Statement:
Derivation: Physical origin of each term :
Volume term \(a_V A\) :
Surface term \(-a_S A^{2/3}\) :
Coulomb term \(-a_C Z(Z-1)/A^{1/3}\) :
Asymmetry (or Pauli) term \(-a_A (A-2Z)^2/A\) :
Pairing term \(\delta(A,Z)\) :
Notes on coefficients and units :
The Semi‑Empirical Mass Formula (SEMF) gives the nuclear
binding energy \(B(A,Z)\) of a nucleus with mass number \(A\) and
atomic number \(Z\) as the sum of five macroscopic terms derived from
the liquid‑drop model:
Formula :
\[B(A,Z)=a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_A
\frac{(A-2Z)^2}{A} + \delta(A,Z)\]
Sometimes the Coulomb term is written as \(a_C \dfrac{Z^2}{A^{1/3}}\)
and the pairing term \(\delta\) has the usual form given below.
Derivation: Physical origin of each term :
Volume term \(a_V A\) :
The nucleus behaves like an incompressible liquid drop. Each nucleon binds with a roughly constant number of nearest neighbours, so binding energy is proportional to \(A\). Hence \(a_V A\) with \(a_V>0\).
Surface term \(-a_S A^{2/3}\) :
Nucleons at the surface have fewer neighbours and so contribute less binding. Surface area scales as \(A^{2/3}\), giving a negative correction \(-a_S A^{2/3}\).
Coulomb term \(-a_C Z(Z-1)/A^{1/3}\) :
Protons repel by Coulomb force. Approximating the nucleus as a uniformly charged sphere of radius \(R\propto A^{1/3}\) leads to electrostatic energy \(\propto Z^2/R \sim Z^2/A^{1/3}\). The factor \(Z(Z-1)\) corrects for self‑interaction.
Asymmetry (or Pauli) term \(-a_A (A-2Z)^2/A\) :
Quantum mechanically, neutrons and protons fill Fermi levels. Minimum energy occurs when \(N\approx Z\) for small nuclei. Deviation from symmetric \(N=Z\) costs kinetic (Fermi) energy, producing a term quadratic in \(N-Z\): since \(N=A-Z\), this becomes \((A-2Z)^2/A\).
Pairing term \(\delta(A,Z)\) :
Because of pairing, nuclei with even numbers of protons and neutrons are extra stable. Empirical form: \[\delta(A,Z)=\begin{cases} +a_P A^{-1/2} & \text{even-}Z,\,\text{even-}N\\ -a_P A^{-1/2} & \text{odd-}Z,\,\text{odd-}N\\ 0 & \text{if } A \text{ is odd}\end{cases}\] This term is small and alternates sign depending on nucleon parity.
Notes on coefficients and units :
Typical fitted values(approximately): \(a_V = 15.5 \, MeV\)
\(a_S = 17.0 \, MeV\)
\(a_C = 0.71 \, MeV\)
\(a_A = 23.0 \, MeV\)
\(a_P = 11.0 \, MeV\)
These are empirical — obtained by fitting measured nuclear masses.
Short derivation sketch for asymmetry term :Treat protons and neutrons as independent Fermi gases. Fermi energy scales as \(E_F\propto (n)^{2/3}\) where density \(n\) differs when \(N\ne Z\). Expanding the total kinetic energy to second order in the neutron excess gives an energy contribution \(\propto (N-Z)^2/A\), producing the asymmetry term shown above.
Example / application & remarks :Binding energy per nucleon \(B/A\) predicted by SEMF peaks near \(A\sim 56\), explaining iron‑group stability and why heavy nuclei fission while light nuclei fuse. SEMF also predicts approximate mass parabolae and decay energetics (Q‑values) qualitatively well.
