The harmonic force field and ground-state average structure of difluoroborane

An improved harmonic force field of difluoroborane has been calculated using the vibrational wavenumbers and quartic centrifugal distortion constants of four isotopic species. The unidentified vibrational mode ν₅ is predicted at 1049 ± 50 and 775 ± 50 cm⁻¹ for HBF₂ and DBF₂, respectively. The ground-state average structure of HBF₂ has been found to be r_z(BH) = 1.195 ± 0.003 Å; r_z(BF) = 1.315 ± 0.001 Å; (FBF) = 118.0 ± 0.1°.     

The exact solution of an elimination problem in kinetic theory: I. The one-dimensional hard sphere gas

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f⁽¹⁾(z₁; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f⁽ⁿ⁾(z₁;…;z_n; t) in terms of the 1-particle reduced distribution function f⁽¹⁾(z₁; t) for the specified particle by starting with separate expressions for these functions in terms of f⁽¹⁾(z₁; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.     

Finite and infinite systems of nonlinearly-coupled ordinary differential equations,the solutions of which feature remarkable Diophantine findings

We use previous results concerning the time evolution of the zeros x_n(t) of time-dependent polynomials p_N (z;t) or entire functions F(z;t) of the complex variable z, in order to identify lots of nonlinearly-coupled, finite or infinite, systems of Ordinary Differential Equations the solutions of which feature remarkable Diophantine properties.     

Spin-dependent magnetotransport through one or more rings in the presence of the Rashba and Dresselhaus terms of the spin–orbit interaction

Magnetotransport through one or several quasi-one-dimensional rings, in the presence of the Rashba (RSOI) and Dresselhaus (DSOI) terms of the spin–orbit interaction (SOI) and of a magnetic field B, is investigated. The RSOI field and an effective DSOI field are taken as E_R=E_R(sinγ₁e_r+cosγ₁e_z) and E_D=E_D(sinγ₂e_r+cosγ₂e_z), their strengths are denoted by α and β, respectively. The exact one-electron eigenvalues and eigenfunctions are obtained and used to evaluate the transmission as a function of α, β, and of the angles γ₁,γ₂. Because the RSOI term couples the electronic orbit (along the θ direction) with the Pauli matrices σ_z and σ_r while the DSOI term couples it with σ_θ, they affect the electronic spin transport through a ring in distinctly different ways. The resulting transmission shows a considerable structure as a function of the angles γ₁ or γ₂. The same holds for the transmission, versus α or β, with the SOI present only in one arm of the ring and for that through two rings with the same or different radii. Various results of the literature, valid for β=0, are readily recovered. For weak magnetic fields the influence of the Zeeman term on the transmission, assessed by perturbation theory, is negligible.     

Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

Condition for adiabatic passage in the earth’s-field NMR technique

The equation of motion dM/dt=γ M×B(t) is solved for the case B(t)=jB_p(t)+kB_e. The field B_e is a small static field, typically the earth’s field. The field B_p(t) decays exponentially toward zero with time constant T. This decay is produced by an overdamped switching transient that occurs near the end of the rapid cutoff of the coil current used to polarize the sample. It is assumed that B_p is initially large compared to B_e, and that magnetization M is initially along the resultant field B. Exact solutions are obtained numerically for several decay time constants of B_p, and the motion of M is depicted graphically. It is found that for adiabatic passage, the final cone angle β of the precession in field B_e is related to the decay time constant of B_p by β=2e^{−(π/2)ω_eT}. This is confirmed by measurements of the amplitudes of the ensuing free-precession signals for various decay rates of B_p. Near-perfect adiabatic passage (magnetization aligned within 2° of the earth’s field) can be achieved for time constants T2.6/ω_e. For the case of sudden passage, an approximate analytic solution is developed by linearizing the equation of motion in the laboratory frame of reference. For the adiabatic case, an approximate analytic solution is obtained by linearizing the equation of motion in a rotating frame of reference that follows the resultant field B=B_p+B_e.     

Propagating waves and edge vibrations in anisotropic composite cylinders

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f(r) exp i(εz + nθ + ωt) with the radial dependence f(r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number ε or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of ε. For ε specified as real, an eigenvalue problem results for the natural frequencies ω_i for waves propagating along the z-direction of a cylinder of infinite extent. When ε is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding ε as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for ε which may be real, purely imaginary or complex. Complex ε's correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields ε as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venant's principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.     

Resonances in at rest

Data on at rest show two resonant processes: (a) f₀(1370)η,f₀(1370)→σσ and ρρ, (b) η(1440)σ, η(1440)→ηπ⁺π⁻. The branching ratio BR[f₀(1370)→ρρ]/BR[f₀(1370)→σσ]=0.98±0.25 in the mass range available here. Using data on , the ratio Γ_4π/Γ_2π5 for f₀(1370). The effects of the strongly s-dependent width of f₀(1370) are discussed in some detail.The η(1440) is observed decaying to ησ and a₀(980)π, with strong destructive interference between them. In its decay to a₀(980)π, a narrow peak appears in the ηπ mass spectrum, but 30–50 MeV above that usually attributed to a₀(980) and significantly above the KK threshold. This effect is explained naturally by a two-step process: η(1440)→K^*(890)K followed by rescattering of the two kaons through a₀(980) to ηπ above the KK threshold.     

Effect of quartic-phase aberrations on the focal switch of Hermite–Gaussian beams

The focal switch of Hermite–Gaussian beams diffracted at an aperture and subsequently focused by a spherically aberrated lens is studied. Our main attention is focused on the effect of quartic-phase aberrations on the behavior of the focal switch. It is shown that quartic-phase aberrations affect the relative focal shift Δz_f, turning position s_1,t, and relative transition height Δz_sw. Apart from a critical maximum truncation parameter α_c,max, there is a critical minimum truncation parameter α_c,min. Within the region α_c,min<α<α_c,max the focal switch can take place, but quartic-phase aberrations give rise to a decrease of α_c,max-α_c,min in comparison with the aberration-free case.     

Clusters of cycles

A cluster of cycles (or (r,q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r,q)-polycyclic realization P(G) on the plane. An (r,q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q; (ii) all interior faces (denote their number by p_r) are combinatorial r-gons; (iii) all vertices, edges and interior faces form a cell-complex.An example of (r,q)-polycycle is the skeleton of (r^q), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r,q)=(3,3),(4,3),(3,4),(5,3),(3,5). Only for those five pairs, P((r^q)) is (r^q) without exterior face; otherwise, P((r^q))=(r^q).Here we give a compact survey of results on (r,q)-polycycles. We start with the following general results for any (r,q)-polycycle G: (i) P(G) is unique, except of (easy) case when G is the skeleton of one of the five Platonic polyhedra; (ii) P(G) admits a cell-homomorphism f into (r^q); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r^q) and a helicene, otherwise. In [ARS Comb. A 29 (1990) 5], all proper spheric polycycles are given. An (r,q)-helicene exists if and only if p_r>(q−2)(r−1) and (r,q)≠(3,3). We list the (4,3)-, (3,4)-helicenes and the number of (5,3)-, (3,5)-helicenes for first interesting p_r. Any outerplanar (r,q)-polycycle G is a proper (r,2q−2)-polycycle and its projection f(P(G)) into (r^2q−2) is convex. Any outerplanar (3,q)-polycycle G is a proper (3,q+2)-polycycle.The symmetry group Aut(G) (equal to Aut(P(G)), except of Platonic case) of an (r,q)-polycycle G is a subgroup of Aut((r^q)) if it is proper and an extension of Aut(f(P(G))), otherwise. Aut(G) consists only of rotations and mirrors if G is finite, so its order divides one of the numbers 2r, 4 or 2q. Almost all polycycles G have trivial AutG.Call a polycycle G isotoxal (or isogonal, or isohedral) if AutG is transitive on edges (or vertices, or interior faces); use notation IT (or IG, or IH), for short. Only r-gons and non-spheric (r^q) are isotoxal. Let T^*(l,m,n) denote Coxeter’s triangle group of a triangle on S², E² or H² with angles π/l, π/m, π/n and let T(l,m,n) denote its subgroup of index 2, excluding motions of 2nd kind. We list all IG- or IH-polycycles for spheric (r,q) and construct many examples of IH-polycycles for general case (with AutG being above two groups for some parameters, including strip and modular groups). Any IG-, but not IT-polycycle is infinite, outerplanar and with same vertex-degree, we present two IG-, but not IH-polycycles with (r,q)=(3,5),(4,4) and AutG=T(2,3,∞)PSL(2,Z), T^*(2,4,∞). Any IH-polycycle has the same number of boundary edges for each its r-gon. For any r≥5, there exists a continuum of quasi-IH-polycycles, i.e. not isohedral, but all r-gons have the same 1-corona.On two notions of extremal polycycles:

1. We found for the spheric (r,q) the maximal number n_int of interior points for an (r,q)-polycycle with given p_r; in general case, (p_r/q)≤n_int<(rp_r/q) if any r-gon contains an interior point.

2. All non-extendible (r,q)-polycycles (i.e. not a proper subgraph of another (r,q)-polycycle) are (r^q), four special ones, (possibly, but we conjecture their non-existence) some other finite (3,5)-polycycles, and, for any (r,q)≠(3,3),(3,4),(4,3), a continuum of infinite ones.

On isometric embedding of polycycles into hypercubes Q_m, half-hypercubes and, if infinite, into cubic lattices Z_m, : for (r,q)≠(5,3),(3,5), there are exactly three non-embeddable polycycles (including (4³)−e, (3⁴)−e); all non-embeddable (5,3)-polycycles are characterized by two forbidden sub-polycycles with p₅=6.     

Perturbation theory of scattering from irregular bodies

A perturbation solution is derived for the following problem: A time harmonic wave of amplitude ψ, propagating in a medium with wave number k, is incident on an irregular volume V, inside of which the propagation constant k′(r) can be an arbitrary function of | r |, where r is a position vector with origin inside V. The boundary conditions are that both ψ and its normal derivative ∂ψ/∂n may be discontinuous across the surface of V. Special cases of these conditions correspond to acoustic scattering, to B-wave scattering from a dielectric cylinder, or to the classical Dirichlet (ψ = 0) or Neumann (∂ψ/∂n = 0) surface conditions. An integral equation is derived that satisfies the appropriate differential equations both outside and inside the body, and satisfies the boundary conditions as well. This equation is reduced to a set of linear algebraic equations by expansion in a certain basis set and these linear equations are then solved in a perturbation approximation for the case that the surface of the body differs from a sphere or cylinder by a small parameter λ. Comparison is made with formulae in the literature, and except for some minor discrepancies, which are here corrected, there is general agreement.     

Reconstruction of Five-Dimensional Bounce Cosmological Models from Deceleration Factor

In this paper, we consider a class of five-dimensional Ricci-flat vacuum solutions, which contain two arbitrary functions μ(t) and ν(t). It is shown that μ(t) can be rewritten as a new arbitrary function f(z) in terms of redshift z and the f(z) can be determined by choosing particular deceleration parameters q(z) which gives early deceleration and late time acceleration. In this way, the 5D cosmological model can be reconstructed and the evolution of the universe can be determined. PACS: 04.50.+h, 98.80.-k     

Long-range spin correlations in a honeycomb spin model with a magnetic field

We consider the spin-1/2 model on the honeycomb lattice [A. Kitaev, Ann. Phys. 321, 2 (2006)] in the presence of a weak magnetic field h _α ? J. Such a perturbation treated in the lowest nonvanishing order over h _α leads [K.S. Tikhonov, M.V. Feigel’man, and A.Yu. Kitaev, Phys. Rev. Lett. 106, 067203 (2011)] to a powerlaw decay of irreducible spin correlations 《s ^z (t, r)s ^z (0, 0)》 ∝ h _z ² f(t, r), where f(t, r) ∝ [max(t, Jr)]^–4. We have studied the effects of the next order of perturbation in h _z and found an additional term of the order h _z ⁴ in the correlation function 《s ^z (t, r)s ^z (0, 0)》 which scales as h _z ⁴ cosγ/r ³ at Jt? r, where γ is the polar angle in the 2D plane. We demonstrate that such a contribution can be understood as a result of a perturbation of the effective Majorana Hamiltonian by the weak imaginary vector potential A _x ∝ i h _z ² .     

The experimental and theoretical investigation of vibration spectra in ferroelectric semiconductor SbSBrxI1−x crystals

The vapor grown SbSBr_xI_1−x (x=0.1; 0.5; 0.9) crystals with clear mirror surfaces have been used for infrared reflection measurements with Fourier spectrometer. The vibration frequencies along c(z)-axis have been derived from Kramers–Kroning and optical parameters fitting analysis of the experimental reflectivity spectra at T=300 K. The theoretical vibration spectra of SbSBr_xS_1−x (x=0.1; 0.5; 0.9) crystals in paraelectric phase (T=300 K) along c(z)-axis have been determined in quasiharmonic approximation by diagonalization of dynamical matrix. The theoretical vibration spectra of these crystals in a–b(x−y) plane have been determined in harmonic approximation. In this work we discuss the nature of anharmonism in SbSBr_xI_1−x crystals along the c(z)-axis.     

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface ${\mathcal {R} = \mathcal {R}(x,t)}The initial value problem for an integrable system, such as the Nonlinear Schr?dinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface R = R(x,t){\mathcal {R} = \mathcal {R}(x,t)} in the spectral variable, where the space-time variables (x, t) play the role of external parameters. The curves in the x, t plane, separating regions of different genuses of R(x,t){\mathcal {R}(x,t)}, are called breaking curves or nonlinear caustics. The genus of R(x,t){\mathcal {R}(x,t)} is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. The evolution theorem ([10]) guarantees continuous evolution of the asymptotic solution in the space-time away from the breaking curves. In the case of the analytic scattering data f(z; x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function á h(z;x,t){{\Im\,h(z;x,t)}}, which is closely related to the g function. Namely, a break can be caused ([10]) either through the change of topology of zero level curves of á h(z;x,t){\Im\,h(z;x,t)} (regular break), or through the interaction of zero level curves of á h(z;x,t){{\Im\,h(z;x,t)}} with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in the region across the curve.     

An analysis of DC conductivity in terms of degradation mechanisms induced by thermal aging in polypyrrole/polyaniline blends

Measurements of the d.c. electrical conductivity on thermally treated polypyrrole/polyaniline (PPy/PANI) samples, in which the PPy content increased by 10% w.w. starting from pure PANI to pure PPy, followed a σ(t, T) = σ₀(t)exp[−(T₀/T)^1/2] law. This is consistent with a heterogeneous structure of the granular metal type, in which aging is accompanied by the shrinking of the conductive grains causing the decrease of the sample conductivity, a process which is described by the increase of the parameter T₀. The preexponential factor σ₀(t) depends on the intrinsic conductivity of the grains and geometrical factors affecting the carrier paths through the energy barriers, as are the grain size distribution and the mean volume occupied by the conducting grains in the material. It was found that for the samples as a whole the thermal aging law, which predicts ln σ(t, T)∝t^1/2 is followed for a given temperature T, where t is the time of the thermal treatment, in accordance with a granular metal type structure. On the other hand, the preexponential factor σ₀(t) decreases with the aging, following a different law [σ(t = 0, T)−σ(t, T)]/σ(t = 0, T)∝t^1/2, where σ(t = 0, T) is the initial value of σ₀(t), that of the fresh sample. This law reveals an aging caused by a degradation proceeding into the interior of the grains in a diffusion-like manner. So, the two different laws of aging, one from T₀ and the other from σ₀, reveal that the aging does not simply reduce the size of the grains, but affects their interior, this degradation decreases with depth.     

A family of integrable evolution equations of third order

We construct a family of integrable equations of the form v_t = f(v; v_x; v_xx; v_xxx) such that f is a transcendental function in v; v_x; v_xx. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential substitutions involves geometry of a family of genus two curves and their Jacobians.     

Simulation of flux dynamics in a superconducting bulk magnetized by multi-pulse technique

We have simulated the time and spatial dependence of local field B_z(t, r) and temperature T(t, r) on the superconducting bulk during pulsed field magnetization (PFM) using the finite element method (FEM). A modified multi-pulse technique with step-wise cooling (MMPSC) was performed to the cryo-cooled bulk, which was experimentally confirmed to the effective PFM technique to enhance the trapped field B_z higher than 5 T. In the simulation, the B_z value at the center of the bulk surface was enhanced at the 2nd stage of the MMPSC method, in which the results of the simulation reproduced the experimental ones. The enhancement of B_z results from the reduction in the temperature because of the already trapped flux in the bulk at the 1st stage of the MMPSC method.     

Energy, momentum, and force in classical electrodynamics: Application to negative-index media

The classical theory of electromagnetism is based on Maxwell's macroscopic equations, an energy postulate, a momentum postulate, and a generalized form of the Lorentz law of force. These seven postulates constitute the foundation of a complete and consistent theory, thus eliminating the need for physical models of polarization P and magnetization M — these being the distinguishing features of Maxwell's macroscopic equations. In the proposed formulation, P(r, t) and M(r, t) are arbitrary functions of space and time, their physical properties being embedded in the seven postulates of the theory. The postulates are self-consistent, comply with special relativity, and satisfy the laws of conservation of energy, linear momentum, and angular momentum. The Abraham momentum density p_EM(r,t) = E(r,t) × H(r,t) / c² emerges as the universal electromagnetic momentum that does not depend on whether the field is propagating or evanescent, and whether or not the host media are homogeneous, transparent, isotropic, linear, dispersive, magnetic, hysteretic, negative-index, etc. Any variation with time of the total electromagnetic momentum of a closed system results in a force exerted on the material media within the system in accordance with the generalized Lorentz law.     

Large-scale amplification of a magnetic field by turbulent helicity fluctuations

In this paper the procedure of large-scale averaging of the magnetic-field diffusion equation with the α-term curlα(r,t)B(r,t) is used to show that a nonuniform distribution of the turbulent helicity fluctuations (more precisely, the fluctuations of the coefficient α) with a zero average value gives rise to large-scale amplification of the initial magnetic field. A detailed study is carried out of the dependence of the resulting large-scale α effect on the characteristics of the correlator 〈〈α(r, t)α(r″,t″)〉〉 in a rotating medium with a nonuniform distribution of the angular velocity ω=ω(ρ,z) (ρ is the distance for the rotation axis z). The effect of helicity fluctuations and the diffusion coefficient on the turbulent diffusion process is also investigated. Zh. éksp. Teor. Fiz. 116, 85–104 (July 1999)