Editing Semi-empirical mass formula (section)

===Coulomb term===
The term <math>a_\text{C} \frac{Z(Z - 1)}{A^{1/3}}</math> or <math>a_\text{C} \frac{Z^2}{A^{1/3}}</math> is known as the ''Coulomb'' or ''electrostatic term''.

The basis for this term is the [[electromagnetic force|electrostatic repulsion]] between protons. To a very rough approximation, the nucleus can be considered a sphere of uniform [[electric charge|charge]] density. The [[potential energy]] of such a charge distribution can be shown to be
: <math>E = \frac{3}{5} \frac{1}{4 \pi \varepsilon_0} \frac{Q^2}{R},</math>

where ''Q'' is the total charge, and ''R'' is the radius of the sphere. The value of <math>a_\text{C}</math> can be approximately calculated by using this equation to calculate the potential energy, using an [[Nuclear size|empirical nuclear radius]] of <math>R \approx r_0 A^{\frac{1}{3}}</math> and ''Q'' = ''Ze''. However, because electrostatic repulsion will only exist for more than one proton, <math>Z^2</math> becomes <math>Z(Z - 1)</math>:
: <math>E = \frac{3}{5} \frac{1}{4 \pi \varepsilon_0} \frac{Q^2}{R} = \frac{3}{5} \frac{1}{4 \pi \varepsilon_0} \frac{(Ze)^2}{r_0 A^{1/3}} = \frac{3 e^2 Z^2}{20 \pi \varepsilon_0 r_0 A^{1/3}} \approx \frac{3 e^2 Z(Z - 1)}{20 \pi \varepsilon_0 r_0 A^{1/3}} = a_\text{C} \frac{Z(Z - 1)}{A^{1/3}},</math>
where now the electrostatic Coulomb constant <math>a_\text{C}</math> is
: <math>a_\text{C} = \frac{3 e^2}{20 \pi \varepsilon_0 r_0}.</math>

Using the [[fine-structure constant]], we can rewrite the value of <math>a_\text{C}</math> as

: <math>a_\text{C} = \frac{3}{5} \frac{\hbar c \alpha}{r_0} = \frac{3}{5} \frac{R_\text{P}}{r_0} \alpha m_\text{p} c^2,</math>

where <math>\alpha</math> is the fine-structure constant, and <math>r_0 A^{1/3}</math> is the [[Nuclear size|radius of a nucleus]], giving <math>r_0</math> to be approximately 1.25&nbsp;[[femtometer]]s. <math>R_\text{P}</math> is the proton [[reduced Compton wavelength]], and <math>m_\text{p}</math> is the proton mass. This gives <math>a_\text{C}</math> an approximate theoretical value of 0.691&nbsp;[[MeV]], not far from the measured value.