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Both Carbon-14 and Carbon-15 decay by the $\beta^-$ mechanism but the half life for $C^{14}$ is approximately 5500 years whereas that if $C^{15}$ is around 2 seconds.What causes this disparity in half life time?

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  • $\begingroup$ This is pretty common. As another example, half-life of carbon-12 is immeasurably long. $\endgroup$ – The Photon Mar 3 at 2:00
  • $\begingroup$ Yes, but the difference between C-14 and C-15 is just 1 neutron but the half lives vary greatly, I wanted to know what causes this huge difference. $\endgroup$ – Vaishakh Sreekanth Menon Mar 3 at 2:03
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Radioactive decay occurs if there is an available state with lower energy than the current state of the nucleus. For example, outside of a nucleus, neutrons decay into protons because protons are lighter (have lower mass-energy) than neutrons.

Inside a nucleus, however, neutrons can be stabilized by interactions with protons. For example, a nucleus with many more protons than neutrons has a very high amount of electric charge concentrated in one place. The electrostatic potential energy increases the total energy of the nucleus by an amount large enough to offset the proton-neutron mass difference. Thus, the energy of the nucleus will be lower if a proton decays into a neutron, and the nucleus decays.

Nuclear physics is much more complex than this, and a large imbalance between neutrons and protons is bad for nuclear stability for other reasons as well. For example, a high density of neutrons means that neutrons are forced into high-energy states (for intuition on this, compare the particle in a box model). Also, neutrons repel each other via the strong force, as do protons, but neutrons and protons attract each other. Since carbon-15 has a very large imbalance of neutrons and protons — 50% more neutrons than protons — we expect the energy difference $\Delta E$ from carbon-15 to nitrogen-15 to be large.

Beta decay proceeds via the weak interaction, so the rate is strongly dependent on the energy difference (see the answers to Why is the (free) neutron lifetime so long? for more information). Thus, all other things being equal, if $\Delta E$ between two states is 2 MeV, the decay rate is much, much higher than it is for a $\Delta E$ of 1 MeV.

According to the IAEA table of nuclides, $\Delta E$ from carbon-15 to nitrogen-15 is 9873.1 keV - 101.4 keV = 9771.7 keV. Thus, a lot of energy is available to power the decay and it proceeds quickly. On the other hand, $\Delta E$ from carbon-14 to nitrogen-14 is only 3019.9 keV - 2863.4 keV = 157 keV, so decay proceeds slowly.

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