Tamm's surface states and Bose-Einstein condensation

We calculate and discuss the effects on the thermodynamic properties of a 3D Bose gas caused by a gap $Δ$ in the energy of the particles constituting the gas that without the gap behaves like an ideal Bose gas. Explicit formulae with arbitrary $Δ$ values are discussed for: the critical temperature which increases as the gap grows; the condensate fraction; the internal energy; and the constant-volume specific heat found to possess a jump-discontinuity for any $Δ$ different from zero. Three dimensional infinite ideal Bose gas results are recovered when we the energy gap goes to zero.

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