The nucleus of an atom is its central core, which consists of one or more protons and, with the exception only of the lightest form of hydrogen, neutrons as well. There is no charge to a neutron, yet something keeps them from slipping out of the nucleus. In addition, every proton within the nucleus is positively charged; they should repel each other, emptying the nucleus — some energy prevents this, as well. By definition, the energy keeping all these particles within the nucleus is the “nuclear binding energy.” Since Einstein discovered the mathematical relationship that equates matter with energy — E = mc^{2}, where E is the energy, m is the mass and c is the speed of light — the nuclear binding energy may be calculated with relative ease.
Mass within the nucleus comes from two sources. One is the mass each particle would contain if it was isolated, free from charge or gravitational interactions. The second source of mass is the increase directly attributable to the nuclear binding energy. These two sources give rise to the equation m_{(t)} = m_{(fp)} + m_{(nbf)}, where “t” stands for total, “fp” stands for free particle and “nbf” stands for nuclear binding force. Since there is no such thing as negative energy, the mass attributable to the nuclear binding energy must be positive and the energy of a total nucleus, greater than the sum of its neutrons and its protons.
Inserting this form of the mass into the original equation, the total energy of a nucleus is E_{(t)} = m_{(t)}c^{2}. Expanding this equation in full gives E_{(t)} = (m_{(fp)} + m_{(nbf)})c^{2}. Multiplying this out gives E_{(t)} = m_{(fp)}c^{2} + m_{(nbf)}c^{2}. Now, if the energy attributable to isolated individual particles is subtracted out, that equation reduces to E_{(t)} - E_{(fp)} = ΔE = m_{(nbf)}c^{2}, where ΔE is the increase in energy above that of free particles — the nuclear binding energy.
Nuclear fission, or the splitting of the atomic nucleus to produce smaller atoms, each of which has its own binding energy, is of particular importance to the design and operation of power plants. The binding energy of the resultant atoms, subtracted from the binding energy of the starting atoms, gives the net yield that is either applied constructively or destructively. Constructive uses of this nuclear energy include the production of electricity, measuring nearly a fifth of all electric power in the United States and more than three-quarters of the power used in France.