8/3/2019 J.A. Tuszynski et al- Applications of Nonlinear PDE's to the Modelling of Ferromagnetic Inhomogeneities

1/2

APPLICATIONS OF NONLINEAR PDE'S TO THE MODELLING OF FERROMAGNETICINHOMOGENEITIES

J.A. TUSZYNSKIDepartment of Physics, .ih~Edmonton, Alberta, Canada,

~;T1iveI5:.t.y of AlbertaT6G 2JlP. WINTERNITZ AND A.M. GRUNDLANDC.R.M., Universite de MontrealMontreal, Quebec, Canada, H3C 3J7.q. 1.

. ABSTRACT. Several approaches lead to the Landau-Ginzburg free energyfor magnetic inhomogeneities in ferromagnets. Minimization of thisfunctional in both reorientational and spin-ordering processes yields a3-D Nonlinear Klein-Gordon equation for the order parameter. SymmetryReduction Method is used to obtain solutions of this PDE. Physicalinterpretation is given.

Invoking

sys-

The Landau-Ginzbur9: Free Ener9:V FunctionalF = I dn x [A2~2 + A4~4 + A6~6 -h~ + ! D(9~)2] 1

of N=2

where n is spatial dimensionality. A2 = a(T-Tc) .A4. A6 are constants.~ is the order parameter and h is the conjugate field. can be used todescribe both field -(h ; 0) and temperature-induced (h = 0). first(A4 < 0) or second order (A4 ) 0) phase transitions. A more or lessformal justification of its applicability to ferromagnetic materialsnear criticality can be demonstrated either through a continuumapproximation applied to the Heisenberg Hamiltonian [1] .or through anevaluation of the corresponding partition function [2]. It is quitecommon. however. to simply postulate (I) on the basis of symmetryarguments and elementary statements of the catastrophe theory [3].In particular. for spin-ordering transitions (P ~ F) occurring in thepresence of an easy magnetization axis z. ~ represents the z-componentof magnetization. D refers to the nearest-neighbours exchange constantand A2. A4f A6 are "molecular field" constants due to highercoordination spheres. Minimization of F gives

2)9.~ = 2A2~ + 4A4~3 + 6A6~5 -hwhich is a 3-D time-independent nonlinear Klein-Gordon equation.Symmetry reduction method has been recently applied [4,5] to analyzeEq. (2) and its solutions. When h = O and the system is either at thecritical or the tricritical point, the symmetry group is the similitudegroup. Then, the symmetry variables represent: 1,2 or 3-D hyper-spheres, planes, spirals or cones. Otherwise, the symmetry group isEuclidean and only plane or hyperspherical symmetry variables areallowed. The effect of the external field h is, first of all, to

593R. Conte and N. Boccara (eds.), Partially Integrable Evolution Equations in Physics, 593-594.C> 1990 Kluwer Academic Publishers. Printed in the Netherlands.

8/3/2019 J.A. Tuszynski et al- Applications of Nonlinear PDE's to the Modelling of Ferromagnetic Inhomogeneities

2/2

594

of the external field h is, first of all, to destroy the scale invar-iance if the system is at either the critical or the tricritical point.Secondly. h affects the form of the solutions. For translationally-invariant ones kinks are disallowed and are replaced by bumps. Webelieve that this plays a crucial role in producing a curvature of theArrott plot (h/~ vs ~2) for magnetization processes. Moreover, theamplitudes and frequencies of the periodic solutions are shifted as aresult of switching on the field.For reorientational processes occurring in the presence of aneasy-magnetization plane, the order parameter ~ is two-component, i.e.consists of the two directional cosines al and a2. The expansioncoefficients for the free energy analogous to eq. (1) correspond tothe appropriate anisotropy constants. In I-D approximation the min-imization of the free energy results in a very similar type of problemas that described above. Substituting (al' a2) = (sin e, cos e) givesthe first integral of the Euler-Lagrange equation in terms of e whichis identical to that obtained from eq. (2) in I-D space. The result-ant kink solution has been known as the so-called Neel domain wallwhen (al. a2) = (ax(z) , az(z)) and the Bloch domain wall when (al,a2) =(ax(z) , ay(z)) .Periodic solutions in terms of Jacobi elipticfunctions can readily be found and correspond to the variousferromagnetic and anti ferromagnetic spin waves.Eq. (2) represents only magnetic structures ~(~) which minimize thefree energy (I). In order to model thermodynamically stablefluctuations it is required of them to satisfy

A2~2 + A~~~ + A6~6 -h~ + ~ D(V~)2 = E = const

Interestingly, Eq. (3) has the same symmetry group as Eq. (2). More-over, in I-D situations it takes the same form as the first integral ofEq. (2). The only difference there is the sign of D. This means thatthe functional extema of (I) for D ) O correspond to equilibriumfluctuations of (1) for D < O and vice versa. Consequently, the entirespectrum of thermodynamically stable inhomogeneities can be found thisway.Work on the energies of the solutions, the influence of externalfields and the modelling of equilibrium fluctuation is in progress andwill be published elsewhere.REFERENCES[1] Anderson. P.W. (1984) Basic Notions of Condensed Matter Physics.

Benjamin, Menlo Park,[2] Amit, D.J. (1978) Field Theory, the Renormalization Group andCri~ical Phenomena, McGraw-Hill, New York.[3) Poston, T. and Stewart (1978) , Catastrophe Theory and ItsApplications, Pitman, London.

[4) Winternitz, P. , Grundland, A.M. and Tuszynski, J.A. (1987)J. Math. Phys. 28, 2194-2212.[5) Winternitz, P:, Grundland, A.M. and Tuszynski, J.A., (1988)

J. Phys. c ~, 4931-4953.