IJxl in - Science Direct

Physica 92B (1977) 187-200 © North-Holland Publishing Company

MAGNETIC BEHAVIOR OF [(CH3)3NH ] CuCI3 • 2H20. EVIDENCE FOR LATTICE-DIMENSIONALITY CROSSOVERS IN A QUASI ONE-DIMENSIONAL FERROMAGNET H. A. ALGRA, L. J. de JONGH, W. J. HUISKAMP Kamerlingh Onnes Laboratory, University of Leiden, The Netherlands and R. L. CARLIN Department of Chemistry, University of lllinois at Chicago O'rcle, Chicago, lllinois 60680, USA (Communication Kamerlingh Onnes Laboratory No. 430a) Received 5 April 1977

The magnetic behavior of [(CH3)3NH ] CuC13 • 2H20 as observed by specific heat and susceptibility measurements, is explained in terms of a lattice anisotropy in the exchange of the form IJxl ~, IJ, I ~" IJzl. At high temperatures (T > 3 Jx[k) the system behaves as a ferromagnetic (S = ~-) Heisenberg chain, with Jx/k = 0.~[5 -+ 0.02 K. As the temperature is lowered lattice-dimensionality crossovers occur successively from one- to two-dimensional and ultimately to three-dimensional behavior. The ordering temperature is T~ = 0.165 +_0.005 K. The ratio IJ~l:lJ,.l:lJ, I is estimated as 1:0.05:0.015. The influence of the spin-space anisotropy, estimated to be about 10% of the isotropic exchange Jx, is also discussed and a comparison is drawn with published results for the isostructural Co 2 + salt, which has a much larger spin anisotropy (~75%).

1. Introduction A number of reports has appeared recently on the magnetic and thermal behavior of the series of isostructural linear chain compounds [(CH3)3NH] MX 3 • 2H20, which have magnetic ordering temperatures in the liquid 4He region or lower [ 1 - 6 ] . Here M = Mn 2+, Co 2+ or Cu 2+, and X = C1- or Br-. From the crystallographic evidence, one expects the magnetic structure to be describable by a three 1 K in addition to our own measurements below 1 K, as well as the susceptibility (×) data of Stirrat et al. [3] (T < 4 K). Estimates of the various exchange constants will be made from the analysis in terms of a 3-d Heisenberg model with the above lattice anisotropy. Furthermore, the interactions are not fully of the isotropic Heisenberg form, there being present a spin anisotropy of the Ising type amounting to about 10% of the isotropic intrachain exchange (Jx)"

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H. A. Algra et aL /Lattice-dimensionality crossovers in [ (CH3) 3NH] CuCl 3 • 21120

For magnetic systems characterized by a lattice anisotropy IJzl ~ klyl ~ IJxl one expects the thermodynamic behavior to be essentially one-dimensional (l-d) at high temperatures (kT >>klx I), whereas lattice-dimensionality crossovers will occur from 1-d to 2-d behavior, and ultimately to 3-d behavior, as the temperature becomes low enough for the weaker interactions Jy and Jz to manifest themselves. Such crossover effects are of current interest in theoretical work [8, 9], and experimental results on these phenomena seem to be of value. The plan of the paper is as follows. In the next section we shall present a description of the crystal structure and of the experimental set-up used for the specific heat measurements below 1 K. In section 3 the specific heat data will be analyzed, whereas section 4 is devoted to a discussion of the susceptibility data. Some concluding remarks are given in the last section.

2. Crystallographic data and experimental details The crystal structure of [(CH3)3NH ] CuCl 3 • 2H20 has been reported by Losee et al. [1], to which paper we refer for more details. It is monoclinic, space group P21/c, with a 0 = 7.479(10) A, c o = 16.730(23) A and/3 ---91.98(3) 0. Views of the structure parallel to [010] and [001] are shown in figs. la and b, respectively. There are two nonequivalent Cu atoms, coordinated similarly in a square formed by two water molecules and two CI atoms. Two other C1 atoms are at a larger distance and complete the distorted octahedral environment of Cu 2+. The chains are formed by edge-sharing octahedra and run parallel to the a-axis. The third C1 atom is situated in betv~een the chains, and accepts four hydrogen bonds from the H20 molecules belonging to the coordination of the Cu atoms (cf. fig. 1b). As regards the superexchange pathways, the Cu 2+ ions along the chains (Cu-Cu distance 3.74 A) are connected by double Cu-C1-Cu bridges, the angle between the Cu-C1 bonds being of the order of 90 °. Within the (001) layers, the connection between Cu 2+ ions belonging to adjacent chains (mutual distance 7.864 A) involves pathways of the type C u - H 2 0 C I - H 2 0 - C u , so one would expect the corresponding interaction (Jy) to be considerably smaller than the

interaction (Jx) between Cu 2+ ions in the same chain. In the third direction (parallel to c) the Cu 2+ ions in adjacent chains are at still larger distances (8.365 A) and the superexchange paths are of the form Cu-C1[(CH3)3NH ] -C1-Cu, which will be even more unfavourable than the path along b. Thus identifying Jx, Jy and Jz as the interactions along a, b and c, we expect IJzl < klyl < IJxl on the basis of the superexchange paths. For heat-capacity measurements below 1 K an adiabatic demagnetization outfit is available at Leiden, which has the advantages of enabling quick interchanges of samples and easy handling. With this apparatus, shown in fig. 2, temperatures of about 40 mK are usually reached. In cases where still lower temperatures are desirable or long cooling periods necessary, we use an apparatus based upon a commercial 3He-4He dilution refrigerator (SHE company, USA). The cooling power is 450 erg/s at T ---0.1 K, and temperatures down to 20 mK can be reached. A schematic view of the calorimeter in this apparatus is also given in fig. 2. From the mixing chamber, which provides the cooling power, a copper rod extends into the high-vacuum space below it and is surrounded by a copper thermal shield connected to the 1 K-plate of the dilution refrigerator. A "coilfoil" cylinder, thermally anchored to the mixing chamber, provides an additional low-temperature shield. The sample holder, the heater (Evanohm wire, = 0.05 mm, R -~ 150 [2) and the magnetic thermometer are soldered to the copper rod, which is interrupted just below the mixing chamber by a thermal switch, consisting of a wire of superconducting lead or tin (4 -~ 0.2 mm; l --- 30 mm) and operated by a small superconducting coil wound upon the vacuum can.

The calorimeter in the adiabatic demagnetization outfit is based upon the same principles (cf. fig. 2). Thermal isolation from the cooling salt is again accomplished by a superconducting switch. To reduce the heat leak a coil-foil shield is applied, which is cooled separately by the "guard salt". For the guard and the cooling salt we use manganous ammonium sulfate and chromic potassium alum, respectively. Furthermore, all leads to the calorimeters are made of superconducting wire. The heat leaks were determined to be about 0.1 erg/s in both outfits.

H. A. Algra et al./Lattice-dimensionality crossovers in [(CH3) 3NH] CuCl 3 • 2H20

189

CI(2) 0.00

~(2) 0.00

cl(z) -O.lO Fig. 1. Crystal structure of [(CH3)aNH ] CuC13 • 2H20. (a) View parallel to [010]. Heights of atoms, including the copper atoms within the octahedra, are given as fractions of b. (b) View parallel to [ 001 ]. Heights of atoms given in fractions of c. Arrows indicate hydrogen bonds.

Temperatures are measured with a magnetic thermometer o f the type described earlier by Haasbroek [10] (cf. fig. 2). The mutual induction coil has two oppositely wound adjacent secondary coils and a primary coil of superconducting wire extending over the total length of the coil system.

One o f the secondaries is f'flied with cerium magnesium nitrate powder, the thermal link with the cooling rod being provided by a brush of thin copper wires. To increase the contact area between the wires and the salt pill, the latter is warmed up to just above its melting temperature (i.e. it forms a solution with

190

H. A. Algra et al./Lattice-dimensionality crossovers in [(CHa)3NH] CuCI 3 • 2 H 2 0

[b]

lal

MC

1

21 t, 3

:4

YA

5 tS

Y4 ~A

SC [c]

Idl

Fig. 2. Experimental apparatus. (a) Adiabatic demagnetization apparatus. HV = high vacuum pumping line; GS = guard salt; CS = cooling salt; MC = magnetizing coil (0.920 kOe/A); SC = switch coil; TS = thermal shield cooled by the guard salt. (b) Calorimeter attached to the dilution refrigerator outfit. MC = mixing chamber; SC = switch coil; 1 K-S = 1 Kelvin shield; IS = inner shield thermally anchored to'the mixing chamber. In figs. (a) and (b) the numbers 1-5 denote: 1 = superconducting switch; 2 = teflon pole; 3 = copper cooling rod; 4 = sample holder; 5 = magnetic thermometer. (c) Detail of sample-holder. 1 = brush of copper wires; 2 = epibond lid; 3 = gold-plated copper can; 4 = heater. (d) Detail of magnetic (CMN) thermometer.

its own crystal water). After inserting the brush o f copper wires the salt is cooled and recrystallization takes place. The susceptibility o f the CMN is measured b y an a.c. mutual inductance bridge

(operated at 31 Hz), equipped with a decade transformer (ESI-DT 72A) as described b y Maxwell [11 ]. With a salt pill o f about 0.5 g, the sensitivity at T = 4.2 K is about 1 - 2 mK. The CMN thermometer was calibrated against the 4He vapour pressure scale in the region 1.2 < T < 4 . 2 K. Due to the rather low sensitivity in this range, the uncertainty in the absolute value o f the temperature deduced from the bridge readings for T < 1 K was initially estimated to be about 2% (by absolute we mean the temperature related to the 4He vapour pressure scale o f 1958). At present we have improved upon this by using a germanium resistor (cryocall CR50) that was calibrated b y the thermometry group of Dr. M. Durieux at our laboratory against the 4He and 3He vapour pressures, and against an NBS superconducting device. With the latter device the transition temperatures o f five different superconducting materials are used to provide fixed points in the range 0.5 < T < 7 K. By calibrating the CMN thermometer against this Ge-thermometer down to 0.5 K, the accuracy in the temperature is mainly determined by the reproducibility of the latter, estimated to be better than 0.5%. The resistance of the Ge-thermometer is measured by an a.c. resistance bridge which combines a very high accuracy (better than 1:105) with a very low power dissipation in the resistor (less than 10 - 1 2 W). The heat capacity is measured b y the conventional heat-pulse technique. An ultra-stable current is fed through the heater and a standard resistor, the duration o f the heating period being set in advance with a digital clock. The voltages over the heater and a standard resistor are measured (with a digital volt meter) at b o t h the beginning and the end o f a heatpulse, to check a possible increase of the heater resistance. In general we have aimed to reduce as much as possible the errors in the heat capacity measurements introduced by the electronics. For this reason each o f the elements of the measuring system was so chosen (or designed) as to have an accuracy and stability better than 1:10 5 . Furthermore, the heat capacity o f the addenda was kept as small as possible. We mention i n this connection that in our first experiments a large T - 2 term (for T < 0.2 K) was observed in the addenda, which turned out to be due to the manganin wire initially used for the heater. The effect was drastically reduced b y changing to Evanohm wire.

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H. A. Algra et al./Lattice-dimensionality crossovers in [(CH3)3NH] CuCl 3 • 2H20

The thermal contact between sample and cooling rod may also pose problems in experiments below 1 K. In the case of single-crystal specimens we wrap the sample tightly in thin, gold-plated copper foil (soldered to the cooling rod), with Apiezon-N grease as a contact agent. In another method the crystal is placed in a sample holder filled with Apiezon-N grease. For the powdered specimens we use a sample holder like the one shown in fig. 2, which is filled with a mixture of the powder and Apiezon-N grease. A brush of copper wires is inserted into the mixture and provides the thermal contact with the cooling rod. To avoid overshoot of the thermometer, a direct thermal contact between the heater and the cooling rod is avoided (cf. fig. 2). Generally speaking, the thermal contact with the sample is much better in the case of a powdered specimen, so that, in particular for T < 0.2 K, the possible advantages of studying a single-crystal specimen (e.g. for the critical behavior) lose their value. Evidently the above applies to insulathag materials only.

3. Analysis of specific heat data Specific heat data in the region 1 < T < 36 K have been reported by Losee et al. [1] and are used below to estimate the lattice contribution. These authors also performed susceptibility measurements on singlecrystals along the three crystallographic axes, in the region 1.5 < T < 20 K. Curie-Weiss fits to these data yielded the results ga = 2.08 + 0.01, gb = 2.02 + 0.01, a n d g c = 2.11 + 0.01 for t h e g values, and 0 a = 0.38 +0.03 K, 0 b = 0.36 + 0.03 K and 0 c = 0.41 +- 0.03 K for the Curie-Weiss temperatures (0 = 2zJS(S + 1)/3k). The g values may be compared with go = 2.238 -+ 0.010,g b = 2.037 +- 0.010, a n d g c = 2.195 - 0.010 obtained by Stirrat et al. [3] from ESR measurements. The positive 0 values indicate that the main interaction ('Ix) is ferromagnetic. On the other hand the X measurements below 1 K of Stirrat et al. [3] show the occurrence of an antiferromagnetic long-range ordering at T c = 0.157 + 0.003 K. For T > 0.5 K they report their X data to be in better agreement with the behavior predicted for the 2-d Heisenberg ferromagnet (z = 4; S = ½) from the high-temperature series expansion, than with the Bonner-Fisher prediction for a ferromagnetic

Heisenberg chain (z = 2; S = ½). The fit of the twodimensional model yielded J / k = 0 . 2 8 + 0.02 K. Values o f J / k may also be derived from the 0 values, giving J / k ~- 0.4 for z = 2 and J / k ~- 0.2 for z = 4. We will not take these into account, however, since in lower-dimensional systems a determination o f J / k from the Curie-Weiss fit is usually of little value, due to the large amount of short-range order present in these systems. The samples used in our specific heat experiments below 1 K include a single crystal and two powdered specimens, each sample weighing approximately 0.5 g. The powder data, obtained after subtraction of the addenda, are shown on a linear scale in fig. 3. The transition to long-range order is found to appear as a sharp, k-type anomaly at Tc= 0.165 --- 0.005 K, in good agreement with the T c value of Stirrat et al. [3]. This sharp anomaly is superimposed upon a substantial short-range order contribution, having the form of a broad, flat maximum. An immediate conclusion to be drawn is that the observed short-range order maximum agrees more closely with the ferromagnetic Heisenberg chain prediction [12] than with the ferromagnetic

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Fig. 3. Magnetic specific heat of [(CH3)3NH ] CuCi 3 • 2H20. (a) = experiment; (b) = ferromagnetic S = ~Heisenberg chain with Jx/k = 0.80 K; (c) = ferromagnetic S = ~-Heisenberg layer with Jx/k = Jy/k = 0.50 K.

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H. A. Algra et al./Lattice-dimensionality crossovers in [(CH3) 3NH] CuCI3 • 2H20

Heisenberg layer result [13]. The two predictions are included in fig. 3, where they have been fitted to the experiment in the high-temperature region (see below). The experimental maximum value of the heat capacity, C/R =~0.19, is closer to the chain prediction (0.134) than to the value found for the quadratic Heisenberg layer (0.38). Also, the extreme flatness of the experimental maximum indicates that a ferromagnetic intrachain interaction predominates. Therefore the specific heat data contradict the conclusion that J x ~- Jy, drawn by Stirrat et al. [3] from the X data. In fig. 4 the experimental data are plotted on a logarithmic scale. Above 1 K the results join smoothly those of Losee et al. [1 ]. For T < 0.4 K the single-crystal data fall below those of the powdered samples. This is attributed to the stronger relaxation effects observed in the single-crystal experiments. Apart from a poorer thermal contact between sample and cooling rod in the case of single-crystal specimens, we ascribe the observed relaxation to a phonon bottleneck effect. The powder experiments were also hampered by increasing thermal response times as T was lowered below Tc, preventing meaningful measurements to be taken below T -~ 0.06 K. As an illustration we mention that in the case of the single crystal

, i i i il~ i /!!i

it took several hours of continuous cooling to lower the temperature of the magnetic spin system from 300 to 120 mK. For T > 3 K the lattice contribution to the heat capacity becomes important. In analyzing the data in the region above 1 K we have assumed that the total specific heat may be represented by a ferromagnetic chain contribution plus a T 3 lattice term. Curve (a) in fig. 4 is the prediction for the ferromagnetic (S =½) Heisenberg chain, the fit to the experiment yielding J x / k = 0.80 + 0.05 K, whereas curve (b) corresponds to a T 3 dependence with a "Debye temperature" of about 86 K. It is seen that curve (c), which is the sum of both contributions, reproduces rather closely the experimental data of Losee et al. [1 ]. Subsequently we have estimated the magnetic entropy associated with the experimental curves by using: (i) the chain prediction [curve (a)] for T > 3 K; (ii) the experimental data for T < 3 K down to the lowest temperature reached; and (iii) a T 3 dependence as an extrapolation to T = 0, since this behavior is followed by the powder data for 0.06 < T < 0.15 K. We thus obtain a magnetic entropy change of A S / R = 0.69 and 0.56 for the powdered samples and the single crystal, respectively, which may be compared to the

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10 cl> 1) reached at T c [ 2 1 - 2 3 ] . As noted by Stirrat et al. [3], Xa shows the behavior expected for the antiferromagnetic parallel susceptibility (×~) by the fact that it apparently extrapolates to zero for T ~ 0. This indicates that the preferential direction coincides with the chain axis. Along the two perpendicular directions b and c, one expects a susceptibility (×±) that is nearly independent of temperature below T c (×±(T t Te the corrected xb/Cb and Xc/Cc would nearly coincide with xa/Ca. We may now derive estimates for Jar and for the spin anisotropies from the corrected susceptibilities. Simple' molecular field theory for a two-sublattice antiferromagnet will be applied, since this has been shown [21-23] to be quite good an approximation when Jar is small compared to the ferromagnetic intrasublattice interaction. Accordingly, we estimate Jar from the relation:

×AT

=

T~)IG

=

(Zaf~afl) -1

obtained from: xe(Tc) = x±(Tc) = Nog2u2/4Zaf[lafl, and using S = ½. Here Zaf denotes the (unknown) number of magnetic neighbours coupled by Jar" From xa/Ca =" 48 K -1 at T e, one has ZafJaf ~" - 0.024 K. Comparing the antiferromagnetic interaction with the ferromagnetic intrachain exchange Jx (for which z x = 2), one has: Zafkfafl/zxJ x ~' 1.5 X 10 -2. At this stage we cannot yet decide whether Jar is to be identified with the interaction Jy or Jz, or whether

197

perhaps both Jy and Jz are antiferromagnetic (we will argue below that Jy is ferromagnetic, and that Jz = J~f). The amount of spin anisotropy can be estimated from the values of the perpendicular susceptibilities ×b and Xc extrapolated to T = 0. We interpret the spin anisotropy in terms of an effective anisotropy field H A, acting upon the spins. Introducing also the antiferromagnetic exchange field asHaf = 2ZafklaflS/g# B, one has, at T = 0, X±(HA) = X±(H A = 0)I(1 +

HAI2Haf).

Since x±(HA = 0, T = 0) = xe(T = To), it follows that H A / 2 H a f = x A T c ) I X a ( O ) - 1.

With ×ATc)IG~" 48 K -1 and ×±(0)/c, ~ 12 K -1 one obtains HA[Hal ~' 6. Defining the intrachain exchange field as Hx = 2ZxJxS[g# B, we have Haf/Hx = ZafIJafl/ZxJx - 1.5 × 10-2, so that ItA[Hx ~-- 0.09. In the above we have disregarded the fact that the effective fields H x, Haf, H A will be different for the various crystallographic directions through their dependence on the g-values, since the errors involved in the )(4 and D i are too large to justify such refinements. The conclusion is that the spin anisotropy is of the order of 10% of the intrachain exchange. In the remainder of this section we will concentrate on the temperature dependence of the parallel susceptibility. For brevity we use the symbol Xexp for the quantity xaT/Ca, where Xa has been corrected for demagnetizing effects. At high temperatures the behavior of T6~xpwill be essentially 1-d in character. A theoretical prediction for the ferromagnetic chain susceptibility can be obtained for the uniaxial hamiltonian given in eq. (1) from calculations on chains or rings containing a varying but f'mite amount of spins, and from analyses of the high-temperature series expansions. The former calculations were kindly carried out by H. W. J. B16te at our laboratory. Paddapproximant analyses of the high-temperature series for the pure Heisenberg case and for the model with intermediate anisotropy were kindly performed by R. Navarro. The behavior of ~ = xT/C versus k T / J for values of the spin.anisotropy intermediate between the Heisenberg and Ising cases, are shown in fig. 9 down to the temperature at which the estimated

H.A. Algra et al./Lattice-dimensionality crossovers in [(CH3)aNH]CuC13 • 2H20

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Fig. 10. Experimental data (0) for Xexp = XaT/Ca for [(CHa)3NH]CuCI 3 • 2 H 2 0 versus kT/J x (Jx = 0.85 K). XH = ferromagnetic S = ~-Heisenbergchain. XO.9 = idem, but 1

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Fig. 9. Theoretical susceptibilities ~ = xT/C for S = ~-ferromagnetic chains with anisotropic interaction. The curves are

labeled by the parameter J~x/JIx. Curves XHand XI are for the Heisenberg and Ising limits, respectively. For comparison the molecular field prediction, XMF,and the result for the ferromagnetic Heisenberglayer, X2d, have been included.

errors become larger than a few per cent. For the Ising chain, one has the exact solution 21 = exp (J/kT). For the Heisenberg chain the first ten terms in the high-temperature series expansion are given by Baker et al. [24]: 2.H=l

+K_lK2+5K4+ 3

2 K7+1269.8 - 6-'3 ~/~

24

7 K5 40 3737 K9 36288

t33

-6

- 4--8--6/C

334691 _ 10 14~ K ,

where K = J/kT. For intermediate spin anisotropy, seven-terms in the high-temperature series expansion have been obtained by Obokata et al. [25]. The molecular field result for a ferromagnetic chain, X M F = (1 -- K) -1, and the high-temperature series prediction [19, 26] for the 2-d Heisenberg ferromagnet (curve X2d) have been included for comparison in fig. 9. In fig. 10 the data for Xexp are compared with the prediction for the ferromagnetic chain with 10% Ising anisotropy, using for the exchange constant the value Jx/k = 0.85 found from the specific-heat analysis. Only at the highest temperatures is Xcxp seen to be described by this prediction. We ascribe the

with 10% Ising anisotropy in the interaction. Curves (a) and (b) are obtained from xO.9 by considering the effects of the interchain interactions Jy and Jz, as explained in the text (data from ref. 3).

differences observed for kT/J < 3 to the effects Of Jy and (at lower temperatures) o f J z. Although for the specific heat we found the influence Of Jy to be appreciable for kT/J x < 2 only, theoretical results regarding lattice-dimensionality crossovers in fact predict that these would become perceivable at higher values of kT/J for the susceptibility than for the specific heat [8, 22]. In the theory assemblies of weakly coupled chains are considered. The thermodynamic functions are developed in expansions with respect to the parameter J'/Jx, where J ' and Jx denote the interactions between and along the chains, respectively. For the susceptibility the first order term in the expansion could be derived [8, 9] :

X = Xx + 2z'

-~

\Jx ! + . . . .

(2)

where z' is the number of additional magnetic neighbours coupled by J ' . Although this relation was derived for the n-vector model (including the S = ½ Ising model and the classical Heisenberg model), it is has recently been shown [22, 23] that the susceptibility behavior of a number of ferromagnetic layer-type Cu 2+ salts can be well explained on the basis of this prediction. In our present problem we may replace ~ in expression (2) by Xexp, and Xx by the prediction X0.9

H. A. Algra et al./Lattice-dimensionality crossovers in [(CHa)aNH] CuCl 3 • 2H20 for the susceptibility of a Heisenberg chain with 10% Ising anisotropy. For the interaction J ' between the chains we have experimentally the two different interactions Jy and Jz to consider. So far we only know that IJyl ~" IJzl, and that at least one o f the two should be antiferromagnetic with zafJaf ~ - 0 . 0 2 4 K. Now from eq. (2) it follows that an antiferromagnetic interaction between the chains will lower the susceptibility with respect to that o f the isolated chain. Since Y~0.9is already below Xexp, the effect of Jar is to yield a curve that is even farther below the experimental one. The prediction obtained by calculating the susceptibility from the sum: n0 9 + (2ZafJaf/kT) ~02.9 is shown in fig. 10 as curve (a). In our opinion this gives a clear indication that one of the interactions Jy and Yz has to be ferromagnetic. Moreover, this ferromagnetic interaction must exceed the antiferromagnetic Jafin order to produce the net upward shift of the susceptibility from X0.9 required to fit the experiment. From these considerations, therefore, we identify the antiferromagnetic interaction Jaf with Jz, i.e. the weakest interaction. The interaction Jy would be ferromagnetic and an order of magnitude estimate for Jy can be obtained from the differences between the experimental data and X0.9 in the range kT/J x > 1. We thus arrive at a ratio: ZyJy/zafklafl ~- 2 - 4 , so that ZyJy/zxJx would be about 5 X 10 -2. We have therefore in total ZxJ x :ZyJy :zz IJzl = 1:0.05:0.015 for the ratio of the three interactions in [(CH3)3NH ] CuCI 3 • 2H20. The prediction for the susceptibility obtained from eq. (2) by including both the ferroand the antiferromagnetic interchain interactions, is drawn as curve (b) in fig. 10. The agreement with the experimental data is reasonable. However, since the model used is very crude (only first-order terms are taken into account), evidently only an order of magnitude estimate for ZyJy can be made from it. Finally, we mention that in the paper by Liu and Stanley [8] a similar expansion as in eq. (2) is made for the specific heat, for which it is found that the first order term in the parameter (J']Jx) vanishes. This may explain why the dimensionality crossover appears at a lower temperature k T / J x in the specific heat than for the susceptibility. It may also explain why the 2-d character (Jy > IJzl) appears so pronounced in the specific heat, in spite of the relatively small difference in magnitude between Jy and Jz.

199

Acknowledgements

We would like to thank J. N. McElearney and J. A. Cowen for making available to us their data on the specific heat and the susceptibility, respectively. We are also grateful to R. Navarro and H. W. J. B16te for permission to use their calculational results cited in this paper. This investigation is part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie" (F.O.M.), which is financially supported by the "Nededandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (Z.W.O.). References

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