Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 Ferrimagnet with magnetic instability: inÃuence of susceptibility on magnetization curves M.I. Bartashevich!, T. Goto!, I.S. Dubenko", N.P. Kolmakova#,*, S.A. Kolonogii#, R.Z. Levitin", M.Yu. Nekrasova# !Institute for Solid State Physics, University of Tokyo, Tokyo, Japan "Moscow State University, 119899 Moscow, Russia #Technical University, 241035 Bryansk, Russia Received 14 April 1998; received in revised form 30 October 1998 Abstract Magnetic phase diagrams of a two-sublattice ferrimagnet with an unstable subsystem are calculated when the susceptibility is taken into account. Analytical expressions for the critical Ãelds and characteristic values of parameters are derived. Evolution of the magnetic phase diagrams and magnetization curves is analyzed. The magnetization curves of the compounds Y 0.8 Ho 0.2 (Co 0.925 Al 0.075 ) 2 and Y 0.83 Er 0.17 (Co 0.92 Al 0.08 ) 2 are measured in Ãelds up to 100 T. Two metamagnetic transitions associated with the magnetic instability of the d-subsystem are observed in both compounds. The numerical analysis in the framework of the molecular-Ãeld theory provides a determining of the reliable fâd exchange parameters. ( 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 75.30. Kz; 75.50. Gg; 75.30. Cr Keywords: Ferrimagnets; Metamagnetic transitions; Magnetic phase diagrams; Ultra-high magnetic Ãelds; fâd exchange parameters 1. Introduction The phenomenon of magnetic instability â a Ãrst-order phase transition from a weakly (paramagnetic or weakly ferromagnetic) to a strongly ferromagnetic state when changing the external parameters (magnetic Ãeld, pressure, temperature) â has been intensively studied in recent years both in experimental and theoretical aspects. This phenomenon is investigated in detail in the intermetallic compounds of the YCo 2 type, where it is caused by the band structure peculiarities of magnetic d-electrons (see e.g. Ref. [1]). The magnetic instability in other intermetallics YCo 3 , Y 2 Co 7 , ThCo 5 , CeNi 5 , etc. is of analogous nature. As another source of magnetic instability may serve a crossover phenomenon, a crossing of energy levels of *Corresponding author. Fax: #7-0832-562408; e-mail:
[email protected]. 0304-8853/99/$ â see front matter ( 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 6 0 1 - 5 localized magnetic electrons results in a change of magnetic state. It is a possible reason of magnetic instability recently found in thulium subsystem in the TmCo 2 [2]. Of particular interest is an investigation of the magnets with two (or more) magnetic subsystems one of which is magnetically unstable. The case of a two-sublattice ferrimagnet with one magnetically unstable subsystem which experiences at the magnetic (external or eâective) Ãeld the Ãrst-order phase transition from a weakly to a strongly ferromagnetic state was theoretically considered in the paper [3]. All possible phase diagrams were constructed and all possible types of magnetization curves were presented. It was shown that in such ferrimagnets a magnetic Ãeld induces the Ãrst-order magnetic phase transitions connected with the change in a magnetic state of unstable subsystem along with the second-order phase transitions to non-collinear phases usual for ordinary ferrimagnets, and also an interference (superposition) of these two types of phase transitions. Hence, the magnetic phase diagrams together with the magnetization curves of such ferrimagnets are incomparably more complicated and interesting than those of the usual ferrimagnets, both magnetic sublattices of which are stable. The analytical part of theoretical investigation in Ref. [3] was performed when neglecting magnetic susceptibilities. Available experimental data meanwhile provide evid- ences for essential magnetic susceptibility of an unstable subsystem. Besides, a non-collinear phase is known not to appear if a susceptibility is large enough (see e.g. Ref. [4]). In this work the inÃuence of the susceptibility of the unstable subsystem, both in the weak and strong ferromagnetic states, on the magnetic phase diagrams (MPD) and magnetization curves is calculated in the framework of the molecular Ãeld theory for a two-sublattice ferrimagnet with one unstable subsystem. We have tried to answer the questions: how the magnetic susceptibility changes the topology of MPD and shape of magnetization curves and under what circumstances the magnetic susceptibility results in appearance (or disappearance) of magnetic phases. In addition, we present the experimental magnetization curves in Ãelds up to 100 T for the Laves phase intermetallic compounds YR(CoAl) 2 with R"Ho and Er, and the values of fâd exchange parameters determined from their numerical simulations. 2. Magnetic phase diagrams and magnetization curves when the susceptibility is taken into account We consider a two-sublattice ferrimagnet, of which one magnetic subsystem undergoes a metamagnetic transition at the Ãeld H . from a weak magnetic state (w) with spontaneous magnetization m and magnetic susceptibility s 8 , to a strong magnetic state (s) with initial magnetization M and magnetic susceptibility s 4 . M 1 "m#s 8 H %&& for H %&& (H . , M 1 "M#s 4 (H %&& !H . ) for H %&& âH . . (1) This magnetic subsystem experiences the eâective Ãeld due to a negative exchange interaction (the exchange parameter j 12 ): H %&& "H#j 12 M 2 , where M 2 is the magnetic moment of the second sublattice. The magnetization value M 2 is considered to be constant because it is saturated at low temperatures [3]. The thermodynamic potential of the ferrimagnet under consideration can be written as G"F!H(M 1 #M 2 )#jM 1 M 2 ; j"Dj 12 D, j 12 (0. (2) Here F is the thermodynamic potential of the unstable subsystem, which experiences the eâective Ãeld diâerent in the weak and strong magnetic states. We construct it using the relation F":M1 0 H %&& (M 1 ) dM 1 and the magnetization curve of the unstable sublattice (1). F 8 "s 8 H2 %&& /2"(M 1 !m)2/2s 8 , F S "!s 8 H2 . /2#H . (M!m)#s 4 (H2 %&& !H2 . )/2"!s 8 H2 . /2#H . (M!m)#(M 1 !M)2/2s 4 . (3) 126 M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 Let us Ãnd out what contribution the unstable sublattice susceptibility gives to the forming of MPDs. In line with [3] all possible states of the ferrimagnet under consideration are notated as follows: AW"ferrimagnetic weak phase; AS"ferrimagnetic strong phase; FW"ferromagnetic weak phase; FS"ferromagnetic strong phase; NW"non-collinear weak phase; NS"non-collinear strong phase. We Ãnd regions of existence for these phases and critical Ãelds by analyzing the values and signs of the eâective Ãelds acting on the sublattice magnetization and choosing the state with a minimum value of the thermodyn- amic potential. In the molecular Ãeld theory the sublattice magnetizations in a non-collinear phase are known not to depend on the external Ãeld (see e.g. Ref. [4]), and so the value of M 1 , and the energy of unstable sublattice which is determined by M 1 , in the non-collinear phases equal these quantities in the corresponding collinear phases at the appropriate critical Ãelds. In Ref. [3] the MPDs are presented in coordinates: magnetic Ãeld H, concentration of atoms of a stable subsystem t, keeping in mind that by varying the concentration t, i.e. the magnetic moment of the stable subsystem M 2 , one can change the eâective Ãeld acting on the unstable subsystem and realize diâerent parts of the MPDs. Depending on the value of exchange interaction, in these coordinates four MPDs topologically quite diâerent were obtained in Ref. [3]. The following questions should be answered when the susceptibility is taken into account: whether new MPDs appear, how criteria in the exchange parameter for MPDs classiÃcation change, how the phase transition Ãelds and characteristic values of t (or M 2 ) depend on the susceptibilities s 8 and s 4 . In the consideration we follow the terminology of Ref. [3], where the Ãxed parameters of the unstable subsystem are: the metamagnetic transition Ãeld H . , the spontaneous magnetization in the weak phase m and the magnetization in the strong phase at the metamagnetic Ãeld M. Depending on the magnitude of the exchange interaction the MPDs are characterized by a various sequence of initial states (at H"0) as the value of M 2 increases. So it is convenient to classify the MPDs using the magnitude of the exchange interaction as a criterium, as it was done in Ref. [3], and to consider the inÃuence of susceptibility separately for every characteristic interval of j. 2.1. Weak interaction In the case of a weak exchange interaction, when j(H . /M, (4) the MPD is shown in Fig. 1a. The metamagnetic transition Ãelds FW%FS (H .1 ) and AW%AS (H .2 ) do not depend on susceptibility and as before are equal to: H .1 "H . #jM 2 , H .2 "jM 2 !H . . (5) The second-order phase transition Ãelds AW%NW (H 18 ) and NW%FW (H 28 ) depend on s 8 as follows: H 18 "jK m 1!js 8 !M 2K, H28"jA m 1!js 8 #M 2B. (6) The compensation point for a weak magnetic state M 28 also depends on s 8 : M 28 " m 1!js 8 . (7) From Eqs. (5)â(7) one can see that as s 8 increases M 28 approaches M 2. "H . /j, (8) M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 127 Fig. 1. (a) Magnetic phase diagram of a ferrimagnet with an unstable subsystem in the case of a weak exchange interaction (j(H . /M). The Ãrst- and second-order phase transitions are indicated by dashed and solid lines, respectively. (1) H .1 ; (2) H 2. ; (3) H 18 ; (4) H .2 . The second-order phase transitions for large values of susceptibility are shown by dashedâdotted lines. The notations of the phases and formulas for the critical Ãelds and characteristic values of M 2 are given in the text. (b) Magnetization curves for M 2 âM 2. for zero (dashed lines) and non-zero (solid lines) values of susceptibility. H 28 approaches H .1 , and H 18 approaches H .2 . When the susceptibility is s 8 "1/j!m/H . , these points and lines would coincide, but this cannot happen because the value of s 8 is restricted by the condition of existence of a jump in the basic magnetization curve for an unstable sublattice: s w (s#3 8 "(M!m)/H . . (9) Critical Ãelds for s#3 8 are presented in Fig. 1a by dashedâdotted lines. The corresponding magnetization curves for M 2 âM 2. are presented in Fig. 1b. It is seen that MPDs and magnetization curves with regard for susceptibility for the case of a weak exchange interaction are qualitatively the same as they are when neglecting it, only H 18 and H 28 are changed. The situation of s 8 â1/j is prohibited by a more rigorous restriction (9), so NW is allowed to appear if mO0. 2.2. Intermediate exchange interaction If H . M (j( H. m#s 8 H . , (10) 128 M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 this is the case of an intermediate interaction, the compensation point for a strong magnetic state M 24 "M!s4H. 1!js 4 (11) appears at H"0. The MPD complicates essentially, has a diâerent form depending on the sign of the expression j (M!s 4 H . )2 1!js 4 !j m2 1!js 8 !2H . (M!m)#(s 8 #s 4 )H2 . , (12) and is presented in Fig. 2a and Fig. 3 for diâerent signs of this combination of parameters. The Ãelds of the second-order phase transitions AS%NS (H 14 ) and NS%FS (H 24 ) depend only on s 4 : H 14 "jKM2! M!s 4 H . 1!js 4 K, H24"jAM2# M!s 4 H . 1!js 4 B. (13) Fig. 2. (a) The same as in Fig. 1a for the case of an intermediate exchange interaction (jâH . /M, the expression (12) is negative). (5) H 24 ; (6) H 14 ; (7) H .3 ; (9) H .4 ; (10) H .6 ; (11) H .7 . The Ãrst-order phase transitions for large values of susceptibility are given by dotted lines. The rest as in Fig. 1a. (b) Magnetization curves for M 2. (M 2 (Mâ 26 for zero (dashed lines) and non-zero (solid lines) values of susceptibility. M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 129 Fig. 3. The same as in Fig. 1a for the case of an intermediate exchange interaction (expression (12) is positive, j(H . /(m#s 8 H . )). (8) H .5 . The rest is as in Fig. 1a and Fig. 2a. All the other critical Ãelds and characteristic points, new in comparison with the MPD of a weak exchange interaction, depend both on s 8 and s 4 . The Ãelds of the Ãrst order phase transitions FW%AS (H .3 ), AS%AW (H .4 ), NW%AS (H .5 ), FW%NS (H .6 ) and AW%NS (H .7 ) are described by the expressions: H .3 "H . #(2H.!jM!jm)M2#s4H.(H.!jM2)#s8(j2M22!H2.!2jM2H.)/2 M!m!2M 2 #s 8 jM 2 !s 4 (H . !jM 2 ) , H .4 "(H.!jM2)(M!m)!s4(H.#jM2)2/2!s8(H2.!j2M22)/2 M#m!2M 2 #s 8 jM 2 #s 4 (H . #jM 2 ) , H .5 "jA M!s 4 H . 1!js 4 !M 2B !G j 1!js 4 C!2H.(M!m)#j (M!s 4 H . )2 1!js 4 ! j 1!js 8 #(s w #s 4 )H2 .DH 1@2 , (14) H .6 "jAM2# m 4 1!js 8 B#G j 1!js 8 C2H.(M!m)!j (M!s 4 H . )2 1!js 4 # jm2 1!js 8 !(s 8 #s 4 )H2 .DH 1@2 , H .7 "jAM2! m 1!js 8 B!G j 1!js 8 C2H.(M!m)!j (M!s 4 H . )2 1!js 4 # jm2 1!js 8 !(s 8 #s 4 )H2 .DH 1@2 . Equations for H .3 and H .4 are written providing s 8 (1/j, (M!m 4 )/H . , s 4 (1/j. For all these phase transitions the magnetic moments of both sublattices change their orientation with respect to the external Ãeld, in contrast to the metamagnetic transitions at the Ãelds H .1 and H .2 (5), where only the magnitude of magnetic moment of the unstable sublattice changes. Critical magnitudes of M 2 in MPDs of Fig. 2a 130 M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 and Fig. 3 equal: MB 26 "1 2A M!s 4 H . 1!js 4 # m4 1!js 8 B $ 1 2jG j 1!js 8 K2H.(M!m4)!j (M!s 4 H . )2 1!js 4 # jm 2 4 1!js 8 !(s 8 #s 4 )H2 .KH 1@2 , (15) M$ 2$ "1 2A M!s 4 H . 1!js 4 ! m4 1!js 8 B ! 1 2jG j 1!js 8 K2H.(M!m4)!j (M!s 4 H . )2 1!js 4 # jm24 1!js 8 !(s 8 #s 4 )H2 .KH 1@2 . An example of magnetization curves for M 2. (M 2 (Mâ 26 for diâerent susceptibility values in the case of MPD of Fig. 2a is given in Fig. 2b. It is seen that increasing the susceptibility makes magnetization jumps less noticeable. Movement of the critical Ãelds as the s 8 and s 4 increase are shown in Fig. 2a and Fig. 3 by dashed-dotted lines for the second-order phase transitions and by dotted lines for the Ãrst-order phase transitions. So, for a Ãxed value of the exchange parameter jâH . /M, increasing the susceptibilities s 8 and s 4 up to certain values results in matching of the lines H 28 and H .6 , H 18 and H .7 and appearance of the line H .5 . The MPD changes from Fig. 2a to Fig. 3. This happens when the expression (12) changes its sign. When this expression equals zero, the Ãeld H .3 equals H . , and the Ãeld H .4 is described by the equation M 2 "M 2. . It should be pointed out that these equations are derived when analyzing the full expressions for H .3 , H .4 ; in formulas (14) only the main terms are presented. We emphasize that the change of sign of the expression (12) and the transformation of MPD from Fig. 2a to Fig. 3 were made possible by increasing both s 8 and s 4 . With a further rise of s 8 and s 4 the regions of existence of NW, FW and AW (between M 28 and M 2. ) decrease, as is shown in Fig. 3. These regions can disappear only if the exchange parameter increases, and not due to the susceptibility s 8 which is limited for an intermediate exchange interaction (see Eq. (10)) by the relation s 8 (1/j!m/H . . 2.3. Strong exchange interaction When jâH . /(m#s 8 H . ), the situation of a strong exchange interaction, the regions of NW, FW and AW (between M 28 and M 2. ) in Fig. 3 disappear and the phase transition AW%AS at H .2 (5) appears. The rest of MPD coincides with that presented in Fig. 3, with the same evolution of states for increasing s 4 . Hence, the susceptibility of unstable sublattice does not change the form of MPDs, if s 4 (1/j. Further- more, the classiÃcation of the situations with the weak, intermediate and strong exchange interactions does not depend on the susceptibility. This is understandable as far as a spectrum of initial states of the ferrimagnet for diâerent values of magnetic moment of a stable sublattice does not depend on the susceptibility. In the case of an intermediate exchange interaction, which MPD, shown in Fig. 2a or in Fig. 3, takes place depends strongly on the susceptibility. Going from one MPD to another one is made possible by increasing both s 8 and s 4 for a Ãxed magnitude of the exchange interaction. Both susceptibilities restrict the existence region of MPD presented in Fig. 2a. A large value of s 4 (more than 1/j; with keeping in mind that s 4 is limited by the state of saturation) changes qualitatively the MPDs for the strong interaction and the intermediate interaction with MPD in Fig. 3, NS does not appear and the ferrimagnet is magnetized at the expense of s 4 . We note that in the case of the compounds Y 1~t R t (Co 1~x Al x ) 2 the situation of s 4 â1/j is not realized for any R and x. The susceptibility s 4 is maximum for x"0.09, s 4 +0.014 l B /T (see Ref. [5]), the exchange parameter j is maximum when R is Gd, and 1/j+0.016 l B /T (see Ref. [6]), so s 4 (1/j. M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 131 A smearing of the metamagnetic transition in the unstable sublattice over some Ãeld interval (which may be caused by an inhomogeneity of a sample) results in a smearing over the same Ãeld interval of the metamagnetic transitions at H .1 and H .2 . As our numerical calculations show [6,7], a nonlinear magnetiz- ation curve of the unstable subsystem reproduces the appropriate nonlinear magnetization curves in the collinear phases of the ferrimagnet. Besides, if an initial part of the magnetization curve is nonlinear, i.e. s 8 Oconst(H), for certain values of exchange interaction, NW appears even when m"0. This is the case of the Y 1~t Ho t (Co 1~x Al x ) 2 compounds with 0.10)x(0.12. 3. Two-step-like magnetization curves of YR(CoAl)2, R"Ho and Er, in Ãelds up to 100 T The Laves phase intermetallic compounds YR(CoAl) 2 are fâd ferrimagnets, the itinerant d-subsystem of which is magnetically unstable. As was shown previously (see e.g. Ref. [3] and the previous section of this paper), the magnetic phase diagrams and magnetization curves of the ferrimagnets of this type are highly diversiÃed, depending strongly on the relation between the magnitude of the fâd exchange interaction and the parameters of the unstable d-subsystem (see Eqs. (4), (10) and (12)). In the compounds Y(Co 1~x Al x ) 2 the parameters H . , M, s 8 , s 4 depend on the aluminum concentration x [5]. With increasing x the metamagnetic Ãeld H . decreases from H . +70 T for x"0 to H . +14 T for x"0.11. The magnetic moment of the unstable d-subsystem at H"H . depends nonmonotonously on x and is maximum at x"0.07, M+1.1 l B . The nonmonotonous dependence of s 4 on x was discussed in Section 2. The susceptibility of the d-subsystem in a weak magnetic state increases monotonously when x increases: for x"0 s 8 +0.0056 l B /T, for x"0.11 s 8 +0.035 l B /T. An accurate determination of the fâd exchange parameters is a serious task since in interpreting the magnetic behavior of these complicated compounds a series of subtle questions arises as to whether a value of the fâ d exchange parameter depends on the substitution of Co for Al, how diâerent is the Co-subsystem in the Y-compounds and compounds with rare earths, etc. Answering questions like these is possible if a whole magnetization curve including all the possible phase transitions up to the terminal phase is available. From Fig. 1, Fig. 2a and Fig. 3 it is clear that MPD is more simple for the case of a weak exchange interaction. The fâd exchange interaction may be considered as a weak one in the Y 0.8 Ho 0.2 (Co 0.925 Al 0.075 ) 2 and Y 0.83 Er 0.17 (Co 0.92 Al 0.08 ) 2 compounds under investigation. Indeed, if we use the values of the exchange parameters evaluated in Ref. [8] from the Curie temperatures: j &â$ "!23 T ) f.u./l B in HoCo 2 and j &â$ "!17 T ) f.u./l B in ErCo 2 and the values of H . +30 T and M+1.03 l B in Y(Co 0.925 Al 0.075 ) 2 and H . +27 T and M+1.05 l B in Y(Co 0.92 Al 0.08 ) 2 , we obtain the fulÃllment of the condition (4) for the compounds investigated. With that instance we begin determining values of the fâd exchange interaction checking the procedure for both rare earth compounds. The alloy samples YR(CoAl) 2 were prepared by melting in argon atmosphere followed by homogenization annealing. High magnetic Ãelds up to 42 T were produced using a wire-wound pulse magnet with a duration time of 9 ms. Ultra-high magnetic Ãelds up to 100 T were generated by means of a fast capacitor discharge into a single turn coil with a 100 kJ capacitor bank. Duration time of the pulse Ãeld was about 7 ls. The magnetization was measured by an induction method with well-balanced pickup coils. Numerical simulations of the magnetization curves were performed in the molecular Ãeld approxi- mation analogously to our previous papers (see e.g. Refs. [6,7]). Magnetic properties of the unstable d-subsystem were described with the help of the experimental magnetization curves of the Y(Co 1~x Al x ) 2 compounds without magnetic rare earth [5], M%91 $ (H) or H%91(M $ ). This permits taking into account an actual character of metamagnetic transition and the d-subsystem susceptibility. We note that taking into account an adiabatic character of magnetization processes at pulsed Ãelds does not result in a noticeable modiÃcation of the magnetization curves, as the eâective Ãelds are large and Ãuctuations are suppressed. 132 M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 Fig. 4. Experimental (solid) and calculated (dashed) magnetization curves (lower panel) and diâerential magnetic susceptibility dM/dH (upper panel) for Y 0.8 Ho 0.2 (Co 0.925 Al 0.075 ) 2 at 4.2 K. Magnetization curves and diâerential susceptibilities dM/dH of the Y 0.8 Ho 0.2 (Co 0.925 Al 0.075 ) 2 and Y 0.83 Er 0.17 (Co 0.92 Al 0.08 ) 2 compounds are presented in Figs. 4 and 5, respectively. It is seen that both magnetization curves are characterized by two metamagnetic transitions, for the Er-compound the Ãrst metamagnetic transition beginning in a very weak Ãeld. Following the MPD of the previous section for a weak exchange interaction given in Fig. 1a and considering the case of m"0, since the system Y(Co 1~x Al x ) 2 is in paramagnetic state for x(0.12 at H"0, and hence vanishing NW, the magnetic behavior of the compounds under consideration can be successfully understood. The presence of two metamagnetic transitions for both compounds means that M 2 âM 2. (8); for Y 0.8 Ho 0.2 (Co 0.925 Al 0.075 ) 2 M 2 +2 l B , H . +29 T and for Y 0.83 Er 0.17 (Co 0.92 Al 0.08 ) 2 M 2 +1.53 l B , H . +27 T. At the Ãrst step the d-subsystem is demagnetized from a strong to a weak magnetic state, its magnetic moment being antiparallel to the external Ãeld. Thereafter the antiparallel orientation of the f- and d-magnetic moments changes for the parallel one through susceptibility s 8 of the d-subsystem, which equals 0.013 l B /T for the Ho-compound and 0.014 l B /T for the Er-compound. Then, a re-entrant magnetizing of the d-subsystem takes place for parallel orientation of the f- and d-magnetic moments. Thus the ferromagnetic state with the Co-subsystem in a strongly magnetic state serves as a terminal one. Figs. 4 and 5 show good accordance between the experimental magnetization curves and the theoretical ones calculated with the values of j &â$ :!22 T ) f.u./l B for the Ho-compound, and !18 T ) f.u./l B for the Er-compound. A rather good description of the position of metamagnetic transitions is also evident from upper panels of these Ãgures where the experimental and theoretical diâerential susceptibilities (dM/dH)(H) calculated from the appropriate magnetization curves are given. We note that our numerical calculations M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 133 Fig. 5. The same as in Fig. 4 for Y 0.83 Er 0.17 (Co 0.92 Al 0.08 ) 2 . show that in the Er-compound of the composition under investigation the Ãrst metamagnetic transition occurs in a very weak Ãeld, since M 2 +1.53 l B is close to M 2. +1.5 l B (see MPD in Fig. 1a). Besides the initial part of the magnetization curve has a very similar shape to the magnetization curve in the region of the second metamagnetic transition (compare with the theoretical magnetization curve in Fig. 1b). The above- listed values of the fâd exchange parameters in the compounds under consideration practically coincide with those evalulated in Ref. [8] from an analysis of Curie temperatures in the RCo 2 compounds. This fact may be considered as an argument in favor of the independence of the fâd exchange interaction from the substitu- tions in the Co- and R-subsystems in the RCo 2 -type compounds. Distinctions between experimental and theoretical curves consist in a diâerent steepness of metamagnetic transitions. They are less steep in experiment. This discrepancy may be ascribed, in particular, to anisotropy eâects. The rare earth anisotropy, which is not taken into account in our calculations, obviously, should smear the metamagnetic transitions. 4. Conclusions 1. The magnitude of the intersublattice exchange interaction serves as the most adequate parameter for the classiÃcation of magnetic phase diagrams by their topology in a two-sublattice ferrimagnet with an 134 M.I. Bartashevich et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 125â135 unstable subsystem. Four types of MPDs exist. The change of susceptibility, both s 8 and s 4 , results in the transformation of only two magnetic phase diagrams one into another. It is the case of an intermediate exchange interaction, these MPDs are presented in Fig. 2a and Fig. 3. 2. The susceptibility does not inÃuence the Ãelds of metamagnetic transitions for which the magnetic moments do not change their orientation, only the value of magnetization for an unstable sublattice changes. The Ãrst-order phase transitions shift when orientations of magnetic moments change and the second-order phase transitions between collinear and non-collinear phases shift with variations in susceptibility. 3. Shape of magnetization curves in the case of a weak exchange interaction does not change under variations of susceptibility. 4. In the case of the strong interaction and intermediate interaction with MPD in Fig. 3, if s 4 â1/j, the non-collinear strong phase does not appear. On the contrary, an increase of the weak state susceptibility, s 8 , within permissible limits does not give rise to the disappearance of the non-collinear weak phase. 5. Our observation of two-step-like metamagnetic curves for the case of weak exchange interactions permits the determining of the exchange parameters in the RCo 2 -type compounds with R"Ho and Er: j H0~C0 "!22 T ) f.u./l B and j E3~C0 "!18 T ) f.u./l B . Acknowledgements This work was supported, in part, by RBRF, project 96-02-16373 and SSS-96-15-96429 and INTAS- RBRF, project 95-641. References [1] R.Z. Levitin, A.S. Markosyan, Sov. Phys. Usp. 31 (1988) 730. [2] I.V. Golosovsky, B.E. Kvyatkovsky, S.V. Sharygin, I.S. Dubenko, R.Z. Levitin, A.S. Markosyan, E. Gratz, I. Mirebeau, I.N. Goncharenko, F. Bouree, J. Magn. Magn. Mater. 169 (1997) 123. [3] I.S. Dubenko, N.P. 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