4.4 Nuclear mass formula
There is more structure in Fig.
4.1
than just a simple linear
dependence on
. A naive analysis suggests that the following terms
should play a rôle:
- Bulk energy: This is
the term studied above, and saturation implies that the
energy is proportional to
.
- Surface energy:
Nucleons at the surface of the nuclear sphere have less
neighbours, and should feel less attraction. Since the
surface area goes with
, we find
.
- Pauli or symmetry
energy: nucleons are fermions (will be discussed later). That
means that they cannot occupy the same states, thus reducing
the binding. This is found to be proportional to
.
- Coulomb energy:
protons are charges and they repel. The average distance
between is related to the radius of the nucleus, the number
of interaction is roughly
(or
). We have to include the term
.
Taking all this together we fit the
formula
|
(4.8) |
to all know nuclear binding energies with
(the formula is not so good for light nuclei). The
fit results are given in table
4.1
.
Table 4.1: Fit of masses to Eq. (
4.8
)
.
|
|
parameter | value |
|
|
| 15.36 MeV |
| 16.32 MeV |
| 90.45 MeV |
| 0.6928 MeV |
|
|
|
In Fig.
4.3
we show how well this
fit works. There remains a certain amount of structure,
see below, as well as a strong difference between
neighbouring nuclei. This is due to the superfluid nature
of nuclear material: nucleons of opposite momenta tend to
anti-align their spins, thus gaining energy. The solution is to
add a pairing term to the binding energy,
|
(4.9) |
The results including this term are
significantly better, even though all other parameters
remain at the same position, see Table
4.2
. Taking all this together
we fit the formula
|
(4.10) |
Table 4.2:
Fit of masses to Eq. (
4.10
)
|
|
parameter |
value |
|
|
|
15.36 MeV |
|
16.32 MeV |
|
90.46 MeV |
|
11.32 MeV |
|
0.6929 MeV |
|
|
|