Question

I was going through the kinematics of beta minus decay. I understood that the Q value in the beta minus decay is the sum of the KE of electron and anti-neutrino

Q=Eₑ+E_ν.
Here the Ee and E_ν are the KE of the electron and antineutrino.

I derived the basic energy-momenta spectra of beta minus decay and obtained the expression as

N(Eₑ)=K∗(√E²ₑ+2Eₑmₑc²)(Q−Eₑ)²(Eₑ+m_ₑc²)
which later I found is the standard energy momenta spectra shape of beta minus decay.

However, in some papers the beta spectra is given as

P=Kₚ_βE_β(Eₒ−E_β)² and it is given that Eo=Q+mec²−Eₙ where Eₙ is the excitation energy of the daughter nucleus.

I checked many papers and books and found that my formula of beta minus decay is correct. But I do not know how to bring the second form of the expression which I actually needed. Basically I have to calculate the anti neutrino spectra from this which comes as

P_ν¯=KE²_ν¯(Eₒ−E_ν¯)(√[(Eₒ−E_ν¯)²−mₑc²]) replacing Eₑ with (Eₒ−E_ν¯)

Can anyone please help me how I can obtain this form of the expression P=Kₚ_βE_β(Eₒ−E_β)²?

Answer 1

The formula for the energy-momenta spectra in beta minus decay can be derived using conservation principles and relativistic energy-momentum relationships. The derived formula you mentioned is correct, and to obtain the second form of the expression, you can substitute Eₑ with (Eₒ−E_ν¯) in your original formula.

Step-by-step explanation:

The formula for the energy-momenta spectra of beta minus decay can be derived using conservation principles and relativistic energy-momentum relationships. The expression N(Eₑ)=K∗(√E²ₑ+2Eₑmₑc²)(Q−Eₑ)²(Eₑ+m_ₑc²) that you derived is the correct formula. However, if you want to obtain the second form of the expression P=Kₚ_βE_β(Eₒ−E_β)², you can substitute Eₑ with (Eₒ−E_ν¯) in your original formula. This will give you the desired expression for calculating the anti-neutrino spectra (P_ν¯=KE²_ν¯(Eₒ−E_ν¯)(√[(Eₒ−E_ν¯)²−mₑc²])).