9  Phonons II

NoteLearning Objectives

• Use the phonon density of states from Lecture 8 to write the lattice internal energy and heat capacity.
• Derive the Einstein heat capacity and explain why it reproduces the Dulong–Petit limit but fails at low temperature.
• Derive the Debye heat capacity and obtain its low-temperature \(T^{3}\) law and high-temperature limit.
• Interpret the Debye temperature as a physically meaningful energy scale rather than a sharp transition temperature.
• Read low-temperature heat-capacity data and extract \(\beta_{\mathrm{LT}}\), \(\Theta_D\), and the metallic \(\gamma T\) term.