The static fluctuation approximation (SFA) is used to study the condensate fraction and the specific heat capacity of finite two-dimensional spin-polarized atomic hydrogen. It is found that Bose-Einstein condensation occurs in this system. The transition temperature at different densities decreases as the number of particles of the system increases. At low density, a sharp peak in the specific heat capacity is observed at the transition temperature. On the other hand, as the density of the system increases, the transition temperature becomes no longer well-defined, and a hump is observed in the specific heat capacity around the transition temperature. A qualitative comparison of our results to published results for finite Bose systems shows good agreement.
The static fluctuation approximation (SFA) is used to study the condensate fraction and the specific heat capacity of finite two-dimensional spin-polarized atomic hydrogen. It is found that Bose-Einstein condensation occurs in this system. The transition temperature at different densities decreases as the number of particles of the system increases. At low density, a sharp peak in the specific heat capacity is observed at the transition temperature. On the other hand, as the density of the system increases, the transition temperature becomes no longer well-defined, and a hump is observed in the specific heat capacity around the transition temperature. A qualitative comparison of our results to published results for finite Bose systems shows good agreement.
In this paper we use the so-called static fluctuation approximation (SFA) to calculate the thermodynamic properties of spin-polarized atomic hydrogen. This approximation is based on the replacement of the square of the local-field operator with its mean value. A closed set of nonlinear integral equations is derived for neutral many-bosonic systems. This set is solved numerically by an iteration method for two triplet-state potentials: a Morse- and Silvera-type potentials. It is found that the mean internal energy per unit volume, the pressure, the entropy per unit volume, and the specific heat per unit volume increase with temperature and decrease with spin polarization in the low-temperature region (<8 mK ); whereas the condensate fraction increases with an applied magnetic field. It is also found that these quantities are nearly independent of spin polarization for high temperatures (>0.1 K ), and that they are independent of the number density up to 10-3Å-3 in the low-temperature region.
The effect of finite number of trapped atoms is discussed in Bose-Einstein condensation without interaction, according to Thomas-Fermi approximation. We obtain the state density by high-temperature expansion and calculate the critical temperature in one-dimensional(1D) and two-dimensional(2D) trap potentials. Then the temperature dependence of specific heat are analyzed. The results show that: in 2D case, the specific heat c is proportional to T2; in 1D case,c increases linearly with temperature.
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.
In this work, the condensation fraction and transition temperature of neutral many bosonic system are studied within the static fluctuation approximation (SFA). The effect of the potential parameters such as the strength and range on the condensate fraction was investigated. A model potential consisting of a repulsive step potential and an attractive potential well was used. As the potential strength or the core radius of the repulsive part increases, the condensation fraction is found to be decreased at the same temperature. Also, as the potential depth or the range of the attractive part increases, the condensation fraction is found to be increased. The transition temperature is decreased as the potential strength or the core radius of the repulsive part increases, and it increases as the potential depth or the range of the attractive part increases.
Spin-polarized atomic hydrogen continues to be one of the most promising candidates for Bose condensation of an atomic system. In contrast to liquid helium, hydrogen is gaseous and therefore its density can be changed to vary its behavior from the weakly to the strongly interacting Bose gas. Until now, efforts to Bose condense hydrogen have been thwarted by recombination and relaxation phenomena. After a long introduction to the subject, two promising approaches to observe quantum degenerate behavior are discussed: the microwave trap and a two-dimensional gas of hydrogen. Introduction Since the stabilization of atomic hydrogen as a spin-polarized gas (H↓), reported in 1980 [1], there has been a continuous effort to observe Bose–Einstein condensation (BEC) or other effects of quantum degeneracy in this Bose gas. This challenge has not yet been realized due to the instability of H↓ towards recombination to H 2 or relaxation among hyperfine states as the conditions for BEC are approached. Although this difficulty is formidable, hydrogen presents a unique opportunity to study BEC and the related superfluidity, as it remains a gas to T = 0 [2]. By comparison, 4 He, the only experimentally observed boson fluid, is a strongly interacting superfluid Bose liquid, with little flexibility for varying its density. As a result of the gaseous nature of hydrogen, its density, n , can be varied over several orders of magnitude with the possibility of studying BEC and its relationship to superfluidity, ranging from the weakly to the strongly interacting boson gas.
This paper presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation---namely, the fraction of particles in the ground state and the specific heat---are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a three-dimensional box, the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit---here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.