# Stability of Nuclei

Not all combinations of neutrons and protons will form a nucleus. Light nuclei tend to have about equal number of neutrons and protons. Heavier nuclei have an excess of neutrons due to the saturation of the nuclear force and the increasing repulsion from protons.

Light nuclei (A < 20) and heavy nuclei (A >150) are less well bound than nuclei with A around 60. Thus energy is released when light nuclei fuse into heavier nuclei and when heavy nuclei fission (split) into two lighter nuclei. The most well bound nucleus is 56Fe. The typical binding energy of nuclei is 8 MeV per nucleon.

The unified mass unit, u, is defined as 1/12 of the mass of a carbon atom (A = 12). The value is u is given by

1 u = 931.5 MeV = 1.67 $\cdot$ 10-27 kg.

The mass of a hydrogen atom is simply the sum of the mass of an electron and that of a proton:

m(1H) = m(p) + m(e-) = 1.007276 u + 0.000548 u = 1.007825 u

Please note: these two number do not seem to add up exactly right. This is due to rounding at the last decimal. To be extremely precise, there is also binding between the electron and proton in the hydrogen atom, but the binding energy associated with the binding between the electron and the proton is very much smaller than the nuclear binding and has no effect on the calculation of the masses at the precision employed here.

The mass of an atom other than hydrogen can be calculated as follows:

• Suppose the nucleus has Z protons and N neutrons.
• For each proton, there has to be an electron, too, in order to make the atom electrically neutral.
• The mass of the atom is then:
m(atom(Z,N)) = N $\cdot$ mn + Z$\cdot$ m(1H) - BE/c2
= N$\cdot$ 1.008665 u + Z$\cdot$ 1.007825 u - BE/c2
• BE stands for binding energy.
• In general, the mass of a nucleus of an atom and the mass of the atom are related via:
m(atom(Z,N)) = m(nucleus(Z,N)) + Z$\cdot$ me-
= m(nucleus(Z,N)) + Z$\cdot$ 0.000548 u
• Thus we have two equally valid ways of expressing the binding energy of a nucleus:
1. BE = [N$\cdot$ mn + Z$\cdot$ m(1H) - m(atom(Z,N))] c2
2. BE = [ N$\cdot$ mn + Z$\cdot$ mp - m(nucleus(Z,N))] c2