Right Handed or Left Handed? Forbidden X-Ray Diffraction Reveals Chirality

Enantiomers, or stereoisomers, have crystal structures that are mirror images of each other and are thus handed, like our right and left hands. The physical properties of enantiomers are identical except for optical activity, which rotates linearly polarized light by equal amounts but in opposite directions. While conventional x-ray Bragg diffraction can determine crystal structures, it does not distinguish between right- and left-handed crystals. We show resonant Bragg diffraction using circularly polarized x rays reveals the handedness of crystals by coupling x-ray helicity to a crystal screw axis. The intensity of resonantly allowed reflection of $\alpha$-quartz is well described by an admixture of a parity-even and a parity-odd process. Our results are of general importance and demonstrate a new method to directly study chiral motifs in structures that include biomaterials, liquid crystals, magnets, multiferroics, etc.

Figure 1

Views of atomic structure of $R$ quartz (right) and $L$ quartz (left) along the $a^{*}$ axis and the $b^{*}$ axis, respectively. The $a^{*}$ and $b^{*}$ axes are perpendicular to the plane of paper. Blue and red spheres represent Si and O atoms, respectively. Lines show the unit cell with hexagonal axes.

Figure 2

XAS (filled circles) and integrated intensity (open circles) of reflection 001 of $L$ quartz as a function of energy of the incident beam. Reflection 001 was measured at azimuth angle $\Psi =20°$. The detection efficiency of the Si-photodiode is corrected for both curves but sample absorption is not corrected for the data of 001 reflection.

Figure 3

Integrated intensity of reflection 001 of $R$ and $L$ quartz as a function of azimuth angle $\Psi$. Filled (open) circles represent the intensity of reflection 001 of $R$ quartz measured with LCP (RCP) incident beam, and filled (open) triangles represent the intensity of reflection 001 of $L$ quartz measured with RCP (LCP) incident beam. Each line shows a result of fit to data with functions expressed by Eq. (1) or (2). Inset shows the $2\theta –\theta$ scan profiles of reflection 001 observed at $\Psi =0$.

Figure 4

Integrated intensity of reflection 001 and $00\overline{1}$ of $L$ quartz as a function of azimuth angle $\Psi$. Measurements were done with RCP and LCP incident beam. Filled triangles (squares) represent the intensity of reflection 001 ($00\overline{1}$) measured with RCP incident beam, and open triangles (squares) represent the intensity of reflection 001 ($00\overline{1}$) measured with LCP incident beam. Each line shows a result of fit to data with functions expressed by Eq. (1) or (2).