Nuclear Energy and Binding Energies

In the same way that chemical reactions can be classified as exothermic (releasing energy) or endothermic (requiring energy to react), so nuclear reactions may either release or use up energy. The energies involved in nuclear reactions are greater by a huge factor. Thousands of tons of coal would have to be burned to produce as much energy as would be produced in a nuclear power plant by one kg of fuel.

Although nuclear reactions that use up energy (endothermic reactions) can be initiated in accelerators, where one nucleus is rammed into another at high speed, they do not occur in nature, not even in the sun. The amount of kinetic energy required is simply not available.

To find the amount of energy consumed or released in a nuclear reaction, you need to know how much potential energy was stored or released. Experimentalists have determined the amount of potential energy stored in the nucleus of every stable element, as well as many unstable elements. This is the amount of mechanical work that would be required to pull the nucleus apart into its individual neutrons and protons, and is known as the nuclear binding energy.

A reaction occurring in the sun

Self-Check	Why is the binding energy of 1H exactly equal to zero?
Answer	The hydrogen-1 nucleus is simply a proton. The binding energy is the energy required to tear a nucleus apart, but for a nucleus this simple there is nothing to tear apart.

Optional topic: conversion of mass to energy and energy to mass

If you add up the masses of the three particles produced in the reaction n → p + e^- + ν[bar] you will find that they do not equal the mass of the neutron, so mass is not conserved. An even more blatant example is the annihilation of an electron with a positron, e^- + e⁺ → 2γ,in which the original mass is completely destroyed, since gamma rays have no mass. Nonconservation of mass is not just a property of nuclear reactions. It also occurs in chemical reactions, but the change in mass is too small to detect with ordinary laboratory balances.

The reason why mass is not being conserved is that mass is being converted to energy, according to Einstein's celebrated equation E=mc², in which c stands for the speed of light. In the reaction e^- + e⁺ → 2γ, for instance, imagine for simplicity that the electron and positron are moving very slowly when they collide, so there is no significant amount of energy to start with. We are starting with mass and no energy, and ending up with two gamma rays that possess energy but no mass. Einstein's E=mc² tells us that the conversion factor between mass and energy is equal to the square of the speed of light. Since c is a big number, the amount of energy consumed or released by a chemical reaction only shows up as a tiny change in mass. But innuclear reactions, which involve large amounts of energy, the change in mass may amount to as much as one part per thousand. Note that in this context, c is not necessarily the speed of any of the particles. We are just using its numerical value as a conversion factor. Note also that E=mc² does not mean that an object of mass m has a kinetic energy equal to mc²; the energy being described by E=mc² is the energy you could release if you destroyed the particle and converted its mass entirely into energy, and that energy would be in addition to any kinetic or potential energy the particle had.

Have we now been cheated out of two perfectly good conservation laws, the laws of conservation of mass and of energy? No, it's just that according to Einstein,the conserved quantity is E+mc², not E or m individually. The quantity E+mc² is referred to as the mass-energy, and no violation of the law of conservation of mass-energy has yet been observed. In most practical situations, it is a perfectly reasonable to treat mass and energy as separately conserved quantities.

It is now easy to explain why isolated protons (hydrogen nuclei) are found in nature, but neutrons are only encountered in the interior of a nucleus, not by themselves. In the process e^- + e⁺ → 2γ, the total final mass is less than the mass of the neutron, so mass is being converted into energy. In the beta decay of a proton, e^- + e⁺ → 2γ, the final mass is greater than the initial mass, so some energy needs to be supplied for conversion into mass. A proton sitting by itself in a hydrogen atom cannot decay, since it has no source of energy. Only protons sitting inside nuclei can decay,and only then if the difference in potential energy between the original nucleus and the new nucleus would result in a release of energy. But any isolated neutron that is created in natural or artificial reactions will decay within a matter of seconds, releasing some energy.

The equation E=mc² occurs naturally as part of Einstein's theory of special relativity, which is not what we are studying right now. This brief treatment is only meant to clear up the issue of where the mass was going in some of the nuclear reactions we were discussing.

The figure above is a compact way of showing the vast variety of the nuclei. Each box represents a particular number of neutrons and protons. The black boxes are nuclei that are stable, i.e. that would require an input of energy in order to change into another. The gray boxes show all the unstable nuclei that have been studied experimentally. Some of these last for billions of years on the average before decaying and are found in nature, but most have much shorter average lifetimes, and can only be created and studied in the laboratory.

The curve along which the stable nuclei lie is called the line of stability. Nuclei along this line have the most stable proportion of neutrons to protons. For light nuclei the most stable mixture is about 50-50, but we can see that stable heavy nuclei have two or three times more neutrons than protons. This is because the electrical repulsions of all the protons in a heavy nucleus add up to a powerful force that would tend to tear it apart. The presence of a large number of neutrons increases the distances among the protons, and also increases the number of attractions due to the strong nuclear force.

Note the two nuclei, in the bottom row of the figure, with zero protons. One is simply a single neutron. The other is a cluster of four neutrons. This "tetraneutron" was reported, unexpectedly, to be a bound system in results from a 2002 experiment. The result is controversial. If correct, it implies the existence of a heretofore unsuspected type of matter, the neutron droplet, which we can think of as an atom with no protons or electrons.