Binding Energy & Mass Formula

Volume Term Derivation

In the liquid drop model, the nuclear density is approximately constant:$\rho_0 \approx 0.16$ fm$^{-3}$. Since each nucleon interacts only with its nearest neighbors (saturation), the binding energy is proportional to the number of nucleons:

From the Fermi gas model, we can estimate $a_V$. The binding energy per nucleon in infinite nuclear matter is:

where $E_F \approx 37$ MeV is the Fermi energy and $\bar{V}$ is the average potential per nucleon. The empirical value $a_V \approx 15.56$ MeV reflects the balance between kinetic energy (positive) and potential energy (negative).

Surface Term Derivation

Nucleons at the surface of the nucleus have fewer neighbors and are therefore less tightly bound. This correction is proportional to the surface area:

since $R = r_0 A^{1/3}$. This is analogous to surface tension in a liquid drop. The nuclear surface tension is:

This is roughly $10^{18}$ times larger than the surface tension of water, reflecting the enormous strength of the nuclear force.

Coulomb Term Derivation

The electrostatic energy of a uniformly charged sphere of charge $Ze$ and radius$R$ is calculated by bringing charge from infinity shell by shell:

where $\rho_{\rm ch} = 3Ze/(4\pi R^3)$. Evaluating the integral:

Since protons are discrete, we replace $Z^2$ with $Z(Z-1)$ to avoid self-interaction:

Asymmetry Term Derivation from the Fermi Gas Model

Consider a nucleus as two independent Fermi gases (protons and neutrons) confined in the nuclear volume $V = \frac{4}{3}\pi R^3$. The total kinetic energy is:

where the Fermi energies are $E_F^{(p)} = \frac{\hbar^2}{2m}(3\pi^2 Z/V)^{2/3}$ and similarly for neutrons. For a symmetric nucleus ($Z = N = A/2$):

For an asymmetric nucleus with $Z = (A - \delta)/2$, $N = (A + \delta)/2$ where$\delta = N - Z$, expanding to second order in $\delta/A$:

This identifies the asymmetry coefficient:

The empirical value $a_A \approx 23.3$ MeV is about twice the kinetic energy estimate, because the potential energy also depends on the N-Z asymmetry. The symmetry potential contributes roughly equally to the asymmetry coefficient.

Drip Line Conditions

The neutron drip line is reached when the one-neutron separation energy vanishes:

Similarly, the proton drip line is:

Using the SEMF, the most stable Z for a given A satisfies:

For light nuclei $Z_0 \approx A/2$; for $A = 240$, $Z_0 \approx 93$, giving$N/Z \approx 1.6$. This neutron excess in heavy nuclei is driven by the Coulomb term, which penalizes large Z.

Isobaric Mass Parabolas

For fixed A, the atomic mass as a function of Z forms a parabola (or two parabolas for even A, split by the pairing term):

where $\gamma = 4a_A/A + a_C/A^{1/3}$. For odd-A isobars, there is a single parabola with one stable nucleus at the minimum. For even-A, the even-even and odd-odd parabolas are offset by $2\delta$, potentially allowing multiple stable isobars.

This explains why some even-A isobars have 2 or even 3 stable isotopes (e.g.,$^{124}$Sn, $^{124}$Te, $^{124}$Xe for A=124), while odd-A isobars typically have only one stable member.

Neutron and Proton Separation Energies

The one-neutron separation energy is the minimum energy needed to remove a neutron:

The two-neutron separation energy is often smoother (less odd-even staggering):

From the SEMF, the neutron separation energy is approximately:

Sudden drops in $S_n$ or $S_{2n}$ at specific N values (N = 8, 20, 28, 50, 82, 126) provide the classic evidence for nuclear shell closures (magic numbers).

Q-Values for Nuclear Reactions

The Q-value of a nuclear reaction $a + A \to b + B$ is the mass-energy difference:

In terms of binding energies:

Important Q-value applications:

• - Alpha decay: $Q_\alpha = B(\alpha) + B(A-4,Z-2) - B(A,Z)$. Alpha decay is possible when $Q_\alpha > 0$.
• - Beta decay: $Q_{\beta^-} = [M(Z,A) - M(Z+1,A)]c^2$. In terms of atomic masses, $Q_{\beta^-} = [\mathcal{M}(Z,A) - \mathcal{M}(Z+1,A)]c^2$.
• - Fusion: $Q_{\rm DT} = B(^4{\rm He}) + B(n) - B(^2{\rm H}) - B(^3{\rm H}) = 17.6$ MeV for the D-T reaction.

Predicting the Limits of Nuclear Stability

The SEMF predicts that nuclei become unstable to spontaneous fission when the Coulomb energy equals twice the surface energy. The fissility parameter is:

Spontaneous fission becomes dominant when $x \geq 1$, giving:

This predicts that nuclei with $Z^2/A > 49$ should be instantly unstable, setting an upper bound on the periodic table around $Z \approx 120$. Shell effects (not included in the SEMF) create islands of stability for superheavy elements near Z = 114, N = 184.

Alpha Decay Q-Value from SEMF

The SEMF predicts the alpha decay Q-value as:

Using $B(^4{\rm He}) = 28.296$ MeV and expanding to leading order:

This formula correctly predicts that $Q_\alpha > 0$ for heavy nuclei with$A \gtrsim 150$, though shell effects cause significant deviations (e.g., nuclei near $^{208}$Pb have anomalously low $Q_\alpha$ due to the Z=82, N=126 shell closures).

Binding Energy Signatures

Magic nuclei show several distinctive features in their binding energies:

• - Extra binding: The binding energy of a magic nucleus exceeds the SEMF prediction. The deviation $\Delta B = B_{\rm exp} - B_{\rm SEMF}$ peaks at magic numbers.
• - Separation energy drops: $S_n$ drops sharply when going from N = magic to N = magic + 1. For example, $S_n(^{208}{\rm Pb}) = 7.37$ MeV but $S_n(^{209}{\rm Pb}) = 3.94$ MeV.
• - Large first excited state energy: Doubly-magic nuclei like $^{208}$Pb have a first excited state at 2.61 MeV, compared to typical ~0.1 MeV for deformed nuclei.
• - Isotope/isotone abundance: Elements with Z or N equal to a magic number have more stable isotopes/isotones.