Specific heat of NaNO2 near its transition points

J. Phys. Chem. Solids, 1973,Vol. 34, pp. 57-66. PergamonPress. Printedin Great Britain

SPECIFIC

HEAT

OF

NaNO2 NEAR POINTS

ITS

TRANSITION

I. H A T T A D e p a r t m e n t of Physics, Faculty of Science, T o k y o Institute of T e c h n o l o g y , O h - o k a y a m a , Meguro-ku, T o k y o 152, Japan and A. IKUSHIMA T h e Institute for Solid State Physics, T h e University of T o k y o , Roppongi, Minato-ku, T o k y o 106, Japan (Received 17 D e c e m b e r 1971 ; in revised f o r m 25 April 1972) Abstract-- U s i n g an a.c. technique, the specific heat of N a N O ~ was m e a s u r e d as a function of temperature near its antiferroelectric-to-paraelectric phase transition point (Tu). T h e transition was found to be of the second order. T h e critical e x p o n e n t s are; a = 0-38 for E = 2 x 10-4 - 1 × 10-1, and c t ' = 0-18 for e = - 2 x 10-4 - - - 3 x 10-3. T h e critical e x p o n e n t s d e d u c e d from the scaling-law relations are roughly close to the values obtained from a r a n d o m phase approximation for a s y s t e m with an isotropic interaction. H o w e v e r , a difference was recognized between the observed e x p o n e n t for the specific heat and the values theoretically given for T > TM by the r a n d o m phase approximation for a s y s t e m with a short-range interaction or for a s y s t e m with a long-range dipolar interaction. A t h e r m o d y n a m i c a l analysis was made by using the generalized Pippard relation, and the present result was found to be consistent with the pressure d e p e n d e n c e of the antiferroelectric transition point. 1. INTRODUCTION

scaling-law relations[8, 9]. In many ferroelectrics, the precise determination of the critical exponent for the specific heat is a key factor in the verification of the scaling law at present [ 10]. It is believed in many ferroelectrics that the mean-field approximation holds well even in the close vicinity of the transition point[11]. However, some deviations of the exponents from the values given by the meanfield theory are expected very near the transition point. The reason for this is that, when the free energy density is expanded in terms of the polarization density, it is not valid to neglect terms related to the space derivatives of the polarization density as compared to those involving the square of the polarization density [12, 13]. Furthermore, the terms of the space derivatives of the polarization density most sensitively reflect a kind of the interaction which is responsible for a phase transition. From the theoretical view point, it is not adequate to deal with

measurements of the static electric susceptibility and the spontaneous polarization near the phase transition point, and the polarization along the critical isotherms in ferroelectrics have been made by many researchers [ 1-6], most frequently in triglycine sulfate (TGS) crystal. As for specific heat, a rather careful measurement in KH2PO4 was recently done by Reese and May by using a conventional discontinuous heating technique[7]. They have pointed out that the divergence of the specific heat near the transition point is better described by a logarithmic law than by a power law in the region - 1 . 0 ° K < T--Tc
57

58

I. H A T F A and A. I K U S H I M A

ferroelectrics or antiferroelectrics as a system with a simple isotropic interaction. Therefore, the effect of the long-range dipolar interaction will be considered in this paper. While, it is important to measure the specific heat in order to determine possible deviation of a from the value, c t = 0 , given by the mean-field theory. NaNO2 undergoes a phase transition of the first order from the ferroelectric to the sinusoidal antiferroelectric phase at about 437°K (Tc, hereafter), which is followed by a second order transition to the paralelectric phase at about 438°K (TN, hereafter), usually called the N6el temperature. Sakiyama et a/.[14] have carried out a comparatively accurate measurement of the specific heat around these transitions. They obtained the total entropy of the transition as 1.26 +-_0.08 cal/mol from the anomalous part of the specific heat associated with the ferroelectric ordering of dipoles. This is almost Rin2 (-1.38 cal/ mol), the value expected for complete disorder in an Ising-variable system. However, their results did not allow possible divergence from the power-law near the N6el temperature to be confirmed. The present experiment has been performed in order to establish the behavior of the second order phase transition near the transition point, Ts. In this we have used a precise a.c. calorimetry method, which enabled us to clarify the above-mentioned problem. In Sections 2 and 3, the experimental method and the results are respectively described. We discuss in Section 4 the thermodynamical characteristics and the critical exponents derived from the experimental results. 2. EXPERIMENTAL PROCEDURE

The specific heat was measured by the a.c. calorimetry technique, which has been utilized by Handler, Mapother, and Rayl[15] and by Sullivan and Seidel[16]. If a piece of a sample is heated and cooled alternatively by a heat source with negligibly small heat capacity,

the temperature of the sample correspondingly increases and decreases. Then, we can obtain the specific heat of the sample by measuring the amplitude of this temperature oscillation, which is inversely proportional to the specific heat of the sample as shown later. An a.c. signal of this sort can generally be detected more sensitively than the usual d.c. signal by the use of a phase-sensitivedetector. The magnitude of the heat capacity can be determined by this a.c. calorimetry technique some 10~ - 10z times as sensitively compared with traditional methods. As it is not so easy in practice to cool the sample, heat is added instead and then shut off by turns. Therefore, the heat contains both d.c. and a.c. components. The d.c. component must be absorbed through an exchange gas to a heat sink so as to avoid an abnormal increase in the temperature of the sample. On the other hand, the a.c. component of the heat must remain without being affected by the heat sink. In the present experiment, radiation of light from a halogen lamp was used as the heat source as shown in Fig. 1. This radiation

PI

Jp le

TDc TB

~c Fig. 1. A schematic diagram of the measuring system in the present a.c. calorimetry method.

SPECIFIC HEAT OF NaNO2

59

was chopped with the desired frequency, Research HR-8 lock-in amplifier driven f Hz. Helium or air was used to allow heat synchronously with the chopped light. exchange between a sample platelet of L cm Another thermocouple was used to measure thick and a thermal bath made of a thick the dc temperature increase at the sample, copper block which acted as the heat sink. To.c, over the temperature of the thermal An increase of the d.c. temperature of the bath TR. Therefore, the actual temperature of sample decays due to conduction through the sample is given by Td.c.+TB. For the the exchange gas with a relaxation time measurement of the temperature of the thermal "cs= pCLd/kg sec, where p and C are the bath, a platinum resistance thermometer density and the specific heat of the sample, supplied by the Rosemount Eng. Co. with an respectively, d the distance between sam- accuracy of___20mdeg. was used. ple and the thermal bath, and ks the thermal Ta.~. and Td.c. are given as follows[15, 16]: conductivity of the exchange gas. As a result, the sample has a slightly higher temTa.c. 2V2Q0 = T 47rfCpLa' (1) perature than the thermal bath. The chopping frequency f should be higher than 1/2rrrs so as not to dissipate the a.c. component of the Q 0d (2) Td.e. -- 21rkga' heat from the sample. A larger signal is thus obtained, and analysis is simplified. The upper limit of the f r e q u e n c y f must be chosen so that where Q0 is heat in J/sec by the light radiathe thickness of the sample L is smaller than tion on the upper surface of the sample with an area a in cm 2. As the radiation is actually the thermal length in the sample, cm[15, 16], where D is the thermal diffusivity chopped in the form of a square wave, the of the sample in cm2/sec. That is, the fre- factor, 2V~/~r, in equation (1) of the rightquency f should be smaller than L2/~D Hz. hand side corresponds to detection of the The thermal diffusivity of NaNO2 is esti- rms amplitude for the fundamental frequency mated as 0-009 cm2/sec at room temperature f of the square wave. In the present work, from the values of the specific heat[14] and equation (1) was used to deduce the specific the thermal conductivity[17]. In the present heat since the linear relationship between experiment, L was around 0.01 cm, and LZ/~rD Ta.e. and 1[f was found to hold for frequencies was approximately 76 Hz. On the other hand, between 2 and I0 Hz. A maximum temperathe thermal relaxation time from the sample ture modulation of about 4 mdeg was employto the thermal bath was calculated to be 7.7 ed to produce a signal of 150 nV so that the signal to noise ratio was more than 50 and sec under the experimental condition ( d = 0-1 cm, the exchange gas was air at one atmos- the specific heat was determined with a phere) and the relaxation rate was accordingly precision of 2 per cent. The instrumental 0-02Hz. The actual relaxation time was rounding of the specific heat related to the determined to be about 7 sec. Therefore, the modulation was less than e(= ( T - - T,v)/Tu) .~frequency selected was 6 Hz. 1 x 10-5. In equation (2), the temperature Two pairs of chromel-alumel thermocouple dependence of the right-hand side term is wires of 1 mil dia. were used as sensors. They believed to be very small, although a temwere attached to the rear face of the sample perature dependence of Ta.e. was in fact platelet with General Electric 7031 adhesive observed. Td.c. decreased from roughly as shown in Fig. I. One thermocouple was 150 to 9 0 ~ V , as the temperature increased used to detect the amplitude of the a.c. through the whole temperature range of the temperature, response Za.e.. The rms emf measurement. This anomalous effect might be was measured with a Princeton Applied due to the heat convection through the ex-

60

I. H A T T A and A. I K U S H I M A

These data were analyzed with the following power-law formulae,

change gas and/or the radiation from the surface of the thermal bath. As the measurement of Td.e. could be made as accurately as 0.1/zV or 2.5 mdeg and T8 could be determined relative to TN with an accuracy of better than 5 mdeg, the accuracy determining the normalized temperature ~ was also better than 1 x 10-5 . Furthermore, as very small samples were used in the present experiment in comparison with other works, the rounding due to the inhomogeneity of every sample is thought to be very small.

C=`4-lel-~'+(Bo+Be)for T < TN, (3) C = ` 4 + c ~ + ( B o + B e ) f o r T > T N, (4) = ( T - - TN)/TN,

where et and a' are the critical exponents of the specific heat above and below TN, respectively, and A+, ,4_, etc. are temperatureindependent factors. The second term of the right hand-side of equation (3) or (4) corresponds to the lattice part of the specific heat. The present results were analyzed by a leastsquares fit, and the results are summarized in Table 1. In this analysis, we omitted the data for lel ~< 2 x 10-4 because the data were found to be unreproducible from sample to sample. The temperature of the maximum specific heat near T~¢ was 437.203°K, for example, while T~¢ from the least-squares analysis was found to be 437-227°K. This difference might be due to a smearing of the transition caused by some kinds of defects in the sample. It is at least established that c~ > et' and A+ < A_, although the listed values are less accurate for T < T~¢ because the first order transition point Tc lies just below TN and

3. E X P E R I M E N T A L RESULTS AND DATA ANALYSES

Figure 2 shows a typical result of the measured specific heat as a function of temperature. The value of the specific heat is scaled by reference to the measurement at 468°K by Sakiyama et al.[14]. Figure 3 shows the data near the N6el temperature with a more expanded temperature scale. The earlier result by Sakiyama et al. using the traditional method[14] is also plotted by adjusting the peak of the specific heat near TN to lie at the present N6el temperature. These two results agree qualitatively. 80 7O o o

3 4Q J¢ ¢J

3o

a.

•

Ferro -phase

IO

I

0

3O0

~emeeooejme~ m

°°

I

I

t

I

t

I

I

35O

I

f

I

400

Tempero'l'ure,

Para-phase

An'l'ifen

-

t

t

I

phase I

I

f

450

K

Fig. 2. Specific heat of N a N O 2 plotted vs temperature in the very wide temperature-range.

S P E C I F I C H E A T O F NaNO2 5x10 - 3

E = 10 -410 -3

2 xlO -2

10 -2

3 xl0

-a

II

6(1

I

I

o o o o

i

% oo

~

go

4C

I

I

oo

50

m o

61

I

I

I

I

I

I

I

I

I

i

C= "T.4 IE I-°'18 + ( 18.8.1.56.3 e )

T
C=0.77e -°3a

T >T N

+ (18"8+36"3~)

N

• •

o o

S a k i y o m o - Kimo~o-Seki

o . Present

resul$

o u 4-

3G

.ic

~ ° ° ° 1 ° ° ° o o oo o o o o

• o ooo • oo Ioo o o o o loo oo o ooo o oooo ~ o oo ooo ooo< oo oo oo o oooo

2(? C N o n - d ivergent = 18 " 8 + 3 6 " 3 E

G'

I 436

I

i

I

i

I

t

440

I

I

i

445

Tempero'l'ure,

I

I

I

I 450

i

I 453

K

Fig. 3. Specific heat of NaNO~ near its N~.el temperature plotted vs temperature. The straight line shows nondivergent term obtained from the analysis by a least-squares fit.

Table 1. Critical exponents and constants of the specific heat analyzed by equation (3) and (4) near the N~el temperature in NaNOe NAN02 near T~,

T > TN a A+

0.38(e = 2 x 10-4 - 1 x 10-~) 0.77 cal/mol • deg

T
0 - 1 8 ( e = - - 2 X 10-4 -- --3X 10-a) 7-4 cal/mol - deg 437-227"K 18.8 cal/mol • deg 36"3 cal/mol - deg

therefore the temperature interval between Tc and T~; is quite narrow. The m~ignitude of the second term of the right-hand side in equations (3) or (4) is also shown in Fig. 3 by the straight line. The critical exponent of the specific heat above Ts is well deter-

mined as ct= 0.38___0.01 in the temperature range of 2 x 10-4 ~ E ~ 1 × 10-1. Figure 4 is a log-log plot of the divergent part of the specific heat vs the normalized temperature e. The divergence of the specific heat observed here at the N6el temperature appears to show the characteristic features of a second order phase transition. From measurements of the static uniform susceptibility previously made[6], we were unable to establish that the transition was second-order because the susceptibility did not diverge at the N6el temperature. The antiferroelectric-to-ferroelectric phase transition shows behavior typical of a first order phase transition. In the present study the time constant of the lock-in amplifier was 30 msec, which was too long to observe the latent heat which appears within a short time interval. Furthermore, the use of thermocouple detectors is not a very effective means of observing heat released or absorbed

62

I. H A T T A

and

A.

IKUSHIMA

i0 z

o~e'~t~"~-,~.

I

T,:T N

a' : 0 " 1 8

o E

I01

T>'rN

o

~

a=0"38 c

,?, 113

I 10 -s

[

I IIIll

I 10-4

I

I IIII1[

I 10-3

I I IIIIli

I 10-Z

I

I I IIII 10-1

I(I

Fig. 4. A log-logplot of the anomalousspecificheat in NaNO._,above T,vvs the reduced temperature. CANOMALOUS slowsthe divergentpart of the specificheat measured. locally, as at a first-order transition point 4. D I S C U S S I O N the boundaries between the new phase and The second order phase transition at Ts the remaining phase move in the crystal. in NaNOz cannot be classified as an EhrenHowever, the d.c. temperature difference lest type of transition. Rather this transition observed in the present study was thought corresponds to a second order transition of very sensitive to the presence of the first type 2c according to Pippard's classification order phase transition. In the present measure- [18], which appears in the order-disorder ment, we observed a very sharp change of transitions of /3-brass and NH4C1, and in Td.e. at To, and not at TN. This behavior the antiferromagnetic transition of CoO, shows that Tc is the first order transition for example, and is characterized by an point and also that Tu is of the second asymmetric divergence in the specific heat order. above and below the transition point. ThereIn the ferroelectric phase, the specific heat fore, we should apply to this transition in versus T curve could not be expressed by a NaNOz the thermodynamical treatments of simple function of temperature such as equa- Pippard [ 18] and of Buckingham and Fairbank tion (3). Furthermore, these curves showed [19], though in an earlier study of this tranvery different behavior from sample to sample, sition Ehrenfest's equation [20] was used. though in both the antiferroelectric and the For anisotropic materials, the thermodyparaelectric phase no difference could be namical relations between the heat capacity detected. The effect in the ferroelectric phase at constant stress, C, and the thermal expanis probably related to formation and destruc- sion coefficients at constant stress, a~(tz--tion of the ferroelectric domains, since in the a, b, c in Cartesian co-ordinates referred to present study nothing was done in order to the axes of NaNOz), have been given by make a single domain over the whole crystal. Janovec in an extension of the BuckinghamQualitatively, the specific heat in the ferro- Fairbank treatment for the h-transition in electric-to-antiferroelectric phase transition liquid helium [2 ! ]. They are expressed as: increases with increasing temperature over a wide temperature range up to Tc. The latent _(OTs~ C T~ heat appears at Tc and the specific heat a,, = \ 0s~/,,, "-T + ~ ' ' (5) suddenly decreases as shown in Fig. 2.

SPECIFIC

HEAT OF NaNO2

where s . is the tensile stress along the /xdirection, /x' includes all directions except the /z-direction, and a S , is the thermal expansion coefficient along the transition-line in the pressure-temperature diagram which varies relatively slowly with temperature. In the paraelectric region of NaNO2, values of (aTN/as.).~, are derived from the present results combined with the thermal expansion coefficients found by M a r u y a m a and Sawada [22] as follows:

OTN

Os----~= -- (7 -- 2) × 10 -s deg/bar,

OTN_

Osb

(10-- 1) X 10-3 deg/bar,

63

and d is the dimensionality of the system. By using equation (6) and a = 0-38, the exponent u in NaNO~ is found to be 0-54. On the other hand, the electric susceptibility with wave vector q lying along the direction of the staggered polarization P ( q ) is given in the paraelectric phase as B x(qa) = (K2+ (qa_Q)2},_m2,

where Q is the wave vector of the staggered long-range order which grows below TN, and the critical exponent r/ is defined in the pair correlation function of the Ising variables as

OTN OSc = + (8 -- 2) × 10-3 deg/bar.

e-~r

(GoO'r)

As their thermal expansion coefficients are not accurate, the errors given above are rather large. T h e dependence of the N6el temperature on the hydrostatic pressure P is easily obtained from the above values;

rd_2_n,

(8)

r being the distance. When qa is equal to Q in equation (7), the staggered susceptibility

x(Q) ~/(-2+-0

~

(T--TN) -l'(2-rD

( T - - Ts)-L

dTN

(7)

(9a) (9b)

(OTlvq_OTN+ aTN~

dP = -- k aso

as~

as~ /

= (9 +_ 5) x 10-3 deg/bar. This value may be compared with the value of 5 . 6 × 10-s deg/bar determined by Gesi et al.[20]. T h e agreement is satisfactory. T h e linear relation between the specific heat and the thermal expansion coefficient seems to hold well in NaNOz. We next consider the scaling-law relation for the paraelectric-to-antiferroelectric phase transition. According to Kadanoff et al. [23], we have the relation,

dv = 2 - a,

(6)

We have shown in a previous paper a value for the critical exponent y of 1.11 _+0.05 from the static electric susceptibility assuming K > Q [6]. Then, by taking the exponents of equations (9a) and (9b) to be equal, and using the value of u and 3' given above, we have "0 = 0.06 +0-09. T h e s e results on the critical exponents are summarized in Table 2. What is very important in our result is that the critical exponent of the specific heat is

Table 2. Values of the critical exponent in NAN02. a and Y were obtained experimentally, and ~ and v were derived from the above values of s a n d y by using the scalinglaw relations

where v is the critical exponent of the inverse correlation length K, that is, Critical e x p o n e n t s

K ~ ( T - TN)~,

c~

T

r/

u

0-38

1.11

0.06 -----0.09

0-54

64

I. HATTA and A. IKUSHIMA

obviously greater than zero. As mentioned in the second paragraph of Section 1, we need to consider the deviation from the mean-field theory without neglecting terms related to the space derivatives of the polarization density in the free energy density. In the discussion on the deviation of the critical exponent, it should be very suitable to use the random phase approximation (RPA) for a system with an isotropic interaction energy are also summarized in Table 3. As for the specific heat, the value of the exponent is the same even in a system with a short-range interaction. Except a, on which a great interest is focused here, the exponents obtained from the present experiment in Table 2 are found to be close to those by the above R P A listed in Table 3. We emphasize here that, in Table 3, the scaling-law relation of equation (6) cannot be checked in the mean-field theory, whereas it holds for above R P A values. On the other hand, another scaling-law relation a' + 2/3 + y' = 2 holds only in the case of the mean-field theory. It is inherently assumed in the above discussions that the correlation length K-1 is isotropic. H o w e v e r , this is not strictly correct in NaNO2, since the crystal structure of NaNO2 is not cubic, and in addition, electric dipolar interaction is known to show very

Table 3. Comparison o f theoretical values of the critical exponents obtained by the meanfield theory and by the RPA for the system with an isotropic interaction Temperature

T < Tc

Critical exponents

a'

Values from the mean-fieldtheory

0* 1/2 1 - -

Values from RPA for a system with an isotropie interaction

/3 3" v'

peculiar direction-dependence as shown in the following [24]:

J(q) = J ( Q ) - S q 2 - ~ c o s 2 0

(I0)

near Q, where J(q) is the Fourier q-component of the interaction energy, 8 and ~ are constant factors and 0 is an angle between the polarization P(q) and the wave vector q. In the mean-field approximation the second and third terms on the right-hand side in equation (10) are neglected. In case of an isotropic interaction the third term on the right-hand side in equation (10) should be omitted. Yamada and Yamada have observed the X-ray diffuse scattering in NaNO2 with respect to the fluctuation of the staggered polarization [25] and clearly observed that K is anisotropic, that is, the temperature-independent factor of K in equation (7) is smaller in the antiferroelectric b-direction (P(q)//b) than those in other directions, although their result shows essentially the R P A values given in Table 3; v ~ 1/2 and r / = 0. T h e y also concluded on theoretical grounds that this anisotropy is mainly due to dipole-dipole interactions as given in equation (10), though the correction term that should appear in J(q), ~ cos 2 0, have been omitted from their paper. Then, we next derive from R P A the critical exponent of the specific heat for a system with long-range dipolar interaction. F r o m the summation of the temperature-derivatives of the internal energy [26] up to the wave vector corresponding to X / ~ - ~ , an average excitation energy level at T, we deduce the following result;

T = Tc T > Tc 8

"0 a 3' u

3 -- 0* 1 -

1/2 I/2 1 1/2 3

0 1/2 1 1/2

*The specificheat has a discontinuity at Tc.

C = 4kB~1/283/2 ]In(T-- T~)I + const.

(11)

for 1, where 8 is finite and independent of the direction of q. Such a singularity has also been obtained by Levanyuk[27] and by Pytte and Thomas[28]. It is not surprising that the above exponent, a, no longer satisfies equation (6) because we assumed that the interaction was isotropic in

S P E C I F I C HEAT OF NaNO2

the derivation of this equation. However, the value of the critical exponent of the specific heat obtained experimentally here is quite larger than zero (logarithmic divergence). Rather the experimental value is closer to that for a system with a short-range interaction. This may be thought that a great deal of the interaction energy consists of short range interactions due to ordering of near-neighboring ions and the remaining part is accounted for by long-range dipolar interactions. This problem should be settled in future. Finally, we discuss briefly the ferroelectric phase transition. As mentioned before, this transition is of the first order. However, we have suggested that this transition is very close to the second-order one because the observed specific heat in the ferroelectric phase diverges very near to Tc, the asymptotic point being only about 0.4°K higher than Tc. The most remarkable feature of the specific heat in this transition is that the specific heat increases with temperature over a wide temperature range below the transition point and diverges at the asymptotic point, while no anomalous tail appears above this transition point as shown in Fig. 2. These results can be qualitatively explained by using the phenomenological theories of Strukov et al.[29], Pompe et al.[30], and Sawada [31]. In the Devonshire theory[l l] the free energy is expanded in terms of the polarization up to the sixth order. If the coefficient of the fourth order term is positive, a second order phase transition takes place, whereas a first order phase transition occurs if it is negative. One of the most interesting cases is that where the coefficient of the fourth order term is zero[29-31], and is classified as a phase transition of intermediate order between first and second, where C ( T o - - T ) -112 for T < Tc and C = const, for T > Tc. This qualitatively shows t h e same features as the present results. A somewhat similar temperature-dependence has also been observed in the ferroelectric phase transition of AgNa(NO2)2132]. In KH2PO4,

65

the observed anomalous specific heat has been analyzed by using the above relation [29]. Acknowledgements-The authors wish to thank Prof. M. B. Salamon of the University of Illinois for many valuable discussions, both of the experimental and the theoretical aspects, while he was at The Institute for Solid State Physics, University of Tokyo. One of the authors (I.H.) thanks Prof. Sawada of Tokyo Institute of Technology for very useful discussions. REFERENCES 1. C R A I G P. P.,Phys. Lett. 20, 140(1966). 2. G O N Z A L O J. A., Phys. Rev. 144, 662 (1966); ibid. B1, 3125 (1970). 3. N A Z A R I O I. and G O N Z A L O J. A., Solid State Commun. 7, 1305 (1969). 4. N A K A M U R A E., NAGA1 T., I S H I D A K., ITOH K., and M I T S U i T., J. phys. Sac. Japan 28, suppl., 271 (1970). 5. BLINC R., B U R G A R M., and LEVSTIK A.,Solid State Commun. 8, 317 (1970). 6. H ATTA 1., J. phys. Sac. Japan 28, 1266 (1970). 7. REESE W. and MAY L. E., Phys. Rev. 144, 662 ( 1966); ibid. 162, 510 (1967). 8. S T R U K O V B. A., Soviet Phys.-Solid State 6, 2278 (1965). 9. T E L L O M. J. and G O N Z A L O J. A., J. phys. Sac. Japan 28, suppl., 199 (1970). 10. H A T T A I. and I K U S H I M A A., Phys. Lett. 37A, 207 (1971). 11. L A N D A U L, D., Phys. Zeits.f Sovj. 12, 123 (1939). 12. D E V O N S H I R E A. F., Phil. Mag. 3, suppl., 85 (1954), 13. G I N Z B U R G V. L., Soviet Phys.-solid State 2, 1824, (1960). 14. S A K I Y A M A M., KIMOTO A.. and SEKI S., J. phys. Sac. Japan 20, 2180 (1965). 15. H A N D L E R P., M A P O T H E R D. E., and RAYL M.,Phys. Rev. Lett. 19, 356 (1967). 16. S U L L I V A N P. F. and S E I D E L G., Phys. Rev. 173, 679 (1968). 17. Y O S H I D A I. and S A W A D A S.,J.phys. Sac.Japan 16, 2467 (1961). 18. P I P P A R D A. B., The Elements of Classical Thermodynamics, Chapter 9, Cambridge University Press, London (1964). 19. B U C K I N G H A M M. J. and F A I R B A N K W. M., Progress in Low Temperature Physics (Edited by C. J. Garter), Vol. 3, Chapter 3, North-Holland, Amsterdam ( 1961). 20. GESI K., O Z A W A K. and T A K A G I Y., J. phys. Sac. Japan 20, 1773 (1965). 21. J A N O V E C V.,J. chem. Phys. 45, 1874 (1966). 22. M A R U Y A M A N. and S A W A D A S., J. phys. Sac. Japan 20, 811 (1965). 23. K A D A N O F F L. P., G O T Z E W., H A M B L E N D., H E C H T R., LEWIS E. A. S., P A L C I A U S K A S V. V., RAYL M., S W I F T J., A S P N E S D. and K A N E J., Rev. mad. Phys. 39, 395 (1967).

66 24. KRIVOGLAZ M. A., Soviet 5, 2526 (1964). 25. Y A M A D A Y. and YAMADA Japan 21, 2167 (1966). 26. BROUT R., Phase Transitions, min, New York (1966). 27. L E V A N Y U K A. P., Bull. Aead. Ser. 29, 885 (1965).

i. HATTA and A. IKUSH1MA

Phys.-solid State T., J. phys. Soc. Chapter 2, Benja-

Sci. USSR Phys.

28. PYTTE E. and THOMAS H., Phys. Rev. 175, 610 (1968). 29. STRUKOV B. A., AMIN M., and KOPCHIK V. A., Phys. Status Solidi 27, 741 (1968). 30. POMPE G. and H E G E N B A R T H E., Soviet Phys.solidState 12, 357 (1970). 31. SAWADA S., private communication. 32. GESI K.,J. phys. Soc.Japan 28, 1377 (1970).