An Estimation of the Effective Number of Electrons Contributing to the Coordinate Measurement with a TPC

For time projection chambers (TPCs) the accuracy in measurement of track coordinates along the pad row direction deteriorates with the drift distance ($z$): ${\sigma }_{\mathrm{X}}^2\sim D_{\mathrm{X}}^2\cdot z\mathrm{/}{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$, where $D_{\rm X}$isthediffusionconstantand$N_{\rm eff}$ is the effective number of electrons. Experimentally it has been shown that $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is smaller than the average number of drift electrons per pad row ($\bar{N}$). We estimated $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ by means of a simple numerical simulation. The simulation shows that $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be as small as $\sim$ 30% of $\bar{N}$ due to the combined effect of statistical fluctuations in the number of drift electrons ($N$) and in their multiplication in avalanches.