Fermi-Dirac Integral

The case of bare Coulomb interactions, y(r) = jr, is of particular interest [29]. In this case, Eq. (8) can be evaluated analytically and expressed in closed form, after lengthy calculations, in terms of standard Fermi-Dirac integrals... 

The special functions occurring here are the complete Fermi-Dirac integral of order j, the (complementary) incomplete Fermi-Dirac integral... 

M. Goano, Series Expansion of the Fermi-Dirac Integral Tj (x) over the entire Domain of real j and x, Solid-State Electron. 36 (1993) 217-221. 

M. Goano, Computation of the Complete and Incomplete Fermi-Dirac Integral (Algorithm 745), ACM Trans. Math. Softw. 21 (1995) 221-232. 

J. S. Blakemore, Approximations for Fermi-Dirac integrals, especially of order 1/2 used to describe electron density in a semiconductor, Solid State Electron. 25 (1982) 1067-1076. 

X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, An analytical approximation for the Fermi-Dirac integral F1/2(i7), Solid State Electron. 24 (1981) 981-982. 

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. 

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 following procedure may be used if more precision is desired to take into account integration over the Fermi-Dirac distribution in the electrode, which may be necessary for low reorganization energies. ks in equation (1.37) is converted into... 

Replacing the Fermi—Dirac distribution by the Boltzman distribution (for U below the Fermi level) and integrating eqn. (151), a current maximum close to the Fermi level of the electrode is predicted with the major contribution within kB T around UF. 

Taking spin indices into account, all two-electron integrals are of the form (ij u kl) = (ij u kl) - (ij u lk), with the convention that orbitals with different spin indices are orthogonal. It is convenient to truncate summations by the use of occupation numbers // which are in principle determined by Fermi-Dirac statistics. At zero temperature, occupation numbers are determined by the structure of the reference state. Then m = 1, na = 0 for i < N < a. A convention used for double summation indices is ij i < j < N,ab N < a < b. 

The integral / is a standard integral of the Fermi-Dirac statistics with the solution (419) ... 

In writing equation (4.6), we have assumed that the nuclei can be treated as Dirac particles, that is, particles which are described by the Dirac equation and behave in the same way as electrons. This is a fairly desperate assumption because it suggests, for example, that all nuclei have a spin of 1 /2. This is clearly not correct a wide range of values, integral and half-integral, is observed in practice. Furthermore, nuclei with integral spins are bosons and do not even obey Fermi Dirac statistics. Despite this, if we proceed on the basis that the nuclei are Dirac particles but that most of them have anomalous spins, the resultant theory is not in disagreement with experiment. If the problem is treated by quantum electrodynamics, the approach can be shown to be justified provided that only terms of order (nuclear mass) 1 are retained. 

Fermi-Dirac statistics apply to fermionic systems. Fermions are particles with half integral spin and that obey the Pauli exclusion principle. 

Fermi-Dirac (particles with odd (half integral) spin)... 

If we consider a situation in which the levels are broadened to a more or less continuous zone the Fermi-Dirac form given by Eq. (25) l.h.s. is only valid, if we attribute Nd and /it to an infinitely small level interval (ranging from Ed to Ed + dEd) then in order to obtain the total concentration, the molar density of states D (Ed) has to be considered, and the result for nd follows by integration ... 

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