Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions

The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the n-dimensional relations and the relations written out for any m-dimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a three-dimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension.     

Photoabsorption cross section of the Thomas-Fermi atom

The photoabsorption cross section σ(ω) and the distribution of oscillator strengths df/dω [these values are related as σ=(2π²e²/mc)(df/dω)] were determined for an atom with a large Z value using the semiclassical approach. These values were found for low frequencies with the use of the Vlasov kinetic equations, which were numerically solved by the method of particles. The asymptotic behavior of the distribution of oscillator strengths at high frequencies was determined by semiclassical equations for the photoabsorption cross section of electron shells in a Coulomb potential. The asymptotic equations were used to suggest an interpolation equation for the distribution of oscillator strengths over the whole Thomas-Fermi frequency range 27 eV ? ?ω ? 27Z² eV. This equation was used to calculate the logarithmic mean excitation energy, which appears in problems of ionization loss of charged particles. The distribution of oscillator strengths in a neutral atom allows the radiative properties of dense matter to be determined.     

Lie algebra contractions on two-dimensional hyperboloid

The Inönü-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E ₂ and eight on E _1,1. The text was submitted by the authors in English.     

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the d-dimensional Euclidean space with \(d\ge 2\). Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected mean-field model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.     

Coulomb deexcitation of muonic hydrogen within the quantum close-coupling method

The Coulomb deexcitation of muonic hydrogen in collisions with the hydrogen atom has been studied in the framework of the fully quantum-mechanical close-coupling method for the first time. The calculations of the l-averaged cross sections of the Coulomb deexcitation are performed for (μp)_n and (μd)_n atoms in the initial states with the principal quantum number n = 3–9 and at relative energies E = 0.1–100 eV. The obtained results for the n and E dependences of the Coulomb deexcitation cross sections drastically differ from the semiclassical results. An important contribution of the transitions with Δn > 1 to the total Coulomb deexcitation cross sections (up to ～37%) is predicted.     

Orbital fractional parentage coefficients for the harmonic oscillator shell model

The Hamiltonian ofn particles moving in a common harmonic oscillator potential has as its symmetry group the unitary groupU(3n) in 3n dimensions,n particle states of the harmonic oscillator shell model can be characterized as bases of irreducible representations (BIR) of the groupU(3n) and of certain subgroups of this group. Use is made of these subgroups for the factorization and calculation of 2, 3, and 4 particle fractional parentage coefficients (fpc) of the harmonic oscillator shell model. Recoupling coefficients for subgroup chains of the symmetric groupS (n) appear as factors in the fpc. These coefficients are analyzed and calculated explicitly. The 2, 3 and 4 particle fpc of the 1s 1p shell configuration are obtained as products of these recoupling coefficients with known reduced Wigner coefficients of the unitary groupU(3) in 3 dimensions. Possible applications are indicated.     

Coulomb Rydberg electron <Emphasis Type="Italic">L</Emphasis>-mixing in collisions with atomic ions

The cross sections of the Rydberg electron L-mixing in a hydrogen atom and a hydrogen-like ion are calculated for slow collisions with atomic ions H*(n, L) + A⁺ = H*(n, L′) + A⁺ without variation of the principal quantum number n. The probability of the L-mixing L → L′ is associated with the quantum interference of the wave functions of adiabatic states, i.e., with the mixing of the time phases of these functions exp(?i∫E _k (t)dt). The effective cross section of such L-mixing for the states with n = 28 are 4–5 orders of magnitude greater than the cross sections determined in previous investigations. The expansion coefficients of spherical Coulomb wave functions in terms of parabolic ones and vice versa, which are necessary for determining cross sections, are calculated on the basis of a comprehensive analysis of the spatial properties of these functions.     

Multiple exchanges in lepton pair production in high-energy heavy ion collisions

As an archetype reaction for pQCD multigluon hard processes in collisions of ultrarelativistic nuclei, we analyze generic features of lepton pair production via multiphoton processes in peripheral heavy ion scattering. We report explicit results for collisions of two photons from one nucleus with two photons from the other nucleus, 2γ + 2γ → l⁺l^?. The results suggest that the familiar eikonalization of Coulomb distortions breaks down for oppositely moving Coulomb centers. The breaking of eikonalization in QED suggests that multigluon pQCD processes cannot be described in terms of collective nuclear gluon distributions. We discuss a logarithmic enhancement of the contribution from the 2γ + 2γ → l⁺l^? process to production of lepton pairs with large transverse momentum; similar enhancement is absent for the nγ + mγ → l⁺l^? processes with m, n > 2. We comment on the general structure of multiphoton collisions and properties of higher-order terms that cannot be eikonalized.     

On the geometric properties of the dispersionless Toda equation

The problem of constructing exact solutions, Noether symmetries, and conservation laws for the (2 + 1)-dimensional dispersionless Toda equation u _xy = exp(?u _zz ) is considered.     

Differential-geometric structures associated with Lagrangians corresponding to scalar physical fields

The paper is devoted to the investigation, using the method of Cartan–Laptev, of the differential-geometric structure associated with a Lagrangian L, depending on a function z of the variables t, x ¹,...,x ⁿ and its partial derivatives. Lagrangians of this kind are considered in theoretical physics (in field theory). Here t is interpreted as time, and x ¹,...,x ⁿ as spatial variables. The state of the field is characterized by a function z(t, x ¹,..., x ⁿ ) (a field function) satisfying the Euler equation, which corresponds to the variational problem for the action integral. In the present paper, the variables z(t, x ¹,..., x ⁿ are regarded as adapted local coordinates of a bundle of general type M with n-dimensional fibers and 1-dimensional base (here the variable t is simultaneously a local coordinate on the base). If we agree to call t time, and a typical fiber an n-dimensional space, then M can be called the spatiotemporal bundle manifold. We consider the variables t, x ¹,...,x ⁿ , z (i.e., the variables t, x ¹,...,x ⁿ with the added variable z) as adapted local coordinates in the bundle H over the fibered base M. The Lagrangian L, which is a coefficient in the differential form of the variational action integral in the integrand, is a relative invariant given on the manifold J ¹ _H (the manifold of 1-jets of the bundle H). In the present paper, we construct a tensor with components Λ⁰⁰, Λ⁰ⁱ , Λ ^ij (Λ ^ij = Λ ^ji ) which is generated by the fundamental object of the structure associated with the Lagrangian. This tensor is an invariant (with respect to admissible transformations the variables t, x ¹,...,x ⁿ , z) analog of the energy-momentum tensor of the classical theory of physical fields. We construct an invariant I, a vector G ⁱ , and a bivalent tensor G ^jk generated by the Lagrangian. We also construct a relative invariant of E (in the paper, we call it the Euler relative invariant) such that the equation E = 0 is an invariant form of the Euler equation for the variational action integral. For this reason, a nonvariational interpretation of the Euler equation becomes possible. Moreover, we construct a connection in the principal bundle with base J ² H (the variety of 2-jets of the bundle H) and with the structure group GL(n) generated by the structure associated with the Lagrangian.     

Die Coulomb-Energien leichter Atomkerne

A compilation of the known data on Coulomb energy differences of isobaric doublets and isobaric triplets is given. Plots of the Coulomb energy differences versus¯Z/A ^1/3 with¯Z=(Z ₁+Z ₂)/2 show an analogous shell structure behaviour for the three series with 2¯Z=A?1,A andA+1 (T=1, 1/2 and 1), i.e. discontinuities at the closed shells atA=4, 16 and 40 and the closed subshell atA=32 and oscillations mainly being due to Coulomb proton-proton pairing energy. A positive energy shift of the lowest states withT=1 of all self-conjugate nuclei withA=4n+2 seems to be indicated by the experimental data. A semi-empirical formula is given that describes the data.     

Drag of ballistic electrons by an ion beam

Drag of electrons of a one-dimensional ballistic nanowire by a nearby one-dimensional beam of ions is considered. We assume that the ion beam is represented by an ensemble of heavy ions of the same velocity V. The ratio of the drag current to the primary current carried by the ion beam is calculated. The drag current turns out to be a nonmonotonic function of velocity V. It has a sharp maximum for V near v _nF/2, where n is the number of the uppermost electron miniband (channel) taking part in conduction and v _nF is the corresponding Fermi velocity. This means that the phenomenon of ion beam drag can be used for investigation of the electron spectra of ballistic nanostructures. We note that whereas observation of the Coulomb drag between two parallel quantum wires may in general be complicated by phenomena such as tunneling and phonon drag, the Coulomb drag of electrons of a one-dimensional ballistic nanowire by an ion beam is free of such spurious effects.     

Symmetry energy of the nucleus in the relativistic Thomas–Fermi approach with density-dependent parameters

We introduce an inhomogeneous term, f(t,x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f(t,x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra s l(2,R). When f is a function of only t, there are five symmetries with the algebra s l(2,R) ⊕ _s 2A ₁. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.     

Electron interaction effects on the thermoelectric power of a quantum dot at T &gt; T<Subscript>K</Subscript>

In the absence of phonon thermal conductivity, we theoretically investigate the output power of an interacting quantum dot thermoelectric setup that is moderately coupled to two electronic reservoirs in the regime T ? T _K . In the noninteracting case, the output power is maximized when the energy level of the dot is around a critical value ε _c . We find that when the energy level of the dot is lower than ε _c , Coulomb interaction can enhance the maximum thermoelectric power that can be achieved by tuning the bias and a wider operating region is also observed. However, when the energy level of the dot is higher than ε _c , Coulomb interaction suppresses the maximum power. Finally when the dot level is around ε _c , Coulomb interaction has minimal effects on the maximum power.     

Non-Fermi liquid theory of quantum Hall effects

Within the Grassmannian U(2N)/U(N) × U(N) nonlinear σ-model representation of localization, one can study the low-energy dynamics of both a free and interacting electron gas. We study the crossover between these two fundamentally different physical problems. We show how the topological arguments for the exact quantization of the Hall conductance are extended to include the Coulomb interaction problem. We discuss dynamical scaling and make contact with the theory of variable range hopping.     

A Harmonic Map of Space-Time

In this paper, we study an n-dimensional space-time (M,g) with Einstein field equation and nondegenerate Ricci tensor. It is shown that taking the Ricci tensor \(\mathit{Ric}=\bar{g}\) as another semi-Riemannian metric on M, the identity map \(i:(M,g)\rightarrow ( M,\bar{g}) \) is a harmonic map. We also obtain a characterization of a vacuum using a differential equation satisfied by the electromagnetic stress tensor on space-time. In addition, we also show that if the Ricci tensor of (M,g) is parallel and the signatures of g and \(\bar{g}\) are same, then the semi-Riemannian manifold \(( M,\bar{g}) \) is an Einstein manifold.     

Optical properties of mechanically compressed K<Subscript>2</Subscript>SO<Subscript>4</Subscript> crystals

The effect of a uniaxial mechanical compression (σ_m ≤ 100 bar) on the spectral dependences (300–800 nm) of the birefringence Δn _i and refractive indices n _i of K₂SO₄ crystals is studied. The electronic polarizabilities, refractions, and parameters (λ_0i , B _1i ) of ultraviolet oscillators of mechanically compressed crystals are calculated. It is shown that the dispersions of Δn _i(λ) and n _i(λ) are normal and sharply increase near the absorption edge. It is found that the uniaxial compression changes the value of the dispersions dΔn _i/dλ and dn _i/dλ rather than their character. It is ascertained that the simultaneous action of the compressions σ_x and σ_z, as well as of σ_y and σ_z, leads to the appearance of new isotropic states in the K₂SO₄ crystal, which manifests itself in the equality of corresponding birefringences. It is shown that the baric dependences n _i(σ) are determined by the change in the density of oscillators (～30%), by the shift of the absorption edge and effective band maximum and by the change in the oscillator strength (～70%).     

Brownian motion on a manifold as a limit of Brownian motions with drift

Let ?ⁿ be n-dimensional Euclidean space and let M ? ?ⁿ be a smooth compact m-dimensional Riemannian manifold (without boundary) embedded in ?ⁿ. By a Brownian motion on M we mean a Markovian process whose transition semigroup is defined by the generator ?½Δ_M, where Δ_M stands for the Laplace-Beltrami operator on M (see, e.g., [2]). This note extends a series of papers in which a measure generated by a Brownian motion on M on the space of trajectories (with values in M) can be represented as the weak limit of measures on the space of trajectories in the ambient space ?ⁿ (see [7–10]). Namely, we claim that a sequence of diffusion processes on ?n which are Brownian motions with drift (in the direction of the manifold) with infinitely increasing modulus converges in distribution to a Brownian motion on the manifold.     

The Study of (<Emphasis Type="Italic">n</Emphasis>, 2<Emphasis Type="Italic">n</Emphasis>) Reaction Cross Sections for Ce and Nd Isotopes up to 20 MeV

Neutron cross section calculations for ¹³⁶Ce(n, 2n)¹³⁵Ce, ¹³⁸Ce(n, 2n)¹³⁷Ce, ¹⁴⁰Ce(n, 2n)¹³⁹Ce, ¹⁴²Ce(n, 2n)¹⁴¹Ce, ¹⁴²Nd(n, 2n)¹⁴¹Nd, ¹⁴⁴Nd(n, 2n)¹⁴³Nd, ¹⁴⁶Nd(n, 2n)¹⁴⁵Nd, ¹⁴⁸Nd(n, 2n)¹⁴⁷Nd, and ¹⁵⁰Nd(n, 2n)¹⁴⁹Nd were done in the incident energy range from 10 to 20 MeV. The calculations were performed using three codes TALYS-1.6 for two-component Exciton model, EMPIRE-3.2 Malta for Exciton model, and ALICE/ASH for the Geometry-Dependent Hybrid (GDH) model. The results of model calculations were compared with the available experimental data and also with the evaluated data in the TENDL-2015 (based on the modified TALYS code), ENDF/B-VII.1 libraries. The calculated cross section data were compared with the available experimental data obtained from EXFOR and also compared with semiempirical formulas around 14–15 MeV. The results of model calculation were found to be in good agreement with the experimental data given in literature and semiempirical data around 14–15MeV.