Diffraction, Fourier Analysis, and the Reciprocal Lattice
Crystal binding, elastic waves
Vibration waves in crystals: dispersion curves, quantization
What are thermal properties?
Heat capacity
Heat transport (Thermal conductivity)
Phonons are important because vibration energies are low (due
to ion mass compared to the electron mass) so that phonons
play an important role at ordinary temperatures
Electronic contribution later
Key points
For total energy, heat capacity one needs ONLY the energies
and number of vibrational states -- the directions of propagations
and atom motions do not matter as long as one knows the energy
for the vibrations and the number of independent vibrations
FUNDAMENTAL LAW: If two states of a system have total energies
E1 and E2, then the ratio of probablilities
of finding the system in state 1 compared to finding it in state 2 is
P1/P2 = exp( - (E1 - E2)/kB T)),
where kB = Boltzman constant. This is true whether the system is
classical or quantum, and whether one is dealing with particles that
obey Bose (like phonons) or Fermi statistics (like electrons)
Quantum mechanics makes it EASIER to calculate the final results
for the properties, with final equations for thermal energy, etc.,
that depend upon whether the particles
obey Bose or Fermi statistics
Planck Distribution
Originally devised by Planck for light (photons)
Start of quantum mechanics
Extended to all types of vibrations and applied to solids
in early days of quantum mechanics
Phonons obey Bose-Einstein distribution (bosons)
Original idea for identical particles in quantum mechanics
For independent bosons, Bose-Einstein statisics lead
dierectly to the Planck distribution n(omega) = 1/(exp(hbar omega/kT) - 1).
Density of states, Internal energy, Heat capacity
Density of states, D(omega) = # states for a given mode per unit energy
U = Integral over omega of D(omega) * n(omega)
C = dU/dT
Normal mode enumeration
Density of state in one dimension
Three dimensions
Debye Model
Debye temperature Theta as a characteristic parameter describing a solid
Density of states
Heat capacity
T3 law at low temperature in 3 dimensions
C ~ 234 N kB (T/Theta)3
High temperature limit - C = classical value = 3 N kB
General quantum expresion must be found numerically (Kittel Figure 7)