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Drift dN/dt in Straw Tubes - Title

Drift Time Distributions in
Straw Tube Wire Chambers
Reflect Fundamental Properties of the Gas

Howard Fenker, October 1996

A straw-tube chamber is made by tensioning a wire along the axis of a conducting cylinder, filling the cylinder with a suitable gas, and applying an electric potential between the wire and the cylinder walls so as to cause electron avalanche multiplication near the wire. In the best case, the gas and electric field are such that electrons drifting toward the wire do so at constant velocity. Then the elapsed time between passage of an ionizing particle through the straw and arrival of the first electron at the wire provides a measurement of the distance between the particle's trajectory and the wire.

One fundamental limitation on the resolution of such a measurement comes from the fact that charged particles passing through the straw leave behind a trail of ionization clusters which are distributed statistically along the path. If an ionization always occurred at the point of closest approach of the particle to the wire then this electron (or these electrons) would be the first to arrive at the wire and the position measurement resolution could be quite good.Instead we have the real case in which the electron nearest the wire often starts some distance away from this minimum. Thus the radial distance from the wire to the track is the minimum drift distance. The actual drift distance will be skewed towards higher values.

By Monte Carlo generation it is easy to exhibit the distribution of electron drift distances in a straw-tube which is uniformly illuminated by a parallel flux of minimum-ionizing radiation (or, more generally, radiation of such type or energy that its characteristics are not significantly perturbed by passage through the straw-tube). The figure shows such a distribution (histogram).

Consider particle tracks parallel to the y-axis. The electron with the minimum drift distance will be the one liberated nearest the x-axis. In rectangular coordinates the probability of generating an electron at y=|Y| is (1/(lambda)) exp(-2y/(lambda)). The '2' in the exponent comes from starting at y=0 and considering that the electron cluster could be at '+' or' -' y. To get the total probability of making a free electron at a distance r from the wire we convert to polar coordinates and integrate over angle at a fixed radius. So we write,in integral form, the probability distribution for this case:

,

where (lambda) is the mean-free-path between ionizations, a characteristic of the gas. While I have not solved this integral, it is easy enough to integrate numerically and to demonstrate that it agrees with the Monte Carlo results. This numerical integration results in the smooth curve shown in the figure.

Two points are clear from the above equation: the length scale is set by (lambda) and the only relevance of the straw diameter is to truncate the above distribution (neglecting the effects of electron absorption for now). So the curve shown in the figure is universal: a variation in (lambda) compresses or expands it horizontally while a variation in straw diameter varies the point at which it is truncated.

It would appear that this characteristic behavior provides a means of quickly measuring two properties of the gas, as the drift velocity is obtained trivially from the maximum drift time observed, knowing this corresponds to drifting one straw radius, and the shape of the time (drift distance) probability distribution provides a measure of the mean collision length. In fact (lambda) may be determined merely from the location of the peak in the distribution using the numerically determined result that with C ~ 1.4 (although I expect C is something more fundamental, like e/2 or 2).