Mass defect and binding energy

A remarkable experimental fact underlies the entire concept of nuclear binding energy: the measured mass of any nucleus is always slightly less than the sum of the masses of its individual, separated constituent nucleons.

Worked case: carbon-12. The carbon-12 nucleus contains 6 protons and 6 neutrons. Adding up their individual masses separately: 6 neutrons contribute $6\times1.00866=6.05196$ u, and 6 protons contribute $6\times1.00727=6.04362$ u, for a combined constituent mass of $12.09558$ u. But mass spectroscopy shows the actual measured mass of the carbon-12 atom is exactly 12 u; subtracting the mass of the 6 orbiting electrons ($6\times0.00055=0.0033$ u) gives the carbon-12 nuclear mass as $11.9967$ u - which is indeed less than the summed constituent mass of $12.09558$ u, by a difference of $\Delta m=0.09888$ u.

Mass defect. In general, for any nucleus $^{A}_{Z}X$ with measured mass $M$, built from $Z$ protons (mass $m_p$) and $N$ neutrons (mass $m_n$), this shortfall is called the mass defect:

Binding energy. Einstein's mass-energy relation $E=mc^2$ explains where the missing mass goes: when the separate nucleons combine to form the nucleus, an amount of mass equal to $\Delta m$ effectively converts into energy that is released, called the binding energy of the nucleus, $BE=(\Delta m)c^2$. Equivalently, to pull the nucleus back apart into its separate free nucleons, exactly this much energy would need to be supplied:

It is more convenient in practice to work with tabulated atomic masses (which already include the electrons) rather than bare nuclear masses; adding and subtracting the mass of the $Z$ atomic electrons converts this formula into

$BE=[Zm_H+Nm_n-M_A]c^2\qquad (8.24)$ …