The electron-longitudinal acoustic phonon interaction is an interaction that can take place between an electron and a longitudinal acoustic (LA) phonon in a material such as a semiconductor.

Displacement operator of the LA phonon

The equations of motion of the atoms of mass M which locates in the periodic lattice is where u_{n} is the displacement of the nth atom from their equilibrium positions. Defining the displacement u_{\ell} of the \ellth atom by, where x_{\ell} is the coordinates of the \ellth atom and a is the lattice constant, the displacement is given by Then using Fourier transform: and Since u_{\ell} is a Hermite operator, From the definition of the creation and annihilation operator Then u_{\ell} expressed as Hence, using the continuum model, the displacement operator for the 3-dimensional case is where e_{q} is the unit vector along the displacement direction.

Interaction Hamiltonian

The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as H_\text{el} where D_\text{ac} is the deformation potential for electron scattering by acoustic phonons. Inserting the displacement vector to the Hamiltonian results to

Scattering probability

The scattering probability for electrons from |k \rangle to |k' \rangle states is Replace the integral over the whole space with a summation of unit cell integrations where, \Omega is the volume of a unit cell.