Bose-Einstein condensate energy

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. 

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... 

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. 

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. 

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]

Fig. 1-1. Probability density functions of partiele energy distribution (a) Fermi function, (b) Bose-Einstein function, e = particle energy f(i) - probability density function cp = Fermi level sb - Bose-Einstein condensation level.

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. 

Comparable low temperatures have also been obtained by trapping gaseous particles in magnetic fields and lowering their velocity by absorption and reemission of laser energy. Using these methods, a new state of matter, the Bose-Einstein condensate has been created. 

The Doppler-sensitive line gives a second clear signature for Bose-Einstein condensation. Because the lowest energy state is the lowest momentum state, the condensate appears as a relatively narrow peak at the center of the Gaussian spectrum. Its width is given by the cold collision frequency shift and is the same as in the case of Doppler free spectrum. 

PROBLEM 5.1.3. If fermions, such as the neutrons (7=1/2) that make a hot neutron star, merge to form a massive Bose particle (cold black hole), then Bose-Einstein condensation occurs, the whole star loses energy massively, and a new minimum energy state is reached (cold black hole) (see also Problem 2.12.4). 

The details were significant. In July 1924 Einstein read a paper before the Prussian Academy in which he applied the Bose statistical method to an ideal gas and drew an analogy between a quantum gas and a molecular gas. Over the following few months, Einstein wrote what Martin Klein has called another of his masterful works, which was published in January 1925. In this paper, Einstein predicted that the particles of an ideal quantum gas could collect together in the lowest energy state and form what is now called a Bose-Einstein condensate. At the time, physicists regarded Einstein s prediction as a curiosity with litde or no physical significance. 

So far, we have considered the Bose-Einstein condensation of Coulomb excitons and thus we neglected the influence of the retardation. Such a consideration is correct only for excitons with small (or zero) oscillator strengths. Thus the above-described consideration on the Bose-Einstein condensation can be applied, for example, to the case of triplet excitons, or to singlet excitons for which the exciton-photon interaction energy is smaller than the exciton level width, caused by, for example, scattering by phonons. 

However, if a singlet exciton has a large oscillator strength, Bose-Einstein condensation is nevertheless possible for excitons but only if the minimum energy in the lowest excitonic band does not correspond to the center of the Brillouin zone. In the opposite case we have to take into account that the retardation drastically modifies the spectrum of excitons, transforming excitons into polari-tons. In such 3D crystals the condensation of excitons is impossible because the spectrum of polaritons in the region of small k (see Ch. 4) coincides with the spectrum of photons with energy E(k) —> 0 at k —> 0. However, for polaritons in a microcavity Bose-Einstein condensation is possible because in a microcavity there exists a so-called cut-of frequency (see Ch. 10)... 

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