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Section 14.5: Simple Nuclear Models: Finite and Woods-Saxon Wells
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In describing nuclear material with quantum-mechanical models, there are two standard descriptions for the radial potential:2
The Finite Well: V = ∞ for r = 0, V(r) = −V0 for r < a, and V(r) = 0 for r > a. The potential depth, V0, describes the depth of the well and a describes its extent.
The Woods-Saxon Potential: V = ∞ for r = 0 and V(r) = V0/(1 + e(x−a)/t) for r > 0. The potential depth, V0, 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 V0 to 0. The shape of the Woods-Saxon 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 r2R2(r) = u2(r)) are shown in Animation 1 for the finite well and Animation 2 for the Woods-Saxon 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 click-drag in the energy level diagram on the left to select a level. The selected level will turn red.
2Recall that in the case of a radial energy eigenfunction, the time-independent Schrödinger equation for a spherically-symmetric potential is
[ −(ħ2/2μ) d2/dr2 + l(l + 1)ħ2/2μr2 + V(r) ] u(r) = E u(r) ,(14.9)
where μ is the mass and u(r) = rR(r). Note also that Veff(r) = V(r) + l(l + 1)/2μr2.