The mass of an isotope of an element expressed in atomic mass units, which are defined as one-twelfth of the mass of an atom of carbon-12. (An atomic mass of 1 is equivalent to about 1.66 x 10-27 kg.)
Assuming that the differences in the average binding energy of electrons in different atoms are negligible, the mass of an atom is given by
MA,Z = Z ·MH + N ·Mn − BE/c2,
where MA,Z, MH, and Mn are the masses of a nuclide with mass number A and atomic number Z, a hydrogen atom, and a neutron respectively. BE is the binding energy holding the nucleus together.
Nuclear masses are expressed in atomic mass units (u), based on the definition that the mass of a neutral atom of 12C is exactly 12.0000 u. The atomic mass unit (u) is defined as 1/12 the mass of a 12C atom. Relative masses of nuclei can be determined from the results of nuclear reactions or nuclear decay. If a nucleus is radioactive and emits an α particle, from conservation of energy, its mass must be greater than that of the decay products by the amount of energy released in the reaction. If the masses of the reaction products are measured, the mass of the initial (radioactive parent) system can be determined.
As an example, consider the decay of 238U, i.e.
238U → 234Th + α + Q.
The energy released in the α decay, Q, is the difference in mass energy between the parent nucleus and the final products and appears as kinetic energy shared between the product particles. Hence
Q = [MU238 −MTh234 −Mα] c2 = ETh234 + Eα,
where c is the speed of light and and ETh234 + Eα are the kinetic energies of the daughter 234Th and the α particle. If the decaying parent is initially at rest, the daughter must recoil in the opposite direction to that of the α particle and with the same momentum, i.e. MTh234 · vTh234 = Mα · vα. It follows that the ratios of their kinetic energies is given by ETh234/Eα = Mα/MTh234 .
This can then be used to obtain the decay energy Q, i.e.
Q = Eα[1 + ETh234/Eα] = Eα[1 +Mα/MTh234] .
Alternatively, if the energy of the alpha particle is measured, then this relation can be used to obtain the mass of the parent or daughter. In the example considered, the energy of the α particle is 4.196 MeV. From this the mass of the daughter 234Th can be obtained using
MTh234 = MU238 −Mα − Q/c2 = MU238 −Mα − Eα[1 +Mα/(MU238 −Mα)]/c2
where the approximation MU238−Mα = MTh234 is used in the factor multiplying Eα/c2. Using the values Eα/c2 = 4.196MeV/(931.5MeV/u) = 0.0045 u, and MU238 = 238.0508 u and Mα = 4.0026 u, the mass of the daughter MTh234 = 234.0436 u is obtained.