} Succeptibilitatea in cazul aplicarii unui camp extern paralel cu axa de usoara magnetizare

Campul H actioneaza asupra subretelelor MA si MB este paralel cu HmA si antiparlel cu HmB. =>

1

2A j B J AM Ng JB x ;

1

2B j B J BM Ng JB x

Unde

0J BA A

B

Jgx H

k T

; 0J BB B

B

Jgx H

k T

;

Iar

2 1 2 1 1

2 2 2 2

AJ A A

xJ JB x cth x cth

J J J J

;

2 1 2 1 1

2 2 2 2

BJ B B

xJ JB x cth x cth

J J J J

;

0 ( )J BA ii A AB BB

Jgx H N M N M

k T

0 ( )J BB AB A ii BB

Jgx H N M N M

k T

Pentru H=0 avem MA = -MB = M0 ; 00 0[ ) ]J B

A B AB ii

B

g Jx x x N N M

k T

;

Pentre valori obisnuite ale lui H, dezvoltam functiile Brillouin in serie Taylor retinand termenii de ordin I ai dezvoltarii:

00 0 0 0[ ( ) ( )] 'J B

J A J ii A AB B J

B

g JB x B x H N M M N M M B x

k T

00 0 0 0[ ( ) ( )] 'J B

J B J AB A ii B J

B

g JB x B x H N M M N M M B x

k T

Unde

0

0

0

'J

J

B xB x

x

Stiind ca Supsceptibilitatea magnetica este M

H =>

||A BM M

H

; Impreuna cu relatiile

1

2A j B J AM Ng JB x ;

1

2B j B J BM Ng JB x

||

1 1

2 2j B J A j B J BNg JB x Ng JB x

H

0 00 0 0 0 0 0 0 01 1

{ [ ( ) ( )] ' } { [ ( ) ( )] ' }2 2

J B J Bj B J ii A AB B J j B J AB A ii B J

B B

g J g JNg J B x H N M M N M M B x Ng J B x H N M M N M M B x

k T k T

H

00 0 0 01 1

[ ( ) ( )] '2 2

J Bj B J j B ii A AB B J

B

g JNg JB x Ng J H N M M N M M B x

k T

H

00 0 0 01 1

[ ( ) ( )] '2 2

J Bj B J j B AB A ii B J

B

g JNg JB x Ng J H N M M N M M B x

k T

H

2 2 2

00 0 0 0

1 1[ ( ) ( )] '

2 2J B

j B J ii A AB B J

B

Ng JNg JB x H N M M N M M B x

k T

H

2 2 2

00 0 0 0

1 1[ ( ) ( )] '

2 2J B

j B J AB A ii B J

B

Ng JNg JB x H N M M N M M B x

k T

H

2 2 2 2 2 2

0 00 0 0 0 0 0 0 0

1 1 1 1[ ( ) ( )] ' [ ( ) ( )] '

2 2 2 2J B J B

j B J ii A AB B J j B J AB A ii B J

B B

Ng J Ng JNg JB x H N M M N M M B x Ng JB x H N M M N M M B x

k T k T

H

2 2 2 2 2 2

0 00 0 0 0 0 0

1 1[ ( ) ( )] ' [ ( ) ( )] '

2 2J B J B

ii A AB B J AB A ii B J

B B

Ng J Ng JH N M M N M M B x H N M M N M M B x

k T k T

H

2 2 2

00 0 0 0 0

1' {[ ( ) ( )] [ ( ) ( )]}

2J B

J ii A AB B AB A ii B

B

Ng JB x H N M M N M M H N M M N M M

k T

H

2 2 2

00 0 0 0 0

1' [ ( ) ( ) ( ) ( )]

2J B

J ii A AB B AB A ii B

B

Ng JB x H N M M N M M H N M M N M M

k T

H

2 2 2

00 0 0

1' [( )( ) ( )( )]

2J B

J A ii AB AB ii B

B

Ng JB x M M N N N N M M

k T

H

2 2 2

00 0 0

1' [( )[( ) ( )]

2J B

J ii AB A B

B

Ng JB x N N M M M M

k T

H

2 2 2

00 0 0

1' [( )[ ]

2J B

J ii AB A B

B

Ng JB x N N M M M M

k T

H

2 2 2

00

1' [( )[ ]

2J B

J ii AB A B

B

Ng JB x N N M M

k T

H

2 2 20 0' [( )[ ]1 1

2

J B J ii AB A B

B

Ng J B x N N M M

k T H

Efectuand calcule in continuare rezulta ca

2 2 2

0 0

||2 2 2

0 0

'

1( ) '

2

B j J

B ii AB B J J

N g J B x

k T N N g NJ B x

DAR NU STIU CUM

Aceasta relatie arata ca la T=0 avem || 0 , cea ce inseamna ca toate momentele magnetice sunt aliniate ( paralel sau antiparalel) la campul

aplicat iar magnetizarea indusa este nula. Odata cu cresterea temperaturii || creste pana la TN iar pentru T > TN succeptibilitatea este data de

relatia ||C

T

.

Daca consideram Nii =0 ; si calculam succeptibilitatea || ( )N

TfT

pentru valori ale lui j=1/2, 3/2, 5/2 se observa ca aceasta (succeptibilitatea )

creste de la 0 la Tn odata cu temperatura (Cazul considerat de B.Liliard ) .