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On the magnetization and magnetostriction of polycrystalline cobalt
Physica 31 1063-1068
Alberts, H. L. Alberts, L. 1965
ON THE MAGNETIZATION AND MAGNETOSTRICTION OF POLYCRYSTALLINE COBALT by H. L. ALBERTS Department
of Physics,
University
and L. ALBERTS
of the Orange South
Free State,
Bloemfontein,
Republic
of
Africa.
synopsis Extending the work of Bamier e.a. on a single crystal of cobalt, an attempt is made to predict the magnetization and magnetostriction curves of polycrystalline cobalt. Simplifying assumptions on the mechanical and magnetic interactions of the crystallites are made and the final results compared with experimental values.
1. Ilztrodzcctiolz.Calculations by Leel) of the magnetostriction curve for polycrystalline cobalt do not agree with the experimental results of L. and H. L. Albertss) (hereafter referred to as paper I). In Lee’s calculations it is assumed that domain wall movement is completed before rotation sets in. The work of Barnier e.u.3) on the magnetization of a single crystal of cobalt clearly indicates that rotation of the magnetic vectors and wall motion takes place simultaneously. In the present work these ideas are applied to polycrystalline material and curves are obtained for the magnetization and magnetostriction. For the sake of clarity it is necessary to review the work of Barnier e.a. briefly here. Consider a single, unit volume, crystal of cobalt, consisting of two domains, the domain wall being in the direction of the c axis. Let the volumes of the two domains be x and y so that x + y = 1. If there were several domains present with total volume x magnetized in the +c direction and volume y in the -c direction the results would still be the same. Consider an external field He making an angle I#Jwith the c axis. By considering the total energy of the system the equilibrium condition becomes (2Kr + 1212)sin 8 + 4Ks sina 8 - H, JB sin 4 = 0 He cos +
x-y=
2Js cos 8 ’
(1)
(4
where 8 is the angle that either domain vector makes with the c axis, K1 and Ks are the anisotropy constants, Js is the saturation magnetization and 12 is the demagnetizing factor for the crystal as a whole. The magnetization -
1063 -
1064 component
H. L. ALBERTS
AND
L. ALBERTS
in the direction of H, is given by: J/I = x.Ts cos (4 -
0) -
YJS cos (4 + 0).
(3)
When wall displacements are completed at a field value H, = H, say, for a given value of I#J,and further magnetization proceeds by rotation only, the equilibrium condition becomes (Kr + 2Ks sin%) sin 20 and the magnetization
in the direction
H, Js sin (4 -
agree remarkably
(4
of the applied field now becomes
JII = Js 44 These expressions same authors.
0) = 0
- 0).
well with the measured
(5) values of the
2. Calculations for polycrystalline material. From the work of Andra.4) Wyslocki5) Takata6) and others one can safely conclude that domains in a cobalt crystal are essentially always parallel to the c axes even if they are of a closure type near the surface of the crystal. Bloor and Martin’s7) investigations on polycrystalline silicon iron indicated that single crystal domain configurations are perpetuated in polycrystals. This was further borne out by Alberts and Shepstone’ss) measurements on the eddy current anomaly factor in polycrystalline iron. In view of its high crystalline anisotropy one would expect that this will almost certainly also be the case for cobalt. In the ensuing calculations it is thus assumed that the domains in the cobalt grains are initially parallel to the c axis in each grain. For a random distribution of grains the average magnetization in the direction of the field is given by 42
J
llaverege
=
$ JII sin4 d4
where C#is still the angle between H, and the c axis of a given grain and sin rj now represents the appropriate weighting factor. As is well known from the work of Holstein and Primakoffa), Law t o n and Stewart 10) and others a difficult problem in accounting for the magnetic behaviour in a polycrystalline material lies in estimating the demagnetising field for any given grain. This is dependent on the shape of the grain and the degree of magnetization of the material surrounding the given grain. In order to make the calculations possible over the whole range of magnetization we have taken the resulting field acting on a grain to be given by: H resultant where n is the demagnetization
--
He
-
~JII
factor for the whole specimen
and J/J has
MAGNETIZATION
AND
MAGNETOSTRICTION
1065
OF CO
the same significance as before. This is tantamount to assuming that each grain is surrounded on the average by material, throughout the whole specimen, in a state of magnetization similar to itself. Clearly the surrounding material will be less magnetised on the average than those grains favourably oriented with respect to the applied field and more so for those unfavourably oriented. The above assumption can be regarded as a fair approximation. On evaluating equation (6) in terms of the previous relations, by graphical method, the curve shown in figure 1 was obtained. It is shown with experimental values taken from paper I. In calculating the magnetostriction the four constant equation for a single crystal, as given by Mason 11), was used viz. 1 = Ji{(c~% +
a282j2
+
nz{(l
-
ai) (1 -
+
na{(l
-
ai)
+
4J4(alBl
+
Pi
-
(al/h Pi)
-
a2B2)
-
(a$1
+
a2B2)
(a& +
+ a2B2)
a3B3)
+
a2P2J2} a3B3)
a3B3.
+
+ (7)
In this equation cylindrical symmetry is assumed, the symbols have the usual meanings and the z axis is taken to coincide with the c axis of the crystal. The values of the 1’s were taken from B o z o r t hiz). Referring again to the case of two adjacent domains in a single crystal as described in paragraph 1. the magnetostriction in the x domain, as measured in the direction of the applied field, in terms of equation (7) now becomes jz = ili(sins 8 sin2 4 -
$ sin 28 sin 24)
+ ils(sin” 8 cos2+ -
$ sin 28 sin 24)
+ &(sin 28 sin 24)
(8)
and a similar expression for the magnetostriction in the y domain. The mean magnetostriction for a given field H, in the polycrystal will be given bY n/2 izaverage=J& sin + d+*
(9)
where I@ is the magnetostriction in a domain for a given value of # and He. Equation (9) was integrated graphically with the aid of equations (1) to (4) and (8). The result is depicted in figures 2 and 3 as a function of field and reduced magnetization, respectively, together with experimental values taken from paper I. 3. Disczlssion of resdts. In these calculations the mechanical interaction between grains has been neglected. The strain in each grain of a polycrystal,
1066
H. L. ALBERTS AND L. ALBERTS
I
2
3 APPLIED
4 FIELD
5 X lO-3
6 7 (OERSTCD)
8
v
IO
Fig. 1. Magnetization curve of polycrystalline cobalt at room temperature. Broken line gives theoretical curve.
0
-10
r
0
2
APPLIED 3
FIELD 4
X IO-’ 5
(OERSTEDJ 6 7
0
IO
t -20
-30 -0 x L
-40
0 : E
-50
i c y
-6C
Y I -70
-SC
Fig. 2. Magnetostriction of polycrystalline cobalt at room temperature, as a function of applied field. Broken line gives theoretical curve.
if isolated, will be different from the strains in its neighbours and obviously the grains exercise a mutual constraint on one another. If it is assumed that uniform stress exists throughout the material one can neglect this mutual interaction. According to Birss 1s) this assumption, as applied to polycrystalline iron and nickel is fairly well substantiated by experiment. It is
MAGNETIZATION AND MAGNETOSTRICTION OF Ca
I067
-801
Fig. 3. Magnetostriction of polycrystalline cobalt as a function of reduced magnetization. -o-o-oExperimental points - - - Theoretical curve ----Lee’s Curve.
also taken to be applicable to cobalt. The assumption made in paragraph 2 as regards the magnetic interaction between the grains is essentially justified by the fact that the theoretical and experimental curves show reasonable agreement. Other lesser causes responsible for the discrepancy between theory and experiment can be sought in factors such as the neglect of stress anisotropy and impurity effects and the limited accuracies of the single crystal anisotropy and magnetostriction constants. As a whole this work can be regarded as a first attempt to account for the magnetic behaviour of polycrystalline cobalt over the whole range of magnetization. Acknowledgement. It is a pleasure to thank the South African Council for Scientific and Industrial Research for financial aid for this investigation. Received 15-Z- 1965
REFERENCES 1) Lee, E. W., Proc. Phys. Sm. 73 (1958) 249. 2) Alberts, L. and Alberts, H. L., Phil. Mag. 8 (1963) 210. 3) Barnier, Y., Pauthenet, R., and Reinet, G., Comptes Rendus ZSZ (1961) 3024. 4) Andrl,
W., Ann. Phys. 15 (1954) 135.
5) W yslocki,
B., Phys. Stat. Sol. 3 (1963) 1333.
1068 6)
Takata,
7)
Bloor,
MAGNETIZATION
Y., J. Phys. Sm. Japan D., and Martin,
8) Alberts, 9) Holstein, 10)
Lawton,
11)
Mason,
12) 13)
Birss,
Bozorth,
AND
H., and Stewart, W. P., Phys.
Phys. Sot. 73 (1959) 694.
B. J., Proc.
T., and Primakoff,
Phys. Sot. 75 (1960) 539.
H., Phys. Rev. 58 (1941) 388. K. H., Proc.
Phys. Sot. A83
Rev. 88 (1954) 302.
R. M., Phys.
Rev.
OF CO
18 (1963) 87.
D. H., Proc.
L., andshepstone,
MAGNETOSTRICTION
88 (1954) 311.
R. R., Phil. Msg. Suppl.
8 (1959) 252.
(1950) 848.
Alberts, H. L. Alberts, L. 1965
ON THE MAGNETIZATION AND MAGNETOSTRICTION OF POLYCRYSTALLINE COBALT by H. L. ALBERTS Department
of Physics,
University
and L. ALBERTS
of the Orange South
Free State,
Bloemfontein,
Republic
of
Africa.
synopsis Extending the work of Bamier e.a. on a single crystal of cobalt, an attempt is made to predict the magnetization and magnetostriction curves of polycrystalline cobalt. Simplifying assumptions on the mechanical and magnetic interactions of the crystallites are made and the final results compared with experimental values.
1. Ilztrodzcctiolz.Calculations by Leel) of the magnetostriction curve for polycrystalline cobalt do not agree with the experimental results of L. and H. L. Albertss) (hereafter referred to as paper I). In Lee’s calculations it is assumed that domain wall movement is completed before rotation sets in. The work of Barnier e.u.3) on the magnetization of a single crystal of cobalt clearly indicates that rotation of the magnetic vectors and wall motion takes place simultaneously. In the present work these ideas are applied to polycrystalline material and curves are obtained for the magnetization and magnetostriction. For the sake of clarity it is necessary to review the work of Barnier e.a. briefly here. Consider a single, unit volume, crystal of cobalt, consisting of two domains, the domain wall being in the direction of the c axis. Let the volumes of the two domains be x and y so that x + y = 1. If there were several domains present with total volume x magnetized in the +c direction and volume y in the -c direction the results would still be the same. Consider an external field He making an angle I#Jwith the c axis. By considering the total energy of the system the equilibrium condition becomes (2Kr + 1212)sin 8 + 4Ks sina 8 - H, JB sin 4 = 0 He cos +
x-y=
2Js cos 8 ’
(1)
(4
where 8 is the angle that either domain vector makes with the c axis, K1 and Ks are the anisotropy constants, Js is the saturation magnetization and 12 is the demagnetizing factor for the crystal as a whole. The magnetization -
1063 -
1064 component
H. L. ALBERTS
AND
L. ALBERTS
in the direction of H, is given by: J/I = x.Ts cos (4 -
0) -
YJS cos (4 + 0).
(3)
When wall displacements are completed at a field value H, = H, say, for a given value of I#J,and further magnetization proceeds by rotation only, the equilibrium condition becomes (Kr + 2Ks sin%) sin 20 and the magnetization
in the direction
H, Js sin (4 -
agree remarkably
(4
of the applied field now becomes
JII = Js 44 These expressions same authors.
0) = 0
- 0).
well with the measured
(5) values of the
2. Calculations for polycrystalline material. From the work of Andra.4) Wyslocki5) Takata6) and others one can safely conclude that domains in a cobalt crystal are essentially always parallel to the c axes even if they are of a closure type near the surface of the crystal. Bloor and Martin’s7) investigations on polycrystalline silicon iron indicated that single crystal domain configurations are perpetuated in polycrystals. This was further borne out by Alberts and Shepstone’ss) measurements on the eddy current anomaly factor in polycrystalline iron. In view of its high crystalline anisotropy one would expect that this will almost certainly also be the case for cobalt. In the ensuing calculations it is thus assumed that the domains in the cobalt grains are initially parallel to the c axis in each grain. For a random distribution of grains the average magnetization in the direction of the field is given by 42
J
llaverege
=
$ JII sin4 d4
where C#is still the angle between H, and the c axis of a given grain and sin rj now represents the appropriate weighting factor. As is well known from the work of Holstein and Primakoffa), Law t o n and Stewart 10) and others a difficult problem in accounting for the magnetic behaviour in a polycrystalline material lies in estimating the demagnetising field for any given grain. This is dependent on the shape of the grain and the degree of magnetization of the material surrounding the given grain. In order to make the calculations possible over the whole range of magnetization we have taken the resulting field acting on a grain to be given by: H resultant where n is the demagnetization
--
He
-
~JII
factor for the whole specimen
and J/J has
MAGNETIZATION
AND
MAGNETOSTRICTION
1065
OF CO
the same significance as before. This is tantamount to assuming that each grain is surrounded on the average by material, throughout the whole specimen, in a state of magnetization similar to itself. Clearly the surrounding material will be less magnetised on the average than those grains favourably oriented with respect to the applied field and more so for those unfavourably oriented. The above assumption can be regarded as a fair approximation. On evaluating equation (6) in terms of the previous relations, by graphical method, the curve shown in figure 1 was obtained. It is shown with experimental values taken from paper I. In calculating the magnetostriction the four constant equation for a single crystal, as given by Mason 11), was used viz. 1 = Ji{(c~% +
a282j2
+
nz{(l
-
ai) (1 -
+
na{(l
-
ai)
+
4J4(alBl
+
Pi
-
(al/h Pi)
-
a2B2)
-
(a$1
+
a2B2)
(a& +
+ a2B2)
a3B3)
+
a2P2J2} a3B3)
a3B3.
+
+ (7)
In this equation cylindrical symmetry is assumed, the symbols have the usual meanings and the z axis is taken to coincide with the c axis of the crystal. The values of the 1’s were taken from B o z o r t hiz). Referring again to the case of two adjacent domains in a single crystal as described in paragraph 1. the magnetostriction in the x domain, as measured in the direction of the applied field, in terms of equation (7) now becomes jz = ili(sins 8 sin2 4 -
$ sin 28 sin 24)
+ ils(sin” 8 cos2+ -
$ sin 28 sin 24)
+ &(sin 28 sin 24)
(8)
and a similar expression for the magnetostriction in the y domain. The mean magnetostriction for a given field H, in the polycrystal will be given bY n/2 izaverage=J& sin + d+*
(9)
where I@ is the magnetostriction in a domain for a given value of # and He. Equation (9) was integrated graphically with the aid of equations (1) to (4) and (8). The result is depicted in figures 2 and 3 as a function of field and reduced magnetization, respectively, together with experimental values taken from paper I. 3. Disczlssion of resdts. In these calculations the mechanical interaction between grains has been neglected. The strain in each grain of a polycrystal,
1066
H. L. ALBERTS AND L. ALBERTS
I
2
3 APPLIED
4 FIELD
5 X lO-3
6 7 (OERSTCD)
8
v
IO
Fig. 1. Magnetization curve of polycrystalline cobalt at room temperature. Broken line gives theoretical curve.
0
-10
r
0
2
APPLIED 3
FIELD 4
X IO-’ 5
(OERSTEDJ 6 7
0
IO
t -20
-30 -0 x L
-40
0 : E
-50
i c y
-6C
Y I -70
-SC
Fig. 2. Magnetostriction of polycrystalline cobalt at room temperature, as a function of applied field. Broken line gives theoretical curve.
if isolated, will be different from the strains in its neighbours and obviously the grains exercise a mutual constraint on one another. If it is assumed that uniform stress exists throughout the material one can neglect this mutual interaction. According to Birss 1s) this assumption, as applied to polycrystalline iron and nickel is fairly well substantiated by experiment. It is
MAGNETIZATION AND MAGNETOSTRICTION OF Ca
I067
-801
Fig. 3. Magnetostriction of polycrystalline cobalt as a function of reduced magnetization. -o-o-oExperimental points - - - Theoretical curve ----Lee’s Curve.
also taken to be applicable to cobalt. The assumption made in paragraph 2 as regards the magnetic interaction between the grains is essentially justified by the fact that the theoretical and experimental curves show reasonable agreement. Other lesser causes responsible for the discrepancy between theory and experiment can be sought in factors such as the neglect of stress anisotropy and impurity effects and the limited accuracies of the single crystal anisotropy and magnetostriction constants. As a whole this work can be regarded as a first attempt to account for the magnetic behaviour of polycrystalline cobalt over the whole range of magnetization. Acknowledgement. It is a pleasure to thank the South African Council for Scientific and Industrial Research for financial aid for this investigation. Received 15-Z- 1965
REFERENCES 1) Lee, E. W., Proc. Phys. Sm. 73 (1958) 249. 2) Alberts, L. and Alberts, H. L., Phil. Mag. 8 (1963) 210. 3) Barnier, Y., Pauthenet, R., and Reinet, G., Comptes Rendus ZSZ (1961) 3024. 4) Andrl,
W., Ann. Phys. 15 (1954) 135.
5) W yslocki,
B., Phys. Stat. Sol. 3 (1963) 1333.
1068 6)
Takata,
7)
Bloor,
MAGNETIZATION
Y., J. Phys. Sm. Japan D., and Martin,
8) Alberts, 9) Holstein, 10)
Lawton,
11)
Mason,
12) 13)
Birss,
Bozorth,
AND
H., and Stewart, W. P., Phys.
Phys. Sot. 73 (1959) 694.
B. J., Proc.
T., and Primakoff,
Phys. Sot. 75 (1960) 539.
H., Phys. Rev. 58 (1941) 388. K. H., Proc.
Phys. Sot. A83
Rev. 88 (1954) 302.
R. M., Phys.
Rev.
OF CO
18 (1963) 87.
D. H., Proc.
L., andshepstone,
MAGNETOSTRICTION
88 (1954) 311.
R. R., Phil. Msg. Suppl.
8 (1959) 252.
(1950) 848.