Saxon-Woods

Fig. 2.1. Approximate potentials for the nuclear shell model. The solid curve represents the 3-dimensional harmonic oscillator potential, the dashed curve the infinite square well and the dot-dashed curve a more nearly realistic Woods-Saxon potential, V(r) = — V0/[l + exp (r — R)/a ] (Woods Saxon 1954). Adapted from Cowley (1995).

Fig. 2.3. At left, energy levels for a Woods-Saxon potential with Vo — 50 MeV, R = 1.25A1/3 fm and a = 0.524 fm, neglecting spin-orbit interaction. At right, the same with spin-orbit term included. Adapted from Krane (1987).

As it was shown in ref [3], this potential is more suitable in the numerical computation because it does not lead to divergence (under r—> 0) of the spin-orbit interaction -25f(l,j) F /r . In this respect, it differs advantageously from the well-known Woods-Saxon potential. 

The central potential can be a simple harmonic oscillator potential/(r) kr2 or more complicated such as a Yukawa function f(r) (e a,/r) 1 or the Woods-Saxon function that has a flat bottom and goes smoothly to zero at the nuclear surface. The Woods-Saxon potential has the form... 

The nuclear potential is frequently represented by a Woods-Saxon form (Chapter 5) as ... 

Random Phase Approximation Calculations of Gamow-Teller /3-Strength Functions in the A = 80-100 Region with Woods-Saxon Wave Functions... 

We discuss some features of a model for calculation of p-strength functions, in particular some recent improvements. An essential feature of the model is that it takes the microscopic structure of the nucleus into account. The initial version of the model used Nilsson model wave functions as the starting point for determining the wave functions of the mother and daughter nuclei, and added a pairing interaction treated in the BCS approximation and a residual GT interaction treated in the RPA-approximation. We have developed a version of the code that uses Woods-Saxon wave functions as input. We have also improved the treatment of the odd-A Av=0 transitions, so that the singularities that occured in the old theory are now avoided. 

Figure 3 shows p-strength functions calculated with Woods-Saxon wave functions for a sequence of Rb nuclei. Above all but the last of the calculated strength functions, which have the words BETA STRENGTH along the vertical axis, there are plots of experimental results from [KRA83] and [KRA81]. 

The experimental results have the label B (GT) along the vertical axis. The results with Woods-Saxon wave-functions are similar to the results obtained with the oscillator model, and also agree fairly well with experiment. 

Due to circumstances beyond our control, we have only been able to present very few initial results with the Woods-Saxon wave functions here. We hope, however, to explore the model more fully in the near future, in particular to run 97Rb and 99Rb with a more appropriate value of p and to explore... 

We are grateful to J. Dudek and W. Nazarewicz for making a copy of their Woods-Saxon code available to us. 

The Woods-Saxon potential. As a test for the efficiency of our methods we consider the case of the numerical solution of the Schrodinger eqn (64) with / = 0 in the well-known case where the potential V r) is the Woods-Saxon one (65). 

The numerical results obtained for the five methods, with several number of function evaluations (NFE), were compared with the anal5hic solution of the Woods-Saxon potential resonance problem, rounded to six decimal places. Fig. 20 show the errors Err = -logic calculated - analytical of the highest eigenenergy 3 = 989.701916 for several values of NFE (Fig. 21-23). 

We use as potential the well known Woods-Saxon potential (65). 

For some well known potentials, such as the Woods-Saxon potential, the definition of parameter v is not given as a function of x but based on some critical... 

We compute the approximate positive eigenenergies of the Woods-Saxon resonance problem using ... 

In this section we present some numerical results to illustrate the performance of our new methods. Consider the numerical integration of the Schrodinger equation (1) using the well-known Woods-Saxon potential (see 1, 4-6, 7,8) which is given by... 

The true solutions to the Woods-Saxon bound-states problem were obtained correct to nine decimal places using the analytic solution and the numerical results obtained for the six methods mentioned above were compared to this true solution. The results are similar with those of resonance problem. 

Theoretical and numerical results obtained for the radial Schrodinger equation and for the well known Woods-Saxon potential and for the coupled differential equations of the Schrodinger type show the efficiency of the new methods. 

The potential used by Knight was the so-called Woods-Saxon potential, although the conclusions are relatively insensitive to the precise form of the potential. The Schrodinger equation for this system is separable into radial and angular parts, and the wavefunctions are given by ... 

Extensions to the spherical jellium model have been made to incorporate deviations from sphericality. Clemenger [15] replaced the Woods-Saxon potential with a perturbed harmonic oscillator model, which enables the spherical potential well to undergo prolate and oblate distortions. The expansion of a potential field in terms of spherical harmonics has been used in crystal field theory, and these ideas have been extended to the nuclear configuration in a cluster in the structural jellium model [16]. 

The non-spherical part is treated as a perturbation of the system. The spherical part is similar in form to the Woods-Saxon potential, but now contains specific information about atomic positions. 

It has been proven [21] that the spectrum obtained with the Q30 model closely resembles that of the modified harmonic oscillator of Nilsson and Clemenger. In both cases, the effect of the 1(14-1) term is to flatten the bottom of the harmonic oscillator potential, making it resemble the Woods-Saxon potential [18]. 

As shown in [22], the QZO model successfully describes the magic numbers of metal clusters. On the other hand, it is known that metal clusters are well described by Ekardt potentials [16] (for which analytical expressions are lacking), while these have recently been parameterized in terms of symmetrized Woods-Saxon and wine-bottle potentials [19] (for which analytical expressions are known). 

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