Degeneration of gases

Just recently the degeneration of gases foreseen by me from the quantum theory has filled this gap and allowed the immediate application of my Theorem even to gases thus chemical constants have been referred to thermal quantities, namely, specific heats. 

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... 

The latter condition indicates a definite line for any special theory of the degeneration of gases. 

II Planck, in an interesting attempt to develop the theory of the " degeneration of gases on the assumption that the impacts of atoms take place according to the laws of ordinary mechanics, arrives at apparently impossible results (cf. Sitzungsber. d. Preuss. Akad. d. Wiss., 1916, p. 665). 

I have recently (108) developed an essentially different conception of what the degeneration of gases is according to this the translatory motion of the molecules of gas is converted more and more at low temperatures, under the influence of the absolute zero radiation (Nullpunkts-strah-lung) of the ether, into a circular movement, such that at the zero itself the molecules of gas revolve with constant velocity about equilibrium positions, uniformly distributed in space. 

But even if our formulae describe the phenomenon of the degeneration of gases to a first approximation only, equation (148) may serve, at any rate, for preliminary exploration. 

As concerns solid solutions (e.g. in the case of dilute solutions) it is of course clear that, if the gas laws are regarded as valid here down to the lowest temperatures, my Heat Theorem cannot hold, for the same reasons as it does not hold for gases in the corresponding case (cf. p. 191). If the theory of the degeneration of gases is taken as a basis there can be no doubt that, since the Heat Theorem is then true for gaseous mixtures, it must hold also a fortiori for solid solutions. 

That the degeneration of gases may show some peculiari-... 

Page 200.—The question of the degeneration of gases cannot yet be regarded as settled as shown in the first edition of this book, it is a consequence of the idea that our Heat Theorem is applicable also to gases, and this is definitely fixed if the variation... 

In the text on page 205 the factor f is given instead of, which was only due to a mistake, but leads to essentially different formulae for the degeneration of gases, as was shown by Dr. Bennewitz in a very important study of the degeneration of gases and the energy at absolute zero (Nemst Jubilee volume," Zeitschr. physik. Chem., 110, p. 725, 1924). 

For an attempt at an indirect test of the degeneration of gases, cf. my note, Ber. d. preuss. Akad " 1919, p. 118. In the examination of the viscosity of gases at very low temperatures, P. Gunther ( Zeitschr. physik. Chem., 110, p. 626, 1924) found what was essentially the predicted remarkable behaviour, when he succeeded in measuring the viscosity of hydrogen and helium down to 160 abs. 

The concept of a mobility edge has proved useful in the description of the nondegenerate gas of electrons in the conduction band of non-crystalline semiconductors. Here recent theoretical work (see Dersch and Thomas 1985, Dersch et al. 1987, Mott 1988, Overhof and Thomas 1989) has emphasized that, since even at zero temperature an electron can jump downwards with the emission of a phonon, the localized states always have a finite lifetime x and so are broadened with width AE fi/x. This allows non-activated hopping from one such state to another, the states are delocalized by phonons. In this book we discuss only degenerate electron gases here neither the Fermi energy at T=0 nor the mobility edge is broadened by interaction with phonons or by electron-electron interaction this will be shown in Chapter 2. 

A dual ion beam collector developed by Nier2 is illustrated in Figure 4.1 b. Both collectors are connected to two amplifiers for the simultaneous and direct measurement of ion currents in the dual mode. Amplifier 1 works with degeneration whereas amplifier 2 works without. Such a dual ion beam collector is applied, for example, for precise and accurate measurements of isotope ratios, especially of gases in commercial stable isotope ratio mass spectrometers.2... 

DEGENERACY. In the kinetic theory of gases, a gas that does not obey the ideal gas laws is referred to as a degenerate gas. The greater the deviation of the real gas from the ideal, the greater is its degeneracy. 

Dalton s law of partial pressures for a mixture of gases in a container the total pressure exerted is the sum of the pressures that each gas would exert if it were alone. (5.5) Degenerate orbitals a group of orbitals with the same energy. (12.9)... 

Liu, I.-S., Muller, I. Extended thermodynamics of classical and degenerate ideal gases. Arch. Ration. Mech. Anal. 83, 285-332 (1983)... 

Castin, Y., Basic theory tools for degenerate Fermi gases, in Ultracold Fermi Gases, Inguscio, M., Ketterle, W., and Salomon, C. Eds., Proceedings of the International School of Physics Enrico Fermi , Course CLXIV, Varenna, 20-30 June 2006, to appear, lOS Press, Amsterdam, 2008 e-print cond-mat/0612613. 

The realization of Bose-Einstein condensates (BEC) and quantum degenerate Fermi gases with cold atoms has been one of the highlights of experimental atomic physics during the last decade [1]. In view of recent progress in the experimental work on the production of cold molecules we expect a similarly spectacular... 

The cooling of atoms to ultracold temperatures has resulted in spectacular discoveries. The realization and study of new states of matter like Bose Einstein Condensates/ degenerate Fermi gases and (Bardeen,... 

The trapping of cold neutral atoms is a powerful tool in experimental atomic physics that has made it possible to conduct many fundamental experiments, such as Bose Einstein condensation and the production of Fermi-degenerate quantmn gases. It is therefore one of the most vivid demonstrations of the capabilities inherent in the laser control of atoms. 

The methods of trapping cold atoms, considered in Chapter 5 and the present chapter in a very brief and retrospective fashion, have become a very powerful tool in experimental physics. They have led to the development of atom optics, the observation and investigation of dilute quantum gases (Bose-Einstein condensation, atom lasers, Fermi-degenerate quantum gases, and ultracold molecules), and probably many other discoveries in the physics of ultracold atoms. These will be discussed in Chapters 7 and 8. But it would be expedient to consider at the end of this chapter a few examples of applications that lie beyond the mainstream, but are of physical interest. 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

---