Relativistic Saturation of Coulomb-Limited Electron Coherence

TL;DR

This paper extends the non-relativistic theory of electron coherence in Coulomb-disordered media to relativistic regimes, deriving a unified framework that explains coherence saturation at high energies relevant for TEM.

Contribution

It derives a relativistic extension of the coherence theory starting from the Dirac equation, revealing a saturation of the effective coupling constant at ultra-relativistic energies.

Findings

The effective coupling constant saturates at 1/(2ħc) for ultra-relativistic electrons.

The transverse coherence length relates to localization length via a universal relation.

Standard TEM energies are near optimal for minimizing Coulomb decoherence.

Abstract

We show that the non-relativistic theory of mutual coherence and localization in Coulomb-disordered media can be extended to relativistic electron beams used in transmission electron microscopy (TEM). Starting from the Dirac equation, we derive a paraxial Schr\"odinger-like equation for the envelope spinor and obtain an effective coupling constant $A_{\rm rel}=(\gamma+1)/(2\gamma\hbar v)$ that governs the disorder-induced phase fluctuations. In the non-relativistic limit $\gamma\to1$ this reduces to $1/(\hbar v)$, while for ultra-relativistic electrons it saturates at $1/(2\hbar c)$. The universal relation between the transverse coherence length $\rho_c$ and the single-particle localization length $\ell$, namely $\rho_c\sim\lambda_D\sqrt{\ell/L}$, remains unchanged. We compare the asymptotic behaviour of the phase structure function $D_\phi(\rho)$ and the localization length in the non-relativistic and relativistic regimes, and show that the emergent algebraic decay of mutual coherence at large separations, analogous to the wave-structure-function asymptotics in turbulent media, persists in both cases. The results imply that standard TEM energies (100--300~keV) are already close to the optimal regime for minimizing Coulomb decoherence, and that further increasing the beam energy yields diminishing returns. While the asymptotic coherence decay is algebraic rather than exponential, the corresponding exponent can still be large for realistic experimental parameters, so the effect is primarily of conceptual and asymptotic significance.