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Two-step BEC coming from a temperature dependent energy gap

Juan José Valencia Acevedo, M. Atala

Published Jan 16, 2024• 0 citations
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Abstract

Abstract We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap ∆ (T) at the lower edge of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is removed. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps that abruptly go to zero at temperature TB, as well as for the damped counterparts whose drop to zero we have smoothed. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature Tc is equal to that of the ideal Bose gas T0, for all TB ≤ T0; surprisingly, the condensate fraction presents two different filling rates of the ground state at Tc = T0 and at TB < T0 while the specific heat shows a finite jump at Tc as well as a divergence at TB. Three-dimensional infinite Bose gas results are recovered when the temperature independent gap is either a constant or equal to zero. PACS numbers: 03.75.Fi; 05.30.Jp; 67.40.Kh

DOI
10.21203/rs.3.rs-3852351/v1

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Abstract We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap ∆ (T) at the lower edge of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is removed. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps that abruptly go to zero at temperature TB, as well as for the damped counterparts whose drop to zero we have smoothed. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature Tc is equal to that of the ideal Bose gas T0, for all TB ≤ T0; surprisingly, the condensate fraction presents two different filling rates of the ground state at Tc = T0 and at TB < T0 while the specific heat shows a finite jump at Tc as well as a divergence at TB. Three-dimensional infinite Bose gas results are recovered when the temperature independent gap is either a constant or equal to zero. PACS numbers: 03.75.Fi; 05.30.Jp; 67.40.Kh
We calculate and discuss the effects on the thermodynamic properties of a 3D Bose gas caused by a gap ΔΔΔ in the energy of the particles constituting the gas that without the gap behaves like an ideal Bose gas. Explicit formulae with arbitrary ΔΔΔ 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 ΔΔΔ different from zero. Three dimensional infinite ideal Bose gas results are recovered when we the energy gap goes to zero.
Abstract 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 Δ between the ground state energy ε 0 and the first excited state, i.e. ε = ε 0 for k = 0 and ε = ε 0 + Δ + c s k s , for k > 0, where ℏ 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 Δ 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 Δ 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 Δ > 0, for any d > 0 exists a Bose–Einstein critical temperature T c ≠ 0 where 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 functions of d / s . At sufficiently high temperatures Δ-independent classical results are recovered while for temperatures below the critical one the gap effects are predominant. For Δ = 0 we recover previous reported results.
In a recent paper [Int. J. Mod. Phys. B 14, 405 (2000)] we discussed the Bose-Einstein condensation (BEC) in the framework of Tsallis’s nonextensive statistical mechanics. In particular, we studied an ideal gas of bosons in a confining harmonic potential. In this memoir we generalize our previous analysis by investigating an ideal Bose gas in a generic power-law external potential. We derive analytical formulas for the energy of the system, the BEC transition temperature and the condensed fraction. PACS numbers: 05.30-d; 03.75.Fi 1 Recently we analyzed the consequences of weak nonextensivity on Bose-Einstein Condensation for an ideal Bose gas 1 by using Tsallis’s nonextensive statistical mechanics (NSM). 2 From the nonextensive Bose-Einstein distribution 3−5 we derived the BEC transition temperature, the condensed fraction and the energy per particle in three different cases: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. In this short contribution we generalize our previous results by considering the ideal Bose gas in a generic power-law potential. All the calculations are performed by assuming a D-dimensional space. The NSM predicts that, for an ideal quantum gas of identical bosons in the grand canonical ensemble, the weak nonextensivity formula of the average number of particles with energy ǫ is given by 〈n(ǫ)〉q = 1
Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.
Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.