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Journal of Crystal Growth 297 (2006) 419–425

www.elsevier.com/locate/jcrysgro

Structural and magnetic properties of CdxInyCrzSe4

D. Skrzypeka,�, E. Malickab, A. Waskowskac, S. Widucha, A. Cichona, T. Mydlarzd

aA. Che!kowski Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, PolandbInstitute of Chemistry, University of Silesia, Bankowa 14, 40-007 Katowice, Poland

cInstitute of Low Temperature and Structure Research, Polish Academy of Sciences, 50 422 Wroclaw, PolanddInternational Laboratory of High Magnetic Fields and Low Temperatures, 53 529 Wroclaw, Poland

Received 2 June 2006; received in revised form 4 October 2006; accepted 10 October 2006

Communicated by D.W. Shaw

Available online 28 November 2006

Abstract

Mixed selenide spinels of (Cd)[Cr1.81In0.19]Se4 and (Cd0.745In0.255)[Cr1.94In0.06]Se4 have been studied using X-ray diffraction, magnetic

measurements and ESR spectroscopy. In3+ ions accommodate both tetrahedral and octahedral sites in the spinel structure. The magnetic

short-range order appears in the critical region from Tc to about 200K for the selenide spinels. The linear temperature dependence of the

resonance linewidth at T4250K is interpreted by an occurrence of one-phonon process in a spin-lattice relaxation.

r 2006 Elsevier B.V. All rights reserved.

PACS: 76.30.�v; 61.10.�i

Keywords: A1. Crystal structure; A1. X-ray diffraction; A2. Single-crystal growth; B2. Magnetic materials; B2. Semiconducting ternary compounds

1. Introduction

Ternary selenide spinels, exhibiting interesting structur-al, magnetic and electrical transport properties have beenthe subject of numerous studies. It was shown thatreplacement of the di- or tri-valent cation by a third metalresulted in new chemical compounds with essentiallychanged physical properties [1–13]. However, experimentalresults reported in the literature so far refer mostly topolycrystalline forms of the quaternary selenides. We haveprepared the Cd–In–Cr–Se4 single crystals with various Inconcentrations for the purpose of studying the influence ofthe In3+ admixtures on the cation distribution andmagnetic ordering in this spinel system.

The parent CdCr2Se4 crystallises in the normal spinelstructure (Fd3m) [14]. Cadmium and chromium in thecubic close packing of selenium atoms are tetrahedrally(A-type sites) and octahedrally (B-type sites) coordinated,respectively. The compound is a ferromagnetic p-typesemiconductor [3] with lattice parameter around 10.7 A [1].

e front matter r 2006 Elsevier B.V. All rights reserved.

rysgro.2006.10.104

ing author. Tel./fax: +48322588431.

ess: dskrzyp@us.edu.pl (D. Skrzypek).

The Curie temperature reported in literature was in therange of 128K [15]–142K [3].The solid solutions CdxInyCrzSe4 were synthesised in

polycrystalline form by Shabunina et al. [16]. X-raystructure determination showed that indium, In3+, couldaccommodate both tetrahedral and octahedral sites in thespinel structure [16].In the present paper, we study the structural and

magnetic properties of single crystals, CdCr2Se4, substi-tuted with In ions in the A- and B-sites using X-raydiffraction, magnetic measurements and ESR spectro-scopy.

2. Experimental procedure

2.1. Preparation of the substrates and crystal growth

The single crystals were grown by chemical vapourtransport method in closed quartz ampoules with anhy-drous chromium chloride (purity 98%) as a transportingagent and with the selenides CdSe and In2Se3 as the solidphases. The starting materials, binary selenides, weresynthesised from elemental cadmium, indium and selenium

ARTICLE IN PRESSD. Skrzypek et al. / Journal of Crystal Growth 297 (2006) 419–425420

(purity 99.999%). The stoichiometric mixtures of theelements were pulverised in an agate mortar and sealed inevacuated quartz ampoules. After heating at the 1075K for7 days, the selenides were ground in an agate mortar andheated once more for 7 days at the same temperature asbefore. X-ray powder analysis showed that all productscontained only the synthesised phase. The mixture ofselenides with transporting agent CrCl3 was sealed inquartz tubes (length E200mm, inner diameter d ¼ 20mm)evacuated to E10�3 Pa. These ampoules were heated in ahorizontal zone furnace to about 1123K at the solutionzone, maintaining the temperature gradient of 10K alongthe ampoule. After 10 days, the furnace was cooled toroom temperature.

Growing of quaternary spinel-type chromium selenidesingle crystals requires special conditions of the chemicaltransport reactions, which are very important in thecrystallisation of the spinels. The equilibrium constants ofchemical transport reactions (Ka) were calculated as afunction of the temperature to determine, for example, thetransport ability of CrCl3. The results of transport dependon logKa value. When the logKa value is close to zero, onemay expect great transport ability of chemical reactions,even for small temperature difference. The transportingagent CrCl3 dissociated to CrCl3, CrCl4 and Cl2 above773K [17]. The transporting reactions, with CrCl3, CrCl4and Cl2 as the transporting agents, were used to calculatethe equilibrium coefficients in heterogeneous system (a gasphase and some solid phases). For the CdSe–In2Se3–CrCl3system, values of logKa for transport reactions are similarto the equilibrium state values (logKaE0). Calculatedvalues of logKaE0 for CrCl3 and CrCl4 confirmed thattransport of CdSe and In2Se3 is carried out by these agents.

The obtained single crystals with well-formed luster faceshave the shape of a regular octahedron of 1–3mm in theedge lengths (see Fig. 1).

The XPS survey spectrum confirmed the presence ofindium in both compounds.

Fig. 1. The single crystals of (Cd)[Cr1.81In0.19]Se4.

2.2. Experimental techniques

2.2.1. X-ray diffraction

The X-ray diffraction measurements were performedwith a four-circle diffractometer Xcalibur/CCD OxfordDiffraction, operating in k geometry, using graphitemonochromated MoKa radiation and o-scan techniqueand Do step of 1.21. A set of 900 images was taken in nineruns of 100 exposures with different orientations in thereciprocal space. The exposure time per image was 30 s.Crystal and instrument stability was controlled by oneimage, selected as a standard and measured after each 50images [18]. The intensity data were integrated andcorrected for Lorentz and polarisation effects with theCrysAlis software [19]. Numerical absorption correctionbased on the crystal shape was applied [19].

2.2.2. Magnetic measurements

The high-field magnetisation measurements were carriedout at T ¼ 4.2K with a ballistic magnetometer in a Bitter-type magnet.The electron spin resonance spectra were recorded with a

standard EPR spectrometer operating at X-band (9GHz)frequency, using 100 kHz field modulation. The microwavefrequency was measured using Hewlett Packard 534microwave frequency counter. The temperature-depen-dence measurements were performed in the temperaturerange from 90 to 400K. The values of the ESR parameters:DB-linewidth and Br-resonance field were obtained on thebasis of the best fit for the simulated Lorenzian profile incomparison with the experimentally observed spectra.In both methods (XRD and ESR), the same octahedral,

as-grown single crystals were measured.

3. Results and discussion

3.1. Crystal structures and cation distribution

Two single-crystal samples representative for the indium-substituted CdCr2Se4 spinel, with nominal In concentra-tion y ¼ 0.2 and 0.25 have been selected for the X-raydiffraction measurements. The aim was to determine thelocation and concentration of the In ion over thetetrahedral (A) and octahedral (B) sites in the cubic-close-packed selenium sublattice. The structure refinementwas performed using the SHELXL-97 program package[20]. The crystal data and the details of experimentalconditions are summarised in Table 1.The crystal structures have been refined in the space

group Fd3m (No. 227) with the origin of the unit cell takenat the point 3m. Similarly as in parent CdCr2Se4, the Cd

2+

ion was located at the tetrahedral position 8a: (1/8, 1/8, 1/8), and Cr3+ at the octahedral position 16d: (1/2, 1/2, 1/2).The anion took the position 32e: (u, u, u). For each sample,two models have been considered: (1) indium was sharingthe tetrahedral position (A) with cadmium and (2) theindium ion was substituting Cr3+ at the octahedral

ARTICLE IN PRESSD. Skrzypek et al. / Journal of Crystal Growth 297 (2006) 419–425 421

[B]-sites. In each model, the cations located in the same sitewere refined with the coupled site occupancy factors. Theresulting values enabled writing the chemical formulae forthe two crystals.

It appeared that in the two selected samples theadmixture has been accommodated in a different way.For the sample I with y ¼ 0.2 the best convergence in thestructure refinement was obtained for the model 2) with Inlocated at the [B]-sites. The chemical formula can thus be

Table 1

Crystal data, experimental details and structure refinement results for the

Cd–Cr–In–Se spinel system

Crystal data (Cd)[Cr1.81In0.19]Se4

(Cd0.745In0.255)

[Cr1.94In0.06]Se4Temperature (K) 297

Crystal system, space group Cubic, Fd3m

Unit cell dimensions (A)

a 10.7767(12) 10.7577(12)

Volume (A3) 1251.58(3) 1159.18(3)

Z 8

Calculated density (Mg/m3) 5.716 5.274

Crystal size (mm) 0.08� 0.08� 0.09 0.08� 0.08� 0.09

Data collection

Wavelength (A) 0.71073

2y max for data collection 94.01 92.90

Limiting indices

h �14, 17 �21, 15

k �14, 17 �13, 20

l �14, 17 �20, 21

Reflections collected 5788 6801

Reflections unique 259 317

Reflections 42 s (I)] 207 276

Absorption coefficient (mm�1) 30.10 30.55

Refinement

Refinement method Full-matrix

least-squares on F2

Number of refined parameters 10

Goodness-of-fit on F2 0.956 0.999

Final R indices [I42s(I)]R1 0.024 0.023

wR2 0.035(4) 0.040(5)

Extinction coefficient 0.0006(4) 0.00110(5)

Largest diff. peak and hole (e A�3) 0.91 and �0.87 0.87 and �1.15

Table 2

Atomic coordinates, site occupation factors and equivalent isotropic displacem

Campound Anion positional paramaeter (u) Si

A

(Cd)[Cr1.81In0.19]Se4 0.26380(2) 1.

(Cd0.745In0.255)[Cr1.94In0.06]Se4 0.26397(2) 0.

Note: The atomic positions are as follows:

Cd (A) site

Cr/In (B) site

Se Anion sit

written as (Cd)[Cr1.81In0.19]Se4. The results of the structurerefinement are given in Tables 2 and 3.The sample II with the nominal y ¼ 0.25, appeared to be

a mixed spinel, as some portion of In can be found inoctahedral sites, but majority of the indium ions havemigrated to the tetrahedral sites leading to the formula(Cd0.745In0.255)[Cr1.94In0.06]Se4. The ionic radius of In withthe tetrahedral coordination RIn ¼ 0.63, while for thesixfold coordination the value is R ¼ 0.80 [21]. This featureexplains the fact that despite the higher value of y in thissample, the unit cell dimension is not increasing whencompared with parent CdCr2Se4 having the unit cellparameter a ¼ 10.745(2) A [22,23].

3.2. Magnetic and ESR studies

In the chromium spinels, the Cr3+ ions always occupythe B-site of the spinel structure. The local symmetry onthis octahedral site leads to a non-degenerate orbitalground state with S ¼ 3

2. The lattice built upon the B-siteconsists of tetrahedra of chromium ions. Each chromiumion is common to two tetrahedra, which are defined by thepositions of their six first-nearest neighbours. The magneticproperties of CdCr2Se4 were analysed by Baltzer et al. [22]in terms of competing interactions: ferromagnetic betweenfirst-nearest neighbours and antiferromagnetic betweenhigher-order neighbours. The field-dependent magneticmoment for both obtained crystals is shown in Fig. 2.The saturation effects have been observed at relatively low

ent parameters for the Cd–Cr–In–Se4 spinel system

te occupation Uiso (A2� 103)

B Cd Cr/In Se

0 0.905:0.095(7) 14.1(1) 11.5(2) 12.3(1)

745:0.255(9) 0.970:0.030(5) 11.3(1) 9.5(1) 9.23(9)

8a 181818

� �

16d 121212

� �

e 32e (u u u)

Table 3

Selected interatomic distances (A) and angles (deg.) for the Cd–Cr–In–Se

spinel system in the tetrahedral A and octahedral B sites

(Cd)[Cr1.81In0.19]Se4 (Cd0.745In0.255)[Cr1.94In0.06]Se4

A–Se 2.5909(5)� 4 2.5893(4)� 4

Cr/In–Se 2.5541(3)� 6 2.5481(3)� 6

Se–Cr–Se 180.00(1)� 3 180.00(1)� 3

Se–Cr–Se 96.86(1)� 6 96.95(1)� 6

Se–Cr–Se 83.14(1)� 6 83.03(1)� 6

Se–Cd–Se 109.47(1)� 6 109.47(1)� 6

ARTICLE IN PRESS

Fig. 2. The magnetic moment versus the external magnetic field (T ¼ 4.2).

K (Cd)[Cr1.81In0.19]Se4 ; J (Cd0.745In0.255)[Cr1.94In0.06]Se4.

Fig. 3. The temperature evolution of the ESR spectrum of

(Cd)[Cr1.81In0.19]Se4 single crystal.

D. Skrzypek et al. / Journal of Crystal Growth 297 (2006) 419–425422

magnetic fields. Admixtures of In3+ only in [B]-sitesaffected strongly the chromium ion interactions, reducingthe saturation magnetic moment. Recently, similar resultsfor the Cd[Cr2�xGax]Se4 were reported by some of us [23].In the system (Cd0.745In0.255)[Cr1.94In0.06]Se4, when themajority of the indium ions have migrated to thetetrahedral sites, the magnetic saturation moment is closeto the theoretical value of 6mB/molecule.

Within the paramagnetic (PM) region, the ESR spectraof both obtained crystals showed a single Lorenzian linewith g ¼ 1.99 which is attributed to Cr3+ ions. The ESRspectra for (Cd)[Cr1.81In0.19]Se4 and (Cd0.745In0.255)[Cr1.94In0.06]Se4 and their temperature evolution belowT ¼ 140K are displayed in Figs. 3 and 4, respectively.Clearly, the ESR spectra for both obtained crystals showthe same varying tendency. Apart from the dramaticmodifications of the spectra at low temperatures areobserved: (a) deviation from Lorenzian line-shape; (b)shift of the resonance field (Br); (c) broadening of thelinewidth (DB); (d) the anomalous increasing of theintensity;and (e) the splitting of the spectra. The spectralmodifications also suggest that there is a structuralphase transition from cubic symmetry to tetragonal ororthogonal one at Tc, in addition to the ferromagnetictransition.

The plots of the temperature dependence of the linewidthand resonance field are shown in Figs. 5 and 6 for(Cd)[Cr1.81In0.19]Se4 and (Cd0.745In0.255)[Cr1.94In0.06]Se4,respectively. Starting from the T ¼ 400K, the linewidthvalues linearly decreased as the temperature was reduced toT ¼ 250K. The ESR linewidth is related to the relaxationof the spin system. For individual spins DB�1/t, where t isthe spin relaxation time. In a dense magnetic material, thisrelationship is modified since the magnetisation relaxestowards an effective field instead of the external field. Thereview of the mechanisms for the spin-lattice relaxation

(and associated temperature dependencies of ESR line-widths) for concentrated magnetic systems was reported byHuber and Seehra [24] and Seehra et al. [25]. For non-S-state systems with SX1, it is expected that the relaxation ofmagnetisation is dominated by the phonon modulation ofthe crystalline field. Following the analysis in Ref. [24], theESR linewidth of the exchange-coupled PM materials canbe described by the expression:

DB ¼ DBss þ DBs�ph,

where DBss is described by the exchange narrowing theory[26]:

DBss ¼ ½ðDBddÞ2�=Bex,

(i.e. is proportional to the square of the dipolar producedlinewidth DBdd divided by the rate of exchange) and DBs-ph

represent the contribution of the spin–phonon interaction.The linear temperature dependence of ESR linewidth,

which was observed for both obtained crystals shows inaccordance with [24] that one-phonon relaxation may

ARTICLE IN PRESS

Fig. 4. The temperature evolution of the ESR spectrum of

(Cd0.745In0.255)[Cr1.94In0.06]Se4 single crystal.

Fig. 5. The temperature dependence of the ESR parameters for

(Cd)[Cr1.81In0.19]Se4.

Fig. 6. The temperature dependence of the ESR parameters for

(Cd0.745In0.255)[Cr1.94In0.06]Se4.

D. Skrzypek et al. / Journal of Crystal Growth 297 (2006) 419–425 423

prevail in PM materials even at high temperatures. Thiseffect is caused by a broad band of phonons that mayparticipate in the spin relaxation process.

In PM state, there are the interactions between spins andby lowering of the temperature, short-range order progres-sively occurs, which can correspond to very small clusters.Near the transition temperature Tc, the clusters grow andcoalesce to create an infinite magnetic matrix at Tc. ESRhas been recognised as a powerful tool for probing the spinstructure and dynamics. Its high sensitivity to both minormagnetic phases and short-range interactions permits aselective disclosure of the subtle changes in spin systems.That is why our measurements of the ESR spectra reveal(below T ¼ 160K for (Cd)[Cr1.81In0.19]Se4 and T ¼ 140Kfor (Cd0.745In0.255)[Cr1.94In0.06]Se4) apart from strong PMline, the presence of the new lines (see Figs. 3 and 4), whichmeans the appearance of the clusters of Cr3+. It is worthnoting that the short-range order, which suggests anincreased magnetic inhomogeneity of the systems, isdepended upon indium distribution. The confirmation ofthis suggestion is in the calculations, which were done byBakrim et al. [27]. These authors calculated, from theresults of the random phase approximation (RPA), thecorrelation functions for a Heisenberg ferromagneticmodel having both nearest-neighbour (nn) and nextnearest-neighbour (nnn) exchange integrals. The theoreti-cal results obtained were used to study the PM state ofspinels Cd[Cr2�xGax]Se4. It can be noted that all correla-tion functions persist far into PM region (see Figs. 1–6 inRef. [27]).The appearance of the short-range magnetic ordering

was confirmed by the temperature dependence of thespectrum intensity. In Figs. 7 and 8, the temperatureevolution of intensity of ESR spectrum for both obtainedcrystals is shown. The adequate values were calculated asdouble integration of the spectrum (DI). This definedintensity should be proportional to the spin susceptibilityof the sample. It can be seen, that below T ¼ 300K, thedeviation of the inverse susceptibility from the Curie–Weisslaw appears. As the temperature is reduced from aboutT ¼ 170K, the values for the intensity rapidly increase.

ARTICLE IN PRESS

Fig. 7. The relative ESR susceptibility versus temperature for

(Cd)[Cr1.81In0.19]Se4.

Fig. 8. The relative ESR susceptibility versus temperature for

(Cd0.745In0.255)[Cr1.94In0.06]Se4.

D. Skrzypek et al. / Journal of Crystal Growth 297 (2006) 419–425424

This rise is more marked than theoretically predicted forthe Heisenberg-type ferromagnets, where magnetic suscept-ibility wE[(T�Tc)/Tc]

�p with p ¼ 43[28]. The effect ob-

served is due to the presence of external magnetic field,which orients the clusters in the field direction.

The spectra below T ¼ 130K (see Figs. 3 and 4) areattributed to the ferromagnetic resonance (FMR). In FMRequation written by Kittel [29], the contributions of thedemagnetisation and magnetocrystalline anisotropy fieldsoccur. The complexity of resonance spectra for both theobtained crystals is due to the inhomogeneity of theinternal magnetic field which is caused by the demagnetis-ing effects brought about by the octahedron sample shape[30] and due to the domain structure [31]. It can be seen,from Figs. 5 and 6, that the transition region is broad forboth the obtained crystals. The original PM line broadensgradually, shifts from PM resonance and finally vanishes.

In addition, the ESR spectra analyses also indicate that thePM–ferromagnetic transition is incomplete and the PMphase may still survive in a certain temperature rangebelow Tc.

4. Conclusions

In this work, we presented the X-ray diffraction analysisfor two types of diluted ferromagnetic CdCr2Se4 singlecrystals: (Cd)[Cr1.81In0.19]Se4 in which In substitutes Crand (Cd0.745In0.255)[Cr1.94In0.06]Se4 where In substitutes Cdand Cr.The magnetic properties of the same as-grown single

crystals were examined by ESR spectroscopy. ESR wasfound to be extremely sensitive for detecting minormagnetic phases and short-range order. In resonancespectra, the ferromagnetic clusters signals are presented:from TE160K for Cd)[Cr1.81In0.19]Se4 and from TE140Kfor (Cd0.745In0.255)[Cr1.94In0.06]Se4. The transition regionappears to be broad for both obtained crystals and PMphase was still present up to T ¼ 125K. The strongermagnetic inhomogeneity occurs for (Cd)[Cr1.81In0.19]Se4crystal. The linear variation in resonance linewidth withtemperature for T4250K is an experimental proof ofHuber’s–Seehra’s theory. It describes the occurrence ofone-phonon processes up to high temperatures in spin-lattice relaxation in magnetically concentrated systems.

References

[1] P.K. Baltzer, P.J. Wojtowicz, M. Robbins, E. Lopatin, Phys. Rev.

151 (1966) 367.

[2] C. Haas, A.M.J.G. van Run, P.F. Bongers, W. Albers, Solid State

Commun. 5 (1967) 657.

[3] H.W. Lehmann, Phys. Rev. 163 (1967) 488.

[4] H.W. Lehmann, G. Harbecke, J. Appl. Phys. 38 (1967) 946.

[5] A.R. von Neida, L.K. Shick, J. Appl. Phys. 40 (1969) 1013.

[6] H. von Philipsborn, J. Crystal Growth 9 (1971) 296.

[7] H. Hahn, K. Schroder, Z. Anorg. Allg. Chem. 269 (1952) 135.

[8] N. Menyuk, K. Dwight, J. Arnott, W. Wold, J. Appl. Phys. 37 (1966)

1387.

[9] J. Tang, T. Matsumoto, T. Naka, T. Furubayashi, S. Nagata, N.

Matsumoto, Physica B 259–261 (1999) 857.

[10] F.K. Lothering, R.P. Van Stapele, G.H.A.M. Van der Steen, J.S. Van

Wiesingen, J. Phys. Chem. Solids 30 (1969) 799.

[11] J. Alvarez, V. Sagredo, J. Mantilla, J. Magn. Magn. Mater. 196–197

(1999) 407.

[12] J. Krok-Kowalski, J. Warczewski, T. Mydlarz, A. Pacyna, I.

Okonska-Kozlowska, J. Magn. Magn. Mater. 166 (1997) 172.

[13] I. Okonska-Kozlowska, E. Malicka, A. Waskowska, T. Mydlarz,

J. Solid State Chem. 148 (1999) 215.

[14] V.T. Kalinnikov, T.G. Aminov, V.M. Novotortsev, Inorg. Mater. 39

(2003) 1159.

[15] M. Nouges, J.L. Dormann, J. Magn. Magn. Mater. 54–57 (1986) 87.

[16] G.G. Shabunina, R.A. Sadykov, T.G. Aminov, Inorg. Mater. 36

(2000) 1208.

[17] V. Plies, Z. Anorg. Allg. Chem. 556 (1988) 120.

[18] Oxford Diffraction, Crys. Alis CCD. Data Collection software,

Oxford Diffraction Ltd., Wroc"aw, Poland, 2001.[19] Oxford Diffraction, CrysAlis RED. Data Reduction program. Issue

171.26, Oxford Diffraction Ltd., 2004.

ARTICLE IN PRESSD. Skrzypek et al. / Journal of Crystal Growth 297 (2006) 419–425 425

[20] G.M. Sheldrick, 1999 SHELXL-99, Program for Crystal Structure

Refinement, University of Gottingen.

[21] R.D. Shannon, Acta Crystallogr. A 32 (1976) 751.

[22] P.K. Baltzer, P.J. Wojtowicz, M. Robbins, E. Lopatin, Phys. Rev.

151 (1966) 367.

[23] I. Okonska-Kozlowska, E. Malicka, A. Waskowska, J. Heimann, T.

Mydlarz, J. Solid State Chem. 158 (2001) 34–39.

[24] D.L. Huber, M.S. Seehra, J. Phys. Chem. Solids 36 (1975)

723–725.

[25] M.S. Seehra, M.M. Ibrahim, V.S. Babu, G. Srinivasan, J. Phys.:

Condens. Matter 8 (1996) 11283.

[26] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of

Transition Ions, Clarendon Press, Oxford, 1970.

[27] H. Bakrim, K. Bouslykhane, M. Hamedoun, A. Hourmatallah, N.

Benzakour, J. Magn. Magn. Mater. 285 (2005) 327.

[28] W. Zarek, Acta Phys. Pol. A 52 (1977) 657–663 and references

therein.

[29] C. Kittel, Phys. Rev. 73 (1948) 155;

C. Kittel, Phys. Rev. 76 (1949) 743.

[30] D.S. Schmool, J.M. Barandiaran, J. Magn.Magn.Mater. 191 (1999) 211.

[31] R. Szymczak, A. Szewczyk, M. Baran, V. Tsurkan, J. Magn. Magn.

Mater. 8 (1990) 481.