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

## TL;DR

This paper estimates the effective number of electrons contributing to coordinate measurement in a TPC, revealing it can be significantly less than the average due to statistical fluctuations, impacting measurement accuracy.

## Contribution

The study introduces a simple numerical simulation to estimate the effective electron number, accounting for fluctuations affecting TPC coordinate accuracy.

## Key findings

- $N_{eff}$ can be as low as 30% of $ar{N}$ due to fluctuations.
- Measurement accuracy deteriorates with drift distance, influenced by $N_{eff}$.
- Simulation provides a practical way to estimate $N_{eff}$ in TPCs.

## Abstract

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