Bose-Einstein

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The fiinction N (T) is sketched in fignre A2.2.7. At zero temperature all the Bose particles occupy the ground state. This phenomenon is called the Bose-Einstein condensation and is the temperature at which the transition to the condensation occurs. 

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... 

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

Recent research (1995-) has produced at very low temperatures (nanokelvins) a Bose-Einstein condensation of magnetically trapped alkali metal atoms. Measurements [41] of the fraction of molecules in the ground... 

Ensher J R, Jin D S, Mathews M R, Weman C E and Cornell E A 1996 Bose-Einstein condensation in a dilute gas measurement of energy and ground-state occupation Phys. Rev. Lett. 77 4984-7... 

Pure magnetic traps have also been used to study cold collisions and tliey are critical for tire study of dilute gas-phase Bose-Einstein condensates (BECs) in which collisions figure importantly. We anticipate, tlierefore, tliat magnetic traps will play an increasingly important role in future collision studies in and near BEC conditions. 

Mewes M-0, Andrews M R, van Druten N J, Kurn D M, Durfee D S and Ketterle W 1996 Bose-Einstein oondensation in a tightly oonfining DC magnetio trap Phys.Rev.Lett. 77 416-19... 

Anderson M H, Ensher J R, Matthews M R, Wieman C E and Cornellm E A 1995 Observation of Bose-Einstein oondensation in a dilute atomio vapor Science 269 198-201... 

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

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

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). 

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). 

Permutations of this type have to be considered in PIMC simulations if a full account of the quantum statistics is intended in the study and required by the physical effect under consideration, which means that additional permutation moves have to be done in the simulation. In this way quantum statistics has been included in a few PIMC simulations, in particular for the study of superfluidity in He [287] and in adsorbed H2 layers [92], for the Bose-Einstein condensation of hard spheres [269], and for the analysis of... 

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. 

All these multifarious activities took a lot of Einstein s energies but did not keep him from his physics research. In 1922 he published Ins first paper on unified field theoiy, an attempt at incorporating not only gravitation but also electromagnetism into a new world geometry, a subject that was his main concern until the end of his life. He tried many approaches none of them have worked out. In 1924 he published three papers on quantum statistical mechanics, which include his discoveiy of so-called Bose-Einstein condensation. This was his last contribution to physics that may be called seminal. He did continue to publish all through his later years, however. 

Brueckner, K. A., and Sawada, K., Phys. Rev. 106, 6, Bose-Einstein gas with repulsive interactions. ... 

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. 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

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. 

---