Semi-Empirical Mass formula

Semi-Empirical Mass formula (SEMF): a formula to estimate the total mass of an atomic nucleus based on the number of protons and neutrons. It includes binding energy.

$$ M(Z,A)=ZM(^1_1 {\rm H})+(A-Z)M_n -E_B/c^2 $$

where $Z$ is the atomic number, $A$ is the mass number, $M(^1_1 {\rm H})$ the mass of a $\rm ^1 _1 H$ atom and $M_n$ the mass of a neutron

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The binding energy $E_B$ is composed of five terms:

$$ E_B= a_v A - a_s A^{2/3} -a_c \frac{Z(Z-1)}{A^{1/3}}-a_a \frac{(A-2Z)^2}{A} \pm a_p A^{-1/2} $$

$a_vA$ - The volume term

Since $A$ is the number of nucleons and the strong force is short ranges, we assume that nucleons only interact with their nearest neighbours. This means that each new nucleon will contribute a constant term $a_v$ thus we write

$$ a_v A $$

🧲 Attractive term: it increases the total binding energy (positive)

$-a_s A^{2/3}$- The surface term

To correct the volume term for the edges of the nucleon we create a term which scales with the surface of the nucleus. Since the radius scales with $A^{1/3}$ and surface is $4\pi r^2$ then we get $\propto A^{2/3}$

$$ -a_s A^{2/3} $$

🦊 Repulsive term: decreasing the binding energy (negative)

🗒️ Note: this term is analogous to surface tension in liquid drops

$-a_c \frac{Z(Z-1)}{A^{1/3}}$ - The coulomb term

This term accounts for electromagnetic repulsion between protons within the nucleus re-written ie

$$ V=\frac 35 \frac{1}{4\pi \epsilon_0} \frac{q^2}{r}=a_c\frac{Z(Z-1)}{A^{1/3}} $$

where all the constants are in $a_c$.

For $Z\gg1$ we write

$$ a_c\frac{Z(Z-1)}{A^{1/3}}\approx a_c\frac{Z^2}{A^{1/3}} $$

🦊 Repulsive term: decreasing the binding energy (negative)

🗒️ Note: we would see the same effect in a charged liquid drop

$a_vA$ - The volume term

$-a_s A^{2/3}$- The surface term

$-a_c \frac{Z(Z-1)}{A^{1/3}}$ - The coulomb term

$-a_a\frac{(A-2Z)^2}{A}$- The asymmetric term