In this section, the coupling of the electrons with acoustic phonons is analyzed. Displacement of the
atoms from their lattice sites are induced by crystal vibrations, which induces a modification of the
bandstructure. For electrons in the conduction band, the variation of the conduction band edge E_{c} can
be induced by acoustic phonons and the corresponding interaction Hamiltonian Ĥ_{e-ph}^{AC} is given
by

| (5.19) |

For small displacements δE_{c} can be written as

| (5.20) |

where Ξ_{ac } denotes the acoustic deformation potential and δV is the variation of the crystal volume V .
The local variation of the volume results from the lattice displacement U = x^{′}- x. The volume of a
cube generated by the orthogonal vectors a = (δx,0,0), b = (0,δy,0) and c = (0,0,δz), is given
by

| (5.21) |

The cube is distorted according to the transformations

and the new volume can be written as

| (5.22) |

Since

| (5.23) |

where the lattice displacement is given by [85]

| (5.24) |

the interaction Hamiltonian reads

| (5.26) |

The electron scattering rate with the assistance of acoustic phonons can be written in the following form [87] (see Appendix B.2)

| (5.27) |

where E_{ac } is the acoustic deformation potential, ρ is the density of the material, and v_{s} stands for the
sound velocity. This equation is only valid for ℏω_{q} ≪ k_{B}T, i.e. when the thermal energy is much larger
than the energy of the phonon involved in the transition, and in the elastic approximation limit
ℏω_{q } → 0 (see Appendix B.2).