The scattering of quantized solitons in the non-linear Schrödinger theory

The scattering of quantized solitons in non-linear Schrödinger theory is treated using the collective coordinate method of Gervais, Jevicki and Sakita. The phase shift for soliton-soliton scattering is calculated up to the one-loop level. We find that the quantum correction vanishes. This result coincides in the first two terms of an expansion in $\text{h}\text{?}$ with the exact amplitude calculated from a quantum mechanical N-body problem.     

Pion production and the higher resonances in pion-nucleon scattering

The reaction π + N → 2π + N has been studied in the vicinity of the higher resonances in the pion-nucleon cross section. The Low equation for the production amplitude is transformed into an integral equation by isolating the true one-meson intermediate states and discarding higher order contributions. The only part kept in the inhomogeneous term corresponds to the collision of a pion in the nucleon cloud with the incident pion in the resonant T = J = 1 state, which is simulated by an unstable vector Boson. Crossed terms are neglected and the 2π-N state is described by the static model. The terms kept in the sum over states describe the rescattering (off the nucleon) of one of the outgoing pions. The required off-the-energy-shell elastic scattering amplitude is approximated by the 3-3 resonance formula of Chew and Low. With these simplifications the Low equation for the production amplitude reduces to an easily soluble linear integral equation. The rescattering amplitude, which dominates the inhomogeneous term in the resonance region, is proportional to the 3-3 scattering amplitude of one of the outgoing pions. Although the result provides some support for the conventional isobar model, it is important to note that the largeness of the rescattering term arises from scattering far off the energy shell, rather than by “real” excitation as in the phenomenological isobar model. Quantitative calculations for the $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ channel leading to a p-wave ($\text{J =}\text{3}\text{2}$) and an s-wave pion produce a maximum in the cross section near 600 Mev incident pion lab energy. For a π-π resonance energy squared S = 10, agreement with experiment is obtained with a width about one third that suggested by nucleon electromagnetic structure. In our approximation, the well known 600 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{1}\text{2}$ resonance occurs at the same energy as the 800 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{3}\text{2}$ resonance. It is assumed, but not proved, that the neglected terms are responsible for the splitting of the resonance energies. When this splitting is taken into account, the predicted charge state ratios near the second resonance agree well with existing data. The “third” resonance occurs for the state having two p-wave pions, according to the present theory, although no numerical calculations were made for this case. This point of view suggests that the $\text{F}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, $\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, and $\text{P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ incident channels contribute to the third resonance.     

Calculation of nuclear reaction parameters with the generator coordinate method and their interpretation

Collisions between complex nuclei are described variationally in terms of the GCM with the aim of providing evidence that it is a manageable calculational procedure. The variational principle of Kohn and Kato is used to derive the expression for the K-matrix. The space of scattering states is spanned entirely by antisymmetrized products of shell-model wave functions describing separate clusters; the generator coordinate is the separation between the two shell-model potentials. Scattering boundary conditions are enforced by solving an integral equation for the channel GC amplitude in each open channel separately. The main part of evaluation of collision parameters is performed by calculating double integrals of a form factor between channel GC amplitudes. A theorem about a property of the form factors is proved which allows one to reduce the amount of work needed to calculate double integrals.Application of the method to elastic ³H to ⁴He scattering has shown the feasibility of the calculation.It is shown how an analysis of calculated scattering parameters and corresponding scattering states in terms of quasibound states enables one to make a consistent comparison with experiment and to extract some knowledge of the reaction mechanism.Finally a comparative list of the calculational procedures of the GCM and RGM for reactions is made.     

Path integral methods for inelastic scattering

Corrections to the primitive semi-classical amplitude for multiple inelastic scattering are obtained from a path integral formulation of scattering theory. The path integrals are calculated by making an expansion about a classical orbit describing elastic scattering. Terms are collected to give a series in inverse powers of the reduced mass m of relative motion of the target and projectile. The leading term is the primitive semi-classical amplitude for multiple excitation and explicit formulae are given for the corrections of order $\text{1}\text{m}$. These are calculated in detail for a one-dimensional model. It is shown that some, but not all, of the corrections can be included by evaluating the primitive amplitude with a symmetrized orbit.     

Analysis of intermediate energy nucleon-deuteron elastic scattering

We present a theoretical analysis of a broad range of aspects of intermediate energy nucleon-deuteron scattering. This analysis is based on a multiple scattering approach using knowledge of the deuteron's structure and nucleon-nucleon interactions. Conversely, comparison of this theory with experiment can yield information about low and intermediate energy strong interactions. The relationship of this multiple scattering type of approach to the complementary Faddeev equation approach is discussed. Our program consists of calculating the single scattering and one nucleon exchange contributions in a realistic way then parametrizing the remaining contributions as an S-wave. We argue that the largest error in this analysis is the P-wave part of the double scattering and we give estimates of its size. The single scattering integral is evaluated numerically. Coulomb effects are neglected. We derive the relativistic expressions for single scattering and nucleon exchange and discuss the approximations made, including the off-mass-shell extrapolation of the nucleon-nucleon scattering amplitude. Fits are made to experimental measurements of differential cross sections, nucleon polarizations, and total elastic cross sections. Unitarity is maintained. We tabulate the partial waves for $\text{J ?}\text{5}\text{2}$, L ? 2. They are consistent with recent Faddeev calculations. We argue that with the additional calculation of double scattering the deuteron D-state percentage can be determined to the same relative uncertainty as the differential cross section. Even without the calculation of double scattering, our results indicate a D-state percentage around 8%. In an effort to provide benchmarks for future work, we have tried to be conscientious in describing our techniques and in tabulating numerical results. Comparisons are also made with earlier analyses.     

Optical phase shifts from low-energy neutron cross sections

The Lorentz-weighted average of the S-matrix introduced by G.E. Brown is used in the Feshbach theory of the generalized optical potential to show that the average many-body S-matrix for elastic scattering is exactly equal to the two-body S-matrix of an optical potential. However, the optical potential S-matrix must be evaluated at the complex energy $\text{E}\text{= E + iI,}\text{where}\text{I}$ is the half-width of the lorentzian. The resulting equation for the optical phase shifts (OPS) δ_c, exp $\text{[2iδ}{}_{\mathrm{c}}\text{(}\text{E}\text{)] = 〈S}{}_{\mathrm{cc}}\text{(E)〉}$, holds even when the level spacing D forces the use of an averaging half-width I > D which is comparable to the energy E, providing that the OPS are also evaluated at the complex $\text{E}$ instead of being approximated by their values on the real energy axis at E. An appendix discusses briefly the conditions on a potential necessary for the result obtained by Brown that $\text{〈S}{}_{\mathrm{cc}}\text{(E)〉 = S}{}_{\mathrm{cc}}\text{(}\text{E}\text{)}$ when Lorentz-weighted averaging is used.     

A computation of Adler's β sum rule

We try to check Adler's β sum rule for q² = μ². The integral of β can be divided into two parts: a first part which, if q² = μ², involves only an integral of the experimental πN cross section, and a second one which is a priori of order μ² compared to the first. This second term can be approximated by an explicit calculation of the $\text{N}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>3</mtext><mtext>2</mtext>}}{}^1$ contribution in the framework of Adler's model for the $\text{ν}\text{N}\text{→ μ}\text{N}{}^1$ reactions. The sum rule is not too well balanced and different hypotheses are presented to explain the defect.     

Variation of the local field through the liquid-vapour interface

We derive an integral equation for the self-consistent local field E^loc(z) within an inhomogeneous non-polar fluid, with particular application to the liquid-vapour interface. Approximate solutions are given for the cases of induced atomic dipoles oriented perpendicular and parallel to the interface. For the perpendicular case we relate the average field to the local field and thus obtain an equation for the static dielectric constant ?(z) in terms of the density profile n(z). The departures of the local field from Lorentz form $\text{E}{}^{\mathrm{<mtext>ext</mtext>}}\text{/(1 + (}\text{8}\text{3}\text{)παn(z))}$ and of the dielectric constant from the Clausius-Mossotti form ($\text{1 + (}\text{8}\text{3}\text{)πan(z))/(1 ? (}\text{4}\text{3}\text{)παn(z))}$ are shown to be small. For the parallel case we discuss fringing of the external field and show that the dipoles align themselves with the average field, not the external field. The departure of the local field from $\text{E}{}^{\mathrm{<mtext>ave</mtext>}}\text{/(1 ? (}\text{4}\text{3}\text{παn(z))}$ is shown to be small.     

Effect of the frustration to the ground state energy and entropy of the spin-glass in the random bond Ising model on the square lattice

The energies and the entropies of the spin-glass state and the paramagnetic state at T = 0 of the random-bond Ising mixture of the ferromagnetic bond (concentration p) and the antiferromagnetic bond (concentration 1 ? p) on the square lattice are calculated by the method of the square approximation in the simple version. A self-consistent relation that the partial trace of the normalized density matrix of the square cluster is equal to that of the vertex ($\text{tr}\text{(}{}_{\mathrm{jkl}}\text{?}\text{?}{}^{\mathrm{(4)}}\text{(i,j,k,l) =}\text{?}\text{?}{}^{\mathrm{(1)}}\text{(i)}$) leads to an integral equation for the distribution function of the effective fields, and it is solved exactly at T = 0. The symmetric solution of the integral equation contains the paramagnetic state and two spin-glass states, SG1 and SG2. The energies and the entropies of these states are obtained as functions of the concentration p. The values of the energies per spin at $\text{p =}\text{1}\text{2}$ are -0.75|E_F|, -0.72746|E_F|, -0.72543|E_F|, and correspond to a minimum, a saddle point, and a maximum, respectively, and the values of the entropies are 0, 0.082886k_B, and 0.054457k_B, respectively. The present results are compared with those of the pair approximation and discussed.     

Elastic backscattering of electrons from surfaces

A Monte Carlo algorithm is proposed for calculating the elastic reflection coefficient, η_E, for elements. The algorithm accounts for the multiple elastic scattering of electrons in solids. The calculated values of η_E compare well with the literature data for elements with atomic number up to 47 and at primary energies above 2 keV. The proposed Monte Carlo method makes it possible to determine the functional relation between η_E and the inelastic mean free path, λ. This relation turned out to be non-linear, arid it deviates from a similar relation based on published earlier single elastic scattering model. The deviation is especially pronounced for elements with medium atomic numbers. The calculated function $\text{η}{}_{\mathrm{<mtext>E</mtext>}}\text{=}\text{f}\text{(λ)}$ offers a convenient method for determining the inelastic mean free path. The values of λ derived in the present work from published experimental values of η_E compare very well with the literature data.     

Role of the 12C-core exchange in the 16O-12C elastic scattering

The exchange of the ¹²C core is taken into account explicitly in ¹⁶O-¹²C elastic scattering to explain the backward rise of the angular distributions. The equation for the relative motion of the colliding particles contains the non-local kernel due to the exchange effect. The equation is solved by the iterative method. The angular distributions are calculated at $\text{E}{}_{\mathrm{<mtext>L</mtext>}}\text{(}{}^{16}\text{O}\text{) = 24, 42, 65}\text{and}\text{80}\text{MeV}$. The results are compared with those of finite-range DWBA calculations. The effects of the multi-step transfer process, the α¹ transfer process and of the non-orthogonality term are studied. The equivalent local potential to the non-local kernel is derived under the adiabatic approximation. The resulting local potential shows the parity dependence with the exponentially decreasing behaviour at the asymptotic regions and is found to describe the low-energy scattering phenomena quite well.     

Light-cone expansions,the parton model and virtual γγ scattering

It is shown that it is sufficient to use the light-cone algebra of currents and the algebra of bilocal operators to find the asymptotic behaviour of the γγ scattering amplitude when one (or two) of the photon masses q_1,2² is large, and for an arbitrary value of the energy squared s = (q₁+q₂)². A general form of this asymptotic behaviour is obtained. The box-diagram is dominant over the wide region in $\text{s(μ}{}^2\text{« s « q}{}_1{}^2\text{q}{}_2{}^2\text{/μ}{}^2\text{,μ ～ 1}\text{GeV}\text{)}$ and so the asymptotic amplitude is known completely. It is shown that the parton model of the type of ref.[8] gives the same predictions for the asymptotic behaviour of the γγ amplitude.     

The quantum mechanical toda lattice

The aim of this paper is to establish the exact quantization conditions for the three-body Toda lattice. The Hamiltonian consists of the kinetic energy for three particles in one dimension, and of the potential energy which couples each particle to its two companions through an exponential spring. After eliminating the center of mass motion, one is left with a system of two degrees of freedom and two constants of motion, the total energy E and a third integral A which commute. Nevertheless, no transformation has been found to separate the classical equations of motion or Schrödinger's equation. The wave function is written as a double Laurent series. Its coefficients have to satisfy two sets of recursion relations on a triangular grid where each set insures that we have a simultaneous eigenfunction of E and A. The condition for the convergence of this series can be expressed as the vanishing of a tridiagonal infinite determinant with 1 in the diagonal and the inverse of a third-order polynomial in the first off-diagonals. The coefficients in this polynomial are E and A, and the variable corresponds to a component of the wave vector associated with the wave function. This determinant can be treated exactly as Hill's, and yields the 3 components. The condition for the square integrability of the wave function requires the phase angle of the principal minors to be equal to 0, $\text{π}\text{3}$, or $\text{2π}\text{3}$ according as the representation of the cyclic groups, for each component of the wave vector. But the third condition follows from the two others. The analogy with the corresponding two-body problem is pointed out.     

Intersections of quasi-energy levels in the relativistic theory of multiply charged ions

The relativistic generalization of the quasi-energy method is given with the help of which the influence of an alternating electric field on the energy levels (2 $\text{1}\text{2}$ 0), (2 $\text{1}\text{2}$ 1) of multiply charged H-like ions is investigated. Intersections are found of quasi-energy levels in external fields with frequencies ω₀ ? ΔE_L and definite values of the field amplitude.     

The fixed scatterer approximation and nucleon deuteron scattering

A method of solving the Schroedinger Equation for the scattering from two fixed local potentials is presented. The solutions are used within the framework of the fixed scatterer approximation to perform model calculations of N-D scattering using both effective range theory potentials and a “semi-realistic” potential with a strong repulsive core. For smooth potentials approximate solutions to the fixed scatterer problem are proposed and found to be quite accurate.Other procedures for calculating elastic scattering were compared with the exact fixed scatterer approximation. The results show that the neglect of longitudinal momentum transfer in the Glauber multiple diffraction theory is a severe effect except in the forward direction. For small angle scattering the Glauber prediction for the double scattering amplitude is quite accurate, and does not depend strongly upon either the extent of potential overlap, or the ratio $\text{V}\text{E}$. Comparisons with the Agassi and Gal results for nonoverlapping potentials indicate that the effects of potential overlap are important, at least for the lowest partial waves. Conclusions about the overall importance of off-energy shell effects in nucleon-deuteron scattering, and about the interference of these effects with the determination of the correlation structure of nuclei are not free from ambiguities.     

Effect of magnetic field on heat flow in a polyatomic gas in intermediate pressure range

The effect of a magnetic field, B, on heat flow in a gas in an intermediate pressure range has been studied. The ratio of the heat flow changes in the fields $\text{B}\text{⊥ ?T}$ and $\text{B}\text{6 ?T}$ was found to change nonmonotonically with pressure in N₂ and CO. With the decreasing pressure, a difference is observed between the dependence of heat flow on field orientation and the corresponding angular dependence in the limiting case Kn→0 ($\text{Kn}\text{=}\text{l}\text{/L}$, $\text{l}$ is the mean free path, L is the geometric size). An expression has been obtained for the heat flow in a magnetic field for Kn ? 0.1 from the solution of an integral kinetic equation. In particular, it has been shown that the special features of the Senftleben-Beenakker effect observed with the decreasing pressure arise not only due to spherically symmetric molecule-surface interaction, but also to nonspherical scattering on walls.     

Rigorous estimates of general mayer graphs and applications: II: Long range behaviour of graphs with two root-points

We find the asymptotic behaviour of graphs with two root-points for neutral, polar and ionized systems, we prove that the pair correlation function h(r) decays at least as fast as the potential ?(r) at small activities, when $\text{?(r) ? r}{}^{\mathrm{?n}}\text{(n>3)}$. We also describe a new approximate integral equation for h(r), in this case.     

N-soliton quantum scattering in the sine-Gordon theory

The quantum treatment of soliton scattering in the sine-Gordon model, using the path integral collective coordinate method is generalized to N solitons. The solitions. The first quantum correction to the phase shift of N-soliton scattering is equal to the zero-point energy of an effective multi-soliton Hamiltonian. The energies of the oscillators of this Hamiltonian are shown to be equal to the stability angles of a complete set of solutions of the Schrödinger equation for small fluctuations around a classical N-soliton. Consequently, calculating the fluctuations and their stability angles by the inverse scattering method, we obtain the energies of the oscillators. The first quantum correction to the phase shift (the O(1) part in a development in powers of γ) is evaluated by summing the stability angles. This result is in agreement with the “exact” scattering amplitude conjectured by Faddeev, Kulish and Korepin.     

Semiempirical determination of the atom-surface interaction

We discuss three methods of determining V(z), the lateral average (G = 0 Fourier component) of the atom-surface interaction, from the bound state spectrum found in beam scattering. One method uses the Rydberg-Klein-Rees technique, which yields the width of the potential (i.e., separation of classical turning points) as a function of energy. This method incorporates also the known asymptotic form V ～ ?C₃z^?3, whereC₃ is derived from the polarizability and dielectric function of atom and solid, respectively. The second method uses a hybrid potential, constructed from a Morse potential with shifted zero of energy connected to the asymptotic form,?C₃z^?3, requiring continuity of V and $\text{d}\text{V}\text{d}\text{z}$. The third potential is a Lennard-Jones 3–9 interaction. Results are presented for H and He scattering from LiF and NaF.     

Evidence for assignment of 14.0 MeV state in 11B from 10B(n,n)10B

The differential cross section and polarization for neutrons scattered from ¹⁰B have been measured at E_n = 2.63 MeV (E_x = 13.85 MeV). The results of this experiment and other available neutron scattering data in the range 1 < E_n < 4 MeV are interpreted through a single-level R-matrix calculation over the region 12 < E_x < 15 MeV. Based on this analysis the most probable J^π assignment for the 14.0 MeV level in ¹¹B is $\text{11}\text{2}{}^{\mathrm{+}}$. The anomaly near E_x = 13.1 MeV can only be explained in terms of two overlapping levels having assignments of ($\text{5}\text{2}$, $\text{7}\text{2}$)^? and ($\text{3}\text{2}$, $\text{5}\text{2}$, $\text{7}\text{2}$)⁺.