Crystal Lattice Vibrations: Phonons. Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html. Lattice dynamics above T=0.
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Introduction to Solid State Physicshttp://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
→ Thermal conductivity of insulators is due to dispersive lattice vibrations (e.g., thermal conductivity of diamond is 6 times larger than that of metallic copper).
→ They reduce intensities of diffraction spots and allow for inellastic scattering where the energy of the scatter (e.g., neutron) changes due to absorption or creation of a phonon in the target.
→Electron-phonon interactions renormalize the properties of electrons (electrons become heavier).
→Superconductivity (conventional BCS) arises from multiple electron-phonon scattering between time-reversed electrons.
Classical Theory:Normal Modes
Quantum Theory: Linear Harmonic Oscillator for each Normal Mode
The non-adiabatic term can be neglected at T<100K!
Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with
Center of the unit cell is not moving!
L — longitudinal
→Relevant symmetries:Translational invariance of the lattice and its reciprocal lattice, Point group symmetry of the lattice and its reciprocal lattice, Time-reversalinvariance.
→All harmonic lattices, in which the energy is invariant under a rigid translation of the entire lattice, must have at least one acoustic mode (sound waves)
←3 acoustic modes (in 3D crystal)
→The most general solution for displacement is a sum over the eigenvectors of the dynamical matrix:
→Second Quantization applied to system of Linear Harmonic Oscillators:
→Hamiltonian is a sum of 3rN independent LHO – each of which is a refered to as a phonon mode! The number of phonons in state is described by an operator:
→Creating and destroying phonons:
→Arbitrary number of phonons can be excited in each mode → phonons are bosons:
→Lattice displacement expressed via phonon excitations – zero point motion!
In contrast to quasiparticles, collective excitations are bosons, and they bear no resemblance to constituent particles of real system. They involve collective (i.e., coherent) motion of many physical particles.
Bragg or Laue conditions for elastic scattering!
Phonon absorption is allowed only at finite temperatures where a real phonon be excited: