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Physical Review A

Effects of an attractive- δ -potential impurity center on Bose-Einstein condensation

S. Goulart Rosa, Oscar Hipólito, R. Lobo

Published Apr 1, 1975• 7 citations
Publisher

Abstract

The exact density of states for one particle in the field of an attractive $\ensuremath{\delta}$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The "Bose-Einstein phase transition" is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature ${T}_{c}$. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, ${E}_{b}$. The transition temperature and the discontinuity of the specific heat at ${T}_{c}$ are also obtained as a function of ${E}_{b}$.

DOI
10.1103/physreva.11.1454

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Cambridge University Press eBooks•Nov 29, 2001•0 citations
The exact density of states for one particle in the field of an attractive \ensuremathδ\ensuremath{\delta}\ensuremathδ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The "Bose-Einstein phase transition" is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature Tc{T}_{c}Tc​. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, Eb{E}_{b}Eb​. The transition temperature and the discontinuity of the specific heat at Tc{T}_{c}Tc​ are also obtained as a function of Eb{E}_{b}Eb​.
The thermodynamic behavior of the one-dimensional Bose-gas–attractive-δ impurity system is studied in this paper. The system is shown to undergo the Bose–Einstein condensation and the cause of the phase transition is attributed to the bound state introduced by the impurity in the free-particle energy spectrum. The condensed phase is composed of particles captured by the impurity, forming a drop of particles well localized in space. This gives to the Bose–Einstein condensation in this system the appearance of the ordinary vapor–liquid phase transition. The expression for the pressure is obtained and is plotted as a function of the temperature and the system length. The shapes of the isothermal curves reinforce the interpretation of a vapor–liquid transition. The evaluation of the heat capacity at constant length shows the existence of a finite discontinuity at the transition temperature.
We study the thermodynamics of a one-dimensional system of noninteracting bosons in the presence of random fixed finite-range repulsive potentials. It is shown that a Bose---Einstein condensation into the lowest state occurs. The critical temperature increases with the number of impurities. The specific heat exhibits a discontinuity and is infinitely differentiable from both sides of the critical point. For a suitably defined pressure the isothermals exhibit a flat portion below the "transition length."
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
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.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/ k , where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).