## Neutrinos (Part 1): Sneaking into Beta decay

September 18, 2019 by anon

Radioactivity was a popular topic of discussion during the early 20th century, scientists wanted to understand this mysterious phenomenon where elements would spontaneously decay into other elements by releasing an electron or a positron (anti-electron). Most of the elements which behaved in this peculiar manner are heavy isotopes of simple elements.

$_{19} K^{40}\rightarrow _{20}Ca^{40}+e^-$$_{19} K^{40}\rightarrow _{20}Ca^{40}+e^-$

The above interaction is an example of Beta- Decay, but this is not exactly what’s happening. If we have a stationary sample which breaks into two parts and starts moving they should move apart from each other in opposite directions and hence (for identical samples) the velocity of the particles must be identical due to the Law of Conservation of Momentum. As electrons were relatively easy to observe they should have had the same kinetic energy for any sample of Potassium-40, but this was not observed. In fact the observed energy was always less than the theoretical calculation. This anomaly drove scientists crazy, Neils Bohr almost disregarded the Law of Conservation of Energy but in 1930 Wolfgang Pauli came up with a solution. He proposed a neutral particle called ‘neutron’ which compensated for the energy difference and left no trace as it was neutral.

In 1932 James Chadwick discovered a neutral particle within the nucleus of the atom and called it neutron, many believed this was the mysterious particle released in the Beta-Decay but it was soon realized that the particle proposed by Pauli had very little energy while the one proposed by Chadwick was relatively higher in energy. Finally, Chadwick’s particle was named the neutron. A few years later an Italian physicist, Enrico Fermi coined the term neutrino to Pauli’s proposed particle, the suffix ‘-ino’ indicates something small in Italian, so neutrino literally meant the smaller neutral particle (neutron+ino). Further research into Beta-Decay showed that the phenomenon was actually the spontaneous conversion of a neutron into a proton or vise-versa.

Neutrinos are of 3 flavors, the electron, muon and tau ($\nu_e, \nu_{\mu}, \nu_{\tau}$). Muon and tau particles have similar properties to electrons (all of them are negetively charged), only their masses are different. A muon is almost 207 times heavier than an electron and tau is 17 times heavier than a muon. The flavor of neutrinos are determined based on their interactions, let’s say a neutrino interacts with a neutron, then an electron flavored neutron will produce an electron, muon neutrino a muon and tau neutrino a tau.

$n + \nu_l \rightarrow p^+ + l^- \;\;\; (l=e,\mu,\tau)$$n + \nu_l \rightarrow p^+ + l^- \;\;\; (l=e,\mu,\tau)$

Neutrinos, electron, muon and tau are collectively called leptons and to distinguish them from their anti particles (anti-leptons) they are assigned a numerical value called the lepton number. Leptons have a lepton number +1(L=+1) whereas anti-leptons have a lepton number -1(L=-1). In any interaction the overall lepton number must sum up to 0, this is called the Law of conservation of lepton number.

The spontaneous conversion of Potassium-40 to Calcium-40 can now be simplified and written in terms of interactions of elementary particles

$n \rightarrow p^++e^-+\overline{\nu_e}$

which is the $\beta^-$ decay in the familiar form.

Using the Law of conservation of lepton number, we can rearrange a particle interaction (just like how we rearrange any mathematical equation) to determine other possible interactions we may observe in nature. For example we can shift the anti-neutrino to the other side, making it a neutrino and shift the electron, making it the positron (notice the sign of lepton number is changed in a way the overall lepton number is still 0) this forms the familiar $\beta^+$ decay

$p^+ \rightarrow n+ e^+ + \nu_e$

Guest post written by Rohan VJ
Rohan VJ is a student at St.Joseph’s College, Bengaluru. He is interested in theoretical physics, and also handles @our.inexplicable.universe on instagram.

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