Timing Resolution Required In Particle Detector

[BOAS]

Homework Statement

A beam on Pions, Kaons and Protons, all with momentum $\mathrm{P} = 10 \mathrm{GeV}$ and negligible angular divergence travels $100 \mathrm{m}$ before hitting a target. What is the required timing resolution of the detector so that pions and kaons can be distinguished with a 10% precision.

Homework Equations

The Attempt at a Solution

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I have calculated the time difference between the pions and kaons as follows:

A particle with mass $m$ and momentum $p$ has velocity $\beta = \frac{p}{\sqrt{p^2 + m^2}}$. For a path length $L$, the time of flight $T = \frac{L}{\beta c}$.

Therefore two particles with different masses, but the same momentum arrive with a time difference of $T_1 - T_2 = \frac{L}{c}(\sqrt{1 + \frac{m^2_1}{P^2}} - \sqrt{1 + \frac{m^2_2}{p^2}})$

I am unsure of how to take this further because I am confused by what it means to distinguish the two particles with a given precision.

What I am picturing is something like this;

the timing resolution of the detector is basically what determines the width of a 'bin' when counting pions and kaons. I need to make my bin widths small enough so that 10% of a measured arrival time is smaller than the time difference between the two particle's arrivals. Does that make sense?

Thanks for any help you can give!

 

[mfb]

I'm a particle physicist, and I don't understand that "10% precision" statement either. I guess we don't care about actual applications, so let's call everything that is measured closer to the pion time "pion" and everything closer to the kaon time "kaon". That is not what an actual detector would do, but that is beyond the scope of this problem. 10% could refer to:

- the timing uncertainty could be 10% of the time difference.
- a pion gets misidentified as kaon in 10% of all cases. Due to symmetry, it also means a kaon gets misidentified as pion in 10% of the cases.
- in the context of particle identification, the "identification power" is often defined as 1-2*(misidentification rate) or the square of that value. This quantity could be 0.1. Here 0 refers to random guessing while 1 means perfect identification.
- the time difference could be 10% of the timing uncertainty

Roughly sorted by probability. If this would appear in an actual scientific context, option (3) would be by far the most likely, but as homework question I don't see how they could expect that.