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The influence of the local Co anisotropy on ferromagnetic resonance in YCo3 and Y2Co17
Journal of Magnetism and Magnetic Materials 62 (1986) 205-208 North-Holland, Amsterdam
205
THE INFLUENCE OF THE LOCAL Co ANISOTROPY ON FERROMAGNETIC RESONANCE IN YCo3 AND Y2Co17 * K. TUREK and A. KOLODZIEJCZYK Departement of Solid State Physics IM, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Cracow, Poland Received 7 March 1986; in revised form 20 August 1986
It is shown that the difference in the results of the macroscopic anisotropy field measurements in YCo 3 and Y2C017 and these obtained by FMR technique can be explained by the local Co anisotropy. In spite of the ferromagnetic ordering of the magnetic moments in these compounds the magnetic resonance in the microwave region cannot be treated as the classical ferromagnetic one. The simplified resonant condition is effected by the anisotropy in other ways than that for an ordinary ferromagnetic material with one magnetic sublattice.
1. Introduction The intermetallic compounds YCo3 and Y2Co17 were investigated in a number of papers [1-14]. They both have a rather large anisotropy energy. The anisotropy fields of these compounds measured by FMR technique, HAF, are much lower than the HAMextracted from magnetisation curves [14-16] which is shown in table 1. The differences in HAF and HAM were observed in Tb and Dy by Vigren and Liu [17] and were explained by the magnetoelastic effects in the so-called "frozen lattice" model. Nevertheless, in the case of Tb and Dy the anisotropy field HAF is higher than HAM in contradictiction to YCo3 and Y2Co17 where HAF is lower than HAM. Brooks and Egami [18] showed that for cylindrical symmetry
* This paper was sponsored by the Physics Institute of Polish Academy of Sciences.
Table 1 The anisotrpy fields HAM and HAF for YCo 3 and Y2Co17 Compound
HAM (10 6 A / m )
H~ (10 6 A / m )
YCo 3 Y2Co17
1.17 0.66
0.45-0.75 0.46
the macroscopic anisotropy constants influenced by the frozen lattice effect must always be lower than these constants which are measured by the resonance method in the easy direction. This means that this effect cannot be responsible for the significant differences in HAF and HAMof YCo3 and of Y2C017, which have both uniaxial symmetry. In this paper we show that this difference can be explained by the local Co anisotropy of these compounds.
2. The crystalline and magnetic structure of YCo3 and Y2C01~ The investigated compounds have complicated crystalline structures which are reflected in their interesting magnetic properties. YCo3 has the elementary unit cell of PuNi 3 type which belongs to the R3m space symmetryl group. There are three different crystallographic ;positions of Co ions in such cell. Three of them are in the b, six in the c and eighteen in the h position in Wyckhoff's notation. Y2C017 crystallizes in the rhombohedral or hexagonal structure. Our samples have proved to have the rhombohedral symmetry belonging to P63/mmc symmetry group with the elementary cell of Th2Nil7 type. The Co ions in the cell occupy four different sites. Nine ions are in the d,
0304-8853/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
206
K. Turek, A. Kolodziejczyk / Ferromagneticresonancein YCo compounds
eighteen in the f, eighteen in the h and six in the e position. Kakol et al. [13] calculated the Co sublattice contributions to the net magnetocrystalline anisotropy in the compounds of Y - C o system using the point charge model with account on shielding effects. According to these calculations, YCo 3 has two sublattices (6c and 18h) with positive and one (3b) with negative anisotropy energy. In the case of Y2Co17 three Co sites (9d, 18h and 6e) have negative and one (180 positive anisotropy energy. The net anisotropy energy of YCo 3 is positive and that of Y2Co17 is negative. This means that, neglecting intersublattice exchange interactions, the magnetisations of these sublattices would be orientated perpendicularly one to another, along the c axis and in the basal plane. Nevertheless, because of the strong exchange coupling between the sublattices all magnetic moments are aligned along the c axis in YCo 3 and in the basal plane in Y2Co17. The materials having the magnetic structure like YCo 3 and Y2Co17 we will call "orthogonal anisotropic ferromagnets".
3. Ferromagnetic resonance in orthogonai anisotropic ferromagnets
anisotropy field /'/A~ y of the sublattice P2 rotates in the plane together with the projection of magnetisation which precesses about the c axis with a frequency to and then:
tt~Y = 2K12m2x e i~°t.~ + 2K12m2y
p,
(1)
where K12 is the first anisotropy constant of the sublattice P2, M20 is the static component of the magnetisation of this sublattice, and m2x and m2y are the dynamic components of the magnetisation in x and y directions induced by the microwave power. The anisotropy field of the first sublattice can be written in the form: 2Kit ^ H A 1 - M, ° z,
(2)
where Kit is the first anisotropy constant of the sublattice P~ and M~0 is the static component of the magnetisation of this sublattice. Then the classical equations of motion of magnetisation for both sublattices with the additional anisotropy terms are:
dMa dt
dM 2 Let us consider the orthogonal anisotropic ferromagnet with two sublattices P1 and P2, the first with the easy axis and the second with the plane of the easy magnetisation perpendicular to this axis. Let us assume that the anisotropy energy of the sublattice P1 is much higher than the anisotropy energy of the sublattice P2 but the exchange coupling between the magnetisations of these sublattices is large enough to pull out the magnetisation of the sublattice P2 from the plane and to keep it parallel to the magnetisation of the sublattice P r Additionally, to simplify the calculations let us assume that the difference between the g factors of both sublattices is negligible small, so that g~ = g2 = g and that the conductivity and the damping terms can be neglected in the equation of motion of magnetisation. Let us apply a dc field along the c axis. Because of the rotational symmetry of anisotropy in the basal plane we are allowed to assume that the
e iwt
dt
= #°yMI × ( H + HA1 + lM2),
(3a)
= #°7M2 × ( H + H~, y + IM1),
(3b)
where M 1 and M 2 are the total magnetisations of the sublattices P1 and P2, respectively, H is the sum of the applied de field, the demagnetising field and the dynamic magnetic component of the rf field h and l is the molecular field coefficient. Rewriting eqs. (3) in the component form and introducing the new complex variables h~ = hnx - ihny and rn~ = rnnx - im~y(n = 1,2) we obtain:
(-- t° + Ha + HAI + I M 2 0 ) m l - I M l o m 2 #oY
(4a)
= Mloh{,
-'M2om{ + (= M20h2 , .
to + H a - HA2 + IMlo)m2
/x0Y
(4b)
where H a is the static component of the field H and HA2 = 2K12/M2o. To find the resonant condi-
207
K. Turek, A. Kolodziejczyk / Ferromagnetic resonance in YCo compounds
tion the determinant of the coefficients in eqs. (4) is to be equal to zero. As a result we get the expression: H = Ha + ½ H A +
Y
½IMo
(5) where H A = HAt -- HA2 is the net anisotropy field, M 0 -- M10 + M20 is the net magnetisation and a = (HA1 + HA2)/IM o. From the two eqs. (5) only the equation with the minus sign has its solution in the microwave area. From the fact that in the investigated compound (Mlo - M20) < (H^I + HA2) it follows that 2
(M 0 -- M20) < Or2.
Y
= Ha + ½HA - ½a(M~o - M2o ).
+/-/,,:) (7)
In this formula the third term at the right side is two orders and the fourth three orders of magnitude less than the first and second terms. So, the resonant condition (7) can be simplified within a good approximation to the form: w = Ha + ½HA. Y
H-y = Ha + HA .
(8)
It follows from this formula that in the orthogonal anisotropic ferromagnet the resonant field is shifted by only half of the anisotropy field towards lower fields in comparison with an ordinary ferromagnet with one magnetic sublattice, for
(9)
Thus, the net anisotropy field calculated from eq. (9) is only half of the true anisotropy field H A, which is measured statically, whereas the anisotropy calculated from eq. (8) compared nicely with the static value. To get a better physical insight into our model let us calculate the ratio of the dynamical components of the magnetisations mi- and m~. For this purpose let us put h~ =khi- where k is some unknown coefficient, and then solve the linear eqs. (4). As a result we find relations between mi-, mE and ha: m1=
(6)
In the case of YC% g#BH~x = 160 K, where Hex is the exchange field and g/%Hex = 500 K for Y2Co17, Mxo + 3420 = 1.87 X l0 s A / m for YCo 3 and 10 6 A / m for Y2COxT, H^I + HA2 > 11.7 × l0 s and 6.6 × l0 s A / m , respectively [13]. Using these values we get a -~ 10- 3 for these compounds. Owing to the inequality (6) we can expand eq. (5) to the series with the accuracy to the first order term, "
which the resonant condition is
_~+H #0T
kM2o
a+H^l
+l(kM~o + M2oMlo)] h i / D ,
[(o -
#oY
)
(lOa)
i
+H.-HA2
mxo (10b)
where D is the determinant of the matrix of coefficients of mi- and m~ in eqs. (4). In the linear approximation we assume that mi- and m~ are small and then the coefficients of i in eqs. (10) have to be small also. Let us assume that they are equal to zero. Then it follows from eqs. (10) that
k = -Mlo/M2o and m1
m2
= (H^I + HA2)/(HAi + 3HA2 ) < 1.
(11)
If we t;~ke into account that M20 < M10 then the cone of precession of the magnetisation M 2 is more obtuse than that of M 1 as is shown in fig. 1. Thus, in the case of an orhogonal anisotropic ferromagnet we have quite another mode of precession than in the case of the ordinary F M R in which the net magnetisation precesses about the effective field contributed by the net anisotropy field. That is why the net anisotropy field in eq. (8) contributes to the effective field in a way other than in the resonant condition (9) for the ferromagnet.
208
K. Turek, A. Kolodziejczyk / Ferromagnetic resonance in YCo compounds
\
/
at least qualitatively. Y2Co17 has the easy plane of magnetisation and the outlined above model for the easy axis of mangetisation cannot be strictly used for this compound. W e can only suggest that the difference in H ~ and H f f in Y2Co17 has the same origin as in Y C o 3.
References
Fig. 1. The cones of precession of the magnetisations M] and M 2. The dynamical components mi- and m2 are enlarged with respect t o m 1 and M e.
4. Conclusions The resonant condition (8) is limited to the ferromagnetic, n o n c o n d u c t i n g sphere. The values of H ~ in table 1 were extracted from the F M R records in the powdered magnetic aligned samples using the model of the so-called parallel geometry [15, 16]. Nevertheless, we expect that also in the case of the parallel geometry only a part of anisotropy field H A contributes to the effective magnetic field in the resonant condition due to the orthogonal anisotropic ferromagnetism. The exact analysis of the resonance in the conducting orthogonal anisotropy ferromagnet in the parallel geometry leads to the secular equation of eighth degree in the square of the reduced wave vector instead of the fourth degree equation for the ordinary ferromagnet with one magnetic sublattice (see eq. (15) in ref. [19]). The presented explanation of the difference in HAF and HAM supports the picture of the C o local anisotropy in Y C o 3 presented b y Kakol et al. [13]
[1] W.R. James, R. Lemaire and F. Bertaut, C.R. Acad. Sci. Paris 255 (1962) 896. [2] K.H. Buschow, J. Less Common Metals 11 (1966) 204. [3] 17,.Lemaire, Cobalt 33 (1966) 201. [4] K. Strnat, G.I. Holler and A.E. Ray, IEEE Trans. Magn. MAG-2 (1966) 489. [5] D. Bloch and F. Chaisse, C.R. Acad. Sci. Paris 68 (1969) 660. [6] E. Kren, J. Schweizer and F. Tasset, Phys. Rev. 186 (1969) 479. [7] K.H. Buschow, Les elements de terres rare I (CNRS, Grenoble, 1970). [8] F. Givord and R. Lemaire, Solid State Commun. 9 (1971) 341. [9] K.H. Buschow, J. Less Common Metals 33 (1973) 311. [10] A.V. Deryagin and N.V. Kudrevatykh, Phys. Stat. Sol. (a) 30 (1975) K129. [11] H. Figiel, E. Oppelt, E. Dorman and K.H. Buschow, Phys. Stat. Sol. (a) 36 (1976) 275. [12] M.R. Kirchmayr and A.C. Poldy, J. Magn. Magn. Mat. 8 (1978) 1. [13] Z. Kakol, H. Figiel and K. Turek, IEFA~Trans. Magn. MAG-20 (1984) 1605. [14] K. Turek, Thesis, University of Mining and Metallurgy, Cracow (1984). [15] K. Turek, Z. Kakol, A. Kolodziejczyk and K. Krop, Physica 130B (1985) 314. [16] K. Turek, Z. Kakol and A. Kolodziejczyk,J. Magn. Magn. Mat. (submitted). [17] D.T. Vigren and S.H. Liu, Phys. Rev. B5 (1972) 2719. [18] M.S. Brooks and T. Egami, J. Phys. C6 (1973) 513. [19] R.F. Soohoo, Phys. Rev. 120 (1960) 1978.
205
THE INFLUENCE OF THE LOCAL Co ANISOTROPY ON FERROMAGNETIC RESONANCE IN YCo3 AND Y2Co17 * K. TUREK and A. KOLODZIEJCZYK Departement of Solid State Physics IM, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Cracow, Poland Received 7 March 1986; in revised form 20 August 1986
It is shown that the difference in the results of the macroscopic anisotropy field measurements in YCo 3 and Y2C017 and these obtained by FMR technique can be explained by the local Co anisotropy. In spite of the ferromagnetic ordering of the magnetic moments in these compounds the magnetic resonance in the microwave region cannot be treated as the classical ferromagnetic one. The simplified resonant condition is effected by the anisotropy in other ways than that for an ordinary ferromagnetic material with one magnetic sublattice.
1. Introduction The intermetallic compounds YCo3 and Y2Co17 were investigated in a number of papers [1-14]. They both have a rather large anisotropy energy. The anisotropy fields of these compounds measured by FMR technique, HAF, are much lower than the HAMextracted from magnetisation curves [14-16] which is shown in table 1. The differences in HAF and HAM were observed in Tb and Dy by Vigren and Liu [17] and were explained by the magnetoelastic effects in the so-called "frozen lattice" model. Nevertheless, in the case of Tb and Dy the anisotropy field HAF is higher than HAM in contradictiction to YCo3 and Y2Co17 where HAF is lower than HAM. Brooks and Egami [18] showed that for cylindrical symmetry
* This paper was sponsored by the Physics Institute of Polish Academy of Sciences.
Table 1 The anisotrpy fields HAM and HAF for YCo 3 and Y2Co17 Compound
HAM (10 6 A / m )
H~ (10 6 A / m )
YCo 3 Y2Co17
1.17 0.66
0.45-0.75 0.46
the macroscopic anisotropy constants influenced by the frozen lattice effect must always be lower than these constants which are measured by the resonance method in the easy direction. This means that this effect cannot be responsible for the significant differences in HAF and HAMof YCo3 and of Y2C017, which have both uniaxial symmetry. In this paper we show that this difference can be explained by the local Co anisotropy of these compounds.
2. The crystalline and magnetic structure of YCo3 and Y2C01~ The investigated compounds have complicated crystalline structures which are reflected in their interesting magnetic properties. YCo3 has the elementary unit cell of PuNi 3 type which belongs to the R3m space symmetryl group. There are three different crystallographic ;positions of Co ions in such cell. Three of them are in the b, six in the c and eighteen in the h position in Wyckhoff's notation. Y2C017 crystallizes in the rhombohedral or hexagonal structure. Our samples have proved to have the rhombohedral symmetry belonging to P63/mmc symmetry group with the elementary cell of Th2Nil7 type. The Co ions in the cell occupy four different sites. Nine ions are in the d,
0304-8853/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
206
K. Turek, A. Kolodziejczyk / Ferromagneticresonancein YCo compounds
eighteen in the f, eighteen in the h and six in the e position. Kakol et al. [13] calculated the Co sublattice contributions to the net magnetocrystalline anisotropy in the compounds of Y - C o system using the point charge model with account on shielding effects. According to these calculations, YCo 3 has two sublattices (6c and 18h) with positive and one (3b) with negative anisotropy energy. In the case of Y2Co17 three Co sites (9d, 18h and 6e) have negative and one (180 positive anisotropy energy. The net anisotropy energy of YCo 3 is positive and that of Y2Co17 is negative. This means that, neglecting intersublattice exchange interactions, the magnetisations of these sublattices would be orientated perpendicularly one to another, along the c axis and in the basal plane. Nevertheless, because of the strong exchange coupling between the sublattices all magnetic moments are aligned along the c axis in YCo 3 and in the basal plane in Y2Co17. The materials having the magnetic structure like YCo 3 and Y2Co17 we will call "orthogonal anisotropic ferromagnets".
3. Ferromagnetic resonance in orthogonai anisotropic ferromagnets
anisotropy field /'/A~ y of the sublattice P2 rotates in the plane together with the projection of magnetisation which precesses about the c axis with a frequency to and then:
tt~Y = 2K12m2x e i~°t.~ + 2K12m2y
p,
(1)
where K12 is the first anisotropy constant of the sublattice P2, M20 is the static component of the magnetisation of this sublattice, and m2x and m2y are the dynamic components of the magnetisation in x and y directions induced by the microwave power. The anisotropy field of the first sublattice can be written in the form: 2Kit ^ H A 1 - M, ° z,
(2)
where Kit is the first anisotropy constant of the sublattice P~ and M~0 is the static component of the magnetisation of this sublattice. Then the classical equations of motion of magnetisation for both sublattices with the additional anisotropy terms are:
dMa dt
dM 2 Let us consider the orthogonal anisotropic ferromagnet with two sublattices P1 and P2, the first with the easy axis and the second with the plane of the easy magnetisation perpendicular to this axis. Let us assume that the anisotropy energy of the sublattice P1 is much higher than the anisotropy energy of the sublattice P2 but the exchange coupling between the magnetisations of these sublattices is large enough to pull out the magnetisation of the sublattice P2 from the plane and to keep it parallel to the magnetisation of the sublattice P r Additionally, to simplify the calculations let us assume that the difference between the g factors of both sublattices is negligible small, so that g~ = g2 = g and that the conductivity and the damping terms can be neglected in the equation of motion of magnetisation. Let us apply a dc field along the c axis. Because of the rotational symmetry of anisotropy in the basal plane we are allowed to assume that the
e iwt
dt
= #°yMI × ( H + HA1 + lM2),
(3a)
= #°7M2 × ( H + H~, y + IM1),
(3b)
where M 1 and M 2 are the total magnetisations of the sublattices P1 and P2, respectively, H is the sum of the applied de field, the demagnetising field and the dynamic magnetic component of the rf field h and l is the molecular field coefficient. Rewriting eqs. (3) in the component form and introducing the new complex variables h~ = hnx - ihny and rn~ = rnnx - im~y(n = 1,2) we obtain:
(-- t° + Ha + HAI + I M 2 0 ) m l - I M l o m 2 #oY
(4a)
= Mloh{,
-'M2om{ + (= M20h2 , .
to + H a - HA2 + IMlo)m2
/x0Y
(4b)
where H a is the static component of the field H and HA2 = 2K12/M2o. To find the resonant condi-
207
K. Turek, A. Kolodziejczyk / Ferromagnetic resonance in YCo compounds
tion the determinant of the coefficients in eqs. (4) is to be equal to zero. As a result we get the expression: H = Ha + ½ H A +
Y
½IMo
(5) where H A = HAt -- HA2 is the net anisotropy field, M 0 -- M10 + M20 is the net magnetisation and a = (HA1 + HA2)/IM o. From the two eqs. (5) only the equation with the minus sign has its solution in the microwave area. From the fact that in the investigated compound (Mlo - M20) < (H^I + HA2) it follows that 2
(M 0 -- M20) < Or2.
Y
= Ha + ½HA - ½a(M~o - M2o ).
+/-/,,:) (7)
In this formula the third term at the right side is two orders and the fourth three orders of magnitude less than the first and second terms. So, the resonant condition (7) can be simplified within a good approximation to the form: w = Ha + ½HA. Y
H-y = Ha + HA .
(8)
It follows from this formula that in the orthogonal anisotropic ferromagnet the resonant field is shifted by only half of the anisotropy field towards lower fields in comparison with an ordinary ferromagnet with one magnetic sublattice, for
(9)
Thus, the net anisotropy field calculated from eq. (9) is only half of the true anisotropy field H A, which is measured statically, whereas the anisotropy calculated from eq. (8) compared nicely with the static value. To get a better physical insight into our model let us calculate the ratio of the dynamical components of the magnetisations mi- and m~. For this purpose let us put h~ =khi- where k is some unknown coefficient, and then solve the linear eqs. (4). As a result we find relations between mi-, mE and ha: m1=
(6)
In the case of YC% g#BH~x = 160 K, where Hex is the exchange field and g/%Hex = 500 K for Y2Co17, Mxo + 3420 = 1.87 X l0 s A / m for YCo 3 and 10 6 A / m for Y2COxT, H^I + HA2 > 11.7 × l0 s and 6.6 × l0 s A / m , respectively [13]. Using these values we get a -~ 10- 3 for these compounds. Owing to the inequality (6) we can expand eq. (5) to the series with the accuracy to the first order term, "
which the resonant condition is
_~+H #0T
kM2o
a+H^l
+l(kM~o + M2oMlo)] h i / D ,
[(o -
#oY
)
(lOa)
i
+H.-HA2
mxo (10b)
where D is the determinant of the matrix of coefficients of mi- and m~ in eqs. (4). In the linear approximation we assume that mi- and m~ are small and then the coefficients of i in eqs. (10) have to be small also. Let us assume that they are equal to zero. Then it follows from eqs. (10) that
k = -Mlo/M2o and m1
m2
= (H^I + HA2)/(HAi + 3HA2 ) < 1.
(11)
If we t;~ke into account that M20 < M10 then the cone of precession of the magnetisation M 2 is more obtuse than that of M 1 as is shown in fig. 1. Thus, in the case of an orhogonal anisotropic ferromagnet we have quite another mode of precession than in the case of the ordinary F M R in which the net magnetisation precesses about the effective field contributed by the net anisotropy field. That is why the net anisotropy field in eq. (8) contributes to the effective field in a way other than in the resonant condition (9) for the ferromagnet.
208
K. Turek, A. Kolodziejczyk / Ferromagnetic resonance in YCo compounds
\
/
at least qualitatively. Y2Co17 has the easy plane of magnetisation and the outlined above model for the easy axis of mangetisation cannot be strictly used for this compound. W e can only suggest that the difference in H ~ and H f f in Y2Co17 has the same origin as in Y C o 3.
References
Fig. 1. The cones of precession of the magnetisations M] and M 2. The dynamical components mi- and m2 are enlarged with respect t o m 1 and M e.
4. Conclusions The resonant condition (8) is limited to the ferromagnetic, n o n c o n d u c t i n g sphere. The values of H ~ in table 1 were extracted from the F M R records in the powdered magnetic aligned samples using the model of the so-called parallel geometry [15, 16]. Nevertheless, we expect that also in the case of the parallel geometry only a part of anisotropy field H A contributes to the effective magnetic field in the resonant condition due to the orthogonal anisotropic ferromagnetism. The exact analysis of the resonance in the conducting orthogonal anisotropy ferromagnet in the parallel geometry leads to the secular equation of eighth degree in the square of the reduced wave vector instead of the fourth degree equation for the ordinary ferromagnet with one magnetic sublattice (see eq. (15) in ref. [19]). The presented explanation of the difference in HAF and HAM supports the picture of the C o local anisotropy in Y C o 3 presented b y Kakol et al. [13]
[1] W.R. James, R. Lemaire and F. Bertaut, C.R. Acad. Sci. Paris 255 (1962) 896. [2] K.H. Buschow, J. Less Common Metals 11 (1966) 204. [3] 17,.Lemaire, Cobalt 33 (1966) 201. [4] K. Strnat, G.I. Holler and A.E. Ray, IEEE Trans. Magn. MAG-2 (1966) 489. [5] D. Bloch and F. Chaisse, C.R. Acad. Sci. Paris 68 (1969) 660. [6] E. Kren, J. Schweizer and F. Tasset, Phys. Rev. 186 (1969) 479. [7] K.H. Buschow, Les elements de terres rare I (CNRS, Grenoble, 1970). [8] F. Givord and R. Lemaire, Solid State Commun. 9 (1971) 341. [9] K.H. Buschow, J. Less Common Metals 33 (1973) 311. [10] A.V. Deryagin and N.V. Kudrevatykh, Phys. Stat. Sol. (a) 30 (1975) K129. [11] H. Figiel, E. Oppelt, E. Dorman and K.H. Buschow, Phys. Stat. Sol. (a) 36 (1976) 275. [12] M.R. Kirchmayr and A.C. Poldy, J. Magn. Magn. Mat. 8 (1978) 1. [13] Z. Kakol, H. Figiel and K. Turek, IEFA~Trans. Magn. MAG-20 (1984) 1605. [14] K. Turek, Thesis, University of Mining and Metallurgy, Cracow (1984). [15] K. Turek, Z. Kakol, A. Kolodziejczyk and K. Krop, Physica 130B (1985) 314. [16] K. Turek, Z. Kakol and A. Kolodziejczyk,J. Magn. Magn. Mat. (submitted). [17] D.T. Vigren and S.H. Liu, Phys. Rev. B5 (1972) 2719. [18] M.S. Brooks and T. Egami, J. Phys. C6 (1973) 513. [19] R.F. Soohoo, Phys. Rev. 120 (1960) 1978.