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Analysis of energy dependence [math]T_{\gamma}\left( T_n\right)[/math]
four-vectors algebra
[math] E = T + m[/math]
[math] E = p^2 + m^2[/math]
writing four-vectors:
[math] p_{\gamma} = \left( T_{\gamma},\ T_{\gamma},\ 0,\ 0 \right) [/math]
[math] p_D = \left( m_D,\ 0,\ 0,\ 0 \right) [/math]
[math] p_{n} = \left( E_n,\ p_n\cos(\Theta_n),\ p_n\sin(\Theta_n),\ 0 \right) [/math]
[math] p_{p} = \left( E_p,\ p_p\cos(\Theta_p),\ p_p\sin(\Theta_p),\ 0 \right) [/math]
Doing four-vector algebra:
[math] p^{\mu}_{\gamma} + p^{\mu}_D = p^{\mu}_p + p^{\mu}_n \Rightarrow [/math]
[math] p^{\mu\ 2}_p = \left(p^{\mu}_{\gamma} + p^{\mu}_D - p^{\mu}_n\right)^2 =
p^{\mu\ 2}_{\gamma} + p^{\mu\ 2}_D + p^{\mu\ 2}_n +
2\ p^{\mu}_{\gamma}\ p^{\mu}_D - 2\ p^{\mu}_{\gamma}\ p^{\mu}_n - 2\ p^{\mu}_D\ p^{\mu}_n [/math]
[math] m_p^2 - m_{\gamma}^2(=0) - m_D^2 - m_n^2 = [/math]
[math] = 2\ T_{\gamma}\ m_D - 2\left( T_{\gamma}\ E_n - T_{\gamma}\ p_n\cos(\Theta_n)\right) - 2\ m_D\ E_n [/math]
[math] = 2\ T_{\gamma}\left( m_D - E_n + p_n\cos(\Theta_n) \right) - 2\ m_D\ E_n [/math]
Detector is located at [math]\Theta_n = 90^o[/math], so
[math] T_{\gamma} = \frac {2\ m_D\ E_n + m_p^2 - m_D^2 - m_n^2} {2\left( m_D - E_n \right)} =
\frac {2\ m_D\ (T_n + m_n) + m_p^2 - m_D^2 - m_n^2} {2\left( m_D - (T_n + m_n) \right)}[/math]
and visa versa
[math] T_n = \frac {2\ T_{\gamma}\ m_D + m_D^2 + m_n^2 - m_p^2} {2\left( T_{\gamma} + m_D \right)} - m_n[/math]
how it looks
low energy approximation
As we can see from Fig.2 for low energy neutrons (0-21 MeV)
energy dependence of incident photons is linear
Find that dependence. We have:
[math] T_{\gamma}(0\ MeV) = 1.715360792\ MeV [/math]
[math] T_{\gamma}(21\ MeV) = 44.78703086\ MeV [/math]
So, the equation of the line is:
[math] T_{\gamma}
= \frac{T_{\gamma}(21\ MeV) - T_{\gamma}(0\ MeV)}{21\ MeV - 0\ MeV}\ T_n + T_{\gamma}(0\ MeV) [/math]
Finally for low energy neutrons (0-21 MeV):
[math] T_{\gamma} = 2.051\ T_n + 1.715 [/math]
example of error calculation
example 1
Say, we have, 10 MeV neutron with uncertainty 1 MeV,
the corresponding uncertainly for photons energy is:
[math] \delta T_{\gamma} = 2.051\ \delta T_n = 2.051\times 1\ MeV = 2.051\ MeV [/math]
example 2
Say, the neutron's time of flight uncertainly is 1 ns
The neutron's kinetic energy is:
[math]T_n = m_n (\gamma - 1) = m_n\left[ \frac{1}{\sqrt{1-\left(\frac{l}{c\ t}\right)^2}} - 1 \right][/math]
And:
[math]\delta T_n \left(\delta t_n\right) = -\ \frac{m\ l^2}{\left(1-\left(\frac{l}{c\ t}\right)^2\right)^{3/2}c^2 t^3}[/math]
Also we need the neutron's time of flight as function of neutron's kinetic energy:
[math]t:=\frac{l}{c\ \beta_n} = \frac{l}{c\ (p_n/E_n)} =
\frac{l\ (T_n + m_n)}{c\sqrt{T_n^2+2m_nT_n}}[/math]
Say, we have 10 MeV neutron, 1 m away detector, and neutron's time of flight uncertainty is 1 ns
Using formulas above:
[math]t(T_n = 10\ MeV) = 23\ ns[/math] neutron time of flight
[math]\delta T_n(\delta t = 1\ ns) = 0.88\ MeV[/math] absolute neutron kinetic energy error
[math]\frac{\delta T_n}{T_n} = \frac{0.88\ MeV}{10\ MeV} = 8.8\ %[/math] relative neutron kinetic energy error
[math]\delta T_{\gamma} = 2.051\cdot \delta T_n = 2.051\cdot 0.88\ MeV = 1.81\ MeV [/math] absolute photon energy error
[math]\frac{\delta T_{\gamma}}{T_{\gamma}} = \frac{1.80\ MeV}{(2.051\cdot 10 + 1.715)\ MeV} = 8.1\ %[/math] relative photon energy error
Some other calculations for different detector distance and neutron kinetic energy are:
| [math]\delta t_n[/math]
|
[math]l[/math]
|
[math]T_n[/math]
|
[math]\beta_n[/math]
|
[math]t_n[/math]
|
[math]\delta T_n[/math]
|
[math]\frac{\delta T_n}{T_n}[/math]
|
[math]\delta T_{\gamma}[/math]
|
[math]\frac{\delta T_{\gamma}}{T_{\gamma}}[/math]
|
| 1 ns |
1 m |
5 MeV |
0.103 |
32 ns |
0.31 MeV |
6.2 % |
0.64 MeV |
5.3 %
|
| 1 ns |
1 m |
10 MeV |
0.145 |
23 ns |
0.88 MeV |
8.8 % |
1.81 MeV |
8.1 %
|
| 1 ns |
1 m |
20 MeV |
0.203 |
16 ns |
2.51 MeV |
12.6 % |
5.16 MeV |
12.1 %
|
| 1 ns |
1 m |
0.5 MeV |
0.033 |
102 ns |
0.010 MeV |
1.9 % |
|
|
| 1 ns |
1 m |
1 MeV |
0.046 |
72 ns |
0.028 MeV |
2.8 % |
|
|
| 1 ns |
1 m |
2 MeV |
0.065 |
51 ns |
0.078 MeV |
3.9 % |
|
|
| 1 ns |
1 m |
4 MeV |
0.092 |
36 ns |
0.22 MeV |
5.5 % |
|
|
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