The blueprint. When Rutherford discovered the proton in 1918, he was still convinced about the possible existence of some other subatomic particle. He conducted several experiments to illustrate this prediction, and in 1920, concluded that there should, indeed, be another subatomic particle; particularly, one with zero charge and mass number of 1, without which the description of the atom would be incomplete.
He conceived the possible existence of the neutron, but he would not be the one to make the discovery.
The misrepresentation
Working independently from Rutherford were Bothe and Herbert. In 1930, they discovered an unusual type of radiation: one that occurred when beryllium was irradiated with alpha particles from a polonium source. Although this radiation was penetrating and neutral, they described it as (what they thought was) high-energy gamma photons.
And yet another misrepresentation
Two years later, Irene Joliot-Curie and her husband, Jean Frederic-Curie, decided to investigate the work done by Bothe and Herbert. They followed the same procedure, except that they got alpha particles from a very strong polonium source and directed it at a paraffin target. The result of this experiment was spectacular; protons were emitted from the paraffin target. They attributed this result to Compton effect. This effect accounted for the particle nature of light which they believed occurred in the experiment.
As a result, they gave the conclusion that high energy photons were striking the hydrogen atoms in the paraffin which resulted in the ejection of protons. This misrepresentation would later cost them the Nobel Prize for that year.
The discovery
That same year, in 1932, a former pupil of Rutherford, James Chadwick, was working at the Cavendish Laboratory in Cambridge. (Rutherford was the director of this lab, and Chadwick later became his assistant.) He came across the work done by the Joliot-Curies and decided to investigate their result. So he did a repeat experiment, but this time he used many different elements as radiation targets instead of paraffin. He compared the energies of the particles ejected from the various targets and concluded that the radiation which caused the ejection of these particles was much more energetic than the high-energy photons.
Chadwick explained that what really occurred when beryllium was bombarded with alpha particles were the formation of a carbon-12 nucleus and the emission of a neutron particle. With that, he was able to give further explanations of the neutron. This explanation won him the Nobel Prize in Physics for that year.
The answer
With the discovery of the neutron, the description of the atom was now complete. It carries no electric charge, which allows it to split the nucleus of even the heaviest element. It also helps to stabilize the atomic nucleus by pairing off with the protons.
The breakthroughs in nuclear science, the availability of a clean and reliable source of energy, the imminent threat of a nuclear war, the nuclear weapons and the nuclear accidents— all wouldn’t have been possible without the discovery of the neutron.