The Charge Distribution on the Cathode of a Straw Tube Chamber

A straw tube chamber is a low-cost version of a proportional counter. These devices consist of a pair of coaxial conducting cylinders with the region between the cylinders filled with a gas such as argon. The inner cylinder of radius a is the anode, and is held at potential V ; the outer cylinder of radius b is the cathode, and is grounded. If a penetrating charged particle passes through the chamber, it will ionize about two gas molecules per mm of path length. The ionization electrons are pulled by the electric field towards the anode. Close to the anode, the field is strong enough that the electrons gain enough energy during one mean free path to ionize the molecule they hit next, liberating one or more additional electrons. In a proportional chamber, the field is kept low enough that the resulting Townsend avalanche involves 10-10 molecules. What is the time dependence, I(t), of the current that flows off the anode due to the avalanche of a single initial electron? What is the spatial dependence, q(z) of the charge distribution induced on the anode during the time when the current is large, where the z axis is the chamber axis? You may restrict your attention to values of z far from the ends of the tube of length l. Measurement of the charge distribution via a segmented cathode permits localization in z of the ionization, and hence, of the initiating charged particle [1]. You may ignore the tiny current that flows while the electron drifts towards the anode. The avalanche takes place so close to the anode, that the small remaining drift time for the electrons to reach the anode may also be ignored. In this approximation, the situation at t = 0 is that electrons of total charge −q0 reside on the anode in close proximity to positive ions of total charge +q0. Current flows off the anode only when some of the field lines from the positive ions detach from the electrons on the anode, and extend to the cathode where charge is induced to terminate these field lines. This occurs only as the positive ions move away from the anode, with velocity related by