Microscopic approach to the quadrupolar-glass problem

Valery B. Kokshenev*Departamento de Fisica, ICEB, Universidade Federal de Ouro Preto, Minas Gerais, Brazil

and Departamento de Fisica, ICEX, Universidade Federal de Minas Gerais, C.P. 702, CEP 30161-970, Belo Horizonte,Minas Gerais, Brazil

~Received 22 November 1993; revised manuscript received 2 November 1995!

We report results of the microscopic approach to the quadrupolar-glass problem in mixed molecular crystalsformed by linear and spherical molecules. The temperature-concentration analysis of the orientational orderparameter derived from NMR spectroscopy data is given on the two-component vector model of site-dilutedortho-para-hydrogen-type mixtures. Two features of the short-range orientational ordering, characteristicallydifferent from the spin-glass case, namely, incomplete freezing and residual ordering, respectively, at low andhigh temperatures are discussed. It is established that the first effect is due to spatial correlated local molecularand reaction fields and the second caused by quadrupolar intrinsic field. A zero-field freezing temperature isestimated taking into account the threshold concentration for quadrupolar glass formation.

Ortho-hydrogen~or para-deuterium! molecules possess aunit angular momentum,J51, which remains a good quan-tum number even in solid state and thus can be treated asquantum rotators or pseudospins.1 In ortho-para hydrogenmixtures (o-pH2) spherical symmetric para-hydrogen mol-ecules play a role of almost perfect dilutants2 and provide astriking effect of random substitution. Molecular hydrogens~H2, D2 , and HD! being the simplest of all molecular solidsgive a rich testing ground for the investigations of randomsubstitution and frustration effects on cooperative phenom-ena in different materials.3,4 The work of Sullivanet al.5

stimulated active experimental,3,4,6 theoretical,3,6–10 andMonte Carlo simulation11,12 investigations of orientationallydisordered hydrogens in the temperature region of short-range-order frozen states originally named by the quadrupo-lar glass~QG!. The QG orientational order parameter, as adirect analog of the Edwards-Anderson spin-glass~SG! mag-netic order parameter,13 can be derived from the second mo-ment of the NMR absorption signal line shape.3 Its tempera-ture behavior indicates some characteristic features, whichmake it possible to distinguish between two types of glasses.First, at low temperatures the QG order parameter, observedin a wide concentration region, being extrapolated to zerotemperature is far from its maximum value equal tounity,14,15 which is assumable16 due to zero-point motion oflinear quadrupoles. On the other hand, such a kind of unsat-urated orientational order effect of the same order in magni-tude also occurs in Ar-N2 mixtures

17,18and thus is specific toboth quantum and classical QG’s. Secondly, even in the ab-sence of external fields the orientational order parameter isdifferent from zero at high enough temperatures.14 This factpoints to the existence of some intrinsic field conjugate to thelocal parameter order3 and characteristic of QG’s. The mainobject of this paper is to draw a fundamental distinction be-tween quadrupolar orientational and dipolar spin glasses re-lated to the short-range-order freezing of suitable degrees offreedom.

The pseudospin reformulation of intermolecular interac-tion originally termed in tensorial quadrupolar dynamicvariables19 has been given for classical20 and quantum21 QG

systems. In the latter case the QG Hamiltonian for the(J51)c(J50)12c system can be presented in the followingform:

H52 (f. f 8

cfcf 8 (m,m8522

2

Jf fmm8Sm fSm8 f 8

2(fcf(

mHm f

~ Cr!Sm f . ~1a!

The orientational degrees of freedom of a given quadrupole(J51) are described by the set of five spherical componentsof the quadrupolar moment tensor given in the local spheri-cal coordinate system, quantization axes of which coincidewith the principal-axis frame of the quadrupolar moment ten-sor ~for details see Refs. 20–26!. Thermal expectation valuesof the dynamic variables are the two-component QG localorder parameter:3,27 ^Sm f&T5dm0s f1dm2h f . An applicationof microscopic statistical consideration to distinguish be-tween random-bond and random-site effects — both are dueto random-substitutional dilution—makes it possible to

present the exchange interactionJf f 8mm8 and crystalline field

Hm f(Cr) given in ~1! in explicit form @see, respectively, Eqs~6!

and ~7! in Ref. 21 and Appendix in Ref. 22#. Their nonzeroexpectation values are

^Jf f 8mm8Jf f 9

nn8&B5G2d f 8 f 9dmm8dnn8;

^Hm f~Cr! Hm8 f 8

~Cr! &B 5 hCr2 d f f 8dmm8, ~1b!

where G is electrostatic quadrupole-quadrupole~EQQ!nearest-neighbor coupling constant.28 The random-bond av-erage, labeled byB, includes the uniform integration overrandom local-axis directions and the hcp-lattice summationover intermolecular directions, i.e.^•••&B5^^•••&L&R . Theconfigurational average, labeled byC, in turn additionallyincludes the average over the random-site variables(^cf&s5c,^cfcf 8&s5cd f f 81c2(12d f f 8), c is concentrationof quadrupoles!, i.e., ^•••&C5^^•••&B&S . In the frameworkof the two-component (m50,2→m,s) vector model the lo-cal order parameters are21

PHYSICAL REVIEW B 1 FEBRUARY 1996-IVOLUME 53, NUMBER 5

530163-1829/96/53~5!/2191~4!/$06.00 2191 © 1996 The American Physical Society

s f5123cosh@~A3/2!«h f /T#

2cosh@~A3/2!«h f /T#1exp~ 32«s f /T!

,

h f5A3sinh@~A3/2!«h f /T#

2cosh@~A3/2!«h f /T#1exp~ 32«s f /T!

. ~2!

The configurational average can be performed by means ofthe effective two-component vector local fielde f , which inturn can be estimated within various mean-field~MF! typeapproximations. We will calculate it on the base of the MF-type Hamiltonian

HMF52(fcf (

m5s,h~em f1Hm f

~Cr!!Sm f12hm f~Sm f2sm f !2,

~3a!

following from ~1! taking into account thermal and spatialfluctuations through the trial molecularem f and fluctuationhm f fields. The desired effective molecular field conjugate tothe local order parameterssm f ~2! can be represented in theform22

«m f5em f1pm f1Hm f , pm f524hm fsm f ,

Hm f5dms (n5s,h

anhn f , am5dms2dmh . ~3b!

Herepm f is a QG analog of the reaction Onsager field, in-troduced in SG theory in Ref. 29;Hm f stands for the intrinsicquadrupolar Zeeman-type field and is due to the kinematicproperties of the dynamic quadrupolar variables:30

2Sm f2 511amSs f . The mean and variance of the isotropic on

the average fluctuation field~3!, respectively,h1 andh2 havebeen calculated in Ref. 22. The variance of the effectivemolecular field components («s and«h) are reestimated hereand given by the relationships

«m~c,T!25e22H 12S JTD 2 12q

q@2q~pm2qm!2~11k2!

3qm~12q!#J 1J4k2

8 S 12q

T D 2dms , ~4!

with e225^em f

2 &C5J2q and J(c)5GAzc, where z512 isnearest-neighbor number on the hcp lattice. The second termcomes from the reaction-field effects and the last term iscaused solely by the quadrupolar intrinsic field. Here neworientational order parameter components are introduced:qm5^^Sm&T

2&C ,pm5^^Sm2 &T&C , which obey the relations

qs1qh5q and ps1ph51. The fluctuation-field parameterk(c) appears in~4! to describe the random-substitution ef-fects through the random-bond correlation coefficientKB ,namely,

k~c!5h2~c,T!

h1~c,T!5KBS 12c

zc D 1/2, KB5A^~Jf f 8

mm8Jf f 8nn8!2&B

^~Jf f 8mm8!2&B

.

~5!

One can expect that for the QG phase bounded in the con-centration rangec0<c<cM(cM50.55 andc0;0.1, seeo-p-H2 diagrams in Refs. 1 and 3! the QG fluctuation

fields introduced through the Hamiltonian~3! should be alsorestricted, i.e.,k(cM)<k(c)<k(c0). Formally, we intro-duce the lower critical concentrationc0 as a singular point ofthe amplitude of the effective molecular field@«(c,T)5(«s

21«h2)1/2# ~4! at which the leading terms of the

reaction-field effects disappear in high-T and low-T asymp-totics of the QG order parameter. We see that this criticalcondition is k(c0)51 which gives c05(11z/KB

2)21.Adopting a Gaussian estimation for the random-bond corre-lation coefficient31 ~5!, KB5A3, and comparing with thefindings of dynamic NMR experiments5,32,33 for the QGlower critical concentration mentioned above, we concludethat low-T consideration requires accounting of the next-nearest-neighbor interactions.

The self-consistent equations for the orientational orderparameterq(c,T)5^s f

21h f2&C follow from ~2! and should

be completed by the equations(c,T)5^s f&C , namely,

q~c,T!5123K 112 exp~ax!cosh~by!

@exp~ax!12 cosh~by!#2 Lx,y

,

a~c,T!5 32 «s~c,T!/T , ~6a!

s~c,T!5123K cosh~by!

exp~ax!12 cosh~by! Lx,y

,

b~c,T!5 ~)/2! «h~c,T!/T , ~6b!

where^•••&x,y denotes nonsymmetrical Gaussian integration(x15«1 /«s) over the local fields («s f5x«s and«h f5y«h). To give low-T consideration in the explicitform34 we introduce the following asymptotic representa-tions:

q~c,T!;123^wq~x,y!&x,y , T→0,

with

wq~x,y!5 exp~ax2by!/@11exp~ax2by!#2 , ~7a!

s~c,T!; 121 3

2 ^ws~x,y!&x,y , T→0,

with ws~x,y!5 exp~ax2by!/@11exp~ax2by!# ~7b!

in the range of variablesuxu,` and y,`. As a furthersimplification, we define the followingT→0 asymptoticequivalent functions: wq(x,y);d(ax2by) andws(x,y);Q(ax2by), with usual notations for the Diracdfunction and the Heaviside step function. After performingintegration the order-parameter-equation system can be pre-sented in the explicit form

q~c,T!512A3/2p T/«LT~c,T! 1O~T/J!3,

«LT5 12A3«s

21«h2, ~8a!

s~c,T!51

4 S 12«h~c,T!

«s~c,T! D13

4A2p

«1~c,T!

«s~c,T!1OS TJ D

2

,

«1~c,T!524s~c,T!h1~c,T!. ~8b!

2192 53BRIEF REPORTS

To illuminate the low-T behavior of the effective MF char-acteristic energy«LT5e2AgLT we introduce the reaction-fieldfactorgLT by means of relations

g LT~c,T!512J2~c!

2$@12q~c,T!#/T%2lLT~c,T!;

lLT~c,T!512k2~c!2 3k2~c!/16q~c,T! , ~9a!

which in turn can be formally interpolated to low tempera-tures:

gLT~c,T!51

11 @J2~c!/2# @12q0~c!/T#2lLT~c,0!,

q05q~c,0!. ~9b!

For the MF characteristic energy«LT one has the followingasymptotics:

«LT~c,T!;TA2q0~cM !/@12q0~c!#, T→0. ~9c!

Unusual for SG’s reaction-field behavior,35 this leads to thelow-T effect of the disappearing of the molecular field~9c!.The latter has been derived from NMR data14 by Li et al.~see Fig. 9 of Ref. 16 and recently discussed in Ref. 25. Toexamine a correspondence with experiment in concentrationsand in a way to verify the microscopic description of spatialcorrelations we have used~9c! to find a nontrivial zero-temperature asymptotic solution for the QG orientational or-der parameter36 ~8a!, namely,

q02/qM 2@12k2~c!#q01

316 k2~c!50, qM5q~cM,0! ~10!

with qM53/4p @ the solution s(c,T);T2 ~8b! is takeninto account#. Analysis of ~10! and its comparison with ex-periment is given in Fig. 1.

The high-T asymptotics for the orientational order param-eter follows immediately from~6!:

q~c,T!5hCr2 1h2

2~c,T!

2T21

«HT2 ~c,T!

2T22OS hCr2 h2

2~c,T!

T4 D ,~11a!

«HT~c,T!5J~c!Aq~c,T!F12J2~c!

2 S 12q~c,T!

T D 23@12k2~c!#G1/2. ~11b!

In contrast to the low-T case small-amplitude crystalline-field effects~1! should be taken into account.22 Formally, thefirst term in ~11! is due to ‘‘external’’ random fields. In thespirit of mean-field theory of SG’s in an external field onecan estimate a zero-field freezing temperatureT0F . Omittingformally hCr andh2 in ~11a! we have the following equationfor the freezing temperatureTF now improved by thereaction-field effects:

T̄0F4

TF2~c!

2T̄0F2 1@12k2~c!#T̄F

2~c!50, TF~c!5GAzc/2.

~12a!

This square equation has a nontrivial solution, which for in-structiveness can be approximated asT̄0F(c);GAc2c0. Itholds in the narrow concentration range,c0<c<5c0/4,where the value@12k2(c)# is small. Considering formallythe latter as a small parameter the iteration corrections to thelast term in~12a! have been found in all orders of its mag-nitude to extend the applicability of the solution of Eq.~12a!to the whole QG concentration range, namely,

FIG. 1. Quadrupolar glass orientational order parameter against ortho-concentration in ortho-para-hydrogen mixtures. Closed and open points cor-respond to the extrapolated to zero-temperature NMR data of, respectively,Refs. 14 and 15. Solid and dashed-dotted lines show the solutions of Eq.~10!, respectively, in the nearest-neighbor (z512) and effective mean-distance-neighbor (zeff) interaction approximations @qM50.72 andk(cM)50 are adopted#. Inset: reduced number of the effective mean-distance neighborszeff(c)/125(R̄/R0)

3 with R̄(c)5(3R0/4pA2c)1/3 andR0 are, respectively, the mean and nearest-neighbor distances on the hcplattice.

FIG. 2. Phase diagram for ortho-para-hydrogen mixtures. Lines: mean-field estimations for zero-field freezing temperature within the quadrupolarglass vector model:~1! TF ~12a!, in the absence or the reaction and instrinsicfields; ~2! T̄0F ~12a! and~3! T0F ~12b! in the presence of the reaction field,respectively;~4! Almeida-Thouless instability line within the quadrupolarglass infinite-range axial model in the presence of the quadrupolar intrinsicfield ~Ref. 37!. Points: the dynamic NMR anomalies observed by Sullivanet al. (d) ~Ref. 5!, Ishimoto et al. (m) ~Ref. 32!, and Husa and Daunt(.) ~Ref. 33!. ~See also comments in Ref. 3.!

53 2193BRIEF REPORTS

T0F4 /TF

2~c! 2T0F2 1S 11

1

12k2~c!

T̄0F2 ~c!

TF2~c!

D 21

T̄0F2 ~c!50,

~12b!

where T̄0F2 (c) stands for one of the solutions of the first-

order-approximation Eq.~12a!. Both solutions are illustratedin Fig. 2. In the next step one should include the ‘‘external’’fields mentioned above to reveal the para-rotational phaseinstability37 in the spirit of that obtained for SG’s in a con-stant external field.38

The microscopic analysis of the competing ordering fieldsgiven above permits one to conclude that the glasses underdiscussion are characteristically different rather in the localreaction-polarization effects that in the kinematic propertiesof their dynamic variables. We summarize the main results ofthe microscopical approach to the QG problem as follows.~1! Orientational ordering in QG’s has a collective character

and is not caused by the intrinsic quadrupolar field as earliersuggested in Ref. 3. In contrast to SG’s where the reaction-field effects can be reduced to corrections~see, e.g., Ref. 39and discussion in Ref. 25!, the QG orientational freezing isdue to the correlated reaction and molecular fields.~2! Non-zero variance of the fluctuation field specific of QG systemsgives rise to incomplete low-temperature orientational order-ing ~Fig. 1!. ~3! The QG phase formation requires a certainthreshold concentrationc0 ~Fig. 2!, above which the varianceof the fluctuation field is relatively small.~4! The quadrupo-lar intrinsic field dominates at high temperatures and can beself-consistently treated as an effective external field.

The author is grateful to Professor Neil S. Sullivan fornumerous stimulating discussions. Professor Alaor S. Chavesand Professor Eduardo C. Valadares are acknowledged forcritical reading of the manuscript. The work was supportedby the Brazilian State Foundations FAPEMIG and CNPq.

*Electronic address: VALERY@oraculo.lcc.ufmg.br,KOKS@brufmg.bitnet; permanent address: Institute for LowTemperature Physics and Engineering Academy of Science ofUkraine, Kharkov 310164, Ukraine.

1I. F. Silvera, Rev. Mod. Phys.52, 393 ~1980!.2J. Van Kranendonk,Solid Hydrogen: Theory of Properties ofSolid H2, HD, D2 ~Plenum, New York, 1983!.

3A. B. Harris and H. Meyer, Can. J. Phys.63, 3 ~1985!; 64, 890~E!~1986!.

4N. S. Sullivanet al., Can. J. Phys.65, 1463~1987!.5N. S. Sullivanet al.Phys. Rev. B17, 5016~1978!.6U. T. Hochli et al.. Adv. Phys.39, 405 ~1990!.7V. B. Kokshenev, Solid State Commun.44, 1593~1982!.8E. A. Luchinskayaet al., J. Phys C17, 665~1984!; E. A. Luchin-skaya and E. E. Tareeva, Teor. Mat. Fiz.87, 669 ~1991!.

9D. Chowdhury, Spin Glasses and Other Frustrated Systems~Princeton University Press, Princeton, NJ, 1986!.

10V. A. Moskalenkoet al., Teor. Mat. Fiz.71, 417 ~1987!.11M. Devoret and D. Esteve, J. Phys. C16, 1827~1983!.12M. Klenin, Phys. Rev B28, 5199~1983!.13S. F. Edwards and P. W. Anderson, J. Phys. E5, 965 ~1975!.14H. Meyer and S. Washburn, J. Low Temp. Phys.57, 31 ~1984!.15The completion of orientational freezing does not occur even at

0.037 K, which has been achieved in the NMR experiment by C.M. Edwardset al., J. Low Temp. Phys.72, 1 ~1988!.

16X. Li et al., Phys. Rev. B37, 3216~1988!.17N. S. Sullivanet al., Mol. Cryst. Liq. Cryst.139, 365 ~1986!.18L. Jin and K. Knorr, Phys. Rev. B47, 14 142~1993!.19K. Binder and J. D. Reger, Adv. Phys.41, 547 ~1992!.20V. B. Kokshenev, Solid State Commun.55, 143 ~1985!.21V. B. Kokshenev and A. A. Litvin, Sov. J. Low Temp. Phys.13,

195 ~1987!.22V. B. Kokshenev, Phys. Status Solidi B164, 83 ~1991!.23K. Walasek, Phys. Rev. B46, 14 480~1992!.24K. Walasek and K. Lukierska-Walasek, Phys. Rev. B49, 9460

~1994!.

25V. B. Kokshenev, Solid State Commun.92, 587 ~1994!.26K. Lukierska-Walasek and K. Walasek, Phys. Rev. B50, 12 437

~1994!.27Y. Lin and N. S. Sullivan, Mol. Cryst. Liq. Cryst.142, 141

~1987!.28Experimental findings for quadrupole-quadrupole coupling con-

stant areGH250.827 K ~Ref. 1!, GH250.82 K ~Ref. 2!,GD251.04 K ~Ref. 2! (GN253.2 K! which are close to the cal-culated valuesGH250.90 K ~Ref. 2! andGD251.13 K ~Ref. 2!.

29D. J. Thoulesset al., Philos. Mag.35, 593 ~1977!.30Kinematic and commutation relations for pseudospins~1! have

been obtained within the Hubbard’s representation in V. B. Kok-shenev, Sov. J. Low Temp. Phys.6, 667 ~1980!.

31The valueKBexp51.8 has been derived from the high-T experimen-

tal data~Ref. 14! in Ref. 22.32H. Ishimotoet al., J. Phys. Soc. Jpn.35, 300 ~1973!.33D. Husa and J. G. Daunt, Phys. Lett.65A, 354 ~1978!.34To reduce low-T consideration to Eq.~6! we have adopted the

isotropic approximation for the reaction-field term in~4! by themeans of@•••#→(12k2)q/2.

35The QG reaction-field results given in~4!, ~9a!, and~11b! can becompared with therandom-bondSG case of theS5

12 Ising

model, namely,« SG(T)5e2(T)„11J2{ @12q(T)#/T} 2…1/2, withe2(T)5JAq(T). One can speculate on its qualitative correspon-dence with the QG case atc,c0 .

36The trivial solutions of the QG order parameter [q(c,0)51] havebeen analyzed in Refs. 21 and 24. The existence of nontrivialsolutions@q(c,0),1# has been claimed recently in Refs. 25 and26.

37The Almeida-Thouless instability problem~Ref. 38! has beenconsidered within the QG axial~Ising-type! model by V. B.Kokshenev and A. A. Litvin, Sov. J. Low Temp. Phys.13, 246~1987!.

38J. R. de Almeida and D. J. Thouless, J. Phys. A11, 983 ~1978!.39C. M. Soukouliset al., Phys. Rev. B28, 1495~1983!.

2194 53BRIEF REPORTS