On the Overestimation of Efficiency in Relativistic Electron Scattering

With the rapid global advancement of relativistic ultrafast electron diffraction systems, the impact of relativistic effects on electron scattering efficiency has attracted renewed attention. Recent reviews emphasize the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>γ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> scaling of the differential elastic-scattering cross-section with increasing electron energy. Although such observations are not formally incorrect, they can lead to a misperception of the relative scattering efficiency of relativistic electrons compared to that of their nonrelativistic counterparts. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>γ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> scaling originates from analyses conducted in the angular domain, where the compression of the scattering angle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>θ</mml:mi> </mml:math> (or solid angle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Ω</mml:mi> </mml:math> ) with increasing energy creates an enhancement of the differential elastic-scattering cross-section, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="italic">dσ</mml:mi> <mml:mo>/</mml:mo> <mml:mi mathvariant="italic">dΩ</mml:mi> </mml:math> . In this work, we recast the problem in momentum-transfer space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>q</mml:mi> </mml:math> , where scattering is accurately accounted for. This transformation eliminates the angular compression artifact and reveals that high-energy scaling follows a simple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>β</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> dependence, with no intrinsic relativistic gain. We demonstrate this by directly integrating relativistic differential elastic-scattering cross-sections from ELSEPA and by applying a straightforward transformation of the well-known Mott–Massey formalism into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>q</mml:mi> </mml:math> -space. The results are general, with calculations performed for elements from carbon to gold and for energies between 50 and 5,000 keV. They reproduce the long-established trend in total elastic-scattering cross-sections, in which scattering strength decreases with increasing electron kinetic energy. Practically, at energies above roughly 50 keV, scattering is already dominated by the forward direction, and most of the scattered intensity falls within the acceptance range of typical ultrafast electron diffraction detectors.