3.3 Anisotropic exchange interaction Dzyaloshinsky-Moriya interaction (antisymmetric

exchange interaction)[ ]jiDM SSD ´×=H Antisymmetric interaction

Dzyaloshinsky presumed this interaction from the crystal symmetry.Moriya lead the exact formula using a perturbation theory.

◆When the middle point of a pair of magnetic ions is the inversion center of the crystal field,

0=DTwo magnetic ions are A and B, the middle point of the lineAB is C, (i) A mirror plane is perpendicular to the lineAB at C,

(ii) A mirror plane contains the lineAB,

(iii) A diad rotation symmetry axis is perpendicular to the lineAB at C,

(iv) A n-fold rotation （n>2） axis is the lineAB,

AB^D

^D 鏡映⾯

^D 2回軸

AB//D

Crystal symmetry and D vector

VJ jijiijjjiiji ++=×-×+×++= HHHHH SSSLSL 2ll

nmjmin =jj

!+-

+-

+= ååm jmjn ini

ji EEVm

mEE

Vnn

00

0000

000000jj

!+-

+-

+= ååm iimn iin

jiji EEVmmV

EEVnnV

VV00

000000000000jjjj

å

å

-

×-×+××-×+×+

-

×-×+××-×+×+»

m iim

jiijjjiijiijjjii

n iin

jiijjjiijiijjjii

EEJmmJ

EEJnnJ

V

0

0

00200200

00200200

SSSLSLSSSLSL

SSSLSLSSSLSL

llll

llll

jiijjjiijiijjjii nJnnnJ SSSLSLSSSLSL ×-×+×=×-×+× 02000000000200 llll

{ }{ } jiijjzjzjyjyjxjx

iziziyiyixix

nJSnLSnLSnL

SnLSnLSnL

SS ×-+++

++=

0200000000000

000000000

l

l

jiijjjii nJnn SSSLSL ×-×+×= 0200000000 ll

DM interaction：2nd order perturbation theory

!+-

+-

+= ååm jmjn ini

jiji EEVmmV

EEVnnV

VV00

000000000000jjjj

[ ][ ]

[ ][ ]å

å

-

×-×+××-×+×+

-

×-×+××-×+×+»

m jmj

jiijjjiijiijjjii

n ini

jiijjjiijiijjjii

EEJmmmmJmm

EEJnnnnJnn

V

0

0

00020000000002000000

00020000000002000000

SSSLSLSSSLSL

SSSLSLSSSLSL

llll

llll

( ) ( ) ( ) ( ) ( ) 2110212

2

2010 **0,00000 rrrrrr ddrenJnJ jinjiij jjjjòò==

( ) ( )[ ] ( ) ( )[ ] !+××-

-××-

-= ååm

jiijjmjn

jiiiini

mEEmJn

EEnJV SSSLSSSL ,000,02,00,002

00

ll

( )[ ] ( )[ ]

( )[ ] ( )[ ]å

å

-

×-××-×+

-

×-××-×+»

m jmj

jijjjijj

n ini

jiiijiii

EEmJmmJm

EEnJnnJn

V

0

0

00,0200,0020

00,0200,0020

SSSLSSSL

SSSLSSSL

ll

ll

( )[ ] ( ) ( ) ijijiijii SSSSSSSSS ×-×=×,

( ) ( ) ijzizjyiyjxixjzizjyiyjxixi SSSSSSSSSSSS SS ++-++=

[ ] [ ]{ } [ ] [ ]{ } [ ] [ ]{ } zjyiziyjxixizyjxiyixjziziyxjzixizjyiyix SSSSSSSSSSSSSSSSSS eee ,,,,,, -+-+-=

( ) ( ) ( ) zjyixjxiyyjxizjzixxjziyjyiz SSSSiSSSSiSSSSi eee -+-+-=

jii SS ´-=

( ) ( )[ ] ( ) ( )[ ]

( ) ( ) [ ]jim

jjmjn

iini

mjiij

jmjnjiii

ini

mEE

mJnEEnJi

mEEmJn

EEnJ

SSLL

SSSLSSSL

´×úúû

ù

êêë

é

--

-=

××-

-××-

-

åå

åå

000,000,002

,0000,02,00,002

00

00

l

ll

[ ]ji SSD ´×=

Parasitic ferromagnetism of α-Fe2O3（hematite）explained by DM interaction

945 K以下： antiferromagnetism260-945 K： ferromagnetism appears in AF phase

（ parasitic ferromagnetism ）[ ] 02

,,<´×+×-= åå JJ

jiji

jiji SSDSSH

( ) qqq sincos2 22 SSJE D--=

Canting angle of spins

qp -

JD2

tan =q

ji SSD ´- //

４. Molecular field thoery

å å><

-×-=ji i

izBjiij SHgJH,

2 µSSå><

×-=ji

jiijJH,

2 SS

Temperature dependence of magnetization

magnetic susceptibility specific heat

å å><

-×-=ji i

izBjiij SHgJH,

2 µSS

å><

×-=ji

jiijJH,

2 SS

4.1 Heisenberg model

(4.1)

(4.2)

Hamiltonian of magnetic material

( )H,0,0=H

・no magnetic anisotropy, but z-axis is easy axis

HSgg zBB µµ =×=×- HSHμ HSg zBµ-

：external field

：pair of nearest neighbor lattice pointsji,

・sign of spin

Hereafter, S: spin magnetic moment

Equation of (4.1) or (4.2) cannot be solved exactly.

iS：total spin of a atom at i-th lattice point

Heisenberg model Ising model @S=1/2

4.2 Molecular field theory - finite temperature

( ) å -×-= +k

izBkii HSgJiH µSS2 (4.3)

Consider iʼth spin

・Sum of k is from one to the number of the nearest neighbor atoms.・Exchange integrals of the nearest neighbor spins are assumed to be equal.

◆Weissʼs molecular field theorySpin around is replaced by the averaged value.iSki+S

zzkiykixki SSSS === +++ ,0

( ) izBizz HSgSSzJiH µ--= 2 ( ) izBz SHgSzJ µ+-= 2

( ) åå -==i

izzi

SSzJiHH 2

(4.4)

( ) ( )åå +-==i

izBzi

SHgSzJiHH µ2

Z(i) of iʼth spin,

( ) ( )å-=

úû

ùêë

é+=

S

Si B

izBz Tk

SHgSzJiZ µ2exp

[ ] ( ) ( ) ASSASAASS

Siiz eeeeAS ----

-=

+++== å 11exp !Tk

HgSzJA

B

Bz µ+=2

( )

TkHgSzJ

STk

HgSzJ

iZ

B

Bz

B

Bz

22

sinh

212

sinh

µ

µ

+

úû

ùêë

é÷øö

çèæ +

+

=

Total magnetic momentN：total number of magnetic atomszB SNgM µ=

{ }ASAAAS eeee 221 ++++= -!

( )

A

SAAS

eee-

-=

+-

11 12

( ) ( )

2/2/

2/12/1

AA

SASA

eeee-

+-+

--

=A

AS

21sinh

21sinh ÷øö

çèæ +

=

(4.5)

(4.6)

(4.7)

◆Temperature dependence of magnetization( )

( )iZ

eSS

S

Si

TkiH

iz

z

Bå-=

-

=

TkHgSzJ

AB

Bz µ+=2

( )( )

ååå-=-=

+

-=

-

===S

Si

ASS

Si

STk

HgSzJS

Si

TkiH

iziz

B

Bz

B eeeiZµ2

( ) å-=

=¶¶ S

Si

ASiz

izeSAiZ

( )( )AiZ

iZSz ¶

¶=1

( )22

21

21

AA

SASA

ee

eeiZ-

÷øö

çèæ +-÷

øö

çèæ +

-

-=

úúú

û

ù

êêê

ë

é

-

-¶¶

-

-=

-

÷øö

çèæ +-÷

øö

çèæ +

÷øö

çèæ +-÷

øö

çèæ +

-

22

21

21

21

21

22

AA

SASA

SASA

AA

ee

eeA

ee

ee

2

22

21

21

222221

21

21

21

22 21

21

÷÷ø

öççè

æ-

÷÷ø

öççè

æ-÷÷

ø

öççè

æ+-÷÷

ø

öççè

æ-÷

÷ø

öççè

æ+÷

øö

çèæ +

-

-=

-

÷øö

çèæ +-÷

øö

çèæ +--÷

øö

çèæ +-÷

øö

çèæ +

÷øö

çèæ +-÷

øö

çèæ +

-

AA

SASAAAAASASA

SASA

AA

ee

eeeeeeeeS

ee

ee

22

22

21

21

21

21

21

212

AA

AA

SASA

SASA

ee

ee

ee

eeS-

-

÷øö

çèæ +-÷

øö

çèæ +

÷øö

çèæ +-÷

øö

çèæ +

-

+-

-

++=

2coth21

21coth

212 ASAS

-úû

ùêë

é÷øö

çèæ +

+=

TkSHgSzJS

ASxB

Bz µ+==2

úû

ùêë

é-÷øö

çèæ ++

=Sx

Sx

SS

SSSSz 2

coth21

212coth

212

( )xSBS=

( )Sx

Sx

SS

SSxBS 2

coth21

212coth

212

-÷øö

çèæ ++

= Brillouin function

[ ] ( ) xxxxB tanhcoth2coth221 =-=

[ ] ( ) SxSx

SxSx

SS eeee

SxxB 2/2/

2/2/

21limcoth -

-

¥®¥® -+

-= ( )11

21limcoth /

/

-+

-=¥® Sx

Sx

S ee

Sx

( ) ( )( ) !

!

+++++

-=¥® 2///

2///221limcoth 2

2

SxSxSxSx

Sx

S

( ) ( )!

!

+++++

-=¥® Sxx

SxSxxS 2/

2///221limcoth 2

2

( )x

x 1coth -= Langevin function( )xL=

(4.8)

→ Ising model

(4.9)

(4.10)

(4.11)

Brillouin function

●H=0，<Sz> is sufficiently small, Magnetization around transition T

!+-+=®<<453

1coth13xx

xxx

[ ]úúú

û

ù

êêê

ë

é

÷øö

çèæ-+-

úúú

û

ù

êêê

ë

é

÷øö

çèæ +

-+

++

+»

33

2451

231

2

121

212

451

212

31

2121

212

Sx

Sx

SxS

xSSx

SS

xSSS

SxBS

34

3422

21

451

212

451

21

31

212

31 x

Sx

SSx

Sx

SS

÷øö

çèæ+÷

øö

çèæ +

-÷øö

çèæ-÷

øö

çèæ +

=

úúû

ù

êêë

é÷øö

çèæ+÷

øö

çèæ +

úû

ùêë

é÷øö

çèæ+÷

øö

çèæ +

úû

ùêë

é÷øö

çèæ-÷

øö

çèæ +

-+

=223

21

212

21

212

21

212

4531

SSS

SSS

SSSxx

SS

2

23

21221

4531

SSS

SSxx

SS +++

-+

= ( ) ( )[ ] 33

22

9011

31 x

SSSSx

SS +++

-+

=

122

<<=+

=TkSzJS

TkSHgSzJS

xB

z

B

Bz µ

(4.12)

( ) ( )[ ] 33

22

9011

31 x

SSSSx

SS +++

-+

=( )xSBS Sz =

( ) ( )[ ] 3

3

22 290112

31

÷÷ø

öççè

æ+++-

+=

TkSzJS

SSSS

TkSzJS

SS

B

z

B

z

( ) ( )[ ]( )

zB

BBz ST

kJSzS

zJk

zJTk

SSSSS ÷÷

ø

öççè

æ-

+÷øö

çèæ

+++=

312

21145 2

223 (4.13)

0=zS

( )B

C kJSzST

312 +

= (4.14) Curie temperature

From eq(4.13)

◆CTT ³

CTT £◆ ( )( )[ ] C

C

Cz T

TTTT

SSSSS -

÷÷ø

öççè

æ

+++

=2

22

222

11

3100=zS

( )( )

( )( ) C

C

C

C

Cz T

TT

SS

SSTTT

TT

SS

SSS -

++

+±»

-÷÷ø

öççè

æ

++

+±=

2222 1

1310

1

1310

● T 〜TC

(4.15)

zB SNgM µ=Magnetization vanishes at Tc

( ) !+-=®>> - SxS e

SxBx /111

CTT <<●

(4.16)

0»T ( ) !+-==-

TkzJS

SzBeSxSBS2

(4.17)

0@ == TSNgM Bµ Magnetization is saturated.

◆ at arbitrary TzS

( )xBSS

Sz =

TkSzJS

xB

z2= x

zJSTk

SS Bz

22=

( )xBSS

Sz =

xzJSTk

SS Bz

22=

xzJSTk

SS CBz

22=

Temperature dependence of magnetization

zB SNgM µ=

P：free energy minimum pointO：free energy maximum point

◆ Temperature dependence of susceptibility

TkSHgSzJS

xB

Bz µ+=2

0=÷øö

çè涶

=HH

Mc

0=÷÷ø

öççè

涶

=H

zB H

SNgµ

(4.18) susceptibility

( )Hx

dxxdBSNg S

B ¶¶

= µc

TkSg

HS

TkzJS

Hx

B

B

H

z

B

µ+÷÷

ø

öççè

涶

=¶¶

=0

2

(4.19)

( )÷÷ø

öççè

æ+=

TkSg

NgTkzJS

dxxdBSNg

B

B

BB

SB

µµcµ 2

( ) ( )

( )dxxdBzJSTk

dxxdBSgN

SB

SB

2

2

2-=

µc (4.20)

0=zS◆CTT ³

CTT £◆

( )( )

SSzJSTkS

SSgNTT

B

B

C

312

31

2

2

+-

+

=³µ

c

From (4.12) ( ) ( ) ( )[ ] 33

22

9011

31 x

SSSSx

SSxBS

+++-

+=

( )S

SBS 310' +

=

( ) ( )CB

B

TTkSSgN

-+

=1

312µ (4.21)

( ) ( ) ( )[ ] 23

22

3011

31' x

SSSS

SSxBS

+++-

+=

( ) ( )[ ] 2

3

22 23011

31

÷÷ø

öççè

æ+++-

+=

TkSzJS

SSSS

SS

B

z

( )( ) C

Cz T

TT

SS

SSS -

++

+»

221

1310Around TC

From (4.15)

( ) ( ) ( )[ ] ( )( ) ÷÷

ø

öççè

æ-

+++

÷÷ø

öççè

æ+++-

+=

CBS T

TSS

SSTkzJS

SSSS

SSxB 1

11

3102

3011

31'

22

222

3

22

÷÷ø

öççè

æ -÷øö

çèæ+

-+

=C

CC

TTT

TT

SS

SS 2131

( )( ) ( )

( )dxxdBzJSTk

dxxdBSgN

TTS

B

SB

C2

2

2-=£

µc

( ) ( )TTk

SSgN

CB

B

-+

=1

612µ

CT

(4.22)

12 2qq =

( )

úúû

ù

êêë

é÷÷ø

öççè

æ -÷øö

çèæ+

-+

-

úúû

ù

êêë

é÷÷ø

öççè

æ -÷øö

çèæ+

-+

=

C

CC

C

CCB

TTT

TT

SS

SSzJSkT

TTT

TT

SS

SSSgN

22

22

1312

131µ

c

( )

( )TTkTTTSSgN

C

C

CB

-

úû

ùêë

é÷÷ø

öççè

æ --

+

=2

313)1(2µ

( )( )

( )C

B

C

B

TkSSgN

TTkSSgN 3

6)1(

6)1( 22 +-

-+

=µµ( )

úû

ùêë

é-

-+

=CC

B

TTTkSSgN 31

6)1(2µ

TkSHgSzJS

ASxB

Bz µ+==2

xx

SS

MM zz 1coth -==

÷÷ø

öççè

æ-

-+

= -

-

±¥®±¥® xeeee

SS

xx

xx

x

z

x

1limlim 1111lim 2

2

±=÷÷ø

öççè

æ-

-+

= -

-

±¥® xee

x

x

x

Vector model（spins rotate continuously）

S→∞，<Mz> →∞

But，<Mz>/M is finite, and the magnitude of spin is normalized.

( )xSLNgM Bµ=

In the vector model of the finite magnitude of spin, 古典的モデルで有限のスピンの⼤きさをもつとき，後からスピンの⼤きさをかける．

( )HxLSNg B ¶

¶= µc

( ) ( ) qqpqµp

dTkSHgSzJiZB

Bz sin2cos2exp0ò ú

û

ùêë

é+=

[ ] aap dASò-=1

1exp2

( )( )

úû

ùêë

é-

-+

=--+

=¶¶

= -

-

-

-

ASeeeeS

AeeeeS

AiZ

iZS ASAS

ASAS

ASAS

ASAS

z111

ベクトル模型（古典的モデル）での分配関数

aq =cos

aqq dd -=sin

[ ]ASAS eeAS

--=p2

( ) [ ] [ ]ASASASAS eeA

eeSAA

iZ -- ++--=¶¶ pp 22

2[ ] [ ]ASASASAS ee

Aee

SA-- ++--=

pp 222

( )ASLAS

AS =úûù

êëé -=

1coth

( ) [ ] aaap dASSAiZ

ò-=¶¶ 1

1exp2

úû

ùêë

é-

-+

= -

-

ASeeee

SS

ASAS

ASASz 1

◆Temperature dependence of susceptibility of a paramagnetic material

Paramagnetism

Hamiltonian of paramagnetic material

åå å -=-×-=>< i

izBji i

izBjiij SHgSHgJH µµ,

2 SS

( ) ( )B

B

kSSgNC

312 +

=µ

(J=0)

(4.23)

Curie constant

( ) ( ) ( )TC

TkSSgNT

B

B =+

=1

312µc

(4.24)

( )TCT =c

( )CTT

CT-

=c

Curieʼs law

Curie-Weissʻs law

Paramagnetism

Ferromagnetism

◆Arrott plot

How is M(T) determined experimentaly?

Since ferromagnetic material is generally in multidomain structure, ( ) ( )TMTMr <

M(T,0) is determined by Arrott plot.

( )TM

H

rM 残留磁化

( )TM 2

MH

M(T,0)

Arrott plot

CTT <

CTT >

CTT =

，from eq(4.12)

TkSHgSzJS

xB

Bz µ+=2

( ) ( ) 0,0 ¹<< HMTM

( ) ( ) ( )[ ] 32

22

9011

31 x

SSSSxSxSBS Sz

+++-

+==

( ) ( )[ ] 3

2

22 290112

31

÷÷ø

öççè

æ+

+++-÷÷ø

öççè

æ+

+=

TkSHg

TkSzJS

SSSS

TkSHg

TkSzJSS

B

B

B

z

B

B

B

z µµ

TkSHg

TkSzJS

B

B

B

z µ>>

2

( ) ( ) ( )[ ] 332

22

9011

31

zz SAS

SSSBHSAS +++-+

+= Tk

SgBTkzJSA

B

B

B

µ== ,2

( )[ ] ( ) ( ) zz S

HAB

SSS

SASSASS 322

2

222

22

130

131

130

+++

þýü

îíì

+-

++=

( ) ( ) ( ) zB

BBBB

SSH

Tkg

zJTk

SSSzJSTk

zJTk

SS ÷÷ø

öççè

æ÷øö

çèæ

+++

þýü

îíì

+-÷

øö

çèæ

++=

µ3

22

2

22 2130

1231

2130

( ) ( ) ( )[ ]( )

( )( ) M

HTT

zJgN

SSSS

TT

TT

SSSSNgSNgTM

C

B

CC

BzB

2443

22

22

22

222

211

3101

11

310

÷÷ø

öççè

æ÷÷ø

öççè

æ

+++

+÷÷ø

öççè

æ-÷÷

ø

öççè

æ

+++

==µµµ

( )B

C kJSzST

312 +

=(4.23)

次の32

TkSzJS

B

z 次の1TkSHg

B

Bµ

( )TM 2

MH

( ) ( )[ ]( ) ÷÷

ø

öççè

æ-÷÷

ø

öççè

æ

+++

=CC

B

TT

TT

SSSSNgTM 1

11

310

2

22

22 µ ref. (4.15)

( ) ( )( ) CC

B

TT

TT

SS

SSNgTM -÷÷ø

öççè

æ

++

+= 1

1

1310

22

µ(4.24)

◆Temperature dependence of specific heat

( ) ( )iZTkiF B ln-=

In the same way of

(4.26)

Free energy

( ) ( )TiF

TTiU

¶¶

-= 2

Partition function Z(i) of iʼth spin( )

TkHgSzJ

STk

HgSzJ

iZ

B

Bz

B

Bz

22

sinh

212

sinh

µ

µ

+

úû

ùêë

é÷øö

çèæ +

+

=

(4.25)

( )iZT

TkB ln2

¶¶

=( )

( )TiZ

iZTkB

¶¶

=2

zS

( ) 22 zSzJiU -=

Energy of a material

(4.27)( ) 2

21

zSNzJiNUU -==

Specific heat at constant volume

V

zz

VV T

SSNzJ

TUC ÷÷

ø

öççè

涶

-=÷øö

çè涶

= 2 (4.28)

(4.29)

●from eq (4.17) at T ≈ 0，

( ) TkzJS

TkzJS

BV

BB eeSTkzJSNzJTC

22

222

--

÷÷ø

öççè

æ-÷÷

ø

öççè

æ»

( ) !+-==-

TkzJS

SzBeSxSBS2

TkzJS

BB

BeTkzJSNk

22

4-

÷÷ø

öççè

æ»

( )TkzJS

BBTVT

BeTkzJSNkTC 2

2

00

14limlim ÷÷ø

öççè

æ=

®®!+÷÷

ø

öççè

æ++

÷÷ø

öççè

æ=

® 2

2

0 22121

14lim

TkzJS

TkzJSTk

zJSNk

BB

BBT

02

61

21

22

lim 20=

+÷÷ø

öççè

æ++÷

øö

çèæ+÷

øö

çèæ

=®

!TkzJS

zJSTk

zJSTk

Nk

B

BB

BT (4.30)

satisfy the third law of thermodynamics

(4.31)

( )( ) C

C

Cz T

TTTT

SS

SSS -÷÷ø

öççè

æ

++

+=

221

1310

● from eq (4.15) at T≈TC ,

( )( ) ú

úû

ù

êêë

é÷÷ø

öççè

æ --

-

++

+=

¶¶

- 2/1

222 21

1

1310

C

C

CC

C

C

z

TTT

TT

TTT

TSS

SSTS

( ) ( )( ) ÷÷

ø

öççè

æ-

+++

=¶¶

-= 123

11

31022 222

22

CC

zzV T

TTT

SSSSNzJ

TS

SNzJTC

( ) ( )( ) C

VTT TSSSSNzJTC

C

111

352lim

22

22

0 +++

=-®

( )B

C kJSzST

312 +

=( )

( ) 22115SS

SSNkC BV +++

=D (4.32)

( ) 0lim0

=+®

TCVTT C

At T>TC ， 0=zS

(4.33)( )TCV

T0 CT

CT32

÷÷ø

öççè

æ-1

23

2CC TT

TTa

TkzJS

B

BeTkzJS 22

-

÷÷ø

öççè

æb ( )

( ) 22115SS

SSNkB+++

Temperature dependence of specific heat

21

=S BV NkC23

=D

¥=S BV NkC25

=D

4.3 Molcular field theory of antiferromagnetism

antiferromagnetism

( ) kBkzz gSSJzkH SH ×-= -+ µ2

( ) lBlzz gSSJzlH SH ×-= +- µ2

In the same way as MFT of ferromagnetism, ＋lattice −lattice

kʼs spin of ＋lattice

Consider two sublattice

(4.34)

(4.35)

H//z axisH

( ) kzBkzz HSgSSJzkH µ-= -+ 2

( ) lzBlzz HSgSSJzlH µ-= +- 2

(4.36)

(4.37)

kʼs spin of −lattice

( )

( )kZ

eSSS

S

Sk

TkkH

kz

zkz

B

+-=

-

+å

+

==

Average value of Skz

÷÷

ø

ö

çç

è

æ --=

-+

TkHSgSSJz

SBSB

BzSz

µ2

Average value of Slz

Since Bs(x) is an odd function of x at H=0,

(4.38)

(4.39)

(4.40)

zzz SSS º-= -+ (4.41)

÷÷ø

öççè

æ=

TkSSJz

SBSB

zSz

2 (4.42) Same temperature dependence as that of ferromagnetic material

÷÷

ø

ö

çç

è

æ --=

+-

TkHSgSSJz

SBSB

BzSz

µ2

( )B

N kJSzST

312 +

= (4.43)

0=zSat

Neel temperature

Spontaneous magnetizations of both lattices are antiparallel, and their magnitudes are the same.

(4.45)

◆Temperature dependence of susceptibility

( )0

// 2=

-+

úúû

ù

êêë

é

¶

+¶=

H

zzB H

SSgN µc (4.44)

From (4.39)，(4.40)

÷÷

ø

ö

çç

è

æ +-=±

TkHSgSSJz

SBSB

BzSz

µ!2( )úú

û

ù

êê

ë

é+÷

÷

ø

ö

çç

è

æ

¶

¶-=÷

÷

ø

ö

çç

è

æ

¶

¶

==

±

TkSg

HS

TkSJz

dxxdBS

HS

B

B

H

z

B

S

H

z µ

00

2 !

( ) ( )úû

ùêë

é+-=

TkgNS

TkSJz

dxxdBS

B

B

B

S2

////

2 µcc

( ) ( )

( )dxxdBSJzTk

dxxdBSgN

SB

SB

2

2

//

2+=

µc (4.46)

◆ NTT ³ 0=±zS ( ) ( ) ( )[ ] 3

3

22

9011

31 x

SSSSx

SSxBS

+++-

+=

( )

SSSJzTkS

SSgN

B

B

312

31

2

2

// ++

+

=µ

c( )B

N kSSJz

T3

12 +=( ) ( )

NB

B

TTkSSgN

++

=1

312µ (4.47)

◆ NTT £

( ) ( )

( )dxxdBSJzTk

dxxdBSgN

SB

SB

2

2

//

2+=

µc

( )( ) ( )

( ) 0121

1

lim2

2

//0=

+=

®

dxxdB

TkSJz

dxxdB

TkSgN

TS

B

S

BB

T

µc

( )( ) ( )

( )( )

NB

B

SB

SB

TT TkSSgN

dxxdBSJzTk

dxxdBSgN

TN 2

13

)1(

2lim

2

2

2

//+

=+

=®

µµc

(4.48)

(4.49)

NT=q

÷÷

ø

ö

çç

è

æ +-=±

TkHSgSSJz

SBSB

BzSz

µ!2( ) ( ) ( )[ ] 3

2

22 290112

31

÷÷

ø

ö

çç

è

æ -+++-÷÷

ø

ö

çç

è

æ -+==±

TkSSJz

SSSS

TkSSJzSxSBS

B

z

B

zSz

!!

( )[ ]( )

33

22

22

11013

zN

zN

z STT

SSSSS

TTS ÷

øö

çèæ

+++

-÷øö

çèæ=

( )B

N kSSJz

T3

12 +=

( ) ( )[ ] 3

2

22 290112

31

÷÷ø

öççè

æ+++-÷÷ø

öççè

æ+=

TkSSJz

SSSS

TkSSJzSS

B

z

B

zz

( )( )

2/3

221

1

1310

÷÷ø

öççè

æ-

++

+=

N

Nz T

TTT

SS

SSS

( ) ( ) ( )[ ] 23

22

3011

31' x

SSSS

SSxBS

+++-

+=

( ) ( )[ ] ( )( ) ÷

øö

çèæ -÷÷

ø

öççè

æ

+++

÷÷ø

öççè

æ+++-

+= 1

11

3102

3011

31

3

22

222

3

22

TT

TT

SSSS

TkSJz

SSSS

SS N

NB

( )( )[ ] ÷

øö

çèæ -÷÷

ø

öççè

æ

+++

= 113

1103

22

222

TT

TT

SSSSS N

Nz

( )÷øö

çèæ -÷÷ø

öççè

æ+-

+= 1131

TT

TT

SS

SS N

N÷÷ø

öççè

æ-

+=

321

NTT

SS

( ) ( )

( )dxxdBSJzTk

dxxdBSgN

SB

SB

2

2

//

2+=

µc

( )

÷÷ø

öççè

æ-

++

÷÷ø

öççè

æ-

+

=

3212

321

2

2

NB

NB

TT

SSSJzTk

TT

SSSgN µ

( )N

NB

TTTT

zJgN

--

=2

234

2µ (4.50)

◆near TN NTT £

External field is applied to the direction perpendicular to z-axis.

from (4.36), ＋lattice is interacted with the nearest neighbor spins. kzS

( ) kzz SSJzkH -+ = 2Magnetization of ＋lattice

++ = SM BgN µ2

Consider effective field produced by the nearest neighbor spins

÷øö

çèæ×÷÷

ø

öççè

æ=×- +-+

+ SSMH BB

gNgJz

µµ 22

+M

-H

+H

^H

-M

( ) ( )-+-+^ +=+-= SSHHH

BgJzµ2

( )-+-+^ +=+= SSMMM

2BNgµ

B

B

gJz

Ng

µ

µ

c22==

^

^^ H

M ( )Jz

gN B

4

2µ=

( ) ( )NB

BN Tk

SSgNT21

3)1(2

//+

=µc

( )Jz

gN B

4

2µ=

constant

(4.51)

(4.52)

(4.53)

(4.54)