Results

We have calculated the resolution as a function of crossing angle for:

• strip pitches of 50,60,70,80,90,100$\mu$m;
• with and without intermediary strips;
• for signal-to-noise ratios of 10:1 and 13:1, which were chosen as worst case and average case scenarios, based on the likely cable lengths and chip parameterisation at our disposal as of 25.11.98.

Fig.  shows the resolution for a $50\mu$m pitch detector with S:N of 13:1, with and without an intermediary strip. Four points are noteworthy:

• firstly, for angles $<\approx 7^o$, the intermediary strip plays an important role;
• secondly, for angles $>\approx 7^o$, the intermediary strip would degrade the resolution by 1 to $2\mu$m;
• thirdly, the effect of the Lorenz angle is evident.
• fourthly, for angles greater than about 12^o, the resolution degrades rapidly.

Fig.  shows the same plot but repeats it for $50,60,70,80\mu$m detectors as indicated. The behaviour in each case is similar. The angular range, over which an intermediary strip is useful, broadens as the pitch increases. In all cases, the resolution degrades rapidly for angles greater than 15^o.

Fig.  shows the same plots for $50,60,70,80,90,100\mu$m pitch, with and without intermediary strips, but this time for a signal-to-noise of 10:1. It is noted that for small angles the degradation (with respect to S/N of 13:1) is small, about $1\mu$m, but for larger angles it is much bigger, up to $5\mu$m. This is a consequence of the fact that in spreading itself over more strips, the charge per strip begins to be obscured by the noise. If we are led to mechanical designs which require longer cable lengths, as now seems likely, then the noise will be worse than had been envisaged, and the resolutions at large angles will be particularly affected.

Fig.  summarises these plots for each pitch with and without an intermediary strip. We note an important point:

• Above about 15^o, the resolution is insensitive to the strip pitch, and furthermore, it is better not to use an intermediary strip here.

With these plots, we can now calculate the response of any mechanical design in terms of the resolution we would expect. We have done this for three cases:

• Firstly, for an 8-fold detector with $50\mu$m read-out pitch with an intermediary strip. This was the favoured design as of 25.11.98.
• Secondly, for a 12-fold detector with $70\mu$m read-out pitch with an intermediary strip. This would be the naive basic design if we were to go to a 12-fold modularity, whether to gain resolution, reduce readout, or for triggering purposes.
• Thirdly, for a 12-fold detector with varying pitches. We have used our results to perform an optimisation for the best average resolution, subject to the constraint that 128 channels span 9432$\mu$m, the width of detector required for a 12-fold geometry with 10% overlaps. Our solution requires starting off with $90\mu$m pitch-no-intermediary for the first 3 strips, changing to $80\mu$m pitch-no-intermediary for the next 15, changing to $80\mu$m pitch-with-intermediary for the next 25, changing to $70\mu$m pitch-with-intermediary for the next 6, changing to $60\mu$m pitch-with-intermediary for the next 5, and finally $50\mu$m pitch until the centre of the detector. The pattern is repeated on the other side.

Fig.  shows the result of these configurations. 8-fold modularity with $50\mu$m pitch is clearly the worst. Although having the greatest number of tracks with $5\mu$m resolution it has long tails. 15% of all tracks have resolutions greater than $15\mu$m and 45% have resolutions greater than $10\mu$m. This is due principally to the poor resolution at large crossing angles. 12-fold modularity with $70\mu$m pitch is better. 5% of tracks have resolutions worse than $15\mu$m and 38% of tracks have resolutions worse than $10\mu$m. However, the best solution is obtained with our optimisation technique. Now less than 5% of tracks have resolutions worse than $10\mu$m and in addition the response is more uniform over the detector.