Woods—Saxon potential

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 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... 

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... 

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]. 

Nishioka, Hansen and Mottelson [71] extended also the calculations to very large clusters using a Woods-Saxon potential with the goal to analyze what would happen in the classical limit. The total binding energy E(n) in this approximation is given by the sum of eigenvalues [153] E n) = This... 

The Resonance Problem Woods-Saxon Potential. - Consider the numerical solution of the Schrodinger equation ... 

The numerical results obtained for the thirty-three methods were compared with the analytic solution of the Woods-Saxon potential. Figure 1 shows the maximum absolute error Err — — log10 ivcurate — Ecomputed in the computation of all resonances En, n = 1(1)4, for step length equal to A = The nonexistence of a value indicates that the corresponding maximum absolute error is larger than 1. 

Modified Woods-Saxon Potential Coulombian Potential. - In Figure 2 the maximum absolute error, defined as Err = -log10 Eaccurate - Ecomputed, in the computation of all resonances E ,n — 1(1)4 obtained with another potential in (121), for step length equal to = and for the methods mentioned above, is shown. This potential is... 

The numerical results obtained for the twelve methods were compared with the analytic solution of the Woods-Saxon potential. Figure 4 shows the maximum absolute error... 

The first supershell node occurs at iV as 850), Calculations by Nishioka et al. [17], using a nonselfconsistent Woods-Saxon potential (instead of the spherical jellium model) give N 1000. This node has been observed, although the experiments also show some internal discrepancies the first node is located at IV 1000 in Ref. [15] while it is at iV 800 in [16]. The experimental discovery of supershells confirms the predictions of nuclear physicists. However, supershelis have not been observed in nuclei due to an insufficient number of particles. In summary, the existence of supershelis is a rather general property of a system form by a large number of identical fermions in a confining potential. 

Fig. 9. Three confining potentials for the electrons in a cluster of radius L. One expects that L scales with the size n of the cluster as L = Lon - D is the well depth. The Woods-Saxon potential V[r) = —0/(1 + exp((R — L)d)), where d is the thickness of the boundary layer, is shown as a heavy solid line and R is the distance from the center of the cluster. The equivalent spherical harmonic potential, —O + kR /2 is shown as a light line. Its depth and force constant are given by a fit to the Woods-Saxon potential. The square well potential is shown as a dashed line.

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