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Macroscopic behaviour of ferromagnets with random and statistically vanishing anisotropy
Solid State Communications, Vol. 42, No. 2, pp. 113-117. 1982. Printed in Great Britain.
0038-1098/82/140113-05503.00/0 Pergamon Press Ltd.
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS WITH RANDOM AND STATISTICALLY VANISHING ANISOTROPY P. Averbuch Centre de Recherches sur les Tr6s Basses Tempdratures, CNRS, B.P. 166 X, 38042 Grenoble Cddex, France
(Received 21 September 1981 ; in revised form 24 November 1981 by E.F. Bertaut) A model is defined in which the anisotropy tensor is a random function of space point characterised by its mean square value o and a correlation length ae, and the exchange density A is uniform. If the magnetic moment density is M, it is shown that two dimensionless numbers can be defined at~as, where aS is a typical Bloch wall width (A/o) 1/2, and 4zrM2/o; they define four types of macroscopic behaviour. For 4~rM2/tr < 1, the pole fields are a perturbation; i f a c < ap, as in rare earth amorphous alloys, there is some frustration in the ground state and ifap < ae, as in inhomogeneous weak ferromagnets the magnetization direction is mainly fixed by the local anisotropy. For 41rM2/o > 1, as in iron alloys, ifap > ac one has the case of soft alloys, the shape anisotropy is the dominant effect and if ap < a e the usual domain theory applies.
1. INTRODUCTION THE STANDARD DESCRIPTION of a macroscopic ferromagnet assumes the system to be made of polycrystals, with a given anisotropy field and the magnetization aligned by the exchange force in this anisotropy field. Then the pole fields is introduced and its effects on the system, the Bloch wall behaviour . . . . are studied. There is in this way no answer to the question of describing the behaviour of a system whose bulk anisotropy is vanishing. In this note, an attempt is made in this direction; as the systems with vanishing anisotropy (and with vanishing magnetocrystalline effects, which play roughly the same role) are alloys [1,2], one can assume that locally the anistropy field is non vanishing, that it vanishes only when its mean value is taken in some volume, and thus study the consequences of this model. So in Section 2 a model of a random anisotropy field is described that is an anistropy tensor whose value and orientation varies from point to point within the sample. To make the calculation easier, one assumes a second rank tensor. Equally, one keeps only the isotropic part of the exchange interaction. Then as a random function of time has a correlation time, one has here a correlation length ac which is one of the parameters of the model. With a further "generalized isotropy" assumption, the model is restricted to two parameters, the second one being o the mean square anisotropy energy. Then, neglecting for the time being the pole field,
one tries to find the nature of the classical ground state of such a system, in Section 3, assuming an uniform exchange energy density A, in the spirit of the Harris, Plischke and Zuckermann model [3]. It is rather intuitive that, if the anisotropy field has a long spacememory, the magnetization will be aligned along it. The critical parameter is the ratio ofae to a second length ap = (A/o) 1/2, the width of a Bloch wall in an homogeneous system whose anisotropy is equal to the mean square value o. The calculation diverges when ac/ap goes to one, showing something like the introduction of some frustration in the system. Furthermore in Section 4 we deal with effects of the pole field neglected till now. It is shown that there is another dimensionless critical parameter. The important quantity if 47rM2, the internal demagnetization energy density, where M is the magnetic moment density. This quantity is to be compared to o, and their ratio is the second dimensionless parameter. Finally a classification is attempted in terms of the two dimensionless parameters in order to characterise the macroscopic behaviour of materials, as long as shape effects of the samples are not taken into account. 2. THE RANDOM ANISTROPY MODEL The micromagnetism problem is to fred the minimum value of the total energy of a magnetic system. One can take as variables the components u~ of a unit vector parallel to the local magnetization M, knowing that they are related by
113
114
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
~u~ :
I.
(1)
i
Another variable is the magnetic potential qJ, the magnetic field being
n
--V~.
=
(2)
Introducing, as described in Section 1, a second rank anisotropy tensor K~j(r) varying with position and a uniform isotropic exchange density, the potential to be minimized is the space integral of the energy density V
A
X (Vu,) 2 +
~ Kij(r)uiu j + M "V@ -- -~ (V@) 2,
'
iJ
(3)
where the last term bears the minus sign, to avoid counting twice the self energy. It can be checked easily by writing the Euler equation of the energy V vs the field ~ which is, as it must be, B being the induction V.B
=
V.(47rM--V~,)
0.
=
Let us now discuss the random tensor K/j. First it can be taken traceless if one introduces at each point the unitarity of the vector ui that is equation (1). Secondly as ui is a classical vector K 0 can always be taken as symmetrical K~i = Kji. So one is left with a tensor belonging to the representation-~2 of the rotation group and what follows is a generalization of a trick used to describe the statistics of a deformation field [4]. Let us introduce
K+_2 = x/6/4(Kxx --Ky v + 2iKxy), K+_, = X/6/2(Kxx +-iKyz) Ko
=
K = --
K*(r)Km'(r + p) = tS6mra'O2(Ipl).
(6)
The analogy with random function is now clear and it is now possible to define a correlation tength a c which fixes the scale o f spatial decrease of oZ(IPl). In the following we shall denote by o(0) by o. 3. MINIMUM IN THE ABSENCE OF POLE FIELD In order to find the ground state of equation (3), the standard procedure is to minimize the space integral of V, taking equation (1) into account by multiplying it by a Lagrange multiplier field. Instead of using this technique, another possibility is to study the transformation of equation (3) if one uses a different axis system to describe the ui for each space point r. If then this axis rotation - a non-cummuting gauge field - is such that the third axis is along the u vector in the minimum energy solution, one has solved the problem. Let us start without the pole field terms, using equation (3) reduced to its first two terms. To make a rotation of the axis system in each point one can define a matrix aij = e i ' f j ,
(7)
where ei and f1 are the old and the new unit vectors. The components of the u vector in both systems are related by
u~ = ~ a~ivj; l the a~s are, of course, functions of r. One must transform into those local axis the expression of the exchange energy and one finds
(8)
+
The correlation properties of the anisotropy tensor are given by the quantities K*(r)Km'(r + p),
Vol. 42, No. 2
(m, m'
=
0, -+ 1, +- 2),
+ vhVv i "aoVaik + vjvkVaij "Vaik ].
(4)
where the mean values are taken over the whole sample or over many equivalent samples. To simplify one assumes the quantities (4) to depend only on the absolute value IPl although there are no symmetry reasons for that. With that assumption, by a rotation of the axis of Euler angles,a,/3, 7, one gets
(9)
Using the relation = ~]k i and its space derivative
Z O[ijOlik
Z V(aij°tik) = X (aij VOLik + aikVaij ) = 0 i i and introducing them in equation (9), one gets
K~(r)Ku'(r + p) = ~ ( # ~ * ( a , / 3 , 7)lm) tutti'
x K,~(r)Km'(r + p)
Z VUi'VUi = X [ai.iOtikVV]'V~)k + V.iVVk'OtikVaij i ijk
(5)
Now the isotropy of the simple implies equality of the matrix K ~ K u, with the matrix K*Km' and from the Shur lemma commutation with all-~2 matrices. It is thus a scalar and
Y.
.Vu
= Z w,j-w,j + £ j
x ~ ~ikVaii + ~. vjvkVa~i"Vaik. i
- vkvvj)
ik
ijk
(10)
Vol. 42, No. 2
115
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
The anisotropy energy is more easily written in the new axes, defining K~) = ~ Ktaaikajt.
E /e'u2 Ol
(1 I)
kI
Ol
lXmn a m j a n k
ran
~"
V nO nO ~.01 ~,01 lxiI Utim ttln ~rajt~nk
ilmn
4-a
Z
VenOm • t"F~O ~01 ^,01 VtqntXmjCtnk
iron
So now, the first two terms of the energy density (3) can be written in the new axes
is diagonal. Let us define the new transformation matrix
v, = A ZCw,) + Y
the new tensor
i
+ a Z(vao'val,)l
jk
jvk
a 1 = ot°a°l;
(18)
i
~rt2
'l- j~k (~i aik ~Olij) " (Vj~k -- Pk~73j).
(12)
If the local axis transform is such that the local third axis is aligned on the magnetization direction of the minimum energy configuration, one has in each point
• ,jk
= X Ki,aOa:n * ' + AXV~b'Va~k i!
i
-----E
1 1 Knaifllk +A ~
i!
~'/nO .w^O nO1 nOl
v ~ira
v ttin~rrti~nk
imn
--]-m E
_0 _0 ,-/,_01. ctiraOtinVOtm j V a n01 h
(19)
iron
vl = v2 = 0,
v3 = 1,
(13)
the first and third terms of equation (I 2) vanish, the second is a diagonalized tensor with its third eigenvalue the smallest. Conversely if one has a continuous change of axis ao(r ) such that K/~ + A ~ i ( V O t i j " V a i k ) is diagonal, equation (13) gives an extremal solution for the total energy as can be checked by using small variations around. The problem is replaced by that of finding transformations of axes which diagonalize a matrix function of those transformations themselves. There is a self-consistency problem. An iterative procedure can be tried. Let us start by diagonalizing K(r) in its primitive form as extracted from equation (3). This means, for the first approximation step,
= E Kit aijatk, o o
(14)
i
to be diagonal and defines the corresponding transformation matrices a°(r). Let us now consider the sum of the true anisotropy and of the part of the exchange which is anisotropy shaped in this axis system: Ky~* = E K/ta~a~k + A E V a ° .Va°k, ilil
(15)
1
which is not diagonal. As in order of magnitude, one has 2rid term 1st term
A(a°/ac)~ ~
A =
(ac) -~ =
(an~~ \ac] ,
(16)
a perturbation calculation should converge if
aria e < 1.
(17)
This calculation should give another rotation of the axis, given by ot°'(r) and
is the sum of a diagonal part made of its first two terms and a non-diagonal one, which can be treated as a perturbation, thanks to equation (16). So an iterative procedure has been defined, showing that the solution where in each point the magnetization is aligned along the anisotropy field can be taken as starting point towards a self consistent diagonalization of anisotropy plus exchange. On the other hand, when ac < av,
(20)
the reverse of equation (17) is fulfilled, there is no simple iteration. One cannot inverse the iterative perturbation procedure, because one should have to start with a uniformly aligned solution which is degenerate; this degeneracy should when lifted simply give again the anisotropy aligned solution; and furthermore the exchange part of the energy would be the principal part of the matrix which would change from one order of iteration to the next. This difficulty is implicit in the Harris, Plischke and Zuckermann model of amorphous rare earth alloys, which of course fulfill equation (20), and probably is the source of the frustration which should be found in its solutions [5, 6]. In any case, when the pole field effects can be neglected, at least in first order, one can define two regimes; the first, where the random anisotropy varies rapidly from one point to another, the condition (20) being fulfilled corresponds mainly to amorphous alloys of rare earth and had been well studied theoretically and experimentally; the second regime, where contrarily equation (17) is fulfilled can be assigned as pseudoamorphous; here the local anisotropy dominates and as it varies slowly with the distance the magnetization is only a little bit further twisted by the exchange torque.
116
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
Vol. 42, No. 2
.Table 1. Classification o f the different types o f macroscopic behaviour o f random anisotropy f erromagnets as function o f the two dimensionless parameters ap/ae and 4rrM2/o 4nM 2 --< o
a~
ap
a_£>l ao
41rM 2
1
- - > t7
1
Rare earth amorphous alloys.
Soft magnetic materials.
Existence of some frustration, although probably less than in spin-glasses
Usually shape induced anisotropy
Pseudo-amorphous (case of weak ferromagnets?)
Polycrystals of standard transition metals and their alloys
Magnetization not much deviated from local anisotropy
Classical domain theory of polycrystals.
axis
4. EFFECTS OF THE POLE FIELD In the conventional description of ferromagnets, one considers uniform domains so the only source of pole field is at the sample boundaries and sometimes at the domain boundaries. Nevertheless it cannot be neglected in many problems such as Block wall kinetics [7] and coercive field theory [8] for instance. So one must here, especially when studying non-uniform solutions, not forget the third and fourth terms of equation (3), assuming no external sources of field. Let us start by looking for the conditions needed if one wants those magnetostatic terms to be small enough for a perturbation. In the standard minimization procedure of the space integral of equation (3), the Euler-Lagrange equation for ff does not depend on the condition (1) which does not imply ft. As written above it is simply V-OV $ - V" M - -
V~O = O,
(21)
or
A~k = 4rrM-u.
(21')
If one designates by ar the spatial rate of rotation of u, that is the correlation length of the magnetization in the solution of the micromagnetic problem, one will have approximately A ~ "~ 41rMar 1,
(22)
IVffl ~ 4rrM,
(22')
so the order of magnitude of the magnetostatic terms in equation (3) will be 4rrM 2 for any spatial variation of the magnetization direction. Comparing this term to the anisotropy, we can define in supplement to ap/a e another dimensionless parameter 4rrM2/o.
It is now possible to attempt a classification of all situations given in Table 1. If 47rM2/o < 1,
(23)
the pole field effect is a simple perturbation of the solutions described in the preceding paragraph. For a e "( ap, that is for rare earth amorphous alloys one has tricky partially frustrated systems and for a o < a e the pseudo-amorphous situation where the local anisotropy dominates. It is suggested that such a situation occurs in weak a ferromagnets like ZrZn2. In contrast for 4¢rM2/o > 1
(24)
the anisotropy is to be introduced after the pole field. It is a rather common situation this ratio being of the order of 10 in iron alloys. One has clearly two different regimes corresponding to the value ofao/a e. Ifao/a e, this is the case of polycrystals, where ac is a typical length scale of compounding monocrystals and the mostly studied situation; one must remark that this condition is given in Ndel's papers on Rayleigh laws [9]. At the contrary, if a e ( a p , the situation can be very complex; it is suggested that soft materials, peculiarly F e - N i alloys with K = 0 are in those conditions. As in practical cases, they occur in sheets, the shape magnetostatic effect plays the dominant role of a uniform anisotropy tensor and classical model applies. Acknowledgements - The author is indebted to Drs J.F. Tiers and G. Couderchon from Imphy Steel Company for many discussions from which the definition of the problem arose. He is also indebted to Dr R. Rammal lbr illuminating points of view.
1.
REFERENCES F. Pfeifer & R. Boll, IEEE Trans. Magn. Mag-5,365 (1969).
Vol. 42, No. 2 2. 3. 4. 5.
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
G.Y. Chin,IEEE TransMagn. Mag-7,102(1971). R. Harris, M. Plischke & M.J. Zuckermann, Phys. Rev. Lett. 31,160 (1973). P. Averbuch, C.R. Acad. Sci. 253, 2674 (1961). E. Callen, Y.J. Liu & J.R. Cullen, Phys. Rev. B16, 263 (1977).
6.
7. 8. 9.
117
J.D. Pattemson, G.R. Gruzalski & D.J. Sellmyer, Phys. Rev. B18, 1377 (1978). W. D6ring, Z. Naturf. 3a, 373 (1948). L. N6el, Cahiers de Physique 25, 21 (1944). L. N~el, Cahiers de Physique 12, 1 (1942).
0038-1098/82/140113-05503.00/0 Pergamon Press Ltd.
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS WITH RANDOM AND STATISTICALLY VANISHING ANISOTROPY P. Averbuch Centre de Recherches sur les Tr6s Basses Tempdratures, CNRS, B.P. 166 X, 38042 Grenoble Cddex, France
(Received 21 September 1981 ; in revised form 24 November 1981 by E.F. Bertaut) A model is defined in which the anisotropy tensor is a random function of space point characterised by its mean square value o and a correlation length ae, and the exchange density A is uniform. If the magnetic moment density is M, it is shown that two dimensionless numbers can be defined at~as, where aS is a typical Bloch wall width (A/o) 1/2, and 4zrM2/o; they define four types of macroscopic behaviour. For 4~rM2/tr < 1, the pole fields are a perturbation; i f a c < ap, as in rare earth amorphous alloys, there is some frustration in the ground state and ifap < ae, as in inhomogeneous weak ferromagnets the magnetization direction is mainly fixed by the local anisotropy. For 41rM2/o > 1, as in iron alloys, ifap > ac one has the case of soft alloys, the shape anisotropy is the dominant effect and if ap < a e the usual domain theory applies.
1. INTRODUCTION THE STANDARD DESCRIPTION of a macroscopic ferromagnet assumes the system to be made of polycrystals, with a given anisotropy field and the magnetization aligned by the exchange force in this anisotropy field. Then the pole fields is introduced and its effects on the system, the Bloch wall behaviour . . . . are studied. There is in this way no answer to the question of describing the behaviour of a system whose bulk anisotropy is vanishing. In this note, an attempt is made in this direction; as the systems with vanishing anisotropy (and with vanishing magnetocrystalline effects, which play roughly the same role) are alloys [1,2], one can assume that locally the anistropy field is non vanishing, that it vanishes only when its mean value is taken in some volume, and thus study the consequences of this model. So in Section 2 a model of a random anisotropy field is described that is an anistropy tensor whose value and orientation varies from point to point within the sample. To make the calculation easier, one assumes a second rank tensor. Equally, one keeps only the isotropic part of the exchange interaction. Then as a random function of time has a correlation time, one has here a correlation length ac which is one of the parameters of the model. With a further "generalized isotropy" assumption, the model is restricted to two parameters, the second one being o the mean square anisotropy energy. Then, neglecting for the time being the pole field,
one tries to find the nature of the classical ground state of such a system, in Section 3, assuming an uniform exchange energy density A, in the spirit of the Harris, Plischke and Zuckermann model [3]. It is rather intuitive that, if the anisotropy field has a long spacememory, the magnetization will be aligned along it. The critical parameter is the ratio ofae to a second length ap = (A/o) 1/2, the width of a Bloch wall in an homogeneous system whose anisotropy is equal to the mean square value o. The calculation diverges when ac/ap goes to one, showing something like the introduction of some frustration in the system. Furthermore in Section 4 we deal with effects of the pole field neglected till now. It is shown that there is another dimensionless critical parameter. The important quantity if 47rM2, the internal demagnetization energy density, where M is the magnetic moment density. This quantity is to be compared to o, and their ratio is the second dimensionless parameter. Finally a classification is attempted in terms of the two dimensionless parameters in order to characterise the macroscopic behaviour of materials, as long as shape effects of the samples are not taken into account. 2. THE RANDOM ANISTROPY MODEL The micromagnetism problem is to fred the minimum value of the total energy of a magnetic system. One can take as variables the components u~ of a unit vector parallel to the local magnetization M, knowing that they are related by
113
114
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
~u~ :
I.
(1)
i
Another variable is the magnetic potential qJ, the magnetic field being
n
--V~.
=
(2)
Introducing, as described in Section 1, a second rank anisotropy tensor K~j(r) varying with position and a uniform isotropic exchange density, the potential to be minimized is the space integral of the energy density V
A
X (Vu,) 2 +
~ Kij(r)uiu j + M "V@ -- -~ (V@) 2,
'
iJ
(3)
where the last term bears the minus sign, to avoid counting twice the self energy. It can be checked easily by writing the Euler equation of the energy V vs the field ~ which is, as it must be, B being the induction V.B
=
V.(47rM--V~,)
0.
=
Let us now discuss the random tensor K/j. First it can be taken traceless if one introduces at each point the unitarity of the vector ui that is equation (1). Secondly as ui is a classical vector K 0 can always be taken as symmetrical K~i = Kji. So one is left with a tensor belonging to the representation-~2 of the rotation group and what follows is a generalization of a trick used to describe the statistics of a deformation field [4]. Let us introduce
K+_2 = x/6/4(Kxx --Ky v + 2iKxy), K+_, = X/6/2(Kxx +-iKyz) Ko
=
K = --
K*(r)Km'(r + p) = tS6mra'O2(Ipl).
(6)
The analogy with random function is now clear and it is now possible to define a correlation tength a c which fixes the scale o f spatial decrease of oZ(IPl). In the following we shall denote by o(0) by o. 3. MINIMUM IN THE ABSENCE OF POLE FIELD In order to find the ground state of equation (3), the standard procedure is to minimize the space integral of V, taking equation (1) into account by multiplying it by a Lagrange multiplier field. Instead of using this technique, another possibility is to study the transformation of equation (3) if one uses a different axis system to describe the ui for each space point r. If then this axis rotation - a non-cummuting gauge field - is such that the third axis is along the u vector in the minimum energy solution, one has solved the problem. Let us start without the pole field terms, using equation (3) reduced to its first two terms. To make a rotation of the axis system in each point one can define a matrix aij = e i ' f j ,
(7)
where ei and f1 are the old and the new unit vectors. The components of the u vector in both systems are related by
u~ = ~ a~ivj; l the a~s are, of course, functions of r. One must transform into those local axis the expression of the exchange energy and one finds
(8)
+
The correlation properties of the anisotropy tensor are given by the quantities K*(r)Km'(r + p),
Vol. 42, No. 2
(m, m'
=
0, -+ 1, +- 2),
+ vhVv i "aoVaik + vjvkVaij "Vaik ].
(4)
where the mean values are taken over the whole sample or over many equivalent samples. To simplify one assumes the quantities (4) to depend only on the absolute value IPl although there are no symmetry reasons for that. With that assumption, by a rotation of the axis of Euler angles,a,/3, 7, one gets
(9)
Using the relation = ~]k i and its space derivative
Z O[ijOlik
Z V(aij°tik) = X (aij VOLik + aikVaij ) = 0 i i and introducing them in equation (9), one gets
K~(r)Ku'(r + p) = ~ ( # ~ * ( a , / 3 , 7)lm) tutti'
x K,~(r)Km'(r + p)
Z VUi'VUi = X [ai.iOtikVV]'V~)k + V.iVVk'OtikVaij i ijk
(5)
Now the isotropy of the simple implies equality of the matrix K ~ K u, with the matrix K*Km' and from the Shur lemma commutation with all-~2 matrices. It is thus a scalar and
Y.
.Vu
= Z w,j-w,j + £ j
x ~ ~ikVaii + ~. vjvkVa~i"Vaik. i
- vkvvj)
ik
ijk
(10)
Vol. 42, No. 2
115
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
The anisotropy energy is more easily written in the new axes, defining K~) = ~ Ktaaikajt.
E /e'u2 Ol
(1 I)
kI
Ol
lXmn a m j a n k
ran
~"
V nO nO ~.01 ~,01 lxiI Utim ttln ~rajt~nk
ilmn
4-a
Z
VenOm • t"F~O ~01 ^,01 VtqntXmjCtnk
iron
So now, the first two terms of the energy density (3) can be written in the new axes
is diagonal. Let us define the new transformation matrix
v, = A ZCw,) + Y
the new tensor
i
+ a Z(vao'val,)l
jk
jvk
a 1 = ot°a°l;
(18)
i
~rt2
'l- j~k (~i aik ~Olij) " (Vj~k -- Pk~73j).
(12)
If the local axis transform is such that the local third axis is aligned on the magnetization direction of the minimum energy configuration, one has in each point
• ,jk
= X Ki,aOa:n * ' + AXV~b'Va~k i!
i
-----E
1 1 Knaifllk +A ~
i!
~'/nO .w^O nO1 nOl
v ~ira
v ttin~rrti~nk
imn
--]-m E
_0 _0 ,-/,_01. ctiraOtinVOtm j V a n01 h
(19)
iron
vl = v2 = 0,
v3 = 1,
(13)
the first and third terms of equation (I 2) vanish, the second is a diagonalized tensor with its third eigenvalue the smallest. Conversely if one has a continuous change of axis ao(r ) such that K/~ + A ~ i ( V O t i j " V a i k ) is diagonal, equation (13) gives an extremal solution for the total energy as can be checked by using small variations around. The problem is replaced by that of finding transformations of axes which diagonalize a matrix function of those transformations themselves. There is a self-consistency problem. An iterative procedure can be tried. Let us start by diagonalizing K(r) in its primitive form as extracted from equation (3). This means, for the first approximation step,
= E Kit aijatk, o o
(14)
i
to be diagonal and defines the corresponding transformation matrices a°(r). Let us now consider the sum of the true anisotropy and of the part of the exchange which is anisotropy shaped in this axis system: Ky~* = E K/ta~a~k + A E V a ° .Va°k, ilil
(15)
1
which is not diagonal. As in order of magnitude, one has 2rid term 1st term
A(a°/ac)~ ~
A =
(ac) -~ =
(an~~ \ac] ,
(16)
a perturbation calculation should converge if
aria e < 1.
(17)
This calculation should give another rotation of the axis, given by ot°'(r) and
is the sum of a diagonal part made of its first two terms and a non-diagonal one, which can be treated as a perturbation, thanks to equation (16). So an iterative procedure has been defined, showing that the solution where in each point the magnetization is aligned along the anisotropy field can be taken as starting point towards a self consistent diagonalization of anisotropy plus exchange. On the other hand, when ac < av,
(20)
the reverse of equation (17) is fulfilled, there is no simple iteration. One cannot inverse the iterative perturbation procedure, because one should have to start with a uniformly aligned solution which is degenerate; this degeneracy should when lifted simply give again the anisotropy aligned solution; and furthermore the exchange part of the energy would be the principal part of the matrix which would change from one order of iteration to the next. This difficulty is implicit in the Harris, Plischke and Zuckermann model of amorphous rare earth alloys, which of course fulfill equation (20), and probably is the source of the frustration which should be found in its solutions [5, 6]. In any case, when the pole field effects can be neglected, at least in first order, one can define two regimes; the first, where the random anisotropy varies rapidly from one point to another, the condition (20) being fulfilled corresponds mainly to amorphous alloys of rare earth and had been well studied theoretically and experimentally; the second regime, where contrarily equation (17) is fulfilled can be assigned as pseudoamorphous; here the local anisotropy dominates and as it varies slowly with the distance the magnetization is only a little bit further twisted by the exchange torque.
116
MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
Vol. 42, No. 2
.Table 1. Classification o f the different types o f macroscopic behaviour o f random anisotropy f erromagnets as function o f the two dimensionless parameters ap/ae and 4rrM2/o 4nM 2 --< o
a~
ap
a_£>l ao
41rM 2
1
- - > t7
1
Rare earth amorphous alloys.
Soft magnetic materials.
Existence of some frustration, although probably less than in spin-glasses
Usually shape induced anisotropy
Pseudo-amorphous (case of weak ferromagnets?)
Polycrystals of standard transition metals and their alloys
Magnetization not much deviated from local anisotropy
Classical domain theory of polycrystals.
axis
4. EFFECTS OF THE POLE FIELD In the conventional description of ferromagnets, one considers uniform domains so the only source of pole field is at the sample boundaries and sometimes at the domain boundaries. Nevertheless it cannot be neglected in many problems such as Block wall kinetics [7] and coercive field theory [8] for instance. So one must here, especially when studying non-uniform solutions, not forget the third and fourth terms of equation (3), assuming no external sources of field. Let us start by looking for the conditions needed if one wants those magnetostatic terms to be small enough for a perturbation. In the standard minimization procedure of the space integral of equation (3), the Euler-Lagrange equation for ff does not depend on the condition (1) which does not imply ft. As written above it is simply V-OV $ - V" M - -
V~O = O,
(21)
or
A~k = 4rrM-u.
(21')
If one designates by ar the spatial rate of rotation of u, that is the correlation length of the magnetization in the solution of the micromagnetic problem, one will have approximately A ~ "~ 41rMar 1,
(22)
IVffl ~ 4rrM,
(22')
so the order of magnitude of the magnetostatic terms in equation (3) will be 4rrM 2 for any spatial variation of the magnetization direction. Comparing this term to the anisotropy, we can define in supplement to ap/a e another dimensionless parameter 4rrM2/o.
It is now possible to attempt a classification of all situations given in Table 1. If 47rM2/o < 1,
(23)
the pole field effect is a simple perturbation of the solutions described in the preceding paragraph. For a e "( ap, that is for rare earth amorphous alloys one has tricky partially frustrated systems and for a o < a e the pseudo-amorphous situation where the local anisotropy dominates. It is suggested that such a situation occurs in weak a ferromagnets like ZrZn2. In contrast for 4¢rM2/o > 1
(24)
the anisotropy is to be introduced after the pole field. It is a rather common situation this ratio being of the order of 10 in iron alloys. One has clearly two different regimes corresponding to the value ofao/a e. Ifao/a e, this is the case of polycrystals, where ac is a typical length scale of compounding monocrystals and the mostly studied situation; one must remark that this condition is given in Ndel's papers on Rayleigh laws [9]. At the contrary, if a e ( a p , the situation can be very complex; it is suggested that soft materials, peculiarly F e - N i alloys with K = 0 are in those conditions. As in practical cases, they occur in sheets, the shape magnetostatic effect plays the dominant role of a uniform anisotropy tensor and classical model applies. Acknowledgements - The author is indebted to Drs J.F. Tiers and G. Couderchon from Imphy Steel Company for many discussions from which the definition of the problem arose. He is also indebted to Dr R. Rammal lbr illuminating points of view.
1.
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MACROSCOPIC BEHAVIOUR OF FERROMAGNETS
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