# Four-vector Algebra

Consider two body reaction :

Write down four-momentum vectors before and after reaction:





Now apply the law of conservation of four-momentum vectors:




Squaring both side of equation above and using the four-momentum invariants we have:




# Detector located at case

Detector is located at , and the formula above is simplified:




We can easily solve the equation above with respect to incident photon energy:




For non-relativistic neutrons and the formula above is become:




Substituting the corresponding masses, we get finally:




and visa versa:




Here I derived the formula [2] just inversing the formula [1]. I can as well start from exact solution above, solve this equation with respect to neutron energy, do the non-relativistic approximation and get exactly the same formula [2]. But anyway we ended up with two useful non-relativistic formulas we can analyze now:

1) from formula [1] above we can predict the threshold of reaction in direction:




comment: it's not true because by momentum conservation there are no neutron with zero kinetic energy.

2) from formula [1] above we can predict the incident photon energy based on the detected neutron energy (neutron polarimeter).

3) from formula [2] above we can predict the detected neutron energy based on the incident photon energy.

 -  for the incident photons up to  we can detect neutrons up to

 -  for the incident photons up to  we can detect neutrons up to


4) we can do the error calculations.

# Example of error calculation

## simple error calculation

Say, we have, 10 MeV neutron with uncertainty 1 MeV, the corresponding uncertainly for photons energy is:




## photon kinetic energy error as function of TOF uncertainty

In the calculations below I attempted to predict the uncertainty in photons energy based on uncertainty in neutrons time of flight.

The neutron kinetic energy as function of time of flight is:




By taking derivative of the expression above we can find the relative error for neutron energy:




In that formula for we need to know the neutron time of flight which is:




And now we can calculate the relative error for photon energy using the formula derived before:




Say, the detector is 1.5 m away and neutron's time of flight uncertainty is:




And below are y numerical calculations and plot with error bars:

And in the plot below I have overlay my error calculations using the formulas above:

Now say, the detector is 1.5 m away and neutron's time of flight uncertainty is:




# dependence

Starting from general kinematic relations:




we can solve the equation above with respect to as function of neutron energy and neutron angle




and using





finally




Let's do some plots using the equation derived above. Below is presented the photon energy as function of neutron angle for several values of neutron energy:

And below are the photon energy as function of neutron angle and neutron energy: