PHYSICAL BEVIEW B VOLUME 8, NUMBER 11 1 DECEMBER 1973

Ultrasonic Investigation of the LamMa Transition in NH, B

Carl W. Garland and Carolyn K. ChooDepartment of Chemistry and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge,

Massachusetts 02139(Received 30 April 1973)

The ultrasonic-attenuation coefficient a has been measured in the disordered phase of NH4Br at 10, 30,and 50 MHz as a function of temperature for the [100) longitudinal wave and the [110]transverse wave

associated with the c' shear. By combining these data with available ultrasonic- and hypersonic-velocity datait is possible to determine the dynamical behavior near the order-disorder transition if one assumes asingle-relaxation model. The critical relaxation time for the longitudinal wave is well represented by 1/r (inunits of 10 sec ') = 1.8 + 1780m'0, where c is the reduced temperature (T —T„)/T„. The transverserelaxation time is quite similar. The critical attenuations themselves can be approximately represented by apower law of the form a, = Sco'e ', where l 1.35. It was found possible to obtain an essentially singledomain NH4Br crystal in the tetragonal ordered phase by cooling the crysta1 slowly with a smalltemperature gradient along one of its axes. Longitudinal and shear velocities were measured in the orderedphase as a function of temperature from 120'K to T„=234.5 'K. Critical attenuation measured at 10 MHzfor the longitudinal wave was well described by a power law with the exponent l = 0.75.

I. INTRODUCTION

Both NH4Cl and NH4Br crystals undergo order-disorder transitions of the X type involving therelative orientations of the tetrahedral NH4 ions.At room temperature, both crystals have a dis-ordered CsC1-type cubic structure with the NH4'

ions distributed at random with respect to twoequivalent orientations in the cubic cell. However,there are major differences between the types ofordering observed in the chloride and in the bro-rnide. In the case of NH4Cl, there is a single X

line marking the transition between the disorderedcubic phase and a "parallel" ordered cubic phase.The phase diagram for NH4Br is much more com-plicated, ' and the X transition at low pressuresinvolves an ordered tetragonal phase with "anti-parallel" chains of NH4' ions. Although the X tem-peratures are fortuitously similar at 1 atm (T„= 242 'K for NH, C1 and 234. 5 'K for NH4Br), the& lines have slopes of opposite sign. Recenttheoretical works' indicate that there are two com-peting interactions between neighboring ammoniumions —a direct octopole-octopole interaction andan indirect interaction via the polarizable halideion. This octopole-dipole term appears to bedominant in NH4Br and explains the "antiferro-magnetic" ordering in that case.

Thus it is of interest to compare the dynamicalbehavior of NH4Br near its X transition with theanalogous behavior recently reported for NH4Cl. 'Acoustic attenuation and dispersion measurementsin NH4Cl have been carried out at 1 atm and athigh pressures, and these data have been analyzedin terms of a single-relaxation formalism toevaluate the critical variation of the relaxationtime 7 for long-range ordering. A similar analy-

sis will be presented here for the acoustic behaviorof NH4Br in its disordered phase at 1 atm. Unlikethe case of NH4Cl, where only longitudinal wavesexhibit dispersion and critical attenuation„ the&' shear wave propagating in the [110]directionand polarized perpendicular to [001] is also strong-ly attenuated near T„ in NH4Br. In addition tothis qualitative difference, the critical exponentsfor the divergence of ~ (as obtained from data on[100] longitudinal waves) are quite different in thetwo crystals.

Ultrasonic-velocity measurements in NH4Br havebeen made previously over a wide range of tem-peratures and pressures. ' Thus the low-fre-quency limiting values of all three independentelastic constants c», c44, and c' = (c»-c,z)/2 aremell known in the disordered phase. The velocityand attenuation of hypersonic acoustic waves havebeen determined at room temperature as a functionof frequency from Brillouin scattering rneasure-ments; in addition, the velocity of the [110]longi-tudinal wave has been measured as a function oftemperature at - 16 GHz. ' Qn the basis of thesedata it is possible to estimate the temperature de-pendence of the velocity dispersion for the c»and c' waves. In this paper we report the ultra-sonic attenuation of c» and c' waves in disorderedNH48r as a function of temperature and frequencyat 1 atm. By assuming a single-relaxation model,we are then able to determine the critical behaviorof the relaxation time for T &T~.

Previous ultrasonic investigations of NH4Br havebeen largely confined to the disordered phase. Whena single crystal undergoes the transition from thecubic disordered phase to the tetragonal orderedphase, many ordered domains are formed withtheir tetragonal axes oriented randomly along the

5143

5144 C. %. GARLAND AND C K CHOO

original cubic axes. Such a multidomain crystalscatters sound very strongly, and the echo pat-tern disappears for longitudinal waves and the c''

shear wave. In the present investigation, it masfound possible to obtain an essentially single-domain ordered crystal. Thus we were able tomeasure the velocities of both longitudinal andshear waves and the anomalous attenuation of thelongitudinal wave as a function of temperature inthe ordered phase.

II. EXPERIMENTAL PROCEDURES

The ultrasonic techniques used in this investiga-tion were very similar to those used previously tostudy ammonium chloride. Attenuation measure-rnents were carried out with a single-transducer,pulse-echo method using Matec ultrasonic equip-ment. Velocity measurements were made at 10MHz with a pulse-superposition method.

A Lauda refrigerating circulator and dry-iceheat exchanger were used for measurements in thedisordered phase. Liquid nitrogen and a variable-pressure transfer gas were used for measurementsin the ordered phase. In both cases, regulation ofthe temperature of a copper sample holder wasachieved with a Bayley model-250 proportionaltemperature controller. The resulting ternpera-ture stability, as measured with a calibrated Rose-rnount platinum resistance thermometer, was+5m K.

The NH48r single crystals used in this experi-ment were from the same batch as those studied inan earlier high-pressure investigation. ~ Theywere at least 99. Yg pure, were free from visibledefects, and had well-developed (100) faces whichwere parallel to within+0. 0001 cm. Some pre-liminary measurements in the [100]direction weremade using a slightly imperfect crystal with alength I.= 0. 7066 cm. Final measurements in thatdirection mere made on a larger and more perfectcrystal, denoted as crystal I. VYith the exceptionof attenuation values in the ordered phase, dataobtained on these two crystals agreed well witheach other. Crystal Ia was used as grown forlongitudinal attenuation measurements in the dis-ordered phase; the length of this crystal in the

[100] direction was 0. 8928+ 0. 0001 cm at 20 'C.Subsequently, one face of this crystal was waterdamaged and a new face was flycut parallel to theundamaged (100) face; the new length was 0. 81'79+0.0005 cm at 20 C. This crystal, designated ascrystal Ib, was then used for measurements in theordered phase. Crystal II was flycut to obtain apair of (110) faces; the parallelity of these faceswas within a 0.0003 cm and the length along the[110]direction was 0. 1562 cm at 20 C. It shouldbe noted that NH4Br crystals are soft and hygro-

scopic and must be handled with considerable care.The transition temperature T, for each crystal

was determined by measuring the shear velocitycorresponding to the elastic constant c~4. Thisshear wave is not attenuated near T, and can evenbe propagated in a polydomain ordered crystal.Furthermore, this shear velocity undergoes avery rapid variation at the transition. Indeed, thetemperature coefficient (Bc,JBT)~ changes abrupt-ly from a small negative value in the disorderedphase to a very large positive value, and the transi-tion point is easily established. The X tempera-ture was found to be (234. 3+ 0. 1) 'K in crystal Iand (234. 8 + 0. 1) 'K in crystal II. These valuesagree reasonably with the value 234. 5 K cited byGarland and Yarnell.

Dow resin 2'76-V9 was used to bond the &-in. —

diam quartz transducers to the NH48r crystals.However, this bond broke when the sample wascooled below 160 K, and the less viscous Dow-Corning 200 silicone fluids were used for mea-surements in the ordered phase.

It was found possible to obtain an effectively"single-domain" NH48r crystal in ~he tetragonalordered phase by cooling the crystal slowlythrough the transition after a temperature gra-dient has been established along one of the [100]axes. This gradient was obtained by placing asmall heating element in good thermal contactwith the face opposite from the transducer (whichis itself in good thermal contact with the coppersample holder). The optimal gradient, as mea-sured with a gold-chromel difference thermo-couple, was 4 to 5 'K cm"'. The gradient heat-er was turned on and the sample holder wasmaintained at -237'K for several hours to al-low the system to achieve a steady-state tem-perature distribution. The sample holder wasthen cooled slowly to about 210'K. The rateof cooling was not crucial, but a constant rateof 4'K h ' seemed to give the best results. Thecriterion for deciding whether a single domainhad been formed was the ability to propagatelongitudinal waves and obtain a good exponen-tial echo pattern with at least five echoes at-200'K. Ultrasonic pulses were not propagatedin the sample during the cooling process, andthe gradient heater was turned off before theecho pattern was inspected. Although there isno direct evidence about the tetragonal domainstructure formed below T„, the observed ultra-sonic behavior is consistent with a single dom-inant domain and we shall consider the orderedtetragonal phase to be a single-domain crystal.After the gradient is turned off, there is noindication of any change in the domain struc-ture with time or with temperature variations(as long as the temperature is kept below T„)

U LTRASONIC INVESTIGATION OF THE LAMBDA. . . 5145

III. RESULTS

A. Attenuation in the Disordered Cubic Phase

TABLE I. Smooth-curve values of n,/~ (in units of10 Np sec cm ) at different values of AT = T —T&.

Longitudinal u aves. The attenuation 0'. of longi-tudinal waves propagating in the [100]direction(c» wave) has been measured as a function of tem-perature at 10, 30, and 50 MHz. Plots of the ob-served + values versus f at various constant tem-peratures between 236 and 255 'K gave a family ofstraight lines with a common intercept 0= 0. 3+ 0. 1 dB cm '. Thus the total attenuation consistsof a large critical attenuation at, related to theorder-disorder transition and a small backgroundattenuation ao which is independent of frequencyand temperature. This background attenuation iscaused by various effects such as beam spreadingdue to the finite size of the transducer, scatteringfrom imperfections in the sample, and energylosses in the bond between the transducer and thesample.

The behavior of the critical attenuation e,= n,„,—ao is shown in Fig. 1, where logo,', isplotted against log(T —T„). The straight linesshown in this figure represent the best fit to thedata with a power-law of the form

1.52.02. 53.03.54.04. 55.56.57.58.5

10.512.513.515.517.520. 5

96.574. 360. 2

48. 540. 935.630.824. 620. 2

16.714.010.37.826.905.644.653.60

c~~ wave~T ( K) e,/~'

0. 8l. 01.21.51.82. 02. 53.03.54.04, 55.05.56.06.57.07.58.09.0

10.012.0

77.465. 2

59.847. 240. 435.728. 323. 2

18.414.612.310.59.348.097.216.425. 845. 284. 233.502.63

c waveb T ('K) 0,,/~2

-1.57—1.89-2.44-3.06—4. 21—5.99—7.45—9.46

-11.27-12.84—15.09-17.74-20.18—22. 98—24. 98—28. 15—32. 51—36, 56-42. 07—52. 21-63.29

5304713893322381911701451331179992796766595150464036

cgoing waveb, T ('K) e,/co

o.', (Npcm ') =S&u'e ',where ~=2wf and e is the reduced temperatureIT —T, I/T„. The values of the parameters are

S= (1.4+0. 3)X10 '9 sec'cm ' and l = 1.35+0.05.In fact, the data show a definite systematic devia-tion from a simple power law. Smooth-curve val-ues of u, /ru' as obtained from the slopes of0.',~,-vs-f plots are given in Table I and will beused in further analysis of these data.

IOO

IO

EO

Q3

I

O. l

0.5 IO IOO

PIG. 1. Log-log plots of e~ vs ET (in 'K) for [100]longitudinal waves in the disordered cubic phase of NH48r.The lines represent a simple power-law fit to the datausing Eq. (1).

The most important factor limiting the accuracyof the attenuation is a slight nonexponential char-acter in the echo pattern. The average deviationin the &,„,values obtained from adjacent pairs ofechoes was - 1+ in regions of small or moderateattenuation. Since the character of the echo pat-tern remained essentially the same throughout arun, changes in 0.'„,with temperature could bedetermined with somewhat higher accuracy. Thescatter in the data indicates that 5% would be areasonable estimate of the random error.

The attenuation of longitudinal waves propagat-ing in the [110]direction was measured only at10 MHz since good echo patterns could not be ob-tained at higher frequencies. The (110) faces ofcrystal II had deteriorated during an extensive setof earlier measurements on the c' shear wave,and recutting this crystal did not improve theparallelity of the faces or reduce the distortionin the high-frequency echo pattern. The criticalattenuation for this wave could be shown to agreewith that for the c «wave at 10 MHz if the back-ground attenuation &0 was assumed to have therather large value of 1.8 dB cm ' in the [110]direction. However, these limited data did notpermit us to make any conclusive determinationof the dependence of n,/uP on the direction ofpropagation of longitudinal waves in NH4Br. At-tenuation measurements on NH4C1 have shown that0.', values in the disordered phase of that crystalare essentially the same for longitudinal wavespropagating in the [100] and [110]directions,

5146 C. %. GARLAND AND C. K. CHOO

and it is likely that the values in these two direc-tions are very similar, if not identical, in NH4Bralso.

Transverse waves. There is no appreciableattenuation for ultrasonic shear waves correspond-ing to the elastic constant c44. However, thetransverse wave propagating in the [110]direc-tion and polarized perpendicular to the [001] di-rection (c' wave) does exhibit critical attenuation.Initially it mas difficult to obtain a satisfactoryecho pattern at high frequencies, but proper im-pedance matching and improvement of the paral-lelity of the (110) faces by repeated flycutting ofcrystal II resulted in a reasonably good exponen-tial pattern of more than 15 echoes at room tem-perature. Because the attenuation of this c' wavewas smaller than that of the c» wave, it was pos-sible to carry out 30 and 50 MHz measurementscloser to T„ than before. Plots of o.'vs f' werelinear at all temperatures, but the intercept masnot independent of temperature as in the case ofth e e» wave. The background attenuation 0.'0 hadan essentially constant value of 1.3 dBcm ' in therange 4. 5& AT & 12. 5 'K but increased steadily onapproaching the transition (for example, o.o= 2. 1at n. T= 2. 37'K and 4. 8 at n, T= 0.80'K). Smooth-curve values of ajw as obtained from the slopesof these o.',b;vs-f~ plots are given in Table L Itis possible to approximate these transverse n, /&a~

values with the power law given in Eq. (1), butsince a log-log plot of u, /&o vs e shows distinctcurvature, one is not able to determine an un-ambiguous value for the exponent l. Values of lbetween 1.35 and 1. 5 are compatible with thedata: for E = 1.35, one finds S= 0. 6x10 ' seccm '; for l=1.5, the corresponding $=0.3X10 'sec cm '.

exponential as those in the disordered phase. Farfrom the transition, five echoes mere measured andthe average deviation of the attenuation values ob-tained from different echo pairs was less thanI(Pq. (Two preliminary runs on the somewhat im-perfect crystal with path length 2I = 1.4132 cmgave much larger e values and more scatter inthe data as a function of temperature. This sug-gests the necessity of having a very good singlecrystal in order to achieve a single domain in thetetragonal phase. )

A simple power-law dependence of 0'.,= 0',„,—oo on hT could be obtained when 0 was as-signed any constant value between 0. 3 and 0. 7dBcm '. In viem of the fact that &0 was 0. 3 dBcm ' in the disordered phase, a value of 0. 5 dBcm ' was assumed to be a reasonable choice forthe ordered phase since some additional back-ground attenuation would be expected owing to im-perfections in the domain structure. The o', val-ues shown in Fig. 2 and the a,/~ values given inTable I mere calculated using this choice of &0.As indicated by Fig. 2, the 10-MHz longitudinalattenuation can be well represented by Eq. (1)with S= (130+30) && 10 ' sec cm ' and l = 0. 75+0.05.

C. Velocity Measurements

The ultrasonic velocity u in the tetragonalordered phase has been measured at 10 MHz bybonding a transducer to one of the (100) faces ofa disordered cubic crystal and cooling the crystalslowly mith a temperature gradient applied paral-lel to the direction of sound propagation (see Sec.II). When a good echo pattern was observed be-

B. Attenuation in the Tetragonal Ordered Phase

Since no measurements were made below T„on crystal II, we observed only the longitudinalattenuation corresponding to that measured for thee» wave in the disordered phase. It is not knownfor certain that the ordered crystal consisted ofa single domain. Even if it did, the tetragonalaxis could have been oriented either parallel orperpendicular to the direction of sound propaga-tion (see Sec. IIIC). Since the longitudinal at-tenuation and velocity measurements were madein the same run, without heating the crystal aboveT„ the 0.'values reported here and the c~, val-ues reported in Sec. DIC definitely correspondto the same configuration of the ordered crystal.

The longitudinal attenuation in the tetragonalphase was only measured at 10 MHz since echoescould not be obtained at higher frequencies between200 'K and the transition. The data reported heremere obtained from an echo pattern which was as

30—

IO-ED

3

IO

b,T

I

30 I 00

FIG. 2. Log-log plot of ~~ vs ET (in 'K) for a 10 MHzlongitudinal wave in the tetragonal ordered phase of NH48r.These 0.~ values were obtained using no=0. 5 dBcmThe line represents a power-law fit to the data using Eq.(i).

U LTRASONIC INVESTIGATION OF THE LAMBDA ~ ~- 5147

low T„for longitudinal waves, it was assumed thatthe crystal consisted of a single tetragonal domain.However, there could have been multidomain re-gions near the surface or a few large domainswith their tetragonal axes perpendicular to eachother. The velocity measurements cited below

strongly support the view that the crystal waseffectively a single domain along the acousticpath, but the data are not definitive. Even if thecrystal does become a single domain, the orienta-tion of the tetragonal axis is not known and couldbe different each time the crystal is cooled belowT~. Depending on the orientation of the tetragonalaxis, different thermal-expansion correction fac-tors are needed to calculate the elastic stiffnessc =pu as a function of temperature. There aretwo possible cases: (a) The tetragonal axis isparallel to the direction of propagation of thesound wave;

oJl

E .77

oO

.75

O

.73

00

a &'(2L')' (&i/&i)'(2)

(b) The tetragonal axis is perpendicular to thedirection of propagation of the sound wave:

.71120 I 50

l

200w( K)

I

250

2 p (2L ) ~~PQ =

p a3

In Eqs. (2) and (3), 6 is the round-trip transittime for the acoustic pulse, p is the mass density,2L is the round-trip path length in the crystal, a,is the lattice parameter of one of the two equiva-lent axes, and a~ is that of the tetragonal axis.The superscript zero denotes the vat. ues at a con-venient reference temperature, which is 20 C.The values p = 2. 4336 gem and a, =a~= 4. 05SO Awere taken from Garland and Yarnell. The valuesof a, and a3 as functions of temperature were takenfrom Bonilla, Garland, and Schumaker, ' sincetheir values fall between those obtained in two in-dependent measurements by Hovi and co-workers. "

The elastic constants obtained from shear-wavemeasurements are shown in Fig. 3, and those ob-tained from longitudinal waves in Fig. 4. Smooth-curve values are given in Table II. In the case ofshear ~aves, two independent s~ts of measure-ments on the same crystal were carried out underexactly the same conditions except that the Y-cutquartz transducer (and thus the direction ofpolarization) was rotated by 90 . The elasticconstants calculated from these two sets of datawere in excellent agreement if it were assumedthat the tetragonal axis was parallel to the propa-gation direction for the first run and perpendicularto it for the second run. If either Eq. (2) or (3)were used for both sets of data, the resultingp& values showed a systematic difference of about0.8%. Thus the shear elastic constant measuredin this experiment is almost certainly c4~. As can

FIG. 3. Temperature dependence of c,~~ as deter-mined from a "single-domain" crystal in the tetragonalordered phase. It is probable, but not certain, that thesedata represent c44 values in the ordered phase (see text).The solid line represents the well-characterized varia-tion of c44 in the disordered cubic phase above Tz andthe average shear stiffness 2 reported previously frommeasurements on multidomain ordered crystals (Ref. 4).

be seen from Fig. 3, the variation of c~,~ is veryrapid just below T~. Indeed, the value changesfrom 0. 783 at 234'K to 0. "I97 at 234. O'K. It ispossible that there is a discontinuous jump in theshear stiffness and hysteresis in the transitiontemperature, as is seen in NH4Cl at 1 atm. How-ever, such effects must be very small if they dooccur. e estimate that &c = 0. 032 x 10 dyn cmand &X=40 m 'K are upper l.imits on the possiblediscontinuity and hysteresis.

In the case of longitudinal waves, measure-ments were made on two different crystals butthe orientations must have been fortuitously thesame since the pu values agree on using eitherEq. (2) or (3). Thus we do not know which longi-tudinal elastic constant has been measured in thisexperiment. It was arbitrarily assumed that thetetragonal axis was parallel to the propagat:ion di-rection, ' and Eq. (2) was used in calculating thec&„,values in Fig. 4 and Table II. If Eq. (3) hadbeen used instead, the resulting values of c~would all be lower by roughly 0. 8%.

The scatter in the data indicates that the randomerrors in the elastic constants are about 0. 1%.

5148 t-". %. GARr. AND AND e. K. CHOO

$4—

3.0—

EO

2.6—O

c 2,2O

O

Cargo ~oXQ

X

FIG. 4. Temperaturedependence of c&~ as de-termined from a single-domain crystal in the te-tragonal ordered phase.The solid line shows thesmooth-curve variation ofcgf in the disordered phase(Ref. 4). The circles in-dicate data obtained on crys-tal Ib, and the crosses de-note data obtained on thecrystal with path length2L = 1.4132 cm.

I.SIOO

I

I 50I

200To(K)

I

250I

500

Possible systematic errors arise not only fromthe ambiguity in the orientation of the tetragonalaxis but also from the choice of the n=0 condi-tion for the pulse-superposition method. Forshear waves, where the echo pattern could befollowed through the transition region, the choiceof n = 0 was made by matching the present e«values above T„with those reported previously. ~

For longitudinal waves, it was necessary to estab-lish the choice of n = 0 independently in the orderedphase. As a check on this choice, c~„,was mea-sured on two crystals of different lengths, and

Fig. 4 shows that the results are in good agree-ment.

IV. DISCUSSION

There are no elastic constant data in the tetrag-onal phase with which to compare our presentresults except for the average shear stiffness ~obtained from measurements on multidomain crys-tals. As shown in Fig. 3, the temperature varia-tion of ~ is qualitatively similar but quantitativelyvery different than the e~,~ variation determinedin a single-domain crystal. It was thought pre-viously that 5 represented the average (2c«+c~8)/3, which would result from a random orienta-tion of the tetragonal axes of the domains alongthe directions of the three equivalent axes of thedisordered cubic crystal. This now seems ques-tionable. If the assignment of c~, as c« is cor-rect, then c6~ would have rather low values (aminimum value of 0. 637 & 10"dyn cm at - 190 Kand 0. 668x 10"at 130 K). Such a large tetragonal

"splitting" between c «and c«seems unlikely.Another possible explanation for the low ~ valuescould be that the effective acoustic pathlength islarger in the multidomain crystal owing to reflec-tions of the wave at domain boundaries. It shouldalso be noted that the minimum in c~, occurs at- 210'K, whereas the minimum in ~ is at - 190 K.A minimum value near 210'K is more reasonablein view of the lattice-parameter variation. ' It isknown that the shear stiffness c« is a sensitivefunction of volume in the case of NH&cl. ' Byanalogy, the rapid decrease inc~, for NH4Br

120130140150155160165170175180185190195

c1ong

2.9412. 9202.8982.8702.8542.8382. 8132.7902.7672.7452.7182.6852.648

0.80560.79810.79100.78760.78420.78100.77790.77510.77260.77030.76830.7665

200205210215220225229230231231.5232233234

c1oag

2.6052. 5552. 4982.4292. 3422. 2252. 0922. 0481.9981.980

0.76540.76480.76430.76450. 76600.76890.77300.77420.77580.77680.77750.77960.7825

TABLE II. Smooth-curve values of the elastic con-stants c~~andc, | (in units of 10 dyn cm ) fora single-domain crystal of NH43r in the tetragonal phase. Thenumber of significant figures does not indicate the abso-lute accuracy.

U LTRASONIC INVESTIGATION OF THE LAMBDA. . . 5149

just below T„would be expected because of theanomalous expansion which takes place on order-ing. The fact that NH48r expands on cooling intothe tetragonal ordered phase whereas NH4C1 under-goes an anomalous contraction on transforminginto its cubic ordered phase presumably explainsthe observation that c~„,far below T„ is smallerthan c» in the disordered phase for NH4Br (seeFig. 4), while the opposite is true for NH4Cl. ~

Our attenuation data can be compared with therecently reported results of Velichkina and co-workers, "who measured the [100] longitudinalattenuation at 5, 15, and 25 MHz in the range5~ T —T„~ —6 'K. Their method involves mea-suring the relative change with temperature inthe height of a single pulse transmitted through athin crystal. This is advantageous when the at-tenuation is very high but ~ot too accurate in re-gions of low to moderate attenuation. (Indeed, asystematic correction of -4. 5 dBcm ' must bemade to all their data owing to an unfortunatechoice of the reference point from which changesin attenuation were measured. ) The values of&,/& estimated from their data are larger thanthose given in Table I by a roughly constant factorof about l. 5. The reason for such a multiplicativedifference is unclear. It should be pointed outthat their attenuation data in the multidomainordered phase appear, rather surprisingly, to bemore reliable than those in the disordered phase;+, varies with frequency in a somewhat erraticway above T& but shows a good quadratic depen-dence below. Incidentally, the presence of multipledomains in a crystal cooled below T, in the ab-sence of a thermal gradient was demonstrated bythe depolarization of linearly polarized light. '

Velichkina et al. have interpreted their ordered-phase attenuation in terms of the Landau-Khalat-nikov model involving a critical relaxation timev=2. 2x10 /(T„- T) This does. not seem compat-ible with the quadratic frequency dependence oftheir data or with the fact that both their o.',/&o

values and the values in Table I vary like (hT)If one ignores the multiplicative factor of - 1.5 be-tween our results and theirs and considers onlythe ratio of o, in the ordered phase to that in thedisordered phase, the critical attenuation in thetetragonal ordered phase would not seem to de-pend on the domain structure. Yet our experienceindicates that two crystals which show the sameattenuation in the disordered phase can give quitedifferent 0.', values in the ordered phase, depend-ing on the perfection of the crystal and the mannerof cooling. Thus the interpretation of +, in theordered phase is puzzling.

Although the critical attenuation in the disorderedphase can be approximately fit by a simple powerlaw (see Fig. 1), the g' value is rather large (5. 9)

u (~)-u (0) e"-c2u((u)u (0) 2u(0)co ' (5)

The analysis is based on using hypersonic-velocitydata to estimate the high-frequency limiting stiff-ness c" and ultrasonic data for the low-frequencylimiting values u(0) and c = pu (0). The resultingvalue of C at a given temperature is then combinedwith the corresponding experimental value ofa,/~ to yield vz, „. One could, of course, alsoanalyze the data in terms of the adiabatic relaxa-tion time at constant stress v~ „. However, thesetwo times are simply related to each other byvz, r/r~„=c "/c, and the behavior of 7 ~ ~ does notdiffer in any significant way from that of 7~ „.

The temperature variations of e» and c' arewell known from ultrasonic measurements at 20MHz. Values of &» and & are known at 24 Cfrom an analysis of Brillouin scattering data atangles corresponding to frequencies between 6. 5and 24 GHz. ' In order to estimate c'"values nearthe transition, we have assumed a linear tempera-ture dependence with a slope equal to the value ofd(c'0)/dT at room temperature (see Fig. 5). Thisis a plausible but essentially arbitrary assumption.Fortunately, c» near the transition can be esti-mated with much greater certainty. The hypersonicvelocity of the [110j longitudinal wave is known as

and the data show systematic deviations from Eq.(1). These deviations are seen to become evenlarger for the 25-MHz data of Velichkina et al.when A, T & 1 K. Thus it is of interest to analyzethe data in terms of a relaxation model. whethera single-relaxation model is adequate to describethe low-frequency dynamical behavior of the order-disorder transition in NH&Br is unclear. Accord-ing to theoretical predictions for magnetic'4 andfluid" systems, a spectrum of relaxation timeswould be expected. However, none of the currenttheories seem to be directly pertinent to the analy-sis of NH~Br. Moreover, the extensive hyperson-ic attenuation and dispersion data at room tem-perature show that, at least for temperatures farfrom T„, NH4Br is well described by a single-re-laxation model. In any event, the assumption ofsuch a model leads to an empirical analysis ofthe ultrasonic attenuation data in which attractive-ly simple behavior of both the relaxation strengthand the relaxation time is obtained.

Since the critical attenuation data exhibit an ffrequency dependence, we shall adopt the low-fre-quency form of the single-relaxation expression:

+c CK ~g gy

where u, is the amplitude attenuation (in Np cm ),v ~ „is the adiabatic relaxation time at constantstrain, and the appropriate relaxation strength Cis defined by

5150 C. W. GARLAND AND C. K. CHOO

1.35—

I.3 I—

3.60—

c'II

o 320230

I

250l

270 290l

3IO

FIG. 5. Variation of the limiting values of c~~ and c'.The smooth curves for c1~ and c' are taken from Ref. 4.The crosses are infinite-frequency elastic constants re-ported in Ref. 5; see text for a discussion of the choiceof the c11 and c'" smooth curves. Note that the verticalscale for c' is 10 times larger than that for c~1.

C '( )e=C '(0)+be (6)

~here E is the reduced temperature and m is anempirical exponent equal to 1 at 1 atm. Theparameters C '(0) and b are, respectively, 6. 4and 120 (in units of 10 cm sec ) for NH4Br; the

persion for the [100] shear wave, c«can be re-placed by the known c«. ) Near the transitionthese c11values should correspond to c «sincethe relaxation times becomes very long. Thec 11 values between 242. 5 and 297 K were obtained

by linear interpolation (see Fig. 5). Uncertain-ties in c11 due to ambiguities in the c "values areless than 1%, which would correspond to less than

a 10% possible systematic error in the relaxationstrength for the c „wave.

The values of the relaxation strength C calculatedat various temperatures from the smooth-curvevalues of c and c are shown in Fig. 6, where1/C is plotted versus n T= T —T„. The relaxa-tion strength for the c» wave in NH4Cl at 1 atm isalso shown for comparison. This empirical plotindicates that 1/C varies linearly with n. T for the[100] longitudinal wave in both NH~C1 and NH4Br.Indeed, this linear variation extends out to d T= 50 for NH4C1. The inverse relaxation strengthfor the longitudinal wave can be represented inthe form

a function of temperature at - 16 GHE, and theelastic stiffness corresponding to this wave isCI c» c +c gg The open circles shown in Fig.5 represent C11 values calculated from c11=cz(16 6Hz)+c' —c«. (Since there is no dis-

16

14

12

NHg Br(C' data) 10

40—

4p 30lh

E

o 20

NHgBr(c„data)

I

8

O

10

l

IO

QT

l

l5l

20 25 0 10AT

15 20

FIG. 6. Inverse relaxation strengths as a function of4T= T -T„(in 'K). The results for NH4Cl are takenfrom Ref. 3.

FIG. 7. Inverse relaxation times for the f.100] longi-tudinal wave and the f110] transverse wave as a functionof 4T (in K).

U LTRASONIC INVESTIGATION OF THE LAMBDA. . . 5151

corresponding values are 5. 8 and 6'7. 7 for NH4Cl. 'In the case of the c' wave, the relaxation strengthis less well established and it does not seemjustifiable to attempt an analytical representation.

The relaxation times obtained from Eq. (4) byusing the experimental u, /uP data in Table I andthe above choices of C values are shown in Fig. 7,where 1/v is plotted versus n. T. This figure in-dicates that the longitudinal relaxation times canbe well represented by

I/v= 1/r(0)+as",

where the subscript 8, x has been dropped forconvenience. The parameters I/v(0) and a cor-responding to the line in Fig. 7 are 1.8 and 1780(in units of 10 sec ) and the exponent n=1. How-

ever, the limiting value of v at T„should not beconsidered as well established, since a reasonablygood fit can also be achieved with I/v(0) = 0 and

a = 1830. The transverse relaxation times arequite similar in magnitude to the longitudinal v.

values but appear to vary in a systematically dif-ferent manner. Indeed, the best fit to these I/rvalues gives an exponent n = 1.25. A decision asto whether such differences are real or not willrequire reliable values of c'"as a function of tem-perature.

It is important to note that the fit to the attenua-tion using Eq. (4) with experimental C values andthe form for v given in Eq. (V) is much betterthroughout the entire temperature range than thefit with Eq. (1). The X value is quite good (0. 9)and there are no systematic deviations. Even ifa single relaxation formalism is not valid, theanalysis we have carried out for low-frequencyattenuation data is an appropriate way to treat theslowly varying strength C and obtain the exponentn for the critical relaxation frequency. That is,we can reinterpret Eq. (4) as the low-frequencyform of a more general relaxation theory involvinga spectrum of relaxation times. If the sound waveis coupled to the order-parameter fluctuations viaa dependence of the NH4'-ion interactions on thelattice parameter (volume magnetostrictive couplingin the analogous magnetic case), the major con-tributions to o at small should come from fluc-tuations with k = x, where K is the inverse cor-relation length. ~~ In this case, our I/r ~ 0„,

which is the characteristic frequency for order-parameter fluctuations with wave vector ~. Sincethe ammonium halides are formally quite similarto magnetic systems in their ordering, we willbriefly state a few theoretical results for antifer-romagnets. For an isotropic Heisenberg antifer-romagnet'~ n would equal ev, where 8= -', and v isthe exponent for the inverse correlation length(K = Kpe ), Thus, the conventional choice of v = 3

would be consistent with our value of n = 1. How-

ever, the Hamiltonian for NH, Br is complicatedowing to the presence of two competing interac-tions, and it does not seem that one should expectagreement with theories based on an isotropicHeisenberg model. The behavior of the criticalspin relaxation rate for an anisotropic Heisenbergantiferromagnetic seems to be best described by"conventional" (van Hove-type) theory, '7'" ac-cording to which I/v should vary like the inversesusceptibility. Thus the exponent n should equaly, which on the basis of scaling relations is givenby y=(2-q) v. The usual choice of 0 = 0 and v = —',

gives y —„ in poor agreement with n. In muchprevious acoustic work, the variation of the re-laxation strength was unknown and ignored, whichled to the assumption that the exponent l in Eq. (1)was the same as n. In this view, the value l = 1.35would appear to be consistent with @=+3, but weconsider this to be a fortuitous agreement. Indeed,if the conventional magnetic theory is appropriateto NH4Br, the value of n would be consistent withthe classical (Landau) exponents 0= 0, v=-,', y= 1.Further progress requires a better understandingof the effective Hamiltonian for NH4Br and greaterdevelopment of the dynamical theories of criticalphenomena.

As a final point, let us compare the dynamicalaspects of the acoustic behavior of NH4Br withthose of NH4Cl. Although the behavior of the max-imum dispersion (i. e. , the longitudinal relaxationstrength C) is very similar in the disordered phaseof NH4Cl and NH48r, the variation in the relaxa-tion time is strikingly different. For NH4CI at1 atm I/a= 414x10' eL6 sec '

up to n. T=+ 50'K. 'Furthermore, there is no dispersion or criticalattenuation associated with the c' wave in NH4Cl.Thus the dynamical critical behavior in these twoisomorphous crystals differs considerably.

~Work supported in part by the National Science Foundation.'C. W. Garland and R. A. Young, J. Chem. Phys. 49, 5282

(1968).'Y. Yamada, M. Mori, and Y. Noda, J. Phys. Soc. Jap. 32, 1565

(1972); A. Huller, Z. Phys. 254, 456 (1972).3C. W. Garland and R. J. Pollina, J. Chem. Phys. 58, 5002(1973).

'C. W. Garland and C. F. Yarnell, J. Chem. Phys. 44, 1112(1966).

'G. J. Rosasco, C. Benoit, and A. Weber, in Proceedings of theSecond International Conference on Light Scattering of Solids,edited by M. Bakanski (Flammarion, Paris, 1971), p. 483.

C. W. Garland and D. D. Snyder, J. Phys. Chem. Solids 31, 1759(1970).

'C. W. Garland and R. Renard, J. Chem. Phys. 44, 1130 (1966).C. K. Choo and C. W. Garland, J. Chem. Phys. 59, 1541(1973).

'According to C. H. Wang and R. B. Wright IJ. Chem. Phys.

5152 C. %. GARLAND AND C. K. CHOO

57, 4401 (1972)], the tetragonal axis was aligned parallel to thedirection of the temperature gradient in their Ramaninvestigation. However, there is no evidence that this shouldalways be true, and the present shear velocity data indicate thatit is not.

' A. Bonilla, C. W. Garland, and N. E. Schumaker, ActaCrystallogr. A 26, 156 (1970).

"V. Hovi, K. Heiskanen, and M. Vartera, Ann. Acad. Sci. Fenn.No. A144 (1964);V. Hovi, K. Paavola, and O. Urvas, Ann. Acad.Sci. Fenn. No. A291 (1968).

"C. W. Garland and R. A. Young, J. Chem. Phys. 48, 146 (1968)."T. S. Velichkina, O. N. Golubeva, O. A. Shustin, and I. A.

Yakovlev, Bull. Acad. Sci. USSR Phys. Ser. 35, 932 (1971)."K. Kawasaki, Int. J. Magn. 1, 171 (1971)."K. Kawasaki, Phys. Rev. A 1, 1750 (1970).', B. I. Halperin and P. C. Hohenberg, Phys. Rev. j.77, 952 (1969)."G. E. Laramore and L. P. Kadanoff, Phys. Rev. 187, 619 (1969)."E.Riedel and F. Wegner, Phys. Rev. Lett. 24, 730 (1970); E. K.

Riedel, J. Appl. Phys. 42, 1383 (1971).