Phonon dispersion in lattices

Definition

Let $\Lambda = \set{R_n}$ be a lattice of translations of $\R^d$, and let a crystal have $r$ atoms in each cell, with masses $m_1, \dots, m_r > 0$. Write $u_{n\kappa}(t) \in \R^d$ for the displacement of atom $\kappa$ of cell $n$ from its rest position. The harmonic approximation gives these equations of motion: $$m_\kappa \, \ddot u_{n\kappa\alpha} = -\sum_{n', \kappa', \beta} \Phi_{\kappa\alpha, \kappa'\beta}(R_{n'} - R_n) \, u_{n'\kappa'\beta}.$$ The force constants $\Phi$ are real, depend only on the difference of the cells, and satisfy $\Phi_{\kappa\alpha,\kappa'\beta}(R) = \Phi_{\kappa'\beta,\kappa\alpha}(-R)$. For a wave vector $k \in \R^d$ the dynamical matrix $D(k)$ is the $dr \times dr$ matrix with entries $$D_{\kappa\alpha, \kappa'\beta}(k) = \frac{1}{\sqrt{m_\kappa m_{\kappa'}}} \sum_{R \in \Lambda} \Phi_{\kappa\alpha, \kappa'\beta}(R) \, e^{i k \cdot R}.$$ The matrix $D(k)$ is Hermitian. The phonon dispersion relation is the family of maps $k \mapsto \omega_j(k)$, $j = 1, \dots, dr$, where $\omega_j(k)^2$ are the eigenvalues of $D(k)$. For each $j$, the motions $u_{n\kappa} = m_\kappa^{-1/2} e_\kappa \, e^{i(k \cdot R_n - \omega_j(k) t)}$, with $e$ an eigenvector, are the plane-wave solutions.