Vacancies effect on the BEC critical temperature of an ideal Bose gas within an imperfect crystal

After we have shown [1] that an ideal Bose gas in one, two or three-dimensional crystal with one structural vacancy presents Bose-Einstein condensation (BEC) at higher finite critical temperature than that when the crystal is perfect; here we report the BEC critical temperature as a function of the number of random vancancies. The particle energy spectrum is obtained using finite difference numerical method, which we use to calculate the critical temperature as the temperature at which the isobaric specific heat has its maximum [2]. When our finite system is taken to the thermodynamic limit, the critical temperature starts at the one vacancy value, it increases reaching a maximum when we have about 17 % of vacancies and then decays almost linearly to zero when the crystal has been removed.
[1] J. García, et al., Raising the Bose-Einstein critical temperature with vacancies, in progress; V.E. Barragán, et al., Int. J. of Mod. Phys. B Vol. 31, 1750100 (2017). [2] H.R. Pajkowski and R.K. Pathria, J. Phys. A: Math. Gen., Vol. 10, No. 4, 1977.