Superfluidity of a perfect quantum crystal II

Abstract

This paper continues the work begun in a previous paper [Eur. Phys. J. B 71, 85 (2009)]. To treat the equations that describe a crystal with condensate that can be superfluid, a method termed the Kirkwood approximation is used. Earlier, the method was found to be rather seminal when applied to a classical crystal. In the case of a simple cubic lattice, solutions to the equations under study can be expressed in terms of the well-known Mathieu functions. A more realistic case of the face centered cubic lattice is also considered although in this case the three-dimensional equations cannot be reduced to one-dimensional ones. Condensate crystals without superfluidity are studied first and then the same crystals in a superfluid state. It is shown in particular that a crystal in which the condensate is formed is energetically preferable with respect to the same quantum crystal without condensate at absolute zero of temperature. Therefore, on lowering the temperature there must somewhere occur Bose-Einstein condensation in the crystal. In the concluding section, we discuss various physical aspects of the problem.