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2-3: Mass of Nuclei, Binding Energy

A nucleus consists of Z protons and N neutrons. Let the mass of the nucleus be M (Z, N ).
If a nucleus is assumed to be a simple collection of Z protons and N neutrons, the mass of the nucleus would be just the sum of the masses of these constituent nucleons, i.e. , where is the mass of proton and is that of neutron.
However, a nucleus is not a simple collection of protons and neutrons (nucleons), but they strongly combine with each other through a strong interaction named the nuclear force.

[Mass Defect of Nuclei]
In general, if two or more particles interact to combin together, then the total mass of the system would decrease to be less than the sum of the masses of the individual particles. The stronger the interaction becomes, the more the mass decreases. This decrease of the mass of the system is called the mass defect.
The mass defect of a nucleus of proton number Z and neutron number N is defined by
[Mass-Energy Equivalence, Unit of Mass]
According to the Special Theory of Relativity proposed by A. Einstein (Germany, USA: 1879-1955), it was made clear that mass and energy are equivalent (the Mass-Energy Equivalence), which is expressed by an equality

On the right-hand side of this equality, mass is multiplied by the square of the speed of light. This implies that very small amounts of mass may be converted into a very large amount of energy, and vice versa.

In the World of the Atomic Nucleus, a usually treated mass is, in general, very small. Therefore mass in this world is sometimes expressed in units of energy by being converted into energy with the above-mentioned Einstein's mass-energy equivalence.
For example, the proton mass is which is a quite small amount. Hence, this is converted into energy by being multiplied by the square of c i.e.

Accordingly, the proton mass is expressed as

In this way, the unit of mass

is often used in the World of the Atomic Nucleus.

[Nuclear Binding Energies]
Since a nucleus consists of A nucleons (Z protons and N neutrons; A = Z + N ) being bound together, the mass defect occurs as explained at the top of the present page. The binding energy B (Z, N ) is what this mass defect is converted into with the mass-energy equivalence. It is therefore given by

where M (Z, N ) is the mass of a nuclide of Z protons and N neutrons. Namely, the binding energy denotes the degree how strongly the nucleons bind together, and its value can be obtained by measuring the mass of the nuclide.
Nowadays, we can obtain the precise value of the mass of an individual nuclide using the mass spectrometer for example.
Experiments to measure the nuclear binding energies of wide range of nuclides have been carried out. The resultant data are shown in the figure below. From this figure, we can see that, the binding energy per nucleon is nearly constant ( = about 8 MeV) for the nuclei of A > 20, though there are some irregularities in the region of light nuclei.
The property that the binding energy per nucleon is nearly constant is called the saturation of the nuclear binding energy. This is one of the remarkable properties of the atomic nuclei.
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