On the Bose-Einstein condensation of free relativistic bosons with or without mass

Abstract

The Bose-Einstein condensation of free relativistic particles [ε=(M²c⁴+c²p²)^1/2−Mc²] is studied rigorously. For massless bosons (ε=cp), the condensation transition of third (second) order occurs in2 (3) dimensions (D). The molar heat capacity follows the T² (T³) law below the condensation temperature T_c [k^BT_c=(2πħ²c²n/1.645)^1/2 [(π²ħ³c³n/1.202)^1/3], reaches4.38 (10.8) R at T=T_c, and approaches the high-temperature-limit value2 (3) R with no jump (a jump equal to6.75R) in2 (3)D. For finite-mass (M) bosons, the phase transition occurs only in3D with the condensation temperature T_c always smaller than that of the corresponding nonrelativistic bosons [ε=(2M)⁻¹p²]. If the mass M is reduced to zero, the condensation temperature T_c grows monotonically and reaches eventually that of massless relativistic bosons. This mass-dependence of T_c is therefore distinct from the case of nonrelativistic bosons, where T_c grows to infinity as M →0. A brief discussion is given for a possible connection with the normal-to-super transition of the independently moving Cooper pairs (bosons).