Lattice Vibrations

Summary

In this chapter, we discussed the vibrations of the atoms in solids. Quantummechanical treatment of lattice dynamics and dispersion curves of the normalmodes are described.

The Hamiltonian of a linear chain is written, in the harmonic approximation,as $H = \sum\nolimits_i {\frac{{P_i^2 }}{{2M}} + } \frac{1}{2}\sum\nolimits_{i,j} {c_{ij} u_i u_j ,}$ where Pi is the momentum and $u_i = R_i - R_i^0$ is the deviation of the ith atom from its equilibrium position. A general dispersion relation of the normal modes is $M\omega _q^2 = \sum\nolimits_{l - 1}^N {c(l)e^{iqla} }$. The normal coordinates are given by

$$qk = N^{ - 1/2} \sum\limits_n {u_n e^{1kna} ;pk = N^{ - 1/2} } \sum\limits_n {P_n e^{ - ikna}.}$$

The inverse of qk and pk are $u_n = N^{ - 1/2} \,\sum\nolimits_k {qke^{ikna}}$; $P_n = N^{ - 1/2} \,\sum\nolimits_k {pke^{ - ikna} }$