Introduction
At the time when the nucleus was discovered by Rutherford, only two atomic particles, proton, and electron were known. If only photons are inside the nucleus then the charge of the nucleus would be A units, where A is the mass number of the atom. But, it was already observed that except for hydrogen, the nuclear charge Z is found to be always less than A, usually less than A/2. To get around this difficulty, it was assumed early that in addition to the protons, atomic nuclei contain just enough electrons to cancel the positive charge of the extra protons; that is, if A is the number of Protons then they were supposed to contain A-Z electrons. These electrons would contribute only a small amount to the mass of the nucleus, but together with the protons, they would make the net charge equal to Z units, as required.
So, it seemed plausible to consider the atom as consisting of a nucleus made up of A protons and A-Z electrons, with Z additional electrons outside the nucleus to make the entire atom electrically neutral. For example, an oxygen atom would have a nucleus with 16 protons and 8 electrons, with 8 additional electrons outside the nucleus. This model of the nucleus is known as the proton-electron hypothesis of nuclear composition.
The proton-electron hypothesis seemed to be consistent with the emission of alpha and beta particles by atoms of radioactive substances. Since it was assumed that the nucleus contained electrons, the explanation of beta decay was no problem. It also seemed reasonable that an alpha particle could be formed, in the nucleus, by the combination of four protons and two electrons.
Violation of the Heisenberg’s Uncertainty Principle
Although the proton-electron hypothesis was satisfactory in some respects, it led to serious difficulties and had to be given up. One of the most serious difficulties arose from Heisenberg’s uncertainty principle in quantum mechanics.
If the electron 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 we can write the uncertainty in the position of the electron as,
\begin{aligned}\Delta x &= 10^{-15} \rm \ m\\\end{aligned}
Now if \Delta p is the uncertainty in momentum, then from the Heisenberg 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 \times1 \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 electron 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+ (0.511\rm \ MeV)^2}\\&\approx98.7\rm \ MeV\\\end{aligned}
And the Kinetic energy of the electron will be:
\begin{aligned}K &= E – m_0 c^2\\&=98.7\rm \ MeV-0.511\rm \ MeV \\&\approx 98.2\rm \ MeV\\\end{aligned}
And finally, the speed of the electron is calculated as
\begin{aligned}v&=\sqrt{\dfrac{2K}{m_0}}\\&=\sqrt{\dfrac{2\times98.2\rm \ MeV}{0.511\rm \ MeV/c^2}}\\&\approx20\rm \ c\\\end{aligned}
So, the confinement of an electron to a space as small as the nucleus would result in the circumstance that the electron’s speed would be around 20 times the speed of light, which is not possible according to special relativity theory.
But how could scientists account for the circumstance that electrons cannot be confined within the nucleus, yet they emerge from the nucleus in beta decay?
There is a story behind this problem.
One day, Heisenberg and his assistants were contemplating this problem while sitting in a café across from a building housing a swimming pool. Heisenberg suggested a possible approach to the problem. He told his assistant
You see people going into the building fully dressed and you also see them coming out fully dressed. But does that mean that they also swim fully dressed?
He actually wanted to say
you see electrons coming out of the nucleus in beta decay, and occasionally being captured by the nucleus, but that doesn’t mean that electrons remain in the nucleus. Perhaps the electrons are created in the process of emission from the nucleus.
Violation of the Angular Momentum Coupling Rule
It was also observed that the proton-electron hypothesis violates the angular momentum coupling rule. Let’s understand it by taking an example of the deuteron. The deuteron is the nucleus of one of the isotopes of Hydrogen, Deuterium.
The deuteron has, A=2 and Z=1. So from the proton-electron hypothesis, there would be two protons and one electron.
Since proton and electron both have \rm spin=\pm\frac{1}{2}\hbar which can simply be written as \pm\frac{1}{2} because in natural unit \hbar = 1 so, for the proton and electron, we have \rm spin = \pm \frac{1}{2}. Here + sign is for the spin along the +z-axis and the – sign is for the –z-axis.
When you add the spin vectors of two protons and one electron then the possible spins of deuteron would be either \pm \frac{1}{2} or \pm \frac{3}{2}. But the measured ground state spin of the deuteron is calculated as +1. And it is also a measure problem for the proton-electron hypothesis.
Wrong Calculation of Magnetic Moment Value of Nuclei
Now let’s prove the proton-electron hypothesis wrong by calculating the magnetic moment value of the nucleus.
The magnetic moment of the electron is
\begin{aligned}\mu_e=\dfrac{e\hbar}{2m_e}\end{aligned}
The magnetic moment of the proton is
\begin{aligned}\mu_p=\dfrac{e\hbar}{2m_p}\end{aligned}
Since the mass of the proton is almost 1836 times the mass of the electron. So, the magnetic moment of the electron would be almost 1836 times the magnetic moment of the proton.
Now, if an electron would exist in the nucleus then the nuclear magnetic moment would be of the order of the magnetic moment of the electron but the measured magnetic moment of nuclei is very much less than the magnetic moment of the electron. Thus the electron cannot reside inside the nucleus.
Also, the presence of some electrons in the nucleus and rest as peripheral around the nucleus exhibit the dual role of electrons in an atomic structure which is difficult to understand.