In a mixed valence impurity system the distribution of impurity charges can be adjusted to minimize the Coulomb energy of inter-impurity interactions. In this paper we discuss the possibility of extending the methods of analytical evaluation of the pair correlation function for classical liquids to apply to a system with a built-in disorder, where the occupation probability is governed by the Fermi-Dirac statistics.
The investigation of electrical conductivity, coefficient of thermal electromotive force, Hall coefficient, microhardness and mobility in Pb_{1-x}Ge_{x}Te (x = 0 ÷ 0.1) alloys in the temperature range of 77-300 K was carried out. Anomalies were detected in isotherms of properties in the vicinity of x = 0.008. The anomalies were treated as a manifestation of concentration phase transitions occurring in solid solutions of any kind and associated with existence of critical concentration (percolation threshold) at which the uninterrupted chain of interactions between impurity atoms is formed.
A classification scheme of quantum states for the system of two free electrons in a cubic box, confined to a single star of quasi-momentum is proposed within a Racah-Wigner type of approach. Coupling of angular momenta of the atomic case is here substituted by Mackey theorem for transitive representations, which provides a crystalline analogue of orbital angular momentum - the resultant orbit of the geometric symmetry group. The action of the Pauli group is combined with that of the octahedral group which yields the connection between spin (i.e. the singlet or triplet pairing of electrons) and statistic of the positional factor. Resultant singlets and doublets - irreducible representations of the octahedral group, exhibit an ordinary Landau diamagnetic behaviour, whereas triplets are paramagnetic. A relation between the Mackey star and the star of resultant momentum is discussed.
A version of Wigner-Racah type of approach is proposed for the model of a free-electron gas in a cubic box. The approach bases on the structure of fibre bundle in description of states of a single electron. Contrary to the tight binding model, the base of the bundle is the reciprocal space rather than the positional one. An elementary quantity of the approach is the star of quasi-momentum. Such a treatment reveals the degeneracy of energy levels of the system of N electrons, in particular of the ground Fermi level, for an uncomplete filling of stars. The associated classification of energy levels and corresponding states can be performed within an atomic-like Wigner-Racah scheme.
The first experimental study of the Compton profiles of Zn_{1-x}Mg_{x}Se for x=0.25, 0.47, 0.56 mixed crystals is presented. The Compton profiles were measured with the use of the ^{241}Am radioactive source with a resolution of 0.57 a.u. The experimentally obtained Compton profiles were compared with the theoretical ones based on the free-atom model. The results are interpreted in terms of outermost electrons of Zn and Mg being promoted to the higher momentum states, and 4p-electrons of Se becoming more delocalised in a solid, being thus promoted to the lower momentum states.
The magnetic translation group was introduced as a set of operators T(R)=exp[-iR·(p-eA/c)/h]. However, these operators commute with the Hamiltonian for an electron in a periodic potential and a uniform magnetic field if the vector potential A (the gauge) is chosen in a symmetric way. It is showed that a local gauge field A_{R}(r) on a crystal lattice leads to operators, which commute with the Hamiltonian for any (global) gauge field A = A(r). Such choice of the local gauge determines a factor system ω(R,R') = T(R)T(R')T(R+R')^{-1}, which depends on a global gauge only. Moreover, for any potential A a commutator T(R)T(R')T(R)^{-1}T(R')^{-1} depends only on the magnetic field and not on the gauge.
The interacting electron Hamiltonian H = H_{D}+∑_{K,ζ}H_{K,ζ} is considered in the Hilbert space spanned by Slater determinants of Bloch wave functions. H_{D} consists of the diagonal part of H in this basis. K and ζ=0, ±1 stand for the total momentum and projected spin of electron pairs and H_{K,ζ} is the off-diagonal part of H describing the most general two-electron scattering process conserving K and ζ. It is shown that the eigenspectrum of H includes all eigenvalues of H_{D}+H_{K,ζ} for every K and ζ value. The associated eigenvectors of H are shown to have off-diagonal long-range order.
The Hubbard Hamiltonian is projected onto a representation consisting of electron pairs characterised by the momentum of their centre of mass. Within this approximation the electron gas can be viewed as a collection of subsets, each of which contains a constant number of electron pairs, all having the same centre of mass momentum. As these subsets are decoupled, the Hubbard Hamiltonian is diagonalised to give two types of many-body eigenstates: correlated and uncorrelated. The uncorrelated pairs build up an ideal Fermi gas. Excellent agreement is found for the uncorrelated energy calculated at zero temperature in one dimension between this model and the Bethe ansatz, for arbitrary electron concentration and magnitude of the electron interaction. The correlated states turn out to be of the BCS type.
For three-dimensional charge carriers described by the dispersion law with quartic terms of the wave vector, the density of states function similar as in the one-dimensional case was determined. This similarity allows the Pekar and Dejgen condenson states in the continuum approximation to exist. The calculated phonon spectrum reveals optical vibrations of a very low frequency, which favours the electron-phonon interaction and creation of the condenson states.
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