Mass excess

The following is modified from w:mass excess.

The mass excess of a nuclide is the difference between its actual mass and its mass number in atomic mass units.^[1] It is one of the predominate methods for tabulating nuclear mass. The mass of an atomic nucleus is well approximated by its mass number, which indicates that most of the mass of a nucleus arises from mass of its constituent protons and neutrons. Thus, the mass excess is an expression of the nuclear binding energy, relative to the binding energy per nucleon of Carbon-12 (which defines the atomic mass unit). If the mass excess is negative, the nucleus has more binding energy than ¹²C, and vice versa. If a nucleus has a large excess of mass compared to a nearby nuclear species, it can radioactively decay, releasing energy.

Energy Scale of Nuclear Reactions

The ¹²C standard makes it useful to think about nuclear mass in atomic mass units for the definition of the mass excess. However, its usefulness arises in the calculation of nuclear reaction kinematics or decay. From Einstein's work on mass-energy equivalence (E=mc²), it is well known that nuclear reactions transform energy into mass and vice versa. However, most of the energy remains in the mass of the nuclei involved, and only a small fraction of the total energy, on the order of 0.01 to 0.1% of the total mass, may be absorbed or liberated. Thus by working in terms of the mass excess, one has effectively removed much of the mass changes which arise from the mere transfer or release of nucleons, making more obvious the scale of the net energy difference.

Nuclear reaction kinematics are customarily performed in units involving the electron volt, a consequence of accelerator technology. The combination of this practical point with the theoretical relation E=mc² makes units of mega electron volts over the speed of light squared (MeV/c²) a convenient form to express nuclear mass. However, the numerical values of nuclear masses in MeV/c² are quite large (even the proton mass is ~938.27 MeV/c²), while mass excesses range in the tens of MeV/c². This makes tabulated mass excess less cumbersome for use in calculations. One trivial point worth noting that the 1/c² term is typically omitted when quoting mass excess values in MeV, since the interest is more often energy and not mass; if one wanted units of mass, one would simply change the units from MeV to MeV/c² without altering the numerical value.

Example

Consider the nuclear fission of ²³⁶U into ⁹²Kr, ¹⁴¹Ba, and three neutrons.

²³⁶U → ⁹²Kr + ¹⁴¹Ba + 3 n

The mass number of ²³⁶U is 236 u (atomic mass units), but the actual mass is $236.045563$ u, so the mass excess is $+0.045563$ u. Calculated in the same manner, the mass excess for ⁹²Kr, ¹⁴¹Ba, and a neutron are $-0.073843$ u, $-0.085588$ u and $0.0086648$ u, respectively. The mass excess of the reactant is $+0.045563$ u, and the mass excess of the products is $-0.073843+(-0.085588)+3*(0.0086648)=-0.1334366$ The difference between reactants and products is $0.045563-(-0.133436)=0.1789996$ u, which shows that the mass excess of the products is less than that of the reactants, and so the fission can occur.