The Semi-Empirical Mass Formula and the Valley of Stability

The constancy of nuclear density and the near-constant binding energy per nucleon are the properties of an incompressible charged liquid drop. Weizsäcker turned the analogy into a formula: write the binding energy as a sum of terms, each with a physical origin and one fitted coefficient, and the result reproduces the masses of hundreds of nuclides to about $1\%$.¹ This lesson derives each term, assembles the mass parabola, and extracts the two structural predictions the formula makes without further input: the most stable isobar and the fissility limit.

The five terms

Volume term $a_{V}A$. Each nucleon interacts only with its nearest neighbours (the force saturates), so every interior nucleon contributes the same fixed binding. The bulk energy is therefore proportional to the number of nucleons, and since the volume $\frac{4}{3}\pi R^3\propto A$, this is also proportional to volume. Left alone this term would give a constant $B/A=a_V$; the other terms are corrections that bend the binding-energy curve.

Surface term $-a_S{A}^{2/3}$. A nucleon at the surface has fewer neighbours than one in the interior, so it is less bound. The number of surface nucleons scales with the area $4\pi R^2\propto A^{2/3}$, and the correction is negative because surface nucleons are undercounted by the volume term. This is the nuclear surface tension, and it dominates the light-nucleus rise of $B/A$: small nuclei are nearly all surface.

Coulomb term $-a_{C}Z(Z-1)/A^{1/3}$. The $Z$ protons repel. Treating the charge $Ze$ as a uniformly charged sphere of radius $R=r_0{A}^{1/3}$, the electrostatic self-energy is

Replacing $Z^2$ by $Z(Z-1)$ removes each proton's unphysical self-interaction. The coefficient is then predicted, not merely fitted:

in agreement with the fit. This is the only term that grows with size relative to the volume, and it is what eventually destabilizes heavy nuclei.

Asymmetry term $-a_A(N-Z{)}^2/A$. Protons and neutrons fill separate ladders of levels in a shared well, two per level. At fixed $A$ the total kinetic energy is minimized when the two ladders are filled to the same height, that is $N=Z$. Moving $\nu =(N-Z)/2$ neutrons up from the proton Fermi level into empty neutron levels costs an energy proportional to the number moved times the level spacing times the shift, all of which combine to give a cost quadratic in the imbalance and inversely proportional to $A$,

The full Fermi-gas derivation fixes the coefficient; here the quadratic form and the $1/A$ suppression are the content. This term pulls light nuclei toward $N=Z$ and, combined with the Coulomb term, sets the neutron excess of heavy nuclei.

Pairing term $\delta$. Two identical nucleons in the same spatial orbital with paired spins overlap strongly and bind an extra amount. The correction is

with $a_P\approx 12\mathrm{MeV}$. Even-even nuclei are the most bound and odd-odd the least, which is why only four stable odd-odd nuclides exist.

A representative fit gives, in MeV,

The mass parabola at fixed A

Fixing $A$ and writing $N=A-Z$, the atomic mass is a quadratic in $Z$:

collecting the $A$-only terms into $\alpha$ and reading off

The isobars of a given $A$ lie on this parabola. Its minimum, obtained from $\partial M/\partial Z=0$, gives the most stable charge:

For light nuclei the denominator is near $2$, giving $Z_A\approx A/2$ and $N\approx Z$. For heavy nuclei the $A^{2/3}$ term (the Coulomb contribution) grows, pushing $Z_A$ below $A/2$: uranium ($A=238$) has $Z_A\approx 92$ against $A/2=119$, a large neutron excess. The curve of $Z_A$ against $A$ is the valley of stability.

A nuclide displaced from $Z_A$ slides down the parabola by beta decay, which changes $Z$ by one at fixed $A$. On the neutron-rich (low-$Z$) side, ${\beta }^{-}$ decay converts a neutron to a proton and raises $Z$; on the proton-rich (high-$Z$) side, ${\beta }^{+}$ decay or electron capture lowers $Z$. Each step moves toward $Z_A$, and the decay stops at the isobar nearest the minimum.

Even-A isobars and double beta decay

For even $A$ the pairing term splits the isobars into two parabolas. Even-even nuclides (both $Z$ and $N$ even) carry $\delta =+a_P{A}^{-1/2}$ and lie on the lower curve; odd-odd nuclides carry $\delta =-a_P{A}^{-1/2}$ and lie on the upper curve, displaced by $2\delta \approx 24A^{-1/2}\mathrm{MeV}$. Successive isobars alternate between the two parabolas as $Z$ steps by one.

Two consequences follow. First, an even-$A$ chain can have two or three stable even-even isobars, because an even-even nuclide may be unable to reach a lower even-even neighbour without passing through a higher-lying odd-odd intermediate, which single beta decay cannot skip. Second, when a would-be single beta decay is energetically forbidden by the odd-odd hump but the even-even nuclide two steps away lies lower, the transition can proceed only by double beta decay, changing $Z$ by two and emitting two electrons and two antineutrinos. This second-order weak process, with half-lives beyond $10^{19}\mathrm{yr}$, is the slowest known radioactive decay; its neutrinoless variant is treated in the double beta decay lesson.

The fissility parameter and the onset of fission

Deform the drop into a prolate ellipsoid at fixed volume, parametrized by a small elongation $\epsilon$. The surface area increases while the mean charge separation grows, so to second order the surface and Coulomb energies shift by

where $E_S=a_S{A}^{2/3}$ and $E_C=a_C{Z}^2/A^{1/3}$ are the spherical values. The net change in energy is

The sphere is stable against deformation while $\Delta E>0$, that is while $E_C<2E_S$. The stability boundary $E_C=2E_S$ defines the fissility parameter:

When $Z^2/A$ approaches $50$ the fission barrier vanishes and the nucleus is unstable to immediate breakup; the heaviest nuclei ($Z^2/A\approx 36$ for uranium) sit below the limit but with a barrier low enough that fission competes with alpha decay. The dynamics of the barrier and the scission path are developed in the fission module.

The mass formula is a smooth surface: it captures the bulk trends and the average valley of stability but has no shell structure, so it misses the extra binding at magic numbers and cannot predict ground-state spins. Those require the single-particle models, and the static shape degrees of freedom the drop supplies here become the collective coordinates of the deformed-nucleus models. The remaining property the drop cannot supply, the angular momentum and electromagnetic moments of the ground state, is the subject of the next lesson.