I have a question about energy reconstruction. I'm writing up a paper, and I'm trying to understand(adsbygoogle = window.adsbygoogle || []).push({}); whythis is the reason, before putting it in my paper. I know from experimental evidence that it is the case, though I would like to know if there is a reason.

Let's say you have a sufficiently long crystal-scintillator detector, on the order of 50 cm or so. On each end of the crystal is one photo tube. Now, the energy that each photo tube receives can be written as the following formula (I know why, but I'd rather not explain if I don't have to)

[itex]E_{1}=\widetilde{E}_{T} \cdot e^{-x \cdot \alpha}[/itex]

[itex]E_{2}=\widetilde{E}_{T} \cdot e^{-\alpha \cdot (\ell-x)}[/itex]

Where [itex]E_{1,2}[/itex] are the corresponding energies for each detector 1 and 2, [itex]\alpha[/itex] is the light attenuation factor for said detector, [itex]\ell[/itex] is the total length of the detector, [itex]x[/itex] is the position of the incident photon before it interacts with the crystal, and [itex]\widetilde{E}_{T}[/itex] is the theoretical value for the incident gamma - ie the total energy of the gamma.

Now, my question is, why when recreating the energy based off of the energy received from each detector is the equation

[itex]E_{T}^{\prime} = \sqrt{E_{1} \cdot E_{2}}[/itex]

instead of

[itex]E_{T}^{\prime} = E_{1} + E_{2}[/itex]

where [itex]E_{T}^{\prime}[/itex] is the reconstructed energy based on the two PMT's

Can anyone help me understand this?

Thanks in advanced.

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# Energy Reconstruction in Gamma Spectroscopy

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