Chap 10 Magnetism in solids

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Chap 10 Magnetism in solids
10.1 Introduction
Source of magnetic moment Spin of electrons
S - Paramagnetic Orbital angular momentum L -
Paramagnetic Induced change of L by B-field -
Diamagnetizm
Magnetic susceptibility
Magnetic moment Angular momentum
a
e
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10.2 Hunds rules
How do we determine L, S, and J of an
atom? Hunds rule Emprical rule to determine
the ground state of an atom with unfilled
shell 1. Maximize the total spin S allowed by
exclusion principle Mn2 (Ar) 3d5
S5/2
2. Maximize Orbital angular momentum L,
consistent with Rule 1.
3
3
Ce3 (Xe) 4f1
L3
3. JL-S if the shell is less than half-filled
LS if the shell is more than
half-filled (Spin-orbit interaction)
3
0
-1
-2
2
-3
1
Ho3 (Xe) 4f10
S2
L3216 JLS8
5I8
2S1 LJ
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• Orbital Momentum Quenching
• - Rare Earth (4f system)
• J by Hunds rule is in good agreement with
  experiment
• both in vapor and in the solid phase
• -Iron Group (3d system Fe, Co, Ni)
• L, S, J given by Hunds rule is in good agreement
  with
• experiment in vapor phase, but in solid
  compounds, Curie constant is much better
  agreement with S.
• As if L0 in solid phase

Crystal field The electrostatic field effect
from neighboring atoms non central, symmetry of
a crystal
Crystal Field gt Spin-Orbit interaction
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Orbital momentum quenching In a noncentral
field the angular momentum components are no
longer constant and may average to zero. When Lz
averages to zero, the orbital angular momentum is
said to be quenched.
The expectation value of L for any nondegerate
state is 0
Px, Py
Pz
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• LS coupling versus JJ coupling

Spin-orbit interaction Electron-electron
interaction

• j-j coupling First consider spin-orbit
  interaction of a single
• electron jls.
• Total angular momentum Jj1j2.jn

• L-S coupling First consider total spin and
  orbital angular
• momentum, (Russell-Saunders )
• Ss1s2sn, Ll1l2l3ln
• Total angular momentum JLS

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10.2 Paramagnetism
Atoms with unfilled shell non zero J
Hamiltonian Magnetic moment in a uniform field
Energy level
H
H
ltQuantumgt
ltClassicalgt
Continuous
2J1 possible direction
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Curies law
Rare Earth Ions Orbital momentum quenching
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Pauli Paramagnetism
Conduction electron has spin Spin interacts with
magnetic field
Magnetic energy
Curies law with Fermi-Dirac statistics
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10.3 Langevin (Lamor) Diamagnetizm
Atoms with filled shell LSJ0 -gt There is no
localized magnetic moment which can give
paramagentic contribution. When a magnetic field
is applied, Faradays law gives induced current,
which gives rise to the magnetic moment opposing
B field. (Lentzs law)
Larmors Theorem In a magnetic field, the
motion of electrons around the nucleus is (to
1st order in B) the same as a possible motion in
the absence of B field, except for the
superposition of a precess of the electron with
angular frequency weB/2mC
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Landau Diamagnetism (Conduction electron)
Orbital motion of conduction electron also
gives diamagnetic contribution.
10.4 Ferro magnetism
Spontaneous magnetic moment in the absense of
the external magnetic field below Tc (Curie
Temperature) gt Spins line up ( Ordered State)
T gtTc
T ltTc
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Exchange Interaction
electron cloud
J is the result of difference in the
electrostatic energy (Coulomb enregy ) due to the
Pauli principle
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• Estimate of magnetic Dipolar Interaction Energies

H
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Exchange Interactions
1. Direct Exchange
magnetic ion magnetic ion
2. Super Exchange
magnetic ion non-magnetic ion magnetic ion
3. Indirect Exchange
e
e
e
e
e
e
e
magnetic ion conduction-electron magnetic ion
3. Itinerant exchange
e
e
e
e
e
e
conduction-electron conduction electron
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lt Mean Field Approximation gt
Assume each magnetic ion experience a field
proportional to the net magnetization
Bint l M
ltMagnetizationgt
In paramagnet,
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ym
t0.5
m
t1
t2
TTc
Mtb
Graphic solution
TT-Tc/Tc
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ltAnti Ferromagnetism gt
Exchange interaction J gt 0 Favors antiparallel
spin
Total magnetization is Zero.
Sublattice magnetization is not zero
sublattice A
sublattice B
Field perpendicular to spin
c
Field parallel to spin
c
0
-TN
TN
Neil Temperature
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10.5 Spin Wave
Elementary excitation of spin states Consider a
linear chain of N spins with exchange interaction
between nearest neighbor only,
.

• The first excited state

UU08JS2
4JS2
We can form an excitation of much lower energy if
we let all spins share the reversal
Spin wave
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ltDispersion relation of spin wavegt
The term that involves p-th spin can be written
as,
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Equation of motion
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Magnon Dispersion Relation
(Quantization of spin wave)
Prove this H.W.
Specific Heat Contributionn C T 3/2
Amplitude v -iu