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ESTHER ONG PECK SUAN_A19SC0485_MINDMAP 3, Lattice Vibration III …
ESTHER ONG PECK SUAN_A19SC0485_MINDMAP 3
Lattice Vibration III
(Diatomic)
For M:
For m:
Dispersion Relation;
Upper branch: Optical branch
Lower: Acoustical branch
Normal mode:
pattern of motion
all parts move sinusoidal
same frequency
A: M&m anti phase, COM at rest
B: m oscillate, M at rest
C: M oscillate, m at rest
Optical Branch:
Forbidden band
smaller if m/M close to 1
no gap when m=M
1st Brillouin Zone
Acoustic Branch:
M=m (degenerate)
CO is extension of CA
Lattice Vibration I
(General-Basic Knowledge)
Approximation before: Lattice point (Static)
Valid: At zero temperatures
Reality: Each atom has thermal energy
vibration occurs
Apply harmonic approximation:
lattice vibration at small amplitude
Atomic Motions
Forces exerted - displaced from equilibrium position
SHM & Hooke's law
Thermal energy
heat capacity
Dulong-Petit Law
At high temperature,
~ Consistent with Classical Gas Theory
At low temperature,
~ Explained by quantum theory
Debye Model
Exact at high and low T
Born Von Karman Model
Introduce periodic boundary condition
Einstein Model
relate to quantum theory
Good for high T behaviour
Not good for low T behaviour
Lattice Vibration II (Monoatomic)
Atoms move only in a direction parallel to the chain.
Only nearest neighbours interact (short-range forces).
w-K Relation: Dispersion Relation
First brillouin zone
Phase Velocity
Group Velocity
Long Wavelength Limit
Linear dispersion
Sound velocity for 1D lattice
Lattice Vibration IV
(Phonon)
Phonons
Quantum of lattice vibration
Energy of mode
Total vibrational energy of crystal
Generation
Piezoelectric
Thermal Excitation
Electron tunnelling
In 3D
(Bravais)
1 longitudinal
2 transverse
Dispersion Curve - Ax, Ay and Az
High symmetry
[100],[110],[111]
Pure longitudinal
Pure trasnverse
In 3D
(non-Bravais)
3z dispersion curve
Acoustic: 3
Optical: 3z-3
3zN degree of freedom
1st Brillouin zone
bcc direct -> fcc reciprocal
fcc direct -> bcc reciprocal
