Mass of a three-body complex on a lattice

According to the ‘‘Mattis-Gallinar effect’’ the mass of an exciton in a lattice depends upon the kinetic energy of the exciton, and, thus, may be different from the sum of the masses of the electron and hole. Here, I show how a formalism used to derive this effect can be applied to obtain a plausible approximate formula for the mass of a trion on a lattice (a three-body complex involving a Frenkel exciton and an extra hole). This approximate formula becomes exact when the mass of the Frenkel exciton is equal to the mass of the hole and/or when the mass of the electron is infinite. A distinction is made between the cases with singlet or triplet pairing of the spins of the two holes in the trion. The approximation used is seen to be appropriate in the singlet case for the bonding bound states of the trion, and for the antibonding states in the triplet case. Finally, a comparison is given between the approximate formula for the trion mass and the analogous, but exact, one for an exciton.