Lattice Vibration

Lattice Vibration I

Hooke's Law

  • Extension of the spring is directly proportional to the pulling force
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Inelastic Limit

  • permanent deformation occurs
  • will not return to its original size and shape

Proportional Limit

  • deformation no longer directly proportional to the applied force

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Sample Periodic Motion

  • motion which body moves back and forth
  • return to each position and velocity
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Definition

  • periodic motion in the absence of friction and produced by restoring force and directly proportional to displacement and oppositely directed

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Thermal energy and lattice vibration

  • atoms vibrate about their equilibrium position
  • produce vibration waves
  • increased as the temperature is raised

Heat Capacity

Definition

  • the amount of heat needed to raise an unit amount substance by unit in temperature

Heat capacity constant volume
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Heat capacity constant pressure
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Classical gas theory

Kinetic energy
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Models Lattice Vibration

Einstein Model

  • assume all the oscillators oscillate with common frequency

Average thermal energy
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  • examine behavior this expression at high and low temperature
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Debye Model

  • recovers the Dulong-Petit law at high temperature
  • The maximum vibration frequency determined by Debye frequency

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  • exact at low and high temperature and an interpolation formula in between

Born Von Karman Model

  • periodic boundary condition which impose the restriction that wave function must be periodic on a certain Bravais lattice

Lattice Vibration II

Monoatomic Chain

  • one dimensional chain of identical atoms
  • consists of very large number of identical atoms with identical mass
  • atoms separated by distance of "a"
  • nearest neighbours interact

Force to the right:
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Force to the left:
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Equation of motion for nth atom
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Dispersion relation of the monatomic
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Transverse wave:

  • displaced perpendicular to the direction the wave travels

Longitudinal wave:

  • displaced parallel to the direction the wave travels

ω-K Relation
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Phase velocity and Group velocity

  • Phase velocity


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  • Group velocity


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Lattice Vibration III

1D diatomic chain of 2N atoms

  • N atoms of mass m and N atoms of mass M
  • mass m has two near neighbours of mass M
  • two atoms per unit cell
  • repeat distance is a

Equation of motion mass M (nth)
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Equation of motion mass m (n-1)th
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ω versus k relation for diatomic chain
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Normal mode

  • two atoms in a molecule vibrate with respect to each other
  • atom make small displacement, minima of bond potentials can be approximated as parabolas
  • possible motion of a spring system

Definition:

  • pattern of motion which all parts of the system move in a sinusoidal fashion, with the same frequency
  • 2N normal mods of vibration as this is the total number of atoms

Acoustic/Optical Branches

Optical branch:

  • a higher energy vibration

Transverse optical mode for diatomic chain
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Transverse acoustical mode for diatomic chain
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At optical branch:
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Lattice Vibration IV

Real Crystal System

  • a 3D crystal, the atoms vibrate in three dimensions
  • obtain simultaneous equations
  • three different dispersion relations or three dispersion curves
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Transverse mode

  • degenerate on only special high-symmetry direction
  • two transverse modes and one longitudinal mode all have different frequencies
  • distinction between longitudinal and transverse waves no longer

Number of branches

  • three branches are acoustic and remaining ( 3z - 3 ) optical
  • allowed k-values in any single branch is N
  • 3z different vibrational

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Simple crystal system

  • Pb has FCC crystal structure
  • z=1 three branches are expected in any one k-direction
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  • Sodium has BCC crystal structure
  • z=1 was expect same number if modes as in Pb
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  • KBr, z=2 has the NaCl structure
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Brillouin zone

Brillouin Zones of Reciprocal Lattice
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Wigner-Seitz Cell construction

  • region bounded by all such planes
  • primitive unit cell for the lattice
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1st Brillouin Zone

  • region in reciprocal space containing all information about the lattice vibrations of the solid
  • k-value corresponding to unique vibration modes
  • k outside is mathematically equivalent to a value
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1st Brillouin for 3D lattices
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Phonons

Phonon:

  • Quantum of lattice vibration
    Photon:
  • Quantum of electromagnetic radiation

Phonons
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Photon:
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Total vibrational energy of crystal is sum of the energies of the individual modes
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Photon Momentum

  • phonon of wavevector k will interact with particles
  • phonon does not carry physical momentum
  • the center of mass of the crystal does not change
  • elastic scattering of crystal is governed by wavevector selection rule K'=K+G

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