The electron-LA 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 is the displacement of the nth atom from their equilibrium positions. Defining the displacement of the th atom by , where is the coordinates of the th atom and is the lattice constant, the displacement is given by Then using Fourier transform: and . Since is a Hermite operator, From the definition of the creation and annihilation operator is written as Then expressed as Hence, using the continuum model, the displacement operator for the 3-dimensional case is , where is the unit vector along the displacement direction. Interaction Hamiltonian The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as , where 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 to states is Replace the integral over the whole space with a summation of unit cell integrations where , is the volume of a unit cell. See also Phonon scattering Umklapp scattering Notes References Atomic physics