Magnetic properties of PrX2 compounds (X = Pt, Rh, Ru, Ir) studied by hyperfine specific heat, magnetization and neutron-diffraction measurements

Physica 119B (1983) 215-227 North-Holland Publishing Company

MAGNETIC PROPERTIES OF PrX2 COMPOUNDS (X = Pt, Rh, Ru, Ir) STUDIED BY HYPERFINE SPECIFIC HEAT, MAGNETIZATION AND NEUTRON-DIFFRACTION M E A S U R E M E N T S F . J . A . M . G R E I D A N U S * , L.J. de J O N G H a n d W.J. H U I S K A M P Kamerlingh Onnes Laboratorium der Rijksuniversiteit, Leiden, The Netherlands

P. F I S C H E R a n d A. F U R R E R E. T.H. Ziirich, Institut fiir Reaktortechnik, Wiirenlingen, Switzerland

K.H.J. B U S C H O W Philips Research Laboratories, Eindhoven, The Netherlands

Received 20 October 1982 Magnetic ordering phenomena in rare-earth intermetallic compounds can be unravelled most advantageously in the case of simple crystallographic structure and when a combination of microscopic and macroscopic techniques is applied. Here we shall present the temperature and magnetic field dependence of the magnetic moment of the cubic PrX2 compounds (X= Pt, Rh, Ru, Ir), as inferred from hyperfine specific-heat, magnetization and neutron-diffraction measurements. The results are compared with a mean-field calculation, taking crystalline electric field and bilinear (dipolar) exchange interactions into account. Adopting experimental values of the Lea, Leask and Wolf parameters x and W from inelastic neutron scattering results, we find satisfactory agreement between our magnetic data and the mean-field theory. An observed discrepancy of about 15% between the calculated and measured saturation values of the spontaneous magnetization can be explained by the presence of quadrupolar interactions.

1. Introduction In a p r e v i o u s p a p e r [!] we have r e p o r t e d inelastic scattering e x p e r i m e n t s o n the cubic Laves phase c o m p o u n d s PrX2 (X = Ir, Pt, Rh, Ru, Ni). F r o m the positions a n d intensities of the observed p e a k s in the inelastic n e u t r o n scattering spectra we were able to derive crystal-field splittings ( C E F ) of the 3H4 electronic g r o u n d state of the Pr -~+ ion. T h e s e splittings are r e l a t e d to x a n d W p a r a m e t e r s as d e f i n e d by Lea et al. [2]. Leaving PrNi2 aside o u r m e a s u r e m e n t s i n d i c a t e d x values r a n g i n g from 0.68 up to 0.93 a n d W values in b e t w e e n - 0 . 3 3 m e V a n d - 0 . 5 7 m e V . A s u m m a r y of the results for the v a r i o u s comp o u n d s is given in table I. T h e region c o v e r e d by the a b o v e values is in fact an i n t e r e s t i n g part of the L L W d i a g r a m since at x = 0.86 the F~, F4 a n d * Present address: Philips Research Laboratories, Eindhoven, The Netherlands.

F3 levels cross, for x < 0.86 the F3 level a n d for x > 0.86 the ['1 level b e i n g the g r o u n d state. Both the singlet (El) a n d the d o u b l e t (F3) are n o n magnetic. It is a n t i c i p a t e d that the n a t u r e of the g r o u n d state, as well as the e n e r g y s e p a r a t i o n s from the higher levels will be reflected in the m a c r o s c o p i c t h e r m o d y n a m i c properties. A t p r e s e n t n e u t r o n inelastic scattering is the most direct m e t h o d p r o v i d i n g microscopic inf o r m a t i o n o n the splittings b e t w e e n the C E F levels. Since it is, u n f o r t u n a t e l y , n o t always possible to d e t e r m i n e the C E F p a r a m e t e r s u n i q u e l y from inelastic n e u t r o n scattering e x p e r i m e n t s , t h e r m o d y n a m i c d a t a are e x p e d i e n t in o r d e r to check the c o n c l u s i o n s from the n e u t r o n data. In addition, i n f o r m a t i o n o n the n a t u r e of the interactions b e t w e e n the m a g n e t i c ions can be obtained. In the p r e s e n t p a p e r we accordingly report hyperfine specific-heat, high- a n d low-field magn e t i z a t i o n a n d n e u t r o n - d i f f r a c t i o n d a t a o n the

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F.J.A.M. Greidanus et al. / Magnetic properties of PrXz compounds ( X = Pt, Rh, Ru, Ir)

above PrX2 compounds. The results refer, in most cases, to a t e m p e r a t u r e region in which the PrX2 are magnetically ordered. In a following p a p e r we will discuss ac susceptibility, resistivity and specific-heat data above 1 K. We obtain from the hyperfine specific-heat data the average value of the z - c o m p o n e n t of the angular m o m e n t u m (Jz) at low temperatures. Neutron-d~f[raction data at various temperatures lead to the t e m p e r a t u r e dependence of the magnetic m o m e n t (t~) = gJp~B(,/) (where gj is the Land6 factor, gs(Pv~+) = 0.8, and p-a the Bohr magneton). From bulk magnetization data we can infer the field dependence of the average magnetization per ion in the sample, which we shall denote by m. The results will be c o m p a r e d with theoretical calculations, based on the x and W values obtained from inelastic neutron scattering. We shall use the molecular field approximation to account for the magnetic interactions between the PC+ ions, which we will assume to be Heisenberg-like: Y ~ = Z u J u Ji .Jj. The molecular-field approximation is appropriate, because the interactions in the PrX2 compounds, which all have metallic character, are long-ranged and hence involve an appreciable n u m b e r of interacting atoms.

217

W e deal with the interactions between the Pr 3+ ions in the molecular-field approximation, and we write the exchange Hamiltonian as: ~ ( ~ = -gs~BHM "J + 1AM2

(2)

with HM = gj/ZBA ( J ) ,

(3)

where A is the molecular-field parameter. The latter is directly related to the transition temperature. If the phase transition is of second order the molecular-field constant A can be calculated from the relation X = Xo'(T~)

(4)

in which X0 is the single-ion susceptibility in the absence of magnetic interactions. The spontaneous magnetization below the transition t e m p e r a t u r e Tc can be calculated by solving selfconsistently the following relation: N

(Jz) = ~_,(ilJzli) e x p ( - E J k B T ) / i-1 N

e x p ( - Ei/ kn T ) ,

(5)

i=l

2. Molecular-field theory for the magnetization As outlined in [2] the crystal-field potential in cubic symmetry with the axis of quatization along the cube edge is given by

~c~ =

W{F--~)[O°+504] +~

[O~-21 064]}

(1)

in which W and x are the L L W parameters, which determine the crystal-field strength, F(4) and F(6) are numerical factors given by L L W and the O~ are "Stevens o p e r a t o r equivalents". The W and x p a r a m e t e r s for the PrX2 compounds as determined from inelastic neutron scattering experiments are tabulated in table I.

where Ei and li) are the eigenvalues and eigenstates of the Hamiltonian ~ ' = ~aCE F Jr- ff~aMF. Magnetizations in external fields below and above the transition t e m p e r a t u r e can also be calculated with formula (5) by adding the Z e e m a n energy term to the Hamiltonian (2). In eq. (5) we assumed the spontaneous magnetization to be pointing along one of the cube edges, as has been d e m o n s t r a t e d experimentally for PrA12 and Prlr2 [3, 4]. W e will assume it to be valid for the other PrX2 compounds as well. In the present experiments (Jz) is determined as a function of t e m p e r a t u r e or external magnetic field. In the following sections the experimental results will be c o m p a r e d to (Jz), calculated on the basis of formula (5), using x and W as inferred from inelastic neutron scattering experiments, and A as calculated from the values of To.

218

F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 compounds (X - Pt, Rh, Ru, Ir)

3. Sample preparation The neutron-diffraction experiments were p e r f o r m e d on t h e s a m e s a m p l e s as in t h e inelastic n e u t r o n s c a t t e r i n g study. T h e s a m p l e s for t h e m a g n e t i z a t i o n a n d h y p e r f i n e specific-heat m e a s u r e m e n t s w e r e p r e p a r e d s e p a r a t e l y , following t h e m e t h o d s o u t l i n e d in [1], a n d t h e y w e r e c a r e f u l l y c h e c k e d by X - r a y t e c h n i q u e s . A f t e r t h e c o m p l e t i o n of t h e h y p e r f i n e specific-heat a n d low-field m a g n e t i z a t i o n m e a s u r e m e n t s on t h e small ingots, t h e l a t t e r w e r e g r o u n d to v e r y fine p o w d e r s ( 5 - 5 0 / z m ) in o r d e r to b e used in highfield (pulsed) m a g n e t i z a t i o n m e a s u r e m e n t s .

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4. High-field magnetization measurements M a g n e t i z a t i o n e x p e r i m e n t s up to v e r y high fields ( 4 0 T ) w e r e p e r f o r m e d in t h e p u l s e d - f i e l d m a g n e t of t h e K a m e r l i n g h O n n e s L a b o r a t o r y [5], at t e m p e r a t u r e s of 4.2 K a n d 77 K. T h e p u l s e d field m e t h o d has t h e d i s a d v a n t a g e that the c h a r a c t e r i s t i c t i m e c o n s t a n t i n v o l v e d is r a t h e r short, since t h e d u r a t i o n of t h e s i n e - s h a p e d pulse is o n l y 20 ms. T h e m e a s u r e m e n t s on o u r s a m p l e s , having a small electric resistivity at 4.2 K, w e r e h a m p e r e d by t i m e effects, arising f r o m t h e finite s k i n - d e p t h of t h e m e t a l l i c crystallites (even for o u r finely p o w d e r e d samples). T h e r e f o r e a s t r o n g d e p e n d e n c e of t h e signal on d H / d t was o b s e r v e d at 4.2 K, which r e n d e r e d t h e d a t a t a k i n g at 4.2 K imprecise. In fig. 1 t h e p u l s e d - f i e l d m a g n e t i z a t i o n m e a s u r e m e n t s in fields up to 4 0 T , at a t e m p e r a t u r e of 77 K, a r e s h o w n as d a s h e d curves. D a t a on PrA12 a n d PrMg2 a r e i n c l u d e d as well. T h e solid lines a r e t h e o r e t i c a l c a l c u l a t i o n s b a s e d on t h e x a n d W v a l u e s listed in t a b l e I. T h e curves l a b e l e d (a) a n d (b) a r e c a l c u l a t e d on t h e basis of t h e crystal-field H a m i l t o n i a n (1), i g n o r i n g i n t e r a c t i o n s b e t w e e n t h e m a g n e t i c ions. F o r t h e c a l c u l a t i o n of t h e curves (b) w e a s s u m e d t h e e x t e r n a l field to b e p o i n t i n g a l o n g t h e c u b e e d g e . T h e curves (a) a r e o b t a i n e d b y a v e r a g i n g o v e r all p o s s i b l e angles b e t w e e n t h e q u a n t i z a t i o n axis a n d t h e e x t e r n a l * We thank G.J. Nicuwcnhuys for making available a computer program to perform the averaging.

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Fig. 1. Magnetization versus magnetic field at 77 K for PrPt2, PrRu2, Prlr2, PrRh2, PrAI2 and PrMg2. The experimental magnetization is indicated by the dashed line, whereas the solid lines are theoretical calculations, as explained in the text. The experimental data on PrRh2 are indicated twice. The calculations, however, are based on different level schemes (A) x = 0.93, W - - 0 . 3 5 m e V and (B) x-0.75, W= 0.33 mcV.

field direction.* F i n a l l y t h e curves (c) a r e o b t a i n e d by i n c l u d i n g b o t h t h e crystal-field H a m i l t o n i a n (1) a n d t h e e x c h a n g e H a m i l t o n i a n (2) in t h e calculation. H e r e a g a i n t h e e x t e r n a l m a g n e t i c field is t a k e n a l o n g o n e of t h e cubic axes. F o r PrRh2 two

F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 compounds ( X = Pt, Rh, Ru, lr)

calculations are presented based on the two different level schemes, tabulated in table I. Even at the highest fields applied, the magnetization is still far from saturation, w h i c h would amount to 3.2/~B (mm~=gjl~BJ, where gj = 0.8 and J = 4). On the other hand, with the exception of PrPt2 and PrRh2 in fields above 25 T, the measured magnetization curves always lie above "the crystal-field-only" prediction, which indicates the presence of exchange interactions. The differences are largest for PrAI2 and PrRu2, in agreement with the relatively large value of the molecular-field parameter )t for these compounds (table I). For all compounds the measured values are lower than those obtained from a calculation including the exchange interaction and assuming the external field to be oriented along the easy axis, i.e. the (001) direction (curve c). The latter may seem an improper assumption for a polycrystalline material. One should, however, keep in mind that our samples are ground to powders fine enough to justify the assumption that a considerable fraction of the crystal grains will be in

219

the form of monocrystals. Upon applying a magnetic field these monocrystals will tend to be oriented with the easy axis along the magnetic field direction. So we may expect the experimental results to be not far off the (001) prediction, the magnitude of the deviation depending on the fraction of monocrystals. From the similarity of the calculations for the two different crystal-field schemes of PrRh2 it follows that the magnetization at the temperature considered (77 K) is rather insensitive to changes in the crystalline electric field. This, together with the circumstance that the fraction of monocrystallites is unknown, makes these measurements unsuitable as a tool for determining crystal-field parameters. The results, however, are clearly in concord with the L L W parameters obtained with inelastic neutron scattering.

5. Low-field magnetization measurements

Magnetic isotherms at 4.2 K in field strengths up to 2 . 0 T have been studied by means of a

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220

F.J.A.M. Greidanus et al. I Magnetic properties of PrX2 compounds ( X = Pt, Rh, Ru, lr)

P A R vibrating sample magnetometer. For all the c o m p o u n d s the transition temperatures, as tabulated in table I, are above this temperature. The experimental data are shown in fig. 2. The dashed lines are obtained by extrapolating the curves in the region H > 0.7 T to H = 0, and the values of the zero intercept, m, are listed in table I. The behaviour of Prlr2 and PrRu2 is peculiar, in regard to the fact that in all but the lowest applied fields the initial magnetization falls below the corresponding hysteresis loop. The same behaviour has been found in PrCo2, and an explanation of this effect is given in ref. [6].

the cooling rod, except for PrRh2, where we used a pressure contact. The results of the specific-heat m e a s u r e m e n t s below ~ 2 K on the PrX2 ( X = Ir, Pt, Rh, lr) compounds are shown in fig. 3, from which the addenda have already been subtracted. The lower limit of the t e m p e r a t u r e range covered is about 50 inK. The high t e m p e r a t u r e limit ranges from 1.1 K (PrRh2) to 2.3 K (PrPt2). The latter could be reached by removing the superconducting switch connecting sample and cooling salt. Below 1 K the data are smooth, and in general the scatter is less than 3%, with the exception of

6. Hyperfine specific-heat measurements ,

The internal field developed below the transition t e m p e r a t u r e will remove the nuclear spin degeneracy (I = 5/2) of Pr 3+. The interaction between the nuclear quadrupole m o m e n t , and the 4f shell can also contribute to the splitting between the nuclear magnetic sublevels. These splittings can be detected by resonant techniques such as N M R or M6ssbauer spectroscopy. If these methods are not suitable, information can be obtained from the hyperfine specific heat, which probes the thermal population of the nuclear sub-levels. Several accurate hyperfine specific-heat m e a s u r e m e n t s have been performed, e.g. at Helsinki, on the rare-earth metals. For a review see refs. [7,8]. Particularly in metallic samples, which are often difficult to p r o b e with NMR, heat-capacity m e a s u r e m e n t s prove to be a very suitable tool in this respect. In spite of this, hyperfine specific-heat measurements on rare-earth intermetallic compounds have until now been relatively scarce. Once the splittings of the nuclear sublevels are known, the mean value of the angular m o m e n t u m operator (J~) can be derived, as outlined below. The hyperfine specific-heat data were obtained in the low-temperature ( T < 2 K ) specific-heat apparatus, described extensively elsewhere [9]. Both magnetic (CMN) and germanium therm o m e t r y have been applied. Heat contact to the samples was provided by electrolytically plating them with copper, and indium soldering them to

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F.J.A.M. Greidanus et al. I Magnetic properties of PrX2 compounds (X = Pt, Rh, Ru, It)

the PrRh2 data below about 140 mK, in which the scatter rises up to 10%. This is a result of the fact that the heat contact to the PrRh2 is worse than for the other samples. In the magnetically ordered state, and for ions occupying cubic symmetry sites, the hyperfine interaction can be written as [10]

221

The nuclear specific heat is given by the formula

[8] R CN -- ( k s T ) 2 +I ~,,j=_, (E 2, - E,E/) e x p [ - ( E , + E/)/kaT] × +i

Y( = -tx(H~,,dI)Iz

~,,/=_, e x p [ - ( E i + Ei)/kBT ]

(6)

(10) in which ~ = gNIxNI is the nuclear magnetic m o m e n t , and H¢a is the internal field. T h e eigenvalues of this Hamiltonian can be written as G = a0/~

(7)

in which

ao = tzH~n,z/I = A(Jz).

(8)

F r o m these formulas it follows that a determination of the energy eigenvalues leads to the expectation value of the z - c o m p o n e n t of the angular m o m e n t u m (Jz), provided that the constant A is known from other sources. The hyperfine field at the rare-earth nucleus, He~, which is proportional to the hyperfine constant A, is usually assumed to be c o m p o s e d of three terms H e , = H4f q- Hsp+ Htr

(9)

in which Hal = A4f(Jz)I/tx is the field created by the 4f electrons, consisting or orbit, spin and core polarization contributions. T h e value of A4f, as deduced from E S R m e a s u r e m e n t s on salts, is Aaf/h = 1093 + 10 M H z [10, 11]. Hsp is the "selfpolarization" field, arising from polarization of the conduction electrons by 4f-electron-conduction-electron exchange, which reflects itself through mechanisms such as conduction-electron contact interaction, core polarization and orbital polarization [12, 13]. The term Htr accounts for the transferred hyperfine field, produced by polarization of the conduction bands by neighbour and distant magnetic ions. Berthier et al. [13] showed that in PrA12 Hsp a m o u n t s to about 10% of H4f, while Htr is negligibly small (0.56 T).

where E/,E/ are given by eq. (7). The solid curves in fig. 3 are the best fits to the formula (10), obtained by shifting the Schottky curve along the t e m p e r a t u r e axis. The hyperfine specific-heat m e a s u r e m e n t s can be fitted reasonably well in this way. A b o v e -~1 K the data exceed the T -2 Schottky curve at all temperatures. This can be explained by the lattice and electronic contributions to the specific heat which are expected to be present at these temperatures. Furthermore, the accuracy of our apparatus is rather limited in this t e m p e r a t u r e region, and the same effect has been observed in a m e a s u r e m e n t on a standard terbium sample [9]. A second discrepancy between the measurements and the calculations, is found near the m a x i m u m of the Schottky anomaly. Except for PrIr2 the m e a s u r e m e n t s fall below the calculated curves, the discrepancy being largest for PrPt2 (about 10%). From the nature of the deviations it may be concluded that they cannot be due to the presence of impurity phases. In any case, it is very unlikely that impurity phases would amount to more than 2 - 3 % as follows from the X-ray analysis. The presence of nuclear quadrupole interactions might cause deviations from the calculations, shown in the figures. The nuclear quadrupole Hamiltinian can be written as ~ o = P[I 2 - ½ I ( I + 1)l ,

(11)

where

P = 3 e Q V = / 4 I ( 2 I - 1)

(12)

in which eQ is the quadrupole m o m e n t of the

F.J.A.M. Greidanus et al. / Magnetic properties of P r X 2 compounds ( X = Pt, Rh, Ru, lr)

222

nucleus, and Vzz is the z - c o m p o n e n t of the electric-field gradient at the nuclear site. T o discuss the influence of quadrupolar interactions on the hyperfine specific heat we will review the possible contributions to the quadrupole constant P. Because the p r a s e o d y m i u m ions in the Laves phase c o m p o u n d s occupy sites of cubic symmetry, the lattice contribution to the quadrupole constant vanishes. The electric-field gradient at the nuclear site due to the localized 4f electrons is given by V4~ = - e ( r ~ 3 } a j ( 3 J 2 - J ( d + 1)}

(13)

in which (rq 3) is the mean inverse cube of the distance between the 4f electrons and the nucleus, taking into account the screening effect of the other electrons and crj is a n u m b e r characteristic of each 4f" configuration, given in ref. [14]. When the saturation magnetization is attained (3J~ - J ( J + 1)} = J ( 2 J - 1). The 4f contribution to the quadrupole constant, p4f, assuming (Jz) = J, is given by Bleaney [10] and amounts for Pr 3+ to P4f/h = - 2 . 6 2 M H z , which is smaller by about a factor 400 than the hyperfine constant A. Higher-order magnetic hyperfine interaction effects, usually referred to as pseudo-quadrupolar interactions, can also contribute to the quadrupole interaction p a r a m e t e r P, depending on the symmetry properties of the electronic wave functions. These contributions can be written as [11, 15] pm

= A 2v

(ilJ~lg) 2

(14)

Schottky anomalies. In this way the observed data, especially for PrPt2 can be explained quite well. W e will discuss the possible origin of such an effect in section 8. Assuming the hyperfine constant A to be equal to A 4 J h = 1093 MHz, the average z-component of the angular m o m e n t u m (Jz) can be determined from the position of the Schottky anomaly on the t e m p e r a t u r e axis. T h e values obtained are listed in table I. T h e presence of a "self polarization" contribution to the hyperfine constant, would change the (Jz)-values. Unfortunately, this contribution is not known, except for PrAI2, in which A w is determined to be about 10% of A4f [13]. In table I the values of (Jz}lJ are listed as well.

7. Neutron diffraction

The neutron-diffraction experiments were carried out on two axis spectrometers at the nuclear reactor Saphir in Wiirenlingen, Switzerland, using neutrons of wavelength A = 2.34/k. In a neutron-diffraction experiment the neutron intensity is measured as a function of the scattering vector Q, yielding well-defined Bragg peaks in the nuclear contribution to the scattered intensity (do-/dJe),ud. The influence of lattice vibrations on the scattered intensity is expressed in the following way: (do-/dO).u~, ~ e -2w ,

(15)

where W is the D e b y e - W a l l e r factor where the summation is over all the excited states. These higher-order effects are small, typically by about a factor 10-2-10 3, as compared to the first-order hyperfine splitting. W e conclude that it is unlikely that nuclear quadrupole or pseudo-quadrupole interactions are responsible for the difference between the observed and calculated hyperfine specific heat. T h e presence of a distribution of hyperfine fields is the most probable explanation for the discrepancies between calculation and experiment as found near the m a x i m u m of the

W = B sin e O/a 2

(16)

in which B is a proportionality constant, 20 is the scattering angle, and A is the wavelength of the incoming neutrons. Absorption and form factor corrections can also be included in this expression. When the ionic magnetic m o m e n t s are distributed in r a n d o m directions as in the paramagnetic state, the magnetic scattering will be incoherent, and will only contribute to the background scattering of the powder pattern. In

F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 compounds ( X = Pt, Rh, Ru, Ir) 15.00

I

I

i:.

B=O

~o

I!

an ordered magnetic substance, however, there is a magnetic contribution to the differential cross section do-/dO, which leads to an additional pattern of peaks in the powder spectrum. For a ferromagnet these coincide with the nuclear peaks, and the magnetic contribution to the neutron scattering appears as an enhancement of the nuclear peaks, which decreases with increasing 0, according to the square of the neutron magnetic form factor. In fig. 4 the neutron diffraction diagram of PrRh2 at 4.2 K is shown as a typical example. The solid curve is a theoretical fit to the data. In the fitting procedure we applied the profile refinement method introduced by Rietveld [16]. By comparison to a diagram at higher temperatures (above To) it became clear that no additional peaks are present. This indicates that the magnetic and chemical unit cells are identical and that the structure is ferromagnetic. From the fit of all the lines in the diagram the value of the magnetic m o m e n t at 4 . 2 K can be obtained.

i

(:311)

10.00

× >,

I (222) o~(4°°~(33~ 0

3.00 2-$heta

6.00 (degrees)

~(42~l 9,00 xiO 1

/

I

Iobs-Icolc

223

'

Fig. 4. Scattered neutron intensity versus scattering angle at 4.2 K and in zero field for PrRh2. The solid curve is a fit as explained in the text. The a g r e e m e n t values for the fit [16] are R,. = 0.046, R . = 0.053.

1.5 Pr Pt 2 ( 2 2 0 )

PP Rh2 ( 1 1

I)

1.0

0.~

F

m

~

0~ 0

i

i

5 ,

co

=l

i

i

J

,

i

~

~

3

,

P

,

,

i

i

I re

r

i

I

OI

10 ,

bl

i

i . . . . . . .

0

i

!

h

5 i

[

i

i

P r Ru2 ( 1 1 1 )

I 1 1)

2

k

41

%,c

.

.

i i. J i. J ~I

5

10

i

JOo d

I0

20

30

40

Fig. 5. Magnetization /.t, from neutron-diffraction data, versus temperature T, for PrPt2, PrRh2, PrRu2, and Prlr2. The solid lines are the results of molecular-field calculations. T h e scatter in the data for PrPt2 is due to the low m o m e n t observed.

224

F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 compounds ( X - Pt, Rh, Ru, Ir)

These values are listed in table I. We also measured the temperature dependence of one of the magnetic reflections. At each temperature the intensity is obtained by the subtraction of a background which is assumed to be linear. The magnetic intensity (proportional to the magnetic moment squared) is obtained by the subtraction of the nuclear contribution. The temperature dependence of the magnetic moment extracted in this way is shown in fig. 5 for the various compounds. For the interpretation of the measurements discussed in this paper it is of importance to know whether the magnetic phase transition is accompanied by a structural transition. To check this, A.C. Moleman, at the University of Amsterdam, performed an X-ray study on the Prlr2 sample at various temperatures. His measurements indicate an increase in the lattice parameter in the liquid helium range of about 5 x 10 3 A but no evidence could be found for a structural phase transition, accompanying the magnetic phase transition. At 4.2 K broadening effects were observed in some reflections which might indicate the presence of strains. In this context we remark that Pourarian et al. [17] have recently found that PrAI2 exhibits a huge magnetostriction. Lattice distortions to a tetragonal or rhombohedral structure have been observed in other rare-earth aluminium compounds by Barbara et al. [18]. TbX2 (X = Fe, Co, Ni) compounds are also well known for the occurrence of structural distortions and magnetostrictive effects, see e.g. ref. [19].

8. Discussion A m o n g the present results neutron-diffraction measurements may offer the best possibility to compare theory and experiment. The solid lines in fig. 5 are calculations based on eq. (5), adopting the values of x, W and A, listed in table I. For PrRh2 two curves are presented, calculated on the basis of the two possible crystal-field schemes. The level scheme with the doublet F3 ground state gives rise to a first-order phase transition with a transition temperature of 8.7 K, if we adopt a value of the molecular-field con-

stant A, based on the estimate A = , ) ( o l ( T c ) (eq. 4), where X0 is the susceptibility at the actual transition temperature. This estimate of A is only valid in case of a second-order phase transition. In order to obtain a more precise estimate in the presence of a first-order transition one should start from a value A = Xo~(T0) with T o < Tc and determine T0 and hence A in such a way that the actual first-order transition takes place at T = To. We did not find it necessary to evaluate this more precise estimate of A, since the neglect of quadrupolar interactions might introduce a much larger uncertainty as discussed below. This firstorder behaviour is more generally found for F3 ground state systems and the nature of the phase transition (first- or second-order) depends on the ratio of magnetic and crystal-field interactions. Comparison of the calculated saturation moments with the extrapolations of the measurements to T = 0 shows the latter to be generally smaller by about 10-20%. This will also hold for PrRh2 if we assume the scheme x = 0.75 and W = - 0 . 3 3 meV to be appropriate. Of course we do not expect the molecular-field approximation to reproduce the experimental results precisely, but in view of the long-range character of the interactions this can be considered as a fairly good approach. Part of these discrepancies between theory and experiment may be explained by J-admixture effects, which may reduce the magnetic moment by 5-10% [20]. A n o t h e r contribution to the magnetization as measured by neutron diffraction arises from the conduction-electron polarization. However, we believe that the most likely explanation for the systematic discrepancy is the presence of quadrupolar, i.e. biquadratic exchange interactions. By inclusion of a quadrupolar term in the molecular-field Hamiltonian, Loidl et al. [21] could explain the measured magnetization of PrMg2 quite well. As compared to the calculation without these quadrupolar effects, the magnetization was reduced by about 20%, which is in good agreement with the values we observe. The influence of quadrupolar interactions will also be considered in the discussion in a forthcoming paper on the high-temperature specific-heat and resistivity measurements. It can be seen in fig. 5c (Prlr2), that the cal-

F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 compounds ( X = Pt, Rh, Ru, It)

culated magnetic moment at T = 0 reaches the maximum value 3.2 I.Lb for Pr 3+ (J = 4). This is a direct consequence of the fact that the calculation is based on an x-value x = 0.86, at which the ground state is a linear combination of the F3, FI and /'4 states with coalescing energy values. As discussed in ref. [1] there is evidence that the x-value for PrIrz falls somewhere in the vicinity of the crossing point. We anticipate the saturation moment to be strongly dependent on the distance from the crossing point. We therefore calculated saturation moments for several combinations of x and W values, always determining A in such a way that Tc is located at 11.2 K (in case of first-order transitions small deviations may occur, but these are expected to be small for the range of x and W values considered, and can therefore be neglected). The results are plotted in fig. 6, in which lines of equal magnetization, expressed in values of (J~), are plotted in an x, W diagram. The measured m o m e n t at T = 0 for PrIr2, obtained by extrapolation, amounts to 2.6 + 0.3/~B or (Jz) = 3.3+0.4. This would only be consistent with an x-value at a fair distance from the crossing-point. In the above, however, we have seen that on average the measured m o m e n t is about 15% below the calculated value. If we take this to hold for the PrIr2 results as well, we arrive at a moment of 3.0 ~B, which, assuming W = - 0 . 5 7 m e V , indicates x-~ 0.83 or x = 0.88 (see fig. 6). The uncertainties are of course rather large, but nevertheless it is clear that the results obtained are in good agreement with the conjecture that PrIr2 is situated in the vicinity of the level crossing point. A similar situation has been discussed by Kim et al. [22] for some holmium compounds in terms of the "cubic model". Next we will compare the values of p,/3.2/~B, m/3.21~B and (Jz)/J, listed in table I, which should be essentially the same. The ratios m/3.2l~B, however, are substantially smaller than the ratios (Jz)/J deduced from the heat-capacity measurements. This points to the presence of a domain structure, as well as a strong anisotropy, causing cancellation effects in the measured m o m e n t in the polycrystalline material. The values of (Jz)/J inferred from hyperfine specific

I O8

-08

>~ E -

O4

-0.2

I

I

225 I

I

3.5,

!ii I

0.82

I

0.84

I 086 ×

,

I 088

,

I 090

Fig. 6. Iso-magnetization (at T = 0) lines in an x - W diagram. The transition t e m p e r a t u r e has been taken to be 11.2 K, as explained in the text. The numbers of the various curves indicate the angular m o m e n t u m expectation value (Jz).

heat measurements are in good agreement with the values of /M3.2~tB obtained by neutron diffraction, with the exception of PrPt2. For this c o m p o u n d the value of (Jz)/J is about 2.5 times as large as ~t/3.2/~B. It is conceivable that there is a relatively large contribution of the conductionelectron polarization to the hyperfine constant in this case, but this will definitively not be sufficient to explain the discrepancy. It has been found by Taylor et al. [23] that compounds of the Laves phase in the P t - G d alloy system form as a single phase over a range of concentrations from 23 at% to 33 at% Gd. These authors could explain their X-ray diffraction measurements in terms of a model in which Pt atoms substitute on G d sites progressively and they suggested the occurrence of vacancies in the stoichiometric composition GdPt2. This conjecture has been studied by D o r m a n n et al. [24] in a series of N M R measurements on the GdPtx (2 --
226

F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 comlxmnds ( X - Pt, Rh, Ru, It')

hyperfine specific-heat measurements on PrPt2 (cf. section 6). The presence of vacancies at the Pr sites in PrPt2 may lead to an erroneous interpretation of the neutron-diffraction data, in this respect, that ignoring these effects, the moment inferred from the data is lower than the moment actually present. Hence the presence of these vacancies may give an explanation for the different low temperature values of (Jz)/J from hyperfine specific heat and /X/3.2/~B from neutron diffraction in PrPt2. On the other hand the value of /x/3.2/XB of PrPt2 observed in neutron diffraction agrees well with the calculated value based on crystal-field parameters determined by inelastic neutron scattering [1].

9. Conclusion We have shown that consistent information on the magnetic properties of the PrX2 compounds has been obtained with quite different techniques. The high-field magnetization measured at 77 K is rather insensitive to the details of the crystal field, but the results obtained are in agreement with calculations based on known x, W and A values. Further information may be obtained once single crystal samples become available. Except for PrPt2, the results from hyperfine specific heat and neutron diffraction are mutually consistent. A molecular-field calculation of the spontaneous magnetization, assuming only dipolar interactions and based on the x and W values as inferred from inelastic neutron scattering and A, from the A-To relation (eq. (4)), yields results which are 10-20% larger than obtained in the experiment. We consider this to be an indication for the possible presence of quadrupolar interactions.

Acknowledgements We thank G.A. Vermeulen for his assistance in the hyperfine specific heat measurements and Prof. H.W. Capel for several stimulating dis-

cussions on the interpretation of the measurements. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research ) .

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F.J.A.M. Greidanus et al. / Magnetic properties of PrX2 compounds ( X = Pt, Rh, Ru, lr) [21] A. Loidl, K. Knorr, M. Milliner and K.H.J. Buschow, J. Appl. Phys. 53 (1981) 1433. [22] D. Kim, P.M. Levy and L.F. Uffer, Phys. Rev. BI2 (1975) 989. [23] R.H. Taylor, I.R. Harris and W.E. Gardner, J. Phys. F6 (1976) 1125. [24] E. Dormann, M. Huck and K.H.J. Buschow, Z. Phys. B27 (1977) 141.

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[25] Th. Frauenheim, W. Matz and G. Feller, Solid-State Commun. 29 (1979) 805. [26] K.H.J. Buschow, R.C. Sherwood, F.S.L. Hsu and K. Knorr, J. Appl. Phys. 49 (1978) 1510. [27] A. Eyers, A. mike, A. Leson, D. Kohake and H.-G. Purwins, J. Phys. C15 (1982) 2459.