Physica B 319 (2002) 90–104 Crystalline-electric-field effect in some rare-earth intermetallic compounds Nguyen Hoang Luong* Center for Materials Science, Faculty of Physics, College of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Hanoi, Viet Nam Received 12 March 2002; received in revised form 18 March 2002 Abstract The results of research on the crystalline-electric-field (CEF) effect in RCu2, R2Fe14B and RFe11Ti compounds are presented In the study of the CEF effect in the RCu2 compounds, attention is paid to the combined analysis of specific heat and thermal expansion An attempt has been undertaken to investigate the systematic behavior of CEF interactions by comparing different compounds with the same crystallographic structure From the analysis of spinreorientation phenomena in R2Fe14B and RFe11Ti compounds the sets of CEF parameters are derived r 2002 Elsevier Science B.V All rights reserved Keywords: Crystalline-electric-field effect; Rare-earth intermetallic compounds Introduction Rare-earth intermetallic compounds are in a prominent situation not only from a fundamental point of view but also because of the important applications, in particular in the field of permanent magnets Magnetic properties of rare-earth intermetallics result to a large extent from the interplay of crystalline-electric-field (CEF) and exchange interactions The CEF removes the degeneracy of the ground state multiplet of the rare-earth ion This results in specific magnetic properties of the corresponding compound The study of CEF effects is an important subject in the field of magnetism and magnetic materials *Corresponding author Fax: +84-4-8589496 E-mail address: luongnh@vnu.edu.vn (N.H Luong) In this work, we present the results of research on the CEF effect in some rare-earth intermetallic compounds Firstly, we discuss the RCu2 (R=rare earth) compounds, the magnetic properties of which are largely affected by the CEF interactions Unlike the Rn Tm compounds, where T is a magnetic transition metal, 4f magnetism can be investigated in the RCu2 compounds without disturbing the effects of the 3d magnetism An attempt has been undertaken to investigate the systematic behavior of CEF interactions by comparing different compounds with the same crystallographic structure Particular attention is paid to the CEF effect in ErCu2, in which the combined analysis of specific heat and thermal expansion is proved to be a valuable tool Then, we discuss the CEF effect in the R2Fe14B and RFe11Ti compounds on which spin-reorientation phenomena are studied In these compounds, both 0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V All rights reserved PII: S - ( ) 1 1 - N.H Luong / Physica B 319 (2002) 90–104 the rare-earth sublattice and the transition-metal one are magnetic Spin-reorientation transitions have been observed in many of these compounds From the analysis of the spin-reorientation phenomena, we derive sets of CEF parameters for the R site It is shown that the study of spinreorientation phenomena in intermetallic compounds is very useful in obtaining information on the CEF parameters The crystalline-electric-field effect 2.1 General formalism of the crystalline electric field The 4f electrons of a rare-earth ion in a solid, being considered as well localized and separated from other charges, experience an electrostatic potential V ðrÞ that originates from the surrounding charge distribution The CEF Hamiltonian, describing the electrostatic interaction of the aspherical 4f charge distribution with the aspherical electrostatic field arising from its surrounding, can be written as X HCEF ¼ À eV ðri Þ: ð1Þ i Here, the summation is taken over all 4f electrons The Hamiltonian may be expanded in spherical harmonics Ynm ; since the charges of the CEF are outside the shell of the 4f electrons: N X n X X Am rni Ynm ðyi ; ji Þ; ð2Þ HCEF ¼ n n¼0 m¼Àn i where Am n are coefficients of this expansion Their values depend on the crystal structure considered and determine the strength of the CEF interaction The value of n in expression (2) is limited to np6 for the rare-earth series The calculation of the matrix elements of the Hamiltonian (2) can be performed by direct integration However, the method called the Stevens operator equivalent method is much more convenient and is widely used This method of Stevens [1] is described in detail by Hutchings [2] In this method, the x; y; z coordinates of a particular electron are replaced by the components 91 Jx ; Jy ; Jz of the multiplet J: The CEF Hamiltonian (2) then takes the form N X n X m HCEF ẳ Bm 3ị n On Jị: nẳ0 mẳn Here, the coefficients Bm n are called the CEF parameters and Om are the Stevens equivalent n operators [1] The parameters Bm n can be written as n m Bm n ¼ yn /r4f SAn : ð4Þ In this expression, the factor related to the 4f ion, yn /rn4f S; and the factor related to the surrounding m charges, Am n ; are separated The coefficients An are known as the CEF coefficients yn is the appropriate Stevens factor of order n which represents the proportionality between the operator functions of x; y; z and the operator functions of Jx ; Jy ; Jz : The parameter yn is denoted as aJ ; bJ ; gJ for n ¼ 2; 4, and 6, respectively The sign of yn represents the type of asphericity associated with each Om n term describing the angular distribution of the 4felectron shell In particular, the factor aJ describes the ellipsoidal character of the 4f-electron distribution For aJ > 0; the electron distribution associated with Jz ¼ J is prolate, i.e elongated along the moment direction whereas for aJ o0 the 4f-electron-charge distribution is oblate, i.e expanded perpendicular to the moment direction For aJ ¼ (which is the case of the Gd+3 ion) the charge density has spherical symmetry /rn4f S is the mean value of the nth power of the 4f radius Values for the average value /rn4f S over the 4f wave function have been computed on the basis of Dirac–Fock studies of the electronic properties of the trivalent rare-earth ions by Freeman and Desclaux [3] Values for yn /rn4f S have been ! collected by Franse and Radwanski [4] The computation of the CEF coefficients, Am n; from microscopic, ab initio, calculations is a difficult problem A full band-structure calculation of the charge distribution over the unit cell and, consequently, of the full set of CEF coefficients, is lacking for almost all compounds In some cases, the point-charge model, with electron charges centered at the ion positions in the lattice, can give the correct sign of the leading second-order CEF coefficients However, this model is questioned, especially in metallic systems where the 92 N.H Luong / Physica B 319 (2002) 90–104 contribution of valence electrons is expected to be significant [5] This has been confirmed by bandstructure calculations by Coehoorn [6] who concluded that the second-order CEF coefficient A02 is mainly determined by the asphericity of the valence-shell electron density of the rare earth under consideration There is only a limited number of compounds for which the CEF interactions have been quantified It is due to the lack of experimental information as well as the complexity of the crystallographic structures Discussions are still going on, even for the best-known systems like the cubic Laves-phase RT2 compounds The situation becomes more complex for the systems with a lower crystal symmetry The orthorhombic RCu2 and the tetragonal R2Fe14B and RFe11Ti compounds, which we are dealing with in this work, belong to these latter cases For cubic symmetry (in case of the RAl2 compounds, for instance) the CEF is described by only two parameters B4 and B6 HCEF ẳ B4 O04 ỵ 5O44 ị ỵ B6 ðO06 À 21O46 Þ: ð5Þ For tetragonal symmetry, five CEF parameters are needed Bm n HCEF ẳ B02 O02 ỵ B04 O04 ỵ B44 O44 ỵ B06 O06 ỵ B46 O46 ð6Þ and for orthorhombic symmetry, nine CEF parameters Bm n are needed HCEF ẳ B02 O02 ỵ B22 O22 þ B04 O04 þ B24 O24 þ B44 O44 þ B06 O06 ỵ B26 O26 ỵ B46 O46 ỵ B66 O66 : ð7Þ Usually the CEF parameters Bm n are evaluated from the analysis of experimental data The methods include the fitting of the magnetization curves, inelastic neutron scattering, measurement of the temperature dependence of the specific heat and susceptibility, Mossbauer spectroscopy, and so on Below, we will discuss the methods of analysis of experimental data that we use for studying the CEF effects in RCu2, R2Fe14B and RFe11Ti compounds These methods comprise the Gruneisen analysis and the spin-reorientation analysis 2.2 Gruneisen analysis In the study of magnetic systems, the specific heat and the thermal expansion are very important The combined analysis of specific heat and the thermal expansion can give valuable information on the system under consideration Here, we briefly describe a procedure that has successfully been applied to several different systems The specific heat is written as the sum of electronic (ce ), lattice (cph ) and magnetic (cm ) contributions c ẳ ce ỵ cph ỵ cm : 8ị Similarly, the thermal expansion contains electronic (be ), lattice (bph ) and magnetic (bm ) contributions b ẳ be ỵ bph ỵ bm : 9ị In Eqs (8) and (9) we neglect a nuclear contribution The electronic part of the specific heat is written as ce ẳ gT; 10ị where g is called the electronic coefficient The electronic part of the thermal expansion is also a linear function of temperature, i.e be ẳ aT: 11ị The phonon part of the specific heat (for a compound with r atoms per formula unit) is approximated by Z yD =T x ex cph ẳ 9rRT=yD ị3 dx; ð12Þ ðex À 1Þ2 where yD is the Debye temperature, R the gas constant The phonon contribution to the thermal expansion, like the contribution to the specific heat, is approximated by Z yD =T x4 ex bph ¼ bðT=yD Þ3 dx: ð13Þ ðex À 1Þ2 An arbitrary contribution to the specific heat, ci ; is related to a corresponding contribution to the thermal expansion, bi ; by a so-called Gruneisen relation Gi ¼ V bi =kci : ð14Þ N.H Luong / Physica B 319 (2002) 90–104 Here V is the molar volume, k the compressibility and Gi the appropriate Gruneisen parameter For the electronic Gruneisen parameter we have (see Eqs (10) and (11)) Ge ¼ Va=kg: ð15Þ Using Eqs (12) and (13) for the lattice Gruneisen parameter, we obtain Gph ẳ Vb=9rRk: 16ị In treating the magnetic contributions to the specific heat and to the thermal expansion, a straightforward approach is to calculate an effective Gruneisen parameter, Geff ; by the relation Geff ¼ V bm =kcm ð17Þ and to follow its variation with temperature A pronounced temperature dependence of Geff ðTÞ indicates the presence of several contributions ErCu2 can serve as an example, in which a change in sign of the parameter Geff is observed upon increasing the temperature [7] In this case, at least two different contributions to cm and bm can be distinguished Therefore, we write cm ẳ clr ỵ ccf and bm ẳ blr ỵ bcf : 18ị Here, cm and bm are split into two contributions, the ‘long-range’ magnetic order contributions clr and blr ; and the contributions ccf and bcf associated with the CEF splitting of the energy levels Assuming that the contributions ci and bi are related by Gruneisen parameters Gi ; we have X Geff ẳ fi Gi ; i ẳ lr; cf: 19ị i Here fi ¼ ci =cm : For any choice of Glr and Gcf ; the separated contributions can be calculated as Geff À Gcf cm ; Glr À Gcf 20ị ccf ẳ Glr Geff cm ; Glr Gcf 21ị blr ẳ Gcf =Geff b À Gcf =Glr m ð22Þ Gcf À Geff =Glr bcf ¼ b : Geff À Gcf =Glr m 23ị clr ẳ 93 This analysis and its application to ErCu2 is described in Refs [8,9] Brommer and Franse [10] have generalized this method, including Gruneisen relations between specific-heat and linearexpansion contributions, as well as focusing attention to the criteria to decide whether the chosen Gi parameters can be considered as genuine Gruneisen parameters They successfully applied this analysis to a variety of materials (see Ref [10]) 2.3 Spin-reorientation analysis For a uniaxial crystal, the magnetocrystalline anisotropy energy may be described by the phenomenological expression E ¼ K1 sin2 y ỵ K2 sin4 y: 24ị Here y is the polar angle of the magnetization with respect to the c-axis K1 and K2 are the anisotropy constants Minimizing anisotropy energy (24) with respect to y gives the orientation of the magnetization A sudden change from easy axis to easy plane may occur However, when, for K2 > 0; K1 changes sign at a certain temperature, a gradual spin reorientation will start at that temperature [11,12] The angle between the moment direction and the c-axis is given by sin2 y ¼ ÀK1 =2K2 : ð25Þ For a deeper understanding of the spin-reorientation phenomena, one has to consider a microscopic model As mentioned in the introduction, the magnetic properties of 3d–4f compounds are governed by a combination of the 3d–4f exchange and CEF interactions The Hamiltonian of a 4f ion in 3d–4f compounds is usually given in the form HR ¼ HCEF ỵ Hex : 26ị Here, HCEF and Hex are the CEF and exchange Hamiltonians, respectively The CEF Hamiltonian for a tetragonal structure is expressed by Eq (6) The exchange Hamiltonian is given by Hex ¼ gmB JB m ¼ 2ðg À 1ÞmB JB ex ; ð27Þ where B m is the molecular field acting on the rareearth magnetic moment which is related to the N.H Luong / Physica B 319 (2002) 90–104 94 exchange field Bex acting on the rare-earth spin by Bm ẳ ẵ2g 1ị=gB ex : J and g are the total angular momentum and the Lande! factor of the R3+ ion, respectively The field Bm is related to the exchange constant nRT ; which links the rare-earth (R) and transition-metal (T) sublattices by Bm ¼ ½2ðg À 1Þ=gŠnRT /M T S: ð28Þ Here /M T S is the average transition-metal sublattice magnetization The relation between K1R and Bm n is [13] K1R ẳ 3=2ịB02 /O02 S À 5B04 /O04 S À ð21=2ÞB06 /O06 S: /Om nS ð29Þ are the thermal averages of the Here Stevens operators The ground state of the 4f ion is calculated by diagonalizing Hamiltonian (26) The temperature dependence of the rare-earth sublattice anisotropy, K1R ðxÞ; which normally dominates, is then calculated according to Eq (29) with K1R xị ẳ À xÞK1R ð0Þ; where K1R ðxÞ and K1R ð0Þ are the rare-earth sublattice anisotropy constants for the substituted and unsubstituted compounds, respectively In the case of R2Fe14B compounds, in this work K1R has been calculated as K1R ¼ E>c À E8c ; where E>c and E8c represent the ground state energy of Hamiltonian (26) for B m being perpendicular and parallel to the c-axis, respectively For calculating the temperature dependence of the anisotropy energy of the rare-earth ion, the Boltzmann distribution function is used The temperature dependence of the transition-metal sublattice anisotropy K1T is taken from the study of the isostructural compound in which R is nonmagnetic In case the data on the temperature dependence of the angle y in aligned powder samples are available, the analysis is proceeded as follows For aligned powder samples of a material with axial anisotropy, the crystallites (powder particles) are oriented in such a way that their c-axis are parallel to each other, whereas the a- and b-axis are randomly distributed in the plane perpendicular to the alignment direction Therefore, it is appropriate to confine the exchange field to the x2z plane [14] Such an approximation leads to the following exchange Hamiltonian: Hex ¼ gmB Bm J z cos y ỵ J y sin yÞ: ð30Þ The rare-earth energy is obtained by diagonalization of Hamiltonian (26) and by calculating the partition function Zðy; TÞ: The rare-earth energy is given by FR y; Tị ẳ ÀkB T ln Zðy; TÞ: ð31Þ For a mixed system such as R1ÀxRx0 Fe11Ti, the total free energy is expressed as F y; Tị ẳ xịFR y; Tị þ xFR0 ðy; TÞ þ K1T sin2 y: ð32Þ Here the last term represents the contribution from the transition-metal sublattice For the calculation of F ðy; TÞ; we need to know the CEF parameters Bm n and the molecular field Bm : The temperature dependence of the transition-metal sublattice anisotropy K1T is again taken from the study of an isostructural compound in which R is nonmagnetic The angular dependence of the total free energy was then calculated and the minimum in the free energy gives the orientation of the total magnetization vector In this way, the temperature dependence of the magnetic structure is determined CEF effect in RCu2 compounds The RCu2 intermetallic compounds have an orthorhombic CeCu2 structure Early magnetization and magnetic susceptibility measurements performed by Hashimoto et al [15] and Hashimoto [16] demonstrated the importance of the CEF in these compounds During the last decade substantial progress has been achieved in the study of the magnetic properties of the RCu2 compounds It is of interest to make an attempt to see some systematic behavior by comparing different compounds with the same crystallographic structure This attempt was undertaken in Ref [9] In this work, we focus our study to the CEF effect in the RCu2 compounds, taking into account the recent results For doing this, we recall also some works by other authors N.H Luong / Physica B 319 (2002) 90–104 CeCu2 is a Kondo compound In Ref [9] some results on the study of CEF effects are mentioned Sugiyama et al [17] have measured the high-field magnetization of CeCu2 in various temperatures and have analyzed it on the basis of the quadrupolar interaction and CEF PrCu2 shows a nearly temperature-independent Van Vleck paramagnetic behavior below 4.2 K and exhibits cooperative nuclear antiferromagnetic order below 54 mK [18] From the measured paramagnetic Curie temperatures, Hashimoto et al [19] have estimated the following values for the secondorder CEF parameters for PrCu2: B02 ¼ 4:27 K and B22 ¼ 2:97 K By using a point-charge model, Hashimoto et al [19] have also calculated the values of B02 and B22 and arrived at B02 cal ¼ 4:1 K and B22 cal ¼ 3:48 K, in good agreement with experiment The CEF and the metamagnetic transition in PrCu2 has been studied in detail by Ahmet et al [20] and Settai et al [21,22] We recall here the set of CEF parameters for PrCu2, which has been derived by Settai et al [22]: B20 ¼ 4:93 K, B22 ¼ 3:50 K, B04 ¼ 5:08  10À2 K, B24 ¼ 5:02  10À2 K, B44 ¼ À3:82  10À1 K, B06 ¼ À1:33  10À3 K, B26 ¼ À1:80  10À2 K, B46 ¼ À2 À2 À2:98  10 K, and B6 ¼ À4:97  10 K For the NdCu2 compound the second-order CEF parameters were first estimated by Hashimoto et al [15] from measurements of the paramagnetic susceptibility in single-crystalline sample: B02 ¼ 0:8 K and B22 ¼ 1:1 K The values of B02 cal ¼ 1:17 K and B22 cal ¼ 1:01 K were obtained by Hashimoto et al [15] by point-charge model calculations A more detailed study of the CEF effect in this compound has been carried out by Gratz et al [23] These authors have derived the following set of CEF parameters which best describe the inelastic neutron-scattering data: B20 ¼ 1:35 K, B22 ¼ 1:56 K, B04 ¼ 2:23  10À2 K, B24 ¼ 1:01  10À2 K, B44 ¼ 1:96  10À2 K, B06 ¼ 5:52  10À4 K, B26 ¼ 1:35  10À4 K, B46 ¼ 4:89 10À4 K, and B66 ¼ 4:25  10À3 K Gratz et al [24] have shown that the coefficient of thermal expansion of SmCu2 exhibits a minimum at 45 K, caused by the CEF effect These authors have used the position of the temperature where the minimum occurs to estimate the splitting energy between the ground-state doublet and the 95 first-excited state doublet (see, for instance, Ref [8]) They have derived a value of about 110 K for this CEF splitting Turning to the heavy RCu2 compounds, first we discuss TbCu2 Again from the values of paramagnetic Curie temperatures, Hashimoto et al [19] have estimated the following values for the second-order CEF parameters: B02 ¼ 1:23 K and B22 ¼ 1:23 K These authors have also calculated the second-order CEF parameters on the basis of the point-charge model Their calculated values are: B02 cal ¼ 1:35 K and B22 cal ¼ 1:12 K Experiments and calculations are in reasonable agreement with each other, indicating that the anisotropy observed in the paramagnetic state for the paramagnetic Curie temperatures along the crystallographic axes can be explained mainly by the CEF effect Measurements of the specific heat and thermal expansion were performed by Luong et al [7] Apart from a l-type of anomaly at TN ; apparently a broad anomaly is observed around 30 K This anomaly can be discussed in terms of Gruneisen parameters For the DyCu2 compound, also from the values of the paramagnetic Curie temperatures, Hashimoto et al [19] have estimated values for the second-order CEF parameters: B02 ¼ 0:43 K and B22 ¼ 0:72 K A point-charge calculation by the same authors results in B02 cal ¼ 0:89 K and B22 cal ¼ 0:71 K, in satisfactory agreement with the experimental values Specific heat and thermal expansion of DyCu2 have been measured by Luong et al [7] Kimura et al [25] have calculated the specific heat of DyCu2 in terms of the molecular-field model including CEF interaction They have obtained the temperature dependence of the specific heat which is similar in trend with the experimental one of Luong et al [7] They have used three CEF parameters: B02 and B22 experimentally obtained by Hashimoto et al [19] and B04 equal to 2.37  10À3 K By calculating the magnetic susceptibility and the magnetization and comparing these calculated properties with the experimental ones, Sugiyama et al [26] and Yoshida et al [27] have derived the full set of CEF parameters for DyCu2 The set of CEF parameters obtained in Ref [27] is the following: B02 ¼ 0:708 K, B22 ¼ 0:99 K, B04 ¼ À0:28 10À4 K, B24 ¼ 2:37  10À2 K, B44 ¼ 0:26  10À5 K, N.H Luong / Physica B 319 (2002) 90–104 96 B06 ¼1:1  10À5 K, B26 ¼ 0:79  10À5 K, B46 ¼ 1:6 10À4 K, B66 ¼ À1:7  10À4 K Like in other RCu2 compounds, from the values of the paramagnetic Curie temperatures, Hashimoto et al [19] have estimated the following second-order CEF parameters for HoCu2: B02 ¼ 0:14 K and B22 ¼ 0:12 K The values of B02 cal ¼ 0:28 K and B22 cal ¼ 0:23 K were obtained by Hashimoto et al [19] on the basis of the pointcharge model The CEF effect in ErCu2 will be discussed in detail below The CEF effects in TmCu2 have extensively been studied The values for all nine CEF parameters obtained by different methods and by combining the results of different experiments for this compound are collected in Ref [9] (see Table 4.1 of Ref [9]) It can be inferred from the above discussion and from Ref [9] that information about the CEF interaction in RCu2 is not complete Due to the orthorhombic structure, nine CEF parameters are needed to describe the CEF Hamiltonian In Table 1, we collect the second-order coefficients A02 and A22 for the RCu2 compounds These coefficients are related to the CEF parameters B02 and B22 by (see Section 2) A02 ¼ B02 =aJ /r24f S; A22 ẳ B22 =aJ /r24f S: 33ị Here, the values for the quantity aJ /r24f S are ! taken from Franse and Radwanski [4] To our knowledge, CEF parameters for SmCu2 are not available Rather low values of A02 and A22 have Table CEF coefficients, in units of KaÀ2 ; for the RCu2 compounds R A20 A22 Ref Pr À168 À195.3 À112 À188 À148 À85.7 À141 À83.5 À190 À153 À134.7 À134.7 À117 À138.7 À154 À218 À148 À143.4 À197 À71.6 À194 À120 À186.4 À176.3 [16] [22] [16] [23] [19] [19] [27] [19] [19] [28] [30] [29] Nd Tb Dy Ho Er Tm been obtained by Trump [31] for the Kondo compound CeCu2, in which anomalous properties are observed Except for CeCu2, as can be seen from Table 1, the coefficients A02 and A22 have the same sign and are of the same order of magnitude We note that the same cannot be said for the higher-order CEF coefficients From the similarity of the lowest-order CEF parameters, it seems that the CEF model can be used for describing the behavior of isostructural RCu2 compounds At the same time, there is no reason why higher-order CEF parameters should be neglected The necessity to take these higher-order terms into account is indicated in Ref [9] and below in a discussion of CEF effects in ErCu2 The importance of the higher-order CEF terms is also revealed from studies on substituted RCu2 compounds Analyzing the data obtained on a Tb(Cu0.7Ni0.3)2 sample, Divis et al [32] have shown that the step-like appearance in the magnetization curves along the b-axis in this compound cannot be explained by using second-order terms in the CEF Hamiltonian only These authors have shown that in order to account for all features of the magnetization data, the higher-order terms should be included into the Hamiltonian Divis et al [33] have also used nine CEF parameters for describing the specific-heat data on Tm(Cu1ÀxNix)2 We discuss below in detail the CEF effect in ErCu2 Luong et al [7] reported on the specific heat and thermal expansion of RCu2 compounds and discussed the excess contributions of both quantities arising from magnetic ordering and CEF effects Apart from some sharp features indicating transitions between different types of antiferromagnetic order, and apart from a lambda type of anomaly characteristic for disordering the antiferromagnetic state, broad anomalies were observed for several compounds in the specific heat and thermal expansion In the case of ErCu2, Luong et al [7] could preliminarily analyze the excess contributions to the specific heat and thermal expansion by applying Gruneisen relations This analysis has been applied and discussed in more detail in Refs [8,9,34], showing that they consist of a ‘long-range’ magnetic and a CEF part (see above) The CEF term yields Schottky-type DCexp curve, shown in Fig Luong et al [7] N.H Luong / Physica B 319 (2002) 90–104 Fig Calculated and experimental results for the CEF contribution to the specific heat of ErCu2 DC1 and DC2 curves are calculated using CEF data in Refs [19,28], respectively DCexp (from Refs [7,8]) is obtained from specific heat and thermal expansion measurements showed that the energy difference between the ground-state doublet and the first excited doublet amounts to 76 K Using the values for the two lowest-order CEF parameters, B20 ¼ À0:35 K and B22 ¼ À0:36 K, obtained by Hashimoto et al [19] from an analysis of the paramagnetic susceptibility, the energy levels in ErCu2 have been calculated, and a splitting of 13 K between the two lowest-order doublets has been derived [8] Apparently, higher-order CEF terms have to be taken into account in order to bring the splitting closer to the experimental value of 76 K As pointed out earlier, due to the low symmetry in RCu2 compounds, nine CEF parameters are needed to describe the CEF of the rare-earth ion It is difficult to derive the full set of CEF parameters A combination of different techniques, experimental and theoretical, is used in order to overcome this difficulty Gubbens et al [28] have measured the ErCu2 compound with 166Er Mossbauer spectroscopy These authors also reported the results of inelastic neutron scattering, which show a not yet definitively determined level sequence of doublets above TN at 0, 61, 78, 88, 124, 142, 148 and 160 K Gubbens et al [28] have determined a tentative set of all nine CEF parameters This set is: B20 ¼ À0:28 K, B22 ¼ À0:22 K, B04 ¼ À0:30 10À2 K, B24 ¼ À0:14  10À2 K, B44 ¼ 0:30  10À2 K, B06 ¼ À0:20  10À4 K, B26 ¼ À0:47  10À4 K, B46 ¼ À0:97  10À4 K, and B66 ¼ À2:96  10À4 K 97 We have performed calculations of the CEF contribution to the specific heat of ErCu2 using CEF data obtained by Hashimoto et al [19] and Gubbens et al [28] Results of the calculations are shown in Fig 1, and are compared with the experimental data DCexp : On calculating DC1 ; using two lowest-order CEF parameters reported by Hashimoto et al [19], we derived the following level scheme of doublets: 0, 13, 25, 33, 40, 49, 61 and 75 K As can be seen from Fig 1, the CEF contribution to the specific heat (curve DC1 ), given by this energy scheme, disagrees with our experiments The curve DC2 in Fig was obtained by taking into account all eight doublets reported by Gubbens et al [28] From this figure it can also be seen that the temperature dependence of DC2 has a behavior similar to the experimental one, but the calculated value of DC2 is larger in the high temperature range This difference between the calculated DC2 and the experimental specific-heat curves has been discussed in more detail in Ref [34] One of the reasons for the above-mentioned discrepancy could be an overestimation of the non-magnetic contribution to the specific heat In order to evaluate the magnetic contribution to the specific heat, non-magnetic (electronic and phonon) contributions have to be subtracted from the total specific heat It is well known that a proper evaluation of the non-magnetic contribution of magnetic compounds is a difficult problem In the case of RCu2, the non-magnetic contribution to the specific heat is obtained from measuring the specific heat of YCu2 [35] We note that YCu2 is taken as the non-f reference material instead of LaCu2 because LaCu2 possesses a different crystallographic structure and crystallizes in the hexagonal AlB2-type of structure A value of 194 K for the Debye temperature of ErCu2, yD(ErCu2), was derived [34] Using the more sophisticated approach of Bouvier et al [36], which accounts for the different molar masses of the components, a value of 197 K for yD (ErCu2) has been determined, i.e very close to the value 194 K obtained above Another possible reason of the discrepancy between the experimental curve DCexp and the calculated curve DC2 could be that higher energy levels not substantially contribute to the N.H Luong / Physica B 319 (2002) 90–104 98 specific heat We have performed calculations of the CEF contribution to the specific heat of ErCu2 taking into account only the four lowest doublets reported by Gubbens et al [28] (see above) The calculated curve is in close agreement with the experimental one This result could suggest (in case of a proper estimate of the phonon contribution to the specific heat) that the energy spectrum in ErCu2 is divided into two groups The first group consists of the four lowest doublets located below 88 K The second group, consisting of four higher doublets, is separated from the first one We note, as mentioned above, that the energy level scheme in ErCu2 is not definitively determined and that the set of CEF parameters for this compound is not unique at the present stage of investigations [28] One of the features of the RCu2 compounds is that the values of the Ne! el temperature for these compounds are not simply proportional to the de Gennes factor ðgJ À 1ị2 JJ ỵ 1ị and reach a maximum for TbCu2 This fact suggests that the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction alone is not sufficient to fully understand the magnetic interactions in the RCu2 compounds In spite of a substantial progress in the study of the magnetic properties of the RCu2 compounds, the above-mentioned exception to the de Gennes rule remained unexplained for a long time Luong et al [37] have performed calculations in order to explain the anomalous behavior of the Ne! el temperature of the RCu2 compounds (R=Tb– Tm) The calculations are based on the model of Noakes and Shenoy [38] When considering only the exchange Hamiltonian, the de Gennes rule can be derived However, when the CEF effects are significant, the de Gennes behavior is not to be expected Adding the CEF Hamiltonian to Hexc leads to the following expression for the ordering temperature: TM ẳ 2GgJ 1ị2 /Jz2 ðTM ÞSCEF ; /Jz2 ðTM ÞSCEF ð34Þ where is the expectation value of Jz2 under the influence of the CEF Hamiltonian alone, at the temperature TM : The exchange parameter, G; can be evaluated from the ordering temperature of the Gd compound when modeling a series of rare-earth compounds, because Gd, an L ¼ ion, is essentially unaffected by CEF For the calculation of the Ne! el temperatures, TN ; of the RCu2 compounds expression (34) is used, in which TM stands for TN : For evaluating G in these compounds, Luong et al [37] took TN (GdCu2)=41 K [7,39,40] In the coordinate system of b ¼ z; c ¼ x and a ¼ y; the orthorhombic CEF Hamiltonian of a CeCu2 type of structure is given by Eq (7) In the calculations, Luong et al [37] first used the two lowest-order terms in the CEF Hamiltonian Values for B02 and B22 were taken for TbCu2, DyCu2 and HoCu2 from Ref [19], ErCu2 from Ref [28] and TmCu2 from Ref [29] In TbCu2, DyCu2 and HoCu2 the magnetic moments lie along the a-axis, whereas in ErCu2 and TmCu2 the magnetic moments are oriented along the b-direction (see Ref [37] and references therein) The TN values for ErCu2 and TmCu2 were calculated directly using the CEF Hamiltonian (7) with only the two lowest-order terms For the RCu2 compounds with R=Tb, Dy and Ho we used the CEF Hamiltonian transformed in the new coordinate system of a ¼ z; b ¼ x; c ẳ y as follows [41]: HCEF ẳ 1=2ịB02 B22 ịO02 ỵ 1=2ị3B02 B22 ịO22 : 35ị The calculated values of TN are compared with experimental data in Table and also in Fig As one can see from Table and Fig 2, addition of CEF interaction enhances TN over the de Gennes values in the RCu2 compounds Moreover, calculations predict that TbCu2 has the highest Ne! el temperature, in good agreement with experiments The calculated values for the Ne! el temperatures across the series are in good agreement with the experimental ones Calculations gave the value of TN for HoCu2 somewhat higher than the one obtained from experiments Luong et al [37] tried also to derive the values for the Ne! el temperature of ErCu2 and TmCu2 with the full Bm n set taken from Refs [28,29], respectively The results of these calculations, using the full CEF Hamiltonian (7), are also shown in Table and Fig As it can be seen, the use of the full CEF Hamiltonian gives better results than the use of the two lowest-order N.H Luong / Physica B 319 (2002) 90–104 99 Table Values for the N!eel temperatures in the heavy RCu2 compounds R TN exp (K) TN cal (K) Gd 40 [7] 41 [39] 42 [40] Tb 54 [39] 53.5 [19] 48.5 [7] 48 [42] 46.4 Dy 24 [39] 31.4 [19] 26.7 [7] 27 [28] 28.7 Ho [39] 9.8 [19] 9.6 [7] 11 [43] 17.2 Er 11 [39] 13.5 [19] 11.5 [7] 9.1, 11.7a Tm 6.3 [44] 4.3, 6.7a a N!eel temperature predicted by the full CEF Hamiltonian with the Bm n sets from Refs [28,29] for ErCu2 and TmCu2, respectively CEF terms only Thus, the magnetic ordering temperatures in the RCu2 compounds can be explained by a combination of the RKKY interaction and CEF effects CEF effect in R2Fe14B compounds In the tetragonal R2Fe14B compounds, the rareearth sublattice consists of two inequivalent rareearth sites The anisotropy at room temperature is uniaxial in the compounds with rare-earth elements for which the Stevens factor aJ is negative, while it is planar in the compounds with rare-earth elements for which aJ is positive [45] In the compounds with Sm, Er, Tm, and Yb, for which aJ is positive, a competition between rareearth and iron sublattice anisotropies is expected and can lead to a spin reorientation Such Fig Comparison of experimental and calculated N!eel temperatures for the RCu2 compounds The open circles represent experimental data The solid circles (solid line) represent calculations using a CEF Hamiltonian with two lowest-order terms, and the solid squares represent calculations with the full CEF Hamiltonian as discussed in the text [37] The dashed line represents the de Gennes rule reorientations were observed in Er2Fe14B, Tm2Fe14B [46] and Yb2Fe14B [47] In Sm2Fe14B a spin reorientation has not been found Nd2Fe14B undergoes a different type of spin reorientation, namely from a conical to an axial arrangement with increasing temperature The temperature dependence of the rare-earth contribution to the magnetocrystalline anisotropy energy of the R2Fe14B compounds (R=Nd, Sm, Er, Tm, Yb) have been calculated [48,49] The results of these calculations in combination with the data on the iron sublattice anisotropy enable us to derive the spin-reorientation temperatures in R2Fe14B with R=Nd, Er, Tm, and Yb, and to show that a spin reorientation is not expected in Sm2Fe14B In the R2Fe14B compounds investigated the CEF parameter B02 is assumed to be dominant in the CEF Hamiltonian The value of B02 is considered here as a mean value for the two inequivalent rare-earth sites, as has been suggested by several groups [50,51] Thus, for calculating the rare-earth magnetocrystalline anisotropy we use the following 100 N.H Luong / Physica B 319 (2002) 90–104 Hamiltonian which is simplied from (26) as H ẳ B02 O02 ỵ gmB J Á Bm : ð36Þ The procedure of analysis has been described in Section The temperature dependence of the iron sublattice anisotropy, K1Fe ; is deduced from the magnetocrystalline anisotropy study of Y2Fe14B by Sagawa et al [52] Results of calculations are presented in Figs and In these figures we also plot the temperature dependence of the experimental erbium and thullium sublattice anisotropies, KlEr and KlTm respectively, which we derived from the study of Y2F14B [52] and of Er2Fe14B and Tm2Fe14B [46] From these figures, it is clear that in the R2Fe14B compounds (R=Er, Tm, Yb) the rare-earth and iron anisotropies are competing and lead to spin reorientations The deduced spin-reorientation temperatures are presented in Table The experimental and calculated values of TSR reported by several groups are also shown in Table By using a simple Hamiltonian (36) with only a second-order CEF parameter, B02 we could attain good fits to the experimental rare-earth anisotropy constants K1R for Sm, Er, and Tm Fig Temperature dependence of rare earth and iron anisotropy coefficients K1 in Tm2Fe14B and Yb2Fe14B K1Fe as in Fig For the experimental K1Tm ðTÞ curve see text - - - - experimental, —— calculated Table CEF coefficients Am n ; molecular-field parameters gmB Bm ; and spin-reorientation temperatures TSR in the R2Fe14B compounds A20 (K) A04 (K) gmB Bm (K) TSR (K) Fig Temperature dependence of erbium and iron anisotropy coefficients K1 in Er2Fe14B K1Fe has been taken from Ref [51] For the experimental K1Er ðTÞ curve see text - - - - - experimental, —— calculated Nd2Fe14B 175 Er2Fe14B 175 65 91.36 67 Tm2Fe14B 130 55.8 Yb2Fe14B 56 47.9 Sm2Fe14B 310 290 124 122 320 316 316 350 310 310 360 380 118 115 210 — Exp [49] Cal [49] Cal [48] Exp [49] Exp [46] Cal [53] Cal [48] Exp [46] Cal [53] Cal [54] Cal [48] Exp [47] Cal [53] compounds in a wide temperature range, and reproduce the correct values of the spin-reorientation temperatures in the compounds with R=Er, Tm and Yb The CEF coefficients A02 deduced N.H Luong / Physica B 319 (2002) 90–104 from the parameters B02 for the compounds investigated are presented in Table together with the molecular-field parameters As can be seen from this table, A02 in Tm2Fe14B is somewhat smaller than that in Er2Fe14B, while A02 in Yb2Fe14B is significantly small in comparison with that in Er2Fe14B Such a behavior of A02 in the R2Fe14B series is also reported by Cadogan et al [51] In the case of Nd, the following Hamiltonian was employed [49]: H¼ B02 O02 þ B04 O04 þ gmB J Á B m : ð37Þ From the values of B02 and Bex obtained for Er2Fe14B (see Table 3), we derived, after scaling to the Nd case, the corresponding values for B02 and the molecular-field parameter for Nd2Fe14B The value of B04 has been chosen in order to reproduce the observed spin-reorientation temperature in Nd2Fe14B The CEF coefficients A02 and A04 for this compound are presented in Table CEF effect in RFe11Ti compounds We have studied the effect of Y substitution on the spin reorientation in NdFe11Ti [55] by magnetization measurements on bulk and oriented powder samples, and by measurements of the AC susceptibility on bulk samples as a function of temperature The spin-reorientation temperatures, TSR, were determined No data for the temperature dependence of the magnetic structure are available The method of analysis, based on the expression for the anisotropy constant (Eq (29)), was applied to discuss the spin-reorientation phenomena in the Nd1ÀxYxFe11Ti compounds For calculating K1R according to Eq (29) we have chosen the set of coefficients Am n presented in Table The used molecular-field parameter gmB Bm value, expressed as 161 K, was derived from the value of nRT given by Hu et al [56] The temperature dependence of the ion sublattice anisotropy K1Fe ẳ K1T ị is taken from the study of YFe11Ti by the same authors [56] The calculated values of the spin-reorientation temperature are in good agreement with the experimental ones (see Refs [55,57] and Table 5) 101 Table Àn CEF coefficients Am n ; in unit of Ka0 ; for the R1ÀxYxFe11Ti compounds R A02 A04 A06 A44 A46 Nd Tb Dy À57 À54.1 À32 À14.3 À0.9 À9 0 3.2 0 105 0 Table Spin-reorientation temperature R1ÀxYxFe11Ti compounds TSR in the series of R x exp (K) TSR calc TSR (K) Nd Nd Nd Nd Tb Tb Tb Dy Dy Dy Dy Dy 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.0 0.2 0.4 0.6 0.8 187 140 114 70 325 235 120 195 171 141 107 73 180 152 120 75 328 235 70 196 172 142 110 76 For the Dy1ÀxYxFe11Ti compounds, TSR stands for the temperature of the spin reorientation of the axis-to-cone type and In the case of Tb1ÀxYxFe11Ti Dy1ÀxYxFe11Ti compounds, we have measured the temperature dependence of the cone angle y [58,59] The experimental results were analyzed based on a method of calculating the free energy (Eq (32)) In our analysis of the Tb1ÀxYxFe11Ti and Dy1ÀxYxFe11Ti data, the assumption is made that the exchange interaction stabilizes the ferrimagnetic collinearity between the magnetic moments of the transition-metal and rare-earth sublattices In fact, analyzing the data of a single-crystalline DyFe11Ti sample, Hu et al [60] have calculated the canting angle between the iron and dysprosium sublattice magnetizations These authors have reported that the maximum value of this canting angle in DyFe11Ti is about 2.51 for the intermediate values of the cone angle The CEF coefficients Am n used for fitting the experimental results for Tb1ÀxYxFe11Ti and Dy1ÀxYxFe11Ti compounds are presented in Table [61] Values 102 N.H Luong / Physica B 319 (2002) 90–104 for the molecular-field parameter gmB Bm are expressed as 128.7 and 85.6 K for Tb1ÀxYxFe11Ti and Dy1ÀxYxFe11Ti, respectively The temperature dependence of the iron sublattice anisotropy, K1Fe ; was again taken from the study of YFe11Ti [56] The calculated temperature dependencies of the angle y for the Dy1ÀxYxFe11Ti and Tb1ÀxYxFe11Ti compounds are shown together with the experimental ones in Figs and 6, respectively As can be seen from Fig 5, for the Dy1ÀxYxFe11Ti compounds, good agreement between calculations and experiments has been obtained The calculated values for the spin-reorientation temperature are in good agreement with those obtained from the experiments for all substituted compounds investigated (see Table 5) For the Tb1ÀxYxFe11Ti compounds the calculated magnetic structure is also in agreement with the experimental one (see Fig 6) The calculated values for the spin-reorientation temperature are basically in agreement with the experimental ones for the substituted compounds exhibiting a spin reorientation (see Ref [59] and Table 5) Inspection of Table shows that the leading CEF coefficient A02 obtained for the three investigated systems has the same sign and is of the same order of magnitude However, for reproducing the experimental results, higher-order CEF coefficients should also be included From an analysis of single-crystal data of DyFe11Ti, Hu et al [60] Fig Experimental and calculated thermal dependence of the cone angle y between the magnetization direction and the c-axis for the Dy1ÀxYxFe11Ti compounds [58] Fig Experimental (symbols) and calculated (full curves) thermal dependence of the cone angle y between the magnetization direction and the c-axis for the Tb1ÀxYxFe11Ti compounds [59] A04 ¼ À12:4 KaÀ4 deduced: A02 ¼ À32:3 KaÀ2 ; ; À4 À6 A4 ¼ 118 Ka0 ; A6 ¼ 2:56 Ka0 and A46 ¼ 0:64 KaÀ6 : We have used this set of CEF coefficients to study the spin-reorientation phenomena in the systems considered For the Nd1ÀxYxFe11Ti compounds, the calculated values for the spin-reorientation temperature, by using this set of Am n ; are somewhat different from those obtained in our experiments [55] Moreover, the calculations not predict a spin reorientation in the Y-substituted compounds with x > 4; which is not consistent with our experimental data The set of CEF coefficients given in Table predicts that a spin reorientation occurs in the compound with x ¼ 0:6 (in agreement with experiments) No spin reorientation is expected in the compound with x ¼ 0:8 since for this compound the iron anisotropy is dominant at all temperatures Measurements on Nd0.8Y0.2Fe11Ti support this expectation, indeed, see Ref [57] For Dy1ÀxYxFe11Ti compounds, calculations performed in the model described above, using the set of Am n values given by Hu et al [60], gave spinreorientation temperatures which are somewhat different from our experimental ones In addition, this set of Am n of Hu et al [60] does not reproduce the spin-reorientation behavior in TbFe11Ti Ivanova et al [61] have reported on the magnetic anisotropy and spin-reorientation transitions N.H Luong / Physica B 319 (2002) 90–104 103 personality The author would like to thank Prof T.D Hien, Prof N.P Thuy Dr L.T Tai and Dr P.H Quang for close collaboration throughout the course of this work The author wishes to thank Prof K Krop and Dr D Givord for valuable comments, discussions and help, Prof Y Onuki and Prof K Sugiyama for providing their work and discussions The author is grateful to all members of the Cryogenic Laboratory and Dr F.F Bekker for cooperation, Dr P.E Brommer for reading and valuable comments on the manuscript Fig Calculated temperature dependence of K1Tb (solid line) of the terbium sublattice in TbFe11Ti Experimental K1Fe values quoted from Hu et al [56] and experimental K1Tb (dashed line) values extracted from Ivanova et al [62] are also shown in TbFe11ÀxCoxTi single crystals These authors have derived the temperature dependence of K1 for TbFe11Ti From these results and the iron sublattice anisotropy K1Fe derived by Hu et al [56], we are able to separate out the magnetic anisotropy of the terbium sublattice K1Tb : The temperature dependence of K1Tb deduced from experiments in this manner is shown in Fig Using the set of CEF parameters in Table for TbFe11Ti, we have calculated the temperature dependence of K1Tb in this compound The calculated temperature dependence of K1Tb is also plotted in Fig The agreement between the calculated K1Tb ðTÞ curve and experimental data is good in a wide temperature range Our results show that the study of spinreorientation phenomena in intermetallic compounds with substitution of a non-magnetic rare earth for a magnetic one is very useful in obtaining information on CEF parameters This is particularly valuable when data on single crystal are lacking Acknowledgements The author is greatly indebted to Prof J.J.M Franse for all his support and encouragement for many years as well as throughout the course of this work The author deeply admires his physics and References [1] K.W.H Stevens, Proc Phys Soc A 65 (1952) 209 [2] M.T Hutchings, Solid State Phys 16 (1964) 227 [3] A.J Freeman, J.P Desclaux, J Magn Magn Mater 12 (1979) 11 ! 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It is shown that the study of spinreorientation phenomena in intermetallic compounds is very useful in obtaining information on the CEF parameters The crystalline-electric-field effect 2.1 General... can be explained by a combination of the RKKY interaction and CEF effects CEF effect in R2Fe14B compounds In the tetragonal R2Fe14B compounds, the rareearth sublattice consists of two inequivalent... the orientation of the total magnetization vector In this way, the temperature dependence of the magnetic structure is determined CEF effect in RCu2 compounds The RCu2 intermetallic compounds