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Section 14.5: Simple Nuclear Models: Finite and WoodsSaxon Wells

Animation 1: Finite Well  Animation 2: WoodsSaxon
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In describing nuclear material with quantummechanical models, there are two standard descriptions for the radial potential:^{2}
The Finite Well: V = ∞ for r = 0, V(r) = −V_{0} for r < a, and V(r) = 0 for r > a. The potential depth, V_{0}, describes the depth of the well and a describes its extent.
The WoodsSaxon Potential: V = ∞ for r = 0 and V(r) = V_{0}/(1 + e^{(x−a)/t}) for r > 0. The potential depth, V_{0}, describes the depth of the well, a describes its extent, and t describes the thickness of the potential. This thickness describes a distance associated with the potential energy going from V_{0} to 0. The shape of the WoodsSaxon potential is due to the gradual change in the charge density of the nucleus from ρ to 0, which occurs over an associated distance, t, the surface thickness
These models, and their solutions (u(r), R(r), and r^{2}R^{2}(r) = u^{2}(r)) are shown in Animation 1 for the finite well and Animation 2 for the WoodsSaxon potential. The angular momentum quantum number, l, can be changed and the effect can be seen on the effective potential and the resulting wave functions. To see the other bound states, simply clickdrag in the energy level diagram on the left to select a level. The selected level will turn red.
^{2}Recall that in the case of a radial energy eigenfunction, the timeindependent Schrödinger equation for a sphericallysymmetric potential is
[ −(ħ^{2}/2μ) d^{2}/dr^{2} + l(l + 1)ħ^{2}/2μr^{2} + V(r) ] u(r) = E u(r) ,(14.9)
where μ is the mass and u(r) = rR(r). Note also that V_{eff}(r) = V(r) + l(l + 1)/2μr^{2}.
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