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Thermodynamic reciprocity of magnetic anisotropy and deformation induced by thermoannealing
Journal of Magnetism and Magnetic Materials 112 (1992) 47-49 North-Holland
,44,.
Thermodynamic reciprocity of magnetic anisotropy and deformation induced by thermoannealing S.N. Lyakhimets Institute of Metal Physics, AS Ukr.SSR, Vernadslo: str. 36, 252680 Kiev 142, Ukraine
A phenomenological theory of induced anisotropy which applies for magnets with internal freedom degrees in atom structure has been suggested. This theory is formulated for amorphous magnets and shows the thermodynamic reciprocity of magnetic anisotropy and deformation induced by thermoannealing.
I. Introduction The phenomenon of induced anisotropy in magnets is connected with internal freedom degrees (IFD) in the atomic structure. In crystal magnets such type as ferrites IFD are conditioned by uncontrolled dopants as well as special dopants which have the possibility to change their position in the lattice. In amorphous magnets the possibility of the small motions in the unsteady chaotic atom structure determines 1FD. The macroscopic anisotropy induced in amorphous magnets after annealing under the applied external magnetic field and stress is widely investigated recently (see for example refs. [2,3]). In the scope of this theory which is formulated for amorphous magnets some interesting and important consequences may be obtained without concretizing of the IFD type.
2. Variables of IFD and the thermodynamic potently! We will characterize IFD by genera;izing variables x,~, where the index ce numbers the freedom degrees. For our analysis it is not necessary Carrespondence to." Dr. S.N. Lyakhimets, Institute of Metal Physics of Ukr. Acad. Sci., Vernadsky Str. 36, 2526~'0 Kiev 142, Ukraine.
to concretize the type of the change in the atom structure of the amorphous magnet which are responsible for the Induced macroscopic anisotropy and determined by x~,. The simplest example of IFD on the microscopic level is the directional atomic pair ordering. In this case the directional distribution function of the pair mean variables x,~ and the unit vector u defining the pair orientation has the sense of the index a. The thermodynamic potential f corresponding to IFD may be expressed in the following form
f( M, uik, x,, T) =f(im)(M, x,~, T) +f'ie)(Uik, X~, T)
+
v),
(l)
which include interactions of IFD with magnetic and elastic subsystems of the magnet, and proper energy (last term). The thermodynamic potential (1) is in general used for derivation of the magnetostriction (see recent theory for disordered ma+ - . i , - rA'I~ t ~, ,t; .tcxto ['-~I.P. B H t
we
U S e t - t # o.130 I U I
tile
U;St..lllJtltOll
of irreversable processes of thermoannealing, in (1) M is the magnetization vector, u,k corresponds to theelastic deformation tensor, i, k = x, y, z, T is the temperature. The main and important property follows from the fact that the x,~ which defines the state of the atomic structure is the invariance ~f the variable x a under time reflection. Due to this and to the assumption of a
0304-8853/92/$05.00 © 1992 - Elscvier Science Publishers B.V. All rights reserved
48
S.N. Lyakhimets / Anisotrop.v and deformation induced by thermoannealing
weak interaction of IFD with the magnetic subsystem in comparison with exchange interactions the expression for f0,~) may be written as an expansion in components of M and the variables x,,, which gives for the first nonzero terms f(im) __ d i k , a M i Mk Xa.
fined by the nonequilibrium part of the potential (1). There are first two terms in (1) where it is necessary to substitute x" for x,, find = f ( i m ) + f(ie) = D i k , n t M i M k M ~ M t'
(2)
P
+ Gik,nlllikUnl
Similarly if the deformation U~k is not large we may write f ( i e ) = g i k , a u ik X t~"
(3)
We consider the case where the deviation of the IFD state from equilibrium is not large. The sense of the tensors d and ~ becomes more clear if we notice that the values k}~) = dik.~Xa, are the coefficients describing the second order magnetic anisotropy and the values o-~(t~) "-gik,aXa are the tensor components defining internal stresses in the magnet. The tensors k~? and tri~) will completely determine the induced anisotropy. It is necessary also to concretize the expression for f(i) as a function of x~. This function must have a minimum at some values x O°f . Near to this minimum we may write f(i) = 7P,.8 ~ x ,~xt~,
(4)
where we redetermine the variables x , as the deviations from equilibrium values x,~0 which we take as zero (x ° = 0).
3. Magnetic anisotropy and deformation induced by thermoannealing IFD have characterizing temperature of freezing Tf. Above Tf at fixed T', M ' and u~k (prime means the values during annealing) the configuration of IFD reaches an equilibrium state characterized by values x'. which are found from the equality af /'?..¢ ~, = 0
x'~ = - p ~ [
dik,,~ Mi'M/~ + gik,,~U~k 1~
(5)
where p~-~ is the inverse to the P, t3 tcnsor. After fast cooling below Tf this thermodynamically nonequilibrium configuration will be frozen up. The effect of induced anisotropy will be de-
-t- e i k , n l ( M i M k U r n i + U i k M t ; M i ¢ ) ,
(6)
where D,"k,nl -- - d ik a d hi,Pal3 -l, Gik, m = --gik,~ g,,t ,[3 P~-~, P r"K, , ' u- ---d ' . ig,e~6ni,Cll~a[3 o ' , , - t • The first term ~., in (6) was suggested by N~el for the description of the. induced magnetic anisotropy [1]. The second term describes the residual deformation after treatment under the external stress (such a possibility has been theoretically discussed before in ref. [5]). Such effects may exist above the Curie temperature which is consistent with experiment [6]. The last term in (6) describes two effects. The first one describes the formation of the induced magnetic anisotropy after cooling under an external stress. The second effect is the formation of residual "frozen" deformations after cooling in a magnetic field. Both these effects are thermodynamically complementary because they are determined by the same phenomenological constants P,k.,a" The existence of one of them leads to the existence of the other one. Taking into account only relativistic interactions and interesting in the effect of the anisotropy we may write down in the case of an isotropic magnet the following factorization 1
Aik.,,l=A~[Si,,Skt + 8,8k,,],
(7)
where A = D,G,P. The reciprocity of the induced anisotropy and of the residual deformation may be observed as the change of the magnetic anisotropy constants k(i) -,,,4 of +h . . . l.w i . . , o l;A,.,,~ . . . . . . . . o.i(~) _,~_ varia• ik ~attu tll~,, t, JtUal Otlk.,33v~,~ . dltVC-I tion of the annealing conditions, i.e. values of M,' f and U,k ~t.(i)
,
~ik = PSUik,
"
5o'i2 ) = PS(Mi'M/, ).
(8)
In :onclusion we want to remark that this theor~ applies for crystalline magnets. The symmetric:d tensor Xik defines the IFD. For formulating the theory it is important to take into
S.N. Lyakh#nets / Anisotropy and deformation induced b.v thermoannealing
account the crystal symmetry and to use symmetry considerations [7].
References [1] S. Krupicka, Physik der Ferrite und der Verwandten Magnetization Oxide, vol. 2 (Prague, 1973).
4t~
[2] M. Vfizquez, J. Gonzfilez and A. Hernando, J. Magn Magn. Mater. 53 (1q86) 323. [3] J. Gonz~ilez, M. Vfizquez, J.M. Barandiarfin, V. Maclurga and A. Hernando, J. Magn. Magn. Mater. 68 (1987) 151. [4] J. Furtmiiller, M. Rihnle and G. Herzer, J. Magn. Magn. Mater. 69 (1987) 79. [5] S.N. Lyakhimets, Ukrain. Fiz. Zh. 34 (1989) 23. [6] O.V. Nielsen and H.J.V. Nielsen, J, Magn. Magn. Mater. 22 (1980) 21. [7] S.N. Lyakhimets, J. Magn. Magn. Mater. 1 i4 (1992) 45.
,44,.
Thermodynamic reciprocity of magnetic anisotropy and deformation induced by thermoannealing S.N. Lyakhimets Institute of Metal Physics, AS Ukr.SSR, Vernadslo: str. 36, 252680 Kiev 142, Ukraine
A phenomenological theory of induced anisotropy which applies for magnets with internal freedom degrees in atom structure has been suggested. This theory is formulated for amorphous magnets and shows the thermodynamic reciprocity of magnetic anisotropy and deformation induced by thermoannealing.
I. Introduction The phenomenon of induced anisotropy in magnets is connected with internal freedom degrees (IFD) in the atomic structure. In crystal magnets such type as ferrites IFD are conditioned by uncontrolled dopants as well as special dopants which have the possibility to change their position in the lattice. In amorphous magnets the possibility of the small motions in the unsteady chaotic atom structure determines 1FD. The macroscopic anisotropy induced in amorphous magnets after annealing under the applied external magnetic field and stress is widely investigated recently (see for example refs. [2,3]). In the scope of this theory which is formulated for amorphous magnets some interesting and important consequences may be obtained without concretizing of the IFD type.
2. Variables of IFD and the thermodynamic potently! We will characterize IFD by genera;izing variables x,~, where the index ce numbers the freedom degrees. For our analysis it is not necessary Carrespondence to." Dr. S.N. Lyakhimets, Institute of Metal Physics of Ukr. Acad. Sci., Vernadsky Str. 36, 2526~'0 Kiev 142, Ukraine.
to concretize the type of the change in the atom structure of the amorphous magnet which are responsible for the Induced macroscopic anisotropy and determined by x~,. The simplest example of IFD on the microscopic level is the directional atomic pair ordering. In this case the directional distribution function of the pair mean variables x,~ and the unit vector u defining the pair orientation has the sense of the index a. The thermodynamic potential f corresponding to IFD may be expressed in the following form
f( M, uik, x,, T) =f(im)(M, x,~, T) +f'ie)(Uik, X~, T)
+
v),
(l)
which include interactions of IFD with magnetic and elastic subsystems of the magnet, and proper energy (last term). The thermodynamic potential (1) is in general used for derivation of the magnetostriction (see recent theory for disordered ma+ - . i , - rA'I~ t ~, ,t; .tcxto ['-~I.P. B H t
we
U S e t - t # o.130 I U I
tile
U;St..lllJtltOll
of irreversable processes of thermoannealing, in (1) M is the magnetization vector, u,k corresponds to theelastic deformation tensor, i, k = x, y, z, T is the temperature. The main and important property follows from the fact that the x,~ which defines the state of the atomic structure is the invariance ~f the variable x a under time reflection. Due to this and to the assumption of a
0304-8853/92/$05.00 © 1992 - Elscvier Science Publishers B.V. All rights reserved
48
S.N. Lyakhimets / Anisotrop.v and deformation induced by thermoannealing
weak interaction of IFD with the magnetic subsystem in comparison with exchange interactions the expression for f0,~) may be written as an expansion in components of M and the variables x,,, which gives for the first nonzero terms f(im) __ d i k , a M i Mk Xa.
fined by the nonequilibrium part of the potential (1). There are first two terms in (1) where it is necessary to substitute x" for x,, find = f ( i m ) + f(ie) = D i k , n t M i M k M ~ M t'
(2)
P
+ Gik,nlllikUnl
Similarly if the deformation U~k is not large we may write f ( i e ) = g i k , a u ik X t~"
(3)
We consider the case where the deviation of the IFD state from equilibrium is not large. The sense of the tensors d and ~ becomes more clear if we notice that the values k}~) = dik.~Xa, are the coefficients describing the second order magnetic anisotropy and the values o-~(t~) "-gik,aXa are the tensor components defining internal stresses in the magnet. The tensors k~? and tri~) will completely determine the induced anisotropy. It is necessary also to concretize the expression for f(i) as a function of x~. This function must have a minimum at some values x O°f . Near to this minimum we may write f(i) = 7P,.8 ~ x ,~xt~,
(4)
where we redetermine the variables x , as the deviations from equilibrium values x,~0 which we take as zero (x ° = 0).
3. Magnetic anisotropy and deformation induced by thermoannealing IFD have characterizing temperature of freezing Tf. Above Tf at fixed T', M ' and u~k (prime means the values during annealing) the configuration of IFD reaches an equilibrium state characterized by values x'. which are found from the equality af /'?..¢ ~, = 0
x'~ = - p ~ [
dik,,~ Mi'M/~ + gik,,~U~k 1~
(5)
where p~-~ is the inverse to the P, t3 tcnsor. After fast cooling below Tf this thermodynamically nonequilibrium configuration will be frozen up. The effect of induced anisotropy will be de-
-t- e i k , n l ( M i M k U r n i + U i k M t ; M i ¢ ) ,
(6)
where D,"k,nl -- - d ik a d hi,Pal3 -l, Gik, m = --gik,~ g,,t ,[3 P~-~, P r"K, , ' u- ---d ' . ig,e~6ni,Cll~a[3 o ' , , - t • The first term ~., in (6) was suggested by N~el for the description of the. induced magnetic anisotropy [1]. The second term describes the residual deformation after treatment under the external stress (such a possibility has been theoretically discussed before in ref. [5]). Such effects may exist above the Curie temperature which is consistent with experiment [6]. The last term in (6) describes two effects. The first one describes the formation of the induced magnetic anisotropy after cooling under an external stress. The second effect is the formation of residual "frozen" deformations after cooling in a magnetic field. Both these effects are thermodynamically complementary because they are determined by the same phenomenological constants P,k.,a" The existence of one of them leads to the existence of the other one. Taking into account only relativistic interactions and interesting in the effect of the anisotropy we may write down in the case of an isotropic magnet the following factorization 1
Aik.,,l=A~[Si,,Skt + 8,8k,,],
(7)
where A = D,G,P. The reciprocity of the induced anisotropy and of the residual deformation may be observed as the change of the magnetic anisotropy constants k(i) -,,,4 of +h . . . l.w i . . , o l;A,.,,~ . . . . . . . . o.i(~) _,~_ varia• ik ~attu tll~,, t, JtUal Otlk.,33v~,~ . dltVC-I tion of the annealing conditions, i.e. values of M,' f and U,k ~t.(i)
,
~ik = PSUik,
"
5o'i2 ) = PS(Mi'M/, ).
(8)
In :onclusion we want to remark that this theor~ applies for crystalline magnets. The symmetric:d tensor Xik defines the IFD. For formulating the theory it is important to take into
S.N. Lyakh#nets / Anisotropy and deformation induced b.v thermoannealing
account the crystal symmetry and to use symmetry considerations [7].
References [1] S. Krupicka, Physik der Ferrite und der Verwandten Magnetization Oxide, vol. 2 (Prague, 1973).
4t~
[2] M. Vfizquez, J. Gonzfilez and A. Hernando, J. Magn Magn. Mater. 53 (1q86) 323. [3] J. Gonz~ilez, M. Vfizquez, J.M. Barandiarfin, V. Maclurga and A. Hernando, J. Magn. Magn. Mater. 68 (1987) 151. [4] J. Furtmiiller, M. Rihnle and G. Herzer, J. Magn. Magn. Mater. 69 (1987) 79. [5] S.N. Lyakhimets, Ukrain. Fiz. Zh. 34 (1989) 23. [6] O.V. Nielsen and H.J.V. Nielsen, J, Magn. Magn. Mater. 22 (1980) 21. [7] S.N. Lyakhimets, J. Magn. Magn. Mater. 1 i4 (1992) 45.