Neil S. Sullivan- NMR Studies of the Dynamics of Orientational Glasses

8/3/2019 Neil S. Sullivan- NMR Studies of the Dynamics of Orientational Glasses

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Vol. 18, No. 3/4 265NMR Studies of the Dynamics of Orientational Glasses1

Nei l S. SullivanDepartment of PhysicsUniversity of Florida21 5 Williamson HallGainesville, Florida 32611-8440

ContentsI. Introduction 265II . Orientational Order Parameters 266III. Stimulated Echo Studies of Low Frequency Dynamics 268IV. Low Energy Exc itations 269V. Conclusion 271V I. Acknowledgments 271VII. References 271

I. IntroductionThere are a number of simple well-definedmolecular systems for which the centers of massform a periodic lattice structure, but for which

the orientational degrees of freedom of the molec-ular units appear to be "frozen" at random withno long-range correlations (1). These orienta-tional glasses include solid ortho-para H2 mix-tures (2), N2-Ar mixtures (3) and KBri_xK(CN)xmixed crystals (4), and they continue to generateintense interest because of the apparently univer-sal low-temperature properties observed for avery diverse range of examples. Many of the prop-erties observed bear strong resemblance to thoseseen for the general family of spin glasses (5), (di-lute magnetic alloys, mixed crystals [EuxSri-^S (6),Rbi_x(NH4)a;H2PO4 (7), partially doped semicon-ductors (8), Josephson junction arrays (9), and oth-ers).The most striking universal features are an ap -parent freezing of the local degrees of freedom on

1 Based upon lectures presented at the XHIth InternationalSummer School held at the University of Waterloo, Canada,July 1991.

long time scales without any average periodic long-range order, characteristic slow relaxations, historydependence following external field perturbations(magnetic, electric, elastic-strain, etc.) and a verylarge number of stable low energy states (whichincrease monotonically with the number of activeatomic or molecular centers). The existence of atrue thermodynam ic phase transition has been ques-tioned, but a number of observations of the canon-ical spin glasses (such as CuMn) point to a ther-modynamic freezing of the spin degrees of freedomand the transition is not purely kinetic. Detailedstudies of the dynamics in these systems is crucialfor developing a better understanding of the funda-mental physics of these glassy systems, and the ori-entational glasses are particularly interesting as testcases because the dynamics can be probed in detailby techniques such as nuclear magnetic resonance.In th is review we will develop a unified quantita-tive treatment of the low temperature thermal be-havior and the explicit time dependence of the low-frequency relaxations in the orientational glass stateof solid hydrogen. This was the first orientationalglass to be discovered and serves as a prototype forthe other orientational systems.


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266 Bulletin of Magnetic ResonanceII. Orientational Order Parame-

tersThe order parameters needed to specify the de-grees of freedom of interacting quantum rotors suchas ortho-H2 are more complex than those needed todescribe the canonical spin glasses (10,11). For thespin glasses we consider only the dipole moments{S x)i, (Sy)i, (Sz)i of the spin 5 at the lattice sitei. For the quantum rotors with angular momen-tum J = 1, we need to specify the quadrupolar mo-ments {J aJp)i in addition to the vectorial compo-nents (J a)i- Only five of the quadrupole momentsare independent variables; a and /3 represent one ofthe Cartesian components (x, y, z) of the local ref-erence frame.In the absence of interactions that break time-reversal symmetry, the orbital angular momentum

is quenched in the molecular solids, i.e. the (J a)ivanish for all a. In order to determine the five pa-rameters needed to fully describe the molecular de-grees of freedom, we therefore select as local molec-ular reference axes (x, y, z) the principal axes of thequadrupolar tensor1Qaf3 = {-^ (1)

In this principal axes frame, Q xy = Q yz = Q zx = 0,and the only intrinsic quadrupolar order parametersare (12)(i) the alignment

i Qzz ( 1 tyand (ii) the eccentricity

m =

(2)

(3)The five order parameters are the three local ref-erence axes (giving the direction of the molecularorientations) and the two quadrupolar parametersa and 77 which give the degree of alignment withrespect to those axes.There are restrictions on the allowed values ofa and 77. If the single particle density matrices areexpressed in terms of a and 77, the strong positivitycondition that the eigenvalues of the density matrixbe positive-definite, leads to the conditions

(4)

This is discussed in detail in references (11) and(13).All of the experimental data has been analyzedassuming local axial symmetry, i.e., th at 77 = 0.This is not justified on theoretical grounds and theassumption is made for simplicity only. Experimen-tal tests for nonvanishing 77 have been proposed (11),but they have not been carried out.The NMR methods used by several groups tostudy ortho-para-H2 mixtures, N2/Ar mixtures andrelated systems, can provide direct measurementsof the local order parameters. The CW absorptionspec tra give the NM R second moments M2(T) whichprovide the equivalent of the Edwards-Andersonglass order parameter, and the nuclear spin-latticerelaxation rates T and spin-spin relaxation rates

T^1 provide information about the autocorrelationfunctions ((&i(t)


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Vol. 18, No. 3/4 267

(a) LONG-RANGEORDER

(b) GLASSFigure 1: Schematic representation of orientationaldegrees of freedom in (a) the quadrupolar glassphase of solid hydrogen for ortho contents less than55%, and (b) the long range ordered Pa 3 structurefor concentrations above 55%.zero, and each molecule contributes a doublet to theNMR spectrum with components at frequencies

3

The expectation value is evaluated with respect tothe magnetic field axis Z and not the local sym-metry axis z with respect to which the molecularorder param eters are defined. Effecting th e tran s-formation to the local molecular frame (x,y,z), wefind

3i = TD[-aiP2(cos 6i) + - ^ sin 26i cos (8)(ai, (pi) are the polar angles defining the orientationof the applied magnetic field with respect to the lo-cal molecular symmetry axes. With the assumptionof local axial symmetry (rji = 0) and a powder dis-tribution for the angles (9i,


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268 Bulletin of Magnetic Resonancetational degrees of freedom a cooperative processsimilar to that which is believed to occur for thecanonical spin glasses, or is it simply a local thermo-dynamic evolution determined by fixed local fieldswhich vary throughout the sample? It is importantto note that these quenched solid-molecular systemsdo not display broken symmetry. Solid H2 remainshep down to the lowest temperature studied for or-tho concentrations less than 55%, and the randomoccupation of lattice sites leads to the existence oflocal electric field gradients, the field conjugate tothe local order parameter, and this field plays thesame role as the magnetic field for the dipolar spinglasses. Th is random local field therefore makesthe problem of local orientational ordering in ran-dom mixtures equivalent to the local dipolar order-ing in spin glasses in the presence of random mag-netic fields, and thus any phase transition would besmeared out. The answer to these questions hasbeen provided by a systematic study of the low fre-quency dynamics of the glass phase which has pro-vided a unified quantitative explanation of the lowtemperature behavior of the heat capacities and theexplicit time dependence of the glass-like relaxationat low temperatures in terms of the low energy den-sity of states for the system.III. Stimulated Echo Studies of

Low Frequency DynamicsNuclear-spin-stimulated echoes have proved tobe very useful for the study of ultraslow molecu-lar motions both in molecular orientational glasses(14) and in the ordinary ("window pane") glasses(15). The stimu lated echoes are prepared by a three-pulse sequence. Two initial rf pulses separated bya short time r prepare the system with a "finger-print" of the local order parameters 0^. The localnuclear-spin tem pera tures following a 90y r 90ysequence depend on a, and vary as cos(DjCTir) where

Di = DP2(cos8i). The detailed calculations aregiven in reference (11). The single particle nuclearspin density matrix just after the first pulse is(9)

where LOL is the nuclear Larmor frequency. Only theterm involving Ix contributes to the NMR responsesand in the following we consider only the evolution

of the off-diagonal componenth, = - (10)

where we have used the irreducible tensorial opera-tors of eqn. 6 to obtain the last expression. After th efirst pulse, the evolution of the density matrix underthe influence of the intramolecular dipolar Hamilto-nian HOD leads to the production of both vectorialand second rank tensorial components of the nuclearmagnetization

Pi'(t = r) = - kRT [ (11)where

The second pulse applied at time r is used to convertthe transverse components NL, 1 into longitudinalcomponents NL^. If the "waiting" period tw, beforeapplying the third pulse is long compared to thetransverse relaxation time T2, but short comparedto the nuclear spin-lattice relaxation time T\, alltransverse components Nitl die away before theapplication of the third pulse and one is left withonly the secular components N^Q. If we considerthe general case of a second pulse which rotates thespins through an angle \& about an axis in the x yplane making an angle with the y-axis, the secularcomponent of the density matrix after the second rfpulse is

Pio(t = (12)

' j s in sm(Di


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Vol. 18, No. 3/4 269changed during tw, the evolution during a time r af-ter tw will match that during the preparation time,and the spin components will focus to form a stim-ulated echo at time t Tw + 2r. If the parameterschange, however, the echo is attenuated, and thisdamping of the echo amplitude is used to follow thechanges in the order parameters during the waitingperiod.The stimulated echo formed from the secularcomponents of the density matrix is called the pri-mary stimulated echo. If nonsecular components re-main during tw, i.e. if tw < T2, additional echoescan be formed and these are called secondary stim-ulated echoes (11).If the order parameters during the preparationperiod are designated by


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270 Bulletin of Magnetic Resonancea connected cluster. We are interested in the low-frequency or long length-scale excitations which in-volve the coherent collective reorientations of a largenumber of connected molecules. The basic assump-tion of Fisher and Huse (18) that the density of ex-cited states for a cluster of size L scales as L~e with0 < 0 < ~2 allows one to consider the thermallyexcited clusters as dilute, and therefore regarded as"non-interacting two level systems". If the free en-ergy barriers for the formation of the clusters is EB,the tunneling rate for a characteristic cluster at tem-perature T is

(17)where To is the characteristic attempt frequency.

In the long-time limit, To is reasonably well de-fined because it is associated with a characteristiccluster dimension. The dynamics of the clustersmust be hierarchical because the low energy bar-riers (for small clusters) must be overcome beforethe larger ones can be attempted. In this model,therefore, the dynamics associated with the fasterdegrees of freedom constrain the slower degrees offreedom. In a time t, the only barriers crossed willbe those for which 0 < EB < Ema x(t), where fromeqn. 17

) (18)For all barriers crossed in time t there will be sig-nificant changes in the local order parameters, andfor the molecules in the clusters participating in theexcitation, the average correlation functions

cos[ZVi(0)r] cos[Dtai(tw)T})}will vanish. The contribution to the stimulatedechoes from these molecules will therefore also van-ish, while the contribution from molecules withlarger energy barriers will still contribute to theecho. Summing over the density of low energy exci-tations, P(EB), we therefore find for the echo am-plitude at time t

tt'max P(EB)dEB] (19)At very low temperatures, we need only consider

the density of states near E = 0. The density ofstates can be determined from an analysis of theheat capacity measurements at low temperatures.

< 0.2 -

Ia

o//

1-{

~ ^ \

(b)

0. 5

1

\

-

1

\

\

/^yE(K)

-

\

2

-

TEMPERATURE (K )Figure 4: Observed heat capacities in the ori-entational glass phase of solid H2. [diamondsRef.(19,20); squares Ref. (21) line (a) is for a flatdistribution P(E) of energy levels, and line (b) is fora P(E) adjusted to fit the high temperature data.The inset shows the two corresponding distributionsfor P{E).As in almost all glass-like systems, P(E) is approx-imately constant at low energy, and the echo ampli-tude at time t is therefore

Sx(t) = kBT[l-kBTP(O)\n(t/To) (20)where To = (TT/DTTO). The slopes of logarithmicdecays shown in Figure 3 are therefore directly pro-portional to the density of low energy excitations.The observed decay shown in Figure 3 for an orthofraction x = 0.54 at 0.22 K corresponds to a densityof states P(0) = 0.59 0.05 KT1

In order to test this interpretation, we have com-pared the density of states P(0) inferred from thestimulated echo decay with that deduced from heatcapacity measurements in the orientational glassphase of solid hydrogen. The experimental capac-ities are shown in Figure 4. Lines (a) and (b) inFigure 4 are the calculated heat capacities assum-ing a simple two level system in which each ortho-H2molecules of angular momentum has energy statesJQ = 1 separated from the state J^ = 0 by an en-ergy gap A;. (Q is the local axis of symmetry of themolecules for which |CTJ| is maximum.) Line (a) is fora hypothetical flat distribution, P{E) = constant,and line (b) is a numerical fit to the experimentaldata. The density of states at E ~ 0 deduced fromthe heat capacity data is P(0) = 0.55 0.08 KT1


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Vol. 18, No. 3/4 2714000

2000

1000

9 400 -

0. 0 0.4 0.6 0.8TEMPERATURE (1/K)Figure 5: Temperature dependence of the nuclearspin-lattice relaxation rate in the orientational glassphase of solid H2. [triangle, Ref. (22); diamond,Ref.(16)]

and this is in excellent agreement with the valueP(0) = 0.59 0.05 K~ l inferred from the NMRdata. The consistency of the analysis of the NMRresults with the thermodynamic data confirms theview that the total density of excited states at lowenergies is dominated by the large-scale excitations.An additional test of the model is provided bythe observed temperature dependence of the nuclearspin-lattice relaxation times in the glass phase. Themost effective relaxation process at very low tem-peratures is a Raman process involving the simulta-neous excitation and de-excitation from two excitedstates which differ by the nuclear Zeeman energy.This relaxation can be calculated using the densityof states estimated above from the thermodynamicanalysis and the NMR results, and the compari-son with the observed rates is shown in Figure 5.The calculations are in good qualitative agreementwith the measurements, and it should be noted thatother relaxation mechanisms that have been pro-posed have failed to describe the temperature de-pendencies even semi-quantitatively (22).

V. ConclusionThe excellent agreement between the predictedNMR stimulated echo decays and t he observed vari-ation shows that the low frequency dynamics inthe orientational glass state of solid H2 can be de-scribed in terms of low-energy large-scale cluster ex-

citations. The available density of states for theseexcitations is approximately constant at very lowtemperatures and is consistent with thermodynamicmeasurements. It will be imp ortant to test thismodel by carrying out systematic studies of relatedglass- like systems where the freezing of the degreesof freedom is also determined by frustrated short-range interactions in random mixtures.V I. Acknowledgments

This work has been supported by the NationalScience FoundationLow Temperature Physicscontracts # DMR-9623536 and #DMR-9216785.V I I . References

xFor recent reviews, see "Dynamical Aspects ofStructured Changes in Liquid Glasses," edited byC.A. Angell and M. Goldstein [Ann. N. Y. Acad. ofSci. 484, 130 (1986)], and "Spin Glasses and OtherFrustrated Systems," by D. Chowdhury [PrincetonUniv. Press, Princeton, NJ, 1986].2N.S. Sullivan, M. Devoret, B.P. Cowan and C.Urbina, Phys. Rev. B 17, 5016 (1978).3W. Press, B. Janik and H. Grimm, Z. Phys. B49, 9 (1982); D. Esteve and N.S. Sullivan, Jour.Phys. Lett. 43, L793 (1982).4J.J. De Yores, M. Meissner, R.O. Pohl, J.M.Rowe, J.J. Rush and S. Susman, Phys. Rev. Lett.51 , 1050 (1983); M. Meissner, W. Knaak, J.P.Sethna, K.S. Chow, J.J. De Yores and R.O. Pohl,Phys. Rev. B32, 6091 (1985).5For a review see D. Sherrington in "Proceedingsof the Heidelberg Colloquium on Spin Glasses," p.348.6H. Maletta and W. Felsh, Phys. Rev. B 20,1245 (1979).7E. Courtens, Jour. Phys. Lett. 43, L199(1982).8M . Grunewald, B . Pohlmann, L. Schweitzer andD. Wuertz, J. Phys. C15, L1153 (1982).9C. Ebner and D. Stroud, Phys. Rev. B 31,165 (1985); S. John and T.C. Lubensky, Phys. Rev.Lett. 55 1014 (1985).10A.B. Harris and H. Meyer, Can. J. Phys. 63,3 (1985); erratum, 1986.n Y . Lin and N.S. Sullivan, Mol. Crystals 142,141 (1987).


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27212 N.S. Sullivan, C M . Edw ards, Y. Lin and D.

Zhou, Can. J. Phys. 65, 1463 (1987).1 3C.M. Edwards, D. Zhou, Y. Lin and N.S. Sul-livan, J. Low Temp. Phys. 72, 1 (1988).

14 N.S. Sullivan, D. Esteve and M. Devoret, J.Phys. C 15, 4895 (1982).1 5M . Engelsberg and N.M. Borges, J. Phys. C17, 3633 (1984).1 6I . Yu, S. Washburn, M. Calkins and H. Meyer,J. Low Temp. Phys. 51, 401 (1983).

1 7D.G. Haase, L.R. Perrell and A.M. Saleh, J.Low Temp. Phys. 55, 283 (1984).

18 D.S . Fisher and D.A. Huse, Phys. Rev. Lett.56 , 1601 (1986).1 9B.W.A. Ricketson, Thesis, Oxford University,1956, unpublished.2 0H . Meyer and S. Washburn, J. Low Temp.Phys. 57, 55 (1984).2 1D.G. Haase, L.R. Perrell and A.M. Saleh, J.Low Temp. Phys. 55, 283 (1984).2 2N.S . Sullivan and R.V. Pound, Phys. Rev. A6, 1102 (1972).

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Neil S. Sullivan- NMR Studies of the Dynamics of Orientational Glasses

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