On the theory of elastic scattering of slow particles

A theory describing scattering of particles with low energy E from an arbitrary 3D potential is proposed. An exact expression is derived for the s scattering amplitude in the limit E → 0. Phenomenological constant κ appearing in the expression for the resonance scattering amplitude is expressed in terms of the parameters typical of the given potential.     

Generalizing Ribbons and the Twist of the Lattice Ribbon

For a standard or canonical ribbon from differential geometry the topological White’s theorem connects the linking number, writhe and total twist of the ribbon. Here we provide an integral expression, analog to the total twist of a canonical ribbon, that connects linking number and writhe of two curves that do not necessarily form a canonical ribbon. First, we apply this integral expression to derive an expression for the writhe of a polygonal curve. Second, but importantly, we revisit the lattice ribbon. Lattice ribbons were introduced some time ago to enable simulation of physical systems modeled by double stranded polymers. Application of the integral expression yields an algorithm for determining the twist of the lattice ribbon. An interesting relation between writhe of the center line of a lattice ribbon and its linking number follows.     

Non-perturbative solution of the Bloch-Mathieu hamiltonian system

A non-perturbative wavefunction is obtained for the Bloch-Mathieu hamiltonian system. The explicit expression of the Floquet exponent indicates that the wavefunction depends non-analytically on E and quadratically on V₁, both demonstrating the illegitimacy of the perturbation expansion. The energy band is also given by an analytic expression instead of Kramers' indirect construction.     

Comments on the calculation of the Cabibbo angle

An expression for the Cabibbo angle, θ_c, is derived when the mass matrices for the truncated two left-handed doublet model are hermitian and the (1,1) components are zero. The expression contains an unknown parameter which is the difference between the two phases of two complex members in the mass matrices. When the phase difference is 90°, the observed value of θ_c is reproduced. We also show that the above mentioned form of the mass matrices can be realized in the Weinberg-Salam model if the Higgs radiative corrections are ignored.     

On the T-matrix formula for multichannel coulomb scattering

It is pointed out that in scattering theory for Coulomb like forces the conventional expression for the off-energy-shell T-matrix does not lead to the correct on-energy-shell T-matrix. An alternative expression giving the correct on-energy shell limit is discussed.     

Calculation of the off-diagonal elements of a two-photon laser

An expression for the off-diagonal matrix elements of a two-photon laser is presented, starting directly from the equation of motion for ρ_nm. The cavity loss mechanism has been simulated by a single-photon process. In the case of a two-photon laser well above threshold, an expression for the lowest eigenvalue of the ρ_nm has been found. Saturated two-photon emission has been also treated.     

Temperature dependence of the surface free energies of solid chemical elements

The “surface free energy” of metals is αΔH′, where ΔH′ is the internal enthalpy (heat) of atomization and α is a constant (0<α<1). In this Letter an expression (−R ln m/ΔH′) for the temperature dependence of α has been derived and compared to the measured values. It has been found theoretically that this expression is independent of coordination number and that the m=4−10 upper layers of the surface contribute to the constitution of the properties of the metals (in good agreement with experiments).     

Watson representation for the amplitude of lepton-pair production in nucleus-nucleus collisions

An expression for the amplitude of lepton pair production in nucleus-nucleus collisions is obtained in terms of the amplitudes of lepton-nucleus scattering. By using this representation an approximate compact expression for this amplitude valid to terms of the order (Zα)⁵ is obtained.     

New expression for the partition function of a two-dimentional classical plasma

A new expression is derived for the canonical partition function Q of a two-dimentional Coulomb system. As a by-product Q is shown to exist above the critical temperature T_c for the collapse of a pair of opposite point charges.     

On the structure function in the scaling law for partial-wave amplitudes

An expression is obtained for the structure function β(x) occuring in a scaling law for s-channel partial wave amplitudes which has been recently derived from t-channel unitarity at threshold.     

Finite-size corrections to the energy levels of light muonic atoms

An analytic expression is developed in perturbation theory for the nuclear finite-size correction to the s-state energy levels of light muonic atoms. Using first-, second- and third-order perturbation theory, the finite-size corrections of order (Zα)⁴, (Zα)⁵, and (Zα)⁶ are calculated analytically for an arbitrary charge distribution. Application is made to the case of the μ? ⁴He atom, where the error in our finite-size expression is shown to be less than 10ppm.     

Pole expansion determination of the pair correlation function of a simple fluid

An explicit expression for the pair correlation function is given in terms of an expansion in the poles of the structure factor S(q) in the complex q plane. This is derived when the direct correlation function or structure factor is of known analytical form. An illustrative example is given for a hard-sphere-Yukawa fluid in the mean spherical approximation.     

The representation of thermodynamic properties at high pressures

It is sometimes advantageous to have an expression for the Gibbs energy, G_m(T,P), from which one can analytically derive an expression for the Heimholtz energy, F_m(T, V_m). Such an expression is suggested for solid substances and it is shown how expressions for other physical properties can be derived from it.     

Clebsch-Gordan coefficients of the groups SO(p, 1)

An explicit expression for the Clebsch-Gordan coefficients for the coupling of most degenerate unitary representations of SO(p, 1) is obtained.     

A self-consistent test of the rescaling model of the EMC effect

We examine predictions of the dynamical rescaling model for the dueteron. An alternate expression for F₂^D is derived by explicitly including the six-quark state. This lead to a new self-consistence test for dynamical rescaling which is well established. However, discrepancies remain between the experimental F₂^D and the predictions of the model.     

Spatial dependence of the RKKY interaction in a semi-infinite d-dimensional medium

The expression proposed by Larsen for the form of the indirect exchange interaction in d-dimensional metals is proven to be correct for all d ⩾ 1. Corrections due to the presence of a specular surface are obtained and exhibited explicitly for d = 1, 2, 3.     

The XY-model and the self-avoiding walk approximation

The SAW-approximation is calculated for the quantum-mechanical XY-model for spin $\text{1}\text{2}$ in a parallel magnetic field. For this purpose we derive an exact expression for the internal weight factors of the polygon graphs.In this approximation we obtain an analytic expression for the phase-separation line in the (H, T)-phase. The critical exponents are the same as for classical models.     

On the critical temperature for any strength of the electron-phonon coupling

A simple analytical expression describing T_c for any strenght of the electron-phonon coupling λ is derived directly from the Eliashberg equation. The behavior of the energy gap in the limits of strong coupling is studied.     

Influence of the kinematical constraints on the transverse momentum correlation features

The influence of the kinematical constraints on the transverse momentum correlation features in multiparticle processes is studied by means of thedN/(d r ₁ d r ₂) distribution,r ₁ andr ₂ being the transverse momenta of the particles entering in each pair. Using a limited transverse momentum phase space we obtain for high energy reactions a simple analytical expression fordN/(d r ₁ d r ₂). This expression, similar to a phenomenological parametrization proposed for analyzing¯pp annihilation data, served to define a correlation parameter, the non vanishing value of which is due to the energy-momentum conservation constraints.     

Solution of the problem of indirect spin-spin interaction via the phonon field

A new transformation which differs from the famous one of Fröhlich is introduced to transform the hamiltonian of the interacting spin system and phonon system to a representation which leads to the exact expression for the operator H_SS of the indirect spin-spin interaction via the phonon field. The evident form of the operator H_SS is determined.