 to 1.89 in adjustment 3. We seefrom Table XLIII that this increase in uncertainty hasnegligible consequences for the value of , but it doesincrease the uncertainty of h by about the same factor,as would be expected. Also as would be expected, RB isfurther reduced.

It may be recalled that faced with a similar situation inthe 2002 adjustment, the Task Group decided to use amultiplicative weighting factor of 2.325 in order to re-duce ri of VmSi to 1.50. The reduced weighting factorof 1.5 in the 2006 adjustment recognizes the new valueof K2

JRK now available and the excellent agreement withthe two earlier values.

Adjustment 4. In adjustment 3, a number of inputdata, as measured by their self-sensitivity coefficients Sc,do not contribute in a significant way to the determina-tion of the adjusted constants. We therefore omit in ad-justment 4 those input data with Sc0.01 in adjustment3 unless they are a subset of the data of an experimentthat provides other input data with Sc0.01. The 14 in-put data deleted in adjustment 4 for this reason areB31.1–B35.2, B37, and B56, which are the five low- andhigh-field proton and helion gyromagnetic ratio results;the five calculable capacitor values of RK; both values ofKJ as obtained using a Hg electrometer and a voltagebalance; the Ag coulometer result for the Faraday con-stant; and the atom interferometry result for the quo-tient of the Planck constant and mass of the cesium-133atom. The respective values of Sc for these data in ad-justment 3 are in the range 0.0000–0.0099. Deleting suchmarginal data is in keeping with the practice followed inthe 1998 and 2002 adjustments; see Sec. I.D ofCODATA-98.

Because h /m133Cs, item B56, has been deleted as aninput datum due to its low weight, Ar

133Cs, item B8,which is not coupled to any other input datum, has alsobeen omitted as an input datum and as an adjusted con-stant from adjustment 4. This brings the total number ofomitted items to 15. Table XLIII shows that deleting

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TABLE XXXVIII. Observational equations that express the input data related to R in Table XXVIII as functions of the adjustedconstants in Table XXXVII. The numbers in the first column correspond to the numbers in the first column of Table XXVIII.Expressions for the energy levels of hydrogenic atoms are discussed in Sec. IV.A.1. As pointed out in Sec. IV.A.1.l, EXnLj /h isin fact proportional to cR and independent of h, hence h is not an adjusted constant in these equations. The notation forhydrogenic energy levels EXnLj and for additive corrections X XXnLj have the same meaning as the notations EnLj and nLj in Sec.IV.A.1.l. See Sec. XII.B for an explanation of the symbol .

Type of inputdatum Observational equation

A1–A16 HnLjHnLj

A17–A25 DnLjDnLj

A26–A31,A38,A39

Hn1L1j1−n2L2j2

EHn2L2j2;R , ,Are ,Arp ,Rp ,Hn2L2j2

−EHn1L1j1

;R , ,Are ,Arp ,Rp ,Hn1L1j1 /h

A32–A37 − 1Hn1L1j1− n2L2j2 4Hn3L3j3

− n4L4j4 EHn2L2j2

;R,,Are,Arp,Rp,Hn2L2j2

− EHn1L1j1;R,,Are,Arp,Rp,Hn1L1j1

− 1 ;R,,Are,Ar 4 EHn4L4j4p,Rp,Hn4L4j4

− EHn3L3j3;R,,Are,Arp,Rp,Hn3L3j3

/h

A40–A44 Dn1L1j1− n2L2j2

EDn2L2j2;R,,Are,Ard,Rd,Dn2L2j2

− EDn1L1j1;R,,Are,Ard,Rd,Dn1L1j1

/h

A45,A46 − 1Dn1L1j1− n2L2j2 4Dn3L3j3

− n4L4j4 EDn2L2j2

;R,,Are,Ard,Rd,Dn2L2j2

− EDn1L1j1;R,,Are,Ard,Rd,Dn1L1j1

− 1 ;R,,Are,Ar 4 EDn4L4j4d,Rd,Dn4L4j4

− EDn3L3j3;R,,Are,Ard,Rd,Dn3L3j3

/h

A47 D1S1/2 − 2S1/2 − H1S1/2 − 2S1/2 ED2S1/2;R,,Are,Ard,Rd,D2S1/2

− ED1S1/2;R,,Are,Ard,Rd,D1S1/2

− EH2S1/2;R,,Are,Arp,Rp,H2S1/2

− EH1S1/2;R,,Are,Arp,Rp,H1S1/2/h

A48 RpRp

A49 RdRd

these 15 data has virtually no impact on the values of and h.

Adjustment 4 is the adjustment on which the 2006CODATA recommended values are based, and as suchit is referred to as the “final adjustment.”

Adjustments 5 and 6. These adjustments are intendedto check the robustness of adjustment 4, the final adjust-ment, while adjustments 7–11, which are summarized inTable XLV, probe various aspects of the R data inTable XXVIII.

Adjustment 5 only differs from adjustment 3 in that itdoes not include the input data that lead to the fourmost accurate values of : the two measurements of ae,

items B11.1 and B11.2, the measurement of h /m133Cs,item B56, and the measurement of h /m87Rb, item B57.The ur of the inferred values of from these data are7.010−10, 3.710−9, 7.710−9, and 6.710−9. We seefrom Table XLIII that the value of from adjustment 5is consistent with the 2006 recommended value from ad-justment 4 the difference is 0.8udiff, but its uncertaintyis about 20 times larger. Moreover, the resulting value ofh is the same as the recommended value.

Adjustment 6 only differs from adjustment 3 in that itdoes not include the input data that yield the three mostaccurate values of h, namely, the watt-balance measure-

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TABLE XXXIX. The 39 adjusted constants variables used inthe least-squares multivariate analysis of the input data inTable XXX. These adjusted constants appear as arguments ofthe functions on the right-hand side of the observational equa-tions of Table XL.

Adjusted constant Symbol

electron relative atomic mass

proton relative atomic mass

neutron relative atomic mass

deuteron relative atomic mass

triton relative atomic mass

helion relative atomic mass

particle relative atomic mass16O7+ relative atomic mass87Rb relative atomic mass133Cs relative atomic mass

fine-structure constant

additive correction to aeth

additive correction to ath

additive correction to gCth

additive correction to gOth

electron-proton magnetic moment ratio

deuteron-electron magnetic moment ratio

triton-proton magnetic moment ratio

shielding difference of d and p in HD

shielding difference of t and p in HT

electron to shielded protonmagnetic moment ratio

shielded helion to shielded protonmagnetic moment ratio

neutron to shielded protonmagnetic moment ratio

electron-muon mass ratio

additive correction to Muth

Planck constant

molar gas constant

copper K1 x unit

molybdenum K1 x unit

ångström star

d220 of Si crystal ILL

d220 of Si crystal N

d220 of Si crystal WASO 17

d220 of Si crystal WASO 04

d220 of Si crystal WASO 4.2a

d220 of Si crystal MO*

d220 of Si crystal NR3

d220 of Si crystal NR4

d220 of an ideal Si crystal

Are

Arp

Arn

Ard

Art

Arh

Ar

16O7+Ar

87RbAr

133CsAr

e

C

O

e− /p

d /e−

t /p

dp

tp

e− /p

h /p

n /p

me /m

Mu

h

R

xuCuK1

xuMoK1

Å*

d220ILL

d220N

d220W17

d220W04

d220W4.2a

d220MO*

d220NR3

d220NR4

d220

TABLE XL. Observational equations that express the inputdata in Table XXX as functions of the adjusted constants inTable XXXIX. The numbers in the first column correspond tothe numbers in the first column of Table XXX. For simplicity,the lengthier functions are not explicitly given. See Sec. XII.Bfor an explanation of the symbol .

Type ofinputdatum Observational equation Sec.

B1 Ar1HArp+Are−Eb1H /muc2 III.B

B2 Ar2HArd+Are−Eb2H /muc2 III.B

B3 Ar3HArt+Are−Eb3H /muc2 III.B

B4 Ar3HeArh+2Are−Eb3He /muc2 III.B

B5 Ar4HeAr+2Are−Eb4He /muc2 III.B

B6 Ar16O Ar

16O7+ + 7Are − Eb16O

− Eb16O7+/muc2

V.C.2.b

B7 Ar87RbAr

87Rb

B8 Ar133CsAr

133Cs

B9 AreAre

B10 ee

B11 aeae ,e V.A.1

B12

B13 R−a ,

1+ae ,eme

m

e−

pV.B.2

B14 CC

B15 OO

B16fs

12C5+fc

12C5+ −

gC,C10Are 12 − 5Are

+Eb12C − Eb12C5+

muc2 V.C.2.a

B17fs

16O7+fc

16O7+−

gO ,O14Are

Ar16O7+ V.C.2.b

B18e−H

pH

ge−H

ge−gpH

gp−1e−

pVI.A.2.a

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TABLE XL. Continued.

Type ofinputdatum Observational equation Sec.

B19dDe−D

gdD

gdge−D

ge−−1

d

e−V.A.2.b

B20pHDdHD

1 + dpe−

p−1 d

e−−1

VI.A.2.c

B21 dpdp

B22tHTpHT

1−tpt

p

VI.A.2.c

B23 tptp

B24 e−H

p

ge−H

ge−

e−

pVI.A.2.d

B25 h

p

h

p

B26 n

p

n

p

B27 MuMu

B28 Mu MuR,,me

m

,,Mu VI.B.1

B29,B30 fp fp;R,,me

m

,e−

p,e,,Mu VI.B

B31 Γp−90 lo −KJ−90RK−901 + ae,e3

20R

e−

p−1

VII.A.1

B32 Γh−90 lo KJ−90RK−901 + ae,e3

20R

e−

p−1

h

p

VII.A.1

B33 Γp−90 hi −c1 + ae,e2

KJ−90RK−90Rhe−

p−1

VII.A.2

B34 RK0c

2VII.B

TABLE XL. Continued.

Type ofinputdatum Observational equation

1/28B35 KJ

0ch

4B36 KJ

2RKh

cMuAre2

B37 F90KJ−90RK−90Rh

B38–B40 d220Xd220X

d220X d220XB41–B52 −1 −1

d220Y d220Y

32cMuAre2d220B53 VmSi

Rh

Sec.

VII.C

VII.D

VII.E

VIII.B

λmeasB54d220ILL

2Are Arn+Arp

Rd220ILL Arn+Arp2−A2dr

h Are c2B55

mnd220W04 Arn 2Rd220W04

VIII.C

VIII.D.1

h Are c2B56,B57

mX ArX 2R

B58 RR

λCuK1 1537.400 xuCuK1B59,B62

d220X d220X

λWK1 0.209 010 0 Å*B60

d220N d220N

λMoK1 707.831 xuMoK1B61

d220N d220N

VIII.D

XI.A

XI.A

XI.A

ments of K2JRK, items B36.1–B36.3. The ur of the in-

ferred values of h from these data, as they are used inadjustment 3 that is, after their uncertainties are multi-plied by the weighting factor 1.5, are 5.410−8, 1.310−7, and 3.010−7. From Table XLIII, we see that thevalue of h from adjustment 6 is consistent with the 2006recommended value from adjustment 4 the difference is1.4udiff, but its uncertainty is well over six times larger.Furthermore, the resulting value of is the same as therecommended value. Therefore, adjustments 5 and 6suggest that the less accurate input data are consistentwith the more accurate data, thereby providing a consis-tency check on the 2006 recommended values of theconstants.

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TABLE XLI. The 12 adjusted constants variables relevant tothe antiprotonic helium data given in Table XXXII. These ad-justed constants appear as arguments of the theoretical expres-sions on the right-hand side of the observational equations ofTable XLII.

Transition Adjusted constant

p

p

p

p

p

p

p

p

p

p

p

p

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

4He+: 32,31→ 31,304He+: 35,33→ 34,324He+: 36,34→ 35,334He+: 37,34→ 36,334He+: 39,35→ 38,344He+: 40,35→ 39,344He+: 37,35→ 38,343He+: 32,31→ 31,303He+: 34,32→ 33,313He+: 36,33→ 35,323He+: 38,34→ 37,333He+: 36,34→ 37,33

p

p

p

p

p

p

p

p

p

p

p

p

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

4He+32,31:31,304He+35,33:34,324He+36,34:35,334He+37,34:36,334He+39,35:38,344He+40,35:39,344He+37,35:38,343He+32,31:31,303He+34,32:33,313He+36,33:35,323He+38,34:37,333He+36,34:37,33

Adjustments 7–11. These adjustments differ from ad-justment 4, the final adjustment, in the following ways.In adjustment 7, the scattering-data input values forboth Rp and Rd, items A48 and A49, are omitted; inadjustment 8, only Rp is omitted, and in adjustment 9,only Rd is omitted; adjustment 10 includes only the hy-drogen data, and adjustment 11 includes only the deute-rium data, but for both, the H-D isotope shift, item A47,is omitted. Although a somewhat improved value of the1S1/2–2S1/2 hydrogen transition frequency and improve-ments in the theory of H and D energy levels have be-

[h/(10−34 J s) − 6.6260] × 105

5 6 7 8 9

5 6 7 8 9

−610 h

F90 NIST-80

Γ p−90(hi)

KJ

Vm(Si)

2KJRK

CODATA-02

CODATA-06

FIG. 7. Values of the Planck constant h implied by the inputdata in Table XXX, taken as a weighted mean when more thanone measurement of a given type is considered see Eqs.367–371, in order of decreasing uncertainty from top tobottom.

come available since the completion of the 2002 adjust-ment, the value of R, which is determined almostentirely by these data, has changed very little. The val-ues of Rp and Rd, which are also determined mainly bythese data, have changed by less than one-third of theiruncertainties. The experimental and theoretical H andD data remain highly consistent.

2. Test of the Josephson and quantum Hall effect relations

Investigation of the exactness of the relations KJ=2e /h and RK=h /e2 is carried out, as in CODATA-02,by writing

2e 8 1/2KJ = 1 +

h J = 0ch

1 + J , 372

h cRK = 1 + 0

= 1 + , 373e2 K 2 K

where J and K are unknown correction factors takento be additional adjusted constants determined by least-squares calculations. Replacing KJ=2e /h and RK=h /e2

with the generalizations in Eqs. 372 and 373 in theanalysis leading to the observational equations in TableXL leads to the modified observational equations givenin Table XLVI.

The results of seven different adjustments are pre-sented in Table XLVII. In addition to the adjusted val-ues of , h, J, and K, we also give the normalized re-siduals ri of the four input data with the largest valuesof ri: VmSi, item B53, h /mnd220W04, item B55,d220NR3, item B39, and the NIST-89 value forΓp−9 0lo, item B31.1. The residuals are included as addi-tional indicators of whether relaxing the assumption KJ=2e /h and RK=h /e2 reduces the disagreements amongthe data.

The adjusted value of R is not included in TableXLVII, because it remains essentially unchanged fromone adjustment to the next and equal to the 2006 recom-mended value. An entry of 0 in the K column meansthat it is assumed that RK=h /e2 in the correspondingadjustment; similarly, an entry of 0 in the J columnmeans that it is assumed that KJ=2e /h in the corre-sponding adjustment. The following comments apply tothe adjustments of Table XLVII.

Adjustment i is identical to adjustment 1 of TablesXLIII and XLIV in the previous section and is includedhere simply for reference; all input data are includedand multiplicative weighting factors have not been ap-plied to any uncertainties. For this adjustment, N=150,M=79, =71, and 2=92.1.

The next three adjustments differ from adjustment iin that in adjustment ii the relation KJ=2e /h is relaxed,in adjustment iii the relation RK=h /e2 is relaxed, andin adjustment iv both relations are relaxed. For thesethree adjustments, N=150, M=80, =70, and 2=91.5;N=150, M=80, =70, and 2=91.3; and N=150, M=81,=69, and 2=90.4, respectively.

10

10

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TABLE XLII. Observational equations that express the input data related to antiprotonic helium inTable XXXII as functions of adjusted constants in Tables XXXIX and XLI. The numbers in the firstcolumn correspond to the numbers in the first column of Table XXXII. Definitions of the symbolsand values of the parameters in these equations are given in Sec. IV.B. See Sec. XII.B for anexplanation of the symbol .

Type of input datum Observational equation

C1–C7 p 4He+n , l :n , lp 4He+n , l :n , l

C8–C12 p 3He+n , l :n , lp 3He+n , l :n , l

00 Are Arp

n,l:n,l + ap − 1C13–C19 p 4He+n,l:n,l p 4He+ ¯ 4He+n,l:n,l Arp Are0Are Ar

+ bp 4He+n,l:n,l − 1 + p 4He+n,l:n,lAr Are

00 Are Arp

p n,l:n,l + ap − 1C20–C24 3He+n,l:n,l p 3He+ 3He+n,l:n,lArp Are

0Are Arh+ bp 3He+n,l:n,l − 1 + p 3He+n,l:n,l

Arh Are

It is clear from Table XLVII that there is no evidencefor the inexactness of either KJ=2e /h or RK=h /e2. Thisconclusion is also true if instead of taking adjustment 1of Table XLIII as our starting point, we had taken ad-justment 2 in which the uncertainties of the five x-rayrelated data are multiplied by the factor 1.5. That is,none of the numbers in Table XLVII would change sig-nificantly, except RB would be reduced from 1.14 toabout 1.08. The reason adjustments iii–vii summa-rized in Table XLVII give values of K consistent withzero within about 2 parts in 108 is mainly because thevalue of inferred from the mean of the five measuredvalues of RK under the assumption RK=h /e2, which hasur=1.810−8, agrees with the value of with ur=7.010−10 inferred from the Harvard University measuredvalue of ae.

Table XLVI and the uncertainties of the 2006 inputdata indicate that the values of J from adjustments iiand iv are determined mainly by the input data forΓp−9 0lo and Γh−9 0lo with observational equations thatdepend on J but not on h; and by the input data forΓp−9 0hi, KJ, K2

JRK, and F90, with observational equa-tions that depend on both J and h. Because the value ofh in these least-squares calculations arises primarilyfrom the measured value of the molar volume of siliconVmSi, the values of J in adjustments ii and iv arisemainly from a combination of individual values of J thateither depend on VmSi or on Γp−9 0lo and Γh−9 0lo. Itis therefore of interest to repeat adjustment iv, firstwith VmSi deleted but with the Γp−9 0lo and Γh−9 0lodata included, and then with the latter deleted but with

TABLE XLIII. Summary of some of the least-squares adjustments used to analyze all input data given in Tables XXVIII–XXXI.The values of and h are those obtained in the adjustment, N is the number of input data, M is the number of adjusted constants,=N−M is the degrees of freedom, and RB=2 / is the Birge ratio. See the text for an explanation and discussion of eachadjustment, but, in brief, 1 is all data; 2 is 1 with the uncertainties of the key x-ray and silicon data multiplied by 1.5; 3 is 2 withthe uncertainties of the key electrical data also multiplied by 1.5; 4 is the final adjustment from which the 2006 recommendedvalues are obtained and is 3 with the input data with low weights deleted; 5 is 3 with the four data that provide the most accuratevalues of deleted; and 6 is 3 with the three data that provide the most accurate values of h deleted.

Adj. N M 2 RB −1 ur−1 h / J s urh

1 150 79 71 92.1 1.14 137.035 999 68793 6.810−10 6.626 068 962210−34 3.410−8

2 150 79 71 82.0 1.07 137.035 999 68293 6.810−10 6.626 068 962210−34 3.410−8

3 150 79 71 77.5 1.04 137.035 999 68193 6.810−10 6.626 068 963310−34 5.010−8

4 135 78 57 65.0 1.07 137.035 999 67994 6.810−10 6.626 068 963310−34 5.010−8

5 144 77 67 72.9 1.04 137.036 001219 1.410−8 6.626 068 963310−34 5.010−8

6 147 79 68 75.4 1.05 137.035 999 68093 6.810−10 6.626 07192110−34 3.210−7

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TABLE XLIV. Normalized residuals ri and self-sensitivity coefficients Sc that result from the six least-squares adjustments sum-marized in Table XLIII for the four input data whose absolute values of ri in Adj. 1 exceed 1.50. Sc is a measure of how theleast-squares estimated value of a given type of input datum depends on a particular measured or calculated value of that type ofdatum; see Appendix E of CODATA-98. See the text for an explanation and discussion of each adjustment; brief explanations aregiven at the end of the caption to the previous table.

Adj. 1 Adj. 2 Adj. 3 Adj. 4 Adj. 5 Adj. 6Item Inputnumber

B53

quantity

VmSi

Identification

N/P/I-05

ri

−2.82

Sc

0.065

ri

−2.68

Sc

0.085

ri

−1.86

Sc

0.046

ri Sc

−1.86 0.047

ri

−1.79

Sc

0.053

ri

−0.86

Sc

0.556B55 h /mnd220W04 PTB-99 −2.71 0.155 −2.03 0.118 −1.89 0.121 −1.89 0.121 −1.57 0.288 −1.82 0.123B39 d220NR3 NMIJ-04 2.37 0.199 1.86 0.145 1.74 0.148 1.74 0.148 1.78 0.151 −1.00 0.353B31.1 loΓp−90 NIST-89 2.31 0.010 2.30 0.010 2.30 0.010 Deleted 2.60 0.143 2.30 0.010

VmSi included. These are, in fact, adjustments v andvi of Table XLVII.

In each of these adjustments, the absolute values of Jare comparable and significantly larger than the uncer-tainties, which are also comparable, but the values havedifferent signs. Consequently, when VmSi and theΓp−9 0lo and Γh−9 0lo data are included at the same timeas in adjustment iv, the result for J is consistent withzero.

The values of J from adjustments v and vi reflectsome of the inconsistencies among the data: the dis-agreement of the values of h implied by VmSi andK2

JRK when it is assumed that KJ=2e /h and RK=h /e2

are exact; and the disagreement of the values of im-plied by the electron magnetic moment anomaly ae andby Γp−9 0lo and Γh−9 0lo under the same assumption.

In adjustment vii, the problematic input data forVmSi, Γp−9 0lo, and Γh−9 0lo are simultaneously de-leted from the calculation. Then the value of J arisesmainly from the input data for Γp−9 0hi, KJ, K2

JRK, andF90. Like adjustment iv, adjustment vii shows that J

is consistent with zero, although not within 8 parts in 108

but within 7 parts in 107. However, adjustment vii hasthe advantage of being based on consistent data.

The comparatively narrow range of values of inTable XLVII is due to the fact that the input data that

mainly determine do not depend on the Josephson orquantum Hall effects. This is not the case for the inputdata that primarily determine h, hence the values of hvary over a wide range.

XIII. THE 2006 CODATA RECOMMENDED VALUES

A. Calculational details

As indicated in Sec. XII.B, the 2006 recommendedvalues of the constants are based on adjustment 4 ofTables XLIII–XLV. This adjustment is obtained by ideleting 15 items from the originally considered 150items of input data of Tables XXVIII, XXX, andXXXII, namely, items B8, B31.1–B35.2, B37, and B56,because of their low weight self-sensitivity coefficientSc0.01; and ii weighting the uncertainties of the nineinput data B36.1–B36.3, B38.1–B40, B53, and B55 bythe multiplicative factor 1.5 in order to reduce the abso-lute values of their normalized residuals ri to less than2. The correlation coefficients of the data, as given inTables XXIX, XXXI, and XXXIII, are also taken intoaccount. The 135 final input data are expressed in termsof the 78 adjusted constants of Tables XXXVII, XXXIX,and XLI, corresponding to N−M==57 degrees of free-dom. Because h /m133Cs, item B56, has been deleted as

TABLE XLV. Summary of the results of some of the least-squares adjustments used to analyze the input data related to R. Thevalues of R, Rp, and Rd are those obtained in the indicated adjustment, N is the number of input data, M is the number ofadjusted constants, =N−M is the degrees of freedom, and RB=2 / is the Birge ratio. See the text for an explanation anddiscussion of each adjustment, but, in brief, 4 is the final adjustment; 7 is 4 with the input data for Rp and Rd deleted; 8 is 4 withjust the Rp datum deleted; 9 is 4 with just the Rd datum deleted; 10 is 4 but with only the hydrogen data included; and 11 is 4 butwith only the deuterium data included.

Adj. N M 2 RB R /m−1 urR Rp/fm Rd/fm

4 135 78 57 65.0 1.07 10 973 731.568 52773 6.610−12 0.876869 2.1402287 133 78 55 63.0 1.07 10 973 731.568 51882 7.510−12 0.876078 2.1398328 134 78 56 63.8 1.07 10 973 731.568 49578 7.110−12 0.873775 2.1389309 134 78 56 63.9 1.07 10 973 731.568 54976 6.910−12 0.879071 2.14112910 117 68 49 60.8 1.11 10 973 731.568 56285 7.810−12 0.88028011 102 61 41 54.7 1.16 10 973 731.568 3913 1.110−11 2.128693

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TABLE XLVI. Generalized observational equations that ex-press input data B31–B37 in Table XXX as functions of theadjusted constants in Tables XXXIX and XXXVII with theadditional adjusted constants J and K as given in Eqs. 372and 373. The numbers in the first column correspond to thenumbers in the first column of Table XXX. For simplicity, thelengthier functions are not explicitly given. See Sec. XII.B foran explanation of the symbol .

Type ofinputdatum Generalized observational equation

−1e−KJ−90RK−901 + ae,e3

B31* lo −Γp−90 20R1 + J1 + K p

KJ−90RK−901 + ae,e3

B32* lo Γh−90 20R1 + J1 + K−1e− h

p p

c1 + ae,e2

B33* hi − 1 + JΓp−90 KJ−90RK−90Rh

−1e−

1 + K p

0cB34* RK 1+K

2

1/28B35* KJ 1 + J0ch

4B36* KJ

2RK 1+J21+Kh

cMuAre2

B37* F90 1+J1+KKJ−90RK−90Rh

B62* JJ

B63*

an input datum due to its low weight, Ar133Cs, item B8,

has also been deleted as an input datum and as an ad-justed constant.

For the final adjustment, 2=65.0, 2 /=RB=1.04,and Q65.0 57=0.22, where Q2 is the probabilitythat the observed value of 2 for degrees of freedom would have exceeded that observed value see AppendixE of CODATA-98. Each input datum in the final ad-justment has Sc0.01, or is a subset of the data of anexperiment that provides an input datum or input datawith Sc0.01. Not counting such input data with Sc0.01, the six input data with the largest ri are B55,B53, B39, C18, B11.1, and B9; their values of ri are−1.89, −1.86, 1.74, −1.73, 1.69, and 1.45, respectively. Thenext largest ri are 1.22 and 1.11.

The output of the final adjustment is the set of bestestimated values, in the least-squares sense, of the 78adjusted constants and their variances and covariances.Together with i those constants that have exact valuessuch as 0 and c; ii the value of G obtained in Sec. X;and iii the values of m, GF, and sin2 W given in Sec.XI.B, all of the 2006 recommended values, includingtheir uncertainties, are obtained from the 78 adjustedconstants. How this is done can be found in Sec. V.B ofCODATA-98.

B. Tables of values

The 2006 CODATA recommended values of the basicconstants and conversion factors of physics and chemis-try and related quantities are given in TablesXLVIII–LV. These tables are very similar in form totheir 2002 counterparts; the principal difference is that anumber of new recommended values have been in-cluded in the 2006 list, in particular, in Table XLIX.These are mPc2 in GeV, where mP is the Planck mass;the g-factor of the deuteron gd; b=max/T, the Wiendisplacement law constant for frequency; and, for thefirst time, 14 recommended values of a number of con-stants that characterize the triton, including its mass mt,magnetic moment t, g-factor gt, and the magnetic mo-ment ratios t /e and t /p. The addition of the triton-related constants is a direct consequence of the im-proved measurement of A 3

r H item B3 in Table XXXand the new NMR measurements on, and reexaminedshielding correction differences for, the HT moleculeitems B22 and B23 in Table XXX.

Table XLVIII is a highly abbreviated list containingthe values of the constants and conversion factors mostcommonly used. Table XLIX is a much more extensivelist of values categorized as follows: universal; electro-magnetic; atomic and nuclear; and physicochemical. Theatomic and nuclear category is subdivided into 11 sub-categories: general; electroweak; electron, e−; muon, −;tau, −; proton, p; neutron, n; deuteron, d; triton, t; he-lion, h; and alpha particle, . Table L gives the variances,covariances, and correlation coefficients of a selectedgroup of constants. Application of the covariance ma-trix is discussed in Appendix E of CODATA-98. TableLI gives the internationally adopted values of variousquantities; Table LII lists the values of a number of x-rayrelated quantities; Table LIII lists the values of variousnon-SI units; and Tables LIV and LV give the values ofvarious energy equivalents.

All values given in Tables XLVIII–LV are available onthe web pages of the Fundamental Constants Data Cen-ter of the NIST Physics Laboratory at physics.nist.gov/constants. This electronic version of the 2006 CODATArecommended values of the constants also includes amuch more extensive correlation coefficient matrix. In-deed, the correlation coefficient of any two constantslisted in the tables is accessible on the web site, as wellas the automatic conversion of the value of an energy-related quantity expressed in one unit to the corre-

K K

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TABLE XLVII. Summary of the results of several least-squares adjustments carried out to investigate the effect of assuming therelations for KJ and RK given in Eqs. 372 and 373. The values of , h, K, and J are those obtained in the indicatedadjustments. The quantity RB=2 / is the Birge ratio and ri is the normalized residual of the indicated input datum see TableXXX. These four data have the largest ri of all the input data and are the only data in Adj. i with ri1.50. See the text for anexplanation and discussion of each adjustment, but in brief, i assumes KJ=2e /h and RK=h /e2 and uses all the data; ii is i withthe relation KJ=2e /h relaxed; iii is i with the relation RK=h /e2 relaxed; iv is i with both relations relaxed; v is iv with theVmSi datum deleted; vi is iv with the Γp−90 lo and Γh−90 lo data deleted; and vii is iv with the Vm lo, andSi, Γp−90 lo data deleted.Γh−90

Adj. RB −1 h / J s K J rB53 rB55 rB39 rB31.1

iiiiiiivvvivii

1.14

1.14

1.14

1.14

1.05

1.05

1.06

137.035 999 68793137.035 999 68893137.035 999 68393137.035 999 68593137.035 999 68693137.035 999 68693137.035 999 68693

6.626 068 962210−34

6.626 06821010−34

6.626 069 062510−34

6.626 06811110−34

6.626 06531310−34

6.626 07441910−34

6.626 07229510−34

0

0

161810−9

201810−9

231810−9

241810−9

241810−9

0

−617910−9

0

−778010−9

−2819510−9

40714310−9

23872010−9

−2.82

−3.22

−2.77

−3.27

Deleted

−0.05

Deleted

−2.71

−2.75

−2.71

−2.75

−2.45

−2.45

−2.45

2.37

2.39

2.36

2.39

2.19

2.19

2.19

2.31

1.77

2.45

1.79

0.01

Deleted

Deleted

TABLE XLVIII. An abbreviated list of the CODATA recommended values of the fundamental constants of physics and chem-istry based on the 2006 adjustment.

Quantity Symbol Numerical value UnitRelative std.uncert. ur

speed of light in vacuum c ,c0 299 792 458 m s−1 Exactmagnetic constant 0 410−7 N A−2

=12.566 370 614. . . 10−7 N A−2 Exactelectric constant 1/0c2 0 8.854 187 817. . . 10−12 F m−1 ExactNewtonian constant

of gravitationG 6.674 286710−11 m3 kg−1 s−2 1.010−4

Planck constant h 6.626 068 963310−34 J s 5.010−8

h /2 1.054 571 6285310−34 J s 5.010−8

elementary charge e 1.602 176 4874010−19 C 2.510−8

magnetic flux quantum h /2e Φ0 2.067 833 6675210−15 Wb 2.510−8

conductance quantum 2e2 /h G0 7.748 091 70045310−5 S 6.810−10

electron mass me 9.109 382 154510−31 kg 5.010−8

proton mass mp 1.672 621 6378310−27 kg 5.010−8

proton-electron mass ratio mp /me 1836.152 672 4780 4.310−10

fine-structure constant e2 /40c 7.297 352 53765010−3 6.810−10

inverse fine-structure constant −1 137.035 999 67994 6.810−10

Rydberg constant 2mec /2h R 10 973 731.568 52773 m−1 6.610−12

Avogadro constant NA,L 6.022 141 79301023 mol−1 5.010−8

Faraday constant NAe F 96 485.339924 C mol−1 2.510−8

molar gas constant R 8.314 47215 J mol−1 K−1 1.710−6

Boltzmann constant R /NA k 1.380 65042410−23 J K−1 1.710−6

Stefan-Boltzmann constant2 /60k4 /3c2

5.670 4004010−8 W m−2 K−4 7.010−6

Non-SI units accepted for use with the SIelectron volt: e /C J eV 1.602 176 4874010−19 J 2.510−8

unified atomic mass unit

1 u=mu=112

m12C

=10−3 kg mol−1 /NA

u 1.660 538 7828310−27 kg 5.010−8

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TABLE XLIX. The CODATA recommended values of the fundamental constants of physics and chemistry based on the 2006adjustment.

Quantity Symbol Numerical value UnitRelative std.uncert. ur

speed of light in vacuum

magnetic constant

2electric constant 1/0c

characteristic impedanceof vacuum 0 /0=0c

Newtonian constantof gravitation

Planck constant

in eV s

h /2

in eV s

c in MeV fm

Planck mass c /G1/2

energy equivalent in GeV5/G1/2 /kPlanck temperature c

31/2Planck length /mPc= G /c51/2Planck time lP/c= G /c

elementary charge

magnetic flux quantum h /2e

conductance quantum 2e2 /h

inverse of conductance quantum

Joesphson constanta 2e /hbvon Klitzing constant

h /e2=0c /2

Bohr magneton e /2me

in eV T−1

nuclear magneton e /2mp

in eV T−1

fine-structure constant e2 /40c

inverse fine-structure constant

Rydberg constant 2mec /2h

c ,c0

0

0

Z0

G

G /c

h

mP2mPc

TP

lP

tP

e

e /h

Φ0

G0−1G0

KJ

RK

B

B/h

B/hc

B/k

N

N/h

N/hc

N/k

−1

R

Rc

UNIVERSAL

299 792 458

410−7

=12.566 370 614. . . 10−7

8.854 187 817. . . 10−12

376.730 313 461…

6.674 286710−11

6.708 816710−39

6.626 068 963310−34

4.135 667 331010−15

1.054 571 6285310−34

6.582 118 991610−16

197.326 9631492.176 441110−8

1.220 892611019

1.416 785711032

1.616 2528110−35

5.391 242710−44

ELECTROMAGNETIC

1.602 176 4874010−19

2.417 989 454601014

2.067 833 6675210−15

7.748 091 70045310−5

12 906.403 778788483 597.89112109

25 812.807 55718

927.400 9152310−26

5.788 381 75557910−5

13.996 246 0435109

46.686 4515120.671 7131125.050 783 241310−27

3.152 451 23264510−8

7.622 593 84192.542 623 6166410−2

3.658 26376410−4

ATOMIC AND NUCLEAR

General

7.297 352 53765010−3

137.035 999 6799410 973 731.568 527733.289 841 960 361221015

−1m s

N A−2

N A−2

F m−1

−2m3 kg−1 s

2−2GeV/c

J s

eV s

J s

eV s

MeV fm

kg

GeV

K

m

s

C

A J−1

Wb

S

Hz V−1

J T−1

eV T−1

Hz T−1

−1 T−1m

K T−1

J T−1

eV T−1

MHz T−1

−1 T−1m

K T−1

−1m

Hz

Exact

ExactExactExact

1.010−4

1.010−4

5.010−8

2.510−8

5.010−8

2.510−8

2.510−8

5.010−5

5.010−5

5.010−5

5.010−5

5.010−5

2.510−8

2.510−8

2.510−8

6.810−10

6.810−10

2.510−8

6.810−10

2.510−8

1.410−9

2.510−8

2.510−8

1.710−6

2.510−8

1.410−9

2.510−8

2.510−8

1.710−6

6.810−10

6.810−10

6.610−12

6.610−12

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TABLE XLIX. Continued.

Relative std.Quantity Symbol Numerical value Unit uncert. ur

Rhc 2.179 871 971110−18 J 5.010−8

Rhc in eV 13.605 691 9334 eV 2.510−8

2Bohr radius /4R=402 /mee a0 0.529 177 208 593610−10 m 6.810−10

Hartree energy e2 /40a0=2Rhc Eh 4.359 743 942210−18 J 5.010−8

=2 2mec

in eV 27.211 383 8668 eV 2.510−8

quantum of circulation h /2me 3.636 947 51995010−4 −1m2 s 1.410−9

h /me 7.273 895 0401010−4 2 −1m s 1.410−9

Electroweak

Fermi coupling constantc GF/ c3 1.166 37110−5 GeV−2 8.610−6

weak mixing angled W on-shell sin2 W 0.222 5556 2.510−3

2scheme sin2 W=sW1− mW/mZ2

−Electron, e

electron mass me 9.109 382 154510−31 kg 5.010−8

in u, me=Are u electron 5.485 799 09432310−4 u 4.210−10

relative atomic mass times uenergy equivalent 2mec 8.187 104 384110−14 J 5.010−8

in MeV 0.510 998 91013 MeV 2.510−8

electron-muon mass ratio me /m 4.836 331 711210−3 2.510−8

electron-tau mass ratio me /m 2.875 644710−4 1.610−4

electron-proton mass ratio me /mp 5.446 170 21772410−4 4.310−10

electron-neutron mass ratio me /mn 5.438 673 44593310−4 6.010−10

electron-deuteron mass ratio me /md 2.724 437 10931210−4 4.310−10

electron to particle mass ratio me /m 1.370 933 555 705810−4 4.210−10

electron charge to mass quotient −e /me −1.758 820 150441011 −1C kg 2.510−8

electron molar mass NAme Me ,Me 5.485 799 09432310−7 kg mol−1 4.210−10

Compton wavelength h /mec λC 2.426 310 21753310−12 m 1.410−9

λC/2=a0=2 /4R C 386.159 264 595310−15 m 1.410−9

classical electron radius 2a0 re 2.817 940 28945810−15 m 2.110−9

2Thomson cross section 8 /3re e 0.665 245 85582710−28 2m 4.110−9

electron magnetic moment e −928.476 3772310−26 J T−1 2.510−8

to Bohr magneton ratio e /B −1.001 159 652 181 1174 7.410−13

to nuclear magneton ratio e /N −1838.281 970 9280 4.310−10

electron magnetic moment

anomaly e /B−1 ae 1.159 652 181 117410−3 6.410−10

electron g-factor −21+ae ge −2.002 319 304 362215 7.410−13

electron-muon e / 206.766 987752 2.510−8

magnetic moment ratio

electron-proton e /p −658.210 684854 8.110−9

magnetic moment ratio

electron to shielded protonmagnetic moment ratio

e /p −658.227 597172 1.110−8

H2O, sphere, 25 °Celectron-neutron e /n 960.920 5023 2.410−7

magnetic moment ratio

electron-deuteron e /d −2143.923 49818 8.410−9

magnetic moment ratio

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TABLE XLIX. Continued.

Quantity Symbol Numerical value UnitRelative std.uncert. ur

electron to shielded helionmagnetic moment ratiogas, sphere, 25 °C

electron gyromagnetic ratio 2e /

muon mass

in u, m=Ar u muonrelative atomic mass times u

energy equivalent

in MeV

muon-electron mass ratio

muon-tau mass ratio

muon-proton mass ratio

muon-neutron mass ratio

muon molar mass NAm

muon Compton wavelength h /mc

λC, /2

muon magnetic moment

to Bohr magneton ratio

to nuclear magneton ratio

muon magnetic moment anomaly / e /2m−1

muon g-factor −21+amuon-proton

magnetic moment ratio

etau mass

in u, m=Ar u taurelative atomic mass times u

energy equivalent

in MeV

tau-electron mass ratio

tau-muon mass ratio

tau-proton mass ratio

tau-neutron mass ratio

tau molar mass NAm

tau Compton wavelength h /mc

λC, /2

proton mass

in u, mp=Arp u protonrelative atomic mass times u

energy equivalent

in MeV

proton-electron mass ratio

proton-muon mass ratio

proton-tau mass ratio

e /h

e

e /2

m

2mc

m /me

m /m

m /mp

m /mn

M ,M

λC,

C,

/B

/N

a

g

/p

m

2mc

m /me

m /m

m /mp

m /mn

M ,M

λC,

C,

mp

2mpc

mp /me

mp /m

mp /m

864.058 25710

1.760 859 770441011

28 024.953 6470Muon, −

1.883 531 301110−28

0.113 428 925629

1.692 833 5109510−11

105.658 366838206.768 2823525.945 929710−2

0.112 609 5261290.112 454 5167290.113 428 92562910−3

11.734 441 043010−15

1.867 594 2954710−15

−4.490 447 861610−26

−4.841 970 491210−3

−8.890 597 05231.165 920 696010−3

−2.002 331 841412−3.183 345 13785

Tau, −

3.167 775210−27

1.907 6831

2.847 054610−10

1776.99293477.485716.8183271.893 90311.891 29311.907 683110−3

0.697 721110−15

0.111 0461810−15

Proton, p

1.672 621 6378310−27

1.007 276 466 7710

1.503 277 3597510−10

938.272 013231836.152 672 47808.880 243 39230.528 01286

−1 T−1s

MHz T−1

kg

u

J

MeV

kg mol−1

m

m

J T−1

kg

u

J

MeV

kg mol−1

m

m

kg

u

J

MeV

1.210−8

2.510−8

2.510−8

5.610−8

2.510−8

5.610−8

3.610−8

2.510−8

1.610−4

2.510−8

2.510−8

2.510−8

2.510−8

2.510−8

3.610−8

2.510−8

2.510−8

5.210−7

6.010−10

2.710−8

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

1.610−4

5.010−8

1.010−10

5.010−8

2.510−8

4.310−10

2.510−8

1.610−4

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TABLE XLIX. Continued.

Relative std.Quantity Symbol Numerical value Unit uncert. ur

proton-neutron mass ratio mp /mn 0.998 623 478 2446 4.610−10

proton charge to mass quotient e /mp 9.578 833 9224107 −1C kg 2.510−8

proton molar mass NAmp Mp ,Mp 1.007 276 466 771010−3 kg mol−1 1.010−10

proton Compton wavelength h /mpc λC,p 1.321 409 84461910−15 m 1.410−9

λC,p /2 C,p 0.210 308 908 613010−15 m 1.410−9

proton rms charge radius Rp 0.87686910−15 m 7.810−3

proton magnetic moment p 1.410 606 6623710−26 J T−1 2.610−8

to Bohr magneton ratio p /B 1.521 032 2091210−3 8.110−9

to nuclear magneton ratio p /N 2.792 847 35623 8.210−9

proton g-factor 2p /N gp 5.585 694 71346 8.210−9

proton-neutronmagnetic moment ratio

p /n −1.459 898 0634 2.410−7

shielded proton magnetic momentH2O, sphere, 25 °C

p 1.410 570 4193810−26 J T−1 2.710−8

to Bohr magneton ratio p /B 1.520 993 1281710−3 1.110−8

to nuclear magneton ratio p /N 2.792 775 59830 1.110−8

proton magnetic shieldingcorrection 1−p /pH2O, sphere, 25 °C

p 25.6941410−6 5.310−4

proton gyromagnetic ratio 2p / p 2.675 222 09970108 −1 T−1s 2.610−8

p /2 42.577 482111 MHz T−1 2.610−8

shielded proton gyromagneticratio 2 /pH2O, sphere, 25 °C

p 2.675 153 36273108 −1 T−1s 2.710−8

/2p 42.576 388112 MHz T−1 2.710−8

Neutron, n

neutron mass mn 1.674 927 2118410−27 kg 5.010−8

in u, mn=Arn u neutron 1.008 664 915 9743 u 4.310−10

relative atomic mass times uenergy equivalent 2mnc 1.505 349 5057510−10 J 5.010−8

in MeV 939.565 34623 MeV 2.510−8

neutron-electron mass ratio mn /me 1838.683 660511 6.010−10

neutron-muon mass ratio mn /m 8.892 484 0923 2.510−8

neutron-tau mass ratio mn /m 0.528 74086 1.610−4

neutron-proton mass ratio mn /mp 1.001 378 419 1846 4.610−10

neutron molar mass NAmn Mn ,Mn 1.008 664 915 974310−3 kg mol−1 4.310−10

neutron Compton wavelength h /mnc λC,n 1.319 590 89512010−15 m 1.510−9

λC,n /2 C,n 0.210 019 413 823110−15 m 1.510−9

neutron magnetic moment n −0.966 236 412310−26 J T−1 2.410−7

to Bohr magneton ratio n /B −1.041 875 632510−3 2.410−7

to nuclear magneton ratio n /N −1.913 042 7345 2.410−7

neutron g-factor 2n /N gn −3.826 085 4590 2.410−7

neutron-electron n /e 1.040 668 822510−3 2.410−7

magnetic moment ratio

neutron-protonmagnetic moment ratio

n /p −0.684 979 3416 2.410−7

neutron to shielded protonmagnetic moment ratioH O, sphere, 25 °C

n /p −0.684 996 9416 2.410−7

2

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TABLE XLIX. Continued.

Quantity Symbol Numerical value UnitRelative std.uncert. ur

neutron gyromagnetic ratio 2n / n 1.832 471 8543108 s−1 T−1 2.410−7

n /2 29.164 695469 MHz T−1 2.410−7

Deuteron, d

deuteron mass md 3.343 583 201710−27 kg 5.010−8

in u, md=Ard u deuteronrelative atomic mass times u

2.013 553 212 72478 u 3.910−11

energy equivalent mdc2 3.005 062 721510−10 J 5.010−8

in MeV 1875.612 79347 MeV 2.510−8

deuteron-electron mass ratio md /me 3670.482 965416 4.310−10

deuteron-proton mass ratio md /mp 1.999 007 501 0822 1.110−10

deuteron molar mass NAmd Md ,Md 2.013 553 212 7247810−3 kg mol−1 3.910−11

deuteron rms charge radius Rd 2.14022810−15 m 1.310−3

deuteron magnetic moment d 0.433 073 4651110−26 J T−1 2.610−8

to Bohr magneton ratio d /B 0.466 975 45563910−3 8.410−9

to nuclear magneton ratio d /N 0.857 438 230872 8.410−9

deuteron g-factor d /N gd 0.857 438 230872 8.410−9

deuteron-electonmagnetic moment ratio

d /e −4.664 345 5373910−4 8.410−9

deuteron-protonmagnetic moment ratio

d /p 0.307 012 207024 7.710−9

deuteron-neutronmagnetic moment ratio

d /n −0.448 206 5211 2.410−7

Triton, t

triton mass mt 5.007 355 882510−27 kg 5.010−8

in u, mt=Art u tritonrelative atomic mass times u

3.015 500 713425 u 8.310−10

energy equivalent mtc2 4.500 387 032210−10 J 5.010−8

in MeV 2808.920 90670 MeV 2.510−8

triton-electron mass ratio mt /me 5496.921 526951 9.310−10

triton-proton mass ratio mt /mp 2.993 717 030925 8.410−10

triton molar mass NAmt Mt ,Mt 3.015 500 71342510−3 kg mol−1 8.310−10

triton magnetic moment t 1.504 609 3614210−26 J T−1 2.810−8

to Bohr magneton ratio t /B 1.622 393 6572110−3 1.310−8

to nuclear magneton ratio t /N 2.978 962 44838 1.310−8

triton g-factor 2t /N gt 5.957 924 89676 1.310−8

triton-electronmagnetic moment ratio

t /e −1.620 514 4232110−3 1.310−8

triton-protonmagnetic moment ratio

t /p 1.066 639 90810 9.810−9

triton-neutronmagnetic moment ratio

t /n −1.557 185 5337 2.410−7

Helion, h

helion 3He nucleus mass mh 5.006 411 922510−27 kg 5.010−8

in u, mh=Arh u helionrelative atomic mass times u

3.014 932 247326 u 8.610−10

energy equivalent mhc2 4.499 538 642210−10 J 5.010−8

in MeV 2808.391 38370 MeV 2.510−8

helion-electron mass ratio mh /me 5495.885 276552 9.510−10

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TABLE XLIX. Continued.

Relative std.Quantity Symbol Numerical value Unit uncert. ur

helion-proton mass ratio mh /mp 2.993 152 671326 8.710−10

helion molar mass NAmh Mh ,Mh 3.014 932 24732610−3 kg mol−1 8.610−10

shielded helion magnetic moment h −1.074 552 9823010−26 J T−1 2.810−8

gas, sphere, 25 °Cto Bohr magneton ratio h /B −1.158 671 4711410−3 1.210−8

to nuclear magneton ratio h /N −2.127 497 71825 1.210−8

shielded helion to proton h /p −0.761 766 55811 1.410−8

magnetic moment ratiogas, sphere, 25 °C

shielded helion to shielded protonmagnetic moment ratio

h / −0.761 786 131333p 4.310−9

gas/H2O, spheres, 25 °Cshielded helion gyromagnetic h 2.037 894 73056108 −1 T−1s 2.810−8

ratio 2h /gas, sphere, 25 °C

h /2 32.434 101 9890 MHz T−1 2.810−8

Alpha particle,

alpha particle mass m 6.644 656 203310−27 kg 5.010−8

in u, m=Ar u alpha particle 4.001 506 179 12762 u 1.510−11

relative atomic mass times uenergy equivalent 2mc 5.971 919 173010−10 J 5.010−8

in MeV 3727.379 10993 MeV 2.510−8

particle to electron mass ratio m /me 7294.299 536531 4.210−10

particle to proton mass ratio m /mp 3.972 599 689 5141 1.010−10

particle molar mass NAm M ,M 4.001 506 179 1276210−3 kg mol−1 1.510−11

PHYSICOCHEMICAL

Avogadro constant NA,L 6.022 141 79301023 mol−1 5.010−8

atomic mass constant mu 1.660 538 7828310−27 kg 5.010−8

1mu= 12m12C=1 u=10−3 kg mol−1 /NA

energy equivalent 2muc 1.492 417 8307410−10 J 5.010−8

in MeV 931.494 02823 MeV 2.510−8

Faraday constantf NAe F 96 485.339924 C mol−1 2.510−8

molar Planck constant NAh 3.990 312 68215710−10 J s mol−1 1.410−9

NAhc 0.119 626 564 7217 J m mol−1 1.410−9

molar gas constant R 8.314 47215 J mol−1 K−1 1.710−6

Boltzmann constant R /NA k 1.380 65042410−23 J K−1 1.710−6

in eV K−1 8.617 3431510−5 eV K−1 1.710−6

k /h 2.083 6644361010 Hz K−1 1.710−6

k /hc 69.503 5612 −1 K−1m 1.710−6

molar volume of ideal gas RT /p Vm 22.413 9963910−3 m3 mol−1 1.710−6

T=273.15 K, p=101.325 kPa

Loschmidt constant NA/Vm n0 2.686 7774471025 −3m 1.710−6

T=273.15 K, p=100 kPa Vm 22.710 9814010−3 m3 mol−1 1.710−6

Sackur-Tetrode constantabsolute entropy constantg

5 kT1 /h23/2kT1 /p0+ln 2mu2

T1=1 K, p0=100 kPa S0 /R −1.151 704744 3.810−6

T =1 K, p =101.325 kPa −1.164 867744 3.810−61 0

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Quantity Symbol Numerical value UnitRelative std.uncert. ur

Stefan-Boltzmann constant22 /60k4 /3c

first radiation constant 2hc2

first radiation constant for spectralradiance 2hc2

second radiation constant hc /k

Wien displacement law constants

b=λmaxT=c2 /4.965 114 231. . .

b=max/T=2.821 439 372. . .c /c2

c1

c1L

c2

b

b

5.670 4004010−8

3.741 771 181910−16

1.191 042 7595910−16

1.438 77522510−2

2.897 76855110−3

5.878 933101010

W m−2 K−4

W m2

−1W m2 sr

m K

m K

Hz K−1

7.010−6

5.010−8

5.010−8

1.710−6

1.710−6

1.710−6

aSee Table LI for the conventional value adopted internationally for realizing representations of the volt using the Josephsoneffect.

bSee Table LI for the conventional value adopted internationally for realizing representations of the ohm using the quantum Halleffect.

cValue recommended by the Particle Data Group Yao et al., 2006.dBased on the ratio of the masses of the W and Z bosons mW/mZ recommended by the Particle Data Group Yao et al., 2006.

The value for sin2W they recommend, which is based on a particular variant of the modified minimal subtraction scheme MS, isˆsin2WMZ=0.231 2215.

eThis and all other values involving m are based on the value of mc2 in MeV recommended by the Particle Data Group Yaoet al., 2006, but with a standard uncertainty of 0.29 MeV rather than the quoted uncertainty of −0.26 MeV, +0.29 MeV.

fThe numerical value of F to be used in coulometric chemical measurements is 96 485.340148 5.010−8 when the relevantcurrent is measured in terms of representations of the volt and ohm based on the Josephson and quantum Hall effects and theinternationally adopted conventional values of the Josephson and von Klitzing constants KJ−90 and RK−90 given in Table LI.

3 5gThe entropy of an ideal monatomic gas of relative atomic mass Ar is given by S=S0+ R ln Ar−R lnp /p0+ R lnT /K.2 2

sponding value expressed in another unit in essence, anautomated version of Tables LIV and LV.

As discussed in Sec. V, well after the 31 December2006 closing date of the 2006 adjustment and the 29March 2007 distribution date of the 2006 recommendedvalues on the web, Aoyama et al. 2007 reported theirdiscovery of an error in the coefficient A8

1 in the theo-retical expression for the electron magnetic momentanomaly ae. Use of the new coefficient would lead to anincrease in the 2006 recommended value of by 6.8

times its uncertainty, and an increase of its uncertaintyby a factor of 1.02. The recommended values and uncer-tainties of constants that depend solely on , or on incombination with other constants with ur no larger thana few parts in 1010, would change in the same way. How-ever, the changes in the recommended values of the vastmajority of the constants listed in the tables would lie inthe range 0 to 0.5 times their 2006 uncertainties, andtheir uncertainties would remain essentially unchanged.

TABLE XLIX. Continued.

TABLE L. The variances, covariances, and correlation coefficients of the values of a selected groupof constants based on the 2006 CODATA adjustment. The numbers in bold above the main diagonalare 1016 times the numerical values of the relative covariances, the numbers in bold on the maindiagonal are 1016 times the numerical values of the relative variances, and the numbers in italicsbelow the main diagonal are the correlation coefficients.a

h e me NA me /m F

0.0047 0.0002 0.0024 −0.0092 0.0092 −0.0092 0.0116h 0.0005 24.8614 12.4308 24.8611 −24.8610 −0.0003 −12.4302e 0.0142 0.9999 6.2166 12.4259 −12.4259 −0.0048 −6.2093me −0.0269 0.9996 0.9992 24.8795 −24.8794 0.0180 −12.4535NA 0.0269 −0.9996 −0.9991 −1.0000 24.8811 −0.0180 12.4552me /m −0.0528 0.0000 −0.0008 0.0014 −0.0014 6.4296 −0.0227F 0.0679 −0.9975 −0.9965 −0.9990 0.9991 −0.0036 6.2459

aThe relative covariance is urxi ,xj=uxi ,xj / xixj, where uxi ,xj is the covariance of xi and xj; therelative variance is ur

2xi=urxi ,xi; and the correlation coefficient is rxi ,xj=uxi ,xj / uxiuxj.

715Mohr, Taylor, and Newell: CODATA recommended values of the fundamental …

Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

TABLE LI. Internationally adopted values of various quantities.

Relative std.Quantity Symbol Numerical value Unit uncert. ur

relative atomic massa of 12C 12CAr 12 Exactmolar mass constant Mu 110−3 kg mol−1 Exactmolar mass of 12C M12C 1210−3 kg mol−1 Exact

bconventional value of Josephson constant KJ−90 483 597.9 GHz V−1 Exactconventional value of von Klitzing constantc RK−90 25 812.807 Exactstandard atmosphere

proportional to , h, or R raised to various powers.Thus, the first six quantities are calculated from expres-sions proportional to a, where a =1, 2, 3, or 6. Thenext 15 quantities, h through p, are calculated fromexpressions containing the factor ha, where a =1 or 1

2 .And the five quantities R through are proportional toRa, where a=1 or 4.

Further comments on the entries in Table LVI are asfollows.

i The uncertainty of the 2002 recommended value of has been reduced by nearly a factor of 5 by the mea-surement of ae at Harvard University and the improvedtheoretical expression for aeth. The difference betweenthe Harvard result and the earlier University of Wash-ington result, which played a major role in the determi-nation of in the 2002 adjustment, accounts for most ofthe change in the recommended value of from 2002 to2006.

aThe relative atomic mass ArX of particle X with mass mX is ArX=mX /mu, where mu=m12C /12=Mu /NA=1 u is theatomic mass constant, Mu is the molar mass constant, NA is the Avogadro constant, and u is the unified atomic mass unit. Thus themass of particle X is mX=ArX u and the molar mass of X is MX=ArXMu.

bThis is the value adopted internationally for realizing representations of the volt using the Josephson effect.cThis is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.

101 325 Pa Exact

XIV. SUMMARY AND CONCLUSION

We conclude by i comparing the 2006 and 2002CODATA recommended values of the constants andidentifying those new results that have contributed mostto the changes from the 2002 values, ii presenting someconclusions that can be drawn from the 2006 recom-mended values and analysis of the 2006 input data, andiii looking to the future and identifying experimentaland theoretical work that can advance our knowledge ofthe values of the constants.

A. Comparison of 2006 and 2002 CODATA recommendedvalues

The 2006 and 2002 recommended values of a repre-sentative group of constants are compared in Table LVI.Regularities in the numbers in columns 2–4 arise be-cause many constants are obtained from expressions

TABLE LII. Values of some x-ray-related quantities based on the 2006 CODATA adjustment of thevalues of the constants.

Quantity Symbol Numerical value UnitRelative std.

uncert. ur

Cu x unit: λCuK1 /1 537.400 xuCuK1 1.002 076 992810−13 m 2.810−7

Mo x unit: λMoK1 /707.831 xuMoK1 1.002 099 555310−13 m 5.310−7

ångström star: λWK1 /0.209 010 0 Å* 1.000 014 989010−10 m 9.010−7

lattice parametera of Si a 543.102 0641410−12 m 2.610−8

in vacuum, 22.5 °C220 lattice spacing of Si a /8 d220 192.015 57625010−12 m 2.610−8

in vacuum, 22.5 °Cmolar volume of Si

MSi /Si=NAa3 /8 VmSi 12.058 83491110−6 m3 mol−1 9.110−8

in vacuum, 22.5 °C

aThis is the lattice parameter unit cell edge length of an ideal single crystal of naturally occurringSi free of impurities and imperfections, and is deduced from measurements on extremely pure andnearly perfect single crystals of Si by correcting for impurity effects.

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Mohr, Taylor, and Newell: CODATA recommended values of the fundamental … 717

TABLE LIII. The values in SI units of some non-SI units based on the 2006 CODATA adjustment of the values of the constants.

Quantity Symbol Numerical value UnitRelative std.uncert. ur

electron volt: e /C Junified atomic mass unit

11 u=mu= 12m12C=10−3 kg mol−1 /NA

n.u. of velocity:speed of light in vacuum

n.u. of action:reduced Planck constant h /2

in eV sin MeV fm

n.u. of mass:electron mass

n.u. of energyin MeV

n.u. of momentumin MeV/c

n.u. of length /mecn.u. of time

a.u. of charge:elementary charge

a.u. of mass:electron mass

a.u. of action:reduced Planck constant h /2

a.u. of length:Bohr radius bohr /4R

a.u. of energy:Hartree energy hartreee2 /40a0=2Rhc=2 2mec

a.u. of timea.u. of forcea.u. of velocity ca.u. of momentuma.u. of currenta.u. of charge density

a.u. of electric potentiala.u. of electric fielda.u. of electric field gradient

a.u. of electric dipole momenta.u. of electric quadrupole moment

a.u. of electric polarizability

a.u. of 1st hyperpolarizability

a.u. of 2nd hyperpolarizability

a.u. of magnetic flux density

a.u. of magneticdipole moment 2B

Non-SI units accepted for use with the SIeV 1.602 176 4874010−19

u 1.660 538 7828310−27

Natural units n.u.

c ,c0 299 792 458

1.054 571 6285310−34

6.582 118 991610−16

c 197.326 963149

me 9.109 382 154510−31

2mec 8.187 104 384110−14

0.510 998 91013mec 2.730 924 061410−22

0.510 998 91013C 386.159 264 595310−15

2 /mec 1.288 088 65701810−21

Atomic units a.u.

e 1.602 176 4874010−19

me 9.109 382 154510−31

1.054 571 6285310−34

a0 0.529 177 208 593610−10

Eh 4.359 743 942210−18

/Eh 2.418 884 326 5051610−17

Eh /a0 8.238 722 064110−8

a0Eh / 2.187 691 254115106

/a0 1.992 851 5659910−24

eEh / 6.623 617 631710−3

3e /a0 1.081 202 300271012

Eh /e 27.211 383 8668Eh /ea0 5.142 206 32131011

2Eh /ea0 9.717 361 66241021

ea0 8.478 352 812110−30

2ea0 4.486 551 071110−40

2e a02 /Eh 1.648 777 25363410−41

3 2e a03 /Eh 3.206 361 5338110−53

4 3e a04 /Eh 6.235 380 953110−65

2 /ea0 2.350 517 38259105

e /me 1.854 801 8304610−23

Jkg

−1m s

J seV sMeV fm

kgJMeV

−1kg m sMeV/c

ms

C

kg

J s

m

J

sN

−1m s−1kg m s

AC m−3

VV m−1

V m−2

C mC m2

C2 2 J−1m

C3 m3 J−2

C4 m4 J−3

T

J T−1

2.510−8

5.010−8

Exact

5.010−8

2.510−8

2.510−8

5.010−8

5.010−8

2.510−8

5.010−8

2.510−8

1.410−9

1.410−9

2.510−8

5.010−8

5.010−8

6.810−10

5.010−8

6.610−12

5.010−8

6.810−10

5.010−8

2.510−8

2.510−8

2.510−8

2.510−8

2.510−8

2.510−8

2.510−8

2.110−9

2.510−8

5.010−8

2.510−8

2.510−8

Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

Relative std.Quantity Symbol Numerical value Unit uncert. ur

a.u. of magnetizability

a.u. of permittivity 107/c2

2e a02 /me

e2 /a0Eh

7.891 036 4332710−29

1.112 650 056. . . 10−10

J T−2

F m−1

3.410−9

Exact

ii The uncertainty of the 2002 recommended valueof h has been reduced by over a factor of 3 due to thenew NIST watt-balance result for K2

JRK and because thefactor used to increase the uncertainties of the data re-lated to h applied to reduce the inconsistencies amongthe data was reduced from 2.325 in the 2002 adjustmentto 1.5 in the 2006 adjustment. That the change in valuefrom 2002 to 2006 is small is due to the excellent agree-ment between the new value of K2

JRK and the earlierNIST and NPL values, which played a major role in thedetermination of h in the 2002 adjustment.

iii The updating of two measurements that contrib-uted to the determination of the 2002 recommendedvalue of G reduced the spread in the values and rein-forced the most accurate result, that from the Universityof Washington. On this basis, the Task Group reduced

the assigned uncertainty from u =1.510−4r in 2002 to

ur=1.010−4 in 2006. This uncertainty reflects the his-torical difficulty of measuring G. Although the recom-mended value is the weighted mean of the eight avail-able values, the assigned uncertainty is still over fourtimes the uncertainty of the mean multiplied by the cor-responding Birge ratio RB.

iv The large shift in the recommended value of d220from 2002 to 2006 is due to the fact that in the 2002adjustment only the NMIJ result for d220NR3 was in-cluded, while in the 2006 adjustment this result but up-dated by more recent NMIJ measurements was in-cluded together with the PTB result for d220W4.2a andthe new INRIM results for d220W4.2a and d220MO*.Moreover, the NMIJ value of d220 inferred from

TABLE LIII. Continued.

TABLE LIV. The values of some energy equivalents derived from E=mc2=hc /λ=h=kT, and based on the 2006 CODATA1 2adjustment of the values of the constants; 1 eV= e /C J, 1 u= 2mu= 12m12C=10−3 kg mol−1 /NA, and Eh=2Rhc= mec is

the Hartree energy hartree.

Relevant unitJ kg −1m Hz

1 J 1 J= 1 J /c2= 1 J /hc= 1 J /h=1 J 1.112 650 056. . . 10−17 kg −15.034 117 47251024 m 1.509 190 450751033 Hz

1 kg 1 kgc2= 1 kg= 1 kgc /h= 1 kgc2 /h=8.987 551 787. . . 1016 J 1 kg −14.524 439 15231041 m 1.356 392 733681050 Hz

1 m−1 1 m−1hc= 1 m−1h /c= 1 m−1= 1 m−1c=1.986 445 5019910−25 J 2.210 218 701110−42 kg −11 m 299 792 458 Hz

1 Hz 1 Hzh= 1 Hzh /c2= 1 Hz /c= 1 Hz=6.626 068 963310−34 J 7.372 496 003710−51 kg −13.335 640 951. . . 10−9 m 1 Hz

1 K 1 Kk= 1 Kk /c2= 1 Kk /hc= 1 Kk /h=1.380 65042410−23 J 1.536 18072710−40 kg −169.503 5612 m 2.083 6644361010 Hz

1 eV 1 eV= 1 eV /c2= 1 eV /hc= 1 eV /h=1.602 176 4874010−19 J 1.782 661 7584410−36 kg −18.065 544 6520105 m 2.417 989 454601014 Hz

1 u 21 uc = 1 u= 1 uc /h= 1 uc2 /h=1.492 417 8307410−10 J 1.660 538 7828310−27 kg −17.513 006 671111014 m 2.252 342 7369321023 Hz

1 Eh 1 Eh= 1 Eh /c2= 1 Eh /hc= 1 Eh /h=4.359 743 942210−18 J 4.850 869 342410−35 kg −12.194 746 313 70515107 m 6.579 683 920 722441015 Hz

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Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

TABLE LV. The values of some energy equivalents derived from E=mc2=hc /λ=h=kT, and based on the 2006 CODATA1adjustment of the values of the constants; 1 eV= e /C J, 1 u= 2mu= 12m12C=10−3 kg mol−1 /NA, and Eh=2Rhc= mec2 is the

Hartree energy hartree.

Relevant unitK eV u Eh

1 J 1 J /k= 1 J= 1 J /c2= 1 J=7.242 963131022 K 6.241 509 65161018 eV 6.700 536 4133109 u 2.293 712 69111017 Eh

1 kg 1 kgc2 /k= 1 kgc2= 1 kg= 1 kgc2=6.509 651111039 K 5.609 589 12141035 eV 6.022 141 79301026 u 2.061 486 16101034 Eh

1 m−1 1 m−1hc /k= 1 m−1hc= 1 m−1h /c= 1 m−1hc=1.438 77522510−2 K 1.239 841 8753110−6 eV 1.331 025 03941910−15 u 4.556 335 252 7603010−8 Eh

1 Hz 1 Hzh /k= 1 Hzh= 1 Hzh /c2= 1 Hzh=4.799 23748410−11 K 4.135 667 331010−15 eV 4.439 821 62946410−24 u 1.519 829 846 0061010−16 Eh

1 K 1 K= 1 Kk= 1 Kk /c2= 1 Kk=1 K 8.617 3431510−5 eV 9.251 0981610−14 u 3.166 81535510−6 Eh

1 eV 1 eV /k= 1 eV= 21 eV /c = 1 eV=1.160 450520104 K 1 eV 1.073 544 1882710−9 u 3.674 932 5409210−2 Eh

1 u 1 uc2 /k= 1 uc2= 1 u= 1 uc2=1.080 9527191013 K 931.494 02823106 eV 1 u 3.423 177 714949107 Eh

1 Eh 1 Eh /k= 1 Eh= 1 Eh /c2= 1 Eh=3.157 746555105 K 27.211 383 8668 eV 2.921 262 29864210−8 u 1 Eh

d220NR3 strongly disagrees with the values of d220 in-ferred from the other three results.

v The marginally significant shift in the recom-mended value of g from 2002 to 2006 is mainly due tothe following: In the 2002 adjustment, the principal had-ronic contribution to the theoretical expression for a

was based on both a calculation that included only e+e−

annihilation data and a calculation that used data fromhadronic decays of the in place of some of the e+e−

annihilation data. In the 2006 adjustment, the principalhadronic contribution was based on a calculation thatused only annihilation data because of various concernsthat subsequently arose about the reliability of incorpo-rating the data in the calculation; the calculation basedon both e+e− annihilation data and decay data was onlyused to estimate the uncertainty of the hadronic contri-bution. Because the results from the two calculations arein significant disagreement, the uncertainty of ath iscomparatively large: ur=1.810−6.

vi The reduction of the uncertainties of the magneticmoment ratios p /B, p /N, d /N, e /p, and d /pare due to the new NMR measurement ofpHD /dHD and careful reexamination of the cal-culation of the D-H shielding correction difference dp.Because the value of the product p /ee /d im-

plied by the new measurement is highly consistent withthe same product implied by the individual measure-ments of eH /pH and dD /eD, the changes inthe values of the ratios are small.

In summary, the most important differences betweenthe 2006 and 2002 adjustments are that the 2006 adjust-ment had available new experimental and theoretical re-sults for ae, which provided a dramatically improvedvalue of , and a new result for K2

JRK, which provided asignificantly improved value of h. These two advancesfrom 2002 to 2006 have resulted in major reductions inthe uncertainties of many of the 2006 recommended val-ues compared with their 2002 counterparts.

B. Some implications of the 2006 CODATA recommendedvalues and adjustment for physics and metrology

A number of conclusions that can be drawn from the2006 adjustment concerning metrology and the basictheories and experimental methods of physics are pre-sented here, where the focus is on those conclusions thatare new or are different from those drawn from the 2002and 1998 adjustments.

Conventional electric units. One can interpretthe adoption of the conventional values KJ−90

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Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

TABLE LVI. Comparison of the 2006 and 2002 CODATAadjustments of the values of the constants by comparison ofthe corresponding recommended values of a representativegroup of constants. Here Dr is the 2006 value minus the 2002value divided by the standard uncertainty u of the 2002 valuei.e., Dr is the change in the value of the constant from 2002 to2006 relative to its 2002 standard uncertainty.

Quantity2006 rel. std.

uncert. ur

Ratio 2002 urto 2006 ur Dr

RK

a0

λC

re

e

h

me

mh

m

NA

Eh

c1

e

KJ

F

pB

N

e

p

R

k

Vm

c2

G

R

me /mp

me /m

AreArpArnArdArhArd220

ge

g

p /B

p /N

n /N

d /N

e /p

6.810−10

6.810−10

6.810−10

1.410−9

2.110−9

4.110−9

5.010−8

5.010−8

5.010−8

5.010−8

5.010−8

5.010−8

5.010−8

2.510−8

2.510−8

2.510−8

2.710−8

2.510−8

2.510−8

2.510−8

2.610−8

1.710−6

1.710−6

1.710−6

1.710−6

7.010−6

1.010−4

6.610−12

4.310−10

2.510−8

4.210−10

1.010−10

4.310−10

3.910−11

8.610−10

1.510−11

2.610−8

7.410−13

6.010−10

8.110−9

8.210−9

2.410−7

8.410−9

8.110−9

4.94.94.94.94.94.93.43.43.43.43.43.43.43.43.43.43.2

3.43.43.43.31.01.01.01.01.01.51.01.11.01.01.31.34.52.30.91.45.01.01.21.21.01.31.2

−1.31.3

−1.3−1.3−1.3−1.3−0.3−0.3−0.3−0.30.3

−0.3−0.3−0.30.30.20.2

−0.4−0.40.4

−0.40.00.00.00.00.00.10.00.20.3

−0.1−0.90.70.10.7

−0.4−2.91.3

−1.40.20.20.0

−0.20.2

=483 597.9 GHz/V and RK−90=25 812.807 for theJosephson and von Klitzing constants as establishingconventional, practical units of voltage and resistance,V90 and Ω90, given by V90= KJ−90/KJ V and Ω90= RK/RK−90 . Other conventional electric units followfrom V90 and Ω90, for example, A90=V90/Ω90, C90=A90 s, W90=A90V90, F90=C90/V90, and H90=Ω90 s,which are the conventional, practical units of current,charge, power, capacitance, and inductance, respectivelyTaylor and Mohr, 2001. For the relations between KJand KJ−90, and RK and RK−90, the 2006 adjustment gives

KJ = KJ-90 1 − 1.92.5 10−8 , 374

RK = RK-90 1 + 2.15968 10−8 , 375

which lead to

V90 = 1 + 1.92.5 10−8 V, 376

Ω90 = 1 + 2.15968 10−8 , 377

A90 = 1 − 0.32.5 10−8 A, 378

C90 = 1 − 0.32.5 10−8 C, 379

W90 = 1 + 1.65.0 10−8 W, 380

F90 = 1 − 2.15968 10−8 F, 381

H90 = 1 + 2.15968 10−8 H. 382

Equations 376 and 377 show that V90 exceeds V andΩ90 exceeds by 1.92.510−8 and 2.1596810−8, re-spectively. This means that measured voltages and resis-tances traceable to the Josephson effect and KJ−90 andthe quantum Hall effect and RK−90, respectively, are toosmall relative to the SI by these same fractionalamounts. However, these differences are well within the4010−8 uncertainty assigned to V90/V and the 1010−8 uncertainty assigned to Ω90/ by the Consulta-tive Committee for Electricity and Magnetism CCEMof the CIPM Quinn, 1989, 2001.

Josephson and quantum Hall effects. The study in Sec.XII.B.2 provides no statistically significant evidence thatthe fundamental Josephson and quantum Hall effect re-lations KJ=2e /h and RK=h /e2 are not exact. The twotheories of the most important phenomena ofcondensed-matter physics are thereby further sup-ported.

Antiprotonic helium. The good agreement betweenthe value of Are obtained from the measured values

TABLE LVI. Continued.

Quantity2006 rel. std.

uncert. ur

Ratio 2002 urto 2006 ur Dr

n /p 2.410−7 1.0 0.0d /p 7.710−9 1.9 −0.3

720 Mohr, Taylor, and Newell: CODATA recommended values of the fundamental …

Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

and theoretical expressions for a number of transitionfrequencies in antiprotonic 4He and 3He with threeother values obtained by entirely different methods in-dicates that these rather complex atoms are reasonablywell understood both experimentally and theoretically.

Newtonian constant of gravitation. Although the in-consistencies among the values of G have been reducedsomewhat as a result of modifications to two of the eightresults available in 2002, the situation remains problem-atic; there is no evidence that the historic difficulty ofmeasuring G has been overcome.

Tests of QED. The good agreement of highly accuratevalues of 133 inferred from h /m Cs and h /m87Rb,which are only weakly dependent on QED theory, withthe values of inferred from ae, muonium transitionfrequencies, and H and D transition frequencies, pro-vide support for the QED theory of ae as well as thebound-state QED theory of muonium and H and D. Inparticular, the weighted mean of the two valuesinferred from h /m133Cs and h /m87Rb, −1

=137.035 999 3469 5.010−9, and the weighted meanof the two values inferred from the two experimentalvalues of a , −1=137.035 999 68094 6.910−10

e , differby only 0.5udiff, with udiff=5.110−9. This is a truly im-pressive confirmation of QED theory.

Physics beyond the Standard Model. If the principalhadronic contribution to ath obtained from the e+e−

annihilation-data plus hadronic-decay-data calculationsee the previous section is completely ignored, and thevalue based on the annihilation-data-only calculationwith its uncertainty of 4510−11 is used in ath, thenthe value of inferred from the BNL experimentallydetermined value of aexp, −1=137.035 670916.6 133 8710−7, differs from the h /m Cs−h /m Rbmean value of by 3.6udiff. Although such a large dis-crepancy may suggest “new physics,” the consensus isthat such a view is premature Davier, 2006.

Electrical and silicon crystal-related measurements.The previously discussed inconsistencies involving thewatt-balance determinations of K2

JRK, the mercury elec-trometer and voltage balance measurements of KJ, theXROI determinations of the 220 lattice spacing of vari-ous silicon crystals, the measurement of h /mnd220W04,and the measurement of VmSi hint at possible prob-lems with one or more of these these rather complexexperiments. This suggests that some of the many differ-ent measurement techniques required for their execu-tion may not be as well understood as is currently be-lieved.

Redefinition of the kilogram. There has been consider-able recent discussion about the possibility of the 24thGeneral Conference on Weights and Measures CGPM,which convenes in 2011, redefining the kilogram, am-pere, kelvin, and mole by linking these SI base units tofixed values of h, e, k, and NA, respectively Mills et al.,2006; Stock and Witt, 2006, in much the same way thatthe current definition of the meter is linked to a fixedvalue of c BIPM, 2006. Before such a definition of thekilogram can be accepted, h should be known with a ur

of a few parts in 108. It is therefore noteworthy that the2006 CODATA recommended value of h has ur=5.010−8 and the most accurate measured value of h the2007 NIST watt-balance result has ur=3.610−8.

C. Outlook and suggestions for future work

Because there is little redundancy among some of thekey input data, the 2006 CODATA set of recommendedvalues, like its 2002 and 1998 predecessors, does not reston as solid a foundation as one might wish. The con-stants , h, and R play a critical role in determiningmany other constants, yet the recommended value ofeach is determined by a severely limited number of in-put data. Moreover, some input data for the same quan-tity have uncertainties of considerably different magni-tudes and hence these data contribute to the finaladjustment with considerably different weights.

The input datum that primarily determines is the2006 experimental result for ae from Harvard Universitywith ur=6.510−10; the uncertainty ur=3710−10 of thenext most accurate experimental result for ae, that re-ported by the University of Washington in 1987, is 5.7times larger. Furthermore, there is only a single value ofthe eighth-order coefficient A8

1 , that due to Kinoshitaand Nio; it plays a critical role in the theoretical expres-sion for ae from which is obtained and requireslengthy QED calculations.

The 2007 NIST watt-balance result for K2JRK with ur

=3.610−8 is the primary input datum that determinesh, since the uncertainty of the next most accurate valueof K2

JRK, the NIST 1998 result, is 2.4 times larger. Fur-ther, the 2005 consensus value of VmSi disagrees withall three high accuracy measurements of K2

JRK currentlyavailable.

For R, the key input datum is the 1998 NIST valuebased on speed-of-sound measurements in argon using aspherical acoustic resonator with ur=1.710−6. The un-certainty of the next most accurate value, the 1979 NPLresult, also obtained from speed of sound measurementsin argon but using an acoustic interferometer, is 4.7times larger.

Lack of redundancy is, of course, not the only diffi-culty with the 2006 adjustment. An equally importantbut not fully independent issue is the several inconsis-tencies involving some of the electrical and siliconcrystal-related input data as already discussed, includingthe recently reported preliminary result for K2

JRK fromthe NPL watt balance given in Sec VII.D.1. There is alsothe issue of the recently corrected but still tentativevalue for the coefficient A8

1 in the theoretical expres-sion for ae given in Sec. V, which would directly effectthe recommended value of .

With these problems in mind, some of which impactthe possible redefinition of the kilogram, ampere, kelvin,and mole in terms of exact values of h, e, k, and NA in2011, we offer the following “wish list” for new work. Ifthese needs, some of which appeared in our similar 2002list, are successfully met, the key issues facing the preci-

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Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

sion measurement-fundamental constants and funda-mental metrology fields should be resolved. As a conse-quence, our knowledge of the values of the constants,together with the International System of Units SI,would be significantly advanced.

i A watt-balance determination of K2JRK from a

laboratory other than NIST or NPL with a ur fully com-petitive with ur=3.610−8, the uncertainty of the mostaccurate value currently available from NIST.

ii A timely completion of the current internationaleffort to determine NA with a ur of a few parts in 108

using highly enriched silicon crystals with x28Si0.999 85 Becker et al., 2006. This will require majoradvances in determining the 220 lattice spacing, den-sity, and molar mass of silicon.

iii A determination of R or Boltzmann constant k=R /NA with a u −6

r fully competitive with ur=1.710 ,the uncertainty of the most accurate value of R currentlyavailable, preferably using a method other than measur-ing the velocity of sound in argon.

iv An independent calculation of the eighth-ordercoefficient A8

1 in the QED theoretical expression for ae.v A determination of that is only weakly depen-

dent on QED theory with a value of ur fully competitivewith ur=7.010−10, the uncertainty of the most accuratevalue currently available as obtained from aeexp andaeth.

vi A determination of the Newtonian constant ofgravitation G with a ur fully competitive with ur=1.410−5, the uncertainty of the most accurate value of Gcurrently available.

vii A measurement of a transition frequency in hy-drogen or deuterium, other than the already well-knownhydrogen 1S1/2–2S1/2 frequency, with an uncertaintywithin an order of magnitude of the current uncertaintyof that frequency, ur=1.410−14, thereby providing animproved value of the Rydberg constant R.

viii Improved theory of the principal hadronic con-tribution to the theoretical expression for the muonmagnetic moment anomaly ath and improvements inthe experimental data underlying the calculation of thiscontribution so that the origin of the current disagree-ment between ath and aexp can be better under-stood.

ix Although there is no experimental or theoreticalevidence that KJ=2e /h and RK=h /e2 are not exact, im-proved calculable-capacitor measurements of RK andlow-field measurements of the gyromagnetic ratios ofthe shielded proton and shielded helion, which couldprovide further tests of the exactness of these relations,would not be unwelcome, nor would high accuracy re-sults ur10−8 from experiments to close the “quantumelectrical triangle” Gallop, 2005; Piquemal et al., 2007.

It will be most interesting to see what portion, if any,of this ambitious program of work is completed by the31 December 2010 closing date of the next CODATAadjustment of the values of the constants. Indeed, theprogress made, especially in meeting needs i–iii, maylikely determine whether the 24th CGPM, which con-

venes in October 2011, will approve new definitions ofthe kilogram, ampere, kelvin, and mole as discussed. Ifsuch new definitions are adopted, h, e, k, and NA as wellas a number of other fundamental constants, for ex-ample, KJ, RK assuming KJ=2e /h and RK=h /e2, R,and , would be exactly known, and many others wouldhave significantly reduced uncertainties. The resultwould be a significant advance in our knowledge of thevalues of the constants.

ACKNOWLEDGMENTS

We gratefully acknowledge the help of our many col-leagues who provided us with results prior to formalpublication and for promptly and patiently answeringour many questions about their work.

NOMENCLATURE

AMDC Atomic Mass Data Center, Centre de Spec-trométrie Nucléaire et de Spectrométrie deMasse CSNSM, Orsay, France

AME2003 2003 atomic mass evaluation of the AMDCArX Relative atomic mass of X:

ArX=mX /muA90 Conventional unit of electric current:

A90=V90/Ω90Å* Ångström-star: λWK1=0.209 010 0 Å*

ae Electron magnetic moment anomaly:ae= ge −2 /2

a Muon magnetic moment anomaly:a= g −2 /2

BIPM International Bureau of Weights and Mea-sures, Sèvres, France

BNL Brookhaven National Laboratory, Upton,New York, USA

CERN European Organization for Nuclear Re-search, Geneva, Switzerland

CIPM International Committee for Weights andMeasures

CODATA Committee on Data for Science and Tech-nology of the International Council for Sci-ence

CPT Combined charge conjugation, parity inver-sion, and time reversal

c Speed of light in vacuumcw Continuous waved Deuteron nucleus of deuterium D or 2Hd220 220 lattice spacing of an ideal crystal of

naturally occurring silicond220X 220 lattice spacing of crystal X of naturally

occurring siliconEb Binding energye Symbol for either member of the electron-

positron pair; when necessary, e− or e+ isused to indicate the electron or positron

e Elementary charge: absolute value of thecharge of the electron

722 Mohr, Taylor, and Newell: CODATA recommended values of the fundamental …

Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

F Faraday constant: F=NAeFCDC Fundamental Constants Data Center, NISTFSU Friedrich-Schiller University, Jena, Ger-

manyF90 F90= F /A90 AG Newtonian constant of gravitationg Local acceleration of free fallgd Deuteron g-factor: gd=d /Nge Electron g-factor: ge=2e /Bgp Proton g-factor: gp=2p /N

gp Shielded proton g-factor: gp=2p /Ngt Triton g-factor: gt=2t /NgXY g-factor of particle X in the ground 1S

state of hydrogenic atom Yg Muon g-factor: g=2 / e /2mGSI Gesellschaft für Schwerionenforschung,

Darmstadt, GermanyHD HD molecule bound state of hydrogen and

deuterium atomsHT HT molecule bound state of hydrogen and

tritium atomsh Helion nucleus of 3Heh Planck constant; =h /2Harvard; Harvard University, Cambridge, Massachu-

HarvU setts, USAILL Institut Max von Laue–Paul Langevin,

Grenoble, FranceIMGC Istituto di Metrologia “T. Colonetti,”

Torino, ItalyINRIM Istituto Nazionale di Ricerca Metrologica,

Torino, ItalyIRMM Institute for Reference Materials and Mea-

surements, Geel, BelgiumJINR Joint Institute for Nuclear Research,

Dubna, Russian FederationKRISS Korea Research Institute of Standards and

Science, Taedok Science Town, Republic ofKorea

KR/VN KRISS-VNIIM CollaborationKJ Josephson constant: KJ=2e /hKJ−90 Conventional value of the Josephson con-

stant KJ: KJ−90=483 597.9 GHz V−1

k Boltzmann constant: k=R /NALAMPF Clinton P. Anderson Meson Physics Facility

at Los Alamos National Laboratory, LosAlamos, New Mexico, USA

LKB Laboratoire Kastler-Brossel, Paris, FranceLK/SY LKB and SYRTE CollaborationLNE Laboratoire national de métrologie et

d’essais, Trappes, FranceMIT Massachusetts Institute of Technology,

Cambridge, Massachusetts, USAMPQ Max-Planck-Institut für Quantenoptik,

Garching, GermanyMSL Measurement Standards Laboratory, Lower

Hutt, New ZealandMX Molar mass of X: MX=ArXMuMu Muonium +e− atom

Mu Molar mass constant: M =10−3 kg mol−1u

mu Unified atomic mass constant:mu=m12C /12

mX, mX Mass of X for the electron e, proton p, andother elementary particles, the first symbolis used, i.e., me, mp, etc.

NA Avogadro constantN/P/I NMIJ-PTB-IRMM combined resultNIM National Institute of Metrology, Beijing,

China People’s Republic ofNIST National Institute of Standards and Tech-

nology, Gaithersburg, Maryland, USA andBoulder, Colorado, USA

NMI National Metrology Institute, Lindfield,Australia

NMIJ National Metrology Institute of Japan,Tsukuba, Japan

NMR Nuclear magnetic resonanceNPL National Physical Laboratory, Teddington,

UKNRLM National Research Laboratory of Metrol-

ogy, Tsukuba, Japann NeutronPRC People’s Republic of ChinaPTB Physikalisch-Technische Bundesanstalt,

Braunschweig and Berlin, Germanyp ProtonpAHe+ Antiprotonic helium AHe++p atom,

A=3 or 4QED Quantum electrodynamicsQ2 Probability that an observed value of chi-

square for degrees of freedom would ex-ceed 2

R Molar gas constantR Ratio of muon anomaly difference fre-

quency to free proton NMR frequencyRB Birge ratio: RB= 2 /1/2

Rd; Rd Bound-state rms charge radius of the deu-teron

RK von Klitzing constant: RK=h /e2

RK−90 Conventional value of the von Klitzing con-stant RK: RK−90=25 812.807

Rp; Rp Bound-state rms charge radius of the pro-ton

R Rydberg constant: R=mec2 /2hrxi ,xj Correlation coefficient of estimated values

xi and xj: rxi ,xj=uxi ,xj / uxiuxjri Normalized residual of xi: ri= xi− xi /uxi,

xi is the adjusted value of xirms Root mean squareSc Self-sensitivity coefficientSI Système international d’unités Interna-

tional System of UnitsStanford; Stanford University, Stanford, California,

StanfU USAStPtrsb St. Petersburg, Russian FederationSYRTE Systèmes de référence Temps Espace, Paris,

France

723Mohr, Taylor, and Newell: CODATA recommended values of the fundamental …

Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

T Thermodynamic temperaturet Triton nucleus of tritium T or 3Hth TheoryType A Uncertainty evaluation by the statistical

analysis of a series of observationsType B Uncertainty evaluation by means other

than the statistical analysis of a series of ob-servations

t90 Celsius temperature on the InternationalTemperature Scale of 1990 ITS-90

U.Sussex; University of Sussex, Sussex, UKUSus

UK United KingdomUSA United States of AmericaUWash University of Washington, Seattle, Wash-

ington, USAu Unified atomic mass unit also called the

dalton, Da: 1 u=mu=m12C /12uxi Standard uncertainty i.e., estimated stan-

dard deviation of an estimated value xi of aquantity Xi also simply u

uxi ,xj Covariance of estimated values xi and xjudiff Standard uncertainty of the difference xi

−xj: u2diff=u2xi+u2xj−2uxi ,xj

urxi Relative standard uncertainty of an esti-mated value xi of a quantity Xi:urxi=uxi / xi, xi0 also simply ur

urxi ,xj Relative covariance of estimated values xiand xj: urxi ,xj=uxi ,xj / xixj

VmSi Molar volume of naturally occurring siliconVNIIM D. I. Mendeleyev All-Russian Research In-

stitute for Metrology, St. Petersburg, Rus-sian Federation

V90 Conventional unit of voltage based on theJosephson effect and KJ−90:V90= KJ−90/KJ V

WGAC Working Group on the Avogadro Constantof the CIPM Consultative Committee forMass and Related Quantities CCM

W90 Conventional unit of power: W 290=V90/Ω90

XROI Combined x-ray and optical interferometerxuCuK1 Cu x unit: λCuK1=1537.400 xuCuK1xuMoK1 Mo x unit: λMoK1=707.831 xuMoK1xX Amount-of-substance fraction of XYAG Yttrium aluminium garnet; Y3Al5O12Yale; Yale University, New Haven, Connecticut,

YaleU USA Fine-structure constant: =e2 /40c

1/137 Alpha particle nucleus of 4HeΓX −90lo ΓX −90lo= X A90 A−1, X=p or hΓp−9 0hi Γp−9 0hi= p /A90 Ap Proton gyromagnetic ratio: p=2p /

p Shielded proton gyromagnetic ratio:p=2p /

h Shielded helion gyromagnetic ratio:h=2h /

Mu Muonium ground-state hyperfine splitting

e Additive correction to the theoretical ex-pression for the electron magnetic momentanomaly ae

Mu Additive correction to the theoretical ex-pression for the ground-state hyperfinesplitting of muonium Mu

p He Additive correction to the theoretical ex-pression for a particular transition fre-quency of antiprotonic helium

XnLj Additive correction to the theoretical ex-pression for an energy level of either hydro-gen H or deuterium D with quantum num-bers n, L, and j

Additive correction to the theoretical ex-pression for the muon magnetic momentanomaly a

0 Electric constant: 0=1/0c2

λXK1 Wavelength of K1 x-ray line of element Xλmeas Measured wavelength of the 2.2 MeV cap-

ture ray emitted in the reaction n+p→d+

Symbol for either member of the muon-antimuon pair; when necessary, − or + isused to indicate the negative muon or posi-tive muon

B Bohr magneton: B=e /2meN Nuclear magneton: N=e /2mpXY Magnetic moment of particle X in atom or

molecule Y0 Magnetic constant: 0=410−7 N/A2

X, X Magnetic moment, or shielded magneticmoment, of particle X

Degrees of freedom of a particular adjust-ment

fp Difference between muonium hyperfinesplitting Zeeman transition frequencies 34and 12 at a magnetic flux density B corre-sponding to the free proton NMR fre-quency fp

Stefan-Boltzmann constant:=25k4 /15h3c2

Symbol for either member of the tau-antitau pair; when necessary, − or + is usedto indicate the negative tau or positive tau

2 The statistic “chi square”Ω90 Conventional unit of resistance based on

the quanum Hall effect and RK−90:Ω90= RK/RK−90

Symbol used to relate an input datum to itsobservational equation

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