Advanced processing of differential phase contrast data: Distinction between different causes of electron phase shifts.

Differential phase contrast, in its high resolution modification also known as first moment microscopy or momentum resolved STEM [1-7] , basically measures the lateral momentum transfer to the electron probe due to the beam interaction with either electrostatic and/or magnetic fields, when the probe transmits the specimen. In other words, the result of the measurement is a vector field p→(x,y) which describes the lateral momentum transfer to the probe electrons. In the case of electric fields, this momentum transfer is easily converted to the electric field E→(x,y) causing the deflection, and from ϱ=ɛ∇⋅E→ the local charge density can be calculated from the divergence of the electric field. However, from experimental data it is known that also the calculation of the vector field's curl ∇→×p→ in general yields non-zero results. In this paper, we use the Helmholtz decomposition (Wikipedia contributors, 2022), also known as the fundamental theorem of vector calculus, to split the measured vector fields into their curl-free and divergence-free components and to interpret the physical meaning of these components in detail. It will be shown, that non-zero curl components may be used to measure geometric phases occurring from irregularities in crystal structure such as a screw dislocation.