Level scheme
even l, parity +odd l, parity −Fermi level
Bar length shows how full each level is. [n] is the running total of nucleons; circled numbers mark the magic numbers 2, 8, 20, 28, 50, 82, 126 when a level closes on one.
Bound levels
How the levels are computed
With ψ = [u(r)/r] Ylm (or the spin–angle function [Yl⊗χ1/2]jm when spin–orbit is on), the Schrödinger equation for a central potential becomes one radial equation for each (l, j), on 0 < r < ∞ with u(0) = 0:
⟨l·s⟩ = ½[j(j+1) − l(l+1) − ¾] is +l/2 for j = l + ½ and −(l+1)/2 for j = l − ½. VC is the potential of a uniformly charged sphere of charge (Z−1)e and radius R, used for protons only. The reduced mass is μ = mA/(A+1).
The equation is discretized on a grid of spacing 0.01 fm out to R + 30 fm, which turns it into a symmetric tridiagonal matrix. The number of its eigenvalues below a trial energy E equals the number of negative pivots in the elimination of (H − E). That is the discrete form of counting the nodes of the trial solution, so the n-th eigenvalue is the energy at which the n-th sign change appears. Bisection on that count finds every bound level to 10−8 MeV. The wavefunction is built by recurrence inward from the far wall and outward from the origin, joined at the outer turning point. The n in 1s, 2s, 3s is therefore 1 + the number of radial nodes.
Default parameters are the Bohr–Mottelson global set: V0 = 51 ∓ 33(N−Z)/A MeV (− for neutrons, + for protons), r0 = 1.27 fm, a = 0.67 fm, Vls = 0.44 V0. Krane (Eq. 5.1) quotes R = 1.25A1/3 fm, a = 0.524 fm and V0 ≈ 50 MeV. A single potential is used for all levels of the chosen nucleus, and occupancy follows the extreme independent-particle picture: no pairing, no deformation.
The ground-state Jπ box solves both the proton and the neutron levels and fills each in order. An even–even nucleus is 0+. An odd-A nucleus takes the j and parity (−1)l of its odd nucleon. In an odd–odd nucleus the parity is the product of the two odd nucleons' parities, J can be anything from |jp − jn| to jp + jn, and the empirical rule that the two intrinsic spins lie parallel picks one value (Krane §16.1): 38Cl, with πd3/2 ⊗ νf7/2, comes out 2−. The rule is for two particles; when an odd nucleon is a hole in a nearly full orbit (as in 40K) the tool shows only the allowed range.
- Lecture notes (Yildiz), Lecture 5 (radial equation, Veff) and Lecture 8 (shell model).
- Krane, Introductory Nuclear Physics, §5.1, Eq. 5.1 and Figs. 5.4–5.6; §16.1 for odd–odd nuclei.
- A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. I (1969), Ch. 2.
- Python version of the same solver:
ws_levels.py(Recitation 5 folder).