Lattice model

Unlike static lattice model , which deals with average positions of atoms in a crystal, lattice dynamics extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion. This motion is not random but is a superposition of vibrations of atoms around their equilibrium sites due to the interaction with neighbor atoms. A collective vibration of atoms in the crystal forms a wave of allowed wavelength and amplitude.
Just as light is a wave motion that is considered as composed of particles called photons, we can think of the normal modes of vibration in a solid as being particle-like. Quantum of lattice vibration is called the phonon. The problem of lattice dynamics (LD) is to find the normal modes of vibration of a crystal. In other words, LD seeks to calculate the energies (or frequencies ) of the phonons as a function of their wave vector's k . The relationship between and k is called phonon dispersion .
LD offers two different ways of finding the dispersion relation:

Quantum-mechanical approach

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Phonon's dispersion relation can be obtained directly using the quantum-mechanical approach. The problem here is to find the solution of the Schrodinger equation for the lattice vibrations. The quantum-mechanical operators for the normal mode coordinates are substituted in the phonon Hamiltonian by the two operators, called creation and annihilation operators. The names of these operators arise because their effects are to create and annihilate phonons respectively.
The form of the Hamiltonian operator and the wave function are not shown here, however, some conclusions of the quantum-mechanical approach are listed below:

• Atoms in the crystal move even at the temperature of absolute zero.
  
• The energy of the quantum of lattice vibration (phonon) is given by .
  
• The energy of the normal mode of vibration changes by integral units of .
  
• The amplitude of atom vibrations is quantized and can be changed only by integral units of .

Semiclassical treatment

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Why this method was called semiclassical? Even though it employs classical mechanics it has to use one additional postulate taken from quantum mechanics namely, that the energy of lattice vibrations is quantized. The classical motions of any atom are determined by Newton's law of mechanics: force=mass x acceleration. Formally, if r(t) is the position of atom at time t, then

where m is the atomic mass, and is the instantaneous potential energy of the atom. This potential energy arises from the interaction of the atom with all the other atoms in the crystal.

For the reason of mathematical convenience the discussion of the semiclassical treatment is limited here by the harmonic approximation. In this case the representation of potential energy in the Taylor series is truncated at the quadratic term. In other words we assume that the force affecting the atom is linearly proportional to the displacement of the atom from its equilibrium position (the Hook's law). We first consider monatomic linear chain to describe acoustical phonons . The optical phonons will appear in the discussion of diatomic linear chain.

ROSSANA HERNANDEZ

C.I 19234048

ELECTRONICA DE ESTADOS SOLIDOS

http://www.chembio.uoguelph.ca/educmat/chm729/Phonons/intro.htm

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