Degenerate gas

We shall in this book use the concept of a degenerate gas of small—or at any rate heavy—polarons. Clearly we should not expect these to be formed unless the number of carriers is considerably less than the number of sites. We also remark, as mentioned earlier, that in all metals, at temperatures less than B phonons lead to a certain mass enhancement, of order less than 2. A treatment is given by Ashcroft and Mermin (1976, p. 520). This affects the thermopower some results for an amorphous alloy (Ca AlJ from Naugle (1984) are shown in Fig. 2.2. A theoretical treatment of the range between this situation and the polaron gas has not yet been given. 

The model of a degenerate gas of spin polarons suggests that if the direct or RKKY interaction between moments is weak and EF too great to allow ferromagnetism then the moments might all resonate between their various orientations. This would mean that it is possible in principle to have a heavily doped magnetic semiconductor or rare-earth metal in which there is no magnetic order, even at absolute zero. This possibility is discussed further in Section 8 in connection with the Kondo effect. 

If the spin-polar on model is correct, we must describe the carriers in the antiferromagnetic semimetal formed when the two Hubbard bands overlap as a degenerate gas of spin polarons it should have the following properties. 

Ramirez et al (1970) discussed a metal-insulator transition as the temperature rises, which is first order with no crystal distortion. The essence of the model is—in our terminology—that a lower Hubbard band (or localized states) lies just below a conduction band. Then, as electrons are excited into the conduction band, their coupling with the moments lowers the Neel temperature. Thus the disordering of the spins with consequent increase of entropy is accelerated. Ramirez et al showed that a first-order transition to a degenerate gas in the conduction band, together with disordering of the moments, is possible. The entropy comes from the random direction of the moments, and the random positions of such atoms as have lost an electron. The results of Menth et al (1969) on the conductivity of SmB6 are discussed in these terms. 

Since the strength of the coupling of the moments to the carriers is about r h2jma2, we deduce that, for a temperature above Tthe moments are free, giving entropy NkB In 2. If we think in terms of spin polarons, the polarons have broken up at this temperature. The excitations are mainly magnetic. The carriers will move through an array of disordered spins, and should behave like a non-degenerate gas with entropy... 

In Fig. 5.11 we show the electronic part of the specific heat of uncompensated Si P for various values of n—n0 (Paalanen et al 1988). The full lines show the calculated specific heat for a degenerate gas of electrons with effective mass... 

We conclude from these results that a degenerate gas of small polarons is a valid concept. 

Fig. 6.5 Specific heat of a degenerate gas as a function of temperature when the degeneracy temperature is exceeded (Mott and Jones 1936).

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. 

HO-OH2+.14 This reaction forms two water molecules and a carbenium ion, which is then hydrated to form the protonated alcohol. The degenerate gas-phase reactions between HF and protonated alkyl fluorides RFH+ (R = Me, Et, i-Pr, r-Bu) were investigated by ab initio methods.15 With the exception of MeFFi+, the protonated alkyl fluorides can be viewed as weak complexes of the carbocation R+ and FiF. Both frontside and backside substitutions occur, supporting the proposal (see Introduction)8 that nucleophilic substitution reactions are better understood through such a competition as opposed to the traditional S /S 2 competition. 

In the metals of the Short Periods and of the A-subgroups (main series), such as the alkali metals, positive ions with the inert gas configurations are also produced while the surplus electrons form the degenerate gas of the conduction electrons. Also the configuration with a filled d-subshell is characterized by a certain preference as is demonstrated by the common 18-electron configuration of many positive ions of the B-subgroups (subseries) e.g. 

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 Fermi momentum implies that the de Broglie wavelength of the most energetic electrons in a degenerate gas is comparable with nj 1,/ 3, the average distance between electrons. 

The Bolution of this difficulty is due to Pauli and Sommerfeld (1927), who pointed out that the laws of classical statistics ought not to be applied to the electron gas within a metal, since it is bound to behave as a degenerate gas. Thus, since the mass of the electron is 1840 times smaller than that of the hydrogen atom, it follows that, at room temperature T = 300°) and an electronic density oi n 3T0 , corresponding to a gas density at a pressure of 1 atmosphere, the value of the degeneracy parameter for the electron is... 

The reliability of this equation depends, of course, on whether we have made a correct guess in our model of a degenerated gas. But a different model would only lead to a different, and not very essentially different, factor in place of 4ir. 

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. 

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