The Woods-Saxon Potential

The above equation has linearly independent solutions kxjfkx) and kxnfkx), where jiiktc), nfkx) are the spherical Bessel and Neumann function respectively. Thus the solution of equation (1) has the asymptotic form (when x — oo) 

Since the problem is treated as an initial-value problem, one needs yo and y, before starting a two-step method. From the initial condition, yo = 0. The value yi is computed using the Runge-Kutta-Nystrom 12(10) method of Dormand et al. With these starting values we evaluate at Xi of the asymptotic region the phase shift d from the above relation. 

J The Woods-Saxon Potential. - As a test for the accuracy of our methods we consider the numerical integration of the Schrodinger equation (1) with / = 0 in the well-known case where the potential V(r) is the Woods-Saxon one (121). 

One can investigate the problem considered here, following two procedures. The first procedure consists of finding the phase shift d(E) = d for E e [1, 1000]. The second procedure consists of finding those for E, for E e [1, 1000], at which d equals Jt/2. 

The above problem is the so-called resonance problem when the positive eigenenergies lie imder the potential barrier. We solve this problem, using the technique fully described by Blatt.  

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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 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). 

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... 

Consider the numerical solution of the Schrodinger eqn (64) in the well-known case that the potential is the Woods-Saxon potential (65). In order to solve this problem numerically we need to approximate the true (infinite) interval of integration by a finite interval. For the purpose of our numerical illustration we take the domain of integration as x 6 [0,15]. We consider eqn (64) in a rather large domain of energies, i.e. E e [1,1000]. 

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]. 

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. 

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... 

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.

Figure 3.5 Na clusters (a) Radial part of the dominant /-component of the one-electron eigenfunctions for Na2o, from the lower to higher occupied cluster orbitals Is, Ip, Id and 2s. (b) Decomposition of the / = 0 component of the LDA potential (Vq) into the pseudopotential, electrostatic and exchange-correlation contributions, and comparison with the Wood-Saxon potential used in Ref. [1]. Reprinted with permission from [123]. Copyright 1991 American Institute of Physics...

Fig. 3.7 Nuclear potential (a) - the Bloumkvist-Wahlbom potential (b) - the Woods-Saxon potential (see text)...

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