We now turn to the other type of boson: a particle, such as an atom or molecule, with integer or zero spin, whose number is conserved. As in the case of the ideal gas, we integrate the distribution function over all the states to obtain a relation between particle density n and the chemical potential . The density is, from Equation (8.4), n = g(E)f(E) dE where, from Equation (8.10), f(E) = 1 e#(E ) 1 .
We now turn to the other type of boson: a particle, such as an atom or molecule, with integer or zero spin, whose number is conserved. As in the case of the ideal gas, we integrate the distribution function over all the states to obtain a relation between particle density n and the chemical potential . The density is, from Equation (8.4), n = g(E)f(E) dE where, from Equation (8.10), f(E) = 1 e#(E ) 1 .
This chapter focuses on integrating the distribution function over all the states of conserved Bosons to obtain a relation between particle density and the chemical potential. It describes the phenomenon of Bose-Einstein condensation in a perfect boson gas. The chapter discusses the experimental observation of Bose-Einstein condensation in perfect gases. There are two routes to obtaining Bose-Einstein condensation in a perfect gas. One is to reduce the mass of the particle, thus reducing critical density to a low value at a reasonable temperature. The other route is to lower the temperature to the point where the critical density is so low that the particles are too far apart to interact with each other.
The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.
In this paper, first of all, we proved if the ideal Bose gas with a finite volume and number of particles has a non-degenerate single-particle energy level n, the chemical potential μ can take the value μn = n and there is a phase transition temperature Tp,n, where n = 0,1,2 Taking 0 ≤ n Tp,n+1. When the temperature T > Tp,n or T ≤ Tp,n+1, μ # n and the most probable occupation number Nn = 0. In the temperature interval Tp,n ≥ T > Tp,n+1, μ = n and 0 ≤ Nn = N - NjsupNn, where Nj is the most probable occupation number in the degenerate level j. Thus, if the finite ideal Bose gas has some non-degenerate single-particle levels, there exists a characteristic temperature Tp = Tp,0. The chemical potential μ is quantized when T ≤ Tp, and this leads to the creation of a macroscopic quantum state (pure state) or Bose-Einstein condensation phase. Tp = Tp,0 is a first-order phase transition point, Tp,n#0 is a zero-order phase transition point. Next, we obtained a new expression of the most probable distribution of the finite ideal Bose gas. In this expression Nj is directly proportional to gj - 1, where gj and Nj are, respectively, the degeneracy and the most probable occupation number in the degenerate level j. This property agrees with what chemical potential can be quantized if there is a non-degenerate level for the finite ideal Bose gas. Finally, using this expression, we defined a micro-partition function M, obtained the statistical expressions of some thermodynamical quantities.
Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
In 1924 the Indian physicist Satyendra Nath Bose sent Einstein a paper in which he derived the Planck law for black-body radiation by treating the photons as a gas of identical particles. Einstein generalized Bose's theory to an ideal gas of identical atoms or molecules for which the number of particles is conserved and, in the same year, predicted that at sufficiently low temperatures the particles would become locked together in the lowest quantum state of the system. We now know that this phenomenon, called Bose-Einstein condensation (BEC), only happens for “bosons” – particles with a total spin that is an integer multiple of h, the Planck constant divided by 2π.