Electron Spectrum in Beta Decay

[Pi-Bond]

Homework Statement

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:

[itex]dw=p(E_e)dE_e[/itex]

[itex]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}[/itex]

where E_e is the energy of the resultant. m_e and m_v 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, E_max, p(E_e) has the following forms

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

Homework Equations

[itex]E_e = \Delta Mc^2 - \sqrt{(p_v c)^2 + (m_v c^2)^2}[/itex]

where p_v is the neutrino momentum. (Energy conservation)

The Attempt at a Solution

E_max occurs when p_v is zero. So E_max=ΔMc² if m_v is zero, and E_max=ΔMc²-m_vc² if m_v is non-zero.

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

Taking the m_v case first,

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

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 m_e) ?

 

[BigJDubs]

For the mv=0 case, I can expand the expression in powers of ε, but I'm not sure what to do with the square root.p(E_e) \propto (E_{max}-\epsilon)(E_{max}-\epsilon) \epsilonI'm not sure how to proceed from here.