Recoil Proton Momentum Spectrum in Neutron Decay

[Waleed Khalid]

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: w_p=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 * m_n/(m_n-m_p)²
and a is the electron neutrino correlation.

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

w_mom=w_p[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 (w_mom=w_p[TofP[p]]*p*TofP[p]). Which doesn't make sense to me, so if anyone can explain I would be highly thankful.

 

[Waleed Khalid]

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/m_n

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: w_p=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 * m_n/(m_n-m_p)²
and a is the electron neutrino correlation.

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

w_mom=w_p[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 (w_mom=w_p[TofP[p]]*p*TofP[p]). Which doesn't make sense to me, so if anyone can explain I would be highly thankful.