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12
1
12.1 Overview
Different types of
magnetism are
characterized by the
magnitude and the
sign of the
susceptibility(k)
B  0 H  0 M  0  r H
M kH
r  1  k
(magnetic permeability)
KINDS OF
MAGNETISM
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12.1.1 Diamagnetism
What is diamagnetism?
Lenz’s law :
Current is induced in a wire loop
whenever a bar magnet is moved
toward(or from) this loop.
The induced current will appear in
such a direction that it opposes
change that produce it.
Diamagnetism: 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)- bounded electrons
For metals with free electrons- circulating free electrons can cause a magnetic moment
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Larmor frequency
External field induce a change in the
magnitude of inner-atomic currents,
either accelerate or decelerate the
orbiting electrons
dL  L sin  d
and d  L dt
dL
by torque definition
dt
 B
e
L  m   B  
B
L
2m


dL
 L L sin 
dt
 m B sin   L L sin 
(Torque: tendency of a force to rotate an object about an axis.   r x F)
Again, for metals with free electrons- circulating free electrons can cause a magnetic moment
It has been observed that superconducting materials expel the magnetic flux lines
when in the superconducting state (Meissner effect)
B  0 H  0 M  0
4
 H  M ,
k = M H  1
12.1.2 Paramagnetism
What is paramagnetism?
Paramagnetism in solids is attributed, to a large extent, to a magnetic moment that
results from electrons which spin around their own axes.
(a) Spin paramagnetism: (in most solids)
(b) Electron-orbit paramagnetism: not effective in most solids since the electron-orbits are
essentially coupled to the lattice.
(free atoms(diluted gas) as well as rare earth elements which have “deep lying” 4f-electrons.
In Curie Law, susceptibility is inversely proportional to the absolute
temperature T
k
C
T
In Curie-Weiss Law,
k
5
C
T 
Hund’s rule: In materials with partially filled bands, the electrons are arranged to
have the total spin moment to be maximized.
Ref. Modern Magnetic Materials ( R.C. O’Handley)
The smallest unit (or quantum) of the magnetic moment is called one Bohr magneton.
B 
6
eh
J
 9.274  1024 ( )  ( A  m2 )
4 m
T
12.1.3 Ferromagnetism
-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 to the unfilled 3d and 4f shells
-ferromagnetic transition temperature (Curie)
Hysteresis Loop
Mr = remanent magnetization
Ms = saturation magnetization
Hc = coercive field
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TEMPERATURE-DEPENDENCE OF SATURATION MAGNETIZATION
Figure 15.7 (a) Temperature dependence of the saturation magnetization of ferromagnetic
materials. (b) Enlarged area near the Curie temperature showing the paramagnetic Curie point
(see Fig. 15.3) and the ferromagnetic Curie temperature .
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.
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Piezomagnetism
The magnetization of ferromagnetics is also stress dependent.
A compressive stress increases
M for Ni, while tensile stress
reduces M
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Magnetostriction
When a substance is exposed to a magnetic field, its dimensions change.
This effect called magnetostriction. (inverse of piezomagnetism)
l
l
|  | 105 ~ 106

10
M orientation => change in dimension
DOMAIN
Minimization of magnetostatic energy with changing domain shape
Approximately half
magnetostatic energy
Closed path within crystal to
reduce the magnetostatic energy
Domain
growth
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Domain wall or Bloch wall:
Barkhausen effect:
Kerr effect:
12.1.4. ANTIFERROMAGNETISM
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.
Consider to be divided
into two
interpenetrating
sublattices A and B.
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Antiferromagnetic ordering
A site
B site
A
B
Thus, the magnet moments of the solid cancel each other and the
material as a whole has no net magnetic moment.
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TEMPERATURE-DEPENDENCE OF ANTIFERROMAGNETIC MATERIAL
TN  Neel temperature

14
C
C

T  ( ) T  
ferromagnetic
Antiferromagnetic

15
C
T  ( )

C
T 
The Neel temperature is often below room temperature.
Most antiferromagnetics are found among ionic compounds – insulators or semiconductors.
16
Non-cooperative (statistical)
behavior
Diamagnetism
Ferromagnetism
Kinds of magnetism
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Paramagnetism
Antiferromagnetism
Ferrimagnetism
Cooperative behavior
12.1.5 FERRIMAGNETISM
Different elements, different moments, for an ions situated in crystal
lattice, and thus be described as imperfect antiferromagnetics.
Ionic bonding, localized field theory
• Cubic: MO•Fe2O3
M = Mn2+, Ni2+, Fe2+, Co2+, Mg2+, Zn2+, Cd2+ etc.
(ferrite)
soft magnet except CoO•Fe2O3
• Hexagonal: BaO•6Fe2O3
hard magnet
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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, Tc also called
Curie temperature and they become paramagnetic
There are two types of lattice sites which are available to be occupied by the metal ions.
A sites: tetrahedral site
B sites: octahedral sites
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3d level: Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn (from at. number 21)
They all have ----- 4s2 3dn electrons
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More rapid suppression
Non CurieWeiss
behavior
Thermal
variation
of the magnetic
properties of a
typical
ferrimagnetic
(NiO •Fe2O3)
Example : NiO •Fe2O3 (Nickel Ferrite)
12 B if ferromagnetic ordering (5 Bohr magnetrons for Fe+3 and 2 for Ni+2)
but experimental value is 2.3 B (56 emu/g) at 0 K
Why?
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Fe3+ occupies both A site(tetrahedral) and B site (octahedral) and cancelled out.
Ni2+ occupies B site.
Tetrahedral A site
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Octahedral B site
AB 2O4
8 molecules
Oxygen
B site: 16b
A site: 8a
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Oxygen
Normal spinel: AO(B2O3)
A2+ on tetrahedral sites, B3+ on octahedral sites
(e.g.) ZnFe2O4, CdFe2O4, MgAl2O4, CoAl2O4, MnAl2O4
MO•Fe2O3 (M = Zn, Cd)
non-magnetic (paramagnetic)
8M2+ : A site, 16Fe3+ : B site
Inverse spinel: B(AB)O4
half of the B3+ on tetrahedral sites, A2+ and remaining B3+ on octahedral sites
(e.g.) FeMgFeO4, FeTiFeO4, Fe3O4, FeNiFeO4
MO•Fe2O3 (M = Fe, Co, Ni)
Ferrimagnetic
8M2+ : B site, 16Fe3+ : A,B
sites
(disordered state)
Mixed ferrite: NiO•Fe2O3 + ZnO•Fe2O3   (Ni,Zn)O•Fe2O3
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12.2 Langevin Theory of Diamagnetism
Classical model
Applied field
e
e
ev r 2 evr
m  I  A  A 
A

t
s/v
2 r
2
r
v
Induced field
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e : electron charge
r : radius of the orbit
s : length of the orbit ( 2 r )
v : velocity of the orbiting electron
t : revolution time
E : electric field strength
Ve : induced voltage
L : orbit length
r
  v
Induced field
Thus,
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Ve
d ( 0 HA) Lenz’s law
d
E ,
Ve  

L
dt
dt
where,  is magnetic flux = BA =(0 H  0 M ) A
F  ma  eE
 a
dv eE

dt m
eA0 dH
e r 2 0 dH
er 0 dH
dv eE Ve e





dt m Lm
Lm dt
2 rm dt
2m dt
r
  v
A change in the magnetic field strength from 0
to H yields a change in the velocity of the
electrons :
Induced field

v2
v1
er 0 H
dv  
dH

2m 0
or
Magnetic moment per electron:
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e 2 r 2 0 H
evr
m 

2
4m
e r
since,  m 
(15.5)
2
v  
v2
v1
er 0 H
dv  
2m
Average value of magnetic moment per electron
So far, we tacitly assumes that the magnetic field is perpendicular to the plane of orbiting
electron. In reality, the orbit plane varies constantly in direction with respect to the external
field.
H
r
r  R sin 
 r  R 2  sin 2  
2


R
 /2
 sin  
2

0
sin 2  d
 /2
 2/3
 d
0
2
  r 2  R 2  sin 2   R 2
3
m  
0 e 2 r 2 H
4m
0 e 2 R 2 H
2 0 e 2 R 2 H


3
4m
6m
Average value of magnetic moment per atom
e 2 Z R 2 0 H
m  
6m
Z : atomic number
R : average radius of all electronic orbits
Magnetization caused by this change of magnetic moment
m
e 2 Z R 2 0 H
M

V
6mV
Diamagnetic Susceptibility
 dia
e 2 Z R 2 0
e2 Z R 2 0 N 0
M



H
6mV
6m
W
The susceptibility is temperature-independent.
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N0 : Avogadro constant
 : density
W : atomic mass
12.3 Langevin Theory of (Electron orbit) Paramagnetism
Not treat spin paramagnetism, which is responsible for the paramagnetic
behavior of many metals and which is only slightly temperature dependent.
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
0H
 exp(  E p / kT )

In general, the potential energy is,
vectors representing the
magnetic moments
m
E p   0 μm  H
E p   m 0 H cos 
dn  const.dA exp( E p / k BT )
dA  2 R 2 sin  d
If R=1 ( unit sphere )
dn  const  2 sin  d exp(
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 m 0 H
k BT
cos  )
 
 m 0 H
k BT
Intergrating “dn” to calculate the total magnetic moments in a unit volume

n  2 (const.)   sin  exp( cos  ) d
0
 const. 
n

2  sin  exp( cos  ) d
0
The magnetization (M) is the magnetic moment per unit volume – sum of all
individual magnetic moment in the field direction.
n
 M   m cos  dn
0


 (const.)  2m  cos  sin  exp( cos  ) d 
0
nm  cos  sin  exp( cos  ) d
0


0
sin  exp( cos  )d
This function can be brought into a standard form by setting
x  cos and dx   sin  d
  3  5

1
M  nm  coth     nm  




 3 45 945
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


Langevin function, L()
  3  5

1
M  nm  coth     nm  




 3 45 945
a  mH / kT
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 
 m 0 H
kT



M 0  nm : the maximum possible magnetization
Because
M


M0
3
 para
 
 m 0 H
k BT
nm2 0 H
 M  M0 
3
3k BT

M nm2 0 1
1


C
H
3kB T
T
nm2 0
C
3k B
34
is usually small,
(Curie’s law)
12.4 Molecular Field Theory:
So far, we implied that the magnetic field, which tries to align the magnetic moments,
stems from the external source only.
Weiss observed that some materials obey a somewhat modified Curie law.
Postulate that the magnetic moment of individual electrons interact each other.
Ht  He  H m
where, H m   M ( : molecular field constant)
M
2
then, k 
1
MS
H eC
T C
M
C
C
Finally, k =


He T   C T 
thus, M 
P
E
C
A
0
M
M
C


H t H e  M T
D
B
Hm
Molecular field
Ms: spontaneous magnetization by a molecular field
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Hm   M
Weiss postulated that the above-mentioned internal or molecular field is
responsible for the parallel alignment of spins and considered
ferromagnetics to be essentially paramagnetics having a very large molecular
field.
Molecular field is essentially the exchange force.
For instance, when there is no external magnetic field,
 
m 0 H m
M
k BT
k BT
 m  0

m 0 M
k BT

Magnetization is a linear
function of  with a
temperature as a
proportionality factor.
T3
H m  M
T
T2 Curie temp 1
Langevin function
M
1
 L(a)  coth a 
M0
a
1

T1  T2  T3
T2  Tc
a  mH / kT
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At Curie temperature, there is no spontaneous magnetization, thus, the slope is
identical to the slope of the Langevin function.
k BT
m 0

k BT C
 m 0

n m M

3
3
By rearranging it,
3k BT C

 m 0 M
Hm   M 
37
3k BT C
 m 0
~ 109 A / m
Saturation magnetization
& Curie temperature
38
Short range order
Spontaneous magnetization
Spin states
39
Spin fluctuation
due to thermal agitation