Quantifying hexagonal stacking in diamond

Diamond is a material of immense technological importance and an ancient signifier for wealth and societal status. In geology, diamond forms as part of the deep carbon cycle and typically displays a highly ordered cubic crystal structure. Impact diamonds, however, often exhibit structural disorder in the form of complex combinations of cubic and hexagonal stacking motifs. The structural characterization of such diamonds remains a challenge. Here, impact diamonds from the Popigai crater were characterized with a range of techniques. Using the MCDIFFaX approach for analysing X-ray diffraction data, hexagonality indices up to 40% were found. The effects of increasing amounts of hexagonal stacking on the Raman spectra of diamond were investigated computationally and found to be in excellent agreement with trends in the experimental spectra. Electron microscopy revealed nanoscale twinning within the cubic diamond structure. Our analyses lead us to propose a systematic protocol for assigning specific hexagonality attributes to the mineral designated as lonsdaleite among natural and synthetic samples.


Preparation of the various diamond structures
Using our Stacky program, 1 a range of stacking disordered diamond structures were created with 48 layers.
Stacky requires the target 1 st order stacking probabilities as input and then produces a closest possible structure ensuring periodicity across the unit cell boundary in the c direction of stacking. As described in ref.
1 in detail, the requirement for periodicity across the boundary of the unit cell imposes some restrictions with respect to the possible stacking sequences.
In addition to the stacking disordered structures, the structures of the regular 2H, 3C and 4H polytypes were also produced with Stacky. The structural details of all created structures are listed in Table S1. The structures.zip file contains the cif files of the various structures.

Calculated Raman spectra
The structures listed in Table S1 were used as starting structures for calculating the Raman spectra with CRYSTAL17 as shown in Figure 3 in the main article. 2 The calculated Raman spectra using Gaussian functions with 10 cm -1 half-widths as the profile functions are contained within spectra.zip.

MCDIFFaX fitting of the X-ray patterns
The recorded X-ray diffraction patterns were baseline corrected using shifted Chebyshev background functions within the GSAS software. 3 Great care was taken not to subtract intensity in the angle ranges where stacking disorder leads to diffuse scattering. The diffraction data were then fitted with the MCDIFFaX software by refining up to 2 nd order stacking probabilities, lattice parameters and peak-profile parameters (u, v, w and Gaussian / Lorentzian ratio). 4 During a typical refinement, a starting value of 0.5 was used for the zero-order stacking probability, and the 1 st and 2 nd order stacking probabilities were successively introduced during the refinement once the fits had converged using the lower-order stacking probabilities. Regarding the Cagliotti terms, the refinements were started by optimising w alone, whereas u and v were introduced during the later stages of the refinements. A detailed description of the MCDIFFaX program as well as the equations needed for the calculations of the 1 st order stacking probabilities from the 2 nd order stacking probabilities as well as the cubicity or hexagonality from the 1 st order stacking probabilities are given in ref.
4. The determined stacking probabilities from all analysed Popigai diamond samples are listed in Table S2.

Effect of hexagonality on the crystallographic c/a ratio
Within DIFFaX, the structure of a stacking disordered material is described by defining the structure of a single layer and then implementing the geometric recipes for stacking these layers together with the corresponding stacking probabilities. 5 Since MCDIFFaX deals with individual layers, the c height of a layer needs to be multiplied by 2 before comparing the c/a ratio with the one of the ideal hexagonal close-packed structure for which c/a = √8/3. For fully cubic diamond, the c/a ratio is √8/3 by definition because of the cubic symmetry. Figure S1 shows the c/a ratios of the various Popigai diamond samples as determined with MCDIFFaX and plotted as a function of the hexagonality. Despite some considerable errors, it can be seen that the c/a ratio increases with increasing hexagonality. This implies that the repulsive interactions within the six-membered rings in the boat conformation lead to slight separations of the stacked layers. The red line in Figure S1 shows the best linear fit to the data and the corresponding equation is shown in the legend. It is noted that the deviations from the fitted line do not seem to be connected with the deviations from random stacking, i.e. differences in the 1 st order memory effects within the stacking sequences as shown in the stackogram in Figure 2B in the main article.