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3.2. Results 17
12 C 16 O
proton neutron proton neutron
1s 1/2 36.47 MeV 39.98 MeV 35.09 MeV 39.22 MeV
1p 3/2 11.58 MeV 14.72 MeV 15.29 MeV 19.13 MeV
1p 1/2 - - 7.94 MeV 11.67 MeV
B.E./nucl. 4.90 (7.42) MeV 5.85 (7.72) MeV
Table 3.2 Single particle energies in 12 C and 16 O for all occupied proton and neutron
levels. The theoretical average binding energy per nucleon is compared to the experimental
energy (in brackets).
potential in a Woods-Saxon form, the Dirac equations can be solved iteratively
using a shooting point method, where one first integrates outward from small r to a
chosen match radius r m (the so-called shooting point), and then integrates inward
from large r to r m . Analytic solutions to the equations in the regions of large and
small r allows one to impose the proper boundary conditions. The solutions are
scaled so that G α is continuous at the shooting point r m , and the wave function is
then normalized to unity. The discontinuity in F α is then used to adjust the energy
eigenvalue according to
δE α = −MG α (r m )[F α (r + m) − F α (r − m)] . (3.8)
This shooting procedure is repeated until |δE α | is less than a predefined tolerance.
Once the nucleon wave functions are obtained, the densities and the meson fields can
be reevaluated. Boundary conditions of exponential decay at large r, and vanishing
slope for the fields at the origin were imposed.
For the 12 C and 16 O nuclei, the newly developed C-code SOR performed all integrations
for a radial extension of the nucleus of 20 fm and a stepsize of 0.01 fm.
The coupled Dirac equations were solved for a shooting point lying at 2 fm using
a fourth-order Runge-Kutta algorithm. As a convergence criterion we imposed a
tolerance level as small as 0.001 MeV on all single-particle energy levels.
In Table 3.2 the calculated neutron and proton single-particle energies in 12 C and
16 O are presented. A comparison of single-particle energy levels with experimental
data is very ambiguous, so we only compare the average binding energy per nucleon
with the empirical binding energies. The total energy of the system E is given by
E = ∑ α occ
ɛ α (2j α + 1) − 1 2
∫
d⃗r [−g s φ 0 (r)ρ s (r)
+g v V 0 (r)ρ B (r) + 1 2 g ρb 0 (r)ρ 3 (r) + eA 0 (r)ρ P (r)] . (3.9)
The computed densities for the 12 C and 16 O nuclei are depicted in Fig. 3.1. We