## 2 Materials and Methods
### Multivariate Data Analysis  
The LIB spectral data of the gemstone samples were analyzed using chemometrics. The spectra acquired from different samples were systemized, and several data sets were arranged in a 90 \(\times\) 20,538 matrix with 20,538 features. Unscrambler-X software (CAMO Software India Pvt. Ltd.) was used for PCA. It solved the data matrix and generated the score and loading plot. PCA is a multivariate technique used to discern similar samples based on small variations in the variables (i.e., the wavelength). PCA eliminates the noise and reduces the dimensionality of spectral data while calculating principal components (PCs) by maximizing the variation in variables (wavelength). Principal components are vectors with a combination of linear variables. The first PC contains the most data variation while the second PC is perpendicular to the first and includes the remaining data variability; this is similar for the higher PCs. The three-dimensional PCA score plot is a significant feature of the analysis. In the score plot, spectra of samples sharing certain similarities cluster together and form numerous groups according to their similarities, whereas spectra with different elemental compositions or concentrations are separated. On the other hand, the loading plot, which is another output of the PCA analysis, is helpful to see the contributions of the individual variables. The loading graphs are used to identify which elements are responsible for the differentiation of the observed clusters. The loadings  
Figure 2: Experimental setup to record LIB spectra of gemstone samples. Considering the general chemical formula and the corresponding spectral lines observed in the LIB spectra of the gemstones, it is clear that every LIB spectrum contains spectral lines of constituent elements of the respective gemstone. Sample S_1 represents an amethyst with a purple color. The corresponding LIB spectrum shows lines of Fe, Al, and Be. Previous studies revealed that Al, Fe, Be, and K are present in amethyst as trace elements [23]. [PERSON] et al. [24] used electron microprobe analysis to study amethyst and found that traces of iron may be responsible for amethysts' coloration. [PERSON] et al. [25] and [PERSON] et al. [26] suggested that amethyst is colored by an O\({}^{2-}\)\(\rightarrow\)Fe\({}^{4+}\) charge transfer. Sample S_2 is aqumarine beryl with a pale blue color. In its LIB spectrum, spectral lines of Fe are also observed, along with Si, O, Al, and Be. [PERSON] et al. [27] used Mossbauer spectroscopy for the characterization of similar samples and found that the pale blue color of aqumarine beryl can be attributed to iron in the crystal structure. Others, especially [PERSON] et al. [28] and [PERSON] [29], explained that blue, green, and yellow tones are the result of different Fe\({}^{2+}\)/Fe\({}^{3+}\) ratios. Sample S_3 is a variety of agate (cryptocrystalline silica), possibly a bloodstream. It is dark green to dark bluish-green in color and dotted with red or brown spots. It has been suggested that heliotrope bloodstream is an opaque green jasper with inclusions of chlorite, amphibole, and pyroxene that contribute to its green color and has red spots of hematite. [30; 31]. [PERSON] et al. [32] analyzed a heliotrope (also known as bloodstream) sample by synchrotron radiation X-ray diffraction (SR-XRD) and PIXE techniques and detected the presence of Si, O, Al, Ca, Pb, K, Fe, P, Cu, and Zn. Another study involved the SR-XRF technique to investigate Si and Fe as major elements, while Ti, Mn, Cu, Zn, Rb, Y, Sr, and Zr were identified as impurity elements in bloodstream samples. [PERSON] et al. [33] suggested that the Fe distribution is not homogeneous over the sample and that the green color is related to Fe\({}^{2+}\) while the red spots correspond to hematite (Fe\({}_{2}\)O)\({}_{3}\). The commercial nomenclature of Sample S_4 is given as diopside (CaMgSi\({}_{2}\)O\({}_{6}\)).