  Excitons are compound particles formed from an electron and a hole in
semiconductors. The impact of this substructure on the phonon-exciton
interaction is described by a closed system of microscopic scattering
equations. To calculate the actual excitonic thermalization properties beyond
the pure bosonic picture, this equation is derived directly from an
electron-hole picture within the Heisenberg equation of motion framework. In
addition to the well-known bosonic character of the compound particles, we
identified processes of a repulsive, fermionic type, as well as attractive
carrier exchange contributing to the scattering process. In this analytical
study we give general statements about the thermalization of excitons in two
and three dimensional semiconductors. We give insights on the strong dependence
of the thermalization characteristics of the exciton Bohr radius and the
thermalization wavelength. Above all, we analytically provide arguments why a
bosonic behavior of excitons - such as an enhanced ground state occupation -
requires the dominant phonon scattering to be quasielastic. Acoustic phonons
tend to fulfil this, as each scattering event only takes small amounts of
energy out of the distribution, while optical phonons tend to prevent
macroscopic occupations of the lowest exciton state, since the Pauli repulsion
between the individual carriers will then dominate the thermalization dynamics.
