Application of two-sublattice bilinearly coupled Heisenberg model to the description of certain ferrimagnetic materials
aa r X i v : . [ c ond - m a t . o t h e r] A ug Application of two-sublattice bilinearly coupled Heisenberg model tothe description of certain ferrimagnetic materials
Hassan Chamati [email protected] Diana V. Shopova [email protected] Corresponding authorAddress:Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia,BulgariaKeywords; ferrimagnetism, Landau theory, Heisenberg model, phase diagram.
PACS : 75.10.Dg, 71.70.Gm, 75.50.Gg
Abstract
We study phenomenologically on the basis of two bilinearly coupled Heisen-berg models the phase diagram of some ferrimagnetic substances. Calculationsare performed with the help of Landau energy obtained through applying theHubbard-Stratonovich transformation to the initial microscopic HeisenbergHamiltonian. The phase transitions within the model are of second order withthe emergence of a compensation point at lower temperatures for some valuesof parameters of the system. The main phase is a two-sublattice collinearferrimagnet but also a metastable non-collinear phase is present within theexchange approximation presented here. The numerical results give a detaileddescription of temperature dependence of magnetization on the strength of in-tersublattice interaction and the difference between the effective exchanges oftwo ferromagnetically ordered sublattices.
Ferrimagnets are substances made of various components having different magneticproperties. The differences in magnetic moments lead to a geometric frustration thatmay arise because either different elements occupy the lattice sites or the same ele-ment occupies nonequivalent crystallographic sites surrounded by a different numberor type of non-magnetic ions, which effectively results in different magnetic prop-erties. For complex alloys a combination of both may take place (For an extensivereview see Ref. [1] and references therein). Within mean-field approach it is gen-erally accepted that ferrimagnets can be modeled with the help of several interpen-etrating sublattices each ordered ferromagnetically with effective antiferromagneticcoupling between them. In the pioneering works of N´eel on ferrimagnetism within1he molecular field approach (see e.g. [2]), a two sublattice model is used to computethe thermal magnetization behaviour of ferrimagnets, and six possible magnetiza-tion curves were derived. Special attention there is paid to iron garnets, where thespontaneous magnetization in comparison with experiment can be interpreted byapplying a three-sublattice model. In view of experimental study of magnetocaloriceffect of rare-earth based ferrimagnets [3] which has great potential for technologi-cal applications in environmentally-friendly refrigeration, the theoretical mean-fielddescription of such alloys with three sublattices, is further elaborated [4]. Such stud-ies are based on considering microscopic classical Heisenberg models with differentexchange and spin-orbit interactions depending on the crystal structure, chemicalcomposition of the particular alloy under study.There is another theoretical mean field approach based on considering mixed spinIsing model for description of ferrimagnets, see for example [5, 6, 7], where a very de-tailed review of the literature on this approach is presented. In the present paper wewill consider ferrimagnets which can be described by different magnetic ions sittingon two interpenetrating sublattices in a body centred cubic structure. The interac-tion of ions on each sublattice is supposed ferromagnetic, while ions on the differentsublattices are coupled antiferromagnetically. The magnetic properties will be inves-tigated on the basis of bilinearly coupled Heisenberg classical model in a mean-fieldapproximation which is treated using the Hubbard-Stratonovich transformation forobtaining the respective Landau free energy and its analysis.The rest of the paper is organized as follows: in Section 2 we describe in detail howwe calculate the Landau free energy from classical Heisenberg model with competinginteractions on the basis of previously applied approach [8] for derivation of meanfield approximation. In Section 3 the solutions of equations of state obtained bythe minimization of Landau energy derived in Section 1 are discussed. Section 4summarizes the results both in strong and weak-coupling limit for ferrimagneticsubstances under consideration. Section 5 generalizes the conclusions and possiblefurther development of our study.
The microscopic Heisenberg Hamiltonian which describes two coupled subsystemsconsisting of classical spins with different magnetic properties which antiferromag-2etically between them through a bilinear term can be written in the following form: H = − N X ij h J (1) ij S (1) i · S (1) j + J (2) ij S (2) i · S (2) j + 2 K ij S (1) i · S (2) j i . (1)Here S (1 , j , are n -component classical Heisenberg spins whose magnitude is nor-malized on the unit sphere in spin space through the condition | S (1 , j | = 1. Theexchange parameters J (1 , ij , K ij in the general case are N × N symmetric matriceswith N - the number of lattice sites considered equal for both subsystems. Thiscondition simplifies the consideration as makes the system symmetric with respectto the interchange of the subsystems. The exchange matrices J (1 , ij denote the inter-action between magnetic atoms of the same sort and K ij - between magnetic atomsof different sorts.The above Hamiltonian may be applied to the description of magnetic systemswhich consist of two different magnetic materials and no matter what is the mi-croscopic origin of this difference, it is effectively described by different exchangeinteractions within the two subsystems. There may be other situation when thesubstance is made only of one type of magnetic ions but they occupy two differentcrystallographic positions in the Bravais lattice and are separated by a number ofnon-magnetic atoms. Such substance may also be considered as built of two mag-netic subsystems with different exchange interactions within them.In order to analyse the behaviour of magnetization and the phase transitions in sys-tems that can be described by the above microscopic Hamiltonian we have to findthe mean-field energy for the Hamiltonian (1) by calculating the partition functionwhich in this case is represented by functional integral in n-dimensional spin space,where n - is the number of spin components. To do this we apply the Hubabrd-Stratonovich transformation; see for example [10] and the papers cited therein. Wehave used this approach in [8] for ferromagnetic coupling between the two magneticsubsystems where a detailed description of procedure is given. Here we will justoutline the important steps in the derivation of Landau free energy , especially inrelation of antiferromagnetic coupling between the subsystems.We present the two interacting different magnetic subsystems on a body-centeredcrystal lattice, for which the corners of elementary cube are occupied by one sort ofmagnetic atoms, and at the center of the cube atoms of different magnetic sort arelocated. So the nearest neighbours belong to different magnetic subsystems and thenext-nearest neighbour to subsystems 1 and 2, respectively. Thus the system may bedescribed as two interpenetrating sublattices,consisting of different magnetic atomsand we assume that the interaction within the sublattices J , ij is ferromagnetic and3etween them, K ij , it is antiferromagnetic. The Hubbard-Stratonovich transfor-mation renders the initial microscopic Hamiltonian in new n-component variables Ψ (1) i , Ψ (2) i defined in real space, directly connected with the initial spins (see [8]),namely: H = 12 N X ij (cid:16) J (1) ij Ψ (1) i · Ψ (1) j + J (2) ij Ψ (2) i · Ψ (2) j + 2 K ij Ψ (1) i · Ψ (2) j (cid:17) (2) − ln " N X i I n/ − ( x (1) i )( x (1) i )2 ) − n/ Γ( n − ln " N X i I n/ − ( x (2) i )( x i − n/ Γ( n . Here I n/ − ( x (1 , i ) is the modified Bessel function, and Γ( n ) is the Gamma function.In the above expression the exchange parameters J (1 , ij and K ij are connected tothose in the initial Hamiltonian (1) by the relations: J (1 , ij = J (1 , ij TK ij = K ij T (3)with T - the temperature.We denote by x (1) i and x (2) i in (2) the following expressions: x (1) i = (cid:12)(cid:12)(cid:12) J (1) ij Ψ (1) j + K ij Ψ (2) j (cid:12)(cid:12)(cid:12) ; x (2) i = (cid:12)(cid:12)(cid:12) J (2) ij Ψ (2) j + K ij Ψ (1) j (cid:12)(cid:12)(cid:12) The terms containing Bessel functions in (2) will be further used only in the formof expansion with respect to x (1 , i up to forth order by using the relation:Γ( n x ) ( n/ − I n/ − ( x ) = 1 + ∞ X k =1 ( x / k k !( k + n/ k + n/ − ...n/ k -space , and pass to con-tinuum limit in k as the finite size effects will not be considered at this stage. Thequadratic part of the obtained Hamiltonian again contains a bilinear term with re-spect to Ψ (1 , ( k ) and we have to diagonalise it. This is done with the help of unitarymatrix ˆ S : ˆ S = S ( k ) − S ( k ) S ∗ ( k ) S ( k ) ! . (4)The eigenvalues of the matrix ˆ S read: λ , ( k ) = 12 h J ( k ) + J ( k ) ± p ( J ( k ) − J ( k )) + 4 K ( k ) i , (5)4here J , ( k ) and K ( k ) are the Fourier transforms of J (1 , ij and K ij , respectively. Inorder to compute the integral we use the steepest-descent method, i.e. the integra-tion contour is taken around the maxima of the eigenfunctions (5). The calculationfor bcc structure show that if we take the nearest neigbour interaction betweenatoms of the same sort and the nearest neighbour interaction between the atoms ofdifferent sort, λ , ( k ) has a maximum in the centre of the Brillouin zone that givesferromagnetic ordering for the sublattices with antiferromagnetic K < λ , ( k ) has a maximum also at the border of the Brillouin zone k = π/a ( a is thelattice constant) which supposes antiferromagnetically ordered sublattices. Theremay be also some local maximum inside the Brillouin zone, which may give someincommensurate ordering within the sublattices, but this case is beyond the scopeof the present study.After performing the reverse Fourier transform to real space we obtain the dimen-sionless Landau energy in the following form: FT = t −→ ψ + t −→ ψ + g h ( −→ ψ ) + ( −→ ψ ) i + b −→ ψ −→ ψ + b ( −→ ψ · −→ ψ ) + w ( −→ ψ − −→ ψ )( −→ ψ · −→ ψ ) . (6)The coefficients of the Landau energy are expressed by the components of the ma-trix (4) and its eigenvalues (5) for k = 0: S = 1 D (cid:16) J − J + p ( J − J ) + 4 K (cid:17) ,S = 2 KD (7)where D is introduced to satisfy the condition k ˆ S k = 1, namely S + S = 1: D = √ (cid:2) ( J − J ) + 4 K (cid:3) / h J − J + p ( J − J ) + 4 K i / . (8)We will write here the explicit expressions for the coefficients of landau energy as wewill need them further in solving the mean field equations and discussion of obtainedresults; t , = 1 λ , − n (9a) g = u S + S ) (9b) b = uS S (9c) w = u S S ( S − S ); (9d)5ere u = n ( n + 2) with n - the number of order parameter components. The realvector fields −→ ψ and −→ ψ in the Landau free energy (6) play the role of two coupledorder parameters, and the averaged sublattice magnetizations are related to themby the equations: −→ m = S λ −→ ψ − S λ −→ ψ (10a) −→ m = S λ −→ ψ + S λ −→ ψ (10b) The initial microscopic Hamiltonian is symmetric with respect to the rotation ofall spins through the same angle. The application of Hubbard-Stratonovich trans-formation for derivation of landau free energy , given in previous section, preservesthe symmetry of initial hamiltonian also with respect to field variables −→ ψ , whichmeans that we can find the magnitude and the mutual orientation between order pa-rameters −→ ψ , but not their orientation with respect to crystallographic axes. Thismay be done for particular magnetic substance by including in the initial microscopicHamiltonian terms accounting for the magnetic anisotropy. For pure exchange inter-actions we can introduce the following notations [8]: −→ ψ i = |−→ ψ | β i and −→ ψ i = |−→ ψ | δ i ,where |−→ ψ | = ψ , |−→ ψ | = ψ are the magnitudes of the vector fields, and β i , δ i arethe respective direction cosines, which fulfil the condition: X i =1 β i = 1 and X i =1 δ i = 1 . (11)The equations of state then will be: ∂f∂X i = 0 , where X i = { ψ , ψ , β i , δ i } (12)Solving the above equations with respect to direction cosines β i , δ i gives two possibleorientations between the vector fields −→ ψ , −→ ψ for K < P i β i δ i = −
1, that is, −→ ψ and −→ ψ are antiparallel, and2. The non-collinear phase with P i β i δ i = 0, that is, −→ ψ and −→ ψ are perpendicular.Below we will discuss in detail the non-collinear phase 2. The angle between theorder parameters −→ ψ and −→ ψ , i.e. ,is; X i β δ i = − w ( ψ − ψ )2 bψ ψ ψ = 0 and ψ = 0. For K <
0, the analysis showsthat the non-collinear phase exists only when the order parameters −→ ψ and −→ ψ areof equal magnitudes, meaning that the order parameters −→ ψ and −→ ψ are mutuallyperpendicular. The magnitude ψ = ψ = ψ for the non-collinear phase in analyticalform reads: ψ = − t + t u . (13)Then the sublattice magnetization magnitudes calculated using the above expres-sions for the non-collinear phase will be: |−→ m | = ψ s S λ + S λ , (14a) |−→ m | = ψ s S λ + S λ . (14b)Note that the sublattice magnetizations are not perpendicular but form an angle ∠ ( −→ m , −→ m ) = γ with each other, expressed bycos( γ ) = S S ( λ − λ ) p ( S λ + S λ )( S λ + S λ ) . The calculations show that this non-collinear phase for
K < K between the sublattices.When K > i.e. , the interaction between the sublattices is ferromagnetic there issmall domain in which the respective non-collinear phase is stable [9].For antiparallel −→ ψ and −→ ψ it is obvious that the sublattice magnetizations (10) willbe also antiparallel. We may write the resulting equations for the magnitudes of theorder parameters ψ and ψ of the collinear phase and K < t ψ + gψ + 3 bψ ψ − wψ ( ψ − ψ ) = 0 , (15a) t ψ + gψ + 3 bψ ψ − wψ (3 ψ − ψ ) = 0 , (15b)with the stability conditions given by: t + 3 gψ + 3 bψ + 3 wψ ψ > t + 3 gψ + 3 bψ + 6 wψ ψ )( t + 3 gψ + 3 bψ − wψ ψ ) − w ( ψ − ψ ) + 2 bψ ψ ] ≥ J , J and K . When J < | K | , J < | K | ,7he leading interaction is determined by the antiferromagnetic coupling between thetwo sublattices. This may be called a strong coupling limit for which the eigenvalue λ ( k = 0) λ = 12 h J + J − p ( J − J ) + 4 K i , (18)becomes negative. This is equivalent to the inequality K − J J >
0. The coefficient t in front of ψ becomes positive; see (9), and the field −→ ψ becomes redundant. TheLandau free energy (6) will be:( FT ) s = f s = t −→ ψ + g −→ ψ ) (19)The minimization of above equation gives for −→ ψ the solution:( −→ ψ ) = − t g (20)which exists and is stable for t < −→ m = S λ −→ ψ (21) −→ m = S λ −→ ψ will be antiparallel as S ∼ K/D and
K <
0. The phase described by the aboveequations will be presented by two antiparallel sublattices with different magnitudesof sublattice magnetizations.In the weak coupling limit for antiparallel configuration, i.e. , when J > | K | , J > | K | , or equivalently J J > K , the system of equations (15), together with thestability conditions (18), (19) should be solved. This can be done numerically andthe results will be presented in the next section. The analytical result for sublattice magnetizations in the limiting case of strongcoupling (21) gives for the magnitude of total magnetization |−→ M | = |−→ m + −→ m | thefollowing expression: |−→ M | = S | ψ | λ (cid:12)(cid:12)(cid:12)(cid:12) S S (cid:12)(cid:12)(cid:12)(cid:12) with | ψ | , given by (20). The phase transition is obviously of second order andthe total magnetization behaviour with temperature is smooth resembling the one8f Weiss ferromagnet with the exception that no saturation is reached for T = 0.According to the Neel’s classification of ferrimagnets, see [2], the change of magneti-zation with temperature in the strong coupling limit falls within R-type curve. Forexample, similar curve is obtained theoretically and compared with the respectiveexperiment for Y Fe O [4] where two sublattice model with strong antiferromag-netic coupling is considered.In the limiting case when J = J = J the relation will hold S = − S = 1 / √ −→ m = −−→ m will appear, only if −→ ψ ≡ |−→ ψ | = − t /u . The transition temperature for antiferromagnetic ordering will begiven by: t ac = ( J + | K | ) /n .Further we will present the numerical results for the temperature dependence ofsublattice magnetizations and the total magnetization of the system in the weak-coupling limit which we define here in the following way: J > |K| and J > |K| .Such a situation is present, for example in some ferrimagnetic compounds likeGdCo B [11]. It is experimentally found that the exchange constants within sub-lattices are ferromagnetic and larger than the antiferromagnetic coupling betweenthe sublattices; moreover there the magnetic anisotropy is small.Experiments for some R-T compounds where R is a rare earth element and T isa transition element, show that the exchange in the transition metal sublattice isleading in magnitude, while the exchange in the rare-earth ion sublattice can besafely ignored and considered as negligible. The intersublattice interaction is alsosmall see, for example, DyFe Al [12], ErFe TiH [13], RCo (R = Tb and Gd and R= Er, Ho, and Dy) [14]. In our notations the relation between the exchange integralsin this case will be J > |K| ≫ J , so this does not fall into our assumption of weakcoupling and will not be considered here.In order to solve numerically the equations of state (15) for weak coupling betweensublattices we introduce the following dimensionless parameters. t = T J + J , (22a) α = J − J J + J , (22b) β = KJ + J , (22c)with t – the dimensionless temperature. In the above expression we have supposedthat J > J which in view of symmetry in interchanging the sublattices does notlimit the consideration; then α > β < K < J > |K| and J > |K| may beexpressed by the parameters from (22) by the relation: α + β < . The parameter α is a measure for the difference in exchange parameters of the twosublattices and by its definition 0 < α < −→ M = −→ m + −→ m = S + S λ −→ ψ + S − S λ −→ ψ . (23)Hereafter we will use the following notations for magnitudes of sublattice magneti-zations and total magnetization both in the text and in figures: m = |−→ m | ; m = |−→ m | ; and M = |−→ M | The calculations show that the phase transition to ordered ferrimagnetic state occursat temperature : t c = 16 (1 + p α + β )which grows when either the difference between the exchange interactions in sublat-tices grows, or when the antiferromagnetic coupling is bigger, or both. The phasetransition from disordered to ordered phase is of second order.We want to note that within the exchange approximation used here for the regime ofweak coupling defined above with the decrease of temperature a compensation pointappears no matter how small is the difference between the exchange interactions ofsublattices. At the compensation temperature t comp , the sublattice magnetizations −→ m and −→ m are equal in magnitude and antiparallel, so M = 0. The relation betweenthe order parameters magnitudes there is defined by: ψ = λ λ ( S − S )( S + S ) ψ . (24)As the calculations show ψ < ψ for all values of α and β , but ψ grows with thedecrease of temperature in a monotonic way, while ψ grows more rapidly. Thequantity λ λ ( S − S )( S + S ) = (6 t c ) − α − β p α + β − βα ! is always > β < ∼ K and α > t comp < t c ,and values of ψ , ψ the condition (24) is fulfilled. In the following figure weshow the change of net magnetization magnitude M ( t ) with the temperature for10 .000.020.040.060.00 0.05 0.10 0.15 0.20 a = 0.1 M t b = - 0.08 b = - 0.11 b = - 0.13 b = - 0.20 Figure 1: The dependence of net magnetization M on reduced temperature t for forfixed α and the different values of antiferromagnetic coupling β . α = 0 . i.e. , J = 0 . J and different values of β . It is seen from Fig.1 thatthe increase of antiferromagnetic coupling between the sublattices slightly shifts thecompensation temperature to higher values, and M ( t ) grows more rapidly belowthe compensation temperature and reaches higher values as t −→
0. We supposethat within the exchange approximation and in weak coupling limit the key factorfor the compensation point to appear is the weakness of antiferromagnetic exchangebetween the sublattices compared to the ferromagnetic exchange of sublattices 1and 2 , respectively.We will discuss in more detail the influence of difference between the magnitudesof exchange interaction in the sublattices, represented by the parameter α on M ( t )and sublattice magnetizations m , and m . For α = 0 . i.e. , J = 0 . J , M ( t ), m , and m are shown in Fig. 2. At t c the transition is of second order and whenlowering the temperature a compensation point t comp appears, which is located closeto t c . Sublattice magnetizations change with temperature in monotonic way, and inthe temperature interval t comp < t < t c , the relation between sublattice magnetiza-tions is m > m , as expected as in sublattice 1 the exchange interaction J > J .Below t comp the magnetization of weaker sublattice m becomes bigger than m .For intermediate values of α = 0 . , or J = 0 . J , see Fig.3, with decrease of11 .000.050.100.150.200.250.00 0.05 0.10 0.15 0.20 a = 0.08, b = -0.3 t c t comp. t m m M Figure 2: The dependence of net magnetization M and sublattice magnetizations m and m on reduced temperature t for for small difference between sublatticeexchange parameters.temperature below the compensation point the total magnetization rapidly grows innon-monotonic way as t →
0, similar to V-curve according to Neel’s classification.The behaviour of sublattice magnetizations with temperature is quite different; m , which is the sublattice magnetization with stronger exchange interaction J growsin smooth way with decrease of temperature, while m for weaker sublattice inter-action J , below compensation point grows drastically in non-monotonic way andin the temperature interval 0 < t < t comp , m ≫ m .Such behaviour is described in detail in [15] for ferrimagnets with compensationpoint for many experimentally observed substances. There explanation of M ( t )behaviour below compensation point is done by introducing the notion of weak sub-lattice where depending on the particular substance considered different mechanismsfor explanation of this effect are pointed out. Within our exchange model the effectof weak sublattice is readily seen and mainly depends on the difference J − J ∼ α .When α further grows the compensation point is shifted to lower temperatures andthe magnetization m of the stronger sublattice decreases for t →
0. This is illus-trated in Fig.4 for J = 0 . J . 12 .00.10.20.30.40.00 0.05 0.10 0.15 0.20 0.25 a = 0.4, b = -0.095 t c t comp. t m m M Figure 3: The dependence of net magnetization M and sublattice magnetizations m and m on reduced temperature t for intermediate difference between sublatticeexchange parameters.A qualitatively similar behaviour of the total magnetization can be seen in theexperiments with ErFe [16] and GdCo B [11] although direct comparison with theexperimental curves can hardly be made, as these substances have crystallographicand magnetic structure that differs from the one assumed within our model.The influence of parameter α is summarized in the next figure, see Fig. 5, wherethe net magnetization is displayed for small values of β = 0 .
08 and different valuesof parameter α . As the difference between the magnitudes of sublattice exchangeinteractions grow the transition temperature is shifted to higher values as expectedand the compensation temperature is lowered. Another effect is the change of netmagnetization behaviour below t comp from monotonic to nearly exponential when α increases. Our mean-field analysis of this relatively simple model with competing interactionson a bcc lattice shows that the behaviour of net magnetization of the two-sublatticeferrimagnet depends essentially on the difference in magnetic interactions between13 .00.20.40.60.80.00 0.05 0.10 0.15 0.20 0.25 0.30 a = 0.7, b = -0.1 t c t m m M t comp. Figure 4: Illustration of the dependence of net magnetization M and sublatticemagnetizations m and m on reduced temperature t when the exchange in sublattice2 is very small compared to sublattice 1.the sublattices within the weak coupling limit presented here. The influence of anti-ferromagnetic coupling is more prominent when its magnitude is of the same order,or larger than the difference between the exchange parameters of the sublattices.The model may be generalized to include the expansion of the effective Hamiltonianup to sixth-order terms in x i , i = 1 , J > |K| ≫ J , that is, when one of thesublattices is very weak. 14 .00.20.40.60.00 0.05 0.10 0.15 0.20 0.25 0.30 b = - 0.08 M t a = 0.1 a = 0.4 a = 0.6 a = 0.7 Figure 5: The dependence of net magnetization M for fixed antiferromagnetic ex-change β and growing difference α between the ferromagnetic exchanges of the twosublattices. Acknowledgments
This work was supported by the Bulgarian National Science Fund under contractDN08/18 (14.12.2017).
References [1] R. Skomski, Simple models of magnetism, OxfordGraduate Texts, Oxford University Press, NY, 2008. doi:10.1093/acprof:oso/9780198570752.001.0001 .[2] L. N´eel, Magnetism and Local Molecular Field, Science 174 (1971) 985. doi:10.1126/science.174.4013.985 .[3] A. S. Andreenko, K. P. Belov, S. A. Nikitin, A. M. Tishin, Magnetocaloriceffects in rare-earth magnetic materials, Sov. Phys. Usp. 32 (8) (1989) 649. doi:10.1070/PU1989v032n08ABEH002745 .154] P. J. von Ranke, B. P. Alho, E. J. R. Plaza, A. M. G. Carvalho, V. S. R.de Sousa, N. A. de Oliveira, Theoretical investigation on the magnetocaloriceffect in garnets R Fe O where (R = Y and Dy), J. Appl. Phys. 106 (2009)053914. doi:10.1063/1.3213383 .[5] T. Kaneyoshi, J. Chen, Mean-field analysis of a ferrimagnetic mixed spinsystem, Journal of Magnetism and Magnetic Materials 98 (1991) 201. doi:10.1016/0304-8853(91)90444-F .[6] J. W. Tucker, The ferrimagnetic mixed spin- and spin-1 Ising system, J. Magn.Magn. Mater. 195 (1999) 733. doi:10.1016/S0304-8853(99)00302-9 .[7] F. Abubrig, Magnetic properties of a mixed-Spin-3/2 and spin-2 Ising ferri-magnetic system in an applied longitudinal magnetic field, World J. Condens.Matter Phys. 03 (2013) 111. doi:10.4236/wjcmp.2013.32018 .[8] D. V. Shopova, T. L. Boyadjiev, Mean field analysis of two coupled Heisenbergmodels, J. Phys. Stud. 5 (2001) 341[9] D. V. Shopova and T. L. Boyadjiev, The existence of a stable noncollinear phasein Heisenberg model with complex structure, Phys. Lett. A 311 (2003) 438[10] R. Brout, Phys. Rep.10 , 1 (1974)[11] O. Isnard, Y. Skourski, L. V. B. Diop, Z. Arnold, A. V. Andreev, J. Wosnitza,A. Iwasa, A. Kondo, A. Matsuo, K. Kindo, High magnetic field study of theGd-Co exchange interactions in GdCo B , J. Appl. Phys. 111 (2012) 093916. doi:10.1063/1.4710995 .[12] D. Gorbunov, A. Andreev, N. Mushnikov, Magnetic propertiesof a DyFe Al single crystal, J. Alloys Comp. 514 (2012) 120. doi:10.1016/j.jallcom.2011.11.020 .[13] N. V. Kostyuchenko, A. K. Zvezdin, E. A. Tereshina, Y. Skourski, M. Do-err, H. Drulis, I. A. Pelevin, I. S. Tereshina, High-field magnetic behavior andforced-ferromagnetic state in an ErFe TiH single crystal, Phys. Rev. B 92,104423 (2015) doi:10.1103/PhysRevB.92.104423 .[14] E. Z. Valiev, A. E. Teplykh, Magnetic properties of RCo compounds in theexchange-striction model of ferrimagnets, Phys. Metals Metallogr. 118 (2017)21. doi:10.1134/S0031918X16120164 .1615] Belov K P, Ferrimagnets with a ’weak’ magnetic sublattice, Phys. Usp. 39(1996)623634[16] I. Chaaba, S. Othmani, S. Haj-Khlifa, P. de Rango, D. Fruchart,W. Cheikhrouhou-Koubaa, A. Cheikhrouhou, Magnetic and magnetocaloricproperties of Er(Co − x Fe x ) intermetallic compounds, J. Magn. Magn. Mater.439 (2017) 269. doi:10.1016/j.jmmm.2017.05.033doi:10.1016/j.jmmm.2017.05.033