Binding energy

The binding energy of a physical system is the energy required to remove one of its components. It is typically denoted with the letter $B$. The larger the binding energy is, the more energy needs to be transferred to break a piece off the system, hence the name. It is found in many fields of physics in different forms.

Nuclear binding energy#

The difference between the mass of the system as a whole (the atomic nucleus) and that of the individual components (the nucleons), is called the mass defect. Using the mass-energy equivalence $E=m_\text{defect}c^{2}$ on the defect gives us the binding energy of the system.

It is convenient to express the nuclear binding energy in terms of atomic masses, as they are measurable with much better precision ($\sim10^{-6}$) than nuclear masses using Mass spectrometry. This way, the binding energy is

where

• $A$, $N$ and $Z$ are the atomic mass number, neutron number and atomic number.
• $m_{n}$ is the mass of the neutron.
• $m(\ce{^{1}H})=m_{e}+m_{p}$ is the mass of the hydrogen atom, with $m_{e}$ and $m_{p}$ being the masses of the electron and the proton.
• $m(Z,A)$ is the mass of an atom with electrons and $A$ nucleons.
• $c$ is the speed of light.

Essentially, this formula measures the difference between the mass of the atom and the mass the system would have if it were composed of separate hydrogen atoms and neutrons.

The term separation energy is sometimes preferred over binding energy when referring to the removal of nucleons. The neutron separation energy $S_{n}$ is the energy required to remove a neutron from a nucleus, i.e., the binding energy difference between $\ce{_{Z}^{A}X_{N}}$ and $\ce{_{Z}^{A-1}X_{N-1}}$:

The proton separation energy $S_{p}$ is defined in a similar manner:

The binding energy of a nucleus depends on its atomic mass number $A$. One might expect the ratio of binding energy to the number of nucleons to be constant, but this is not the case.

$B$ increases sharply for the first few atoms (up to $A\simeq20$), after which it begins to flatten out and decline with a slight slope. The vast majority of nuclei reside in a relatively narrow band, ranging from 7.2 to 8.8 MeV/nucleon ($8.0\pm0.8$ MeV/nucleon). The only elements not included in this band are the very light ones, with $A<10$, such as hydrogen and helium. The peak of energy per nucleon is at $A\sim60$, where the nuclei are evidently particularly bound. In particular, iron $\ce{^{56}Fe}$ is a particularly resistant nuclide.

This plot gives us information on which process is energetically viable to release energy:

1. For $A\lesssim 60$, it is more efficient to combine nuclei together through nuclear fusion.
2. For $A\gtrsim60$, it is more efficient to break nuclei to create smaller ones through spontaneous fission.

The breaking point of these processes is around $A\sim60$. For example, nuclear fusion cannot create nuclei heavier than iron ($A=56$) without external energy input.

A theoretical description of the nuclear binding energy behavior across nuclides is provided by the semi-empirical mass formula.