Undergrad Decay constant of Different Elements and Isotopes?

[Aromalsp]

TL;DR

Is there an equation to predict the decay constant of elements and isotopes?

Different elements and isotopes have different rates of beta decay because the half-life of the element or isotope reflects its stability, which is determined by the nuclear force between the protons and neutrons in the nucleus of the atom. The number of protons and neutrons affects the balance of the nuclear force, which in turn affects the stability of the atom and determines its half-life and rate of beta decay. Is there any law to predict the rate of decay?
And how does an electron and anti neutrino get formed when an up quark turns into a down quark?

 

[Vanadium 50]

There is Sargent's Rule.

 

[phyzguy]

Rather than a law, there are patterns of nuclear stability. Referring to this chart from Wikipedia, you can see that stable isotopes fall along the valley of stability. This is a line with equal numbers of protons and neutrons for low atomic numbers, but bends towards larger numbers of neutrons as the atomic number increases. This can be qualitatively understood because as you have more protons the repulsive electromagnetic force gets stronger, so you need more neutrons so the strong force can counterbalance the stronger electromagnetic repulsion. The further you depart from this valley of stability, the shorter the half life of the isotope.

 

[snorkack]

The rate of decays strongly depends on the spin of the mother... and daughter. Mind you, not just the ground state of the daughter but the spins and energies of all accessible excited states of the daughter.
Apart from the 4 light odd-odd nuclei which are daughters of decay of even-even ones (these are D, Li-6, B-10 and N-14), there are 5 primordial odd-odd nuclei - all of them have spin over 3. Remember that at least the ground states of their daughters must have spin 0:

1. K-40 spin 4 half-life 1,25*10⁹ y
2. La-138 spin 5 half-life 102*10⁹ y
3. V-50 spin 6 half-life 271*10¹⁵ y
4. Lu-176 spin 7 half-life 38*10⁹ y
5. Ta-180m spin 9 stable

How do you predict the correct half-lives of V-50, Lu-176 and Ta-180m from their spins? Can you get the correct branching ratios between beta decay, electron capture and positron decay?