Part IV Magnetic Properties of Materials · Part IV Magnetic Properties of Materials Chap. 14...

Part IV Magnetic Properties of Materials

Chap 14 Foundations of Magnetism

Chap 15 Magnetic Phenomena and Their

Interpretation- Classical Approach

Chap 16 Quantum Mechanical Considerations

Chap 17 Applications

KINDS OF MAGNETISM

Different types of magnetism are characterized by the magnitude and the sign of the susceptibility

151 Overview

1511 Diamagnetism

Diamagnetism may then be explained by postulating that the external magnetic field induces a change in the magnitude of inner-atomic currents in order that their magnetic moment is in then opposite direction from the external magnetic field

A more accurate and quantitative explanation of diamagnetism replaces the induced currents by precessions the electron orbits about the magnetic field direction (Larmor precession)

Lenzrsquos law

What is diamagnetism

The induced current will appear in such a direction that it opposes change that produce it

151 Overview

Larmor frequency

sin and sin

by torque definition sin sin

2

L L

m L

mL

dLdL L d d dt Ldt

dLT B Ldt

B eB BL m

θ φ φ ω ω θ

micro θ ω θ

microω γ

= = there4 =

= there4minus =

minusthere4 = = minus = minus

External field induce a change in the magnitude of inner-atomic currents

H M= minus

It has been observed that superconducting materials expel the magnetic flux lines when in the superconducting state (Meissner effect)

1MH

χ = = minus

151 Overview

Paramagnetism in solids is attributed to a large extent to a magnetic moment that results from electrons which spin around their own axes

What is paramagnetism

In Curie Law Susceptibility is inversely proportional to the absolute temperature T

CT

χ =

In Curie-Weiss Law

CT

χθ

=minus

1512 Paramagnetism

151 Overview

Hundrsquos rule

Hundrsquos rules (1) are based primarily on Coulomb repulsion and secondarily on spin-orbit interactions and (2) account for the existence of atomic magnetic moments even in some atoms with an even number of valance electrons

Ref Modern Magnetic Materials ( RC OrsquoHandley)

151 Overview

Hysteresis Loop

-Spontaneous magnetization

-transition metals Fe Co Ni rare-earth Gd Dy

-alignment of an appreciable fraction of molecular magnetic

moment in some favorable direction in crystal

-related o the unfilled 3d and 4f shells

-ferromagnetic transition temperature (Curie)

Mr = remanent magnetization

Ms = saturation magnetization

Hc = coercive field

1513 Ferromagnetism

151 Overview

Above Curie Temperature Tc ferromagnetics become paramagnetic

For ferromagnetics the Curie temperature Tc and the constant θ in the Curie-Weiss law are nearly identical

However a small difference exists because the transition from ferromagnetism to paramagnetism is gradual

TEMPERATURE-DEPENDENCE OF SATURATION MAGNETIZATION

Figure 157 (a) Temperature dependence of the saturation magnetization of ferromagnetic materials (b) Enlarged area near the Curie temperature showing the paramagnetic Curie point (see Fig 153) and the ferromagnetic Curie temperature

151 Overview

Piezomagnetism

The magnetization of ferromagnetics is stress dependent

A compressive stress increases M for Ni while tensile stress reduces M

151 Overview

Magnetostriction

When a substance is exposed to a magnetic field its dimensions change This effect Called magnetostriction (inverse of piezomagnetism)

65 10~10|| minusminus=

∆=

λ

λll

M orientation =gt change in dimension

151 Overview

Minimization of magnetostatic energy with changing domain shape

Approximately half magnetostatic energy Closed path within crystal to

reduce the magnetostatic energy

Grain growth

DOMAIN

151 Overview

What is antiferromagnetism

Antiferromagnetic materials exhibit just as ferromagnetics a spontaneous alignment of moments below a critical temperature However the responsible neighboring atoms are aligned in an antiparallel fashion

1514 Antiferromagnetism

151 Overview

A site

B site

A

B

Antiferromagnetic ordering

151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +

TEMPERATURE-DEPENDENCE OF ANTIFERROMAGNETIC MATERIAL

151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview

Diamagnetism Paramagnetism

Ferromagnetism Antiferromagnetism

Ferrimagnetism Kinds of magnetism

Non-cooperative (statistical) behavior

Cooperative behavior

151 Overview

Different elements different moments

bull Cubic MObullFe2O3 M = Mn2+ Ni2+ Fe2+ Co2+ Mg2+ Zn2+ Cd2+ etc (ferrite) soft magnet except Co bullFe2O3 bull Hexagonal BaO bull6Fe2O3 hard magnet

Ionic bonding localized field theory

1515 Ferrimagnetism

151 Overview

Ferrimagnetic substances consist of self-saturated domains and they exhibit the phenomena of magnetic saturation and hysteresis Their spontaneous magnetization disappears above a certain critical temperature also called Curie temperature and they become paramagnetic

cT


151 Overview

151 Overview

Example NiO bullFe2O3 12 microB if ferromagnetic ordering (5 Bohr magnetrons for Fe+3 and 2 for Ni+2)

but experimental value is 23 microB (56 emug) at 0 K

More rapid suppression Non Curie-Weiss behavior

Thermal variation Of the magnetic Properties of a Typical ferrimagnetic (NiO bullFe2O3)

151 Overview

Tetrahedral A site Octahedral B site (a) (b)

(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview

Normal spinel AO(B2O3) A2+ on tetrahedral sites B3+ on octahedral sites (eg) ZnFe2O4 CdFe2O4 MgAl2O4 CoAl2O4 MnAl2O4

MObullFe2O3 (M = Zn Cd) non-magnetic (paramagnetic) 8M2+ A site 16Fe3+ B site

MObullFe2O3 (M = Fe Co Ni) Ferrimagnetic 8M2+ B site 16Fe3+ AB sites (disordered state)

Inverse spinel B(AB)O4 half of the B3+ on tetrahedral sites A2+ and remaining B3+ on octahedral sites (eg) FeMgFeO4 FeTiFeO4 Fe3O4 FeNiFeO4

Mixed ferrite NiOFe2O3 + ZnOFe2O3 (NiZn)OFe2O3

151 Overview

electron charge radius of the orbit length of the orbit ( 2 ) velocity of the orbiting electron revolution time

ers rvt

π=

r v

Classical model

2mevrI Amicro = sdot =

2

2 2me e ev r evrI A A At s v r

πmicroπ

= sdot = = = =

Applied field

Induced field

152 Langevin Theory of Diamagnetism

0( )

ee

V d HAdVL dt dt

dv eF ma e adt m

microφ= = minus = minus

= = there4 = =

E

EE

e

electric field strengthV induced voltageL orbit Length

E

20 0 0Thus

2 2eV e eA e r erdv e dH dH dH

dt m Lm Lm dt rm dt m dtmicro π micro micro

π= = = minus = minus = minus

E

r v∆+ν


Induced field


A change in the magnetic field strength from 0 to H yields a change in the velocity of the electrons

2 2

1 1

0 00

or 2 2

v H v

v v

er er Hdv dH v dvm mmicro micro

= minus ∆ = = minusint int int

r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus

Magnetic moment per electron

Induced field


Average value of magnetic moment per electron

2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H

m m mmicro micro micromicro∆ = minus minus = minus = minus

sinr R θ=

R

H

θ

r

θ∆

2 2 2sinr R θlt gt= lt gt 2

2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin

3r R Rθthere4 lt gt= lt gt=


Average value of magnetic moment per atom 2 2

0 6m

e Z R Hmmicromicro∆ = minus

atomic number average radius of all electronic orbits

Zr

Magnetization caused by this change of magnetic moment 2 2

0 6

m e Z R HMV mVmicro micro

= equiv minus

Diamagnetic Susceptibility

2 2 2 20 0 0

6 6diae Z R e Z R NM

H mV m Wmicro micro δχ = = minus = minus

0 Avogadro constant density

W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22

00 microρχ minus==∆∆6108518 minustimesminus=Vχ


Assumptions no interaction only m H interaction and thermal agitation

In a state of thermal equilibrium at temperature T The probability of an atom having an energy E follows the Boltzmann distribution

)exp( kTEpminusprop

0 cosp mE Hmicro micro α= minus

exp( )p Bdn dA E k T= Κ minus22 sindA R dπ α α=

If R=1 ( unit sphere )

02 sin exp( cos )m

B

Hdn dk T

micro microπ α α α= Κ sdot 0m

B

Hk T

micro microζ =

153 Langevin Theory of (Electron orbit) Paramagnetism

02 sin exp( cos )n d

ππ α ζ α α= Κ sdot int

0

2 sin exp( cos )

n

dπ

π α ζ α αthere4 Κ =

int

Intergrating ldquodnrdquo to calculate the number of total atoms in a unit volume

The magnetization M is the magnetic moment per unit volume

0

00

0

cos

cos sin exp( cos ) 2 cos sin exp( cos )

sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α

micro α α ζ α απmicro α α ζ α α

α ζ α α

there4 =

= Κ sdot =

intint

intint

3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro

ζ

= minus = minus minus minus

This function can be brought into a standard form by setting cos and sinx dx dα α α= = minus


kTmHa =

Langevin function L(a) means a ζ


ζ



20

0 3 3

m

B

n HM Mk T

micro microζthere4 = =

20 1 1

3m

paraB

nM CH k T T

micro microχ = = equiv sdot

20

3m

B

nCkmicro micro

=

Because is usually small 0m

B

Hk T

micro microζ =

0 3MM

ζ=

0 the maximum possible magnetizationmM nmicro=


M

SM P

2

1

mH0A B

C D

E

Spontaneous magnetization by a molecular field

m γ=H M

A magnetization greater than will spontaneously revert to in the absence of an applied field The substance has therefore become spontaneously magnetized level which is the value of

PP

PSM

Ferromagnetic

Paramagnetic subject to a very large molecular field

Molecular field

154 Molecular Field Theory

kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB

m sdot===microγmicroγmicro

aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ

Law of corresponding states


Saturation magnetization amp Curie temperature


Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++

= Brillouin function

xsmallforx xx coth 31 +asymp

aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB

m microγmicro==primea

MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0


appl m appl γ= + = +H H H H MWhen a magnetic field is applied

0 0 0

`H

kT Haσσ micro γρσ γρσ

= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ

micro σσχmicro γρσ

+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order

Spontaneous magnetization Spin fluctuation due to thermal agitation



슬라이드 번호 2









































KINDS OF MAGNETISM

Different types of magnetism are characterized by the magnitude and the sign of the susceptibility

151 Overview

1511 Diamagnetism



Lenzrsquos law



151 Overview

Larmor frequency

sin and sin


2

L L

m L

mL

dLdL L d d dt Ldt

dLT B Ldt

B eB BL m

θ φ φ ω ω θ

micro θ ω θ

microω γ

= = there4 =

= there4minus =



H M= minus


1MH

χ = = minus

151 Overview




CT

χ =

In Curie-Weiss Law

CT

χθ

=minus

1512 Paramagnetism

151 Overview

Hundrsquos rule



151 Overview

Hysteresis Loop









Hc = coercive field

1513 Ferromagnetism

151 Overview






151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































1511 Diamagnetism



Lenzrsquos law



151 Overview

Larmor frequency

sin and sin


2

L L

m L

mL

dLdL L d d dt Ldt

dLT B Ldt

B eB BL m

θ φ φ ω ω θ

micro θ ω θ

microω γ

= = there4 =

= there4minus =



H M= minus


1MH

χ = = minus

151 Overview




CT

χ =

In Curie-Weiss Law

CT

χθ

=minus

1512 Paramagnetism

151 Overview

Hundrsquos rule



151 Overview

Hysteresis Loop









Hc = coercive field

1513 Ferromagnetism

151 Overview






151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Larmor frequency

sin and sin


2

L L

m L

mL

dLdL L d d dt Ldt

dLT B Ldt

B eB BL m

θ φ φ ω ω θ

micro θ ω θ

microω γ

= = there4 =

= there4minus =



H M= minus


1MH

χ = = minus

151 Overview




CT

χ =

In Curie-Weiss Law

CT

χθ

=minus

1512 Paramagnetism

151 Overview

Hundrsquos rule



151 Overview

Hysteresis Loop









Hc = coercive field

1513 Ferromagnetism

151 Overview






151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order
















































CT

χ =

In Curie-Weiss Law

CT

χθ

=minus

1512 Paramagnetism

151 Overview

Hundrsquos rule



151 Overview

Hysteresis Loop









Hc = coercive field

1513 Ferromagnetism

151 Overview






151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Hundrsquos rule



151 Overview

Hysteresis Loop









Hc = coercive field

1513 Ferromagnetism

151 Overview






151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Hysteresis Loop









Hc = coercive field

1513 Ferromagnetism

151 Overview






151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order


















































151 Overview

Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Piezomagnetism



151 Overview

Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Magnetostriction



∆=

λ

λll


151 Overview




Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order
















































Grain growth

DOMAIN

151 Overview




151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order
















































151 Overview

A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































A site

B site

A

B


151 Overview

Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Neel temperature

( )

NTC C

T Tχ

θ θ

=

= =minus minus +


151 Overview

)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































)( θχ

minusminus=

TC

CT

χθ

=minus

ferromagnetic

Antiferromagnetic

151 Overview

151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































151 Overview






151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order


















































151 Overview




1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order
















































1515 Ferrimagnetism

151 Overview


cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































cT


151 Overview

151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































151 Overview





151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order

















































151 Overview


(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































(c)

151 Overview

Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Oxygen

Oxygen

A site 8a

B site 16b

42OAB8 molecules

151 Overview






151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order


















































151 Overview


ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































ers rvt

π=

r v

Classical model


2


πmicroπ

= sdot = = = =

Applied field

Induced field


0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































0( )

ee

V d HAdVL dt dt

dv eF ma e adt m


= = there4 = =

E

EE

e


E

20 0 0Thus




E

r v∆+ν


Induced field



2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































2 2

1 1

0 00

or 2 2

v H v

v v



r v∆+ν

2 20

2 4me r He vr

mmicromicro ∆

∆ = = minus


Induced field



2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































2 2 2 2 2 20 0 024 3 4 6me r H e R H e R H


sinr R θ=

R

H

θ

r

θ∆


2

2 0 2

0

sinsin 2 3

d

d

π

π

θ θθ

θlt gt= =

int

int2 2 2 22 sin




0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































0 6m



Zr


0 6


= equiv minus


2 2 2 20 0 0




W atomic mass

Nδ

2r

2r

2r

2r


emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































emHrem

432 22

0microminus=∆

summinus=∆i

ie

n rm

Hem 22

0

6micro

emHrZe

ANM

6)(

2200 microρ

minus=∆

eV m

rZeA

NHM6

)(22





)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order















































)exp( kTEpminusprop




02 sin exp( cos )m

B

Hdn dk T


B

Hk T

micro microζ =




0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order















































0

2 sin exp( cos )

n

dπ


int



0

00

0

cos


sin exp( cos )

n

m

mm

M dn

n dd

d

ππ

π

micro α


α ζ α α

there4 =

= Κ sdot =

intint

intint


ζ




kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































kTmHa =



ζ



20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































20

0 3 3

m

B

n HM Mk T


20 1 1

3m

paraB

nM CH k T T


20

3m

B

nCkmicro micro

=


B

Hk T

micro microζ =

0 3MM

ζ=



M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































M

SM P

2

1

mH0A B

C D

E


m γ=H M


PP

PSM

Ferromagnetic


Molecular field


kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































kTmHa =

T1 T2 T3

321 TTT ltlt

cTT =2

Langevin function

aaaL

MM 1coth)(

0

minus==

MH γ=m

γ1


0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































0

0

MM

TkM

TkM

TkHa

BBB


aMTk

MM B sdot

=

00 microγ

)(31

0 ccc

cB

TTf

TT

TT

MTk

=

=

microγ

)(0 c

s

TTf

MM

asymp

cB TTatMTk

== 31

0microγ





Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order















































Ja

Ja

JJ

JJ

MM

2coth

21

212coth

212

0

primeminusprime

++



aJ

JaJB prime

+

=prime3

1)(

TkM

TkHa

BB


MTk

MM B primesdot

=

00 microγ JJ

MTkB

31

0

+=

microγat T=Tc

aTTJ

JMM

c prime

+

= )(3

1

0

+=

cc

cB

TT

JJ

TT

MTk

31

0microγ


0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































0

1 tanh2

MJ aM

primerarr =

aTTJ

JMM

c prime

+

= )(3

1

0

aTTMM

c prime= )(0

=

cTTMM

MM

tanh 0

0



0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order














































0 0 0

`H


= minus

0

1 `3

J aJ

σσ

+ =

0

0

( 1) 3[ ( 1) 3 ]

H

H

J kJH T J kJ


+= =

minus +

( )C

Tχ

θ=

minus

0 ( 1)3

H JCkJ

micro σ += 0 ( 1)

3H J

kJmicro γρσθ +

=


Spin states

Short range order













































Spin states

Short range order













































Part IV Magnetic Properties of Materials · Part IV Magnetic Properties of Materials Chap. 14...

Documents

Transcript of Part IV Magnetic Properties of Materials · Part IV Magnetic Properties of Materials Chap. 14...