Fermionic character of Frenkel excitons in one-dimensional molecular crystals

5 Fermionic character of Frenkel excitons in one-dimensional molecular crystals 

In previous sections we have considered collective properties of Frenkel excitons in three-dimensional molecular crystals only. In one- and two-dimensional molecular crystals the kinematic interaction appearing in the transition from Pauli to Bose operators can be, generally speaking, quite large, so that the description of Frenkel excitons in terms of Bose excitations can be very rough. Below we consider this problem in more detail. 

First we must keep in mind that if we are not interested in collective properties of excitons and, in particular, we not consider exciton-exciton scattering, then in a good approximation we can take into account only those crystal states where the number of excitons is 0 or 1. It is evident that in these states, independent of the crystal dimension, the kinematic interaction identically vanishes so that the Frenkel exciton can be considered both as a boson as well as a fermion (i.e. it can be described by Bose or Fermi operators) and both descriptions will give identical results. 

Quite another situation occurs when we consider collective properties of excitons. In this case, as shown above, the kinematic interaction can play an important role. However, the mutual scattering of elementary excitations, which causes this interaction, is quite different in three-, two- and one-dimensional crystals. In three-dimensional crystals at not very large concentration of excitations this scattering at least does not change the statistics. 

Another situation can be observed in one-dimensional crystals. Chesnut and Suna (30) have shown that if in a one-dimensional paulion gas only interactions with nearest-neighbors are taken into account, the energy operator for this case can be represented as the energy operator of an ideal gas of fermions. 

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