Bose gas with generalized dispersion relation plus an energy gap

• Published in: arXiv.org 2018-12
• Main Authors: Martínez-Herrera, J G, García-Nila, J, Solís, M A
• Format: Article
• Language: English
• Subjects: Bosons; Critical temperature; De Broglie wavelengths; Energy; Energy gap; Equations of state; Internal energy; Momentum; Specific heat; Transition temperature
• ISSN: 2331-8422

Bose-Einstein condensation in a Bose gas is studied analytically, in any positive dimensionality (\(d>0\)) for identical bosons with any energy-momentum positive-exponent (\(s>0\)) plus an energy gap \(\Delta\) between the ground state energy \(\varepsilon_0\) and the first excited state, i.e., \(\varepsilon=\varepsilon_0\) for \(k=0\) and \(\varepsilon=\varepsilon_0 +\Delta+ c_sk^s\), for \(k>0\), where \(\hbar \mathbf{k}\) is the particle momentum and \(c_s\) a constant with dimensions of energy multiplied by a length to the power \(s > 0\). Explicit formula with arbitrary \(d/s\) and \(\Delta\) are obtained and discussed for the critical temperature and the condensed fraction, as well as for the equation of state from where we deduce a generalized \(\Delta\) independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at \(T_c\). When \(\Delta > 0\), a Bose-Einstein critical temperature \(T_c \neq 0\) exists for any \(d > 0\) at which the internal energy shows a peak and the specific heat shows a jump. Both the critical temperature and the specific heat jump increase as functions of the gap but they decrease as of \(d/s\). At sufficiently high temperatures \(\Delta\)- independent classical results are recovered. However, for temperatures below the critical one the gap effects are predominant. For \(\Delta = 0\) we recover previous reported results.