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The determination of anisotropy constants from torque curves — II
Journal of Magnetism and Magnetic Materials 9 (1978) 333-335 0 North-Holland Publishing Company
THE DETERMINATION OF ANISOTROPY CONSTANTS FROM TORQUE CURVES - II
M.HUQ Department of Physics, College of Science, Universityof Mosul, Iraq
and
E.W.LEE Department of Physics, Universityof Southampton, UK
Received 21 June 1978
This is an extension of a previous paper which examined the methods available for the determination of the anisotropy constant from torque curves when the anisotropy is governed by a single constant i.e. E = K1 sin20. The two methods singled out in that paper as being of special merit are briefly reexamined when the anisotropy energy takes the form E = K1 sin20 + K2 sin4e.
easy direction is the c-axis for all ratios K2/K1. (The notation throughout this section is the same as in [ 11). The torque per unit volume exerted by the crystal on the magnetic moment per unit volume is
1. Introduction In a previous paper [l] we analysed the procedures available to experimentalists required to determine magnetic anisotropy constants from torque curves. It was shown that when the anisotropy can be described by a single term K1 sin’0, two methods offer advantages over the others. These are 1) the method of torque correction and 2) the analysis of the torque close to the easy axis. Both methods when properly applied enable accurate values of K1 to be determined from torque measurements in external fields less than the anisotropy field. There are, however, many materials which require more than one term to describe the anisotropy energy and it is of some importance tp see whether these two methods can be safely employed in this case or whether special caution has to be used in applying them.
L(B) = -
aE/M = - (K1tK2)sin 28 t qK2 sin 48 . (2)
The magnetic field extends a torque L(B)=M,Bsina!. In equilibrium sina=
Kl+K2 MB
(3) L(B) + L(B)= 0 and so: sin 2((#l-
s
cx)- &
sin 4(f#~- a) .(4) s
From eqs. (2) and (4) it is possible to calculate the torque L(#, B)asa function of $ and B. This procedure is valid as long as the magnetisation can be represented by a single vector, that is when the crystal is a single domain. When the external field is below a critical value BK it is energetically favourable for two sets of domains to form. From the equations given in [2] it may be shown that
2.Theoretical considerations It is assumed that the crystal, uniaxial as in [l] is in the form of a disc and that the anisotropy
E = K, sin26 + K2 sin40 .
energy is:
BK = N2i@+(BAt
(1)
=2 t M
K1 and K2 are both taken to be positive so that the 333
s
2BANM,) sin20
(BA t NM,)sin48
8 1,
+ $$ z sin68
112
M. Huq and E. W. Lee /Determination
334
of anisotropy constants
Table 1
ApparentanisotropyconstantsKi and K$ obtained from the Fourier analysis of the torque curves calculated for different combinations of Kr and K2 using different values of B in units of BA 2
BIBA
5
(KlIK2)
(Ki/Kl)
(KYK2)
(Ki/Kt)
CKiIK2) tKi/Kt)
(KvK$
(Ki/Kl)
0.01 0.1 1 10 100
1.001 0.999 0.999 0.999 1.ooo
0.999 1.ooo 0.999 0.999 0.999
1.005 0.999 0.999 0.999 0.999
0.999 1.000 1.000 1.000 1.000
0.999 0.998 0.999 1.ooo 0.999
3.379 1.888 1.190 1.025 1.011
1
0.2
0.5
1.004 1 .ooo 1 .ooo 0.999 1.000
(KgK$ -0.059 -0.189 -0.195 -0.323 -4.651
CKi/Kl) 1.223 1.034 0.922 0.902 0.899
0.125 (KgK2) -0.008 -0.064 -0.552 -5.358 -53.39
where BA = 2K1/Ms and N is the demagnetising factor in the plane of the crystal. The torque in the two domain region is *:
2.2. The torque near the easy axis
L(@,B) = (B2/2N) sin 24~- BM, sin 0 cos C/J,
L(e)=-(K1+K2)2(+c+4(#-a)=ZK,($-a),
(5)
(Ki/KI) 0.519 ‘0.527 0.524 0.523 0.523
(KyK$ -0.003 -0.032 -0.315 -3.148 -31.475
In the limit as 4 + 0 eqs. (2) and (3) can be written:
in which 8 and I#Jare related by
(2Kr + NM;) sin 0 + 4Ka sin30 - BM, sin @= 0 . (6) A set of torque curves L(@,B) was calculated from eqs. (3-6) using a single value of Kr = lo7 erg/cm3 (lo6 J/m3) with Kr/K2 = 100, 10, l,O.l, 0.01, MS = 1000 emu/cm3 (106/4n A/m) and N = 1 (1/47r). These calculated curves formed the simulated data that were treated by the two methods specified in section 1.
(7)
and L(B) = M,Bcu .
(8)
From the condition that L(B) t L(B) = 0 it can be shown that
=L+L 2K1
M,B ’
(9)
2.1. The correction of torque curves
so that 2K1 may be determined from measurements of the slope of the torque curve L@, B) evaluated at Q= 0.
In this method one converts L(@,B) to L(B) by recognising that 8 = $ - (Yand sin (Y= I!+(#,B)IM,B. The corrected torque L(e) is then subject to Fourier analysis. This procedure was applied to the calculated torque curves and the apparent values of K1 and K2 thereby determined are shown in table 1. From this it may be seen that the procedure yields correct values of K1 and Ks only when B > BA. Once the external field falls below BA the method begins to falter and although K1 is not affected very much K2 emerges incorrect even as to sign.
The torque will normally be measured at discrete intervals of $Iand the problem now is that of knowing how closely spaced these intervals must be in order to yield accurate values of K1. To examine this L(#, B) was calculated for 9 = l”, 3” and 5’ and the slope dL/d$ was taken to be the quotient L/4 in each case. These values of L/G were analysed graphically in the light of eq. (9) and the value of K1 deduced from the intercept. The results are shown in table 2. The method performs very creditably; only when L(@, B) is measured at 5’ intervals and then in the rather unlikely case that K2 = 10 K1 does the error exceed 10%. Curiously, the error is incurred by using too high a field causing a breakdown of the approximation sin 44 = 4#. This implies that when Ka > K1 the graph of (a/d@)-’ falls below the straight line described by eq. (9) at high fields. This, together with the result [l] that incomplete saturation
l
Eq. (6) of [l] is wrong. The torque curves given in [l] are not affected since they were calculated from correct expressions for the transverse magnetisation.
M. Huq and E. W. Lee /Determination Table 2 Apparent anisotropy constant Ki obtained from the initial slope drawn at different angular position # of the torque curves (K&s)
(Ki/Kl), #at 1” intervals
0.01 0.1 1 10 100
1.052 1.005 1.001 1.000 1.000
1.039 1.004 1.001 1.000 1.000
1.024 1.003 1.000 1.000 1.000
1.010 1.002 1 .ooo 1 .ooo 1 .ooo
1.003 1.001 1 .ooo 1.000 1.000
(B/BA)
5-2
2-l
l-0.5
0.5-0.2
0.2-0.125
(K,fKsJ
(Ki/KI), Qat 3” intervals
0.01 0.1 1 10 100
1.441 1.045 1.003 1.000 0.999
1.318 1.032 1.002 0.999 0.999
1.195 1.020 1.002 0.999 0.999
1.086 1.009 1.001 1.000 1.000
1.029 1.003 1.001 1.000 1.000
@/DA)
5-2
2-l
l-0.5
0.5-0.2
0.2-0.125
U&/K,)
(K;/K,) $ at 5” intervals
0.01 0.1 1 10 100
2.114 1.122 1.009 0.997 0.996
1.681 1.091 1.007 0.998 0.997
1.483 1.056 1.004 0.999 0.999
1.221 1.025 1.003 1.001 1 .ooo
1.079 1.009 1.002 1.001 1.001
(B/BA)
5-2
2-l
l-O.5
0.5-0.2
0.2-0.125
of anisotropy constants
335
torque curves is to use fields for which B/BA is always greater than unity. If this cannot be achieved then every case should be examined separately and particular care should be exercised in connection with the determination of the vulnerable higher order constants. It should be emphasized that the errors arising from the use of the full torque curves which have been analysed and quantified in section 2.1, all originate from the existence of a multidomain state when B < BK. There is,no way of correcting the torque curves adequately in this case since it is known [2] that the discrepancy between the experimentally observed magnetisation curves and those properly calculated using a two phase state is at its worst in the neigh. bourhood of BK. Au the really useful information in a torque curve is contained in the single-domain region for which B > BK and it could be advantageous to restrict the torque measurements to those value of 9 for which this condition is satisfied, preferably by a comfortable margin.
References [l] J. Burd, M. Huq and E.W. Lee, J. Mag. Mag. Mat. 5 (1977) 135. [2] Y. Barnier, R. Pauthenet and G. Rimet, Cobalt 15 (1962) 14.
Erratum causes the graph to rise above the line at low fields means that, in practice, the method is always likely to overestimate the value of Ki .
There is a mis-print in [ 11. The first term of eq. (A.l) should be:
3. Conclusion
- ((1 - kz - ;k;
It is clear that the only certain way to avoid errors in the determination of anisotropy constants from
- 2kr(l
- k;) (K1 + K2)
- 2kz)Kz}
sin 2$.
THE DETERMINATION OF ANISOTROPY CONSTANTS FROM TORQUE CURVES - II
M.HUQ Department of Physics, College of Science, Universityof Mosul, Iraq
and
E.W.LEE Department of Physics, Universityof Southampton, UK
Received 21 June 1978
This is an extension of a previous paper which examined the methods available for the determination of the anisotropy constant from torque curves when the anisotropy is governed by a single constant i.e. E = K1 sin20. The two methods singled out in that paper as being of special merit are briefly reexamined when the anisotropy energy takes the form E = K1 sin20 + K2 sin4e.
easy direction is the c-axis for all ratios K2/K1. (The notation throughout this section is the same as in [ 11). The torque per unit volume exerted by the crystal on the magnetic moment per unit volume is
1. Introduction In a previous paper [l] we analysed the procedures available to experimentalists required to determine magnetic anisotropy constants from torque curves. It was shown that when the anisotropy can be described by a single term K1 sin’0, two methods offer advantages over the others. These are 1) the method of torque correction and 2) the analysis of the torque close to the easy axis. Both methods when properly applied enable accurate values of K1 to be determined from torque measurements in external fields less than the anisotropy field. There are, however, many materials which require more than one term to describe the anisotropy energy and it is of some importance tp see whether these two methods can be safely employed in this case or whether special caution has to be used in applying them.
L(B) = -
aE/M = - (K1tK2)sin 28 t qK2 sin 48 . (2)
The magnetic field extends a torque L(B)=M,Bsina!. In equilibrium sina=
Kl+K2 MB
(3) L(B) + L(B)= 0 and so: sin 2((#l-
s
cx)- &
sin 4(f#~- a) .(4) s
From eqs. (2) and (4) it is possible to calculate the torque L(#, B)asa function of $ and B. This procedure is valid as long as the magnetisation can be represented by a single vector, that is when the crystal is a single domain. When the external field is below a critical value BK it is energetically favourable for two sets of domains to form. From the equations given in [2] it may be shown that
2.Theoretical considerations It is assumed that the crystal, uniaxial as in [l] is in the form of a disc and that the anisotropy
E = K, sin26 + K2 sin40 .
energy is:
BK = N2i@+(BAt
(1)
=2 t M
K1 and K2 are both taken to be positive so that the 333
s
2BANM,) sin20
(BA t NM,)sin48
8 1,
+ $$ z sin68
112
M. Huq and E. W. Lee /Determination
334
of anisotropy constants
Table 1
ApparentanisotropyconstantsKi and K$ obtained from the Fourier analysis of the torque curves calculated for different combinations of Kr and K2 using different values of B in units of BA 2
BIBA
5
(KlIK2)
(Ki/Kl)
(KYK2)
(Ki/Kt)
CKiIK2) tKi/Kt)
(KvK$
(Ki/Kl)
0.01 0.1 1 10 100
1.001 0.999 0.999 0.999 1.ooo
0.999 1.ooo 0.999 0.999 0.999
1.005 0.999 0.999 0.999 0.999
0.999 1.000 1.000 1.000 1.000
0.999 0.998 0.999 1.ooo 0.999
3.379 1.888 1.190 1.025 1.011
1
0.2
0.5
1.004 1 .ooo 1 .ooo 0.999 1.000
(KgK$ -0.059 -0.189 -0.195 -0.323 -4.651
CKi/Kl) 1.223 1.034 0.922 0.902 0.899
0.125 (KgK2) -0.008 -0.064 -0.552 -5.358 -53.39
where BA = 2K1/Ms and N is the demagnetising factor in the plane of the crystal. The torque in the two domain region is *:
2.2. The torque near the easy axis
L(@,B) = (B2/2N) sin 24~- BM, sin 0 cos C/J,
L(e)=-(K1+K2)2(+c+4(#-a)=ZK,($-a),
(5)
(Ki/KI) 0.519 ‘0.527 0.524 0.523 0.523
(KyK$ -0.003 -0.032 -0.315 -3.148 -31.475
In the limit as 4 + 0 eqs. (2) and (3) can be written:
in which 8 and I#Jare related by
(2Kr + NM;) sin 0 + 4Ka sin30 - BM, sin @= 0 . (6) A set of torque curves L(@,B) was calculated from eqs. (3-6) using a single value of Kr = lo7 erg/cm3 (lo6 J/m3) with Kr/K2 = 100, 10, l,O.l, 0.01, MS = 1000 emu/cm3 (106/4n A/m) and N = 1 (1/47r). These calculated curves formed the simulated data that were treated by the two methods specified in section 1.
(7)
and L(B) = M,Bcu .
(8)
From the condition that L(B) t L(B) = 0 it can be shown that
=L+L 2K1
M,B ’
(9)
2.1. The correction of torque curves
so that 2K1 may be determined from measurements of the slope of the torque curve L@, B) evaluated at Q= 0.
In this method one converts L(@,B) to L(B) by recognising that 8 = $ - (Yand sin (Y= I!+(#,B)IM,B. The corrected torque L(e) is then subject to Fourier analysis. This procedure was applied to the calculated torque curves and the apparent values of K1 and K2 thereby determined are shown in table 1. From this it may be seen that the procedure yields correct values of K1 and Ks only when B > BA. Once the external field falls below BA the method begins to falter and although K1 is not affected very much K2 emerges incorrect even as to sign.
The torque will normally be measured at discrete intervals of $Iand the problem now is that of knowing how closely spaced these intervals must be in order to yield accurate values of K1. To examine this L(#, B) was calculated for 9 = l”, 3” and 5’ and the slope dL/d$ was taken to be the quotient L/4 in each case. These values of L/G were analysed graphically in the light of eq. (9) and the value of K1 deduced from the intercept. The results are shown in table 2. The method performs very creditably; only when L(@, B) is measured at 5’ intervals and then in the rather unlikely case that K2 = 10 K1 does the error exceed 10%. Curiously, the error is incurred by using too high a field causing a breakdown of the approximation sin 44 = 4#. This implies that when Ka > K1 the graph of (a/d@)-’ falls below the straight line described by eq. (9) at high fields. This, together with the result [l] that incomplete saturation
l
Eq. (6) of [l] is wrong. The torque curves given in [l] are not affected since they were calculated from correct expressions for the transverse magnetisation.
M. Huq and E. W. Lee /Determination Table 2 Apparent anisotropy constant Ki obtained from the initial slope drawn at different angular position # of the torque curves (K&s)
(Ki/Kl), #at 1” intervals
0.01 0.1 1 10 100
1.052 1.005 1.001 1.000 1.000
1.039 1.004 1.001 1.000 1.000
1.024 1.003 1.000 1.000 1.000
1.010 1.002 1 .ooo 1 .ooo 1 .ooo
1.003 1.001 1 .ooo 1.000 1.000
(B/BA)
5-2
2-l
l-0.5
0.5-0.2
0.2-0.125
(K,fKsJ
(Ki/KI), Qat 3” intervals
0.01 0.1 1 10 100
1.441 1.045 1.003 1.000 0.999
1.318 1.032 1.002 0.999 0.999
1.195 1.020 1.002 0.999 0.999
1.086 1.009 1.001 1.000 1.000
1.029 1.003 1.001 1.000 1.000
@/DA)
5-2
2-l
l-0.5
0.5-0.2
0.2-0.125
U&/K,)
(K;/K,) $ at 5” intervals
0.01 0.1 1 10 100
2.114 1.122 1.009 0.997 0.996
1.681 1.091 1.007 0.998 0.997
1.483 1.056 1.004 0.999 0.999
1.221 1.025 1.003 1.001 1 .ooo
1.079 1.009 1.002 1.001 1.001
(B/BA)
5-2
2-l
l-O.5
0.5-0.2
0.2-0.125
of anisotropy constants
335
torque curves is to use fields for which B/BA is always greater than unity. If this cannot be achieved then every case should be examined separately and particular care should be exercised in connection with the determination of the vulnerable higher order constants. It should be emphasized that the errors arising from the use of the full torque curves which have been analysed and quantified in section 2.1, all originate from the existence of a multidomain state when B < BK. There is,no way of correcting the torque curves adequately in this case since it is known [2] that the discrepancy between the experimentally observed magnetisation curves and those properly calculated using a two phase state is at its worst in the neigh. bourhood of BK. Au the really useful information in a torque curve is contained in the single-domain region for which B > BK and it could be advantageous to restrict the torque measurements to those value of 9 for which this condition is satisfied, preferably by a comfortable margin.
References [l] J. Burd, M. Huq and E.W. Lee, J. Mag. Mag. Mat. 5 (1977) 135. [2] Y. Barnier, R. Pauthenet and G. Rimet, Cobalt 15 (1962) 14.
Erratum causes the graph to rise above the line at low fields means that, in practice, the method is always likely to overestimate the value of Ki .
There is a mis-print in [ 11. The first term of eq. (A.l) should be:
3. Conclusion
- ((1 - kz - ;k;
It is clear that the only certain way to avoid errors in the determination of anisotropy constants from
- 2kr(l
- k;) (K1 + K2)
- 2kz)Kz}
sin 2$.