KAHIBARO
Table of Contents
Using time of flight to identify particles
A time of flight detector helps identify a particle by measuring how long the particle takes to travel a known distance. If the path length $L$ and travel time $t$ are known, the particle speed is
$$
v = \frac{L}{t}
$$
and it is often convenient to write this as
$$
\beta = \frac{v}{c} = \frac{L}{ct}
$$
where $c$ is the speed of light. By itself, speed is not enough to tell exactly which particle was detected, because different particles can move at similar speeds. The key idea of particle identification is to combine the speed from time of flight with the momentum measured by a tracking system. From momentum and speed together, the particle mass can be inferred, and that mass can be compared with known particle species such as pions, kaons, and protons.
Mass from momentum and flight time
In relativistic motion, the momentum is
$$
p = \gamma m v
$$
with
$$
\gamma = \frac{1}{\sqrt{1-\beta^2}}
$$
Solving for the mass gives
$$
m = \frac{p}{\gamma v}
$$
Using $\beta = v/c$, this is often written as
$$
m = \frac{p}{\gamma \beta c}
$$
and therefore
$$
m^2 c^2 = p^2\left(\frac{1}{\beta^2} - 1\right)
$$
This form is especially useful in experiments because the detector measures $p$ and $t$, then computes $\beta$, then obtains $m^2$.
Important identification formula:
$$
\beta = \frac{L}{ct}
$$
$$
m^2 c^2 = p^2\left(\frac{1}{\beta^2} - 1\right)
$$
A particle can be identified by comparing the calculated mass, or mass squared, with known particle masses.
Why different particles arrive at different times
If two particles have the same momentum but different masses, the heavier one usually moves more slowly and arrives later. This difference in arrival time is what makes time of flight particle identification possible.
For a particle of momentum $p$ and mass $m$,
$$
\beta = \frac{p c}{E}
$$
with
$$
E = \sqrt{p^2 c^2 + m^2 c^4}
$$
So at fixed momentum, increasing the mass lowers $\beta$, increases $t$, and makes the particle easier to separate from lighter species.
A simple comparison is shown below.
| Particle type | Mass | At same momentum, speed | Arrival time |
|---|---|---|---|
| Electron | very small | very high | earliest |
| Pion | light | high | early |
| Kaon | medium | lower | later |
| Proton | heavier | lower still | latest |
This ordering is most useful at low and moderate momentum. At very high momentum, all particles approach $v \approx c$, so their times become very similar and time of flight loses separating power.
Time separation between particle species
Suppose two particles travel the same path length $L$. Their arrival times are
$$
t_1 = \frac{L}{v_1}, \qquad t_2 = \frac{L}{v_2}
$$
and the time difference is
$$
\Delta t = L\left(\frac{1}{v_2} - \frac{1}{v_1}\right)
= \frac{L}{c}\left(\frac{1}{\beta_2} - \frac{1}{\beta_1}\right)
$$
If $\Delta t$ is larger than the timing uncertainty of the detector, then the two species can be separated.
Particle identification by time of flight works only if the time difference between species is large enough compared with the detector time resolution.
A rough rule is that better separation requires
$$
\Delta t \gg \sigma_t
$$
where $\sigma_t$ is the timing uncertainty.
Example of pion and kaon separation
Imagine a detector with path length $L = 2.0\ \text{m}$, and two particles each with momentum $p = 1.0\ \text{GeV}/c$. Use approximate rest masses
$$
m_\pi c^2 \approx 0.140\ \text{GeV}, \qquad m_K c^2 \approx 0.494\ \text{GeV}
$$
For the pion,
$$
E_\pi = \sqrt{(1.0)^2 + (0.140)^2}\ \text{GeV} \approx 1.010\ \text{GeV}
$$
so
$$
\beta_\pi = \frac{pc}{E_\pi} \approx \frac{1.0}{1.010} \approx 0.990
$$
Thus
$$
t_\pi = \frac{L}{\beta_\pi c}
\approx \frac{2.0}{0.990 \cdot 3.0\times10^8}
\approx 6.73\ \text{ns}
$$
For the kaon,
$$
E_K = \sqrt{(1.0)^2 + (0.494)^2}\ \text{GeV} \approx 1.115\ \text{GeV}
$$
so
$$
\beta_K = \frac{1.0}{1.115} \approx 0.897
$$
and
$$
t_K \approx \frac{2.0}{0.897 \cdot 3.0\times10^8}
\approx 7.43\ \text{ns}
$$
Therefore,
$$
\Delta t \approx 7.43\ \text{ns} - 6.73\ \text{ns} = 0.70\ \text{ns}
$$
A difference of $0.70\ \text{ns}$, or $700\ \text{ps}$, is large enough for a detector with sufficiently good timing resolution to distinguish these two particles.
Mass squared plots
In many experiments, the result is shown as a plot of calculated $m^2$ versus momentum. Different particle species form separate bands or clusters. A pion band appears near $m_\pi^2$, a kaon band near $m_K^2$, and a proton band near $m_p^2$.
This is practical because particle identification is often statistical. Not every event gives a perfectly exact mass. Instead, many events group around the expected values.
Separation power
A detector does not simply say yes or no for a given particle type. Instead, it has a separation power. If the measured time distributions for two species overlap strongly, identification is poor. If they are far apart compared with the timing spread, identification is good.
One useful estimate is
$$
n_\sigma = \frac{|t_1 - t_2|}{\sigma_t}
$$
where $\sigma_t$ is the timing resolution and $n_\sigma$ tells how many standard deviations apart the two time peaks are. Larger $n_\sigma$ means better discrimination.
| $n_\sigma$ | Interpretation |
|---|---|
| $< 1$ | poor separation |
| $\approx 2$ | moderate separation |
| $\approx 3$ or more | good separation |
A larger flight path $L$ and a better timing resolution $\sigma_t$ both improve particle identification.
Longer path, larger time difference.
Smaller timing uncertainty, clearer separation.
Practical limits
Time of flight identification has natural limits. If the momentum becomes too high, then even heavy particles move very close to the speed of light. Their arrival times differ by only tiny amounts. Also, uncertainty in the path length and uncertainty in the event start time reduce the quality of the identification. In real experiments, time of flight is often combined with other methods such as energy loss or Cherenkov detectors to improve reliability.
Another practical issue is that the particle path is not always a straight line. In a magnetic field, a charged particle curves, so the actual path length used in the calculation must come from the reconstructed track, not just from the geometric distance between two detector points.
Visual picture
The basic idea can be seen as particles with equal momentum traveling the same distance, but arriving at different times because their masses are different.
What is identified in practice
In practice, time of flight identification most commonly distinguishes hadrons such as pions, kaons, and protons over a certain momentum range. It can also help reject lighter particles such as electrons when combined with momentum information. The detector does not directly read the particle name. It measures time, the tracking system measures momentum, and the identification follows from the inferred mass.
The essential idea is simple. Measure how fast the particle moved, compare that speed with its momentum, compute a mass, and match that mass to a known particle species.