Homework Help: Electron Spectrum in Beta Decay

1. Apr 22, 2013

Pi-Bond

1. The problem statement, all variables and given/known data
A nucleus N1 decays through beta decay to nucleus N2. The mass difference between N1 and N2 is ΔM. The differential decay rate may be written as:

$dw=p(E_e)dE_e$

$p(E_e) \propto E_e (E_e^2-(m_ec^2))^{1/2}(\Delta Mc^2-E_e)((\Delta Mc^2-E_e)^2-(m_vc^2)^2)^{1/2}$

where Ee is the energy of the resultant. me and mv represent the mass of the electron and neutrino respectively.

The electron spectrum near the maximum of the electron energy can be used to ascertain if the neutrino has mass. Show that, near the maximum electron energy, Emax, p(Ee) has the following forms

$p(E_e) \propto (E_{max}-E_e)^2 \mbox{ if m_v is zero}$
$p(E_e) \propto (E_{max}-E_e)^{1/2} \mbox{ if m_v is non-zero}$

2. Relevant equations
$E_e = \Delta Mc^2 - \sqrt{(p_v c)^2 + (m_v c^2)^2}$

where pv is the neutrino momentum. (Energy conservation)

3. The attempt at a solution

Emax occurs when pv is zero. So Emax=ΔMc2 if mv is zero, and Emax=ΔMc2-mvc2 if mv is non-zero.

The question gives a hint to use ε=Emax-Ee, and expand in powers of ε.

Taking the mv case first,

$p(E_e) \propto (E_{max}-\epsilon) ((E_{max}-\epsilon)^2-(m_ec^2))^{1/2}\epsilon (\epsilon^2-(0)^2)^{1/2}$

I'm not sure what to do now. The expression with the square root doesn't seem to be expandable without imaginary numbers. Should I neglect the constant term (the one with me) ?