Nonsingular Potentials Research Articles

Deduction of non-singular potentials from scattering phase shifts (II). Charged particles

A practical method is given for charged-particle, elastic scattering, enabling phase-equivalent, non-singular potentials to be deduced from η l ( k) curves. The method is an extension of earlier work on neutral-particle scattering, employing the distorted plane wave approximation. The approximate potential is obtained as a finite series of short-range functions, such as the exponential function, accurate to within several percent. An iterative method previously employed enables the error in the potential to be reduced successively by factors of 2 or more per cycle, where the data so justify it.

VELOCITY-DEPENDENT NUCLEON–NUCLEON POTENTIALS AND SATURATION IN NUCLEAR MATTER

Recently, nonsingular velocity-dependent potentials have been constructed which fit the the two-nucleon data, but do not give saturation in nuclear matter at reasonable densities. In this paper, we have asked what features a potential should have in order to give saturation, and we have found that the short-range wave-function distortion (defined in the text) is important. Reasons are given for the failure of the earlier potentials to saturate, and a new velocity-dependent potential is proposed which gives results similar to the standard hard-core potential model. We speculate on the usefulness of such potentials for future calculations of nuclear properties.

Calculation of regge poles by continued fractions - I

An expansion of the Regge trajectories for large energy is derived for singular and nonsingular potentials. The Pade approximant is introduced in order to sum these asymptotic expansions. Further calculations for a Yukawa potential are reported with emphasis on the analytic properties of the trajectories. All the trajectories investigated have singularities not present in theS-matrix itself. We conclude that one can analytically continue from one trajectory to another.

Distorted wave expansion for scattering problems

The properties of a distorted wave method for the scattering wave function ψ l ( r) and phase-shift δ l are examined. For |δ l|< 1 2 π, tan (1) δ l reduces to a series whose leading term is the first Born approximation for tan δ l , whereas at low energies an approximate form of the shape-independent formula is obtained. For weak potentials (| δ l ( k)|≦ π for all k), the method is valid for all k, but for strong potentials the assured range of validity is |δ l|< 1 2 π , provided the energy is sufficiently high. The overall accuracy is 2–3 % for δ l , using non-singular potentials. Potentials singular at r = 0 may give rather lower accuracy. Higher approximations in the form of series expansion for ψ l ( n) ( r) and tan δ l ( n) are developed, which reduce in the limit n → ∞ to the corresponding Born series, convergent for |δ l < 1 2 π . However, the rate of convergence for small n is much better than for the Born expansions, particularly for potential strengths near the radius of convergence. The expansions are also convergent for weak potentials when π ≧|δ l|> 1 2 π , whereas the Born series are not. It is concluded that ψ l (1)( r) and tan (1) δ l are approximate summations of the corresponding Born series for |δ l|< 1 2 π , and therefore include contributions from the Born expansions of all orders. The summations for ψ l ( n) ( r) and tan δ l ( n) are also valid for weak potentials for π ≧ |δ l| ≧ 1 2 π , having the character of Born expansions in inverse powers. The method is illustrated on zero energy neutron-proton scattering.

An improved approximation for scattering problems (II)

A method of calculating scattering phase-shifts developed in a previous paper 1), hereafter referred to as (I), is improved for long-tailed potentials by using a better form factor g l ( r), which takes into account distortion of the wave function by the potential tail. A numerical comparison for 3S and 1S neutron-proton scattering is made for wells of gaussian, exponential and Yukawa shape. Comparison of results with those of (I) shows a reduction in error of more than half for non-singular potentials for which satisfactory g l ( r) can be constructed, such as the exponential potential. Singular potentials such as the Yukawa well give the least satisfactory results, as satisfactory g l ( r) cannot be constructed in the usual simple way, due to neglect of logarithmic terms in the distorted wave function.

Variational Wave Function for Nuclear Matter

A variational wave function for large nuclear systems is proposed. The function is written as a product of plane wave and deuteron-like functions. The exclusion principle is conveniently introduced by explicitly cutting out overlap between the plane wave and pair states in a manner equivalent to antisymmetrization. The method is limited to nonsingular potentials. Numerical calculations with saturating Yukawa potentials yield a correlation energy of 1.1 Mev per particle at low density, decreasing with increasing density. The method of introduction of the exclusion principle is applied to a qualitative discussion of elastic scattering by a complex projectile, e.g., an alpha particle. The distortion of the projectile wave function due to the exclusion of low-momentum components reduces the depth of an effective optical potential with respect to the sum of the potentials seen by the component nucleons.

Asymptotic expansions of phase shifts at high energies

One gives asymptotic expansions of the phase shifts for large values of the momentum, considering central static potentials only. For non-singular potentials a recurrence relation which allows the calculation of terms of every order in 1/p is established. For potentials which are singular as 1/r at the origin, or regular for all angular momenta, the expansions are explicitly written neglecting terms of the order 1/p5.