**5. Analytical Uncertainties**

The presented data from grain mounts of processing products are based on the analysis of 336,000–599,000 individual particles in a single sample. When the analytical uncertainties are evaluated, the mineral mode appears as the most prominent parameter in comparison to the particle and grain sizes and their shape geometries. An MLA measurement starts with the acquisition of the first frame in the centre of a round epoxy grain mount block. The subsequent frames are then arranged in a single spiral toward the margin of the block [16]. During the preparation of the grain mount block, particle separation may be induced by the stirring of the particles into the liquid epoxy and subsequent gravitational subsidence of high-density particles during epoxy hardening. These effects can lead to heterogeneous particle distributions in the surface of a polished block [42,43], an effect that is particularly prominent for sample materials (mineral mixtures) with large differences in particle sizes and/or densities. For evaluating this potential uncertainty induced by heterogeneous particle distribution in the polished block surface, the full datasets with >300.000 particles were compared to the datasets from the inner part of the spiral. At a scale of 0–35 wt % the data align almost perfectly (Figure 7a). As a consequence the measurement of the inner spiral appears representative of the complete sample. However, when the inner spirals and the outer spirals are compared at the scale 0–5 wt % of mode, it is obvious that minerals with a mode below 1 wt % may be heterogeneously distributed within a sample, as is the case for the REE-Low-Mix group in sample M45 (Figure 7b). This

apparent heterogeneity is attributed to a nugge<sup>t</sup> effect for minor and trace minerals induced during sample preparation. Yet, it is encouraging that the modes of the other mineral groups match very well between the different data sets.

**Figure 7.** Influence of various parameters on the modal mineralogy of processed REE carbonatite ores. Datasets from single samples are based on the analysis of 300,000–600,000 particles in epoxy grain mount blocks of 30 mm in diameter. Samples are: SPC2—second cleaner concentrate; SPC1—scavenger cleaner concentrate; BSC1—scavenger cleaner middlings; BS1—scavenger tailings; M45, M90 are samples of comminution test with 45 and 90 min of milling. See legend insets. (**a**) Comparison of full dataset composed by inner and outer spiral of measurement frames vs. inner spiral, in the scale 0–35 wt %. At this scale a measurement of the inner spiral appears as sufficient. (**b**) Comparison of the inner spiral vs. the outer spiral of measurement frames, in the scale 0–5 wt %. Although some outliers with low modes exist, the epoxy block displays homogeneous distribution of particles. (**c**) Influence of the parameter reliability of conformity of EDX spectra during measurement classification against reference spectra set. The values of reliability of conformity are intermediate (1℮<sup>−</sup>25) and high (1℮<sup>−</sup>10) and refer to grouped spectra. A high reliability of conformity leads to a significant increase of "unknown" (unk) which is only relevant for modes between 0 and 1 wt %.

There are also uncertainties related to the spectra classification. The EDX spectra classification algorithm for the MLA software packages follows the principle of *best match* along a scale of reliability between 1℮<sup>−</sup><sup>10</sup> (absolute conformance) to 1℮<sup>−</sup><sup>100</sup> (no conformance), as outlined by [16]. REE-bearing minerals display a comparably complex pattern of X-ray emission lines, with many peaks and sub-peaks that are marked by considerable interference (Figure 1g–i). Due to the complex X-ray spectra characteristic for REE-bearing there is considerable risk that EDX spectra are not at all classified, if classification is carried out at a high reliability value. The classification algorithm allows no alternative assignment to another EDX reference spectrum or to another mineral in the list. Due to this principle, the spectra which cannot be classified by the higher reliability scale value will remain as unknown and increase the mode of *unknown* grains. For the study presented here the sample EDX spectra were thus classified by the reliability values of 1℮<sup>−</sup><sup>10</sup> (high degree of conformance) and 1℮<sup>−</sup><sup>25</sup> (fair degree of conformance). The latter reliability value is applied to process samples, in an effort to reduce the amount of unknowns below 0.1 wt % mode (by assigning the specific weight of carbon to the unknown spectra).

Applying a reliability value of 1℮<sup>−</sup><sup>25</sup> to the samples of the third case study, the modes of unknowns remained low, ranging between 0.04 and 0.15 wt %. These values increase to 0.21–0.59 wt % when a reliability value of 1e−<sup>10</sup> is applied to the same data sets. For the modes above 1 wt %, the differences between the two classification schemes are negligible and far below 1 wt %, except for dolomite (−1.2 wt %) in sample BSC1. The largest differences are observed for minor and trace minerals (modal abundance <1.0 wt %; Figure 7c). It is obvious that the increase of the unknowns leads to a reduction of the modes of REE mineral groups. This is the case for the REE-Low-Mix and the REE-Ca-F groups in the samples M90, SPC1 and BSC1 (Figure 7c). For the modes above 1 wt % (not shown), the differences are marginal. The samples of the grinding tests display similar trends. Here, one also observes a reduction of the REE-Low-Mix group, presumably induced by the low intensities along the LREE spectra.
