Magnetic properties and critical behaviour of GdAl2: Thermal expansion, magnetization, magnetostriction and magnetocaloric effect

Journal of Magnetism and Magnetic Materials 73 (1988) 289-298 North-Holland, A m s t e r d a m

289

MAGNETIC P R O P E R T I E S AND CRITICAL B E H A V I O U R O F GdA! 2: T H E R M A L EXPANSION, MAGNETIZATION, M A G N E T O S T R I C T I O N AND M A G N E T O C A L O R I C E F F E C T E. du T R E M O L E T de LACHEISSERIE Laboratoire Louis NOel, C N R S - U S T M G , 166X, 38042 Grenoble cedex, France

Received 21 January 1988; in revised form 1 April 1988

The magnetic properties of GdAI 2 have been revisited with a special emphasis on their critical behaviour. Thermal expansion measurement has confirmed the spontaneous volume magnetostriction to be negative ( A V / V = --11.6 × 10-4). The paramagnetic susceptibility has been fitted up to 2 Tc by a modified C u r i e - W e i s s law: X = Xo + ( C / T ) ( 1 - T c / T ) -'e with a paramagnetic m o m e n t equal to the theoretical one for J = 7 / 2 and a critical exponent - / = 1.13 + 0.01. At T = Tc, o - H~/~ with ~ = 3.60+0.05 and just below To, M s - ( T ~ - T ) ~ with fl = 0.44+0.015 in excellent agreement with the scaling laws. Spontaneous and forced magnetostriction have been measured along [111] and [100] directions. ~,,2 is found to exhibit a sign opposite to the one previously given in the literature. The magnetocaloric effect has been deduced from the difference between adiabatic and isothermal strains, and derived exactly in terms of "t above To; it is shown to decrease with temperature more slowly than the parastriction does.

1. Introduction

The rare earth-aluminium alloys with Laves R A I 2 , have been extensively studied these last 25 years, and their main magnetic properties have been summarized by Rossignol [1]. In this series, GdA12 exhibits the highest Curie temperature ( Tc = 168 K) and is considered to be a good example of a Heisenberg ferromagnet. No crystal field effect is expected in this alloy, due to L = 0: its g value is 1.989 _+ 0.005 above 200 K, as derived from ESR data [2]. A number of experiments give its spontaneous magnetic moment to lie between 6.83 and 7.20/z B, its ferromagnetic Curie temperature between 153 and 182 K, its paramagnetic Curie temperature from 168 to 180 K and its effective paramagnetic moment within (7.93 _+ 0.01)/t B. Further on, a neutron diffraction experiment at 4.2 K gave a magnetic moment too small on the Gd sites (6.6/zB) with a 0.6t~B moment in the conduction band [3]. Large exchange effects have been observed on various physical properties, namely the thermal expansion [4], the elastic constants [5], the transphase,

port and thermal properties [6], and the specific heat [7]. The magnetic anisotropy is very small as expected for an S-state and both N M R data [8] and torque measurements [9] seem to indicate an [111] easy magnetization direction. Last, the magnetostriction coefficients •V,2 and ~k''2 have been measured by a strain gauge technique [9] but the authors found a large positive forced magnetostriction near T~ while one should expect a negative one from the negative sign of the spontaneous exchange striction [4]. So we have decided to check both the thermal expansion and the magnetostriction and to try to elucidate this contradiction. In order to scrutinize also the critical behaviour of the magnetoelastic effects in connection with the magnetization, we have undertaken a complete study of both the magnetization and the magnetostriction from liquid helium up to room temperature (--2Tc), taking advantage of the recent works by Souletie et al. concerning the critical behaviour of the susceptibility up to 3Tc in ferromagnets [10]. Please, remind A L P H A will denote hereafter

0304-8853/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

290

E. du Tremolet de Lacheisserie / Properties of GdAI 2

the linear thermal expansion coefficient, in order to avoid any confusion with the critical exponent Or.

2. Experimental The alloy was prepared by melting 27.92 g G d and 9.57 g A1 in a cold crucible by the levitation method. Then the ingot was fused in a sealed tantalum crucible and cooled from the melt in a Bridgman furnace. A single crystal of 18.3 g was obtained in this way and a sphere 5.58 m m in diameter was spark cut from this boule. The mosaicity was within 3 o as checked from X-ray data. The magnetic moments have been measured along a [111] direction by an extraction method from 1.6 to 300 K. The magnetic field was supplied by a superconducting coil previously heated up to room temperature, in order to eliminate any remanent field. Then using a low current power supply and an accurate ~A-meter it was possible to measure the magnetic moments in low applied magnetic fields (80 A m -1 < H a < 48 kAm-X). After a few cycles up to this maximum field, the remanent field was found to be + 280 A m - 1 when the liquid helium level was maximum in the cryostat, and + 230 A m -1 when it was minimum. In order to know the true applied magnetic field, H a, the remanent field was measured after each run, and added to the value of the field derived from the measurement of the current. This procedure was used first at 1.6 K in order to check the linearity of the specific magnetic moment, o, against H a in the whole field range up to 48 kAm -a, and to verify the value N = 1 / 3 of the demagnetizing factor, and later on to plot accurately a against the internal field H i near T~. A second set of experiments was performed with a standard power supply. Most experiments were performed up to 1.6 M A m -a since it was quite sufficient for saturating the sample. At T~ however, data were taken up to 6 M A m -1 in order to plot ln(T) against ln(Hi). The magnetic susceptibility was measured using the same magnetometer up to 300 K. A last set of experiments was performed up to 6 × 10 6 A m -1 after the sam-

ple was annealed (1050°C, 24 h), in order to confirm the values of the spontaneous magnetization and of the high field susceptibility. The thermal expansion was measured using a tube-type dilatometer [11] recently automatized; the heating rate was 0.6 K / m i n . The magnetostrictive strains were also measured by means of this dilatometer but operations were manual in order to separate adiabatic and isothermal strains as will be described later on.

3. Thermal expansion The linear thermal expansion coefficient ALP H A = dL/LdT has been measured up to 280 K. In fig. 1, the experimental data are given, together with a theoretical Griineisen law where we have chosen the Debye temperature O D = 380 K taken from the literature [12]; when fitted to experimental data above 200 K, this gives an asymptotic value of A L P H A = 13.7 x 10 -6 K -a at high temperature. The experimental expansivity is compared in fig. 2 with the one obtained by integrating the above Gri~neisen curve and shifting upwards the resulting curve in order to get the coincidence with experimental data above T~. So, our experiment gives a volume magnetic anomaly at 0 K: (SV/V)M = -- 11.6 × 10 - 4 , which confirms the negative sign given in ref. [4] but our

eLPHA,

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-6

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1

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GdAI 2 • <+.,,

:

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0

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K

Fig. 1. Thermal dependence of the experimental linear expansion coefficient ALPHA = d L / L d T (dots) and of the theoretical Griineisen curve derived with O D = 380 K and ALPHA(T~) = 13.7 × 10 -6 K - 1 (full line). Broken line: theory (see section 7).

E. du Tremolet de Lacheisserie / Properties of GdAI 2 &L/L 10-4 I

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[

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I /

I

In fact, all these data can be fitted by a function

/

gO

GdAI

2

291

Y = (1 +

x )m 2 xm 4, -

(2)

-

"

rather than the expected m 2 function. The best fit is observed with x = 0.5, which gives the full line in fig. 3. The volume a n o m a l y can then be written:

It

8

(~-)M=(--17.4m2+5.8m4)×10

4 0

(3)

= ~a,2 _~_ ~ka,4. so

too

tso

zoo

zso

"r,

Fig. 2. Linear thermal expansion of GdA12 (full line) and shifted upwards theoretical expansion derived by integrating the theoretical Gfiineisen curve of fig. 1 (dotted line).

numerical value is twice as large as the one obtained from strain gauge measurements [4]: the difference could be ascribed to insufficient thermal corrections of the gauge factor and of the drifts of the strain gauges. The pressure dependence of Tc has been given in ref. [13], and using the elastic constants [5] one finds d l n ( T ~ ) / d l n ( V ) = - 3 . 3 6 . In the molecular field approximation, one can expect the exchange contribution to (SV/V)M to be written: X-,2

-4

So, the exchange striction X~'2 which is usually the only origin of (SV/V),vl enters in competition with an other contribution: X"'4 = + 5.8 × 1 0 - a m 4. Such a contribution has already been observed in other G d alloys, e.g. G d Z n (see section 7); its origin is not clear presently, but one can note that subtracting X a'4 f r o m the observed magnetic volume a n o m a l y at 0 K gives X~,2 = _ 17.4 × 10 -4,

+.++~. +~ +

I

I

I

~\.

++

I

°% +

3NkBTcs S

3 O ln(Tc ) + 1 C n + 2C12 0 I n ( V ) m2, (1)

where N is the n u m b e r of G d atoms per unit volume and m = M(T)/M(O) is the reduced magnetization taken f r o m the next section. The negative sign is in agreement with the data, but the predicted strain value, X a ' 2 = - 1 8 . 5 × 10 -4 , is 60% larger than the whole volume a n o m a l y we have observed. Pourarian [4] f o u n d his data to agree with a m 2 temperature dependence above 60 K and observed also a misfit at lower temperatures. We give in fig. 3 the thermal dependence of our reduced volume a n o m a l y (SV/V)T/(SV/V)o, together with the reduced magnetic contributions Ac/Ac o t o C L = ½ ( C l l -4- C 1 2 -4- 2C44), C ' = ½(Cal - - C 1 2 ) and C44 taken from ref. [5], since these four quantities are expected to vary as the same isotropic correlation function Y = (Si. Sj) [14].

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o n

o\-.

+

o~+,

oo

°\+

o 0.4

o

~+ +

_

oo ';.; o

_ o"

°

°oo.

!

Fig. 3. Reduced thermal variation of the volume exchange striction (AV/V)r/(AV/V)o (full circles), of the ma~uetic contribution to various elastic constants, after ref. [5]: C L ½(Cll + C12 +2C44) (dots); C' = ½(Cla - C12) (crosses) and C44 (open squares). The full line represents 1.5m 2 - 0 . 5 m 4. =

E. du Trernolet de Lacheisserie / Properties of GdAI:

292

a value in excellent agreement with the one predicted by eq. (1), namely - 1 8 . 5 × 1 0 - 4 .

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I

I

/', L n ( T ) 6Ln(XT) -- 0.5

GdAI2 / /

4. Magnetization

The specific weight of GdA12 has been given 10300 (5679.3 ___0.2) kg m -3 [5] at 300 K. Using our expansion data, we can derive p at any lower temperature and find P0 = 5727.0 kg m - 3 at 0 K. With a spherical sample, N = 1 / 3 and we expect at 0 K a ratio n a / o = 1909 kg m -3 as long as H i = 0. The experimental value we find at 1.6 K for this ratio is 1912.3 kg m -3 below H a = 48 k A m - i : the two figures agree within 0.17%, so it is possible to accurately derive the internal field H i and the magnetization M from the applied magnetic field H a and the specific magnetic moment a. N e a r T~, a has been measured as a function of H a, from 80 A m - t up to 6 × 10 6 A m - ] with

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Fig. 4. Logarithmic plot of o (Am2kg - 1 ) against H i ( A m - i ) at various temperature near T& - , - : - : T c + 1 2 0 0 inK, [] []: Tc + 1 0 3 m K , o: T¢+27 m K , O: T c - 2 2 m K , O : T ~ - 9 1 inK, - . . . . . : T o - 7 1 0 mK, - - . . . . . . : Tc-1840 mK.

/

--

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200

220

240

260

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T, K

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Fig. 5. - A I n ( T ) / A I n ( x T ) - ( T - Tc)/y ~ vs. T.

temperature increments of about 0.1 K. Various isotherms are given in fig. 4 in terms of In(o) plotted against l n ( H 0 : a perfect straight line is observed for T~, and significant deviations appear, at lower fields, for T - Tc as small as 0.1 K. At T~ = 168.3 K, we can derive the critical exponent 8 from equation:

a-H:/~,

T
-

-

(4)

and we find 8 = 3.60 + 0.05 to be c o m p a r e d with the predicted value 8 = 3 from the molecular field theory and 8 = 4.8 from the 3-d Heisenberg model. A b o v e Tc, we have derived f r o m our data a ( H A, T ) the thermal variation of the paramagnetic susceptibility X taken for H i ~ 0. Following Souletie et al. [10], we analyse x(T) by plotting - A l n ( T ) / A l n ( x T ) against T in fig. 5. W e observe a straight line which defines the same Tc as above; f r o m the slope, we derive the critical exponent: y = 1.13 _ 0.01. GdA12 is usually considered to be a g o o d Heisenberg ferromagnet: y should be equal to = 1.39 in this case. One can note that the observed value is closer to the molecular field prediction 3' = 1: this indicates that long range magnetic interactions do contribute markedly to the ferromagnetism of this alloy.

E. du Tremolet de Lacheisserie / Properties of GdAl2 So, X appears to be suitably fitted in terms of the modified C u r i e - W e i s s law: x=

.,=0 =

+x0

(5)

In fig. 5, A l n ( x T ) was calculated using the experimental X without any correction for Xo: this means that either X0 varies as the main contribution to X or Xo is smaller than 5 x 10 -4 MKS. As it is usually accepted that Xo is temperature independent, we conclude that Xo < 5 × 10 -4 which is consistent with a spin wave analysis leading to X0 = 1.9 x 10 -4 M K S [15]. The best fit leads to a Curie constant C = 2.64 (dimensionless in rationalised M K S A unit), f r o m which we can derive a paramagnetic m o m e n t /% = 7.89ttB per G d a t o m in a perfect agreement with the theoretical value g j J ( J + 1) where gj = 1.989 [2] and J = 7 / 2 . Last, the paramagnetic Curie temperature is given [10] by: 0 = yT~ = 189 K.

(6)

At this point, we can derive the critical expon e n t / 3 f r o m the scaling law: /3 = , / / ( 8 - 1) = 0.435 + 0.015

(7)

and, using these/3 and 7 values, we can plot the isotherms M 1/p v e r s u s ( H i / M ) 1/v (Arrott plots) f r o m which it is easy to derive the spontaneous magnetic moment, o 0 and the spontaneous magnetization M s. Below 30 K, where Arrott plots are not so convenient, o ( h ) is best analysed up to 4 X 10 6 A m -1 in terms of o = o 0 + XHFHi + B / ~ i i

,

(8)

where XHF is the high field susceptibility and B/~i the law of a p p r o a c h to the saturation at moderate fields ( M J 3 < H a < 2Ms). B was f o u n d to be -- 900 M K S A and temperature independent below 30 K. X HF is the sum of a temperature independent term X0 previously discussed [eq. (5)] and a temperature dependent term which should tend towards 0 at 0 K: in fact XHF decreases strongly d o w n to 25 K with an extrapo-

293

lated value at 0 K which is X0 = (0 + 2 x 10 - 4 ) M K S A . But below 25 K, the temperature dependent contribution deviates f r o m this simple behaviour, exhibits a m i n i m u m at 10 K (XHF ----"16 X 10 - 4 ) and increases up to 18 x 10 - 4 at 1.6 K: this anomalous behaviour of XHF seems to indicate the appearance of a slight non-collinearity of the moments below 25 K. We shall n o w check the consistency of fl as derived f r o m eq. (7) with the value deduced from the power law: M s ( T ) - ( T ~ - T ) ~.

(9)

Letting m = M s ( T ) / M s ( O ) be the reduced magnetization, we have plotted y = m / ( T c - T ) t~ against T near Tc for various values of /3. F r o m 0.9 to 0.999T~, the best fit was achieved w i t h / 3 = 0.44 + 0.005 thus proving the consistency: in this temperature range, y was f o u n d to remain equal to 0.101 + 1%. The molecular field model predicts /3 = 0.5, and the 3-d Heisenberg model /3 = 0.365. Last, we tried to check the validity of the spin waves theory: our low temperature data do not follow the expected T 3/2 dependence on the temperature: up to 100 K, our data are best fitted by a law: M s ( T ) = 1.071 x 10611 - 3.44 X 1 0 - 4 T 3/2 X exp( - A / k T ) ]

Am-'

(10)

with A = 1.5 meV. This gap can be associated with the anomalously large XHF observed below 25 K. After annealing ( 1 0 5 0 ° C , 24 h), this effect was not modified. At 1.6 K, the specific magnetic m o m e n t was f o u n d (186.8 _+ 0.3) A m 2 kg -1 thus giving (7.06 _+ 0.01)~t B per G d atom, a value consistent with previous data although somewhat smaller than the more recent determinations.

5. Magnetostriction We have already pointed out that the magnetocaloric effect can contribute significantly to the observed magnetostrictive strains near Tc in G d Z n [16]. In GdA12 also, a large magnetocaloric effect

294

E. du Tremolet de Lacheisserie / Properties of GdAI 2

is observed, but due to the improved performances of our dilatometer, we have observed that this effect does contribute also far above Tc. So, an accurate determination of the magnetostriction necessitates to measure isothermal strains. When the magnetic field has just reached a given value, the observed initial strain (adiabatic strain) is the sum of the isothermal strain and of the thermal expansion due to the magnetocaloric effect. Then, the temperature of the sample decreases slowly down to the assigned value thus giving an exponential decay of the strain: the final strain, observed after a few minutes in the present case, is referred to as the isothermal one. When the field is switched off, the sample is rapidly cooled and the equilibrium zero strain is finally reached after a few additional minutes (see inset of fig. 10). Let All (A~I) and X± (AA) be the isothermal (adiabatic) relative change of length observed when the magnetic field lies, respectively, parallel and perpendicular to the measurement direction. The sum All -F 2A ± gives the volume magnetostriction A~'2 + A~'4 and the difference A~l- X ± gives the anisotropic magnetostriction, Av'~ or A''2 when the direction of the measurement of the strain is along [001] or [111], respectively. The isotherms All - X ± = F ( H a ) for a measurement direction along [111] are given in fig. 6. One observes that the line associated with the null internal field varies as the 3 / 2 power of the external applied magnetic field. Also the forced magnetostriction OA"2//0H at Tc is only twice as large as the one observed at 5 K. Note that isothermal and adiabatic measurements give the same values of All- X± since the magnetocaloric effect is isotropic. The isotherms All + 2X± as functions of H a beyond the saturation give the forced volume magnetostriction, OA'~'2/OH. The thermal variation of the isothermal forced volume magnetostriction is given in the inset of fig. 6. We observe that it is negative, in agreement with the volume anomaly (section 3) and the pressure dependence of Tc [12]. Let us discuss these results: first, extrapolating higher field data in fig. 6 down to null internal field, one gets the A''2 values. Their thermal variation is given in fig. 7. The magnetostriction is positive contrary to the previously published data:

T

r

1

r

o~ (>,#-~..)111 25

GdAI 2 20

~

4

0]5

0.6 H a 106Am 1

Fig. 6. Isotherms giving the anisotropic magnetostriction along [111] versus the applied magnetic field H a. Inset: forced volume magnetostriction vs. T.

the data from ref. [9] measured in 1.6 MAm -1, and changed in sign, are represented by open circles in fig. 7; their wrong sign explains why the forced volume magnetostriction had been found positive instead of negative. The best fit of X"2(T) is given by a function m 3/2 (broken line in fig. 7) instead of the m 2 law predicted by the two-ion localised model [14]. An enlarged plot of A''2 against T near T~ is given in the inset of fig. 7, together with a broken line representing: A', 2 __ m 3 / 2 _ ( T c - T ) 0-66.

Unfortunately, the accuracy of the measurements is poor due to the very large magnetocaloric effect observed in this temperature range; it seems, however, that A''2 varies as (T~ - T ) ~ with x < 1, the value predicted by the molecular field model. Additional data, concerning the [111] measurement direction, have been taken above T~. The isotropic parastriction p " = H i [All + 2X± 1-1/2 is plotted against the temperature in fig. 8. Thermo-

E. du Tremolet de Lacheisserie / Properties of Gd,4l e

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Fig. 7. Thermal variation of %,,2. Full circles: this work; open circles: after Burd and Lee [8], data taken at 1.6×106 Am -1 and changed sign (see text). Inset: enlarged figure near T~. Broken line: 29 X 10-6m3/2.

dynamics predicts a m 2 dependence of )ka'2 above T~. Letting M 0 be the magnetization at 0 K and X be given by eq. (5), one gets above T j

50

75

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100 T-Tc

I

I

Fig. 8. Isotropic parastriction versus T - T ~ . Full line: p ~ = 9.71 × 106 T(1 - T J T ) Lt3 A m - 1. Broken line: asymptote. Full circles: data measured along the [111] direction. Inset: volume forced magnetostriction vs. T below Tc.

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i

106x .~f,2 m2 / I

[2.64] 2 1 [

%11+ 2)~±

T

/

]2VH2"

= Am2= A~--~o I - ~ ~ ~ - T~ 1

tI

(11) The best fit for the isotropic parastriction has been found to be:

GdAI 2

I t

i I

150

/ / •

i

e

p~ = H t ALL+ 2)~ ± [-1/2

= 9.71 × 106T(1 - T J T ) 113 Am -1.

(12)

The full line in fig. 8 represents this latter equation thus giving A = - 1 7 . 5 × 10 -4. One observes a curvature near Tc, instead of the linear dependence p~ - T - Tc predicted by the molecular field model. The value for A is in excellent agreement with the one ( - 17.4 × 10 -4) given in eq. (3). In the same way, we tried to plot the an•sotropic parastriction p' = H 1%11 - %* 1-1/2 against

100

•/ / /

50

/ ¢

,.,,#PIt , - # " .ee"

50

100

150

200

I

[

I

I

T K I

Fig. 9. Thermal variation of %"2/m2.

296

E. du Tremolet de Lacheisserie

T. Unfortunately, it was impossible to fit p ' with an equation similar to (11): we give in fig. 9 the thermal variation of ~k~'2//m2. The m 2 dependence of ~,2 seems to be valid only at lower temperatures, with an increasing departure below Tc (X''2 -- m 3 / 2 ) and above: this means that the magnetoelastic coupling coefficient B c'2 in the Hamiltonian is no longer temperature independent. Along [001], the ~,2 strain is always very small as compared with the forced magnetostriction and with the strain due to the magnetocaloric effect: Burd and Lee [9] found Xv,2_ + 3 × 1 0 - 6 at 100 K. We find also a positive magnetostriction, but one order of magnitude smaller namely Xv ' 2 q'-0.45 × 1 0 - - 6 at this same temperature. A misorientation of about 1 ° of the measuring direction with respect to [001] could induce a strain of this order of magnitude, due to ~,,2 __. 17 × 10 -6 at this temperature: so it would be meaningless to give the thermal variations of such a small magnetostrictive strain, taking into account the observed mosaicity in our sample.

6. M a g n e t o c a l o r i c

effect

We have derived the magnetocaloric effect for the difference between the forced volume magnetostrictions measured adiabatically and isothermally, respectively. Using the thermal expansion coefficient taken from section 3, we have: 8T =

8X~,2ADIA - 8X~,2ISOTH 3 × ALPHA

(13)

Weiss and Forrer [17] have given a general expression for the magnetocaloric effect, namely: dHi

Cu

where o is the specific magnetic moment and CH the specific heat at constant field. Below Tc and beyond the technical saturation, this formula predicts a magnetocaloric effect 8T linear versus H i, while above Tc and for small magnetic fields o is linear versus H i and so 8T is expected to vary as H~. So, we have plotted d T / d H i vs. T below Tc, while above Tc, O =

/

Properties o f GdAI 2

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O-~ _

~r , 10-6 KIAm -I

3H~,

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i0 6Arn'IK~_/~

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•

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t ?

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51o II0 Fig. 10. Thermal variation of the magnetocaloric effect. Full circles: our data. Small dots below To: derived from our magnetization data, using C H values from ref. [7]. Small dots above To: derived from eq. (15) using C M values from ref. [7]. Broken lines are guides for the eye.

vs. T is a more convenient representation for discussing high temperature data which become negligibly small. The full circles in fig. 10 represent our experimental data, while the small dots are derived from eq. (14) using our magnetization data and the values of the specific heat tabulated in ref. [7]. We shall note that, using expression (5) for the susceptibility and integrating eq. (14), one finds the magnetocaloric effect to be written (in kelvin) above T~:

Hi/~

c

ST- c.

+(v-l)

Tc)Hi2,

T

(1~) where C is the specific Curie constant. When T >> Tc, eq. (15) leads to 8 T - T-1 while eq. (11) leads to X - T - 2 . So the magnetostriction vanishes more drastically than the magnetocaloric effect above T~: the perturbation due to this effect

E. du Tremolet de Lacheisserie / Properties of GdAl 2

when measuring the parastriction increases with the temperature. This T - a dependence of 8T gives a ~ dependence of O at T >> Tc and explains the curvature of O ( T ) in fig. 10. Let us note that our determination of the magnetocaloric effect gives always too a small value for ST: this is due to the fact that when the magnetic field is increasing before being stabilized at a given value, the sample is heated and begins to heat the sample holder. So, the so-called "adiabatic strain" i.e. the m a x i m u m of the curve given in the inset of fig. 10 is somewhat smaller than it should be in a truly adiabatic experiment.

7. Conclusions

The critical behaviour of some magnetic properties of GdA12 have been investigated. The values obtained for the critical exponents/3, y and 8 are consistent with the scaling laws and lead to an extremely small ct value: a = 2 - / 3 ( 1 + 8) = - 0 . 0 2 4 _+ 0.04, which explains the absence of any observed X-like peak in the specific heat near Tc: this peak, if it exists, would be associated with a very small positive value of a and could be observed only for Tc - T as small as - say - 10 -2 K. These values of the critical exponents lie between the values predicted by the molecular field theory and the ones expected for a 3d Heisenberg ferromagnet, but are closer to the first set" this indicates the presence of long-range magnetic interactions in GdA12, and this was suggested by a study on hyperfine fields in Dyl_xYxA12 [19]. The thermal expansion coefficient exhibits the same kind of critical behaviour, with a sharp X-like peak in the close neighborhood of T~. ALP H A can be calculated by derivating eq. (3) with respect to T after having replaced m by 0.101(T~ -- T ) 0"435. One gets for T close to Tc: 1 a(AV / A L P H A = -~ - ~ \ - - V } M

= {5.82(Tc- r ) -°13 - 0.13( × 10 - 6 K - 1 .

- r ) °74} (16)

The broken line in fig. 1 represents this equation

297

for Tc - T varying from 10 to 10 - 2 K: due to the heating rate of expansion measurement (0.6 K / m i n ) and to the time constant for thermal stabilisation of the spherical sample (~----3 min) this ),-like peak is hidden. It should be necessary to measure the expansion step by step with thermal increments as small as 10 m K in order to observe this peak. The critical behaviour of the magnetostriction just below Tc is given by (T~ - T ) "B if X - m". We have found n - - 3 / 2 for X"z and 13=0.44. So x = n/3 is smaller than unity, the prediction of the molecular field model. The magnetization of GdA12 had already been extensively studied and a rather large dispersion of the magnetic m o m e n t can be noticed, from 6.83 to 7.20#B per G d atom. The comparison of our data with those of Lee and Montenegro ( L & M ) [15] could clarify this situation: their crystal was grown by the Czochralski method [9] while our crystal was grown by the Bridgman technique with unavoidable stresses due to the crucible, and a further annealing did not modify anything. L & M observed a magnetic m o m e n t o = 7.2/~ B while ours exhibits 7.06/~ B. We find deviations from the Z 3/2 thermal dependence of a below 30 K which can be accounted for by a gap A--. 1.5 meV, while L & M observe the T 3/2 law down to 4.2 K. The value X0 = 1.9 × 10 -4 M K S A they found for the contribution of the conduction electrons to the susceptibility is consistent with our paramagnetic susceptibility data, but we observe a minimum at 10 K for the high field susceptibility Xnv, with a value for Xnv at 1.6 K ten times larger than ×0The same large XHV had been observed also by Rossignol [1] with an other crystal prepared by the same Bridgman technique: a non-collinear structure had been suggested for explaining this effect, and could also account for the smaller moment observed. Unfortunately, L & M have worked with a magnetic field limited to 1.4 × 10 6 A m -1 with disks 5 m m in diameter and 1 m m thick: so it is impossible to estimate from their data the value of Xrte at 4.2 K. An experimental determination of ×HE at 1.6 K with a Czochralski crystal would be worthwhile in order to check whether the high ×HE is a feature associated with the Bridgman crystals or is an intrinsic property of GdA12-

298

E. du Tremolet de Lacheisserie / Properties of GdAl 2

These low t e m p e r a t u r e p r o b l e m s d o n o t affect the m e a s u r e m e n t s a b o v e 50 K. The p a r a m a g n e t i c d a t a are in a close a g r e e m e n t with Souletie's m o d e l [10]: the p a r a m a g n e t i c m o m e n t 7.89/~ B is j u s t the one calculated when using the g factor t a k e n f r o m E S R e x p e r i m e n t [2], a n d Op = 189 K is larger t h a n any previous e s t i m a t i o n as expected since the slope of X - I ( T ) at r o o m t e m p e r a t u r e is s o m e w h a t smaller than the a s y m p t o t derived f r o m eq. (5). T h e value of T~ was f o u n d to b e 168.3 K f r o m m a g n e t i z a t i o n d a t a a n d 167.8 K f r o m m a g n e t o striction m e a s u r e m e n t s : b o t h p l a t i n u m resistances h a d been c a l i b r a t e d using two different reference thermometers. The isotropic m a g n e t o s t r i c t i o n M ' 2 + M '4 derived from A L P H A ( T ) varies b e l o w T~ as A [ m 2 - l m 4 ] a n d this b e h a v i o u r is similar to the one o b s e r v e d in G d Z n [16]. F o r T > Tc, m 4 c a n be ignored as c o m p a r e d with m2: the i s o t r o p i c p a r a s t r i c t i o n is j u s t given b y A x Z H 2, a n d the A coefficient is also consistent with the p r e s s u r e d e p e n d e n c e of Tc [13]. Last, the a n i s o t r o p i c m a g n e t o s t r i c t i o n coefficients ?`~,2 a n d ?`,,2 have b e e n checked: the o r d e r of m a g n i t u d e is consistent with previous d a t a b u t the sign of ?,,,2 in ref. [9] was incorrect p r o b a b l y due to a w r o n g analysis of strain gauge m e a s u r e ments. Both coefficients are positive a n d ?`,,2 is m u c h larger t h a n ?`~,2 as also o b s e r v e d in some o t h e r Laves phases. T h e t e m p e r a t u r e d e p e n d e n c e was e x p e c t e d to vary as m 2, since ?`p,2 c a n n o t arise f r o m single-ion m a g n e t o e l a s t i c c o u p l i n g (S-state). In fact, ?`c'2/m2 r e m a i n s roughly c o n s t a n t up to 100 K a n d increases m o r e a n d m o r e when the t e m p e r a t u r e is raised. This u n e x p e c t e d b e h a v i o u r h a d a l r e a d y been observed in nickel a n d i r o n single crystals [18], a n d p o i n t s o u t that the m a g n e t o e l a s t i c coup l i n g coefficient B c'2 in the H a m i l t o n i a n increase when the t e m p e r a t u r e is raised. A l o n g the [001] direction, it was i m p o s s i b l e to s t u d y the t h e r m a l v a r i a t i o n of ?`~,2 f o u n d to b e as small as a n y spurious strain that could occur from the m i s o r i e n t a t i o n of the crystal.

Acknowledgements

W e are p l e a s e d to gratefully a c k n o w l e d g e B. B a r b a r a , w h o suggested this study, for his stimulating interest a n d to J. Souletie for m a n y fruitfull discussions. W e are i n d e b t e d to R. R a p h e l for the crystal g r o w t h a n d to R. A l e o n a r d for his k i n d assistance.

References

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