Diffraction by Circular and Triangular Apertures as a Diagnostic Tool of Twisted Matter Waves

We study diffraction of twisted matter waves (electrons and light ions carrying orbital angular momentum /=0,±1,±2,… by circular and triangular apertures. Within the scalar Kirchhoff-Fresnel framework, circular apertures preserve cylindrical symmetry and produce ringlike far-field profiles whose radii and widths depend on || but are insensitive to its sign. In contrast, equilateral triangles break axial symmetry and yield structured patterns that encode both the magnitude and the sign of . A transparent Fraunhofer mapping links detector coordinates to the Fourier plane, explaining the (||+1)-lobe rule and the sign-dependent rotation of the pattern. We validate these results for both ideal Bessel beams and localized Laguerre-Gaussian packets, and we cross-check them by split-step Fourier propagation of the time-dependent Schr"odinger equation. From these analyses we extract practical design rules (Fraunhofer distance, lattice pitch, detector sampling) relevant to OAM diagnostics with moderately relativistic electrons with Eₖᵢₙ~0.1 to $5$ MeV and light ions with Eₖᵢₙ~0.1 to $1$ MeV/u. Our results establish triangular diffraction as a simple, passive, and robust method for reading out the OAM content of structured quantum beams.