Fermi condensation

Finally, also in the laboratory the study of the interaction of cavities inside a uniform fermionic background is of importance (Bulgac and Wirzba., 2001). Examples are C6o buckyballs immersed in liquid mercury. The liquid metal itself serves only as free-moving shapeable neutral background which provides the Fermi gas environment by its conductance electrons, in which the buckyballs drill the voids. Another example would be buckyballs in liquid 3He as Fermi gas. Finally, in the future, boson condensate cavities immersed in dilute atomic Fermi condensates could serve as further system with which the effective interactions of cavities inside a Fermi gas can be studied in the lab. 

Quantum mechanics makes it clear that no atom has a fixed size. Electron orbitals extend from the nucleus to a greater or lesser extent, depending upon the chemical and physical environment in the locality of the atomic nucleus. Indeed, recent research on Bose-Einstein and Fermi condensation reveals that a collection of millions of atoms can enter an identical quantum state at temperatures just above 0 K and behave as a single atom, with a single wavefunction that spreads over the whole collection. 

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. 

Fermi (1940) pointed out that as /)—-1 the stopping power would power would approach °° were it not for the fact that polarization screening of one medium electron by another reduced the interaction slightly. This effect is important for the condensed phase and is therefore called the density correction it is denoted by adding -Z<5/2 to the stopping number. Fano s (1963) expression for 8 reduces at high velocities to... 

This topic is relevant to the physics of neutron stars (nuclei or quark bubbles embedded in a neutron gas), to dilute Bose-Einstein-condensate bubbles inside the background of a Fermi-Dirac condensate, to buckyballs in liquid mercury and to superconducting droplets in a Fermi liquid. 

It is quite likely to find dense quark matter inside compact stars like neutron stars. However, when we study the quark matter in compact stars, we need to take into account not only the charge and color neutrality of compact stars and but also the mass of the strange quark, which is not negligible at the intermediate density. By the neutrality condition and the strange quark mass, the quarks with different quantum numbers in general have different chemical potentials and different Fermi momenta. When the difference in the chemical potential becomes too large the Cooper-pairs breaks or other exotic phases like kaon condensation or crystalline phase is more preferred to the BCS phase. 

Our analysis is, however, not complete in any respect. For instance, as the large difference in Fermi momenta renders the BCS-type condensation in the classical 2SC phase difficult, it seems to be worthwhile to consider other possibilities. Among others we thereby think of a crystalline phase, deformed Fermi surfaces [32, 35], spin-1 pairing [20] or the gapless 2SC [33, 34] or CFL phase [60], We conclude that whether quark matter exists in hybrid or... 

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.

The particles we will deal with in this textbook are mainly electrons and ions in condensed solid and liquid phases. In condensed phases ions are the classical Boltzmann particles and electrons are the degenerated Fermi particles. 

As mentioned in Sec. 1.3, the electrochemical potential of electrons in condensed phases corresponds to the Fermi level of electrons in the phases. There are two possible cases of electron ensembles in condensed phases one to which the band model is applicable (in the state of degenera< where the wave functions of electrons overlap), and the other to which the band model cannot apply (in the state of nondegeneracy where no overlap of electron wave functions occurs). In the former case electrons or holes are allowed to move in the bands, while in the latter case electrons are assumed to be individual particles rather than waves and move in accord with a thermal hopping mechanism between the a4jacent sites of localized electron levels. 

Finally, it may be useful to note that the Fermi golden rule and time correlation function expressions often used (see ref. 12, for example) to express the rates of electron transfer have been shown [13], for other classes of dynamical processes, to be equivalent to LZ estimates of these same rates. So, it should not be surprising that our approach, in which we focus on events with no reorganization energy requirement and we use LZ theory to evaluate the intrinsic rates, is closely related to the more common approach used to treat electron transfer in condensed media where the reorganization energy plays a central role in determining the rates but the z factor plays a second central role. 

O.K. Andersen, O. Jepsen and D. Gloetzel, Highlights of Condensed Matter Theory - Varenna notes - Proceedings of The International School of Physics Enrico Fermi, North Holland, New York(1985). 

D. Stefanatos and J. S. Li. Frictionless decompression in minimum time of Bose-Einstein condensates in the Thomas-Fermi regime. Phys. Rev. A, 86(6) 063602—063608(2012). 

Throughout this book, particularly in the later chapters, we assume that a condensed electron gas can be treated as a Fermi liquid of pseudoparticles, for instance dielectric or spin polarons. We recognize that this is an unproved assumption. 

If states at the Fermi energy of a condensed electron gas are localized, two conduction mechanisms are possible. 

Interaction of electrons with phonons, and the fact that the presence of a trapped electron can deform the surrounding material, allows the radius of an empty localized state to change when the state is occupied. Also, in a condensed electron gas phonons lead to a mass enhancement near the Fermi energy, or in some circumstances to polaron formation. For the development of the theory, and comparison with experiment, it is therefore desirable to begin by choosing a system where these effects are unimportant. The study of doped semiconductors provides such a system. This is because the radius aH of a donor is given, apart from central cell corrections, by the hydrogen-like formula... 

The most important assumptions for the applicability of thermodynamical statistics is the independence of the particles from one another and the absence of interchange effects between them. Boltzmann — as well as Bose- and Fermi-statistics consider individual particles without interaction. In the gaseous state, photons, electrons as well as molecules coexist. In applying these theories to condensed phases, the individual particle is to be considered, according to Schrodingerls, either in a continuous medium otherwise the interaction must be taken into account. 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

Following from formula (4.54), the transfer of energy on excitation of molecules has a noticeable probability even in the case where the impact parameter is much greater than their size d. Since the intermolecular spacings in a condensed medium are of order of d, a charged particle interacts with many of its molecules. The polarization of these molecules weakens the field of the particle, which, in its turn, weakens the interaction of the particle with the molecules located far from the track. This results in that the actual ionization losses are smaller than the value we would get by simply summing the losses in collisions with individual molecules given by formula (5.1). This polarization (density) effect was first pointed out by Swann,205 while the principles of calculation of ionization losses in a dense medium were developed by Fermi.206... 

Work Function (WF) plays a key role in the physics and chemistry of materials. Phenomena such as the semiconductor field effect, photo- and thermionic electron emission (Allen and Gobelli, 1962), catalysis (Vayenas et al 1996), and the like are dominated by the WF. This fundamental property of electronic materials is defined as the minimum work required to extract an electron from the Fermi level Ep of a conducting phase, through the surface and place it in vacuum just outside the reach of the electrostatic forces of that phase (Trasatti and Parsons, 1986). The reference level for this transfer is thus called the vacuum reference level. Because even a clean surface is a physical discontinuity, a surface dipole t] with its associated electric field always appears at the surface of the condensed phase. Thus, the work of extracting the electron can be conceptually divided between the work required to... 

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