Modeling of structural and energetic parameters of р-Er1-xScxNiSb semiconductor

The energy expediency of the existence of Er1-xScxNiSb substitutional solid solution up to the concentration x≈0.10 was established by modeling the variation of free energy ΔG(x) values (Helmholtz potential). At higher Sc concentrations, x> 0.10, there is stratification (spinoidal decomposition of phase). It is shown that in the structure of p-ErNiSb semiconductor there are vacancies in positions 4a and 4c of Er and Ni atoms, respectively, generating structural defects of acceptor nature. The number of vacancies in position 4a is twice less than in position 4c. This ratio also remains for p-Er1-xScxNiSb. Doping of p-ErNiSb semiconductor by Sc atoms by substitution of Er atoms is also accompanied by the occupation of vacancies in position 4a. In this case, Ni atoms occupy vacancies in position 4c, which can be accompanied by the process of ordering the p-Er1-xScxNiSb structure. Occupation of vacancies by Sc and Ni atoms leads to an increase of the concentration of free electrons, an enlarge of the compensation degree of semiconductor, which changes the position of the Fermi level εF and the mechanisms of electrical conductivity.

The Sommerfeld problem and inverse problem for the Helmholtz equation

The study of a time-periodic solution of the multidimensional wave equation {\frac{\partial^{2}}{\partial t^{2}}\widetilde{u}-\Delta_{x}\widetilde{u}=% \widetilde{f}(x,t)}, {\widetilde{u}(x,t)=e^{ikt}u(x)}, over the whole space {\mathbb{R}^{3}} leads to the condition of the Sommerfeld radiation at infinity. This is a problem that describes the motion of scattering stationary waves from a source that is in a bounded area. The inverse problem of finding this source is equivalent to reducing the Sommerfeld problem to a boundary value problem for the Helmholtz equation in a finite domain. Therefore, the Sommerfeld problem is a special inverse problem. It should be noted that in the work of Bezmenov [I. V. Bezmenov, Transfer of Sommerfeld radiation conditions to an artificial boundary of the region based on the variational principle, Sb. Math. 185 1995, 3, 3–24] approximate forms of such boundary conditions were found. In [T. S. Kalmenov and D. Suragan, Transfer of Sommerfeld radiation conditions to the boundary of a limited area, J. Comput. Math. Math. Phys. 52 2012, 6, 1063–1068], for a complex parameter λ, an explicit form of these boundary conditions was found through the boundary condition of the Helmholtz potential given by the integral in the finite domain Ω:($*$)u(x,\lambda)=\int_{\Omega}\varepsilon(x-\xi,\lambda)\rho(\xi,\lambda)\,d\xi{}where {\varepsilon(x-\xi,\lambda)} are fundamental solutions of the Helmholtz equation,-\Delta_{x}\varepsilon(x)-\lambda\varepsilon=\delta(x),{\rho(\xi,\lambda)} is a density of the potential, λ is a complex number, and δ is the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region Ω to {\partial\Omega} pass {\partial\Omega} without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in {\mathbb{R}^{n}}, {n\geq 3}, we have proved the problem of reducing the Sommerfeld problem to a boundary value problem in a finite domain. Under the necessary conditions for the Helmholtz potential (*), its density {\rho(\xi,\lambda)} has also been found.

A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials

In this article, we demonstrate the use of a Gibbs-potential-based formulation as a means for developing a thermodynamically consistent model for a class of viscoelastic fluids of the rate type. Since one cannot always use a formulation based on a Helmholtz potential to model rate-type models, the formulation takes on added significance. The salient features of this approach are the following: — this approach provides a thermodynamical rationalization of many commonly used models that are developed on purely phenomenological grounds; furthermore, the study provides a framework for generating other classes of models and allows for a relatively straightforward means for the inclusion of thermal effects, — the approach provides a simple means for including anisotropic effects without the need for directors or other new internal variables, and — the approach does not use any additional variables (such as conformation tensors or elastic strains measured from stress free configurations) other than the current (or Cauchy) stress, the current mass density and the velocity gradient. We also show how the entire structure of the theory is obtained from just two scalar functions, the Gibbs potential and the rate of dissipation function.