text: Competing with the generation of electrons is the downconversion of optical phonons into multiple acoustic phonons. The coupling arises from the crystal Hamiltonian (H) expanded in terms of the ionic displacement () from the equilibrium position () of atom i in direction   in direction, i.e.,
  where  is the ground-state Hamiltonian, the linear term vanishes (as the ground state is found by minimizing the energy in terms of the ionic position), and   and   are the second- and third-order force constants between atoms i, j, and k when moved in along coordinate , , and . The second order term is primarily responsible for the phonon dispersion, while the anharmonic (third order and higher) terms are responsible for thermal expansion as well as the phonon up- (multiple low-energy optical phonons combine to form a high-energy phonon) and downconversion (a high-energy phonon splits into multiple low-energy phonons).
  Typically, up- and down-conversion is dominated by the third-order interaction. Thus, the perturbation Hamiltonian used in Fermi's golden rule for phonon up- and downconversion is 
  where  is the direction of displacement for atom i due to the phonon. The resulting downconversion rate, from Fermi's golden rule, is
  where two phonons are produced with polarization  and  and momentum  and .