# Converting between mass and binding energy in semi-empirical mass formula

The semi-empirical mass formula is given in terms of the binding energy as:

$$B(Z,N) = aA - bA^{\frac{2}{3}} - s \frac{(N-Z)^{2}}{A} - d \frac{Z^2}{A^{\frac{1}{3}}} - \frac{\delta}{A^{\frac{1}{2}}}$$

where $$B(Z,N)$$ is the binding energy of the atom and $$Z$$ is the proton number and $$N$$ is the neutron number.

I wish to rearrange this formula in terms of the mass instead of the binging energy and to write this in terms of $$Z$$, assuming a constant $$A$$. The formula relating binding energy and mass is:

$$M(N, Z) = N m_{n} + Z(m_p + m_e) - \frac{B(Z,N)}{c^2}$$

I am told the result of this rearrangement is supposed to be

$$M(N, Z) = \frac{1}{c^2} (\alpha - \beta Z + \gamma Z^2)$$

where

$$\alpha = Am_nc^2 - aA + bA^{\frac{2}{3}} + sA + \delta A^{-\frac{1}{2}}$$

$$\beta = 4s + (m_n - m_p - m_e) c^2$$

$$\gamma = \frac{4s}{A} + \frac{d}{A^{\frac{1}{3}}}$$

I can understand this result except for the terms involving $$s$$. On the $$\alpha$$ term I would have expected a prefactor of $$1/4$$, and on the $$\beta$$ and $$\gamma$$ terms I don't see the origin of the $$4$$ prefactor.

Looking only at the terms containing $$s$$, the binding energy goes as $$-s\frac{(N-Z)^2}{A}=-s\frac{(A-2Z)^2}{A}\tag{1}$$ and given the relation between mass and binding energy, the mass depends on $$s$$ as $$\frac{1}{c^2}s\frac{(A-2Z)^2}{A}=\frac{s}{c^2}\left(A+4\frac{Z^2}{A}-4Z\right).\tag{2}$$ Now compare with what we get using $$M(N, Z) = \frac{1}{c^2} (\alpha - \beta Z + \gamma Z^2).\tag{3}$$ $$\alpha$$ goes as $$sA$$ with $$s$$, $$\beta$$ goes as $$4s$$ and $$\gamma$$ as $$4s/A$$; so the terms containing $$s$$ in $$(3)$$ are $$\frac{1}{c^2}\left(sA-4sZ+\frac{4s}{A}Z^2\right)=\frac{s}{c^2}\left(A+4\frac{Z^2}{A}-4Z\right),\tag{4}$$ the same as in $$(2)$$.