Thermodynamic properties of two-band superconductors with low carrier density

A theory of superconductivity is constructed for systems with two overlapping energy bands at the Fermi surface and with low carrier density. The phonon superconductivity mechanism is assumed and allowance is made for all possible pairings of electrons both within one band and of electrons from different bands. A bell-shaped dependence of the temperatureT _c of the superconducting transition and a jump of the electron specific heatC _S –C _N atT=T _c as a fuction of the carrier density are obtained in the case of strong hybridization. The possibility of an increase in the value of the chemical potential at which the kink at the pointT=T _c arises is demonstrated. It is also shown that the relative jump (C _S –C _N )/C _N of the specific heat atT=T _c depends on the carrier density and may be both greater or less than 1.43.In memory of Professor D. N. ZubarevInstitute of Applied Physics, Moldavian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 459–472, September, 1993.     

Superconductivity based on a model with three overlapping energy bands and a nonphonon mechanism

A study is made of the thermodynamic properties of a three-band superconductor with an electron (or hole) mechanism of superconductivity. Interband carrier interaction is considered as the main reason for the occurrence of superconducting pairs. Expressions are obtained for the order parameters _n (T) (n=1, 2, 3), the specific heatC _S, and the thermodynamic critical fieldH _c in the region of low temperatures and also the temperatureT _c of the superconducting transition, the jump in the specific heat (C _S–C_N) at the pointT=T _c; and 2 _n (0)/T _c . It is shown that a characteristic feature in the behavior ofT _c, (C _S–C_N), and _n (0) as functions of the carrier concentrationn is the presence of a smeared step. It is also shown that (C _S–C_N)/C _N1.43 and that this ratio depends strongly on the relationship between the densities of the electron states of the considered bands.Institute of Applied Physics, Moldavian Republic Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 483–499, June, 1992.     

The influence of overlapping energy bands on the superconductivity of localized bose-condensate pairs atT=0

A system of equations is obtained for the order parameters _nm and the chemical potential of two-band superconductors with low carrier densities at T=0. The limiting case of _n<0, corresponding to a Bose condensate of localized pairs in both bands, is considered, and all possible inter- and intraband interactions are taken into account. For _nm ² / _n,m ² 1, the above system of equations is reduced to a system of linear equations whose analytical solutions are found. The dependences of _nm and on electron-electron interaction constants and carrier densities are analyzed.Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 105, No. 3. pp. 491–502. December, 1995.     

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

The effect of ionisation loss on the energy spectra of cosmic ray nuclei undergoing fermi acceleration

The effect of ionisation loss, on the energy spectra of cosmic ray nuclei undergoing Fermi acceleration, has been studied using an analytic expression derived for the energy spectra of the nuclei undergoing acceleration. The spectra thus obtained have negative slopes which become steeper with increasing charge of the nucleus. In cases where the rate of acceleration is slow, solar modulation effects have been superimposed on the above spectra and it is found that the resulting spectral shapes can be fitted with recent satellite measurements for the energy spectra of various groups of cosmic ray nuclei at low energies.     

Optimal control of the transverse vibrations of a beam with a bound on the potential energy

The problem of the optimal control of the transverse vibrations of a uniform beam with a bound on the potential energy is considered. Approximation theory is used. It is shown that the optimal control exists, and a simple method for its computation is suggested.     

Port-Hamiltonian representation for pdes with second-order derivatives in the energy density

In this contribution we consider a port-Hamiltonian setting for partial differential equations. A crucial property of this system class is the property to be able to link a power balance relation to the structure of the equations. However, one has to take into account also the effects of energy flows via the boundary. This is straightforward when the Hamiltonian depends on derivative variables of first order, e.g. by using integration by parts. If second-order derivatives appear then integration by parts cannot be used without due care, thus we suggest an approach by using the so-called Cartan-form. We visualize the derivation of a power balance relation by using the Kirchhoff plate as an example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds

Let X ₁, . . . , X _n be a random sample from some unknown probability density f defined on a compact homogeneous manifold M of dimension d ≥ 1. Consider a ‘needlet frame’ ${\{\phi_{j\eta}\}}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on M with corresponding eigenvalues less than 2^2j , as constructed in Geller and Pesenson (J Geom Anal 2011). We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator f _n (j) obtained from an empirical estimate of the needlet projection ${\sum_\eta \phi_{j \eta}\int f \phi_{j \eta}}$ of f. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density f. The confidence bands are adaptive over classes of differentiable and H?lder-continuous functions on M that attain their H?lder exponents.     

The energy density in the planar Ising model

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.     

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.      

The half-widths of the bands due to the skeletal vibrations of polyethylene molecules,free and under an applied load

The third-order phonon interactions in free polyethylene chains and polyethylene chains under load have been analyzed to obtain the theoretical temperature dependence of the half-widths of the bands due to the skeletal vibrations potentially active in the absorption and Raman spectra. The problems of the harmonic and anharmonic vibrational analysis of polymers under load are discussed.A. F. Ioffe Order of Lenin Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 3–11, January–February, 1972.     

Permutation congruent transformations of the freudenthal triangulation with minimum surface density

Linear transformations of the Freudenthal triangulation with permutation congruent simplexes are characterized, and within this class those transformations yielding minimum normalized surface density are identified. This research was partially supported by National Science Foundation Grants SOC 78-16811 and MCS 81-21838; and Department of Energy Contract AM03- 76F00326, PA# DE-AT03-76ER72018.     

Optimization techniques in energy calculations involving the Hartree-Fock density matrix

Three optimization procedures are examined for their utilization in determining the optimum density matrix for a single determinant wave function. The total energy of a molecular system is written as a function of the density matrix and then optimized subject to the constraints of idempotency and total electron population. This direct calculation of the density matrix (DCDM) method was studied in an attempt to have a formalism which would avoid convergence problems associated with the self-consistent field (SCF) cycle, and which would be applicable to large molecular systems. The optimization procedures studied were the Powell algorithm, Gauss-Jordan reduction, and dynamic programming. Computational factors studied include convergence criteria, stepsize, and weight factors for constraint equation penalty functions. The application considered is for HF. An ab-initio SCF method was used to obtain initial values for the density matrix, and its SCF results were compared to corresponding DCDM predictions. Approximations of the quadratic two-electron energy contributions will be necessary to apply dynamic programming, but this appears to be the method most applicable to large molecular systems if an acceptable approximation can be found. Gauss-Jordan reduction is an applicable technique, but probably not for large molecular systems. BOTM appears to be the method most applicable without introducing approximations, but weight factors for the penalty functions will have to be more efficiently determined as Lagrange multipliers, and this addition would result in a technique probably not applicable to large molecular systems.This paper is based in part on the PhD Thesis of C. A. Waggoner, Department of Chemistry, Mississippi State University. Partial support was furnished by the National Science Foundation, Grant No. Rii-89-02064, the State of Mississippi, and Mississippi State University. Other support was furnished by the Physical and Biological Sciences Institute of MSU and the Office of the Dean, College of Arts and Sciences, MSU. Computer time was furnished by the Thomas E. Tramel Computing Center.