Fig. 3. Enhanced bremsstrahlung from heavy-element 2D hexagonal crystal lattices and shaped electron.

a–c illustrate 20 keV, shaped electron wavefunctions with spatial probability distribution localized in the vicinity of the heavier atoms of each crystal lattice unit cell, and their corresponding bremsstrahlung emission profiles at 15 keV. a(i)–c(i) correspond to WS₂, MoS₂, and WSe₂, respectively, where the electron wavefunction for (a(i), b(i)) is an approximate Bessel beam of order 0, and for (c(i)) is an approximate Bessel beam of order 1. a(ii)–c(ii) show the bremsstrahlung differential cross sections for single unit cell against the number of electron momentum states $N_{\mathrm{s}}$ for WS₂, MoS₂ and WSe₂ respectively. Optimum photon emission angles are chosen at ${\theta }_{k^{\prime }}$ = −0.35 [π rad], ${\varphi }_{k^{\prime }}$ = 0.5 [π rad] for a(ii) WS₂ and b(ii) MoS₂ (referred to as off-axis emission) and ${\theta }_{k^{\prime }}$ = 0 [π rad], ${\varphi }_{k^{\prime }}$ = 0 [π rad] for c(ii) WSe₂ (referred to as on-axis emission), respectively. For the off-axis emission cases a(ii) WS₂ and b(ii) MoS₂, coherent emissions (“Coh.”, red) scale up linearly with the number of states $N_{\mathrm{s}}$. Coherent off-axis emission is enhanced by more than two orders of magnitude for 300-momentum-state electron, as compared to a single-state (unshaped) electron. The paraxial and non-recoil approximation (“Para. & non-rec.”, black) also predicts linear $N_{\mathrm{s}}$ scaling but with a higher intensity. In c(ii) WSe₂, starting from 6-state with a relatively low intensity, coherent on-axis emission scales polynomially with the number of states $N_{\mathrm{s}}$, showing an enhancement up to three orders of magnitude for 300 electron momentum states. The paraxial and non-recoil approximation fails to produce meaningful results, as it predicts a vanishing emission intensity. In contrast, in all cases, the incoherent emissions (“Inc.”, blue) remain relatively unchanged. a(iii)–c(iii) show the bremsstrahlung X-ray emission angular profiles (coherent emission) of a 300-momentum-state electron for WS₂, MoS₂ and WSe₂ respectively