## 4 Discussion
### Distribution of REEs  
In order to understand the accumulation of the REEs in the Ti-phases, goethite, and cancrinite, fractions of a single red mud sample (representing the 4-8 m deep interval) were produced. Four mineral concentrates were studied; the magnetic and non-magnetic fractions of the coarse (\(>\)20 microns) grains, as well as a cancrinite-rich and a goethite-rich fraction. According to the semi-quantitative XRD estimation, the proportion of cancrinite decreases from 40-50% down to 20-30%, while that of goethite increases from 10-20% up to 40-50% in the last two fractions, respectively. The rest of both fractions is hematite. Comparing the REE content of the four fractions, the cancrinite-rich sample is the highest in all REEs followed by the goethite-rich one, while the two coarse grain fractions are the lowest in all REEs (Table 4). Concentrations in the cancrinite-rich concentrate are similar to or are slightly higher than those of the total sample proposing that cancrinite is the main REE container phase of the red mud (Figure 6). The non-magnetic coarse fraction is higher in each REE than the magnetic mineral separate. Considering the tight correlation between Ti and the light REEs, anatase, as the only non-magnetic Ti-phase may be responsible for this kind of accumulation. Despite this behaviour of anatase and keeping their low modal proportion in mind, the coarse grains are not important REE container minerals of the red mud studied. Element enrichment between the major phases can be characterized by enrichment factor for each element so that:  
\[\text{E}_{\text{A/B}}(\text{element})=\text{X}_{\text{A}}(\text{element})/ \text{X}_{\text{B}}(\text{element}), \tag{1}\]  
where A and B are the mineral phases studied and X stands for concentration of the element in question. To calculate concentrations in the pure end members based on the mineral mixtures, a linear mixing model was used (Figure 7) based on the following equation system:  
\[\text{a}_{\text{i}}\ \text{*X}_{\text{A}}(\text{element})+(\text{1-a}_{\text{i}}) \ \text{*X}_{\text{B}}(\text{element})=\text{X}_{\text{i}}(\text{element}), \tag{2}\]  
where I = 1, 2 for the two fine-grained mixtures, and a\({}_{1}\) and a\({}_{2}\) are the proportions of cancrinite in the mineral mixtures what are only roughly known from XRD. Figure 5: Highly simplified and idealized rock column suggesting the most essential mineralogical processes inside the red mud pitfall. From this on, there are two equations with four unknowns (a\({}_{1}\), a\({}_{2}\), X\({}_{\rm A}\), X\({}_{\rm B}\)) for each element. Moreover, an equation with an identical form exists for the whole sample so that:  
\[\rm{a_{tot}}\ *X_{\rm A}(element)+(1-a_{tot})*X_{\rm B}(element)=X_{tot}(element), \tag{3}\]  
where a\({}_{\rm tot}\) is the proportions of carcinrite in the whole sample. The parameter triplet (a\({}_{1}\), a\({}_{2}\), a\({}_{\rm tot}\)) can be optimized so that be minimal. For optimization, the trial-and-error method was followed (Figure 8). Finally, based on the estimated a\({}_{1}\) and a\({}_{2}\), both X\({}_{\rm A}\) and X\({}_{\rm B}\) can be calculated:  
\[\rm{\varepsilon}=\sum_{\rm REE}\biggl{(}\frac{X_{tot,measured}-X_{tot,estimated}}{X_{tot,measured}} \biggr{)}^{2} \tag{4}\]  
Figure 6: REE concentrations in the goethite-rich and the carcinrite-rich fractions relative to the original red mud sample. Figure 7: The theoretical background of the linear mixing modeling procedure. The result of the model was accepted at the lowest available \(\varepsilon\) (= 34.47), where the Pearson correlation coefficient between the measured and estimated total concentrations is better than 0.999 and the slope of the regression line between the measured and estimated concentration is as good as 1.01 (Figure 8d., Table 5).