Ziman Principles Of The Theory Of Solids 13 May 2026

$$V_total(\mathbfr) = V_0(\mathbfr) + \delta V(\mathbfr, t)$$

$$\hbar\omega_ph > |E_\mathbfk - E_F|$$

If an ion at position $\mathbfR$ displaces by $\mathbfu(\mathbfR, t)$ due to a phonon, the potential $V(\mathbfr)$ experienced by an electron at position $\mathbfr$ changes. The total potential is: ziman principles of the theory of solids 13

The interaction Hamiltonian $H_e-ph$ does not just scatter electrons; it can create an effective attraction between two electrons. How? One electron emits a virtual phonon; a second electron absorbs it. This process is second-order in perturbation theory. One electron emits a virtual phonon; a second

The perturbation $\delta V$ is the electron-phonon interaction Hamiltonian, $H_e-ph$. For long-wavelength acoustic phonons (sound waves), the lattice is locally dilated or compressed. A change in volume changes the bottom of the conduction band (or top of the valence band). This is captured by the deformation potential constant , $E_1$: the interaction Hamiltonian becomes:

This is the glue of Cooper pairs. Chapter 13 thus provides the microscopic justification for why a lattice—a source of resistance—can paradoxically become the medium for zero-resistance superconductivity below a critical temperature $T_c$. Finally, Chapter 13 extends its reach to ionic semiconductors. In polar crystals (e.g., GaAs, NaCl), an electron moving through the lattice polarizes its surroundings, dragging a cloud of virtual optical phonons with it. The composite entity—electron plus phonon cloud—is called a polaron .

This simple scalar term is the workhorse for understanding scattering of electrons by acoustic phonons in simple metals and semiconductors. To make this quantitative, Chapter 13 introduces the second-quantized form of the interaction. Quantizing both the electron field and the phonon field, the interaction Hamiltonian becomes: