Bose-Einstein system

Bose-Einstein statistics Bose-Einstein systems Boson Bosons... 

Systems containing symmetric wave function components ate called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Ditac systems (130,131). Systems in which all components are at a single quantum state are called MaxweU-Boltzmaim systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Eermi-Ditac statistics (132). 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

Similar principles apply to ortho- and para-deuterium except that, as the nuclear spin quantum number of the deuteron is 1 rather than as for the proton, the system is described by Bose-Einstein statistics rather than the more familiar Eermi-Dirac statistics. Eor this reason, the stable low-temperature form is orriio-deuterium and at high temperatures the statistical weights are 6 ortho 3 para leading to an upper equilibrium concentration of 33.3% para-deuterium above about 190K as shown in Eig. 3.1. Tritium (spin 5) resembles H2 rather than D2. 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

Although we have explained Bose-Einstein condensation as a characteristic of an ideal or nearly ideal gas, i.e., a system of non-interacting or weakly interacting particles, systems of strongly interacting bosons also undergo similar transitions. Eiquid helium-4, as an example, has a phase transition at 2.18 K and below that temperature exhibits very unusual behavior. The properties of helium-4 at and near this phase transition correlate with those of an ideal Bose-Einstein gas at and near its condensation temperature. Although the actual behavior of helium-4 is due to a combination of the effects of quantum statistics and interparticle forces, its qualitative behavior is related to Bose-Einstein condensation. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

Now — L is the Landau-Ginzburg free energy, where m2 = a(T — Tc) near the critical temperature, is a macroscopic many-particle wave function, introduced by Bardeen-Cooper-Schrieffer, according to which an attractive force between electrons is mediated by bosonic electron pairs. At low temperature these fall into the same quantum state (Bose-Einstein condensation), and because of this, a many-particle wave function (f> may be used to describe the macroscopic system. At T > Tc, m2 > 0 and the minimum free energy is at = 0. However, when T [Pg.173]

One of the most amazing phenomena in quantum many-particle systems is the formation of quantum condensates. Of particular interest are strongly coupled fermion systems where bound states arise. In the low-density limit, where even-number fermionic bound states can be considered as bosons, Bose-Einstein condensation is expected to occur at low temperatures. The solution of Eq. (6) with = 2/j, gives the onset of pairing, the solution of Eq. (7) with EinP = 4/i the onset of quartetting in (symmetric) nuclear matter. At present, condensates are investigated in systems where the cross-over from Bardeen-Cooper-Schrieffer (BCS) pairing to Bose-Einstein condensation (BEC) can be observed, see [11,12], In these papers, a two-particle state is treated in an uncorrelated medium. Some attempts have been made to include the interaction between correlated states, see [7,13]. 

This difference between fermions and bosons is reflected in how they occupy a set of states, especially as a function of temperature. Consider the system shown in Figure E.10. At zero temperature (T = 0), the bosons will try to occupy the lowest energy state (a Bose-Einstein condensate) while for the fermions the occupancy will be one per quantum state. At high temperatures the distributions are similar and approach the Maxwell Boltzman distribution. 

Although positron plasmas can be considered to be systems containing many positrons, and as such technically fall within the scope of this section, we will not consider them here. Rather, we will concentrate on the theory of, and the possibilities of observing, assemblages of particles containing both positrons and electrons. These include the positronium molecule and a Bose-Einstein (BE) condensate of positronium atoms. 

The possibility of producing a system of positronium atoms at a sufficiently high density and low temperature to produce Bose-Einstein (BE) condensation has been raised by Liang and Dermer (1988) and Platzman and Mills (1994). The former authors outlined a scheme in which positronium atoms are laser-cooled in vacuum, which seems feasible despite their short lifetimes because of their low mass. The required temperature of the positronium is around 0.1 K and the density is 1015 cm-3. Liang and Dermer (1988) argued that the overall scheme appears possible, but hitherto neither the temperature nor the density condition has been approached and laser cooling of positronium has not yet been attempted. 

COVALENT BONDING involves a pair of electrons with opposite electron spin. The bond (or electron charge distribution) is essentially localized between nearest neighbor atoms that contribute electrons for the bonding. Since these electron pairs follow Bose-Einstein statistics, therefore they are known as boson. In this case the paired particles do not obey the Pauli Exclusion Principle and many electron pairs in the system may occupy the same energy level. 

One of the simplest procedures to get the expression for the Fermi-Dirac (F-D) and the Bose-Einstein (B-E) distributions, is to apply the grand canonical ensemble methodology for a system of noninteracting indistinguishable particles, that is, fermions for the Fermi-Dirac distribution and bosons for the Bose-Einstein distribution. For these systems, the grand canonical partition function can be expressed as follows [12] ... 

To thoughtful persons both in and out of the biological sciences, the existence and evolution of living organisms and their development have posed and continue to pose enormously challenging intellectual problems. Fritz London, who had contributed much to the phenomenological theory of superfluidity and superconductivity, was also fascinated by these problems. From his earlier work on superfluid helium and superconductivity he knew of the unifying effect that the existence of a Bose-Einstein condensate had upon a superfluid system. Today we would describe this as the effect on the system of the existence of an order parameter—in par-... 

Systems containing more than one identical particles are invariant under the interchange of these particles. The permutations form a symmetry group. If these particles have several degrees of freedom, the group theoretical analysis is essential to extract symmetry properties of the permissible physical states. Examples include Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann statistics, Pauli exclusion principle, etc. 

The ortho-para ratio is determined by the statistical calcnlation of the availability of states given by the partition fhnction. Calculated ortho-para equilibria for H2, D2, and T2 are shown in Figure 6 at the temperature range from 0 to 300 K. When the two nuclear spins are parallel, the resnltant nnclear spin quantum number is 1 (i.e. 1 /2 -b 1 /2) and the state is threefold degenerate. When the two spins are opposed, however, the resultant nuclear spin is zero and the state is nondegenerate. Therefore para-H2 has the lower energy and this state is favored at lower temperatures. The equilibrium concentration of H2 approaches three parts ortho to one part para at room temperature. As the nuclear spin quantum number of the deuteron is 1 rather than 1/2 for the proton (see Table 1), the D2 system is described by Bose-Einstein... 

We recall that Bose-Einstein condensation is the macroscopic occupation of the ground state of a system at finite temperature. For a weakly interacting gas, this phase transition occurs when the inter-particle spacing becomes comparable to the thermal de Broglie wavelength A = /2nh /mkBT, where ks is the Boltzmann constant and T is the temperature. A rigorous treatment for the ideal Bose gas yields n > 2.61221 , where n is the density [35]. At a temperature of 50 yuK, for instance, the critical density for hydrogen is 1.8 x 10 cm. ... 

A number of studies on size-sjjecific effects in phase transitions were reported and summarized by Pathria and coworkers. Here we mention only two of them. The first deals with the problem of Bose-Einstein condensation in restricted geometries for an ideal Bose gas. While the general conclusion is that in a finite system the transition is spread over a fractional range of temjjerature AT/7 (oo) L<, where 7 (oo) is the critical temperature of the infinite system and L< is the shortest side of the cuboidal container Li X L2 X L3) enclosing the gas, the actual fraction of the condensate de-jjends on the shajje of the container and on the boundary conditions. Thus in partially infinite geometries (such as L - ao or Li °°) Bose-Einstein... 

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