Fermi quantum gases

Boltzmann gas Is negligible above 1250 K. Below this temperature the deviation between classical and quantum statistics will be significant Mitchell (4) calculates S (298.15 K) = 22.72 -J k" mol" for the Fermi-Dirac gas compared to the classical value of... 

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... 

However, at the molecular level, S3mimetry properties of the quantum mechanical wave function give rise to deviations from the classic behavior as we showed in Section 2.5. These deviations may be interpreted as a net repulsion (Fermi-Dirac gas) or attraction (Bose-Einstein gas) between the molecules. As we emphasized in Section 2.5.3, qucuitum efhscts are maximized in seiniclassic ideal gases. Fi om this point of view, it then seems sensible to address the following que.stions ... 

The thermodynamics of a l-d Fermi system can be perfectly mapped onto the thermodynamics of a two-component classical real gas on the surface of a cylinder. The relationship between these two infrared problems (cf. Zittartz s contribution) is exploited as follows. We treat the classical plasma by a modified Mayer cluster expansion method (the lowest order term corresponding to the Debye Hiickel theory), and obtain an exponentially activated behavior of the specific heat (cf. Luther s contribution) of the original quantum gas by simply reinterpreting the meaning of thermodynamic variables. 

At the center of the approach taken by Thomas and Fermi is a quantum statistical model of electrons which, in its original formulation, takes into account only the kinetic energy while treating the nuclear-electron and electron-electron contributions in a completely classical way. In their model Thomas and Fermi arrive at the following, very simple expression for the kinetic energy based on the uniform electron gas, a fictitious model system of constant electron density (more information on the uniform electron gas will be given in Section 6.4) ... 

The properties of the two helium isotopes in the liquid state are strongly influenced by quantum effects. In Fig. 2.8, the specific heat of 3He, calculated from the ideal gas Fermi model (Tp = 4.9 K) with the liquid 3He density, is compared with the experimental data. The inadequacy of this model is evident. A better fit, especially at the lower temperatures, is obtained by the Landau theory [25]. 

The electrons that occupy the levels of a Fermi gas have energies < and may be considered as confined to a (Fermi) sphere of radius kF in k-space. For large volumes the free-electron quantum numbers may be treated as continuous variables and the number of states in a range dk = dkxdkydkz, is... 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

A Fermi liquid is a quantum-mechanical liquid of fermions at very low temperatures, with properties resembling those of a Fermi gas of noninteracting fermions. 

Y. Kwon, D. M. Ceperley, and R. M. Martin (1994) Quantum Monte Carlo calculation of the Fermi-liquid parameters in the two-dimensional electron gas. Phys. Rev. B 50, pp. 1684-1694... 

I7e. Heat Capacities at High Temperatures.—Although the theoretical treatment of heat capacities requires the limiting high temperature value to be 3/2, i.e., 5.96 cal. deg. g. atom , experimental determinations have shown that with increasing temperature Cv increases still further. The increase is, however, gradual for example, tfie heat capacity of silver is 5.85 cal. deg. g. atom at 300° K and about 6.5 cal. deg. g. atom at 1300° K. This increase is attributed mainly to the relatively free electrons of the metal behaving as an electron gas. By the use of the special form of quantum statistics, viz., Fermi-Dirac statistics, applicable to electrons, the relationship... 

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

We have shown in 4 (p. 209) that the introduction of the principle of indistinguishability into statistics leads to two, and only two, new systems of statistics, one of which, the Bose-Einstein statistics, we have discussed in detail in the last two sections (light quanta, gas molecules). We turn now to the second possible statistics, which is based on Pauli s principle, and was introduced by Fermi and Dirac. We have seen in 4 (p. 209) that this statistics is intimately connected with the employment of Pauli s principle, observing that the proper function of a state in which two electrons have the same partial proper function (with respect to the four quantum numbers, including the spin quantum number) automatically vanishes. 

The reason for this behaviour is the presence of Shockley surface states [176] on the noble metal surfaces. On these surfaces, the Fermi energy is placed in a band gap for electrons propagating normal to the surface. This leads to exponentially decaying solutions both into the bulk and into the vacuum, and creates a two-dimensional electron gas at the surface. The gas can often be treated with very simple quantum mechanical models [177, 178], and much research has been done, especially with regards to Kondo physics [179, 180, 181]. There has also been attempts to do ab initio calculations of quantum corrals [182, 183], with in general excellent results. 

The calculation of exchange functionals is as old as quantum theory itself. The first ab initio functional to be calculated for the exchange was the one for the ground state of an uniform gas of electrons. Given a uniform density of electrons, assumed to have a Fermi distribution, Bloch [22]... 

The subject of statistical mechanics is a branch of mechanics which has been found very useful in the discussion of the properties of complicated systems, such as a gas. In the following sections we shall give a brief discussion of the fundamental theorem of statistical quantum mechanics (Sec. 49a), its application to a simple system (Sec. 496), the Boltzmann distribution law (Sec. 49c), Fermi-Dirac and Bose-Einstein statistics (Sec. 49d), the rotational and vibrational energy of molecules (Sec. 49e), and the dielectric constant of a diatomic dipole gas (Sec. 49/). The discussion in these sections is mainly descriptive and elementary we have made no effort to carry through the difficult derivations or to enter into the refined arguments needed in a... 

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. 

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