Question

Fermi's Law of Beta Decay - the matrix element?

Answer 1

In Fermi's law of beta decay we have the matrix element:

∫ψ∗pψ∗eψ∗νH′ψndr3∫ψp∗ψe∗ψν∗H′ψndr3

on which several assumptions are made, firstly we assume that ψeψe and ψνψν can be treated as plane waves since this holds away from the potential. i.e.

ψe=1V−−√eik⃗ r⃗ (1)(1)ψe=1Veik→r→

and likewise for νν we then assume what is called that allowed approximation i.e. in the range where H′H′ is significant k⃗ r⃗ <<0k→r→<<0 and thus:

ψe≈1V−−√(2)(2)ψe≈1V

To me these seem in conflict. Surly near in the region where k⃗ r⃗ <<0k→r→<<0 our approximation of a plane wave solution fails and thus we should not be able to use (1) to get (2) as the wave function does not take this form. Please can someone explain to me why this we can make these approximations.

Answer 2

Fermi's theory was the first theoretical effort in describing nuclear decay rates forbeta decay. The Gamow–Teller theory was a necessary extension of Fermi's theory. ... However, this did not incorporate parity violation within the matrix element inFermi's Golden Rule seen in weak interactions.