Fermi integrals

It will be intriguing to theoretically examine the possibility of superconductivity in CNT prior to the actual experimental assessment. A preliminary estimation of superconducting transition temperature (T ) for metallic CNT has been performed considering the electron-phonon coupling within the framework of the BCS theory [31]. It is important to note that there can generally exist the competition between Peierls- and superconductivity (BCS-type) transitions in lowdimensional materials. However, as has been described in Sec. 2.3, the Peierls transition can probably be suppressed in the metallic tube (a, a) due to small Fermi integrals as a whole [20]. 

Ef is the Fermi energy, Ec and Ev are the energies of the conduction and valence band edges, respectively, and F1/2(rj) is the Fermi integral of order one-half for the argument 17. The activity coefficients approach values of unity at dilute carrier concentration because the value of Fi/2(t7) approaches exp(Tj) at dilute carrier concentrations. The concentration dependency of Eqs. (5a)-(5d) can be obtained explicitly through analytic expressions relating exp(r7) to F M -50... 

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... 

Instead of the above expression one may use the general approach of Leonard and Michael [50] where the connection between these two parameters is made using the Fermi integral normalized using the intrinsic concentration to determine a correction for the Einstein relation as a derivative of the quasi-Fermi level over a logarithm of electron concentration... 

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

In one of the earliest DFT models, the Thomas-Fermi theory, the kinetie energy of an atom or moleeule is approximated using the above kind of treatment on a loeal level. That is, for eaeh volume element in r spaee, one assumes the expression given above to be valid, and then one integrates over all r to eompute the total kinetie energy ... 

Coulomb blockade effects have been observed in a tunnel diode architectme consisting of an aluminum electrode covered by a six-layer LB film of eicosanoic acid, a layer of 3.8-nm CdSe nanoparticles capped with hexanethiol, and a gold electrode [166]. The LB film serves as a tunneling barrier between aluminum and the conduction band of the CdSe particles. The conductance versus applied voltage showed an onset of current flow near 0.7 V. The curve shows some small peaks as the current first rises that were attributed to surface states. The data could be fit using a tunneling model integrated between the bottom of the conduction band of the particles and the Fermi level of the aluminum electrode. 

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... 

Rice5 used this Thomas-Fermi theory for the metal in the interface. This equation is multiplied by 2 d(n)213/dx9 and then integrated from x = 0 to x = —oo. Since dn(x)/dx - 0 when x -> — oo one gets... 

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