Localization of a quantum particle in a classical one-component plasma.III. Mutual coherence and coherence degradation in Coulomb-disordered media

We derive the mutual coherence function of an electron beam propagating through a static or dynamic Coulomb-disordered medium and show that its decay introduces an intrinsic coherence-reduction mechanism relevant for electron microscopy in Coulomb-disordered media. Using the Efimov path-integral formalism, the coherence length ${\rho }_c$ is expressed through the same disorder correlator that governs the single-particle localization length $\mathrm{\ell }$. For both a static electrolyte and a dynamic plasma we obtain a universal relation ${\rho }_c\sim {\lambda }_D\sqrt{\mathrm{\ell }\mathrm{/}L}$, where ${\lambda }_D$ is the Debye length and $L$ the sample thickness. In the static case $\mathrm{\ell }\propto k^2$ (electron momentum), whereas in the dynamic slow-particle regime $\mathrm{\ell }\propto k$, leading to qualitatively different energy dependences of the coherence scale. The ion thermal velocity cancels out in the final expression, demonstrating a formal connection between transverse coherence decay and longitudinal localization phenomena. Exact analytical results are given for the phase structure function of a model electrolyte, and numerical estimates indicate that disorder-induced phase decorrelation may contribute appreciably to the attenuation of high-spatial-frequency contrast under experimentally relevant liquid-cell electron microscopy conditions. Possible implications for cryo-EM, disordered liquids, soft condensed matter, and biological media are discussed. In an appendix we extend the theory to the relativistic regime relevant for transmission electron microscopy.