• Aucun résultat trouvé

On the free energy of a ferromagnet : anisotropy and rotation

N/A
N/A
Protected

Academic year: 2021

Partager "On the free energy of a ferromagnet : anisotropy and rotation"

Copied!
4
0
0

Texte intégral

(1)

HAL Id: jpa-00210428

https://hal.archives-ouvertes.fr/jpa-00210428

Submitted on 1 Jan 1987

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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�

(2)

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

(3)

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 2

and p == a2a2a2 , as

1 2 3

previously 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 :

(4)

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

Comparing equations (7) and (14), one finds

the following result : the components of the magnetic torque are simply given, for a sphere with isotropic magnetization, by the coefficients of the

wij I s in the expression of the rotational free energy Fr’

The same treatment can be repeated for any

2rystalline symmetry, and leads to the same conclusion. When restricted to the usual anisotropy coefficients, this gives

for a cubic crystal :

I

and for an hexagonal one :

6 it CONCLUSIONS

Using the property of rotational invariance of the free energy, one can expand F as a sum of Fa,

the magnetocrystalline anisotropy energy, Fme the

derivative of Fa with respect to symmetrical

strains, and Fr the derivative of Fa with respect to

rotations. The magnetic torque can be easily derived from the expression for Frt which can be simply

written as :

REFERENCES

[ 1] G. Aubert, Thèse Grenoble (1966) ;

J.M.M. Franse, Thesis Amsterdam (1969).

[2] K.B. Vlasov, B. Kh. Ishmukhametov Sov. Phys. J.E.T.P., 19 (1964) 142-148.

[3] H.F. Tiersten

J. Math. Phys., 5 (1964) 1298-1318.

[ 4] W.F. Brown Jr.

J. Appl. Phys., 36 (1965) 994-1000.

[ 5] D.E. Eastman

Phys. Rev., 148 (1966) 530-542.

[ 6] J. Rouchy, E. du Tremolet de Lacheisserie Z für Physik, B36 (1979) 67-80.

[7] R.L. Melcher

Phys. Rev. Lett., 25 (1970) 1201-1204.

[8] R.L. Melcher

Phys. Rev. Lett., 28 (1972) 165-168.

[ 9] E.R. Callen, H.B. Callen, Phys. Rev., 139A (1965) 455-471 [10] V. Dohm, P. Fulde

Z. Für Physik B21 (1975) 369-379.

[11] G. Aubert, Y. Ayant, E. Belorizky, R. Casalegno Phys. Rev., B14 (1976) 5314-5326.

[12] J.H. Van Vleck

Phys. Rev., 52 (1937) 1178-1198 [13] L. Néel

J. Phys. Radium, 15 (1954) 225-239.

[14] E. du Tremolet de Lacheisserie, P. Morin, J. Rouchy

Ann. de Physique, 3 (1978) 479-501.

Références

Documents relatifs

Nantel Bergeron, Muriel Livernet. A combinatorial basis for the free Lie algebra of the labelled rooted trees.. The pre-Lie operad can be realized as a space T of labelled rooted

Finally, in Section 5 we extend the Main Theorem to the orientation preserving homeomorphisms of the circle and we make some remarks and derive some consequences on the existence

Key-words : Free group - Uniformly bounded representations... parametrized by spherical functions was given. There arises a prob- lem of finding the theorem for a free group related

This wear resistance, which is equivalent in concept to the inverse of wear volume, is directly proportional to hardness, but the constant of proportionality

Abstract: This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure ν associated with such

The quality of delivery score encompassed whether or not the officer spoke at an appropriate speed (i.e., at or less than 200 wpm), delivered each sentence of the right, checked

Development and testing of a high speed SCARA type robotic manipulator with hydraulic actuators: A custom-designed hydraulic actuator was developed and fabricated for high speed

by the family of such vector fields. an arbitrary dilation as a natural extension of the concept of linearity. In fact, classical results valid for linear vector