Determining the uncertainty of Eγ

To determine the uncertainty in Eγ we pick an angle for the neutron within [[math]\theta_n[/math], [math]\theta_n[/math] + Δ [math]\theta_n[/math]] and a momentum of the neutron between [[math]P_n[/math], [math]P_n[/math] + Δ [math]P_n[/math]].

What are reasonable Δ[math]\theta_n[/math] and Δ [math]P_n[/math]?

[math]P_n[/math] is determined by time of flight.

Knowns:

[math]m_n[/math] = 939.565 ± 0.00028 [math]MeV/c^2[/math]

d = 3 ± 0.005 m

t = 50 ± 1 ns

Fractional Uncertainties

[math]\frac{\delta_m}{m}=\frac{0.00028}{939.565}=0.00003%[/math]

[math]\frac{\delta_d}{d}=\frac{0.005}{3}=0.2%[/math]

[math]\frac{\delta_t}{t}=\frac{1}{50}=2%[/math]

[math]v=\frac{d}{t}=\frac{3 +/- 0.2%}{50 +/- 2%}[/math] = 0.2c ± 2.2%

[math]P_n=m_nv[/math] = 188MeV/c ± 2.2%

Δ[math]P_n=4MeV/c[/math]

Δ[math]\theta_n[/math] can be determined knowing that the detector is 3 meters away and the dimensions of the detector are 5cm wide by 5cm tall.

Δ[math]\theta_n=tan^{-1}(\frac{5}{300})=0.0167rads=0.95degrees[/math]

Applying the consevation of energy and momentun to the system we come up with three equations:

[math]E_{\gamma}-1877.9-\sqrt{m_n^2+P_n^2}-\sqrt{m_p^2+P_p^2}[/math]

[math]E_{\gamma}-P_ncos(\theta_n)-P_pcos(\theta_p)[/math]

[math]P_nsin(\theta_n-P_psin(\theta_p)[/math]