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A method of determining the third order anisotropy constant K3 for uniaxial ferromagnets — Application to Nd0.37Sm0.63Co5
Journal of Magnetism and Magnetic Materials 23 (1981) 117-122 North-Holland Publishing Company
117
A METHOD OF DETERMINING THE THIRD ORDER ANISOTROPY CONSTANT K3 FOR UNIAXIAL FERROMAGNETS - APPLICATION TO Nd0.37Sm0.63C05 W.I. KHAN and D. MELVILLE Department
of Physics, University of Southampton,
Southampton,
England
Received 29 October 1980
A method is described which enables the fist three anisotropy constants K1, Ka KS to be extracted from magnetisation curves for uniaxial ferromagnets. The intercept, initial slope and curvature of plots of B&f against M2 are used to provide numerical values of the constants. The method is applied to data of Ermolenko and Rozhda on Ndo.37Sm0.63Co5 and the results indicate that for this material below 100 K both K2 and K3 become large compared to K1. At 4.2 and 40 K the values of the ratios K2/Kl and K3/K1 indicate the existence of a first order magnetisation process, which is in agreement with the sharp rise in magnetisation observed experimentally at these temperatures.
1. Introduction It is convenient to express the anisotropy of a uniaxial ferromagnet as EK = K1 sin26 + K2 sin40 + K3 sin”0 + . . . .
energy
(1)
where the K’s are anisotropy constants and 0 is the angle between the magnetisation vector M, and the caxis. It has been conventional practice to restrict analysis to the first and second order anisotropy constants although this rationale has probably been based on convenience rather than any fundamental principles or experimental evidence. Indeed, the importance of third order anisotropy constants in determining the observable properties of magnetic materials has been stressed both from a theoretical and experimental point of view. Asti et al. [l] has shown that for uniaxial materials the presence of even small values of KS extends considerably the range of anisotropy values over which first order magnetisation processes are expected. In addition Asti et al. [2] observed that the magnetisation curves for a W-type barium zinc hexaferrite could not be explained without a K3 contribution. Atzmony and Dariel [3] observed non-symmetry easy-directions in binary rare earth cubic Laves phase compounds of the type RFe2. Such directions arise quite naturally in phenomenological theory when 0304-8853/8
l/0000-0000/$02.50
0 North-Holland
third order anisotropy constants exist, however small. More recently [4], K3 has been shown to be significant in the compound PrCoskx and Rinaldi and Pareti [5] have presented a potential mechanism for the appearance of finite K3 values in uniaxial materials due to relative canting of sublattice magnetic moments. Considerations such as these have led us to consider simple methods of estimating higher order anisotropy constants without the need for large single cyrstals and resort to torque measurements. We shall, for the present, restrict considerations to uniaxial symmetry. The singular point detection technique [6] is readily extended to any order of anisotropy constant and provides values for the anisotropy field which is in turn a linear combination of anisotropy constants. For example for a hard plane the anisotropy field is BA = (2K, + 4K2 + 6K3)/kls. The major advantages of this technique are its applicability to polycrystalline samples and high anisotropies and in cases where first order magnetisation processes are present it may be possible using aligned polycrystals to estimate all three anisotropy constants independently. In the following section we propose a new technique based on magnetisation measurements on single crystals or aligned polycrystals which enables K1, K2 and K3 to be readily estimated.
Publishing Company
W.I. Khan, D. Melville / Anisotropy
118
constants
of uniaxiaiferromagnets
r
2. Theory
I
I
1
Using the convention of eq. (1) the total energy E when a field B,-, is applied in an arbitrary direction to a uniaxial crystal is given by E=EK
-M;B,.
The equilibrium condition of eq. (2) gives directly for the field along symmetry directions B,/M=P+QM2+~,
(3)
where M is the magnetisation direction and P = 2K,/M,2,
.-“0 % LL s ii
,_
measured along the field
Q = 4K,/@,
R = 6K3/@,
(4)
for Be applied in a hard plane and P = -2(K,
2-
(2)
0
0.5
1.0
Q = 4(Kz + 3Ks)/N,
+ 2K2 + 3K,)/M;,
R = -6KJM&
(5)
for the field applied along a hard axis [2]. Eq. (3) is quadratic in M2 so that a plot of B,/M against M2 has P as intercept, Q as initial slope and a curvature whose sign and magnitude are determined by R. Fig. la shows schematic Be/M vs. M2 plots. From such plots the signs and magnitudes of K1, Kz and K3 can in principle be estimated. It should be
/
P.O.
GO,
P=-0,
-4
Cl>O,R>O
Fig. 1. (a) Schematic form of BdM VS. MZ plots, (b) terms in eq. (6) for hard plane case, (c) terms in eq. (6) for hard axis case
R.0
\
P
noted further that in the case of field applied along a hard plane, because of their direct relation to P, Q and R, the signs and relative magnitudes of K1, K2 and K3 can be determined by immediate inspection of the B,/M vs. M2 plot. The method as outlined above is a direct extension to higher order of the technique used by Sucksmith and Thomson to determine the
W.I. Khan, D. Melville / Anisotropy constants of uniaxialferromagnets
119
magnetic anisotropy of cobalt [7]. It is perhaps worth stressing that eqs. (3) (4) and (5) are expressed in terms of hard directions only. Consequently all possible easy direction symmetries - viz. easy plane, axis and cone are included in the treatment. 2.1. Numerical limitations The precision with which the individual constants can be determined depends on their relative magnitude
M
fG
and the applied field available. In order to estimate the individual contributions in eq. (3) we re-express it as follows: Bo =Boi
+Bo2
+Bo3,
(6)
where the terms on the right hand side of (6) are equivalent fields corresponding to BoI = PM, Bo2 = QM3 and Bo3 = RM5 with P, Q and R given by (4) or (5). Figs. lb and lc show plots ofBo2/BoI, Bo3/Bo1 and Bo3/Bo2 for the situation Kr = K2 = K3, for the hard plane and hard axis cases, respectively. From fig. lb, which is easiest to use since B. 1, etc., are directly proportional to K1, etc., we see that if K2/KI = K3/K1 = 1 then we need iVIM, > 0.4 in order for the contributions made by K2 and K3 to both exceed 10% of the K1 contribution. This would require applied fields in excess of O.lBA. However, if K2/KI = K3/K1 = 0.1 the same conditions require M/MS > 0.76 corresponding to an applied field of B. > 0.6B~. In the same way fig. 1b and lc can be used to estimate the required applied field in order to achieve a given precision for any prevailing conditions.
a 5.0 Field
0’
3. Results The method described above has been applied to the single crystal magnetisation data of Ermolenko and Rozhda [8] on Nd0.3,Sm0.63C05. Magnetisation curves and Be/M vs. M2 plots are shown in fig. 2a and b. Using the ideas outlined above and comparing with fig. la for the hard plane case it can be seen by inspection of fig. 2b that, (i) K1 is positive throughout, (ii) K2 changes sign from positive to negative at a temperature between 120 and 80 K, (iii) K3 becomes significant below 120 K and is positive. It should further be noted that the discontinuous change of slope in the plots for 4.2 and 40 K is associated with
I
1 (Specific
I
2
(Teslal
I
3
magnetisation)2
I L iAm2
I
5 1x103) kg-‘12
Fig. 2. (a) Magnetisation curves for Ndo03+mo.63Cog after ref. [ 81, with curves calculated from measured anisotropy constants, (b) Eo/M vs. M2 plots for Ndo 3,Srno 63Cos after ref. [8].
the rapid rise of magnetisation appearing at these temperatures. The temperature dependence ofK1, K2 and K3 for this alloy derived by the method presented above is shown in fig. 3. MS values at 280 and 4.2 K were taken from ref. [8] and values between these temperatures were obtained by linear interpolation.
120
W.I. Khan, D. Melville / Anisotropy
-2 0
constants of uniaxial ferromagnets
1
1
1
I
1
50
100
150
200
250
Temperature
300
( K)
Fig. 3. Temperature dependence of the anisotropy constants for Ndo.37Smo.63Cos.
4. Discussion The temperature dependence of K1, K2 and K3 shown in fig. 3 indicates that whereas K I rises monotonically with temperature and is similar in magnitude to the values reported by Ermolenko and Rozhda [8], K2 and K3 take on very large values which are negative and positive respectively, when the temperature is taken below 80 K. The curves are of the generally expected monotonic form arising from single ion effects, but the magnitude of K2 and K3 are most likely to be an overestimate associated with extracting the constants from the limited data shown in fig. 2a and b. Since the determination of K3 depends on using the appropriate value of K2, errors in the latter will be balanced by errors of opposite sign in K3. Hence it is likely, since the K3 term just before the transition (MIM, = 0.1) in fig. 2a for the 4.2 K curves contributes only 15% (see fig. lc) to the magnetisation, it may contain an error of the order of 50%. This, however, does not affect our conclusions or the argument which follows.
A q
80K 120K
, Am atL.2K V Am at LOK
-250
Al Easy
Plane
t
Fig. 4. Kz/Kl, K3/K1 plane for a uniaxial ferromagnet (after ref. [ 11). FOMP regions: Al-hard axis, type I; Pl-hard plane, type I; PZ-hard plane, type II. Open symbols refer to measured anisotropy constants. So%?lsymbols show possible Kz/Kl, K3/KI combinations derived from measured Am values.
W.I. Khan, D. Melville / Anisotropy constants of uniaxialferromagnets
Asti and Bolzoni [l] have calculated the conditions under which first order magnetisation processes exist for the hard plane situation found in this alloy. Fig. 4 shows an extension of their fig. 5 to high KS/K1 and K2/K1 ratios. Plotted on this figure is the trajectory of the anisotropy ratios derived for Nd0.37Sm0.63C05 in the present work. It can be seen that for the 40 and 4.2 K curves a FOMP of type II is predicted. Such a process involves a transition to a final magnetisation state which is below saturation. This is indeed the general form of the transition shown in fig. 2a [8]. In addition it is clear that the anisotropy values are such as to maintain the alloy in the easy axis region throughout [8]. Using eq. (3) with the determined values of K1,Kz, K3 we have calculated the whole magnetisation curve corresponding to each of the measurement temperatures. These are plotted with the measured curves in fig. 2a. It can be seen that there is a good fit to the experimental data for temperatures of 280,120 and 80 K. For the 40 and 4.2 K curves values of the anisotropy constants determined from measurements below the FOMP correctly predict its position, both in terms of the critical field B, and initial critical magnetisation M1 . The final critical magnetisation Mz, however, is overestimated in both cases. It can be shown that the reduced magnetisation discontinuity at the transition is [I] Am = [(-5x
- (60~ - 1 lxZ)rn}/6y]‘“,
(7)
where x = K2/KI and y = K3/K1. Experimentally at 40 K, Am = 0.39 is observed while the measured anisotropy constants predict a value of Am = 0.1. The magnetisation discontinuity is sensitive to the ratios of the anisotropy constants and indicates the errors in the estimations made. Similarly at 4.2 K the observed and calculated values of Am are 0.43 and 0.06, respectively. It is not possible to determine K2/K1 and KS/K1 unambiguously from eq. (7). However, for a particular observed value of Am, eq. (7) defines the line -?-(1 - (Am)“x) ’ = 6(Amf *6&d](5+(Am)2x)(5-
ll(Am)*~)]~~.
The lines defined by eq. (8) for Am = 0.39 and Am = 0.43 are plotted on fig. 4. It can be seen that
(8)
121
within the type II FOMP region the experimentally observed values of Am can be obtained if, as is suggested above, the K2/K1 and K3fK1 ratios have been overestimated. For example, if we assume that the error is chiefly in the K3 estimation, then at 40 K a reduction in K3 of SO%, keeping K2/KI constant, would give m = 0.36 which is close to the value 0.39 observed experimentally. Similarly, fig. 4 indicates that at 4.2 K KS/K1 is probably over-estimated by a factor of 2, giving a value KS/K1 = 300 at 4.2K. It is worth considering further the question as to why such large values of K2 and K3 are present in this alloy. SmCos is a strongly uniaxial, easy axis ferromagnet whose moment is associated with the cobalt atoms. The anisotropy energy has been estimated to be [9] 19 X lo6 Jmp3 at 0 K. NdCos on the other hand is an easy plane material in which the moment on the Nd sublattice has an anisotropy with strong planar tendency which is sufficient to overcome the axial tendency of the Co sublattice anisotropy. The anisotropy constant is given by Tatsumoto et al. [lo] as -40 X lo6 Jme3. Rinaldi and Pareti [S] have shown that in circumstances of competing sublattice anisotropies, higher order ‘effective anisotropy constants’ are generated. In the case where only K1 is considered for each sublattice, they have shown that an effective K2 and K3 become apparent due to a varying canting angle between the individual sublattice magnetic moments. Application of this simplified analysis to the Nd 0.37sm0.63 Co5 alloy at 4.2K, however, yields effective K2 and K3 values which are positive and negative, respectively. If this mechanism is responsible for the higher order anisotropy constants observed experimentally it is clear that we need a more complete model which takes the large second order anisotropy constant observed for NdCos into account (K2 = 19 X lo6 Jm-‘). 5. Conclusions For uniaxial ferromagnets, plots of B,/M vs. Mz can be used to estimate the first three anisotropy constants. The precision of such estimations is limited by the maximum applied field available. Application of this technique to Nd0.37Sm0.63C05 shows that a first order rotation of the magnetisation occurs in this material above a certain critical field at low temperatures.
122
W.I. Khan, D. Melville /Anisotropy
Acknowledgements We are grateful to Dr. S. Rinaldi and Professor G. Asti for illuminating discussions and advice.
References [I] G. Asti and F. Bolzoni, J. Magn. Magn. Mat. 20 (1980) 29. [2] G. Asti, F. Bolzoni, F. Licci and M. Canah, IEEE Trans. Magn. MAG-14 (1978) 676.
constants of uniaxial ferromagnets
[ 31 V. Atzmony and M.P. Dariel, Phys. Rev. B 13 (1976) 4006. [4] G. Asti, F. Bolzoni, F. Leccabue, R. Panizzieri, L. Pareti and S. Rinaldi, J. Magn. Magn. Mat. 15-18 (1980) 561. [5] S. RinaIdi and L. Pareti, J. Appl. Phys. 50 (1979) 7719. [6] G. Asti and S. Rinaldi, J. Appl. Phys. 45 (1974) 3600. [ 71 W. Sucksmith and J.E. Thomson, Proc. Roy. Sot. (London) 225 (1954) 362. [ 81 A.S. Ermolenko and A.F. Rozhda, IEEE Trans. Magn. MAG-14 (1978) 676. [9] M.G. Benz and D.L. Martin, J. Appl. Phys. 43 (1972) 4733. [lo] E. Tatsumoto, T. Okamoto, H. Fujii and C. Inoue, J. de Phys. 32 (1971) Cl-550.
117
A METHOD OF DETERMINING THE THIRD ORDER ANISOTROPY CONSTANT K3 FOR UNIAXIAL FERROMAGNETS - APPLICATION TO Nd0.37Sm0.63C05 W.I. KHAN and D. MELVILLE Department
of Physics, University of Southampton,
Southampton,
England
Received 29 October 1980
A method is described which enables the fist three anisotropy constants K1, Ka KS to be extracted from magnetisation curves for uniaxial ferromagnets. The intercept, initial slope and curvature of plots of B&f against M2 are used to provide numerical values of the constants. The method is applied to data of Ermolenko and Rozhda on Ndo.37Sm0.63Co5 and the results indicate that for this material below 100 K both K2 and K3 become large compared to K1. At 4.2 and 40 K the values of the ratios K2/Kl and K3/K1 indicate the existence of a first order magnetisation process, which is in agreement with the sharp rise in magnetisation observed experimentally at these temperatures.
1. Introduction It is convenient to express the anisotropy of a uniaxial ferromagnet as EK = K1 sin26 + K2 sin40 + K3 sin”0 + . . . .
energy
(1)
where the K’s are anisotropy constants and 0 is the angle between the magnetisation vector M, and the caxis. It has been conventional practice to restrict analysis to the first and second order anisotropy constants although this rationale has probably been based on convenience rather than any fundamental principles or experimental evidence. Indeed, the importance of third order anisotropy constants in determining the observable properties of magnetic materials has been stressed both from a theoretical and experimental point of view. Asti et al. [l] has shown that for uniaxial materials the presence of even small values of KS extends considerably the range of anisotropy values over which first order magnetisation processes are expected. In addition Asti et al. [2] observed that the magnetisation curves for a W-type barium zinc hexaferrite could not be explained without a K3 contribution. Atzmony and Dariel [3] observed non-symmetry easy-directions in binary rare earth cubic Laves phase compounds of the type RFe2. Such directions arise quite naturally in phenomenological theory when 0304-8853/8
l/0000-0000/$02.50
0 North-Holland
third order anisotropy constants exist, however small. More recently [4], K3 has been shown to be significant in the compound PrCoskx and Rinaldi and Pareti [5] have presented a potential mechanism for the appearance of finite K3 values in uniaxial materials due to relative canting of sublattice magnetic moments. Considerations such as these have led us to consider simple methods of estimating higher order anisotropy constants without the need for large single cyrstals and resort to torque measurements. We shall, for the present, restrict considerations to uniaxial symmetry. The singular point detection technique [6] is readily extended to any order of anisotropy constant and provides values for the anisotropy field which is in turn a linear combination of anisotropy constants. For example for a hard plane the anisotropy field is BA = (2K, + 4K2 + 6K3)/kls. The major advantages of this technique are its applicability to polycrystalline samples and high anisotropies and in cases where first order magnetisation processes are present it may be possible using aligned polycrystals to estimate all three anisotropy constants independently. In the following section we propose a new technique based on magnetisation measurements on single crystals or aligned polycrystals which enables K1, K2 and K3 to be readily estimated.
Publishing Company
W.I. Khan, D. Melville / Anisotropy
118
constants
of uniaxiaiferromagnets
r
2. Theory
I
I
1
Using the convention of eq. (1) the total energy E when a field B,-, is applied in an arbitrary direction to a uniaxial crystal is given by E=EK
-M;B,.
The equilibrium condition of eq. (2) gives directly for the field along symmetry directions B,/M=P+QM2+~,
(3)
where M is the magnetisation direction and P = 2K,/M,2,
.-“0 % LL s ii
,_
measured along the field
Q = 4K,/@,
R = 6K3/@,
(4)
for Be applied in a hard plane and P = -2(K,
2-
(2)
0
0.5
1.0
Q = 4(Kz + 3Ks)/N,
+ 2K2 + 3K,)/M;,
R = -6KJM&
(5)
for the field applied along a hard axis [2]. Eq. (3) is quadratic in M2 so that a plot of B,/M against M2 has P as intercept, Q as initial slope and a curvature whose sign and magnitude are determined by R. Fig. la shows schematic Be/M vs. M2 plots. From such plots the signs and magnitudes of K1, Kz and K3 can in principle be estimated. It should be
/
P.O.
GO,
P=-0,
-4
Cl>O,R>O
Fig. 1. (a) Schematic form of BdM VS. MZ plots, (b) terms in eq. (6) for hard plane case, (c) terms in eq. (6) for hard axis case
R.0
\
P
noted further that in the case of field applied along a hard plane, because of their direct relation to P, Q and R, the signs and relative magnitudes of K1, K2 and K3 can be determined by immediate inspection of the B,/M vs. M2 plot. The method as outlined above is a direct extension to higher order of the technique used by Sucksmith and Thomson to determine the
W.I. Khan, D. Melville / Anisotropy constants of uniaxialferromagnets
119
magnetic anisotropy of cobalt [7]. It is perhaps worth stressing that eqs. (3) (4) and (5) are expressed in terms of hard directions only. Consequently all possible easy direction symmetries - viz. easy plane, axis and cone are included in the treatment. 2.1. Numerical limitations The precision with which the individual constants can be determined depends on their relative magnitude
M
fG
and the applied field available. In order to estimate the individual contributions in eq. (3) we re-express it as follows: Bo =Boi
+Bo2
+Bo3,
(6)
where the terms on the right hand side of (6) are equivalent fields corresponding to BoI = PM, Bo2 = QM3 and Bo3 = RM5 with P, Q and R given by (4) or (5). Figs. lb and lc show plots ofBo2/BoI, Bo3/Bo1 and Bo3/Bo2 for the situation Kr = K2 = K3, for the hard plane and hard axis cases, respectively. From fig. lb, which is easiest to use since B. 1, etc., are directly proportional to K1, etc., we see that if K2/KI = K3/K1 = 1 then we need iVIM, > 0.4 in order for the contributions made by K2 and K3 to both exceed 10% of the K1 contribution. This would require applied fields in excess of O.lBA. However, if K2/KI = K3/K1 = 0.1 the same conditions require M/MS > 0.76 corresponding to an applied field of B. > 0.6B~. In the same way fig. 1b and lc can be used to estimate the required applied field in order to achieve a given precision for any prevailing conditions.
a 5.0 Field
0’
3. Results The method described above has been applied to the single crystal magnetisation data of Ermolenko and Rozhda [8] on Nd0.3,Sm0.63C05. Magnetisation curves and Be/M vs. M2 plots are shown in fig. 2a and b. Using the ideas outlined above and comparing with fig. la for the hard plane case it can be seen by inspection of fig. 2b that, (i) K1 is positive throughout, (ii) K2 changes sign from positive to negative at a temperature between 120 and 80 K, (iii) K3 becomes significant below 120 K and is positive. It should further be noted that the discontinuous change of slope in the plots for 4.2 and 40 K is associated with
I
1 (Specific
I
2
(Teslal
I
3
magnetisation)2
I L iAm2
I
5 1x103) kg-‘12
Fig. 2. (a) Magnetisation curves for Ndo03+mo.63Cog after ref. [ 81, with curves calculated from measured anisotropy constants, (b) Eo/M vs. M2 plots for Ndo 3,Srno 63Cos after ref. [8].
the rapid rise of magnetisation appearing at these temperatures. The temperature dependence ofK1, K2 and K3 for this alloy derived by the method presented above is shown in fig. 3. MS values at 280 and 4.2 K were taken from ref. [8] and values between these temperatures were obtained by linear interpolation.
120
W.I. Khan, D. Melville / Anisotropy
-2 0
constants of uniaxial ferromagnets
1
1
1
I
1
50
100
150
200
250
Temperature
300
( K)
Fig. 3. Temperature dependence of the anisotropy constants for Ndo.37Smo.63Cos.
4. Discussion The temperature dependence of K1, K2 and K3 shown in fig. 3 indicates that whereas K I rises monotonically with temperature and is similar in magnitude to the values reported by Ermolenko and Rozhda [8], K2 and K3 take on very large values which are negative and positive respectively, when the temperature is taken below 80 K. The curves are of the generally expected monotonic form arising from single ion effects, but the magnitude of K2 and K3 are most likely to be an overestimate associated with extracting the constants from the limited data shown in fig. 2a and b. Since the determination of K3 depends on using the appropriate value of K2, errors in the latter will be balanced by errors of opposite sign in K3. Hence it is likely, since the K3 term just before the transition (MIM, = 0.1) in fig. 2a for the 4.2 K curves contributes only 15% (see fig. lc) to the magnetisation, it may contain an error of the order of 50%. This, however, does not affect our conclusions or the argument which follows.
A q
80K 120K
, Am atL.2K V Am at LOK
-250
Al Easy
Plane
t
Fig. 4. Kz/Kl, K3/K1 plane for a uniaxial ferromagnet (after ref. [ 11). FOMP regions: Al-hard axis, type I; Pl-hard plane, type I; PZ-hard plane, type II. Open symbols refer to measured anisotropy constants. So%?lsymbols show possible Kz/Kl, K3/KI combinations derived from measured Am values.
W.I. Khan, D. Melville / Anisotropy constants of uniaxialferromagnets
Asti and Bolzoni [l] have calculated the conditions under which first order magnetisation processes exist for the hard plane situation found in this alloy. Fig. 4 shows an extension of their fig. 5 to high KS/K1 and K2/K1 ratios. Plotted on this figure is the trajectory of the anisotropy ratios derived for Nd0.37Sm0.63C05 in the present work. It can be seen that for the 40 and 4.2 K curves a FOMP of type II is predicted. Such a process involves a transition to a final magnetisation state which is below saturation. This is indeed the general form of the transition shown in fig. 2a [8]. In addition it is clear that the anisotropy values are such as to maintain the alloy in the easy axis region throughout [8]. Using eq. (3) with the determined values of K1,Kz, K3 we have calculated the whole magnetisation curve corresponding to each of the measurement temperatures. These are plotted with the measured curves in fig. 2a. It can be seen that there is a good fit to the experimental data for temperatures of 280,120 and 80 K. For the 40 and 4.2 K curves values of the anisotropy constants determined from measurements below the FOMP correctly predict its position, both in terms of the critical field B, and initial critical magnetisation M1 . The final critical magnetisation Mz, however, is overestimated in both cases. It can be shown that the reduced magnetisation discontinuity at the transition is [I] Am = [(-5x
- (60~ - 1 lxZ)rn}/6y]‘“,
(7)
where x = K2/KI and y = K3/K1. Experimentally at 40 K, Am = 0.39 is observed while the measured anisotropy constants predict a value of Am = 0.1. The magnetisation discontinuity is sensitive to the ratios of the anisotropy constants and indicates the errors in the estimations made. Similarly at 4.2 K the observed and calculated values of Am are 0.43 and 0.06, respectively. It is not possible to determine K2/K1 and KS/K1 unambiguously from eq. (7). However, for a particular observed value of Am, eq. (7) defines the line -?-(1 - (Am)“x) ’ = 6(Amf *6&d](5+(Am)2x)(5-
ll(Am)*~)]~~.
The lines defined by eq. (8) for Am = 0.39 and Am = 0.43 are plotted on fig. 4. It can be seen that
(8)
121
within the type II FOMP region the experimentally observed values of Am can be obtained if, as is suggested above, the K2/K1 and K3fK1 ratios have been overestimated. For example, if we assume that the error is chiefly in the K3 estimation, then at 40 K a reduction in K3 of SO%, keeping K2/KI constant, would give m = 0.36 which is close to the value 0.39 observed experimentally. Similarly, fig. 4 indicates that at 4.2 K KS/K1 is probably over-estimated by a factor of 2, giving a value KS/K1 = 300 at 4.2K. It is worth considering further the question as to why such large values of K2 and K3 are present in this alloy. SmCos is a strongly uniaxial, easy axis ferromagnet whose moment is associated with the cobalt atoms. The anisotropy energy has been estimated to be [9] 19 X lo6 Jmp3 at 0 K. NdCos on the other hand is an easy plane material in which the moment on the Nd sublattice has an anisotropy with strong planar tendency which is sufficient to overcome the axial tendency of the Co sublattice anisotropy. The anisotropy constant is given by Tatsumoto et al. [lo] as -40 X lo6 Jme3. Rinaldi and Pareti [S] have shown that in circumstances of competing sublattice anisotropies, higher order ‘effective anisotropy constants’ are generated. In the case where only K1 is considered for each sublattice, they have shown that an effective K2 and K3 become apparent due to a varying canting angle between the individual sublattice magnetic moments. Application of this simplified analysis to the Nd 0.37sm0.63 Co5 alloy at 4.2K, however, yields effective K2 and K3 values which are positive and negative, respectively. If this mechanism is responsible for the higher order anisotropy constants observed experimentally it is clear that we need a more complete model which takes the large second order anisotropy constant observed for NdCos into account (K2 = 19 X lo6 Jm-‘). 5. Conclusions For uniaxial ferromagnets, plots of B,/M vs. Mz can be used to estimate the first three anisotropy constants. The precision of such estimations is limited by the maximum applied field available. Application of this technique to Nd0.37Sm0.63C05 shows that a first order rotation of the magnetisation occurs in this material above a certain critical field at low temperatures.
122
W.I. Khan, D. Melville /Anisotropy
Acknowledgements We are grateful to Dr. S. Rinaldi and Professor G. Asti for illuminating discussions and advice.
References [I] G. Asti and F. Bolzoni, J. Magn. Magn. Mat. 20 (1980) 29. [2] G. Asti, F. Bolzoni, F. Licci and M. Canah, IEEE Trans. Magn. MAG-14 (1978) 676.
constants of uniaxial ferromagnets
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