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On the free energy of a ferromagnet : anisotropy and rotation
G. Aubert, E. Du Tremolet de Lacheisserie
To cite this version:
G. Aubert, E. Du Tremolet de Lacheisserie. On the free energy of a ferromagnet : anisotropy and
rotation. Journal de Physique, 1987, 48 (2), pp.169-171. �10.1051/jphys:01987004802016900�. �jpa-
00210428�
169
ON THE FREE ENERGY OF A FERROMAGNET : ANISOTROPY AND ROTATION
G. AUBERT and E. du TREMOLET de LACHEISSERIE*
Service National des Champs Intenses, CNRS-USTMG, 166 X 38042 Grenoble Cedex, France
*Laboratoire Louis Néel, CNRS-USTMG, 166 X 38042 Grenoble Cedex, France
(Reçu Ze 14 octobre 1986, accepté Ze 10 dgcembre 1986)
RESUME.- Mettant à profit la propriété d’invariance rotationnelle de l’énergie libre, on peut développer celle-ci, F, comme la somme de l’énergie magnétocristalline Fa, de sa dérivée par rapport aux déformations symétriques Fme, et de sa dérivée par
rapport aux rotations, Fr. Ce dernier terme, toujours négligé, fournit une expression analytique simple du couple magnétique pour n’importe quelle direction de l’aimantation.
ABSTRACT.- Using the property of rotational
invariance of the free energy, F can be expanded as the sum of Fa the magnetocrystalline energy, Fme its
derivative with respect to the symmetrical strains
and Fr its derivative with respect to the rotations.
This latter term, always ignored, provides a simple analytical expression of the magnetic torque for any given direction of the magnetization.
LE JOURNAL DE PHYSIQUE
Tome 48 N° 2 FÉVRIER 1987
J. Physique 48 (1987) 169-171 FTVRIER 1987,
Classification
Physics Abstracts
75.30G
1 Y INTRODUCTION
Most of the textbooks dealing with the magnetocrystalline anisotropy consider a deformable body that cannot rotate, and define a so-called free energy as :
where Fa is the magnetocrystalline anisotropy energy and Fme is the magnetoelastic coupling energy, i.e.
the derivative of Fa with respect to the six symmetrical components cij of the strain tensor. The torque exerted by the sample on the sample holder,
or "magnetic torque" is then derived a posteriori from equation (1) as
in any symmetry plane, where t denotes the angle between the direction of the magnetization and a given direction in this plane (see e.g.ref. [1]),
but be carefull with the confusion in the literature between the torques exerted by the sample, the fiend-, the apparatus, etc...
Criticizing equation (1), several authors have proposed an improved description taking into account the finite strain theory [2,3,4]. The original work by VLASOV et aL. [2] was devoted to
dynamical problems, and EASTMAN [5] developed a theory for the field dependence of the sound velocity in cubic ferrimagnets where rotational effects were emphasized ; this work has been
revisited later on [6]. A number of further studies also scrutinized these rotational effects in
antiferromagnets and in paramagnets, see e.g. ref.
[7] and [8], but all of them were essentially oriented towards the dynamical aspects.
Here, we would like to discuss a static aspect, namely write the free energy for a single crystal (which is allowed to rotate), by introducing
in equation (1) a rotational energy Fr that can be
built in a similar way as Fme but is a function of the three antisymmetrical components Wi] of the
strain tensor, as defined within the small-deformation theory.
We shall see that the derivation of the magnetic torque for any arbitrary direction of the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802016900
170
magnetization is easier when taking into account this term. It is surprising that higher order rotational effects have been extensively studied these last twenty years, while this much more classical effect has never received, to our knowledge, the same analytical treatment.
2 - SYMMETRY CONSIDERATIONS
CALLEN and CALLEN [9] have constructed the hamiltonian of a single crystal from group theoretical considerations. Along these lines, Fa
must be a sum of harmonic polynomials belonging to
the fully symmetric representation r1 ’
,while Fme is
a sum of scalar invariants of the elastic functions (linear combinations of the six eij’s) and the
harmonic polynominals belonging to the same representation.
The rotational energy, i.e. the derivative of Fa with respect to the wjjts must be constructed in the same way. Considering that the three wij ’s
form a basis for the r’4 irreducible representation of the cubic group, one must look for harmonic polynominals belonging to T4 : : those of the lowest degree that we can find are of fourth degree in the direction cosines ai of the magnetization, and can be written aiaj (a2 - a 2
Hence, the lowest degree contribution to Fr
can be written in a cubic crystal :
The same derivation is possible for any symmetry, but the group theory cannot provide the
numerical value of the rotational coefficients.
3 - ROTATIONAL IIi111ARIJU1CE
The most straightforward method for deriving Fr and obtaining the values of the
rotational coupling coefficients is to write that Fa
is invariant under any arbitrary infinitesimal
rotation of the system. As stated in ref. [10],
this implies simply to replace the magnetic moment in the expression of F by the reversely rotated
moment R-1(M). In the present case where we are only concerned with terms linear in
cij and wij, we
replace in Fa the a s’ by
thus giving for a. hexagonal crystal :
In the case of a cubic crystal, the magnetocrystalline energy can be expanded in terms
of so (120LI
2 3 +a2a2
3 1 +a2a2
1 2and p == a2a2a2 , as
1 2 3previously demonstrated [11] :
where 1 is given by the diophantian equation 1
=4n
+6m . Using equation (4) we can derive :
Restricted to the usual first anisotropy constant, this general expression becomes simply :
in agreement with equation (3) .
This analysis can be repeated for any crystal symmetry, as well. We must only mention that second order effects are not so easy to handle, see e.g. ref. [10] for a discussion of these problems.
4 y PAIR MODEL
The pair model developed by Van VLECK for deriving the magnetocrystalline anisotropy [12] was
extended by NEEL to the magnetostriction [13]. More recently, rotational effects were included in this model, thus giving the magnetic hamiltonian correct to second order in the strains (cij and wij) and to
fourth degree in spin operators for cubic crystals [14].
Fa and Fr may be derived from the
expectation value of this hamiltonian. This rather pedestrian method leads to :
thus providing a check of our equations (3) and (8).
5 - MAGNETIC TORQUE
In the following, we shall omit the magnetoelastic coupling energy, assuming that the magnetostrictive contribution to the magnetocrys- talline energy is already incorporated in the anisotropy constants. Let us now introduce an
g
+external magnetic field Bo = PoHo rigidly oriented
with respect to the laboratory axes. The thermodynamical potential for the magnetic system
can be written as usual :
The magnetic torque rm is derived by setting the differential dg+ equal to zero for
~
Bo,T
any infinitesimal rotation dw :
171
For a sample exhibiting an isotropic magnetization, dM
=dw x M, and equation (11) can be easily rewritten as :
~ ~
This gives the magnetic torque M x Bo, and reduces to the magnetocrystalline contribution when the sample is spherical, namely
In the cubic case, where F a
is given by equation (6), one easily finds that : a
m