Motion of the atoms: Lattice vibrations. • Properties of solids can be divided (
roughly) into phenomena that are related to motion of the atoms (around their ...
Motion of the atoms: Lattice vibrations • Properties of solids can be divided (roughly) into phenomena that are related to motion of the atoms (around their equilibrium position) and those determined by electrons. This division is justifiable since the motion of the nuclei is much Slower (larger mass) than electron motions (e- remain in ground state). • From Heisenberg’s uncertainty principle it is clear that even at absolute zero, the atoms must vibrate around their equilibrium position (“zero point energy”). • For many phenomena electron motion leads to transport phenomena (bands, scattering, thermoelectric effects). These are not considered here. Lattice vibrations can explain: • Sound velocity Thermal properties such as • Heat capacity • Specific heat • Thermal expansion • Thermal conductivity (for insulators and semicoductors) Elastic properties such as • Temperature dependence of elastic constants • Hardness Optical properties such as • Infrared absorption
Linear chain of coupled harmonic oscillators Longitudinal waves
r k un-1
un
un+1
un+2
un+4
a Transversal waves un-1
un
un+1
un+2
un+4
r k
a Key assumptions: Periodic boundary conditions Only nearest neighbor interactions Small displacements (harmonic oscillations: F~x, V~x2)
Dispersion ω(k) for linear atomic chain
r r r 2π m • k’ and k are physically equivalent: k ' = k + a
(m integer)
• Only wavelengths longer than 2a are needed.
• All physical information (all frequencies, group velocities) are π r π contained in the interval of −
