A revised version of the Projected Range ALgorithm with numerical solutions

Abstract

In a previous paper a revised version of the Projected Range ALgorithm (PRAL) was presented. These new equations, called the Kent Range ALgorithm (KRAL), consider fully the motion of the ion in three dimensions. In the current paper we review a simple derivation of the KRAL equations starting from the standard LSS integral equations for moments of the range distribution. The resulting 2^nd order ordinary differential equations are difficult to solve using classical numerical techniques. We present a modified set of equations called the Kent Optimised Range ALgorithm (KORAL) which is designed for iterative solution. Finally, we compare results from KORAL and 1^st and 2^nd order PRAL codes with Monte Carlo data from a TRIM code modified to treat targets of infinite extent. Comparisons are performed using consistent nuclear and electronic energy loss models.     

Self-duality and octonionic analyticity of S7-valued antisymmetric fields in eight dimensions

We exhibit two octonionic extensions of the Kalb-Ramond type fields with rank two and four in eigth dimensions. By analogy to the d = 4 (anti-) self-duality of the SU(2) ≈ S³ quaternionic gauge field we consider the respective d = 8 (anti-) self-duality equations for these nonlinear, S⁷-valued antisymmetric fields. By way of an octonionic 't Hooft ansatz these equations reduce to the same generalized Fueter-Cauchy-Riemann equations over S⁸. Explicit (9n + 8) parameter S⁷ → S⁷ mapping solutions, n being a winding number, are found in terms of holomorphic functions of the spacetime octonion. An infinite number of local continuity equations results.     

Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

We consider U(n + 1) Yang–Mills instantons on the space Σ × S ², where Σ is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n + 1) instanton equations on Σ × S ² are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A,?), where A is a gauge potential of the group U(n) and ? is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g > 1, when Σ × S ² becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.     

Field Equations in Quantum Electrodynamics

A formulation of quantum electrodynamics is presented, based on finite local field equations. These Dirac and Maxwell equations have the usual form except that the current operators f(x) and j^μ (x) are explicitly expressed as local limits of sums of non-local field products and suitable subtraction terms. These limits are shown to exist and to yield finite operators in the sense that the iterative solutions to the field equations are equivalent to conventional renormalized perturbation theory. The various invariance properties of the theory, including Lorentz invariance, gauge invariance, charge conjugation invariance, and renormalization invariance, are discussed and related directly to the field equations and current definitions. Initially only the general forms of the currents, based on dimensional arguments, are given. The electric current, for example, contains the (suitably defined) term :A³(x) :.The corresponding field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions ∑, II^μν, Λ^μ, and X^αβγδ (the four-photon vertex function), etc. Application of the boundary conditions ∑(p̀ = m) = ∑′(p̀ = m) = II(O) = II′(O) = II″(O) = Λ(p̀ = m, o) = X(O, O, O, O) = O completely specifies the current operators. Consistency is established by deriving the same equations from rigorous renormalization theory so that their iterative solutions are proved to reproduce the correct renormalized perturbation expansion. The electric current operator is exhibited in a manifestly gauge invariant form and in a form which is manifestly negative under charge conjugation. It is shown, in fact, that much of j^μ (x) can be determined directly from the requirements of gauge invariance and charge conjugation covariance, without recourse to the integral equations. It is suggested that equal time commutation relations can serve to similarly specify the rest of the current.     

The electromagnetic two-body problem in special relativity

A system of two point charged particles is considered. Each particle moves in the electromagnetic field created by the other particle according to Maxwell's equations. A scheme of successive approximations is developed to study the field and the motion of the charges. The field (potentials and intensities) are exapanded in powers of c^?1 using a retarded time coordinate. The variables of the motion (position vectors, velocities, etc) are expanded in powers of c^?1 with coefficients depending on t only. The field is evaluated in the first three approximations. The equations of motion are derived in the same approximations and the corresponding conserved quantities are explicitly given. Thus, the usual assumption of an action-at-a-distance principle is avoided and the original nonlinear integrodifferential equations are reduced to a sequence of linear equations.     

Complete reduction of fermion-antifermion bethe-salpeter equation with static kernel

The Bethe-Salpeter (BS) equation for a $\text{spin}\text{-}\text{1}\text{2}$ fermion-antifermion bound system is considered for the case in which the kernel is static and is the fourth component (i.e., three-scalar part) of a vector potential. Relative time (or relative energy) dependence can be eliminated easily. The 16 BS bispinor amplitudes are reexpressed in the usual way in terms of corresponding tensor amplitudes which satisfy 16 coupled integrodifferential equations. If Lorentz-, parity-, and charge-conjugation invariance are used, these equations can be reduced through a sequence of transformations to single eigenvalue equations, involving scalar and three-vector wavefunctions for singlet and triplet states, respectively. The effective Hamiltonians obtained in these equations are correct to all orders in the coupling constant and have a simple structure, consisting in general of a scalar, a spin-orbit, and a tensor part, which are explicitly exhibited.Although the equations could well be used for consideration of a general particleantiparticle system (e.g., quark-antiquark), for the present only positronium with a Coulomb interaction kernel is treated as an illustrative example. There exists a singlettriplet splitting in leading order mα⁶ ln α even though no spin-spin forces are directly introduced in the kernel. The splitting is calculated in detail in perturbation theory to order mα⁶ ln α and mα⁶.     

立方体AgCl微晶中［Fe(CN)6］4-引入的浅电子陷阱阱深与俘获截面的动力学模拟

为了描述在晶体生长阶段掺入［Fe(CN)₆］^4-的立方体AgCl微晶中 光电子的产生与 衰减过程，建立了一种由三个固有中心和一个浅电子陷阱(SETs)组成的动力学模型，并引出一组微分方程.通过求解微分方程得到与实验结果相符合的光电子衰减曲线及其寿命.调整相 关模拟参数，于常温下得到由［Fe(CN)₆］^4-引入的SETs阱深为0.1 15eV，电子俘获截面为2.136×10^-17cm². 关键词： 光电子 浅电子陷阱 俘获截面 AgCl微晶     

Theoretical analysis of gain characteristics of Dy-doped fiber amplifiers

The rate equations and power evolution equations of Dy³⁺-doped fiber amplifiers (DDFA) are proposed, and numerically analyzed. Based on the numerical analysis, several major characteristics of the fiber amplifier are calculated and compared with those of Pr³⁺-doped fiber amplifier (PDFA). The results show that DDFA can achieve better efficiency and wider gain bandwidth than PDFA.     

Structure of quantum-mechanical relativistic two-body interactions for spinning particles

Relativistic constraint mechanics yields consistent systems of coupled Dirac equations for pairs of spinning particles. We explicitly connect these equations to the Bethe-Salpeter equation of quantum field theory and to the interactions of classical Fokker-Tetrode dynamics (and hence to classical relativistic field theory) to obtain versions of these equations governed by systems of (possibly noncoulombic) relativistic potentials whose detailed structures contain important relativistic effects like correct Darwin interactions. We recast the defining pair of Dirac equations in a number of equivalent but important forms—“external potential,” Sazdjian, hyperbolic, and Breit— and examine their interconnection. Since the potentials in these equations are no more singular than — 1/4r² we are able to solve appropriate versions of them nonperturbatively for the qˉq system to obtain a very good fit to the entire meson spectrum and for the e ⁺ e ⁻ system to calculate the positronium spectrum of QED correct through order α ⁴ .     

Coulomb forces in the three-body problem with application to direct nuclear reactions

Faddeev equations are considered in the case of three charged particles interacting with both separable nuclear two-body interactions and also including Coulomb forces. Modified Faddeev equations with Coulomb Green's functions are introduced. The three-body amplitudes are given into pure Coulomb and distorted-Coulomb amplitudes. Introducing a decomposition in the angular momentum states, a set of three-body integral equations is obtained. The effect of pure coulomb amplitudes is studied in direct nuclear reactions and found to give a large contribution to the cross sections. The three-body integral equations obtained are applied for direct nuclear reactions. The angular distributions for¹²C(⁶Li,d)¹⁶O,¹⁶O(⁶Li,d)²⁰Ne, and¹²C(⁶Li,α)¹⁴N transfer reactions are calculated as well as for the⁶Li elastic scattering on¹²C. From the good agreement between the theoretically calculated and experimental data, better spectroscopic factors are extracted. The effect of including Coulomb forces in the three-body problem is found to improve the results by about 16.26%.     

SU(2) ×U(1) Gauge theory of bosonic and fermionic fields inS 3 ×R space-time

The tetradic Lorentz-gauge invariant formulation of the SU(2) × U(1) theory in S³ × R space-time is presented and the general gauge covariant Dirac-Klein-Gordon-Maxwell-Yang-Mills equations are derived. A direct comparison of these equations to those of the SU(2) × U(1) gauge theory on Minkowskian background points out major differences effectively induced by the minimally coupling to S³ × R gravity.     

Scattering of fermions on the SU(5) colorless magnetic monopole

We discuss the scattering of fermions (d?, u-quarks and e^c) on the SU(5) colorless magnetic monopole. The radial functions of scattering wave are totally separated from the angular parts in the Dirac equations which then reduce to a set of coupled ordinary differential equations. We solve these equations numerically for the d?e^c group with j = 1, and get the result that only the incoming wave of d with j = 1 and j′ = 0 can be scattered into e^c. The cross section is obtained which is intimately connected with the catalysis effect of the monopole for baryon decay.     

测定温度因子的共振X射线动力学衍射法

提出了一种测定双原子晶体中不同原子温度因子的新方法 .利用原子吸收限附近完整晶体共振X射线动力学衍射的特征，得到了关于温度因子的两个方程.方程中的参数可以由实验和理论计算得到，求解这两个方程便可分别求出两种原子的温度因子.将该方法应用于GaAs完整晶体，300K时，B_Ga=0.4573×10^-20m²，B_As=0.7339×10^-20m².该方法可以推广到多原子化合物晶体情形 关键词：     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity (II)

It is shown that all torsion-free vacuum solutions of the model of de Sitter (dS) gauge theory of gravity are the vacuum solutions of Einstein field equations with the same positive cosmological constant. Furthermore, for the gravitational theories with more general quadratic gravitational Lagrangian (F ² + T ²), the torsion-free vacuum solutions are also the vacuum solutions of Einstein field equations.     

Numerical analysis of vortex dynamics and unsteady turbulent heat transfer in lid-driven square cavity

Based on the solution of unsteady Reynolds equations (URANS) closed with the aid of differential equations for shear stresses transport (MSST), the evolution of the vortex structure in a square cavity with a moving lid and an unsteady turbulent heat transfer in the air medium are computed while maintaining constant temperatures of the hot moving and cold motionless walls (Re = 5·10⁴). The phases of the development of dynamic and thermal processes are analyzed.     

Spinor Wave Equation of Electromagnetic Field

In this paper, we give two spinor wave equations of free electromagnetic field, corresponding to the reducibility and irreducibility representations D ¹⁰+D ⁰¹ and D ¹⁰ of the proper Lorentz group, which are the differential equations of space-time one order. The spinor equations are covariant and are equivalent to Maxwell equations.     

On some properties of multidimensional hyperbolic quasi-gasdynamic systems of equations

We study a multidimensional hyperbolic quasi-gasdynamic (HQGD) system of equations containing terms with a regularizing parameter τ > 0 and 2nd order space and time derivatives; the body force is taken into account. We transform it to a form close to the compressible Navier–Stokes system of equations. Then we derive the entropy balance equation and show that the entropy production is similar to the latter system plus a term of the order of O(τ²). We analyze an equation for the total entropy as well. We also show that the corresponding residuals in the HQGD equations with respect to the compressible Navier–Stokes ones are of the order of O(τ²) too. Finally we treat the simplified barotropic HQGD system of equations with the general state equation and the stationary potential body force and obtain the corresponding results for it.