Structure Preparation

From a mineragraphic analysis (QUEMSCAN PMA) (^{Avino Silver & Gold Mines, 2015}) of a mineral concentrate obtained in Durango, Mexico, crystalline structures were identified and used as a starting point to model through the Visualizer Module of Materials Studio software version 8.0 (Table 1).

Structure Optimization

Each structure was optimized according to an energy evaluation and conformational adjustment. Structural parameters were obtained by moving the constituent atoms to positions where the energy was minimal. This procedure was carried out using the CASTEP module by performing the Quasi-Newton BFGS method (^{Li & Fukushima, 2001}; ^{Pfrommer et al., 1997}; ^{Zhou, 2020}). This method accumulates information about enthalpy in the inverse of the Hess matrix, once the coordinates of the configuration space are defined. The crystalline structure is delimited by the matrix of lattice vectors h={a,b,c} and the coordinates s _i=1…N, relative to h of the N atoms in the unit cell, which has a volume of Ω=det(h). Conveniently, strain tensor Ɛ is chosen in exchange for the lattice vectors and is related to the reference configuration h ₀ in h, as observed in equation 1.

A point in configuration space by the column vector X and its deformation components, which is the negative of the derivative of the enthalpy H for X, is given by equation 2, also known as force vector.

Moreover, from the enthalpy per unit cell (equation 3), the components of the deformation of F are derived (equation 4).

The deformation components of the force vector are multiplied by the metric tensor g =h^Th , resulting in all the deformation components of F (equation 5).

So, the change in enthalpy around the minimum is given by equation 6:

The key of the method is to assume a value of A and to improve it at each step. The dependence of the enthalpy on the strain tensor is governed by the mass, and the deformation of H₀ is multiplied by (3ΩB₀)^-1 for the internal coordinates. Internal coordinates are proposed to start at H₀ as a block diagonal with matrixes of the g0-1 M--1ω-0-1 form, allowing equation 7 to show the final form for the calculation of H₀.

Crystal Creation

Crystalline structures were created in the Crystal Builder Module of Materials Studio version 8.0 software. Corresponding parameters of each crystal such as the space group, crystallization system, as well as the values of the lattice parameters and the measurement of the angles were captured.

Validation of crystalline structures

Through a study of XRD (Rigaku MiniFlex) of mining concentrates, crystalline structures were characterized, obtaining parameters that allowed to compare experimental results against the simulated ones to validate each of the structures that were created. XRD data were obtained with an SC-70 detector, with voltage and current of the source at 40 kV and 15 mA, respectively, using a 2θ scan axis.

Structure Preparation

The structural information of the crystals found in the mineral concentrate is presented in Table 1. All structures were created from their base structure. Representative examples are shown in Figure 1.

Source: Authors’ own elaboration.

Figure 1 Base structures of a) Arsenopyrite, b) Calcite, c) Wittichenite d) Quartz, e) Chalcopyrite. 

Structure Optimization

From the CASTEP module, the universal force field was chosen because it allows to analyze all metallic materials (^{BIOVIA, 2014c}). Optimization of each structure was carried out using the quasi-Newton BFGS iterative method, in which, with each iteration, a value for energy is obtained. Figure 2 shows the energy optimization lines of the different crystalline structures.

Source: Authors’ own elaboration.

Figura 2 Optimización de todas las estructuras cristalinas simuladas. 

Crystal Creation

Each crystal was created according to the specific lattice parameters, as well as the space group (^{BIOVIA, 2014a}). For instance, arsenopyrite crystallizes in a monoclinic system in a space group P 2(1)/c, and it is composed of 46.01% arsenic, 34.30% iron, and 19.69% sulfur (Figure 3a). Calcite is a mineral of calcium carbonate whose crystallization system is trigonal and its space group is R3*c, which is made up of 40.04% calcium and the rest is carbonate (Figure 3b). Wittichenite is a very rare mineral that crystallizes in an orthorhombic system; its space group is P2(1)2(1)2(1); and it is composed of sulfur, copper, and bismuth in 19.40%, 38.45%, and 42.15% respectively (Figure 3c). Quartz is one of the best-known minerals on earth. It is composed of 46.74% silicon and the rest is oxygen, it crystallizes in a trigonal system and its space group is P 3 (2) 21; its structure can be seen in Figure 3d. Chalcopyrite, with a tetragonal crystallization system and an I-42D space group, is composed in equal parts of copper and sulfur (34.78%) and the rest is iron (^{Cambridge Crystallographic Data Center [CCDC], 2020}; ^{Crystallography Open Database, 2020}; ^{Mineralogical Society of America [Minsocam], 2017}) (Figure 3e).

Source: Authors’ own elaboration.

Figure 3 Examples of crystalline structures created in Materials Studio a) Arsenopyrite, b) Calcite, c) Wittichenite, d) Quartz, and, e) Chalcopyrite. 

Validation of Crystalline Structures

The unit cell size and orientation of the crystal determine the position at which the diffraction patterns can be measured at the appropriate wavelength. The analysis of the positions of the diffraction peaks produces the unit cell size and its orientation (^{BIOVIA, 2014b}); for this reason, to validate each one of the structures that were modeled in the simulator, lattice parameters and space groups were compared against those obtained from the XRD study (Table 2). Not all crystals that were simulated appeared in the diffraction study; therefore, only the results of those that appeared in the study are shown.

Results shown in Table 2 indicate that more than 50% of the crystals that appeared in the XRD study agree with the values used for the simulation. Furthermore, the lack of concordance might occur because of the discrepancy of their chemical composition, resulting in a mismatch of space groups, in that several crystals depend on pressure and temperature to define the previously mentioned parameters, which may present variations.

X-ray Diffraction Study

Minerals are crystalline and the spatial distribution of their components is arranged as structured networks (^{Consejo Superior de Investigaciones Científicas [CSIC], 2020}; ^{Servicio Geólogico Mexicano [SGM], 2010}); hence, XRD studies can also be used to obtain the diffraction pattern of the sample (Figure 4) quantitative analyzes.

Source: Authors’ own elaboration.

Figure 4 XDR pattern obtained experimentally 

Subsequently, the simulation of the diffraction patterns of each crystal structure that appeared in the study was performed. This procedure was carried out using the REFLEX module of the same software (^{BIOVIA, 2014d}). The obtained results can be seen in Figure 5. However, it is a considerably different pattern from the one observed in Figure 4, this may be because in the simulation each crystal is individually analyzed, whereas in the study carried out experimentally the sample of the mining concentrate contains all its components at once.

Source: Authors’ own elaboration.

Figure 5 XDR patterns of simulated crystal structures 

To analyze the similarity of the XRD patterns, further research is required for each pure crystal to fully compare Figures 4 and 5.