Magnetic and crystal field properties of DyFe2Si2 from 161Dy Mössbauer study

Journal of Magnetism and Magnetic Materials 119 (1993) 301-308 North-Holland

Magnetic and crystal field properties of DyFe2Si 2 from 161Dy M6ssbauer study P. Vulliet

a,1, K. Tomala a,2, B. M a l a m a n b, G. Venturini b and J.P. Sanchez a

a Centre d'Etudes Nuclgaires de Grenoble, Dgpartement de Recherche Fondamentale sur la Mati~re Condens~e, SPSMS/LIH, 85 X, 38041 Grenoble Cedex, France b Laboratoire de Chimie du Solide Mingral associ~ au CNRS UA158, Universitg de Nancy I. Bo~te Postale 239, 54506 Vandoeuvre les Nancy Cedex, France Received 15 July 1992

A detailed 161DyM6ssbauer study has been performed on OyFe2Si 2. The hyperfine interaction parameters at low temperatures reveal that the crystal field ground state was a fairly pure 1-1-15/2) Kramers doublet. This allowed one to conclude that the modulated spin structure deduced from the neutron data should be squared at saturation, and that the Dya+-ions carry a magnetic moment of 10it B aligned parallel to the tetragonal c-axis. The energies of the low lying excited Kramers doublets, as well as the second- (B °) and fourth-order (B °) crystal field parameters, have been estimated from the analysis of the temperature dependence of the hyperfine field and quadrupole interaction found in the paramagnetic state.

1. Introduction

The ternary intermetallic compounds RT2Si 2 (R = rare earth, M = transition metal) have been extensively investigated, owing to their various and often surprising physical properties [1]. Most of them crystallise in the ThCr2Si2-type body centered tetragonal structure (space group 14/ mmm), and usually order magnetically at low temperatures, exhibiting different types of magnetic structures due to the RKKY type long range interaction. The present study reports on detailed hyperfine interaction measurements on the 161Dy nuclei in DyFe2Si 2. The aim of this work was to shed some light on the electronic structure of the

1 Also at Universit6 Joseph Fourier, Grenoble, France. 2 On leave from Institute of Physics, Jagellonian University, Cracow, Poland. Correspondence to: Dr. P. Vulliet, Centre d'Etudes Nuclraires de Grenoble, Drpartement de Recherche Fondamentale sur la Mati~re Condensre, SPSMS/LIH, 85 X, 38041 Grenoble Cedex, France.

Dy 3+ ions in this intermetallic compound. At low temperatures, both the hyperfine field (Hhf) and the electric field gradient (EFG) induced by the 4f shell electrons at the 161Dy nuclei are fashioned by the crystalline electric field (CEF) ground state wave functions. Moreover the temperature dependence of these parameters can provide information on the position and nature of the excited CEF levels. Finally, the analysis of the hyperfine interaction data is expected to give access to the easy direction of magnetisation of the Dy sublattice, and to allow to choose between different possible magnetic structure models. DyFe2Si 2 was already studied by bulk acand dc-magnetisation measurements [2,3]. They showed that DyFe2Si 2 orders antiferromagnetically at 3.8 K, with moments only localised at the Dy 3+ ions. Mrssbauer results on both 57Fe and 161Dy nuclei were also reported [2,4]. The broadened non-Lorentzian line shape observed for the 57Fe resonance at 4.2 K was attributed to either transferred polarisation produced by slowly relaxing Dy 3+ ions or to short range magnetic order [2]. On the other hand, the 4.2 K 161DyMrssbauer

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

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P. Vulliet et aL / Propertiesof DyFe2Si2 from 161DyM6ssbauerstudy

spectrum was well analysed by a unique set of hyperfine parameters, with nhf close to the Dy3 + free ion value [4]. The neutron diffraction study on DyFe2Si 2 was realised only recently [5]. It confirmed that DyFe2Si 2 orders A F at 3.8 K. The magnetic structure was shown to be of sine-modulated-type, with a propagation vector k = (0.335, 0, 0.136) and Dy magnetic moments parallel to the tetragonal c-axis. The m o m e n t expected for a square modulation at 1.3 K, /.~ = 7.52(5)/zB, was found to be significantly below the free ion value of 10/~ B [5]. The p a p e r has been organised in the following way. In section 2, we give a brief account of the experimental details and describe the experimental results. In section 3, we discuss our results and present our conclusions concerning the C E F level scheme, the magnetic structure a n d the dynamics of the Dy 3+ m o m e n t s in the paramagnetic state.

I . . . .

t

. . . .

I . . . .

I . . . .

I . . . .

I . . . .

I

o

1.00 0.99 0.98 0.97 z

1.000

~

0.995 0.990 0.985 0.980

o

0.975 0.970 I . . . .

2. Experimental procedure and results 2.1. Experimental procedure

A polycrystalline sample of DyFe2Si 2 was prepared from commercially available high purity elements: Fe (powder, 99.9%), Dy (ingot, 99.9%) and silicon (powder, 99.99%). Pellets of starting composition DyFe2Si 2 were compacted using a steel die, were annealed several times (with grinding and compacting each time) at 1250 K in sealed silica tubes under argon (0.2 atm) and were finally quenched in water. Purity of the final samples was determined by an X-ray diffraction technique, using a Guinier camera (CuK~). The lattice p a r a m e t e r s we measured were found to be in good agreement with those reported previously

I . . . .

I , , , , I , , , , [ i , , , I

. . . .

-300 -200 -100 0 100 200 VELOCITY(mm/s)

I

300

Fig. 1. 161DyM6ssbauer spectra of DyFe2Si2 at 1.65 K (a) and 4.3 K (b), i.e. at temperatures close to the Ndel temperature of TN = 3.8 K. Both spectra were fitted with a static hyperfine Hamiltonian.

metallic Dy (Hhf = 5689(3) kOe, e2qQ = 124.9(2) m m / s [6]). The effective-field magnetic spectra were directly least-squares computer fitted to the hyperfine p a r a m e t e r s by constraining the relative absorption energies and intensities of the Lorentzian lines to the theoretical values. Some spectra were analysed using spin relaxation models described in detail in section 3.

[5].

2.2. Experimental data

161Dy M6ssbauer spectroscopy m e a s u r e m e n t s ({_,53, 25.7 keV transition) were performed using a sinusoidal drive motion of a neutron irradiated 161Gd0.6162Dy0.sF3 source kept at room temperature. The absorber was maintained at different temperatures between 1.6 to 300 K. An absorber thickness of 30 mg D y / c m 2 was used. The g a m m a rays were detected with an intrinsic G e detector. The velocity scale was calibrated using

Figs. 1-3 show spectra taken at temperatures between 1.65 and 87 K. In the magnetically ordered state (i.e. at T < 3.8 K), the data are well represented by a unique set of hyperfine parameters, with the magnetic field parallel to the main axis of the electric field gradient (fig. la, table 1). The isomer shift (2.0(1) m m / s relative to the source) is slightly smaller than the one observed

P. Vulliet et aL / Properties of DyFe2Si 2 from 161DyM6ssbauer study I''''

' ' ' ' 1

. . . .

f

. . . .

I

. . . .

I

. . . .

I

I . . . .

a

1.000

m

o

I . . . .

I

303 . . . .

I . . . .

[,,,,I

....

I . . . .

I . . . .

I

1.000 b

0.995

0.995

0.990

0.990

0.985

0.985

0.980

0.980 Z 0 cO cO

0.975

b

-

1.000

-

cO Z < rr F-

0,995 0,990

0.975

0.970 1.00

0.99 o

0.985

0.98

0.980 C

o

1.005

0.97

1.000

I,,,,I

. . . .

I,,,,I,,,,I

-300 -200 -100 0 100 200 VELOCITY(mm/s)

0.995 0.990 0.985

Fig. 3. 161Dy M6ssbauer spectra for DyFe2Si 2 taken at 77 and 87 K. The data were least-squares analyzed using a two-level relaxation model. The fitted hyperfine parameters are given in table 1.

o

o

0.980 I ....

I , , , , I , , , , I j , , , I

....

300

I , , , , I

-300 -200 -100 0 100 200 VELOCITY(mm/s)

300

Fig. 2. 1 6 1 D y M6ssbauer spectra fo DyFe2Si 2 taken at different temperatures, 20.8 K (a), 44.7 K (b) and 58.5 K (c), in the paramagnetic region. The spectra were fitted with the relaxation model described in section 3.2 and with the parameters given in table 2.

in metallic Dy (2.9(1) m m / s ) . Any hyperfine parameter distribution is ruled out from the observed very narrow resonance line of W = 4.5(2) m/n//s.

Above T N, the temperature dependence of the spectra clearly evidences two types of behaviour. First, one should notice that neither the overall hyperfine splitting nor the spectral shape change abruptly when crossing the ordering temperature (fig. 1). From T N up to 60 K (fig. 2) one observes a well developed magnetic hyperfine structure, with some line broadening occurring above 20 K

Table 1 Hyperfine interaction parameters in DyFe2Si 2 at different temperatures obtained from a "static" hyperfine Hamiltonian and a two-level relaxation analysis T (K)

Hhf (kOe)

e2qQ

W

(mm/s) ~)

(mm/s)

1.65 2.03 3.50 4.30 20.8 44.7 58.5 77 87

5666(10) 5695(10) 5695(10) 5717(10) 5560(10) 5390(25) 5314(25) 5175(50) b) 5 1 4 4 ( 1 0 0b) )

109.6(8) 110.2(7) 110.5(6) 110.8(5) 101.6(6) 88.6(8) 82.9(1.8) 72.8(3.0) b) 67.8(6.2) b)

4.4(2) 4.5(2) 4.5(2) 4.4(2) 4.7(2) 5.5(2) 8.2(4) 4.5 e) 4.5 c)

For E~ = 25.7 keV in 161Dy: 1 m m / s = 8.5576× 10 -8 eV = 20.69 MHz. b) From two-level relaxation model least-squares fit. o Fixed value. a)

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P. Vullietet al. / Propertiesof DyFe2Si2 from 161DyM6ssbauerstudy

(table 1, fig. 2). The data can be well fit by a "static" hyperfine Hamiltonian up to 45 K. The severe line broadening observed at 58.5 K (W= 8.2(4) m m / s ) indicates that the spectral shape starts to be influenced by relaxation effects. The continuous decrease of both the magnetic hyperfine field and the quadrupolar interaction when increasing the temperature shows clearly that excited crystal field levels become populated. Thus, a straightforward analysis of the data should be made in the framework of a relaxation model which takes into account transitions involving excited crystal field levels [7]. This model will be discussed in detail in section 3. Above 60 K, the shape of the resonance spectra is strongly modified (fig. 3). The abrupt collapse of the two inner lines into a single line shows distinctly the increasing Dy magnetic moment relaxation rate which takes place between "levels" which on the average have zero-magnetisation. The spectra taken at 77 and 87 K were both analysed by a phenomenological two-level relaxation model [8].

3. Discussion 3.1. Hyperfine interaction parameters and electronic structure

The magnetic hyperfine field acting on rare earth nuclei is commonly described as a sum of several contributions [9]: Hhf = H4f + H~p + Hop + Hn,

(1)

where H4f represents the contribution of the localised 4f electrons (orbital plus spin-dipolar fields); H~p is the core polarisation field, whereas the two other terms stand for the conduction electron contributions due to the own polarisation (Hop) and to the neighbouring magnetic atoms (Hn). The effective quadrupole coupling constant e2qQ consists of an electronic (40 and a lattice (lat) contribution. The possible contribution of the conduction electrons is included in the lattice term.

The lattice EFG contribution (of axial symmetry in our case) can be treated as a perturbation of the 4f component. Its projection along this component gives [10]: z 1 2 -qz' l a t t~" lJ / " z t c°s20 - 1), e2qQ = e2q4fQ + -~e

(2)

where 0 is the polar angle defining the orientation of the lattice EFG principal axis z' with respect to the direction of the hyperfine field (or magnetic moment, taken as the z-axis). In our case, the z'-axis of the lattice EFG is along the tetragonal c-axis. The 4f contribution to e2qQ and to Hhf give useful information concerning the electronic structure of the Dy 3+ ions. Indeed, the (2J + 1)fold degeneracy of the ground state multiplet (6H15/2) is lifted by the crystalline electric field (CEF) and the molecular field (MF) acting of the 4f shell. If I F~) is the wave function of the ith electronic level, the 4f contributions to the zcomponent of the hyl~erfine field and EFG are proportional to (F/I Jz I F~) and (F/13fz2 - J ( J + 1)[F~), respectively. Thus, both hyperfine interaction parameters are sensitive to the wavefunctions of the ground and excited levels [11]. For an S-state ion (i.e. Gd3+), the 4f contribution to Hhf vanishes, thus t Hhf I of 303.4(6) kOe measured in isostructural GdFe2Si 2 [12] provides an estimate of the Hhf contribution in DyFe2Si a, when scaling the Gd data with the spin S = (gy 1)J factor 5/7. With this procedure, I Hcp + Hop + Hnl is evaluated to amount about 220 kOe in DyFe2Si 2. The small difference between the hyperfine coupling constant A ( Z ) for Gd and Dy was not taken into account [13], because it will at most introduce a change of roughly 10 kOe on the scaled value. On the other hand, the values of Hhf in the paramagnetic state contain an additional error connected with the averaging to zero of the transferred field contribution (H,), but even in the ordered state, H, is expected to be small when the transition metal (Fe) does not carry a magnetic moment. This assumption is fully justified, since only a minute change of Hhf is observed when crossing the N6el temperature. When assuming that Hhf measured in GdFeESi 2 should be negative as in the isostructural GdMn 2-

305

P. Vulliet et al. / Properties of DyFe2Si2 from t61DyMgssbauer study

Ge E compound [14], it follows that H4f in DyFe2Si 2 at 4.3 K amounts to 5940(20) kOe. The 4f contribution to eZqQ also vanishes for a Gd 3+ ion. This allows the lattice contribution to be calculated in the isostructural DyFe2Si 2 from the quadrupolar data of GdFeESi z [12]. From eaqQ (155Gd)=-4.08(2) ram/s, measured in GdFeaSi z, and from the known values of the ground state quadrupole moments: Q(155Gd)= 1.30(2) b [15] and Q(161Dy) = 2.35(16) b [16], the lattice contribution to the quadrupolar interaction at the 161Dynuclei was estimated to amount to -24.7(2.3) mm/s. Thus, if one assumes, as shown by the neutron diffraction data, that the Dy moments are parallel to the tetragonal c-axis, the angle 0 in eq. (2) is equal to zero, and the electronic 4f contribution amounts to 135.5(2.8) m m / s at 4.3 K. It turns out that both H4f and e2q4fQ at 4.3 K are equal within the experimental errors to the free ion Dy 3+ estimates of, respectively, 5930(30) kOe and 135(1) m m / s [7]. This indicates that at 4.3 K, i.e. in the paramagnetic state, the ground state Kramers doublet should be of almost pure 1+ 15/2) character. As it will be shown below, this is a consequence of the rather large and negative axial term (B °) fo the crystal field Hamiltonian. Below TN ----3.8 K, the ground state Kramers doublet is exchange splitted by the molecular field, giving the pure [ + 15/2) ground state, thus the Dy 3+ ions carry a magnetic moment of 10jz a at saturation in the ordered state.

3.2. Excited CEF levels of Dy 3+ The monotonic decrease with temperature of the magnetic hyperfine field and quadrupole coupling constant observed in the paramagnetic state (table 1) is attributed to the thermal occupation of the excited CEF levels of the Dy 3+ ions. Therefore, the analysis of the thermal dependence of the hyperfine interaction parameters may be used to gain information on the nature and the excitation energies of these levels. In order to take into account the change of the hyperfine interaction strength, together with the line broadening, observed above 20 K, we used a relaxation model which includes the possible

transitions to excited CEF states [7]. In our analysis we constrained the hyperfine interaction parameters corresponding to each level to values which will be discussed below. The Hamiltonian which describes the electronic behaviour of the Dy 3+ ions in the paramagnetic state of DyFe2Si 2 has the following form: X=B°6

° + B ° 6 ° +B,O4

, ^4 + B 600 6^0 +B606.

(3)

The crystal electric field splits the (2J + 1) degenerate ground multiplet (6H15/2) into a set of 8 Kramers doublets, with wave functions being in general far from pure I + m ) states. However, when B ° is large and negative (see section 3.3), the ground and the low lying excited states may be considered as having almost pure I ___m) character [7,17], with the ground state being I + 15/2) (see section 3.1). Therefore, in the analysis of the 20.8, 44.7 and 58.5 K spectra we assumed the occurrence of the sequence I + 15/2), I+ 13/2) and I+ 11/2) for the CEF states, but transitions induced by spin-spin relaxation can only occur between states with Am = + 1. Thus, our physical system can be divided into two thermodynamic systems, characterized by I + m ) and [ - m ) states, respectively [7]. Each I + m) or I - m) level contributes its corresponding hyperfine field, adjusted from the known value for the I+ 15/2) state and scaled to the quantum number m. The lattice contribution to e2qQ was included as a constant value for each level. The standard formula valid in the so-called "adiabatic" case [8] was used to describe the shape of the resonance spectra, with the relaxation matrix given by [18]:

w(m--+m')--O01(m'lS+ I m) 14 n ( m ' ) , where ~(rn') describes the Boltzmann occupation of the Ira') level and /20 the strength of the dipole-dipole interaction (relaxation rate). The resonance linewidth was constrained to the value of 4.5 m m / s observed at low temperature. Then the spectral shape is only influenced by the relaxation rate/20 and the energies A1 and A2 of the excited CEF levels. The energy A1 of the first excited state ( l + 13/2)) can be estimated straightforwardly from

P. Vulliet et al. / Properties of DyFe2Si2 from

306

Table 2 Excitation energies and average values of hyperfine parameters obtained from least-squares fit of the MSssbauer data using the relaxation model described in section 3.2. Average values of Hhf and e2qQ were calculated from the fitted excitation energies A 1 and A 2 and adjusted hyperfine interaction constants for ground and excited CEF levels (see text)

(e2qa>

T (K)

A1 (K)

A2

~"~0

(K)

(MHz)

(Hhf> (kOe)

(mm/s)

20.8 44.7 58.5

27.8(3) 27.8 a) 27.8 a)

111(4) 103(5)

62(18) 56(7) 25(4)

5560(20) 5386(20) 5307(30)

99.6(1.2) 87.8(1.6) 82.5(3.0)

a) Fixed in the fitting procedure.

the analysis of the 20.8 K spectrum. At this temperature, higher excited levels are almost unpopulated. Information about the position Az of the next ( 1 + 1 1 / 2 ) ) excited level was obtained from the analysis of the 44.7 and 58.5 K spectra (fig. 2, table 2). As expected for spin-spin interactions, O 0 is almost t e m p e r a t u r e independent in this narrow t e m p e r a t u r e range. Notice that the average values of Hnf and e2qQ calculated using the hyperfine p a r a m e t e r s corresponding to each level and the excitation energies A 1 and Az are in good agreement with the results obtained from the analysis of the data with a static hyperfine Hamiltonian (table 1). As shown in fig. 3, the spectral shape changes abruptly above 58.5 K. The collapse of the two inner lines into a single line indicates that the relaxation process is different from the one discussed above. This behaviour was attributed to the occurrence of spin-lattice relaxation via Orbach and R a m a n processes [19] which induce transitions between both subsystems of C E F levels. The spectra taken at 77 and 87 K can be well reproduced by a phenomenological two-level model, but one should rather understand them as resulting from slower relaxation between both subsystems and fast relaxation inside each of them. The fitted relaxation rates were found to amount to 280(10) and 455(20) M H z at 77 and 87 K.

161ByMSssbauer study

field p a r a m e t e r s B m (see eq. (3)). One should in principle be able to evaluate the C E F parameters, knowing the excitation energies At and A 2. Some initial guessed value for the B ° C E F parameter can be obtained from the lattice contribution to the EFG, i.e., from 155Gd M6ssbauer measurements in GdFezSi z [12]. B2° is related to the Gd quadrupolar interaction AEQ = eZqQ by the relation [2,20]: B2° = aj4f(1 - 0-2)A °,

(4)

with A ° = - h E o / 4 e ( 1 -- yo~)Q. With the selected values of the shielding factors (1 - %0) = 60, (1 - 0-2) = 0.4, and tabulated 4f [21] and a j Stevens factors [20], one obtains B ° = - 2 . 4 K. One should however keep in mind that relation (4) is only justified for ionic compounds; its usefulness for intermetallic systems has been recently questioned [22]. Nevertheless, especially for ThCr2Si 2 type structure compounds, eq. (4) seems to predict correctly the sign and magnitude of B2° [17]. As a matter of fact, the main uncertainty on the actual value of B ° is essentially related to the poor knowledge of the shielding factor y~ and o-2. The ratio (1 - y=)/(1 - 0-2) may range from 150 to 300. To perform a quantitative analysis of the data, i.e. reproduce the excitation energies A 1 and A 2, we truncated the C E F Hamiltonian (eq. (3)) to fourth order terms. It was shown that, when B ° is large and negative, the eigenvalues and eigenfunctions are insensitive to the choice of the B44 value [7]. A g r e e m e n t between calculated and experimental results was achieved as follows. B ° was constrained to values ranging from - 2 . 4 to - 3 . 0 K, and B4° was increased step by step. The best agreement with the experimental excitation energies (table 2) was obtained with B ° = - 3.0 K and B4° = 0.0045 K. The truncated C E F Hamiltonian with this set of C E F parameters gives A 1 = 27.7 K and A z = 100.6 K.

3.4. Magnetic structure of DyFe2Si 2

3.3. Crystal-field parameters in DyFe2Si 2 The deduced sequence of the low lying D y 3 + Kramers doublets is directly related to the crystal

As shown in fig. 1 and table 1, the M6ssbauer spectra recorded in the magnetically ordered state can be well analyzed with a unique set of hyper-

P. Vulliet et al. / Properties of DyFe2Si 2 from 161DyMb'ssbauer study fine parameters. This does not necessarily imply, as shown below, that all Dy 3+ ions carry the same magnetic moment. Indeed, the C E F ground state is a I + 1 5 / 2 ) Kramers doublet which is exchange splitted in the magnetically ordered state. This is the only doublet populated at low t e m p e r a t u r e then, neither spin-lattice nor s p i n spin interactions can induce electronic relaxation within the ground state exchange splitte, d doublet (see section 3.2). Thus, the 4f contribution to the magnetic hyperfine field should correspond to its saturation value, and be independent of the splitting A of the Kramers doublet. Notice that for an amplitude modulated spin structure, A is distributed. The situation described above is the common one when B2° is negative and large enough to be the dominating part of the C E F Hamiltonian, i.e., when the C E F ground state is predominantly of I + 1 5 / 2 ) character. This implies that: i) the magnetic moments in the ordered state point along the tetragonal c-axis and reach at saturation their maximum value of 10~ B, ii) the sine-modulated incommensurate magnetic structure deduced from neutron diffraction data should evolve towards a square-wave structure with constant moments. More accurate neutron diffraction measurements, aimed to search for higher harmonics (3k, . . . ) proving squared modulation, are highly desirable. On the other hand, the magnetic m o m e n t of 7.52#B found at 1.3 K by neutron diffraction is too low c o m p a r e d to the value of 10/x B expected from the M/Sssbauer results.

4. Summary and conclusions The present work reports on the results of a study of DyFe2Si 2. The analysis of the saturation hyperfine interaction p a r a m e ters Hhf and e2qQ allowed to conclude that the electronic ground state is fairly pure I_.Y_1 5 / 2 ) Kramers doublet, which is exchange splitted below T N = 3.8 K. The occurrence in the paramagnetic region (from T N up to 60 K) of "static" magnetic hyperfine spectra with t e m p e r a t u r e dependent nhf and e2qQ was explained straightfor-

161DyM/Sssbauer

307

wardly in the frame of a relaxation model which takes into account electronic transitions involving excited crystal field levels. The excitation energies of the low-lying excited I + 1 3 / 2 ) and 1+ 1 1 / 2 ) Kramers doublets, as well as the second (B °) and fourth-order (B °) crystal field parameters, were estimated from the t e m p e r a t u r e dependence of Hhf and e2qQ. It was furthermore concluded that the occurrence of a I + 1 5 / 2 ) ground state implies that the amplitude modulated spin structure should become square-wave modulated, with constant m o m e n t value of 10/z B at saturation. The sign (negative) of B °, which is the leading C E F parameter, determines the tetragonal c-axis as the easy direction of magnetisation.

Acknowledgement One of the authors (K.T.) acknowledges the Centre d'Etudes Nucl~aires de Grenoble for its hospitality and financial support.

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[12] G. Czjzek, V. Oestreieh, H. Schmidt, K. L]tka and K. Tomala, J. Magn. Magn. Mater. 79 (1989) 42. [13] I.A. Campbell, J. Phys. C 2 (1969) 1338. [14] J.P. Sanchez, K. Tomala and A. Szytuh, Solid State Commun. 78 (1991) 419. [15] Y. Tanaka, D.B. Laubacher, R.M. Steffen, E.B. Shera, H.D. Wohlfart and M.V. Hoehn, Phys. Lett. B 108 (1982) 8. [16] J.G. Stevens, in: Handbook of Spectroscopy, vol. 3, ed. J.W. Robinson (Chemical Rubber, Boca Raton, FL, 1981) p. 403.

[17] K. Tomala, A. Blaise, R. Kmie6 and J.P. Sanchez, J. Magn. Magn. Mater. 117 (1992) 275. [18] M. Blume, Phys. Rev. Lett. 18 (1967) 305. [19] R. Orbach, Proc. Roy. Soc. A 264 (1961) 458. [20] M.T. Hutchings, in: Solid State Physics, vol. 16, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964) p. 227. [21] A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mater. 12 (1979) 1. [22] R. Coehoorn, K.H.J. Buschow, M.W. Dirken and R.C. Thiel, Phys. Rev. B 42 (1990) 4645.