How does space charge stop a Geiger discharge?

Question

It seems common in the literature (e.g. McGregor and Shultis) to say that the accumulation of space charge from positive ions around the anode wire lowers the electric field below the critical field strength needed to sustain avalanching (I pretty much paraphrased from page 404).

I have never actually understood that. The ions are positively charged. The anode wire is positively charged. I pulled out Gauss's law, and I don't see how positive charge around the anode wire lowers the field at the anode wire. That is, an infinite line charge for the anode, a cylinder of uniform charge density around it, the field would be like $\lambda/r+\rho r$, which just goes up. How does an accumulation of space charge around the anode reduce the electric field below the critical value?

Answer 1

Because in a Geiger tube the anode is a thin wire at the axis of the tube, the radial electric field, where the avalanche charge carrier multiplication sets in is highest near the anode. For a given anode-cathode voltage, the presence of a positive ion charge cloud around the anode lowers the field strength in this region. This can be easily veryfied by using Gauss' law. When a positive space charge is introduced near the anode, the electric field increases in the region from the cathode to the cloud, but decreases in the region of the cloud to the anode.

That, for a constant applied voltage, the electric field has to decrease near the anode when a positive space charge is introduced can qualitatively be inferred from the increase of the electric field strength between the cathode and the positive charge. As the applied voltage stays the same, the increased slope of the potential between cathode and the space charge must produce a decreased potential slope in the space-charge anode region in order to still comply with the constant applied cathode-anode voltage boundary condition. Note that the constant applied voltage condition implies that the charge on the anode electrode also changes when you introduce the positive charge cloud.