Lakhasly

Consider a simple solid containing $N$ atoms.Let $q_i$ be the (appropriately normalized) amplitude of the $i$th normal mode, and $p_i$ the momentum conjugate to this coordinate.In this case, we can expand the potential energy of interaction between the atoms to give an expression which is quadratic in the atomic displacements from their equilibrium positions.Now, atoms in solids cannot translate (unlike those in gases), but are free to vibrate about their equilibrium positions.Each atom is specified by three independent position coordinates, and three conjugate momentum coordinates.

Consider a simple solid containing $N$ atoms. Now, atoms in solids cannot translate (unlike those in gases), but are free to vibrate about their equilibrium positions. Such vibrations are called lattice vibrations, and can be thought of as sound waves propagating through the crystal lattice. Each atom is specified by three independent position coordinates, and three conjugate momentum coordinates. Let us only consider small amplitude vibrations. In this case, we can expand the potential energy of interaction between the atoms to give an expression which is quadratic in the atomic displacements from their equilibrium positions. It is always possible to perform a normal mode analysis of the oscillations. In effect, we can find $3,N$ independent modes of oscillation of the solid. Each mode has its own particular oscillation frequency, and its own particular pattern of atomic displacements. Any general oscillation can be written as a linear combination of these normal modes. Let $q_i$ be the (appropriately normalized) amplitude of the $i$th normal mode, and $p_i$ the momentum conjugate to this coordinate. In normal mode coordinates, the total energy of the lattice vibrations takes the particularly simple form