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Experimental determination of the kapitza resistance from the relaxation behaviour in antiferromagnetically ordered MnCl2·4H2O and MnBr2·4H2O
33
Physica 112B (1982) 33-41 North-Holland Publishing Company
EXPERIMENTAL RELAXATION
DETERMINATION
BEHAVIOUR
OF THE KAPITZA RESISTANCE
IN ANTIFERROMAGNETICALLY
FROM THE
ORDERED MnCly4Hz0
AND
MnBry4Hz0 H.A.
GROENENDIJK,
Kamerlingh
Received
Onnes Laboratorium
5
J. FLOKSTRA*
and A.J. van DUYNEVELDT
der Rijksuniversiteit
Leiden,
Leiden,
7’he Netherlands
June 1981
The relaxation behaviour of the antiferromagnets MnClr.4HzO and MnBrr.4HzO near their phase transitions have been determined with an ac susceptibility technique. The time constants show a maximum at the field-dependent transition temperature, while also a linear dependence of T upon crystal size exists. A description of the data is given by considering the Kapitza resistance between the crystal and the liquid-helium bath as the dominant factor in the relaxation process. From our results the Kapitza resistance could be determined as 1.9 X lo-‘T’ K mz W-’ and 0.89 x 10d3Tm3K mZ W-’ for the chloride and the bromide, respectively.
1. Introduction Non-resonant relaxation studies have been long a subject of interest in low-temperature magnetism. Already in the early years it was found that the Casimir-Du Pre model for magnetic relaxation [l] provides an adequate method for the derivation of time constants from observed ac susceptibilities. The physical interpretation of these time constants was less simple. On the one hand, resonance experiments usually revealed different time constants for directly comparable systems, while, on the other hand, the observed temperature and field dependences of the time constants rarely agreed with the behaviour as predicted by theory [2]. In the last decade it was possible to account for most of the discrepancies between theory and experiment by exploring the systems at strong external magnetic fields [3]. Simultaneously, extended computer facilities allowed the numerical solution of a more sophisticated model in which all the
* Permanent address: Technische Hogeschool Twente, Afdeling der Technische Natuurkunde, Enschede, The Netherlands.
037%4363/82/0000-0000/$02.75
@ 1982 North-Holland
various kinds of energy exchange between the magnetic spin system and the cooling liquid are considered [4,5]. As a result most of the earlier existing difficulties are understood, and it is now possible to make use of a non-resonant relaxation study for other problems, e.g. the determination of phonon lifetimes [6]. The present paper deals with a study of the relaxation behaviour near the paramagneticantiferromagnetic phase transition. It will .be discussed that the observed time constants are strongly influenced by the Kapitza resistance between the magnetic crystal and the liquid-helium bath,
and that in fact ac susceptibility
ments provide low-temperature
2. Experimental
an elegant Kapitza
measure-
way of determining resistance.
the
results
Dynamic susceptibility measurements have been performed on MnBr,*4H,O and on the o-form MnClz-4Hz0, which is stable at room temperature. The crystal structure of the chloride is monoclinic - space group pi/n - with a = 11.186& b = 9513A, c = 6.186A and p =
34
H.A. Groenendijk et al. I Kapitza resistance and relaxation behaviour in antiferromagnetic crystals
99.74” [7]. There are four molecules in the unit cell. The structure consists of discrete octahedra with a Mn2+ ion in the centre. These octahedra are formed by four oxygens and two chlorines and are slightly distorted. Four of the hydrogen atoms
are
between
involved
neighbouring
in
the
hydrogen
manganese
bonding
ions.
compound
work
orders
can be found
below
in [8,9].
The
T, = 1.62 K as a simple
two-sublattice antiferromagnet with collinear spins. The present measurements were made along the c’ axis (perpendicular to the a and b crystal axes) which is close to the easy axis of the antiferromagnetic alignment [8]. Moreover, the results of our study are not subject to large changes in the relative direction of the magnetic field (section 3). The crystallographic structure of MnBrz*4Hz0 is isomorphous to that of the chloride [lo]. The magnetic structure is not known in detail, but it is not likely to be much different from that of MnC12*4Hz0. The zero-field antiferromagnetic ordering temperature of the bromide is 2.13 K [ll]; more details about, e.g., the magnetic phase diagrams of both compounds can be found in
frequency
MnC12.4H20 was reported in 1958 by Lasheen et al. [13]. Later Soeteman et al. [14-161 performed a study on both MnC12*4H20 and MnBrl*4Hz0, showing that T(T, H) resembled the anomaly in the specific heat cH(T, H). Our present measuring technique is basically the same as those used in the above studies [U-16]. Variation in the mutual inductance of a set of coils is directly detected by means of a PAR 124A with a two-
model
PAR
127 [17]. The oscil-
range
the the
from 0.2 Hz to 15 kHz.
The experiments were performed on single crystals which were grown at room temperature slowly
MnC12.4H20
evaporating or
MnBr2*4H20
grade).
Large
crystal known
faces were morphology
single
crystals
solutions of (Merck, reagent with
easily obtained. of the crystals
well-defined From the [lo] it was
possible to orientate the external magnetic field parallel to the C’ axis within a few degrees. An example of an Argand diagram k” versus x’) as measured for MnBr2*4H20 (H]]c’) is shown in fig. la. The curve is circular, which demonstrates that the relaxation process can be described with a single time constant. In cases where x”/,$ < 0.1 it is often easier to concentrate on the behaviour of only x”(o). An example of the frequency dependence of XI’ is depicted in fig. lb.
o.21
[9,121. For the present study we concentrate upon the behaviour of the ac susceptibility near the antiferromagnetic-paramagnetic phase transition in the temperature range between 1.2 and 2.0 K at external magnetic fields of up to 2T. The first observation that the relaxation time constant r derived from the susceptibility showed an transition in a phase anomaly at Such
accessory
lating field h always remains parallel to external field H. The system operates over
from
The magnetic properties of MnClr.4HrO have been studied extensively, the references to most of the earlier
phase
I./
b
1
F?
=x i
Fig. 1. x(o) for MnBr2.4HzO (Hljc’) at T = 1.90 K and H = OST. (a) Argand diagram; the numbers with the data give the ac frequency in Hz. (b) Normalized out-of-phase component x”/x~ ko, the zero-field susceptibility) as a function of frequency.
35
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaoiour in antiferromagnetic crystals
For calibration purposes the signals have been divided by the ,$ values at zero field ho); the solid line in this figure represents the curve [l]:
(1) where xr is the isothermal susceptibility, xs the adiabatic one, and r the relaxation time constant, derived from fig. lb as the reciprocal value of w at maximal absorption. A comparable set of data points for the chloride is given in fig. 2. In this case the Argand diagram shows more scatter due to the fact that the measurements had to be made at lower frequencies. However, the plot of x” versus w/27r (fig. 2b) closely resembles eq. (1) thus allowing an accurate determination of r; all relaxation times quoted in this paper were obtained from this kind of plots. The relaxation time constants of MnCl*-4H20 were obtained with a 760 mg single crystal. The
I
0.1
\
B
1 0/2~r
time constants for H = 0.4 and 0.6T are plotted as a function of temperature in fig. 3. It is seen that at low temperature in the antiferromagnetic region, r initially decreases slightly with increasing T. Near T,(H) there is a sharp increase of r. A maximum occurs at the field-dependent antiferromagnetic-paramagnetic phase transition T,(H), and then, in the paramagnetic region, the relaxation time decreases sharply. The field dependence of 7, measured at T = 1.44 K, is plotted in fig. 4. The relaxation time increases slowly as a function of the external magnetic field in the antiferromagnetic region. A sharp peak occurs again at the transition field H,(T), and then T decreases sharply in the paramagnetic region. The temperature dependence of the relaxation times for a 1300 mg single crystal of MnBrz*4H20 were measured for three values of the external magnetic field. As an example the data for H = 0.8T are plotted in fig. 5. Although the actual time constants are about a factor of three smaller than those of the chloride, their temperature dependence is quite similar. The field dependence of r is also similar to that of the chloride; we do not include an example in the figures. Our measurements on both MnC12.4H20 and
I
IO (Hz)
Fig. 2. ~(0) for MnC12.4H20 (H//c’) at T = 1.20 K and H = 0.4T. (a) Argand diagram; the numbers with the data give the ac frequency in Hz. (b) Normalized out-of-phase component X”/XOk,~, the zero-field susceptibility) as a function of frequency.
16:;
2.0 T(K)
Fig. 3. Relaxation time as a function of temperature for MnC12.4H20 (Hllc’). 0: H = 0.4 T; 0: H = 0.6 T; the drawn lines are proportional to CHT-~.
36
H.A.
”
Groenendijk et al. I Kapitza resistance and relaxation behaviour in antiferromagnetic
IO
P
4
‘+biV
IO H (T)
Fig. 4. Relaxation time as a function of external magnetic field for MnCly4H20 (Hllc’) at T = 1.44 K. The drawn line is proportional to CHT-~.
MnBr2*4H20 are in good agreement with previous data as regards the temperature and field dependence of 7 [15]. The interesting difference is, however, that the absolute values of r agree for the case of MnC12.4H@ and that our data for MnBrz*4H20 show time constants systematically 30% larger in the antiferromagnetic state. This difference cannot be ascribed to a lack of experimental accuracy, especially not because of the “ideal” Casimir-Du PrC behaviour of x’ and x” versus frequency (figs. 1 and 2). The slower
crystals
relaxation of MnBrz*4Hz0 indicates a possible sample size dependence of the relaxation time constants as the previous experiments on single crystals [15,16] were performed on samples of approximately 700 mg. This mass is roughly identical to that of the chloride crystal we used, but smaller for the case of the bromide. In order to verify this aspect we also performed measurements on a MnBr2.4H20 single crystal of 270 mg. Faster relaxation times (-50%) were indeed obtained for this crystal. The above crystal size dependence of r indicates that the observed relaxation times are not due to a magnetic relaxation process only. In that case it is conceivable that the crystal orientation with respect to H may also not be a factor. Therefore, we measured the relaxation times of a crystal of MnBr2*4H20 (710 mg) with the b axis (which is the hard axis of the antiferromagnetic alignment [12]) oriented parallel to H. In order to get reasonable x” signals in this case the measurements have to be made at stronger external fields (2.0 T). The results (fig. 6) show not only the same behaviour as in the case H]]c’, but also the order of magnitude of the time constants is the same for both field orientations. From the measurements described above it can be concluded that it is unlikely that an intrinsic magnetic relaxation process is responsible for the observed relaxation behaviour. In the following discussion we will show that the
20 T
(K)
Fig. 5. Relaxation time as a function of temperature for MnBr2.4HzO (Hlc’) at H = 0.8 T. curve proportional to c”T-~.
Fig. 6. Relaxation time as a function of temperature MnBry4H20 ‘(Hllb) at H = 2 T.
for
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaviour in antiferromagnetic crystals
Kapitza resistance at the crystal surface governs the relaxation process.
3. Discussion It is known that the Casimir-Du Pre model may give an inadequate description of the relaxation behaviour at liquid-helium temperatures, especially in the case of experiments on single crystals [5,18]. The finite heat conductivity of the crystal lattice and the limitations in the heat transfer to the helium bath may lead to situations where the experimentally determined time constant is no longer related to the intrinsic relaxation processes. In order to analyze our results we will use a model which takes these effects into account. This so-called thermal conduction model [4,5] is basically an extension of the model of Casimir and Du PrC. In the thermal conduction model the cooling system (in our case, liquid helium) is considered as a separate system with thermal conductivity A, and specific heat c,, as is shown schematically in fig. 7. The heat exchange between the crystal lattice and the liquid helium is determined by the Kapitza boundary resistance Rk, the associated relaxation time rR can be written as [19,35] TR = YroC&
,
(2)
with y = l/2 if one approximates the crystal as a cylinder with radius r. and y = l/3 for the case of a sphere. In the lattice system (specific heat cL) it is assumed that the heat transport is characterized by the thermal conductivity hL. The introduction of the extra heat transfer processes in the thermal conduction model leads to situations where the relaxation times as derived from the Argand diagrams can no longer be identified with the intrinsic spin-lattice relaxation times rs. It can even occur that the curves in the Argand diagram become strongly flattened and that the determination of a characteristic time constant is no longer possible [4,5]. The thermal conduction model can be solved exactly. In the resulting expression for the susceptibility, several dimensionless parameters occur which determine the actual frequency dependence of x. These parameters are defined in table I. For the analysis of our data we have to calculate their magnitudes, but before doing so we first discuss the Kapitza boundary resistance Rk. Recent reviews of both experimental and theoretical work on the Kapitza resistance are given by Snyder [20] and Challis [21]. No really adequate theory for Rk exists; we will use the results from the phonon mismatch theory of Khalatnikov [22]. The value of Rk as calculated from Khalatnikov’s theory is often too high, but we will use the theory for an order of magnitude estimate. The following expression is derived: R
_ k
15h3&v:
r2k4pv
T-3
’
Table I Dimensionless parameters of the thermal conduction model [5]
Fig. 7. Thermal conduction model.
37
(3)
38
H.A. Groenendijk et al. I Kapitza resistance and relaxation behaviour in antifewomagneticcrystals
The quantities ps and v, are the density and the velocity of sound in the crystal, respectively; p and v are the corresponding values for liquid helium. We now restrict our calculations to the case of MnC12.4H20, as all the quantities necessary in the calculations are available for this compound; the result of our calculations will undoubtedly be of the correct order of magnitude for the bromide, too. We used ps = 2.0 X lo3 kg rnw3[23], while us was estimated from the Debye temperature (0, = 110 K is calculated from cL in [25]): us = 2.0 x lo3 m/s. For p and u we chose values representative for the temperature range of our experiments: P’ 140 kg mm3[24] and v = 2.4 X lo* m s-’ [24]. As a result we calculated Rk = 23 x 10m3Tm3 K m* W-l. Let us now consider the parameters of table I for the case of our experiments at T = 1.44 K and H = 0.4 T. The thermal conductivity constant AL of MnC12.4H20 was recently determined by Dixon et al. [26], who obtained the value of 50 W K-’ m-l at T = 1.44 K, independent of H. If we approximate our 760 mg chloride crystal as a cylinder, one obtains r. = 3.8 mm, thus yielding Q = 120. The parameter D is found directly from the experimental values for cH (25.2 X lo4 Jmm3K-’ [27]) and cL (48.9 Jme3K-’ [25]) to be 5.2 x 103. The parameter P contains the spinlattice coupling constant (Y,the value of which we do not know. Using the definition of r,(= cH/(Y [l]) one may write P = r&/hLTs and one calculates P = 7.3 x lo-*/~,. The actual time constants determined in our experiments set an upper limit to the possible values of TV. If one considers r = 0.25 s (T = 1.44 K and H = 0.4T) as being representative of this upper limit, we find that P is larger than 0.3. Finally, since the cooling liquid is helium below the A-point (A, is large) it seems reasonable to assume S = 0. With the above-estimated values for the dimensionless parameters we calculated x(o). We will not give the details of this lengthy calculation; all relevant information can be found in [5]. The resulting curves in the Argand diagram were circular, allowing a description with one
time constant. This single time constant, T, consists of two contributions according to T=T,+DT~.
(4)
Expression (4) is similar to the one derived by Stoneham [28] for the case of a phonon-bottlenecked spin system. Using eq. (2) we may write 7=
Ts + Yr&&
.
(5)
Not many theoretical treatments of the spinlattice relaxation TV in antiferromagnetically ordered systems exist. The most solid numerical prediction can be found in the work of Huber [29] (T~S lo-’ s). Apart from the earlier work on manganese chloride and bromide, experiments on the relaxation in antiferromagnetically ordered systems are reported for two cobalt compounds in [30,31]. Both references show T 10e3 s, but it is clear that these time constants are not free from influences as described above, which implies TV< low3s. Considering the above values of TV and our experimentally observed time constants of -0.1 s, it seems reasonable to neglect TV in eq. (5). The parameters of table I are of course temperature and field dependent; nevertheless, in the range of our experiments these changes are not drastic and eq. (5) will remain valid. Furthermore, if neglecting T, alSO remains correct, we can use eq. (5) to calculate T from & One obtains T = 4 s for T = 1.44 K and H = 0.4T, while the experiments show T = 0.25 s (fig. 4). This discrepancy of approximately a factor of ten is not large considering the uncertainty in &. In the above numerical estimates we concentrated on the data of MnClz-4H20, we did not perform a similar analysis for the bromide. If we neglect TV in eq. (5) this would mean that the observed relaxation times show a linear dependence on ro. Our bromide samples were not shaped as cylinders or spheres, but we can of course approximate them as such. We performed such an analysis for the data on the three
H.A. Groemndijk et al. / Kapitza resistance and relaxation behaviour in a&ferromagnetic crystals
different samples at T = 1.70 K and H = 0.8 T. Fig. 8 shows that the time constants are indeed linearly dependent on the crystal size, irrespective of the approximation used to obtain ro, a result which clearly demonstrates that for the case of MnBr2*4Hz0 the observed time constants are also governed by the Kapitza resistance, and the use of eq. (5) (neglecting 7,) is allowed. From the above it is clear that the Kapitza resistance indeed explains the order of magnitude of the observed long relaxation times; let us now analyze whether one can also understand the observed temperature and field dependences of T from the Kapitza resistance. It follows from eq. (5) - still assuming that it is correct to neglect TV- that our time constants should vary with H and T just as the product c&. The relation derived by Khalatnikov (eq. (3)) predicts & to be proportional to Tm3,independent of H. Thus for our experiments where also the product pu hardly varies with T, one expects r a cHT-~ .
(6)
For the case of MnC12*4Hz0 detailed information about cH can be found in the work of Giauque et
6
8
Fig. 8. The observed relaxation time at 1.7 K and 0.8 T for three different MnBr2.4HzO crystals as a function of ro; 0: sample approximated as a sphere; 0: sample approximated as a cylinder.
39
al. [27]. From these data we were able to determine the field dependence of cH and the temperature dependence of CHT-~ at the particular T and H values corresponding to the data plotted in figs. 3 and 4. We scaled these results with our time constants in the antiferromagnetic phase. The resemblance between our data and +Te3 (solid lines in figs. 3 and 4) in the ordered phase is good. In the paramagnetic phase the time constants remain significantly above the curve. This discrepancy may be related to the fact that in the paramagnetic region it is no longer correct to neglect TV.For example, in fig. 4, a T, value of -3 x lo-* s would explain the observed difference. This value is precisely what Soeteman [16] reports for the relaxation time of a powdered sample above T,. For powder measurements the influence of the Kapitza resistance on the relaxation behaviour is small due to the small value of ro, in which case one expects to measure TV directly. The T versus T curves for MnBr2*4H20 were analyzed on the basis of a CHT-~ dependence in a similar way. An example of such a fit, using the specific heat data from ref. [ll], is given by the solid line in fig. 5. Although the agreement between the time constants and the curve CHT-~ is reasonable, it is not as good as for the chloride. In fact one may conclude from fig. 5 that the temperature dependence of RI, is stronger than Te3, an effect also deserved in other materials [32-341. We did not consider such a difference in the temperature dependence of RI, in any detail. There are no physical arguments to account for such a different behaviour between crystals so similar as MnC12*4H20 and MnBr2+4H20. Furthermore, the observed difference may well be an artifact of our type of analysis, based strongly on the reliability of the specific heat data. The specific heat data of the chloride [27] and those of the bromide [11] show a different temperature dependence below T/T, = 0.85. This effect could easily account for a 20% increase of the curve in fig. 5 at T = 1.2 K. Considering the above arguments we have
40
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaoiour in antiferromagneticcrystals
determined the proportionality constant of the Kapitza resistance in eq. (3) without fitting the exponent of T. This can be done if one knows the values of y and r. in eq. (5). In the present research we used unshaped single crystals, so we have to use estimates for y and ro. In table II we give the values for the Kapitza resistance which were calculated from our experimental data approximating our crystals as a cylinder. It is seen from this table that the actual values of Rk resulting from the different T(H) or 7(T) curves mutually agree reasonably well. The difference in magnitude between the Kapitza resistance of manganese chloride and manganese bromide may be related to the difference in the density and the velocity of sound in the crystals (see eq. (3)). We did not consider these differences in any detail, as the results for Rk in table II may still be influenced by slight alterations of the crystal surface [20]. Although our crystals were handled carefully, no particular attention was paid to these possible surface effects, It is interesting to note that our results for RI, are of the same order of magnitude as those obtained for other hydrated single crystals: CeES [35]. RI, - 3 x 10-3T-2.4 K m2 W-’ and PrES: These results were determined from experiments of a pulsed nature. It was suggested that such experiments may be interpretable in terms of Rk, probably under restricted conditions of T and H [34]. Our results show that a steady-state type of experiment does indeed measure the same characteristic boundary resistance. It would be interesting to determine Rk from
Table II Kapitza resistance MnC12.4HzO From r(T) at 0.4T From r(T) at 0.6T From T(H) at 1.44 K
1.8 2.1 1.7
MnBrr.4HrO From r(T) at 0.5 T From T(T) at 0.8 T From r(T) at 1.0 T
0.94 x 10-3T-3 K mz W-’ 0.82 x 10e3Tm3 K m2 W-’ 0.92 x 10v3Tm3 K m* W-’
x x x
10m3Tm3K m2 W-i 10e3T-’ Km* W-’ 10-‘T-’ Km* W-’
the relaxation time over a larger temperature interval, together with the specific heat measurements. It is then possible to emphasize the temperature dependence of Rk more accurately and the modifications of the Khalatnikov model may be verified experimentally.
4. Conclusion The relaxation behaviour of MnC12*4H20 and MnBrra4HzO near the paramagnetic-antiferromagnetic phase transition exhibits an anomaly that resembles the anomaly in the magnetic specific heat. The time constants in the antiferromagnetic state were shown to be determined completely by the Kapitza resistance, involved in the energy exchange between the lattice and the cooling liquid. Finally, it is argued that this type of experiment, if performed on shaped, single crystals of antiferromagnetically ordered compounds, can yield important data about the Kapitza resistance.
Acknowledgements We would like to thank Prof. Dr. N.J. Poulis for his interest in this work.
References 111H.B.G. Casimir and F.K. Du Pr6, 121K.J. Standley and R.A. Vaughan,
Physica 5 (1938) 507. Electron Spin Relax(Adam Hilger, London,
ation Phenomena in Solids 1969). [31C.J. Gorter and A.J. van Duyneveldt, Proc. LT13, Vol. 2 (Plenum Press, New York, 1972) p. 621. r41T.P. Valkering and L.C. van der Marel, Physica 47 (1970) 412. [51J. FIokstra, C.J. Gerritsma, G.A. Hartemink and L.C. van der Mare& Physica 77 (1974) 99. PI A. van der Bilt and A.J. van Duyneveldt, Proc. LT15, J. de Physique C6 (1978) 1002. [7] A. Zalkin, J.P. Forrester and D.H. Templeton, Inorg. Chem. 3 (1964) 529. [8] R.F. Altman, S. Spooner, D.P. Landau and J.E. Rives, Phys. Rev. Bll (1975) 458.
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaoiour in antiferromagnetic crystals [91 J.E. Rives and V. Benedict, Phys. Rev. B12 (1975) 1908. P. Groth, Chemische Kristallographie, Vol. I, W. Engelmann (Leipzig, 1908). J. Appl. Phys. 34 WI J.H. Schelleng and S.A. Friedberg, (1963) 1087. WI C.C. Becerra, A. Paduan Filho and N.F. Oliviera, Jr., Solid State Commun. 16 (1975) 791. J. van den Broek and C.J. Garter, P31 M.A. Lasheen, Physica 24 (1958) 1076. A.J. van Duyneveldt and C.J. Gorter, 1141 J. Soeteman, Physica 45 (1969) 435. J. Soeteman and C.J. Gorter, in WI A.J. van Duyneveldt, Cooperative Phenomena, H. Hagen and M. Wagner, eds. (Springer-Verlag, Berlin, 1973). 1161 J. Soeteman, Thesis (Leiden, 1974). t171 A. van der Bilt, Thesis (L&den, 1978). PhyWI J. Soeteman, J. Bevaart and A.J. van Duyneveldt, sica 74 (1974) 126. P91 J. Flokstra, Thesis (Enschede, 1977). PO1 N.S. Snyder, Cryogenics 10 (1970) 89. WI L.J. Challis, J. Phys. C7 (1974) 481. Zh. Eksper. i. Teor. Fiz. 22 (1952) WI I.M. Khalatnikov, 687.
m
41
[23] Handbook of Chemistry and Physics, J.C. Weast, ed. Fe Chemical Rubber Co, Cleveland). [24] J. Wilks, The Properties of Liquid and Solid Helium (Clarendon Press, Oxford, 1967). [25] S.A. Friedberg and J.D. Wasscher, Physica 19 (1953) 1072. [26] G.S. Dixon, V. Benedict and J.E. Rives, Phys. Rev. B21 (1980) 2865. [27] W.F. Giauque, R.A. Fischer, E.W. Hornung and G.E. Brodale, J. Chem. Phys. 53 (1970) 1474. [28] A.M. Stoneham, Proc. Phys. Sot. 86 (1%5) 1163. [29] D.L. Huber, Phys. Rev. B3 (1971) 836. [30] H. Yamazaki, J. Phys. Sot. Japan 25 (1%8) 1190. [31] J. Flokstra, G.J. Gerritsma, A.J.W.A. Vermeulen and A.C. Botterman, Phys. Lett. 44A (1973) 485; J. Flokstra, G.J. Gerritsma, B. van den Brandt and L.C. van der Mare& Phys. Lett. 53A (1975) 159. [32] L.J. Challis and R.A. Sherlock, J. Phys. C3 (1970) 1193. [33] F. Wagner, F.J. Kollarits, K.E. Wilkes and M. Yaqub, J. Low Temp. Phys. 20 (1975) 181. [34] D.J. Grifiths and W.H. Chen, Phys. Rev. B20 (1979) 616. [35] H. Gliittli, Can. J. Phys. 46 (1968) 103.
Physica 112B (1982) 33-41 North-Holland Publishing Company
EXPERIMENTAL RELAXATION
DETERMINATION
BEHAVIOUR
OF THE KAPITZA RESISTANCE
IN ANTIFERROMAGNETICALLY
FROM THE
ORDERED MnCly4Hz0
AND
MnBry4Hz0 H.A.
GROENENDIJK,
Kamerlingh
Received
Onnes Laboratorium
5
J. FLOKSTRA*
and A.J. van DUYNEVELDT
der Rijksuniversiteit
Leiden,
Leiden,
7’he Netherlands
June 1981
The relaxation behaviour of the antiferromagnets MnClr.4HzO and MnBrr.4HzO near their phase transitions have been determined with an ac susceptibility technique. The time constants show a maximum at the field-dependent transition temperature, while also a linear dependence of T upon crystal size exists. A description of the data is given by considering the Kapitza resistance between the crystal and the liquid-helium bath as the dominant factor in the relaxation process. From our results the Kapitza resistance could be determined as 1.9 X lo-‘T’ K mz W-’ and 0.89 x 10d3Tm3K mZ W-’ for the chloride and the bromide, respectively.
1. Introduction Non-resonant relaxation studies have been long a subject of interest in low-temperature magnetism. Already in the early years it was found that the Casimir-Du Pre model for magnetic relaxation [l] provides an adequate method for the derivation of time constants from observed ac susceptibilities. The physical interpretation of these time constants was less simple. On the one hand, resonance experiments usually revealed different time constants for directly comparable systems, while, on the other hand, the observed temperature and field dependences of the time constants rarely agreed with the behaviour as predicted by theory [2]. In the last decade it was possible to account for most of the discrepancies between theory and experiment by exploring the systems at strong external magnetic fields [3]. Simultaneously, extended computer facilities allowed the numerical solution of a more sophisticated model in which all the
* Permanent address: Technische Hogeschool Twente, Afdeling der Technische Natuurkunde, Enschede, The Netherlands.
037%4363/82/0000-0000/$02.75
@ 1982 North-Holland
various kinds of energy exchange between the magnetic spin system and the cooling liquid are considered [4,5]. As a result most of the earlier existing difficulties are understood, and it is now possible to make use of a non-resonant relaxation study for other problems, e.g. the determination of phonon lifetimes [6]. The present paper deals with a study of the relaxation behaviour near the paramagneticantiferromagnetic phase transition. It will .be discussed that the observed time constants are strongly influenced by the Kapitza resistance between the magnetic crystal and the liquid-helium bath,
and that in fact ac susceptibility
ments provide low-temperature
2. Experimental
an elegant Kapitza
measure-
way of determining resistance.
the
results
Dynamic susceptibility measurements have been performed on MnBr,*4H,O and on the o-form MnClz-4Hz0, which is stable at room temperature. The crystal structure of the chloride is monoclinic - space group pi/n - with a = 11.186& b = 9513A, c = 6.186A and p =
34
H.A. Groenendijk et al. I Kapitza resistance and relaxation behaviour in antiferromagnetic crystals
99.74” [7]. There are four molecules in the unit cell. The structure consists of discrete octahedra with a Mn2+ ion in the centre. These octahedra are formed by four oxygens and two chlorines and are slightly distorted. Four of the hydrogen atoms
are
between
involved
neighbouring
in
the
hydrogen
manganese
bonding
ions.
compound
work
orders
can be found
below
in [8,9].
The
T, = 1.62 K as a simple
two-sublattice antiferromagnet with collinear spins. The present measurements were made along the c’ axis (perpendicular to the a and b crystal axes) which is close to the easy axis of the antiferromagnetic alignment [8]. Moreover, the results of our study are not subject to large changes in the relative direction of the magnetic field (section 3). The crystallographic structure of MnBrz*4Hz0 is isomorphous to that of the chloride [lo]. The magnetic structure is not known in detail, but it is not likely to be much different from that of MnC12*4Hz0. The zero-field antiferromagnetic ordering temperature of the bromide is 2.13 K [ll]; more details about, e.g., the magnetic phase diagrams of both compounds can be found in
frequency
MnC12.4H20 was reported in 1958 by Lasheen et al. [13]. Later Soeteman et al. [14-161 performed a study on both MnC12*4H20 and MnBrl*4Hz0, showing that T(T, H) resembled the anomaly in the specific heat cH(T, H). Our present measuring technique is basically the same as those used in the above studies [U-16]. Variation in the mutual inductance of a set of coils is directly detected by means of a PAR 124A with a two-
model
PAR
127 [17]. The oscil-
range
the the
from 0.2 Hz to 15 kHz.
The experiments were performed on single crystals which were grown at room temperature slowly
MnC12.4H20
evaporating or
MnBr2*4H20
grade).
Large
crystal known
faces were morphology
single
crystals
solutions of (Merck, reagent with
easily obtained. of the crystals
well-defined From the [lo] it was
possible to orientate the external magnetic field parallel to the C’ axis within a few degrees. An example of an Argand diagram k” versus x’) as measured for MnBr2*4H20 (H]]c’) is shown in fig. la. The curve is circular, which demonstrates that the relaxation process can be described with a single time constant. In cases where x”/,$ < 0.1 it is often easier to concentrate on the behaviour of only x”(o). An example of the frequency dependence of XI’ is depicted in fig. lb.
o.21
[9,121. For the present study we concentrate upon the behaviour of the ac susceptibility near the antiferromagnetic-paramagnetic phase transition in the temperature range between 1.2 and 2.0 K at external magnetic fields of up to 2T. The first observation that the relaxation time constant r derived from the susceptibility showed an transition in a phase anomaly at Such
accessory
lating field h always remains parallel to external field H. The system operates over
from
The magnetic properties of MnClr.4HrO have been studied extensively, the references to most of the earlier
phase
I./
b
1
F?
=x i
Fig. 1. x(o) for MnBr2.4HzO (Hljc’) at T = 1.90 K and H = OST. (a) Argand diagram; the numbers with the data give the ac frequency in Hz. (b) Normalized out-of-phase component x”/x~ ko, the zero-field susceptibility) as a function of frequency.
35
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaoiour in antiferromagnetic crystals
For calibration purposes the signals have been divided by the ,$ values at zero field ho); the solid line in this figure represents the curve [l]:
(1) where xr is the isothermal susceptibility, xs the adiabatic one, and r the relaxation time constant, derived from fig. lb as the reciprocal value of w at maximal absorption. A comparable set of data points for the chloride is given in fig. 2. In this case the Argand diagram shows more scatter due to the fact that the measurements had to be made at lower frequencies. However, the plot of x” versus w/27r (fig. 2b) closely resembles eq. (1) thus allowing an accurate determination of r; all relaxation times quoted in this paper were obtained from this kind of plots. The relaxation time constants of MnCl*-4H20 were obtained with a 760 mg single crystal. The
I
0.1
\
B
1 0/2~r
time constants for H = 0.4 and 0.6T are plotted as a function of temperature in fig. 3. It is seen that at low temperature in the antiferromagnetic region, r initially decreases slightly with increasing T. Near T,(H) there is a sharp increase of r. A maximum occurs at the field-dependent antiferromagnetic-paramagnetic phase transition T,(H), and then, in the paramagnetic region, the relaxation time decreases sharply. The field dependence of 7, measured at T = 1.44 K, is plotted in fig. 4. The relaxation time increases slowly as a function of the external magnetic field in the antiferromagnetic region. A sharp peak occurs again at the transition field H,(T), and then T decreases sharply in the paramagnetic region. The temperature dependence of the relaxation times for a 1300 mg single crystal of MnBrz*4H20 were measured for three values of the external magnetic field. As an example the data for H = 0.8T are plotted in fig. 5. Although the actual time constants are about a factor of three smaller than those of the chloride, their temperature dependence is quite similar. The field dependence of r is also similar to that of the chloride; we do not include an example in the figures. Our measurements on both MnC12.4H20 and
I
IO (Hz)
Fig. 2. ~(0) for MnC12.4H20 (H//c’) at T = 1.20 K and H = 0.4T. (a) Argand diagram; the numbers with the data give the ac frequency in Hz. (b) Normalized out-of-phase component X”/XOk,~, the zero-field susceptibility) as a function of frequency.
16:;
2.0 T(K)
Fig. 3. Relaxation time as a function of temperature for MnC12.4H20 (Hllc’). 0: H = 0.4 T; 0: H = 0.6 T; the drawn lines are proportional to CHT-~.
36
H.A.
”
Groenendijk et al. I Kapitza resistance and relaxation behaviour in antiferromagnetic
IO
P
4
‘+biV
IO H (T)
Fig. 4. Relaxation time as a function of external magnetic field for MnCly4H20 (Hllc’) at T = 1.44 K. The drawn line is proportional to CHT-~.
MnBr2*4H20 are in good agreement with previous data as regards the temperature and field dependence of 7 [15]. The interesting difference is, however, that the absolute values of r agree for the case of MnC12.4H@ and that our data for MnBrz*4H20 show time constants systematically 30% larger in the antiferromagnetic state. This difference cannot be ascribed to a lack of experimental accuracy, especially not because of the “ideal” Casimir-Du PrC behaviour of x’ and x” versus frequency (figs. 1 and 2). The slower
crystals
relaxation of MnBrz*4Hz0 indicates a possible sample size dependence of the relaxation time constants as the previous experiments on single crystals [15,16] were performed on samples of approximately 700 mg. This mass is roughly identical to that of the chloride crystal we used, but smaller for the case of the bromide. In order to verify this aspect we also performed measurements on a MnBr2.4H20 single crystal of 270 mg. Faster relaxation times (-50%) were indeed obtained for this crystal. The above crystal size dependence of r indicates that the observed relaxation times are not due to a magnetic relaxation process only. In that case it is conceivable that the crystal orientation with respect to H may also not be a factor. Therefore, we measured the relaxation times of a crystal of MnBr2*4H20 (710 mg) with the b axis (which is the hard axis of the antiferromagnetic alignment [12]) oriented parallel to H. In order to get reasonable x” signals in this case the measurements have to be made at stronger external fields (2.0 T). The results (fig. 6) show not only the same behaviour as in the case H]]c’, but also the order of magnitude of the time constants is the same for both field orientations. From the measurements described above it can be concluded that it is unlikely that an intrinsic magnetic relaxation process is responsible for the observed relaxation behaviour. In the following discussion we will show that the
20 T
(K)
Fig. 5. Relaxation time as a function of temperature for MnBr2.4HzO (Hlc’) at H = 0.8 T. curve proportional to c”T-~.
Fig. 6. Relaxation time as a function of temperature MnBry4H20 ‘(Hllb) at H = 2 T.
for
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaviour in antiferromagnetic crystals
Kapitza resistance at the crystal surface governs the relaxation process.
3. Discussion It is known that the Casimir-Du Pre model may give an inadequate description of the relaxation behaviour at liquid-helium temperatures, especially in the case of experiments on single crystals [5,18]. The finite heat conductivity of the crystal lattice and the limitations in the heat transfer to the helium bath may lead to situations where the experimentally determined time constant is no longer related to the intrinsic relaxation processes. In order to analyze our results we will use a model which takes these effects into account. This so-called thermal conduction model [4,5] is basically an extension of the model of Casimir and Du PrC. In the thermal conduction model the cooling system (in our case, liquid helium) is considered as a separate system with thermal conductivity A, and specific heat c,, as is shown schematically in fig. 7. The heat exchange between the crystal lattice and the liquid helium is determined by the Kapitza boundary resistance Rk, the associated relaxation time rR can be written as [19,35] TR = YroC&
,
(2)
with y = l/2 if one approximates the crystal as a cylinder with radius r. and y = l/3 for the case of a sphere. In the lattice system (specific heat cL) it is assumed that the heat transport is characterized by the thermal conductivity hL. The introduction of the extra heat transfer processes in the thermal conduction model leads to situations where the relaxation times as derived from the Argand diagrams can no longer be identified with the intrinsic spin-lattice relaxation times rs. It can even occur that the curves in the Argand diagram become strongly flattened and that the determination of a characteristic time constant is no longer possible [4,5]. The thermal conduction model can be solved exactly. In the resulting expression for the susceptibility, several dimensionless parameters occur which determine the actual frequency dependence of x. These parameters are defined in table I. For the analysis of our data we have to calculate their magnitudes, but before doing so we first discuss the Kapitza boundary resistance Rk. Recent reviews of both experimental and theoretical work on the Kapitza resistance are given by Snyder [20] and Challis [21]. No really adequate theory for Rk exists; we will use the results from the phonon mismatch theory of Khalatnikov [22]. The value of Rk as calculated from Khalatnikov’s theory is often too high, but we will use the theory for an order of magnitude estimate. The following expression is derived: R
_ k
15h3&v:
r2k4pv
T-3
’
Table I Dimensionless parameters of the thermal conduction model [5]
Fig. 7. Thermal conduction model.
37
(3)
38
H.A. Groenendijk et al. I Kapitza resistance and relaxation behaviour in antifewomagneticcrystals
The quantities ps and v, are the density and the velocity of sound in the crystal, respectively; p and v are the corresponding values for liquid helium. We now restrict our calculations to the case of MnC12.4H20, as all the quantities necessary in the calculations are available for this compound; the result of our calculations will undoubtedly be of the correct order of magnitude for the bromide, too. We used ps = 2.0 X lo3 kg rnw3[23], while us was estimated from the Debye temperature (0, = 110 K is calculated from cL in [25]): us = 2.0 x lo3 m/s. For p and u we chose values representative for the temperature range of our experiments: P’ 140 kg mm3[24] and v = 2.4 X lo* m s-’ [24]. As a result we calculated Rk = 23 x 10m3Tm3 K m* W-l. Let us now consider the parameters of table I for the case of our experiments at T = 1.44 K and H = 0.4 T. The thermal conductivity constant AL of MnC12.4H20 was recently determined by Dixon et al. [26], who obtained the value of 50 W K-’ m-l at T = 1.44 K, independent of H. If we approximate our 760 mg chloride crystal as a cylinder, one obtains r. = 3.8 mm, thus yielding Q = 120. The parameter D is found directly from the experimental values for cH (25.2 X lo4 Jmm3K-’ [27]) and cL (48.9 Jme3K-’ [25]) to be 5.2 x 103. The parameter P contains the spinlattice coupling constant (Y,the value of which we do not know. Using the definition of r,(= cH/(Y [l]) one may write P = r&/hLTs and one calculates P = 7.3 x lo-*/~,. The actual time constants determined in our experiments set an upper limit to the possible values of TV. If one considers r = 0.25 s (T = 1.44 K and H = 0.4T) as being representative of this upper limit, we find that P is larger than 0.3. Finally, since the cooling liquid is helium below the A-point (A, is large) it seems reasonable to assume S = 0. With the above-estimated values for the dimensionless parameters we calculated x(o). We will not give the details of this lengthy calculation; all relevant information can be found in [5]. The resulting curves in the Argand diagram were circular, allowing a description with one
time constant. This single time constant, T, consists of two contributions according to T=T,+DT~.
(4)
Expression (4) is similar to the one derived by Stoneham [28] for the case of a phonon-bottlenecked spin system. Using eq. (2) we may write 7=
Ts + Yr&&
.
(5)
Not many theoretical treatments of the spinlattice relaxation TV in antiferromagnetically ordered systems exist. The most solid numerical prediction can be found in the work of Huber [29] (T~S lo-’ s). Apart from the earlier work on manganese chloride and bromide, experiments on the relaxation in antiferromagnetically ordered systems are reported for two cobalt compounds in [30,31]. Both references show T 10e3 s, but it is clear that these time constants are not free from influences as described above, which implies TV< low3s. Considering the above values of TV and our experimentally observed time constants of -0.1 s, it seems reasonable to neglect TV in eq. (5). The parameters of table I are of course temperature and field dependent; nevertheless, in the range of our experiments these changes are not drastic and eq. (5) will remain valid. Furthermore, if neglecting T, alSO remains correct, we can use eq. (5) to calculate T from & One obtains T = 4 s for T = 1.44 K and H = 0.4T, while the experiments show T = 0.25 s (fig. 4). This discrepancy of approximately a factor of ten is not large considering the uncertainty in &. In the above numerical estimates we concentrated on the data of MnClz-4H20, we did not perform a similar analysis for the bromide. If we neglect TV in eq. (5) this would mean that the observed relaxation times show a linear dependence on ro. Our bromide samples were not shaped as cylinders or spheres, but we can of course approximate them as such. We performed such an analysis for the data on the three
H.A. Groemndijk et al. / Kapitza resistance and relaxation behaviour in a&ferromagnetic crystals
different samples at T = 1.70 K and H = 0.8 T. Fig. 8 shows that the time constants are indeed linearly dependent on the crystal size, irrespective of the approximation used to obtain ro, a result which clearly demonstrates that for the case of MnBr2*4Hz0 the observed time constants are also governed by the Kapitza resistance, and the use of eq. (5) (neglecting 7,) is allowed. From the above it is clear that the Kapitza resistance indeed explains the order of magnitude of the observed long relaxation times; let us now analyze whether one can also understand the observed temperature and field dependences of T from the Kapitza resistance. It follows from eq. (5) - still assuming that it is correct to neglect TV- that our time constants should vary with H and T just as the product c&. The relation derived by Khalatnikov (eq. (3)) predicts & to be proportional to Tm3,independent of H. Thus for our experiments where also the product pu hardly varies with T, one expects r a cHT-~ .
(6)
For the case of MnC12*4Hz0 detailed information about cH can be found in the work of Giauque et
6
8
Fig. 8. The observed relaxation time at 1.7 K and 0.8 T for three different MnBr2.4HzO crystals as a function of ro; 0: sample approximated as a sphere; 0: sample approximated as a cylinder.
39
al. [27]. From these data we were able to determine the field dependence of cH and the temperature dependence of CHT-~ at the particular T and H values corresponding to the data plotted in figs. 3 and 4. We scaled these results with our time constants in the antiferromagnetic phase. The resemblance between our data and +Te3 (solid lines in figs. 3 and 4) in the ordered phase is good. In the paramagnetic phase the time constants remain significantly above the curve. This discrepancy may be related to the fact that in the paramagnetic region it is no longer correct to neglect TV.For example, in fig. 4, a T, value of -3 x lo-* s would explain the observed difference. This value is precisely what Soeteman [16] reports for the relaxation time of a powdered sample above T,. For powder measurements the influence of the Kapitza resistance on the relaxation behaviour is small due to the small value of ro, in which case one expects to measure TV directly. The T versus T curves for MnBr2*4H20 were analyzed on the basis of a CHT-~ dependence in a similar way. An example of such a fit, using the specific heat data from ref. [ll], is given by the solid line in fig. 5. Although the agreement between the time constants and the curve CHT-~ is reasonable, it is not as good as for the chloride. In fact one may conclude from fig. 5 that the temperature dependence of RI, is stronger than Te3, an effect also deserved in other materials [32-341. We did not consider such a difference in the temperature dependence of RI, in any detail. There are no physical arguments to account for such a different behaviour between crystals so similar as MnC12*4H20 and MnBr2+4H20. Furthermore, the observed difference may well be an artifact of our type of analysis, based strongly on the reliability of the specific heat data. The specific heat data of the chloride [27] and those of the bromide [11] show a different temperature dependence below T/T, = 0.85. This effect could easily account for a 20% increase of the curve in fig. 5 at T = 1.2 K. Considering the above arguments we have
40
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaoiour in antiferromagneticcrystals
determined the proportionality constant of the Kapitza resistance in eq. (3) without fitting the exponent of T. This can be done if one knows the values of y and r. in eq. (5). In the present research we used unshaped single crystals, so we have to use estimates for y and ro. In table II we give the values for the Kapitza resistance which were calculated from our experimental data approximating our crystals as a cylinder. It is seen from this table that the actual values of Rk resulting from the different T(H) or 7(T) curves mutually agree reasonably well. The difference in magnitude between the Kapitza resistance of manganese chloride and manganese bromide may be related to the difference in the density and the velocity of sound in the crystals (see eq. (3)). We did not consider these differences in any detail, as the results for Rk in table II may still be influenced by slight alterations of the crystal surface [20]. Although our crystals were handled carefully, no particular attention was paid to these possible surface effects, It is interesting to note that our results for RI, are of the same order of magnitude as those obtained for other hydrated single crystals: CeES [35]. RI, - 3 x 10-3T-2.4 K m2 W-’ and PrES: These results were determined from experiments of a pulsed nature. It was suggested that such experiments may be interpretable in terms of Rk, probably under restricted conditions of T and H [34]. Our results show that a steady-state type of experiment does indeed measure the same characteristic boundary resistance. It would be interesting to determine Rk from
Table II Kapitza resistance MnC12.4HzO From r(T) at 0.4T From r(T) at 0.6T From T(H) at 1.44 K
1.8 2.1 1.7
MnBrr.4HrO From r(T) at 0.5 T From T(T) at 0.8 T From r(T) at 1.0 T
0.94 x 10-3T-3 K mz W-’ 0.82 x 10e3Tm3 K m2 W-’ 0.92 x 10v3Tm3 K m* W-’
x x x
10m3Tm3K m2 W-i 10e3T-’ Km* W-’ 10-‘T-’ Km* W-’
the relaxation time over a larger temperature interval, together with the specific heat measurements. It is then possible to emphasize the temperature dependence of Rk more accurately and the modifications of the Khalatnikov model may be verified experimentally.
4. Conclusion The relaxation behaviour of MnC12*4H20 and MnBrra4HzO near the paramagnetic-antiferromagnetic phase transition exhibits an anomaly that resembles the anomaly in the magnetic specific heat. The time constants in the antiferromagnetic state were shown to be determined completely by the Kapitza resistance, involved in the energy exchange between the lattice and the cooling liquid. Finally, it is argued that this type of experiment, if performed on shaped, single crystals of antiferromagnetically ordered compounds, can yield important data about the Kapitza resistance.
Acknowledgements We would like to thank Prof. Dr. N.J. Poulis for his interest in this work.
References 111H.B.G. Casimir and F.K. Du Pr6, 121K.J. Standley and R.A. Vaughan,
Physica 5 (1938) 507. Electron Spin Relax(Adam Hilger, London,
ation Phenomena in Solids 1969). [31C.J. Gorter and A.J. van Duyneveldt, Proc. LT13, Vol. 2 (Plenum Press, New York, 1972) p. 621. r41T.P. Valkering and L.C. van der Marel, Physica 47 (1970) 412. [51J. FIokstra, C.J. Gerritsma, G.A. Hartemink and L.C. van der Mare& Physica 77 (1974) 99. PI A. van der Bilt and A.J. van Duyneveldt, Proc. LT15, J. de Physique C6 (1978) 1002. [7] A. Zalkin, J.P. Forrester and D.H. Templeton, Inorg. Chem. 3 (1964) 529. [8] R.F. Altman, S. Spooner, D.P. Landau and J.E. Rives, Phys. Rev. Bll (1975) 458.
H.A. Groenendijk et al. / Kapitza resistance and relaxation behaoiour in antiferromagnetic crystals [91 J.E. Rives and V. Benedict, Phys. Rev. B12 (1975) 1908. P. Groth, Chemische Kristallographie, Vol. I, W. Engelmann (Leipzig, 1908). J. Appl. Phys. 34 WI J.H. Schelleng and S.A. Friedberg, (1963) 1087. WI C.C. Becerra, A. Paduan Filho and N.F. Oliviera, Jr., Solid State Commun. 16 (1975) 791. J. van den Broek and C.J. Garter, P31 M.A. Lasheen, Physica 24 (1958) 1076. A.J. van Duyneveldt and C.J. Gorter, 1141 J. Soeteman, Physica 45 (1969) 435. J. Soeteman and C.J. Gorter, in WI A.J. van Duyneveldt, Cooperative Phenomena, H. Hagen and M. Wagner, eds. (Springer-Verlag, Berlin, 1973). 1161 J. Soeteman, Thesis (Leiden, 1974). t171 A. van der Bilt, Thesis (L&den, 1978). PhyWI J. Soeteman, J. Bevaart and A.J. van Duyneveldt, sica 74 (1974) 126. P91 J. Flokstra, Thesis (Enschede, 1977). PO1 N.S. Snyder, Cryogenics 10 (1970) 89. WI L.J. Challis, J. Phys. C7 (1974) 481. Zh. Eksper. i. Teor. Fiz. 22 (1952) WI I.M. Khalatnikov, 687.
m
41
[23] Handbook of Chemistry and Physics, J.C. Weast, ed. Fe Chemical Rubber Co, Cleveland). [24] J. Wilks, The Properties of Liquid and Solid Helium (Clarendon Press, Oxford, 1967). [25] S.A. Friedberg and J.D. Wasscher, Physica 19 (1953) 1072. [26] G.S. Dixon, V. Benedict and J.E. Rives, Phys. Rev. B21 (1980) 2865. [27] W.F. Giauque, R.A. Fischer, E.W. Hornung and G.E. Brodale, J. Chem. Phys. 53 (1970) 1474. [28] A.M. Stoneham, Proc. Phys. Sot. 86 (1%5) 1163. [29] D.L. Huber, Phys. Rev. B3 (1971) 836. [30] H. Yamazaki, J. Phys. Sot. Japan 25 (1%8) 1190. [31] J. Flokstra, G.J. Gerritsma, A.J.W.A. Vermeulen and A.C. Botterman, Phys. Lett. 44A (1973) 485; J. Flokstra, G.J. Gerritsma, B. van den Brandt and L.C. van der Mare& Phys. Lett. 53A (1975) 159. [32] L.J. Challis and R.A. Sherlock, J. Phys. C3 (1970) 1193. [33] F. Wagner, F.J. Kollarits, K.E. Wilkes and M. Yaqub, J. Low Temp. Phys. 20 (1975) 181. [34] D.J. Grifiths and W.H. Chen, Phys. Rev. B20 (1979) 616. [35] H. Gliittli, Can. J. Phys. 46 (1968) 103.