Fermi integral function

Still, the Fermi integral function f Z, p) is very useful even in the general case, because it can help visualize the overall dependence of the decay constant 1 on the beta energy and the atomic number Z For instance, Figs. 42A and B in Chap. 2 (as well as Fig. 7.15) show that (1) the probability for beta decay steeply increases with beta energy (this is equally true for positive and negative beta decay), (2) negative beta decay becomes more probable as the atomic number increases, (3) positive beta decay becomes less probable as the atomic number increases. 

Friedlander et al. (1981) cite empirical formulas for the energy- and atomic-number dependence of the Fermi integral function. The following formulas (see also Fig. 7.15) are valid in the intervals 0 < Z < 100 and 0.1 < Fp < 10 (where Z is the atomic number of the daughter and Fp is the maximum of the beta energy) ... 

The Fermi integral function for electron capture calculated fromO Eq. (7.80). The figure shows that EC is most likely for larger values of Z... 

Find the upper limit of 5. (Hint show that the Fermi integral function for large values of its argument can be approximated by... 

Electrons thermally excited from the valence band (VB) occupy successively the levels in the conduction band (CB) in accordance with the Fermi distribution function. Since the concentration of thermally excited electrons (10 to 10 cm" ) is much smaller than the state density of electrons (10 cm ) in the conduction band, the Fermi function may be approximated by the Boltzmann distribution function. The concentration of electrons in the conduction band is, then, given by the following integral [Blakemore, 1985 Sato, 1993] ... 

The integrand includes factors arising from the magnon occupation number and from a combination of Fermi distribution functions for the conduction electrons it may be evaluated subject to a number of simplifying assumptions. At low temperatures only low wavevector magnons are appreciably excited and we may extend the limit of integration to infinity. Dispersion relations of the type (a and w oc q then yield resistivity contributions proportional to and T, respectively. 

The electronic contribution to the energy is obtained by integrating over all occupied states. To a good approximation, the Fermi-Dirac distribution can be replaced by a step function, and the integral can be performed up to the Fermi level ... 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

First of all we note that the Fermi hole - which is due to the antisymmetry of the wave function - dominates by far the Coulomb hole. Second, another, very important property of the Fermi hole is that it, just like the total hole, integrates to -1... 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

The universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

Equation 4.49 defines the exchange or Fermi hole. It is as if an electron of a given spin digs a hole around itself in space in order to exclude another electron of the same spin from coming near it (Pauli exclusion principle). The integrated hole charge is unity, i.e., there is exactly one electron inside the hole. Likewise, the correlation energy functional can be defined as... 

Integrating out the fast modes, modes far from the Fermi surface and hard gluons, the QCD partition function (1) becomes... 

The concentration of electrons that occupy a part of the total concentration, N, of the energy states available for electrons is obtained by integrating the product of the state density. Die), and the Fermi function, fie), as shown in Eqn. 2-30 ... 

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