On the calculation of lattice sums arising in Bose-Einstein statistics of quasiparticle excitations

A new method for the calculation of the average occupation number of bosonic quasiparticle excitations valid for any type of lattice is proposed. The method is based on the recognition of the connection between lattice Green functions and generalized Watson integrals, on one hand, and on a very simple differentiation technique, which renders unnecessary and artificial to this problem more sophisticated Laplace-transform summation procedures. The mean-field approximation to Green function theories of ferromagnetism arises naturally as the zeroth term in the obtained summation formulae, The results have been specified completely for the three cubic lattices. They are new for the simple cubic and face-centred cases, whereas a certain redundancy is removed from the known body-centred cubic result. Applications of the method to more complex sums, such as, for instance, the thermodynamic sum for the total energy of the quasiparticles, are straightforward. A new three-position recursion relation for the calculation of frequently occurring triple geometric integrals in the face-centred cubic case has also been found. It originates from a corresponding relation for a relevant Heun function.