The semi empirical binding energy formula predicts the binding energies of neon-20, iron- 56, and uranium-238 within about 1\%. Make a $\mathbf{3}\mathbf{\times }\mathbf{9}$ table whose rows are the three isotopes, whose first four columns are numerical values of the four tennis in the formula, whose next four columns are each of these terms divided byA, and whose final column is the sum of the preceding four. Discuss in some detail what the table reveals.

Mass number of neon-20$A_{Ne}=20$

Mass number of iron-56$A_{Fe}=56$

Mass number of uranium-238$A_U=238$

Atomic number of neon$Z_{Ne}=10$

Atomic number of iron$Z_{\text{Fe}}=26$

Atomic number of uranium $Z_{II}=92$

Semi empirical binding Energy is given by the expression,

$\mathit{B}\mathit{E}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathit{A}\mathbf{-}{\mathit{c}}_{\mathbf{2}}{\mathit{A}}^{\mathbf{2}\mathbf{/}\mathbf{3}}\mathbf{-}{\mathit{c}}_{\mathbf{3}}\frac{\mathbf{Z}\mathbf{(}\mathbf{Z}\mathbf{-}\mathbf{1}\mathbf{)}}{{\mathbf{A}}^{\mathbf{1}\mathbf{/}\mathbf{3}}}\mathbf{-}{\mathit{c}}_{\mathbf{4}}\frac{{\mathbf{(}\mathbf{N}\mathbf{-}\mathbf{Z}\mathbf{)}}^{\mathbf{2}}}{\mathbf{A}}$

The semi-empirical binding energy formula is$BE=c_{1}A-c_2{A}^{2/3}-c_3\frac{Z(Z-1)}{A^{1/3}}-c_4\frac{{(N-Z)}^2}{A}$.

For neon$20,A=20,Z=10$:

$$\begin{gathered} N=A-Z \\ N=20-10 \\ N=10 \end{gathered}$$

For iron-$56,A=56,Z=26$:

$$\begin{gathered} N=A-Z \\ N=56-26 \\ N=30 \end{gathered}$$

For uranium-$2358,A=238,Z=92$:

$$\begin{gathered} N=A-Z \\ N=238-92 \\ N=146 \end{gathered}$$

The volume per nucleon contribution is the same: the surface term effect is bigger when the smaller the nucleus is because surface is a bigger proportion of the nucleus if the nucleus is smaller.

The third term effect is bigger for bigger nuclei, because the bigger nucleus has more protons.

The last term effect is bigger for bigger nuclei, because the asymmetry is more severe in bigger nuclei,

So, as number of nucleons increase, the surface effect gets smaller, but the Coulomb term and the asymmetry term effect gets bigger.