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Lattice VibrationsLattice VibrationsPart IIPart II
Solid State PhysicsSolid State Physics
355355
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Three DimensionsThree Dimensions
For each mode in a given propagation direction,the dispersion relation yields acoustic and optical branches:
• Acoustic• Longitudinal (LA)• Transverse (TA)
• Optical• Longitudinal (LO)• Transverse (TO)
If there are p atoms in the primitive cell, there are 3p branches in the dispersion relation: 3 acoustic and 3p -3 optical.
NaCl – two atoms per primitive cell
6 branches:
1 LA
1 LO
2 TA
2 TO
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CountingCounting
This enumeration follows from counting the number of degrees of freedom of the atoms. For p atoms in N primitive cells, there are pN atoms. Each atom has 3 degrees of freedom, one for each of the 3 directions x, y, and z. This gives 3Np degrees of freedom for the crystal.
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q = ±/a
q.
q
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Three DimensionsThree Dimensions
Al Ge
) ( trqin ueu
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Quantization of Elastic WavesQuantization of Elastic WavesThe energy of an elastic mode of angular frequency is
It is quantized, in the form of phonons, similar to the quantization of light, as both are derived from a discrete harmonic oscillator model.
Elastic waves in crystals are made up of phonons. Thermal vibrations are thermally excited phonons.
En n 12
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Phonon MomentumPhonon Momentum
A phonon with a wavevector q will interact with A phonon with a wavevector q will interact with particles, like neutrons, photons, electrons, as if it particles, like neutrons, photons, electrons, as if it had a momentum (the crystal momentum)had a momentum (the crystal momentum)
qp
• Be careful! Phonons do not carry momentum like photons do. They can interact with particles as if they have a momentum. For example, a neutron can hit a crystal and start a wave by transferring momentum to the lattice.
• However, this momentum is transferred to the lattice as a whole. The atoms themselves are not being translated permanently from their equilibrium positions.
• The only exception occurs when q = 0, where the whole lattice translates. This, of course, does carry momentum.
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R
rr R
Proton AProton B
electron
Phonon MomentumPhonon Momentum
H2
For example, in a hydrogen molecule the internuclearvibrational coordinater1 r2 is a relative coordinate and doesn’t have linear momentum.
The center of mass coordinate ½(r1 r2 )corresponds to the uniform mode q = 0 and can have linear momentum.
r1 r2
O
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Phonon MomentumPhonon Momentum
Earlier, we saw that the elastic scattering of x-rays from the lattice is governed by the rule:
Gkk
If the photon scattering is inelastic, with a creation of a phonon of wavevector q, then
Gkqk
qp
If the photon is absorbed, then
k k G q
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Phonon Scattering (Normal Phonon Scattering (Normal Process)Process)
q1
q2
q3 = q1 + q2
q3 = q1 + q2 or q3 = q1 + q2 + G
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Measuring PhononsMeasuring Phonons
Gkqk
scattered neutron
phonon wavevector(+ for phonon created, for phonon absorbed)
incident neutron
reciprocal lattice vector
Stokes or anti-Stokes Process
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Measuring PhononsMeasuring Phonons
q
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Measuring PhononsMeasuring Phonons
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Measuring PhononsMeasuring Phonons
• Inelastic X-ray Spectroscopy• Raman Spectroscopy (IR, near IR, and visible light)• Microwave Ultrasonics
Other Techniques
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Heat CapacityHeat Capacity
VV T
UC
Cv = yT+T3
You may remember from your study of thermal physics thatthe specific heat is the amount of energy per unit mass required to raise the temperature by one degree Celsius. Q = mcT
Thermodynamic models give us this definition:
electrons phonons
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Heat CapacityHeat Capacity
Equipartition Theorem:The internal energy of a system of N particles is
Monatomic particles have only 3 translational degrees of freedom. They possess no rotational or vibrational degrees of freedom. Thus the average energy per degree of freedom is
It turns out that this is a general result.
TNkB23
TNkB21
The mean energy is spread equally over all degrees of freedom, hence the terminology – equipartition.
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Heat CapacityHeat Capacity
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Answer: You need to use quantum statistics to describe this properly.
Bosons and FermionsBosons: particles that can be in the same
energy state (e.g. photons, phonons)Fermions: particles that cannot be in the same
energy level (e.g. electrons)
Heat CapacityHeat Capacity
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Planck DistributionPlanck Distribution
Max Planck – first to come up Max Planck – first to come up with the idea of quantum with the idea of quantum energyenergy
worked to explain blackbody worked to explain blackbody radiationradiation empty cavity at empty cavity at
temperature temperature TT, with which , with which the photons are in the photons are in equilibriumequilibrium
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Planck DistributionPlanck Distribution
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Einstein ModelEinstein Model
1907-Einstein developed first reasonably 1907-Einstein developed first reasonably satisfactory theory of specific heat satisfactory theory of specific heat capacity for a solidcapacity for a solid
assumed a crystal lattice structure assumed a crystal lattice structure comprising comprising NN atoms that are treated as an atoms that are treated as an assembly of 3assembly of 3NN one-dimensional one-dimensional oscillatorsoscillators
approximated all atoms vibrating at the approximated all atoms vibrating at the same frequency (unrealistic, but makes same frequency (unrealistic, but makes things easier)things easier)
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Planck DistributionPlanck Distribution
number of phonons in energy level n
total number of phonons
all possible energy levels 0, 1, 2, etc.
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Fraction of small as n gets large
Phononsat energy n a constant
Planck DistributionPlanck Distribution
n n nhf
/ /
/ /
0 0 0
n
n
kT n kTn
kT n kTn
N e e
N e e
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Planck DistributionPlanck Distribution
average occupied
energylevel
//
0 0 0
1 1 1
1 1 1B
B
n k Tn nk T
n n n
x e ex e e
0 0
1 1
1 1n n
n n
x ex e
0
1
1n
n
xx
0 0
n n
n n
nx xx
20 0
1
1 1
n n
n n
xnx x x x
x x x x
0
n
n
nx
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Einstein ModelEinstein Modelaverage
energyper oscillator
We have 3N such oscillators, so the total energy is
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Einstein ModelEinstein Model
2 and
let
TkdT
dv
Tkv
B
B
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Einstein ModelEinstein Model
T
How did Einstein do?
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Einstein ModelEinstein Model
How did Einstein do?
K0T
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Einstein ModelEinstein Model
The Einstein model failed to identically match the behavior of real solids, but it showed the way.
In real solids, the lattice can vibrate at more than one frequency at a time.
Answer: the Debye Model