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1. Introduction

This translation was undertaken to make accessibleto readers of English a foundational paper on the theo-ry of spectral line shapes, Sullo spettro di assorbimen-to dei gas nobili presso il limite dello spettro darco,U. Fano,Nuovo Cimento,N. S. 12, 154-161 (1935).

The results of this paper are widely known via a sub-sequent publication by the same author in the PhysicalReview in 1961. [1] The 1961 paper has been citedmore than 4500 times, and it was judged to be among

the most influential papers published in the history ofThe Physical Review journal series, according to arecent study that examines both numbers and timeseries patterns of citations. [2] It is, by a considerablemargin, the most-cited paper that has been publishedunder the byline of the National Bureau ofStandards/National Institute of Standards andTechnology (NBS/NIST). [3]

The 1935 paper translated here lacks the generalityof the 1961 paper, but its results are identical for an

important limiting case, and it should be considered tobe the first paper which correctly elucidates the gener-al form of line shapes encountered in the excitation ofmany important atomic and condensed-matter systems.In particular, it treats the case in which a discrete statecoexists in the same energy region as a continuum ofstates, and accounts for the interaction between the dis-crete and continuum states, and the interference

between their separate excitation amplitudes. The keyline-shape formula derived in the 1935 paper is identi-cal in a practical sense to that of the 1961 paper, which

is now famous as the Fano profile: it does not includea shift in the discrete-state energy due to its interactionwith the continuum (as does the 1961 paper), but this isnot a direct observable.

In addition to its historic interest, the 1935 paperpresents its subject in a remarkably clear way, no doubtreflecting the influence of Enrico Fermi, who was UgoFanos supervisor at the time. It does not use the some-what formidable mathematical apparatus of the 1961

paper, and it offers insights which may still seem fresheven to those familiar with the subject matter (for

Volume 110, Number 6, November-December 2005

Journal of Research of the National Institute of Standards and Technology

583

[J. Res. Natl. Inst. Stand. Technol. 110, 583-587 (2005)]

On the Absorption Spectrum of Noble Gases atthe Arc Spectrum Limit

Volume 110 Number 6 November-December 2005

Ugo Fano1, Guido Pupillo,

Alberto Zannoni, and Charles

W. Clark

National Institute of Standardsand Technology,Gaithersburg, MD 20899-0001

Rydberg spectral lines of an atom aresometimes superimposed on the continu-ous spectrum of a different configuration.Effects of interaction among different con-figurations in one of these cases are theo-retically investigated, and a formula isobtained that describes the behavior ofabsorption spectrum intensity. This offersqualitative justification of some experi-mental results obtained by BEUTLER instudies of absorption arc spectra of noblegases andIb spectra of some metal vapors.

Key words: absorption spectrum of noblegases; Fano resonance.

Accepted: October 11, 2005

Available online: http://www.nist.gov/jres

1 Deceased

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example, Fanos observation of how a discrete statewith zero excitation amplitude can cause the total exci-tation probability to vanish at its own energy).

Note on the text: the original publication does notidentify equations by number. Equation numbers have

been added in the translation for readers convenience.The editors are grateful to the Societa Italiana di

Fisica for permission to publish this translation, and toMs. Susan Makar, of the NIST Research Library of the

National Institute of Standards and Technology, formuch helpful assistance.

2. Text of the Translation[Nuovo Cimento 12, 154-161 (1935)]

On the Absorption Spectrum of NobleGases at the Arc Spectrum Limit

Ugo Fano

Rydberg spectral lines of an atom are sometimes superimposed onthe continuous spectrum of a different configuration. Effects ofinteraction among different configurations in one of these cases aretheoretically investigated, and a formula is obtained that describesthe behavior of absorption spectrum intensity. This offers qualitative

justification of some experimental results obtained by BEUTLER instudies of absorption arc spectra of noble gases and Ib spectra ofsome metal vapors.

It is experimentally known that arc spectrum seriesof noble gases do not converge toward a single limit,

but toward two distinct limits. The explanation is thatremoval of the optical electron from a noble gas atomyields an ion whose ground configuration does not con-sist of a single level, but rather a doublet 2Po3/2,

2Po1/2. Theinterval between the doublets levels is about 1500wave numbers for A, 5000 for Kr, and 10000 for Xe.Within this interval, two different kinds of arc spectrumterms can occur: a) continuous spectrum terms repre-sented by the formula (p5)3/2 +free electron; b) discretespectrum terms represented by the (jj coupling) formu-la (p5)1/2nl; the latter belong to series that converge

toward the2

Po

1/2 limit. In a recent work ([1]) BEUTLERinvestigated absorption spectra of noble gases, obtain-ing the following results. At very low pressure of thenoble gas (0.002 mm), continuous absorption with aregular behavior is observed for frequencies greaterthan the 2Po1/2 limit, and also continuous absorption withcharacteristic intensity modulations is observed

between the 2Po3/2 and2Po1/2 limits. Absorption between

the two limits shows maxima that can be classified intotwo groups: a more peaked, and a much less peakedone; positions belonging to each of these two groups

are Rydberg series that converge to the 2Po1/2 limit. Withincreasing noble gas pressure, absorption peaks grow inintensity and width until they overlap. At the pressure

of 0.030 mm the absorption is already continuous andhomogeneous, starting from the 2Po3/2 limit. In any caseabsorption due to energy levels below the 2Po3/2 limit issmaller in magnitude than absorption above the samelimit. The intensity distribution in the absorption spec-trum is shown by BEUTLER in a graph whose charac-teristic appearance is reproduced in Figure 1. He inter-

prets single maxima as lines of the discrete spectrum,much broadened due to the large probability of self-ionization (AUGER effect) (p5)1/2nl (p

5)3/2 +freeelectron. Wide maxima are assigned to the series(p5)1/2nd, and narrow maxima to the series (p

5)1/2ns. Theaim of the present work is to show how it is possible to

justify such an intensity distribution in a qualitativeway, by supposing that positions of discrete terms donot correspond to absorption maxima, but to pointslocated along the steep parts of the curve, which aretherefore slightly shifted with respect to the former.

The intensity distribution in the part of the spectrumof interest is obtained by evaluating the squares ofdipole matrix elements referring to transitions from theground state to states whose energy lies between the2Po3/2 and

2Po1/2 levels of the ion. If we performed this cal-culation starting with zeroth-order eigenfunctions, cor-responding to single electronic configurations, wewould find that absorption is due to superposition of acontinuum of almost constant intensity with lines

belonging to series that converge to the 2Po1/2 limit.In order to obtain a result in agreement with

BEUTLERs experimental data we should start insteadwith better approximate eigenfunctions, which takeinto account interaction between different configura-tions. Ordinarily, eigenfunctions of this type areobtained via perturbation theory; this method is notapplicable to our case, as we deal with states belongingto the continuum whose energies are infinitely close to,and also coincident with, the energies of the discretestates. We therefore have to abandon perturbation

theory and look directly for eigenfunctions of theSCHRDINGER equation. We may assume as a firstapproximation that states which are not close to each

Volume 110, Number 6, November-December 2005

Journal of Research of the National Institute of Standards and Technology

584

Figure 1:(From Zeitschift fr Physik, 93, 181, 1935)

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other do not appreciably interact. Therefore, absorptionin proximity of the position of a particular discrete termis obtained considering the interaction only between the

term itself and the continuum.With the problem so outlined, it will be convenient to

treat the atom as imbedded in a sphere of very largeradiusR, in order to simplify the treatment of the con-tinuum. The continuous spectrum is then replaced by adiscrete one. The interval between two consecutiveeigenvalues is almost constant for small energy varia-tions, and is inversely proportional toR. The correspon-ding eigenfunctions contain a factor related to the freeelectron, which far from the atom takes the form

(1)

so that the normalization coefficient in theR limitis proportional to 1/R

. It follows that if we letR go to

we have to express a factor of

in the normaliza-tion coefficient. By invoking the fact that states withvery different energies interact weakly, we can finallyconsider the spectrum of the (p5)3/2 +free electron con-figuration to be produced by the succession of eigen-values

(2)

where the energy of the discrete state under considera-tion is defined as the zero of energy.

Let be the zeroth-order eigenfunction correspon-ding to the discrete term, and n the one correspondingto the eigenvalueEn. From the above hypothesis it fol-lows that a perturbed eigenfunction whose energy isclose to the discrete term must have the form:

(3)

Let V be the interaction between electrons (which ismainly electrostatic), and define the first-order approx-

imate energy as the energy associated with a given con-figuration (the sum of the eigenvalue of the equationfor independent electrons, the exchange energy, and thediagonal term ofV). The SCHRDINGER equation for is thus decomposed into the infinite system of equa-tions:

(4)

where Vn = (|V|n) is supposed to be real, for the sakeof simplicity.

Let us now introduce a new hypothesis, that is, n isindependent of n at distances from the origin of theorder of the atomic radius. It follows that Vn = q is con-

stant, and the last equation reads:

(5)

while the other equations give:

(6)

Substituting, we obtain:

(7)

which determines the eigenvalues. In order to find b,

we impose the following normalization condition(where dv is the element of volume in configurationspace):

(8)

therefore

(9)

Let us considerXn = (u|x|n) =Xc to be independentof n, where u is the ground state eigenfunction, and

X0 = (u|x|); the square of the x-component of thedipole matrix element is, taking (*) into account:

(10)

Having obtained this result, we have to take the limitR . Matrix elementsX,Xc and q contain a factor of

, because of their definition; it is therefore conven-ient to set X=X

, Xc =X

c

, q = q

. Actually

the quantity we are interested in is X2, since(X2 + Y2 +Z2)dEdetermines the transition probabilityfrom the ground state to a state of energy in the rangedE. Therefore, we have:

Volume 110, Number 6, November-December 2005

Journal of Research of the National Institute of Standards and Technology

585

2

2

1 8sin ( )

2

mW r f

h

+

, ( )n E n n= < <

n n

n

a b

=

= +

,n n n n n n Ea E a bV Eb a V = + =

n

Eba

q=

n

n

bqa

E E=

22 21 1(*) cot

n

q E E q q

E E E n

= = =

22 2 2 2

2

2 2 22 2

2 2

2 2 2 22

2 4 2

1 | | 1( )

11 1

sin

1 1

n

n

En

qdv a b b

E E

q qb b

E E E

q Eb

q

= = + = +

= = +

= + +

2 2 2

2 21

n

q

nE E

q E

q

+ =

+ +

{ } { }2 2 2 4 22

2 2 2 2 2

2 2

0 02

1 1

qEc cq E

q q qE

q E E

X X X X X

+ += =

+ + + +

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(11)

and taking the limit 0

(12)

The same formulae hold for Y2 and Z2. SettingD = (X, Y,Z), Dc = (X

c , Y

c ,Z

c), D0 = (X0, Y0,Z0), we

obtain:

(13)

In a small enough range of frequencies, this quantitycan be regarded as proportional to the absorption inten-sity.

In order to discuss this formula, it is convenient toexamine the three-term expansion. The first two termshave a well defined physical meaning, since |D|2

reduces to them ifD0 orDc goes to zero, respectively.Therefore we observe that if the dipole matrix elementassociated with the continuous spectrum is zero, weobtain as the absorption spectrum a line which is broad-ened due to the AUGER effect, and whose width isgiven by = , as expected. If instead the matrix ele-ment associated with the discrete spectrum term is zero,the term itself affects absorption by the continuum, inthat the latter vanishes at the position of the former. Thethird term is truly distinctive, as it results in net absorp-tion being not simply obtained as a superposition ofabsorptions due to discrete and continuum terms, albeitmutually influencing each other; this term represents ashift of absorption intensity, or, in other words, itdiminishes the intensity on one side of the discrete

terms position, and increases it by the same amount onthe other side.In Figure 2, |D|2 is depicted as a function ofE for

some values of |D0|, |Dc|,D0 Dc, q(the parameters thatdetermine the curve); values have been chosen to showthat the curve itself can have a behavior that justifiestheoretically the results obtained by Beutler.Characteristic features of the curve that readily resultfrom the discussion of its equation are: a) the curvegoes asymptotically to |Dc|

2 forE ; b) the ordinateof the intersection of the curve with the E= 0 axis

depends only on |D0| and q, as it is equal to ; c) thecurve has a maximum and a minimum on oppositesides of theE= 0 axis; in particular, ifDc is parallel to

D0, the minimum is equal to zero; d) the difference

between the abscissa of the maximum and of the mini-mum is of the order ofq2 , where is the lifetimeof the discrete term with respect to the AUGER effect(estimating as 1014 sec, one gets q2/hc 500 wavenumbers).

Obviously, due to the simplifying assumptions thatwe used, the result obtained has merely a qualitativevalue, which is to show the behavior of the curve.

That the derived formula fails to fulfill the sum ruleis to be attributed to the hypothesis adopted, since itindeed should yield:

(14)

In fact, we assumed the presence of a continuousspectrum of infinite extent, with |Dc| constant, which is

physical nonsense as it would result in an infinite num-ber of dispersion electrons. This incorrectness is partic-ularly evident in the limiting case D0 = 0, where itappears that the number of dispersion electrons of thecontinuum is reduced by a factor of in the vicin-ity of the perturbing discrete term, without being corre-spondingly increased in other parts of the spectrum, sothat the total sum of dispersion electrons does notchange.

A trial calculation has shown that the derived formu-la is not even susceptible to a rough numerical evalua-tion, due to the large number of electrons that must beincluded for noble gases, and the poor approximationachievable in evaluating individual integrals.

Application of the obtained formula to line broaden-

ing phenomena in Ib absorption spectra. - Ib spectra,obtained by excitation of an electron belonging to theoutermost closed shell, have been studied by BEUT-LER in a series of important works ([2]). Superpositionof discrete terms ofIb spectra upon continuum terms of

Volume 110, Number 6, November-December 2005

Journal of Research of the National Institute of Standards and Technology

586

4 2 4 2

2 2

2 2 2 22 0 0

2 4 2

02 4 2

| | | | | || |

1 1

2

q

c cE

q q

E E

c

D D D q DD

E q

E qD D

E q

+= = +

+ + +

+ +

{ }2 4 2

2 2

2

02

1

q

c E

q q

E E

X XX

+=

+ +

{ }4 2

2

2

02

1

q

c E

q

E

X XX

+=

+

Figure 2:|Dc| = 2; |D0| = 4.2; q

= 0.6;Dc D0 = 6; = 5 arbitrary units2

02 2

| |D

q

2 2 2

0lim{ | | 2 | | | | } 0.

U

cUU D dE U D D

=22 q

h

20

2 2

| |D

q

2

h

4 2

1

1 qE

+

|D|2

E

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ordinary arc spectra gives rise to the same situation thatoccurs in noble gases spectra between the two limits ofthe arc spectrum. Nevertheless, the phenomena look

different, since, up to the present, in known casesabsorption due to the continuum: a) is much lessintense than absorption due to discrete terms ofIb spec-tra, and b) decreases very rapidly in intensity withincreasing frequency.

BEUTLER observed that someIb series have brightand narrow lines superimposed upon the continuousspectrum. He explains this phenomenon by showingthat interaction among terms belonging to these seriesand to the continuum vanishes. On the other hand, otherseries have diffuse lines, which he describes as asym-metrically broadened. In every diffuse series, lines tendto become symmetric again and to narrow as frequency

increases, as the intensity of the continuum on whichthey are superimposed decreases.

In Figure 3 two different graphs of |D|2 as a functionofEare shown; curve 1 is obtained taking a value of|Dc|

2 small compared to , curve 2 is obtained for thesame values of |D0| and q, and withDc = 0. Line shape

peculiarities in diffuse series ofIb spectra can thereforebe explained, since the data relative to lower frequencylines are those utilized to obtain Figure 1, while byincreasing frequency we get closer to conditions corre-sponding to Figure 2. Narrowing of the lines withincreasing frequency is probably due to the fact that theq interaction tends in general to decrease with increas-ing total quantum number of the corresponding discreteterm.

I want to deeply thank Prof. FERMI who guided andhelped me throughout this work. Rome, Istituto di

Fisica della R. Universita, February 1935-XIII.

References

[1] BEUTLER, Zs. fr Physik, 93, 177, 1935.[2] Zs. fr Physik, 86,495,710; 87,19,176,187; 88,25; 91,143,

202,218.

3. References to the Introductory Text

[1] Effects of Configuration Interaction on Intensities and PhaseShifts, U. Fano, Phys. Rev. 124, 1866-1878 (1961).

[2] Citation Statistics From More Than a Century of Physical

Review, S. Redner, physics/0407137 (2004).[3] Effects of Configuration Interaction on Intensities and PhaseShifts, C. W. Clark, in ACentury of Excellence in Measurements,Standards, and Technology; A Chronicle of Selected Publicationsof NBS/NIST, 1901-2000, ed. David R. Lide, National Instituteof Standards and Technology SP 958 (U.S. Government PrintingOffice, Washington, DC, 2001), pp. 116-119. Online:http://nvl.nist.gov/pub/nistpubs/sp958-lide/116-119.pdf.

About the authors and translators: Ugo Fano (1912-

2001) was a physicist at the National Bureau of

Standards (NBS), now NIST, from 1946 to 1966, and

subsequently a professor of physics at the University of

Chicago. He is the author of the most-cited paper writ-

ten by an NBS/NIST scientist, of which the present

paper is a predecessor. Guido Pupillo was a Guest

Researcher at NIST from 2001 to 2005, while complet-

ing his Ph.D. degree in physics at the University of

Maryland, College Park. He is currently a postdoctor-

al research associate at the University of Innsbruck,

Austria. Alberto Zannoni was a Guest Researcher at

NIST from 1999 to 2000. He is currently the proprietor

of an internet services company in Ravenna, Italy.

Charles W. Clark has been a physicist at NBS/NIST

since 1981, and is currently Chief of the Electron and

Optical Physics Division of the NIST Physics

Laboratory. Ugo Fano was the supervisor of his Ph.D.thesis at the University of Chicago. The National

Institute of Standards and Technology is an agency of

the Technology Administration, U.S. Department of

Commerce.

Volume 110, Number 6, November-December 2005

Journal of Research of the National Institute of Standards and Technology

587

Figure 3:Curve 1: |Dc| = 1; |D0| = 4.2; q

= 0.6;Dc D0 = 3; =5

Curve 2: |Dc| = 0; |D0| = 4.2; q = 0.6;Dc D0 = 0; =5

arbitrary units

20

2 2

| |D

q 2

02 2

| |D

q

20

2 2

| |D

q

|D|2

E

1

2