# Recoil Proton Momentum Spectrum in Neutron Decay

I wish to draw the proton momentum spectrum by transforming the energy spectrum of recoil protons. I have calculated the energy spectrum using Nachtmann's spectrum: wp=g1[T]+a*g2[T]
Where:
g1[T]=(1 - x2/σ[T])2 * Sqrt[1 - σ[T]] * (4*(1 + x2/σ[T]) - (4/3*(σ[T] - x2)/σ[T])*(1 - σ[T]));
g2[T]=(1 - x2/σ[T])2 * Sqrt[1 - σ[T]] * (4*(1 + x2/σ[T] - 2*σ[T]) - 4/3*(σ[T] - x2)/σ[T]*(1 - σ[T]));
and σ[T]=1 - 2 * T * mn/(mn-mp)2
and a is the electron neutrino correlation.

To get the momentum spectrum, I am transforming the functions (non relativistically):
TofP[p]=p2/(2*mn)

wmom=wp[TofP[p]]

However this doesn't yield the correct spectrum for the momentum of the recoiled protons, as far as I have gotten it I have to multiply it with p and TofP[p] to get the shape of the correct spectrum (wmom=wp[TofP[p]]*p*TofP[p]). Which doesn't make sense to me, so if anyone can explain I would be highly thankful.

Nevermind, I was being stupid, the answer was simple, since I was using a non relativistic conversion I had to multiply with a factor of p/mn

I wish to draw the proton momentum spectrum by transforming the energy spectrum of recoil protons. I have calculated the energy spectrum using Nachtmann's spectrum: wp=g1[T]+a*g2[T]
Where:
g1[T]=(1 - x2/σ[T])2 * Sqrt[1 - σ[T]] * (4*(1 + x2/σ[T]) - (4/3*(σ[T] - x2)/σ[T])*(1 - σ[T]));
g2[T]=(1 - x2/σ[T])2 * Sqrt[1 - σ[T]] * (4*(1 + x2/σ[T] - 2*σ[T]) - 4/3*(σ[T] - x2)/σ[T]*(1 - σ[T]));
and σ[T]=1 - 2 * T * mn/(mn-mp)2
and a is the electron neutrino correlation.

To get the momentum spectrum, I am transforming the functions (non relativistically):
TofP[p]=p2/(2*mn)

wmom=wp[TofP[p]]

However this doesn't yield the correct spectrum for the momentum of the recoiled protons, as far as I have gotten it I have to multiply it with p and TofP[p] to get the shape of the correct spectrum (wmom=wp[TofP[p]]*p*TofP[p]). Which doesn't make sense to me, so if anyone can explain I would be highly thankful.