The Application of the Three-Dimensional Patterson Method and the Crystal Structures of Proustite, Ag3AsS3, and Pyrargyrite, Ag3SbS3

It is shown that the three‐dimensional Patterson method can be so simplified by the use of the symmetry properties of the crystal under consideration that its use in determining the positions of atoms in crystals is practicable. This method is then used to determine the positions of the heavy atoms in proustite, Ag₃AsS₃. The positions of the sulfur atoms are found by the use of the covalent atomic radii and assumptions concerning bond directions. The structure so derived is found to be compatible with the observed data. Proustite is found to contain pyramidal ${\mathrm{AsS}}_3^{\mathrm{\prime \prime \prime }}$ groups and silver atoms forming two bonds to sulfur in almost opposed directions. Pyrargyrite, Ag₃SbS₃, is found to have almost the same structure as proustite. The analytical statement of the structures is as follows:— The space group, C_3v⁶ — R3c, is common to both proustite and pyrargyrite. The special positions of C_3v⁶ are (in hexagonal axes) $$\begin{array}{cccc}\text{(2}\mathrm{a})& \text{0,0,Z;}& \text{0,0,}\frac{1}{2}\mathrm{+Z;}& \\ \text{(6}\mathrm{b})& \mathrm{X,Y,Z;}& \overline{Y}\mathrm{,X-Y,Z;}& \mathrm{Y-X,}\overline{X}\mathrm{,Z;}\\ & \overline{Y},\overline{X},\frac{1}{2}\mathrm{+Z;}& \mathrm{X,X-Y,}\frac{1}{2}\mathrm{+Z;}& \mathrm{Y-X,Y,}\frac{1}{2}\mathrm{+Z;}\end{array}$$ and positions derived from these by the operations of the rhombohedral lattice. The parameter values are: $$\begin{array}{cccc}\mathrm{Proustite}& & \mathrm{Pyrargyrite}& \\ a_0\text{=10.74}\mathrm{A},& c_0\text{=8.64}\mathrm{A}& a_0\text{=11.04}\mathrm{A},& c_0\text{=8.71}\mathrm{A}\\ 2\mathrm{As\; in}\text{\hspace{0.17em}2a,}& Z_{\mathrm{As}}\text{=0.000}& 2\mathrm{Sb\; in}\text{\hspace{0.17em}2a,}& Z_{\mathrm{Sb}}\text{=0.000}\\ 6\mathrm{Ag\; in}\text{\hspace{0.17em}6b,}& X_{\mathrm{Ag}}\text{=0.246}& 6\mathrm{Ag\; in}\text{\hspace{0.17em}6b,}& X_{\mathrm{Ag}}\text{=0.250}\\ & Y_{\mathrm{Ag}}\text{=0.298}& & Y_{\mathrm{Ag}}\text{=0.305}\\ & Z_{\mathrm{Ag}}\text{=0.235}& & Z_{\mathrm{Ag}}\text{=0.210}\\ 6\mathrm{S\; in}\text{\hspace{0.17em}6b,}& X_{\mathrm{S}}\text{=0.220}& 6\mathrm{S\; in}\text{\hspace{0.17em}6b,}& X_{\mathrm{S}}\text{=0.220}\\ & Y_{\mathrm{S}}\text{=0.095}& & Y_{\mathrm{S}}\text{=0.105}\\ & Z_{\mathrm{S}}\text{=0.385}& & Z_{\mathrm{S}}\text{=0.355}\end{array}$$