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Timing Resolution

Both the cosmic ray and TRIUMF $\pi / \mu /e$ measurements were accomplished by sandwiching the fiber bundles between two scintillator counters in coincidence mode, definining the event trigger. The setups involved at least four PMT's arranged as shown in Figure 1.

Figure 1: Schematic diagram of experimental setup used for both cosmics and TRIUMF runs. Additional counters, used in some of the measurements, are not shown here.


\includegraphics [height=94.5mm,angle=0,clip=]{timing.eps}

It can be easily shown that the time difference between the fiber left and right PMT's is given by the equation:

\begin{displaymath}
\Delta T = \eta \left[ \frac{n}{c} ( 2 d_1 - L_{fiber} ) +
T_{CONSTANT}\right],
\end{displaymath} (1)

where $d_1$ is the distance from one of the PMT's, $L_{fiber}$ is the total length of the fiber, $c$ is the speed of light, $n$ is the index of refraction of the fibers, and $\eta$ is the TDC conversion factor in channels/ns.

Therefore, a plot of $\Delta T$ vs. $d_1$ (measured in cm) will result in a straight line, with the slope of the line being given by

\begin{displaymath}
m_{\Delta T} =\frac{ 2 \eta}{c} n.
\end{displaymath} (2)

For the Lecroy TDC that was used for the cosmic ray runs, the conversion factor was 200 ps/channel, therefore, $\eta=5~channels/ns$. Thus:

\begin{displaymath}
m_{\Delta T} = \frac {2 \cdot [5~channels/ns] \cdot n}{30~cm/ns} = \frac{n}{3} [channels/cm].
\end{displaymath} (3)

Given that $n \sim 1.7$ [4], it is expected that $m_{\Delta T} \approx
0.57~channels/cm$. Likewise, simply plotting the TDC peak location vs. position for each PMT should result in a slope:
\begin{displaymath}
m_{single} \approx \frac{1}{2}m_{\Delta T}
\end{displaymath} (4)

and therefore a predicted value of $m_{single} \approx 0.285~channels/cm$.


next up previous
Next: Unfolding the Timing Resolution Up: Notes on Timing Resolution Previous: Notes on Timing Resolution
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2001-10-29