6  Lattice Dynamics I

NoteLearning Objectives

• Derive the Born–Oppenheimer approximation in the form relevant for lattice dynamics and explain why it yields an effective potential for the nuclei.
• Set up the displacement field for a crystal with a basis and derive the harmonic expansion of the potential energy.
• Define force constants, state their key symmetry and sum-rule properties, and write the coupled equations of motion.
• Derive the plane-wave reduction of the lattice-dynamics problem and formulate the dynamical-matrix eigenvalue problem.
• Count the vibrational branches of a crystal with \(p\) atoms in the primitive cell and distinguish acoustic from optical modes.