Post on 24-Aug-2020
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
)( 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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
Different elements different moments
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
2mevrI Amicro = sdot =
Induced field
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
152 Langevin Theory of Diamagnetism
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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=
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
152 Langevin Theory of Diamagnetism
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
152 Langevin Theory of Diamagnetism
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
153 Langevin Theory of (Electron orbit) Paramagnetism
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
kTmHa =
Langevin function L(a) means a ζ
3 51coth3 45 945m mM n n ζ ζ ζmicro ζ micro
ζ
= minus = minus minus minus
153 Langevin Theory of (Electron orbit) Paramagnetism
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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=
153 Langevin Theory of (Electron orbit) Paramagnetism
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
kTmHa =
T1 T2 T3
321 TTT ltlt
cTT =2
Langevin function
aaaL
MM 1coth)(
0
minus==
MH γ=m
γ1
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
Saturation magnetization amp Curie temperature
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
0
1 tanh2
MJ aM
primerarr =
aTTJ
JMM
c prime
+
= )(3
1
0
aTTMM
c prime= )(0
=
cTTMM
MM
tanh 0
0
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
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 +
=
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory
Spin states
Short range order
Spontaneous magnetization Spin fluctuation due to thermal agitation
154 Molecular Field Theory