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Lattice Vibration - Coggle Diagram
Lattice Vibration
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1-D Monoatomic lattice
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Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies
total force is
Eqn of motion is
ω-K relation
-In a transverse wave, the particles are displaced perpendicular to the direction the wave travels.
-In a longitudinal wave the particles are displaced parallel to the direction the wave travels.
-The simplest crystal is the one dimensional chain of identical atoms with identical masses M connected by identical springs of spring constant K.
-atom are separted by distance, a, and move only in parallel direction to the chain
phase velocity
the velocity of a particular feature of the wave (such as its maximum)
group velocity
energy is transported by the wave at a generally slower speed, the group
Phonon in 3D
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real crystal
- In a 3-D crystal, the atoms vibrate in three dimensions with two
vibration modes and three vibrational branches, one longitudinal and two transverse
- Force constant may vary - the branches are not degenerate
in 3D,eqn of motion is
Transverse mode
- in real crystals, these two transverse modes are degenerate on only in special high-symmetry directions.
- In general, for directions of propagation, each of these two transverse modes and the one longitudinal mode all have different frequencies.
- in a general k-direction, the distinction between longitudinal and transverse waves no longer has meaning since the displacements are no longer exactly parallel or perpendicular to the wave vector
brillouin zones of reciprocal lattice or 1D monoatomic lattice
wigner seitz cell construction
- unit cellis to connect each lattice point to all its neighboring point with a line segment & bisect each line segment with perpendicular plane
Phonons
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phonon momentum
A phonon of wavevector k will interact with particles such as photons, neutrons and electrons as if it had a momentum ħk but it does not carry physical momentum
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phono generation
piezoelectric
- When an electric field is applied to a piezoelectric material, it experiences strain.
- EM waves (10 GHz) can generate an oscillating electric field
- The oscillating electric field the piezoelectric transducer at the same frequency.
- The transducer emits phonon into the specimen medium
- Conversion from phonon to phonon is inefficient (probability order of ~ 10-7)
thermal excitation
- Current is flow through metal wires causing the electron temperature to rise.
- the hot electrons releases energy into the surrounding by emitting phonon and photons into the metal and its surrounding.
- Above the threshold frequency, only photons are produced.
electron tunneling
- A thin layer of insulator is placed between two thin layers of metal to form a barrier for the electrons.
- At certain energies, the electron can tunnel through the barrier
- The electron is speed up with additional kinetic energy eV.
- This additional energy is released in the form of phonon emission.
- Phonon is emitted by hot electrons that losses energy when returning to equilibrium
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-In reality, atoms are not very heavy and the force
acting between the atoms is not an infinite
-therefore, static lattice models are valid only at zero temperatures
-when T is not equal 0, each atom has thermal
energy and there is vibration around the equilibrium lattice
position
-amplitude of the motion increases as the atoms gain more
thermal energy at higher temperatures.
Hooke's law
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Proportional Limit is the point at which the deformation is
no longer directly proportional to the applied force
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Debye Model
the eqn for the Cv is
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Born Von Karman Model
boundary conditions are periodic boundary conditions which impose the resriction that a wave function must be periodic on a certain Bravais attice