V G Baryakhtar, A G Danilevich - These relaxation terms describes only dissipation of relativistic nature - страница 1

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V.G. Baryakhtar A.G. Danilevich

Donetsk 2010

Landau - Lifshits equation:

Relaxation terms:

DM

~dt

-/[WSR ]+ R

These relaxation terms are postulated for the case when:

m2 — M02

These relaxation terms describes only dissipation of relativistic nature.

It is natural to use such terms with one relaxation constant in case of cubic symmetry of the magnet.

paramagnet:

RB 1(m - m 0) +1m ± Bloch

(1945)

ferromagnet:

R LL — Ml

[m, [h, m]]

r G

[m, m ]

Landau - Lifshits (1935)

Gilbert

(1956)

M

In case when: M2 ф m

we can't use the above relaxation terms

T

Other reasons:

1. The only one relaxation constant does not responding to other

symmetries of the magnets.

2. When using such relaxation terms for the uniaxial ferromagnet one

could collide physical contradiction. In case of degenerated states the condition of spin wave existence didn't realize. There is diapason of absolutely damping spin waves.

3. It is impossible to describe the exchange part of the relaxation in the

magnet.

4. It is impossible to describe the relaxation of the absolute value of

magnet moment.

2

0

To solve above-mentioned problems it is necessary to built the dissipative function of magnet. These dissipative functions are based on

magnetic crystal's symmetry.

We have taken the effective magnetic field as main parameter that characterize quasi-equilibrium states.

-= -2Q =1--dV = -\ h-dV = -\ hrdV    ^> Q = -\ hrdv = -\qdv

dt       *   3 Sm dt 3     dt 3 2J 23

q

Dissipative function:

1 ЛН,Нк+2 X

2

Relaxation term:

SH

Tensor Xik depends on magnetization of the crystal. We have to expand Xjk by magnetization power to take into account the

magnetic structure of the media.

q

1 Xk (0) HH +1 M,k, spHHkMsMp +1 x

2

2

2

v dxi J

2

4

We need to chose a symmetry of the magnet to determine tensors Xikand juiksp.

H invariants for this type of magnet:

 

|4i

0

о 1

H + ну, H

 

xii

0

 

V0

0

X33 J

M and H invariants for this type of magnet: m2x + m]„ m]..

H + H„ ні,

(HyMx + HxMv)2, hvmv + HxMx, hm

y у

x     x?       z z

1 1

1       M\ +1M55 (HyMx + HxMy )2 +1 Мб6 (HxMx + HyMy )2 + Мб7 (HxMx + HyMy )HzM

1

1

+ 2 V42HzMz + 2 V55 (HyMx + HxMy j + 2

x + " y~-y r~ z— z

Dissipative function:

_ 1     2 1

q = 2 XH ±+2 хззн

33- > z + qB +1 (M Hi щ [M, H])+ і X

2

\ (^32 - V41)Hz'M2 + 1 (^32 - ^41 + «42)HM +

^32 0

0

+1 (v31 + v66 )(hx2 Ml + Щ M2 )+ (v31 + 2^55 + v66 )HxHyMxMy

2

+ (M32 + ^67^xMx + HyMy )HzMz

Relaxation term:

+

Лік =      0       M32 0

RBx = (M31 + M66 )HxMl + (M31 + 2^55 + M66 )HyMxMy + (V32 + И 67 )HzMxM RBy = (V31 + ^66 )HyMy2 + (V31 + 2^55 + ^66 )HxMxMy + (^32 + ^67 )H zMyM Z

RBz = (V32 - «41 )HzM2 + (^32 - M41 + M42)HzM + (^32 + ^67 )(HxMx + HyMy К

The full energy density of the tetragonal ferromagnet.

=-(M1 - Mlf +1JM dM -1 ^1M, -1K m4 -1K3 Ml Ml

8^M2       2   dx} dx,    2   1   z   4   2   z   2   3   x y

Spin waves dispersion law for tetragonal ferromagnet.

Basic state of magnetization: "light plane"     q> = 0, 0 =—; k3 < 0, k1 < 0

2

sw

-((X1 - (v31 + V55)M02 + Xк2)(ak2 -K3M02) + (X33 - V41M02 + к2)(ak2 -K1)) ±

2

2

4/2M02 (ak2 - K1 )(ak2 - K3M02) -- [(ak2 - K1 )(X3 - V41M02 + Xек2) - (ak2 - K3M02 )(X 1 - (v31 + M55 )M02 + Xек2)]2

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