Introduction
The discovery of the neutron is an extraordinary development in atomic physics in the first half of the 20th century. Before that protons and electrons had already been discovered. In 1911, Ernest Rutherford developed an excellent model of the atom based on the gold foil experiment of Geiger and Marsden, called the Rutherford model of the atom. In his given model, he concluded that atoms had their mass and positive electric charge concentrated in a very tiny nucleus. But at that time there was a curiosity in every mind that which particles would be inside the nucleus. Initially, the nucleus was viewed as composed of combinations of protons and electrons, but that hypothesis presented several experimental and theoretical contradictions which I have discussed in the last post on Nuclear Physics.
Failure of Proton-Electron Hypothesis
The proton-electron theory failed due to the properties of free electrons. Therefore, in 1920, Rutherford suggested that the electron does not exist in the nucleus in the free-state but is bound with the positively heavy nucleus.
The Discovery Of The Neutron
In 1930, Walther Bothe and Herbert Becker in Giessen, Germany found that if the energetic alpha particles fell on certain light elements, such as beryllium, boron, or lithium, then unusually penetrating radiation is produced. It was observed that the penetrating radiation was not influenced by an electric field and hence it was thought to be gamma radiation. But, the radiation was quite more penetrating than any gamma rays known, and so the details of experimental results were difficult to interpret.
Two years later Joliot and Curie studied the same reaction in Paris. They made an arrangement in which the penetrating radiation was allowed to fall on a hydrogen-containing compound like paraffin wax.
They observed that the penetrating radiation ejected out protons from the paraffin wax which was found to be very energetic having energy around 5\rm \ MeV. But they misinterpreted the phenomenon as a scattering of gamma-rays on protons similar to the scattering of gamma-rays on electrons in the Compton Effect.
On hearing of the Paris results, neither Rutherford nor James Chadwick believed in the gamma-ray hypothesis. At that time they were working at the Cavendish Laboratory, University of Cambridge. Chadwick quickly performed a series of experiments showing that the gamma-ray hypothesis could not really be fitted with the Paris results.
Chadwick repeated the creation of the radiation using beryllium and the alpha particles and the radiation was allowed to fall on paraffin wax. Just like the Paris experiment, the radiation scattered some protons. Chadwick measured the range of these protons and also measured how the radiation impacted the atoms of various gases. He used several target materials like H, He, Li, etc on the way of radiation and observed that the particles ejected from hydrogen behaved like protons with speeds up to 3.2\times 10^7\rm \ m/s. On the basis of his observations, he concluded that if the ejection of a proton is due to the scattering of a photon on the nucleus, then to speed up the proton up to 3.2\times 10^7\rm \ m/s, a 52\rm \ MeV photon is needed. This exceeded all known energies of photons, emitted by nuclei. All difficulties disappeared when he assumed that incident particles are neutral particles with a mass equal to that of a proton.
Chadwick called this neutral particle neutron and published its findings in a letter to Nature Journal in 1932, on the basis of which he got the Nobel Prize in 1935.
Proton-Neutron Hypothesis
The discovery of neutrons led to the presently accepted hypothesis for the constitution of the nucleus. According to this hypothesis, every nucleus consists of protons and neutrons, and the total number of two types of particles is equal to the mass number A.
There are many facets that are in support of the proton-neutron hypothesis. One of the important facts which support the proton-neutron hypothesis is Heisenberg’s uncertainty principle in quantum mechanics.
If the neutron has to be within the nucleus, then its de-Broglie wavelength had to be of the order of the size of the nucleus.
Since the size of a nucleus is in the order of 1 fermi so, the uncertainty in the position of the neutron inside a nucleus is;
\begin{aligned}\Delta x &= 1\rm \ fermi\\ &= 10^{-15}\rm \ m\\\end{aligned}
Now if \Delta p is the uncertainty in the momentum of the neutron, then from the uncertainty principle,
\begin{aligned}\Delta p\times \Delta x &\geq \dfrac{\hbar}{2}\\\end{aligned}
That is;
\begin{aligned}\Delta p &\geq \dfrac{\hbar}{2\Delta x}\\&= \dfrac{197.46\rm \ MeV/c.fermi}{2 \times 1 \rm \ Fermi}\\&= 98.7\rm \ MeV/c\\\end{aligned}
Now from the formula;
\begin{aligned}E^2&=(pc)^2+ (m_0 c^2)^2\\\end{aligned}
The energy of the neutron correspondence to this uncertainty in momentum is;
\begin{aligned}E&=\sqrt{(\Delta pc)^2+ (m_0 c^2)^2}\\&=\sqrt{(98.7\rm \ MeV)^2+ (939.6\rm \ MeV)^2}\\&\approx 945\rm \ MeV\\\end{aligned}
And the Kinetic energy of the neutron will be:
\begin{aligned}K &= E – m_0 c^2\\&=945\rm \ MeV-939.6\rm \ MeV\\&\approx5.4\rm \ MeV\\\end{aligned}
And finally, the speed of the neutron is calculated as;
\begin{aligned}v&=\sqrt{\dfrac{2K}{m_0}}\\&= \sqrt{\dfrac{2\times 5.4\rm \ MeV}{939.6\rm \ MeV/c^2}}\\&\approx 0.107\rm \ c\\\end{aligned}
Which is around 1 percent of the speed of light.
Since the kinetic energy of the neutron is few MeV and the speed of the neutron is around 1 percent of the speed of light. So, it is possible to have a free neutron to be contained in the nucleus.