Verify that the proton recoil energy can be neglected in neutron beta decay. Cal
ID: 2075363 • Letter: V
Question
Verify that the proton recoil energy can be neglected in neutron beta decay. Calculate the density of states rho (E) for neutron beta decay. Using Fermi's golden rule and assuming that the matrix element M_alpha beta = can be averaged over the angle between the electron and the anti-neutrino write down the differential neutron beta decay rate dW_alpha beta integrated over the solid angles d ohm_e d ohm_ v bar. Describe how one might be able to extract the value of the matrix element M_alpha beta from the observed neutron beta decay rates. Here H_w is the weak interaction Hamiltonian and alpha, beta represent the initial and final states respectively .Explanation / Answer
Solution :-
Let us assume
M() = < pe-v (Hw) n>
A free neutron will undergo decay with a half-life of approximately 10 minutes
We might expect to write the decay equation as
(1,0) n -> (1,1) p +(-1,0)e + Q
However,all three particles in this equation are fermions with intrinsic spins S=1/2 . Therefore,we cannot balance the angular momentum in the reaction as written.
The spin of the proton and the electron can be couple to 0 to 1 and can also have relative angular momentum with any integral value from the emission process.
According to above,
The differential neutron beta decay rate from above equation and from the weak interaction Hamiltonian
dW = Hw*M*de*dv *(1,1) p +(-1,0)e + Q
The value of M can be extracted from above values by considering all values mentioned in above equation as a diagonal matrix.
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