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Bose condensation and the Casimir effects of an imperfect Bose gas in a d-dimensional configuration space

Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Most importently, this study enabled us to predict the shift of the density of excited bosons due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be $\frac{1}{d-2}$ for the imperfect Bose gas. This leads to a value of critical exponent $1$ for $d=3$.