KAHIBARO
27.4. Time Resolution
Table of Contents
Timing uncertainty
In a real detector, the measured time of a signal never corresponds exactly to the true physical time of the interaction. Instead, each measurement is affected by random fluctuations that you model as a timing uncertainty or time resolution. In Geant4 you typically start from an ideal time, obtained for example from a particle hit or from an energy deposition, and then apply a statistical smearing procedure to obtain a realistic measured time.
Timing uncertainty is usually characterized by the standard deviation $\sigma_t$ of a Gaussian distribution. If $t_{\text{true}}$ is the ideal time produced by the simulation, the measured time $t_{\text{meas}}$ is modeled as
$$
t_{\text{meas}} = t_{\text{true}} + \Delta t,
$$
where $\Delta t$ is a random number drawn from a Gaussian distribution with mean 0 and width $\sigma_t$. The parameter $\sigma_t$ is the time resolution. Many detector specifications quote time resolution as a full width at half maximum (FWHM), so you often convert between them using the relation
$$
\text{FWHM} = 2.355\,\sigma_t.
$$
Key rule: A realistic detector time is obtained by adding a random Gaussian offset to the ideal simulated time.
Important relation: $\text{FWHM} = 2.355\,\sigma_t$.
Always express the resolution in consistent Geant4 units, for example using ns, ps, or s.
Within a Geant4 application you normally compute or retrieve an ideal time for each event or hit, for example using G4StepPoint::GetGlobalTime() or a hit time stored in your sensitive detector. You then pass this ideal time to your analysis code and apply smearing there. A typical C++ pattern uses the Geant4 random library, such as G4RandGauss::shoot(0., sigma_t), where sigma_t is defined in appropriate units, for example 100.*ps. The result is added to the ideal time to obtain a smeared value that you record in your histograms or ntuples.
It is often useful to allow the time resolution to depend on other quantities, for example the deposited energy or the type of detector. In that case you can compute a per-hit or per-event $\sigma_t$ using a simple formula, such as
$$
\sigma_t(E) = \frac{a}{\sqrt{E}} \oplus b,
$$
where $E$ is energy in suitable units, $a$ is a stochastic term, $b$ is a constant term, and $\oplus$ denotes a quadratic sum. You then use this $\sigma_t(E)$ instead of a fixed value when generating the Gaussian fluctuation.
When you study coincidence timing, time-of-flight, or other time dependent observables, you should apply the same smearing model consistently to all relevant detector channels. The smeared times are then used to build time difference distributions, coincidence windows, or reconstructed time-of-flight. By comparing ideal distributions to smeared ones you can see directly how the time resolution of your detector influences peak widths, background levels, and the precision of time based measurements.
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