Determining the uncertainty of Eγ

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

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

[math]\displaystyle{ P_n }[/math] is determined by time of flight.

Knowns:

[math]\displaystyle{ m_n }[/math] = 939.565 ± 0.00028 [math]\displaystyle{ MeV/c^2 }[/math]

d = 3 ± 0.005 m

t = 50 ± 1 ns

Fractional Uncertainties

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

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

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

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

[math]\displaystyle{ P_n=m_nv }[/math] = 188MeV/c ± 2.2%

Δ[math]\displaystyle{ \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.