Nuclear Models: the Liquid Drop and the Semi-Empirical Mass Formula
Treating a nucleus as an incompressible charged drop gives volume, surface, Coulomb, asymmetry and pairing terms that reproduce the binding energy curve to within a per cent. The same formula says which nuclide in an isobaric chain is stable and why heavy nuclei fission at all.
What a learner can do afterwards
- Names each term of the mass formula and says what physical picture it comes from
- Predicts the most stable nuclide for a fixed nucleon number and compares it with the chart of nuclides
- Uses the formula to argue that fission of a heavy nucleus releases energy while fission of a light one does not
1 · Read
Picture the nucleus as a drop of incompressible fluid with a sharp surface and uniform density. This works because nuclear density stays nearly constant and the force is short ranged. Weigh a nucleus and it comes out lighter than its separate protons and neutrons. The missing mass times c squared is the binding energy. Divide by the nucleon count to compare light and heavy nuclei fairly.
Five terms share the binding. Volume pays for interior nucleons feeling full attraction. Surface subtracts for skin nucleons with fewer neighbours. Coulomb subtracts the proton repulsion, which grows with the square of their number while spreading over the radius. Asymmetry and pairing fine tune the total. Each term maps straight onto one picture in the drop.
For a fixed nucleon number the formula is a parabola in charge, and the bottom of the curve is the most stable nuclide. Plot stability against proton and neutron numbers and you get the band of stability, with the chart of nuclides as the scoreboard. Compare your predicted minimum with that chart. Extra tough nuclei sit at magic numbers, like closed shells in atoms.
The binding per nucleon climbs steeply, peaks near iron, then falls away, and that single shape decides which direction pays. Heavy nuclei gain per nucleon by splitting, so fission releases energy. Light nuclei gain by joining, so fusion releases energy there. Fission of a light nucleus would climb the wrong way and cost energy.
Five drop terms reproduce the binding curve, the parabola bottom picks the stable nuclide, and the iron peak decides whether splitting or joining pays.
More to read
21.1 Nuclear Structure and Stability
A nucleus stays together only if its binding energy beats the forces pulling it apart. Plotting stability against proton and neutron numbers draws the band of stability, with everything outside it doomed to decay. Magic numbers of protons or neutrons give extra tough nuclei, like closed shells in atoms. The whole chart of nuclides is the scoreboard that the mass formula has to explain.
21.4 Transmutation and Nuclear Energy
Splitting a heavy nucleus or joining two light ones can both release energy, because the products bind tighter per nucleon than the starting material. The binding energy curve climbs steeply, peaks near iron, then falls away, and that single shape decides which direction pays. Reactors and stars are just two ways of sliding down that curve. The mass formula turns this picture into numbers you can calculate.
31.6 Binding Energy
Weigh a nucleus and it comes out lighter than its separate protons and neutrons. The missing mass, times c squared, is the binding energy holding it together. Divide by the nucleon count and you get the fair way to compare light and heavy nuclei. The semi-empirical formula predicts exactly this curve with five fitted terms.
31.3 Substructure of the Nucleus
Zoom in and the nucleus shows protons and neutrons, then quarks inside those. Yet for bulk properties like total mass, fine detail hardly matters, which is why picturing the nucleus as a liquid drop works so well. Volume pulls together, surface and charge push apart, and symmetry plus pairing fine tune the total. Each formula term maps straight onto one of these pictures.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
This opens up
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.