Experimental – Method verification

Our FXD-CSD method has recently been applied to a number of cases in geo- and material science using synchrotron X-rays (Chaouachi et al., 2017) and lab X-ray sources (Neher et al., 2018). To systematically verify the principle, as well as the limits of FXD-CSD when used on a common lab X-ray source, the LaB6

standard reference material (SRM) 660a⁵ from the National Institute of Standards and Technology (NIST) - in the following called LaB6-Powder - and four corundum Crystal Size Fractions (CSF I - IV) with different mean crystal sizes are analyzed here. The corundum CSFs are produced via sedimentation in water⁶. All samples are characterized by scanning electron microscopy (SEM) imaging to obtain their volume-based crystal size distribution (CSD) as accurately as possible. Thus, the CSDs of all samples are established and they all can be treated as reference material or as unknown sample. Synchrotron measurements of the LaB6-Powder were already published (Chaouachi et al., 2017) and are available for comparison with the lab X-ray results obtained here.

2.4.1 Sample SEM CSD characterization

For the CSD characterization all samples (see Table 2.1) are dispersed on a specimen holder and investigated via SEM imagery (FEI Quanta 200 FEG) (see Figure 2.4 and Figure 2.5 as well as Figure App.

B-7 and Figure App. B-9). Details on preparation and the subsequent analysis referred to below, are found in the Appendix A 2 and B 4. To analyze their size distribution a Python script is used to manually draw enclosing rectangular boxes around each crystal. The box dimensions (dx, dy), corresponding to the half-axis of the crystal, are automatically stored as text file for the crystal volume calculations. In case of the LaB6-Powder a spherical crystal shape is assumed for volume calculation; the average of dx and dy is used as sphere diameter. The volume calculation of the corundum CSF is calculated using a cuboid model; the used dimensions are derived from dx and dy, including an assumption for the 3^rd dimension (see Appendix A 2).

5 A newer batch of the NIST standard (SRM 660b) turned out to be not suitable because the powder contains large amounts of agglomerated crystals what do not have to be single crystals.

6 In total 10 corundum CSFs are produced (see Appendix B 4). The CSFs with bigger mean crystal sizes are not analysed because the diffraction data shows satellite peaks, an artefact coming from a defective part of the monochromator (see Appendix B 5).

2-37 The resulting CSDs can be presented either as diameter- or volume-based CSD histogram. Figure 2.6 shows the resulting CSDs as diameter histogram with log-normal probability density functions (PDF) fitted to the mid points of the histogram bins; the assumption of a log-normal CSD is supported by the good quality of fit; to avoid duplication this is at first shown in Table 2.4. Table 2.1 shows the SEM-derived CSD mean values calculated from the log-normal fitting parameters. The volume-based presentations of the CSDs are shown in Figure App. B-10.

Figure 2.4: Example SEM image of LaB6-Powder Figure 2.5: Example SEM image of CSF III

Figure 2.6: SEM derived diameter CSDs of all measured samples and their log-normal PDF fit curves. The data entries show the bin midpoints of the histograms. The bin size is 1 µm. The diameters are calculated from the SEM derived crystal volume, assuming spherical crystal shape (see Appendix A 2).

2-38 2.4.2 Sample preparation

For the X-ray measurements the samples are mixed with starch⁷ and filled in Kapton^® capillaries with a diameter of 0.8 mm. The starch is used to dilute the powder and control the number of crystals in the irradiated sample volume. The Kapton^® capillaries are mounted with a brass holder on the goniometer head. This way it is possible to obtain a rotationally symmetric sample with no pronounced preferred crystal orientation. A good starch-to-sample proportion is obtained by starting from a mixture of even weight proportions and gradually adding starch, while repeatedly checking the diffraction pattern on the diffractometer.

2.4.3 FXD-CSD Measurements

All samples are measured with a Bruker AXS Apex II CCD diffractometer (D8 Base) with a fixed χ-angle, equipped with a Molybdenum tube (Mo-Kα, λ= 0.71073 Å) in the sweep-stepwise rotation manner (see Section 2.3.3). Apart from the exposure time all other scan parameters and diffractometer settings are kept the same for all samples measured⁸; the sweep-step size is set to 0.025° in ϕ, the total ϕ-rotation is 10° (see Table 2.1). All samples are measured three times at different positions along the capillary axis. The sample to detector distance is set to 60 mm; this corresponds to a 2ϴ range of 0 – 26° on the detector. All diffraction rings present in this 2ϴ range are listed in Table 2.1. Figure 2.7 shows an example frame of the LaB6-Powder. Visible are six spotty diffraction rings. Noteworthy is the relatively strong background underneath the 100 and the 111 hkl-ring, originating from starch. Figure 2.8 shows example frames from all analysed corundum samples. They show spotty diffraction patterns of six rings with more or less similar spot intensities while having different exposure times (see Table 2.1). Also apparent are significant differences in the number of spots on the diffraction rings and the background level. This is traceable to the differences in the exposure time, the change in the sample-starch proportions and variations in the compaction of the mixture in the capillary.

Visually comparing the LaB6-Powder frame (Figure 2.7) with the frame of the CSF 1 (Figure 2.8A), both measured with the same exposure time, reveals that the LaB6 crystals diffract similar amounts of intensity, even though the LaB6-Powder sample has a smaller mean crystal size (see Table 2.1); this is expected due to the superior scattering power of LaB6.

7 Mondamin® “Feine Speisestärke” Unilever, corn starch. Starch from other manufactures turned out to be crystalline.

8 The LaB6-Powder and the CSF I sample have been measured at different occasions. To correct for possible intensity differences due to a different diffractometer setup, the CSF III was measured at this occasion again and used as reference to link both datasets.

2-39 Table 2.1: Samples and measurement parameters

CSF III is used as reference; all others are used as sample.

Sample SEM derived

mean¹ crystal