Quantum degenerate gases

This section and the next two are very much intertwined, particularly as techniques based on magnetically tunable Feshbach resonances and on a wide variety of optical lattices are increasingly used in studies of quantum degenerate gases. Our goal here... 

Cold Collisions, Quantum Degenerate Gases, Photoassociation, and Cold Molecules... 

The possibility of pairing fermionic atoms (Cooper pairing) to form bosonic atoms has far-reaching consequences, as far as studies into such phenomena as superfluidity and superconductivity under the controlled conditions of quantum degenerate gases are concerned (see Section 8.5). 

Next, the promise of atom-molecule and molecule-molecule photoassociation (not yet experimentally realized) is briefly mentioned. The rapidly expanding fields of photoassociation in a quantum degenerate gas, in an electomagnetic field, and in an optical lattice are briefly discussed as a partial introduction to later chapters. 

Fig. 8.2 A quantum degenerate gas of ultracold atoms reaches degeneracy when the matter waves of neighboring atoms overlap (a) at absolute zero, gaseous bosonic atoms all end up in the lowest energy state (b) fermions, in contrast, fill the states with one atom per state, and the energy of the highest filled state at T = 0 is the Fermi energy Ep ...

The equation of state for a degenerate gas is obtained by evaluating the internal energy. Substituting ep = mc2+p2/2m into an integral over quantum states gives for the non-relativistic case pp -C mec... 

The length R is an intrinsic parameter of a Eeshbach resonance. It characterizes the width of the resonance. From Equations 10.1 and 10.2 we see that small W and, consequently, large R correspond to narrow resonances, whereas large W and small R lead to wide resonances. The term wide is generally used when the length R drops out of the problem, which, according to Equation 10.3, requires the condition kR 1. In a quantum degenerate atomic Fermi gas the characteristic momentum... 

The first step was the production of ultracold molecules from a quantum-degenerate Fermi gas of atoms at a temperature below 150 nK (Regal et al. 2003). The low binding energy of the molecules was controlled by detuning the magnetic field away from the Feshbach resonance. Clear evidence of diatomic molecules was achieved through direct, radio-frequency spectroscopic detection of molecules. 

Thus, a necessary condition for the formation of metal-like refractory compounds is the participation in the bonds of incomplete d and / electron levels or the possibility of their formation in the compounds in other words, this condition is reduced in the majority of cases to the requirement that the metallic components belong to the transition elements. As a qualitative criterion of the degree of participation in the bond and the determination of the distribution of electron concentration in the lattice, it is possible to use the quantity 1/Nn, where n is the number of electrons in the incomplete level and N is the principal quantum number of this level, as proposed by the author in 1953 (the proposal considered the electrons to be a degenerate gas in the Coulomb field of atomic nuclei or atomic cores). Another criterion is the ability of atoms of the nonmetals to give off valence electrons, which may characterize the ionization potentials of these atoms. 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

A gas in which the pressure no longer depends on the temperature is said to be degenerate, an unfortunate term indeed, because the corresponding state borders on perfection. One might call it a state of perfect fullness, since no interstice is left vacant. Electrons occupy all possible energy states and total order prevails. Both the electrical conductivity and the fluidity also attain perfection. Objects made from this sublime form of matter are perfectly spherical. And yet, in quantum circles, this state of nature is obstinately referred to as degenerate ... 

There are two limits (a) a "weakly degenerate" ideal FD gas, where the quantum effects are weak, because the factor /.i/kBT is so small that a series expansion in p/kBT can converge (b) a strongly degenerate ideal FD gas (which is where real metals exist), where the factor p/kBT is large, so no series expansion in p/kBT is advisable. 

If one puts two electrons per quantum state ("spin-up and spin-down"), then the minimum resistance becomes 2R0 = 25.812986 kQ. The internal resistance of a molecule has not yet been measured. Recently, it was shown that a degenerate quasi-one-dimensional electron gas in a GaAs GaAh YAsY system, when interrogated in a four-probe geometry, has zero resistance drop between probes 2 and 3, in contrast to the expected R0 between probes 1 and 4 the transport is ballistic [13]. 

The important fact that the quantum theory explains the extraordinarily small specific heat of electrons at normal temperatures, as is well known, was pointed out by Sommerfeld. Our model likewise explains this fact, though in a somewhat different way. If, as is permissible here, we work with the spins in the same way as we previously did with the free electrons, it follows by what we have said above that we have to deal with a degenerate Bose-Einstein gas, the specific heat of which at low temperatures is, as we know, given by... 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

The Fermi-Dirac distribution law for the kinetic energy of the particles of a gas would be obtained by replacing p W) by the expression of Equation 49-5 for point particles (without spin) or molecules all of which are in the same non-degenerate state (aside from translation), or by this expression multiplied by the appropriate degeneracy factor, which is 2 for electrons or protons (with spin quantum number ), or in general 21 + 1 for spin quantum number I. This law can be used, for example, in discussing the behavior of a gas of electrons. The principal application which has been made of it is in the theory of metals,1 a metal being considered as a first approximation as a gas of electrons in a volume equal to the volume of the metal. 

We have already seen in Chapter V that this cannot be the case if we regard the requirements of the quantum theory as generally valid we have seen, too, that there are already direct observations on helium (p. 71) which indicate an abnormal behaviour of the gas at very low temperatures, and here, apparently, is the key to a closer understanding. We shall now deal more fully with the theory of the "degeneration of gases" with which many eminent theorists, as Tetrode, Sackur, Keesom, Sommerfeld, Planck, have occupied themselves of late years, agreeing... 

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