Eulytine (Bi4(SiO4)3, $S1_{5}$) Structure : A4B12C3_cI76_220_c_e_a

Picture of Structure; Click for Big Picture
Prototype : Bi4O12Si3
AFLOW prototype label : A4B12C3_cI76_220_c_e_a
Strukturbericht designation : $S1_{5}$
Pearson symbol : cI76
Space group number : 220
Space group symbol : $I\bar{4}3d$
AFLOW prototype command : aflow --proto=A4B12C3_cI76_220_c_e_a
--params=
$a$,$x_{2}$,$x_{3}$,$y_{3}$,$z_{3}$


Other compounds with this structure

  • Bi4(GeO4)3, Ca3Bi(PO4)3, and $M$3$L$(PO4)3 ($M$ = Sr, Ba), $L$ = (La, Nd, Gd, Y, Lu, In, Bi)

Body-centered Cubic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & - \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} - \frac12 \, a \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(12a\right) & \mbox{Si} \\ \mathbf{B}_{2} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{z}} & \left(12a\right) & \mbox{Si} \\ \mathbf{B}_{3} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} & \left(12a\right) & \mbox{Si} \\ \mathbf{B}_{4} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} & \left(12a\right) & \mbox{Si} \\ \mathbf{B}_{5} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(12a\right) & \mbox{Si} \\ \mathbf{B}_{6} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(12a\right) & \mbox{Si} \\ \mathbf{B}_{7} & = & 2x_{2} \, \mathbf{a}_{1} + 2x_{2} \, \mathbf{a}_{2} + 2x_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + x_{2}a \, \mathbf{\hat{y}} + x_{2}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 2x_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{2}\right)a \, \mathbf{\hat{y}} + x_{2}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{9} & = & \left(\frac{1}{2} - 2x_{2}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{2}\right)a \, \mathbf{\hat{x}} + x_{2}a \, \mathbf{\hat{y}}-x_{2}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - 2x_{2}\right) \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & x_{2}a \, \mathbf{\hat{x}}-x_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{2}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{11} & = & \left(\frac{1}{2} +2x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +2x_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{12} & = & \frac{1}{2} \, \mathbf{a}_{1}-2x_{2} \, \mathbf{a}_{3} & = & -a\left(x_{2}+\frac{1}{4}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{13} & = & -2x_{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{x}}-a\left(x_{2}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{14} & = & -2x_{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}}-a\left(x_{2}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Bi} \\ \mathbf{B}_{15} & = & \left(y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{17} & = & \left(y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}}-z_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{19} & = & \left(x_{3}+y_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{21} & = & \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & -z_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{22} & = & \left(x_{3}-y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{3}\right)a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{23} & = & \left(x_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{24} & = & \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{25} & = & \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{2} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}}-z_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{3}\right)a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{27} & = & \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & -a\left(y_{3}+\frac{1}{4}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{29} & = & \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}}-a\left(x_{3}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{30} & = & \left(x_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}}-a\left(z_{3}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{31} & = & \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{32} & = & \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}}-a\left(y_{3}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{33} & = & \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -a\left(x_{3}+\frac{1}{4}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{34} & = & \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}}-a\left(z_{3}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{35} & = & \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{36} & = & \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}}-a\left(y_{3}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{37} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}}-a\left(x_{3}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -a\left(z_{3}+\frac{1}{4}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \mbox{O} \\ \end{array} \]

References

  • H. Liu and C. Kuo, Crystal structure of bismuth(III) silicate, Bi4(SiO4)3, Zeitschrift für Kristallographie – Crystalline Materials 212, 48 (1997), doi:10.1524/zkri.1997.212.1.48.

Found in

  • R. T. Downs and M. Hall–Wallace, The American Mineralogist Crystal Structure Database, Am. Mineral. 88, 247–250 (2003).

Geometry files


Prototype Generator

aflow --proto=A4B12C3_cI76_220_c_e_a --params=

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