the number of electrons from charge neutrality. These protons have finite uncertainties, i.e., finite spreads of the momenta due to Rutherford scatterings between charges $$\begin{aligned}
\Delta p={\hbar \over l_{\text{Ru}}},~l_\text{Ru}={1 \over \sigma_{\text{Ru}} n_\text{c}},\end{aligned}$$ where $n_c$ is the density of charged particles and $\sigma_\text{Ru}$ is Rutherford scattering cross section, which is given in Appendix A.

Even if these momentum spreads are small, a system of many protons could give a large uncertainty to an electron owing to the constructive effects. In this case, the whole uncertainty of the momentum becomes large, and this system of the electron and protons violates the translational invariance maximally.

A mean free path of the electron due to Thomson scattering is given by $$\begin{aligned}
l_\text{Th}={1 \over \sigma_\text{Th} n_{\gamma}},\end{aligned}$$ where $n_{\gamma}$ is photon density. A mean free path of the electron due to Rutherford scattering is given by $$\begin{aligned}
l_\text{Ru}={1 \over \sigma_\text{Ru} n_\text{pr}},\end{aligned}$$ where $n_\text{pr}$ is a proton density. Their magnitudes are given in Appendix A. The ratio between two lengths $$\begin{aligned}
N_\text{T}={l_\text{Th} \over l_\text{Ru} }\end{aligned}$$ is the number of average collisions due to Rutherford scattering per Thomson scattering. During a Thomson scattering time, Rutherford scatterings occur $N_\text{T}$ times. The $l_\text{Th}$, $l_\text{Ru}$, and $N_\text{T}$ are given in Appendix 