Mass Defect Calculator

A bound nucleus weighs less than the free particles it is made from, and the missing mass is the energy that holds it together. Adding up Z proton masses of 1.007276 u, N neutron masses of 1.008665 u, and Z electron masses of 0.000549 u, then subtracting the measured neutral atomic mass, gives the mass defect in unified atomic mass units. Because the tabulated atomic mass already includes the electrons, they must be counted on the constituent side as well or the answer is wrong by roughly 0.0005 u per proton. Converting through Einstein mass-energy equivalence with the factor 931.494 MeV per u turns the defect into the total binding energy. Dividing by the mass number A gives binding energy per nucleon, the quantity that actually ranks stability: it climbs steeply through the light elements, reaches about 8.79 MeV at iron-56 and a fractionally higher 8.79 to 8.80 MeV at nickel-62, then declines slowly to about 7.57 MeV at uranium-238. That curve is the reason fusion of light nuclei and fission of heavy ones both release energy. Several practical limits apply. The result is only as good as the atomic mass entered, so masses should be taken to six or more decimal places from an evaluated table; a rounded value such as 4.0026 instead of 4.002602 shifts helium-4 binding energy by tenths of an MeV. The calculation returns atomic rather than strictly nuclear binding energy, since the electron binding energies, on the order of electronvolts for light atoms and up to about 760 keV in total for uranium, are silently folded in. The 931.494 MeV per u factor and the nucleon masses are themselves measured constants carrying small uncertainties. Local structure also breaks the smooth trend: nuclei with magic numbers of protons or neutrons at 2, 8, 20, 28, 50, 82, and 126 sit noticeably above the general curve, which is why helium-4 already reaches 7.07 MeV per nucleon and lead-208 is unusually tightly bound.

Francis William Aston built the first accurate mass spectrograph at the Cavendish Laboratory in Cambridge in 1919 and found that the masses of the elements were nearly but not exactly whole-number multiples of the hydrogen mass. He expressed the discrepancy as the packing fraction, published a systematic survey of it in the early 1920s, and received the 1922 Nobel Prize in Chemistry for the work. Aston recognised that the deficit was not a measurement flaw but the mass equivalent of nuclear binding energy, connecting his data to the relation between mass and energy that Albert Einstein had set out in 1905, and he pointed explicitly to the enormous energy locked in helium formation from hydrogen. Direct experimental confirmation came in 1932, when John Cockcroft and Ernest Walton at the Cavendish bombarded lithium-7 with accelerated protons and detected two alpha particles whose combined kinetic energy of about 17.3 MeV matched the mass lost in the reaction, the first quantitative laboratory test of mass-energy equivalence. James Chadwick had identified the neutron earlier the same year, which finally allowed nuclear masses to be broken down into protons and neutrons rather than protons and nuclear electrons. Carl Friedrich von Weizsaecker proposed the semi-empirical mass formula in 1935, refined by Hans Bethe and Robert Bacher in 1936, modelling binding energy through volume, surface, Coulomb, asymmetry, and pairing terms. Bethe used the binding energy curve in 1939 to work out the stellar fusion cycles, and evaluated masses are now published in the periodically updated Atomic Mass Evaluation.