Particle diffraction by a thin rigid crystal lattice: A formal approach

The diffraction process of a particle by a thin rigid crystal is considered. An integral equation is derived for the particle wave function φ which is quite suitable to obtain physical and mathematical properties. A class of potentials is presented for which the integral equation can be solved by means of the Fredholm theory. The convergence of the Born series for φ is studied, as well as the existence and convergence properties of the transmission and reflection amplitudes T^±. Results are given about φ and T^±: (i) at high energies, (ii) at those special energies such that new diffracted beams appear, and (iii) at glancing incidence on the crystal. Analyticity properties of T^± as functions of the energy are derived and analytic representations for them are presented. The diffraction process when the particle is being simultaneously accelerated by a uniform electric field is also considered. Finally, the generalization to the case of an imperfect thin crystal is treated.     

Vacuum energy ingϕ d /4 -theory forg→∞

In the nonlocalg? _d ^/4 (d≧1) and localg? ₂ ⁴ theory theS-matrix is obtained in a form of the functional integral which is proved to exist. The density of vacuum energy $$E(g) = - \mathop {\lim }\limits_{V \to \infty } \frac{1}{V}\ln \left\langle {0\left| {S_V (g)} \right|0} \right\rangle $$ is investigated. It is proved to be analytic through the whole complexg-plane except for the negative real axis and pointg=0. Its asymptotic behaviour forg→∞ is found.     

The Bethe-Goldstone equation with singular interactions: A fully off-shell solution for the boundary condition model

The mutual interaction of a pair of fermions imbedded in a many-body system of identical particles when they are excited out of the filled Fermi sea, is studied via the T-matrix or transition amplitude specified by the Bethe-Goldstone (BG) equation. The role of the bare two-body interaction is emphasised, and in particular the consequences are elucidated of whether the potential is “well-behaved” (nonsingular) or not. The properties of the BG T-matrix, including generalized orthonormality and completeness relations, are derived both for nonsingular potentials and for singular potentials containing an infinite hard core. General analytic properties are exploited to derive relations that express the fully off-shell BG T-matrix purely in terms of the half-shell amplitude (and the properties of any possible bound states in the medium). The general formalism is illustrated by deriving exact analytic expressions for the fully off-shell BG T-matrices for a pair of particles with equal and opposite momenta interacting via either of two singular model interactions; namely, the pure hard-core interaction and the boundary condition model. Results for both models are expressed in terms of the solution to a simple one-dimensional Fredholm integral equation. The analytic properties of the solutions are discussed and exploited to prove both their uniqueness and that they satisfy the various general relations derived. To our knowledge, these results represent the first exact nontrivial solution to the fully off-shell BG equation for any local potential, or singular limiting case thereof.     

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.     

A path-integral formulation of the Pauli equation using two real spin coordinates

Feynman's path-integral quantum-mechanical formulation is generalised for particles of spin 1/2. In the one-particle case, the path-integral formulation uses paths in a Euclidean real five-dimensional space, two coordinates (u, v) being reserved for spin. The path integral is proven to correspond exactly to the Pauli equation. A canonical density-matrix formulation is also dealt with. Basic ideas are to start with differential spin operators instead of the Pauli matrices and apply them to functions ψ=ψ ₁(r,t)u +gy ₂(r,t)v whereψ ₁,ψ ₂ are the Pauli wave functions. Then a ‘nilpotent’ spin ‘kinetic-energy’ term is added to the Hamiltonian. This enables us to find a non-matrix spin-dependent Lagrangian which is used as usual in the action of a path integral of the Feynman type. Integral relations are derived from which the path integral can be transformed into components of the Pauli matrix Green's function (propagator) or the canonical density matrix. As an example, a path-integral calculation of the normal Zeeman splitting is carried out.     

A nonperturbative foundation of the Euclidean–Minkowskian duality of Wilson-loop correlation functions

In this Letter we discuss the analyticity properties of the Wilson-loop correlation functions relevant to the problem of soft high-energy scattering, directly at the level of the functional integral, in a genuinely nonperturbative way. The strategy is to start from the Euclidean theory and to push the dependence on the relevant variables θ (the relative angle between the loops) and T (the half-length of the loops) into the action by means of a field and coordinate transformation, and then to allow them to take complex values. In particular, we determine the analyticity domain of the relevant Euclidean correlation function, and we show that the corresponding Minkowskian quantity is recovered with the usual double analytic continuation in θ and T inside this domain. The formal manipulations of the functional integral are justified making use of a lattice regularisation. The new rescaled action so derived could also be used directly to get new insights (from first principles) in the problem of soft high-energy scattering.     

Inverse solutions to the inelastic scattering problem

Exact inverse solutions to the integral equation $\text{φ(r}{}_{\mathrm{s}}\text{|r}{}_0\text{, k) = ?}{}_{\mathrm{D}}{}^3\text{f (r, ω)g(r|r}{}_0\text{, k)g(r|r, k)d}{}^3\text{r}$, where g(r|r_j, k); j = 0 or s is the free space Green function, are derived in plane and cylindrical coordinates for fixed ω. These solutions allow an inelastic scattering potential f(r, ω) which is of compact support r ? $\text{D}$³ to be recovered from scattering data collected over the surfaces of a plane and cylinder respectively.     

Phase transition in a linear chain of classical spins with a logarithmic nearest neighbour pair potential

We present the complete calculation of the partition function and correlation functions of a linear array of classical spins coupled by a nearest neighbour logarithmic pair potential. In the case of a ferromagnetic coupling there occurs a phase transition at T_c > 0. The critical exponents of the specific heat C and the magnetic susceptibility χ (in the absence of an external field) are shown to have the non-classical value α = 2 and classical value γ = 1 respectively. The underlying mathematical mechanism of the phase transition is the complete degeneracy of all the eigenvalues of the corresponding integral equation (Kac's mechanism). Below T_c the partition function becomes complex. For antiferromagnetic coupling the free energy is analytic in the whole temperature range and so no phase transition occurs in this case.     

Analytic continuation of massless Feynman amplitudes in the schwartz spaceI'

The main result of the paper is a theorem on the existence of an analytic continuation of massless Feynman amplitudes as tempered distributions in external momenta. Propagators are considered with arbitrary (complex) powers λ_l and the analytic continuation is constructed in the powers λ_l from Speer's domain of absolute convergence of Feynman integral to the whole complex space. Sets are found which contain ultraviolet and infrared series of poles of the analytic continuation. In the proof amplitudes with propagators having cut-offs at the origin or at infinity are introduced and their analytic continuations are investigated. The introduction of the cut-offs permits to divide ultraviolet and infrared series of poles of massless Feynman amplitude.     

Zusammenhang zwischen Paarkorrelationsfunktion und Paarpotential für flüssiges Aluminium

We solved the Percus-Yevick and the hypernetted chain equation for liquid aluminium using a pair potential given byHarrison and investigated how the radial distribution functiong(r) is affected by changing the shape of the potential. It was also tried to calculate a pair potential from experimentalg(r) and structure factor data by the PY and the HNC approximation.     

Analytic solution of the RISM equation for symmetric diatomics with Yukawa closure

We present the site-site direct correlation function c(r) for a fluid of hard diatomic symmetric molecules obtained from Monte Carlo simulation data via the RISM integral equation. This c(r) ensures that the site-site correlation function given by the RISM equation is exact, and thus provides a basis for critically examining the usual closure for the RISM equation. As an example of an improved closure we present the analytic solution of the RISM integral equation with a Yukawa closure for c(r).     

The exclusive interaction

It is proposed that there exists a fifth basic interaction from which the Pauli exclusion principle can be deduced. Such an interaction would affect only particles of half integral spin, whose total number is less than the number of particles which participate directly in the strong interaction. This is the reason for choosing the name exclusive interaction, which, like the strong interaction, would be expected to obey a relatively large number of conservation laws and symmetry principles. The tentative mathematical expression for this interaction $$V(r,\sigma ) = \frac{g}{r}\cos (cr)\exp ({{ - r} \mathord{\left/ {\vphantom {{ - r} R}} \right. \kern-\nulldelimiterspace} R})\left[ {\frac{{1 + \cdot }}{2}} \right]$$ is suggested by a discussion of fermion-fermion scattering in the Born approximation. It is found that a potential energy of this form may be used instead of antisymmetrizing the wave function in the manner required by the Pauli principle. An investigation of the binding of helium-like atoms is expected to lead to a determination of the relative strengthg/?c and the rangeR of this interaction.     

On the existence of Feigenbaum's fixed point

We give a proof of the existence of aC ², even solution of Feigenbaum's functional equation $$g{\text{(}}x) = - \lambda _0^{ - 1} g{\text{(}}g( - \lambda _0 x)),g{\text{(0) = 1,}}$$ whereg is a map of [?1, 1] into itself. It extends to a real analytic function over ?.     

An Integro-Differential Equation for Bose�CEinstein Condensates

We present an integro-differential equation describing systems with large number of bosons. The new equation includes the two-body correlations exactly into account and the kernel has a simple analytic form. The equation has been employed to obtain results for ${A\in\{10,100\}}$ ⁸⁷Rb atoms confined by an externally applied trapping potential V _trap(r). Our results are in excellent agreement with those of the Potential Harmonic Expansion Method (PHEM) and the Diffusion Monte Carlo (DMC) method.     

Exact asymptotic solution of the Della Sala-Görling integral equation for the exchange-only potential for Be-like atomic ions at large Z

Della Sala and Görling (DSG) have written an integral equation for the exchange-only potential Vx(r) in terms of the Dirac density matrix. Here, an exact asymptotic solution of this integral equation is presented, for the ground state of Be-like atomic ions, in terms of γ(r,r^′) plus the 2s HOMO orbital. In the large Z limit of such ions, the DSG integral equation corrects the asymptotic form −e²/r of Vx(r) by exponentially decaying terms. This amounts to setting the polarizability equal to zero.     

Density of states of two interacting holes in a solid

If two holes are suddenly created in the same band and at the same atomic site e.g. by an Auger process in a solid, their density of states N(ω) will depend on their Coulomb interaction. In a tight binding model, we present the exact N(ω), in the limit of zero bandwidth. In the case of a general band, we give an exact integral equation that allows calculating N(ω) once the single electron density of states is known. The interaction is shown to produce a characteristic distortion of N(ω) and hence of the Auger spectrum.     

Functional integral representation of the nuclear many-body grand partition function

A local functional integral formulation of the nuclear many-body problem is proposed which is a generalization of the method previously developed [Ann. Phys. (N.Y.)148 (1983), 436; Phys. Rev. C24 (1981), 1029]. Its most interesting feature is that it allows an expansion of the many-body evolution operator around any arbitrary mean-field which takes into account the pairing correlations between the nucleons. This is explicitly illustrated for the nuclear many-body grand partition function for which special attention is paid to the static temperature-dependent Hartree-Fock-Bogolyubov (H.F.B.) approximation. Indeed, the temperature-dependent H.F.B. configuration appears to be the optimal choice from a variational point of view among all the possible independent quasi-particle motion approximations. An analytic approximation of the energy level density p(E,A) is given using explicitly the arbitrariness in the choice of the mean-field and a possible numerical application is proposed. Finally, a new compact formulation of our functional integral that might be useful for future Monte Carlo calculations is proposed.     

Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

The theory of liquids at equilibrium

No rigorous calculation of the properties of a liquid has yet been made from a knowledge of the forces between its molecules. This review describes the method by which most solutions have been sought and the progress that has been made in the last five years.

The equilibrium properties of a simple liquid can be expressed in terms of the pair distribution function g(r), which measures the probability of finding a second molecule at distance r from a first. Kirkwood, and Born and Green showed that an approximate calculation of g(r) can be made by solving an integral equation. This equation, its solutions and their imperfections are described briefly.

More recently it has been found convenient to write the correlation function, h(r)=g(r) -1, as the sum of two terms, a direct term, c(r), and an indirect term. Four recent approximations can be described and compared by the form they assume for the direct correlation function, c(r). The most promising of these is the approximation of Percus and Yevick and its quantitative predictions are discussed in some detail.