- 14.10.1 Draft: Nuclear radius
- 14.10.2 Draft: von Weizsäcker formula
- 14.10.3 Draft: Explanation of the formula
- 14.10.4 Draft: Accuracy of the formula

14.10 Draft: Liquid drop model

Nucleons attract each other with nuclear forces that are not completely understood, but that are known to be short range. That is much like molecules in a classical liquid drop attract each other with short-range Van der Waals forces. Indeed, it turns out that a liquid drop model can explain many properties of nuclei surprisingly well. This section gives an introduction.

14.10.1 Draft: Nuclear radius

The volume of a liquid drop, hence its number of molecules, is
proportional to the cube of its radius

femtometer,equal to 1

It should be noted that the above nuclear radius is an average one. A
nucleus does not stop at a very sharply defined radius. (And neither
would a liquid drop if it only contained 100 molecules or so.) Also,
the constant

It may be noted that these results for the nuclear radii are quite
solidly established experimentally. Physicists have used a wide
variety of ingenious methods to verify them. For example, they have
bounced electrons at various energy levels off nuclei to probe their
Coulomb fields, and alpha particles to also probe the nuclear forces.
They have examined the effect of the nuclear size on the electron
spectra of the atoms; these effects are very small, but if you
substitute a muon for an electron, the effect becomes much larger
since the muon is much heavier. They have dropped pi mesons on nuclei
and watched their decay. They have also compared the energies of
nuclei with mirror nuclei

that have with

14.10.2 Draft: von Weizsäcker formula

The binding energy of nuclei can be approximated by the “von Weizsäcker formula,“ or “Bethe-von Weizsäcker formula:”

where a MeV (mega electron volt) is 1.602 18 1

Plugged into the mass-energy relation, the von Weizsäcker formula
produces the so-called “semi-empirical mass formula:”

(14.11) |

14.10.3 Draft: Explanation of the formula

The various terms in the von Weizsäcker formula of the previous
subsection have quite straightforward explanations. The

The

The

The last two terms cheat; they try to deviously include quantum
effects in a supposedly classical model. In particular, the

The last

14.10.4 Draft: Accuracy of the formula

Figure 14.9 shows the error in the von Weizsäcker
formula as colors. Blue means that the actual binding energy is
higher than predicted, red that it is less than predicted. For very
light nuclei, the formula is obviously useless, but for the remaining
nuclei it is quite good. Note that the error is in the order of MeV,
to be compared to a total binding energy of about

Near the magic numbers the binding energy tends to be greater than the predicted values. This can be qualitatively understood from the quantum energy levels that the nucleons occupy. When nucleons are successively added to a nucleus, those that go into energy levels just below the magic numbers have unusually large binding energy, and the total nuclear binding energy increases above that predicted by the von Weizsäcker formula. The deviation from the formula therefore tends to reach a maximum at the magic number. Just above the magic number, further nucleons have a much lower energy level, and the deviation from the von Weizsäcker value decreases again.