Non-centro-symmetric electron diffraction pattern of icosahedral quasicrystal induced by combination of linear phason strain and curvature of Ewald sphere

Coherent Lattice Wobbling and Out-of-Phase Intensity Oscillations of Friedel Pairs Observed by Ultrafast Electron Diffraction

“…Unequal intensities of Friedel pairs can be observed. − ,, This phenomenon breaks Friedel’s law, when Friedel’s law is stated directly by equal intensities of Friedel pairs (the first statement), , or the equal intensities of Friedel pairs can be deduced according to the crystal symmetry in the second statement. ,, Currently, most reports about such unequal Friedel pair intensities, or breaking of Friedel’s law, are conducted in static diffractions, caused by either the curvature of the Ewald sphere − or the dynamic scattering effects. ,, After a pump excitation ( e . g ., ultrafast laser pulse), the time-resolved diffraction intensities of a Friedel pair can respond with opposite trends due to lattice dynamics, meaning that time-resolved unequal diffraction intensities of Friedel pairs or time-resolved violation of Friedel’s law can be observed.…”

“…Unequal intensities of Friedel pairs can be observed. [8][9][10]13,14 This phenomenon breaks Friedel's law, when Friedel's law is stated directly by equal intensities of Friedel pairs (the first statement), 8,9 or the equal intensities of Friedel pairs can be deduced according to the crystal symmetry in the second statement. 6,13,14 Currently, most reports about such unequal Friedel pair intensities, or breaking of Friedel's law, are conducted in static diffractions, caused by either the curvature of the Ewald sphere 8−10 or the dynamic scattering effects.…”

“…Only by tilting the sample to a low-symmetry axis, such as the [212] direction in single-crystal TaSeTe (when the Friedel pair that breathes in opposite directions is cut by the Ewald sphere), 17 can out-ofphase intensity oscillations be experimentally observed, because of the breathing phonon and Ewald sphere curvature. [8][9][10]17 Distinctive from the previously studied 2D materials, monoclinic gallium telluride (GaTe) is a lowsymmetry crystal, 21,22 which belongs to the C2/m space group and has only four symmetry operations (i.e., E, C 2 , σ h , and i). It can be expected that distinct lattice dynamics and diffraction features can be found in monoclinic GaTe.…”

“…One form states that the diffraction intensities of a Friedel pair, i . e ., diffraction peaks of ( hkl ) and ( −h–k–l ), are equal. , This statement is empirical and less rigorous, but can be easily compared with experimental results, thus has been widely adopted. − This statement can be derived by Born approximation and the kinematic diffraction theory, which indicates that the scattering amplitudes are Fourier components of the real scattering potential and are thus complex conjugates to each other for a Friedel pair . Another form of Friedel’s law is “the diffraction phenomena of waves by a crystal are invariant under an inversion of the crystal”. ,, This statement involves no ambiguity, but is not straightforward to directly compare with experimental results .…”

“…Taking the oscillations caused by the lowest-order breathing acoustic phonon mode in graphite as an example, the breathing phonon mode is longitudinal, because it is excited by the thermal expansion of crystal lattices, while the transverse acoustic ones are usually hard to achieve. , When the electron beam is incident along the [001] direction, the diffraction pattern explicitly obeys Friedel’s law, because the acoustic breathing phonon of graphite has a displacement vector aligned with the incident electron beam. Only by tilting the sample to a low-symmetry axis, such as the [212] direction in single-crystal TaSeTe (when the Friedel pair that breathes in opposite directions is cut by the Ewald sphere), can out-of-phase intensity oscillations be experimentally observed, because of the breathing phonon and Ewald sphere curvature. − , Distinctive from the previously studied 2D materials, monoclinic gallium telluride (GaTe) is a low-symmetry crystal, , which belongs to the C 2 /m space group and has only four symmetry operations ( i.e. , E , C 2 , σ h , and i ).…”