Bertrandite (Be4Si2O7(OH)2, $S4_{6}$) Structure : A4B7C2D2_oC60_36_2b_a3b_2a_b

Picture of Structure; Click for Big Picture
Prototype : Be4O7(OH)2Si2
AFLOW prototype label : A4B7C2D2_oC60_36_2b_a3b_2a_b
Strukturbericht designation : $S4_{6}$
Pearson symbol : oC60
Space group number : 36
Space group symbol : $Cmc2_{1}$
AFLOW prototype command : aflow --proto=A4B7C2D2_oC60_36_2b_a3b_2a_b
--params=
$a$,$b/a$,$c/a$,$y_{1}$,$z_{1}$,$y_{2}$,$z_{2}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$


  • All sources agree that the space group for Bertrandite is $Cmc2_{1}$ #36 but differ slightly in details. We have chosen to use (Hazen, 1986) as the prototype over (Ito, 1932), which was originally designated $S4_{6}$, and (Solov'eva, 1965), which has unrealistically large Be–O distances.
  • No paper locates the hydrogen atoms in the hydroxide ions, which may indicate that these ions can freely rotate.
  • We use the data taken at 1 atmosphere.

Base-centered Orthorhombic primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & -y_{1} \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & y_{1}b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{2} & = & y_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & -y_{1}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{3} & = & -y_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & y_{2}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{OH I} \\ \mathbf{B}_{4} & = & y_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & -y_{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{OH I} \\ \mathbf{B}_{5} & = & -y_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & y_{3}b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{OH II} \\ \mathbf{B}_{6} & = & y_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{OH II} \\ \mathbf{B}_{7} & = & \left(x_{4}-y_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be I} \\ \mathbf{B}_{8} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be I} \\ \mathbf{B}_{9} & = & \left(x_{4}+y_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be I} \\ \mathbf{B}_{10} & = & \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be I} \\ \mathbf{B}_{11} & = & \left(x_{5}-y_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}+y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be II} \\ \mathbf{B}_{12} & = & \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}-y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be II} \\ \mathbf{B}_{13} & = & \left(x_{5}+y_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be II} \\ \mathbf{B}_{14} & = & \left(-x_{5}-y_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Be II} \\ \mathbf{B}_{15} & = & \left(x_{6}-y_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}+y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O II} \\ \mathbf{B}_{16} & = & \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}-y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O II} \\ \mathbf{B}_{17} & = & \left(x_{6}+y_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O II} \\ \mathbf{B}_{18} & = & \left(-x_{6}-y_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O II} \\ \mathbf{B}_{19} & = & \left(x_{7}-y_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+y_{7}\right) \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O III} \\ \mathbf{B}_{20} & = & \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O III} \\ \mathbf{B}_{21} & = & \left(x_{7}+y_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O III} \\ \mathbf{B}_{22} & = & \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O III} \\ \mathbf{B}_{23} & = & \left(x_{8}-y_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}+y_{8}\right) \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O IV} \\ \mathbf{B}_{24} & = & \left(-x_{8}+y_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}-y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O IV} \\ \mathbf{B}_{25} & = & \left(x_{8}+y_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}-y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O IV} \\ \mathbf{B}_{26} & = & \left(-x_{8}-y_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}+y_{8}\right) \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{O IV} \\ \mathbf{B}_{27} & = & \left(x_{9}-y_{9}\right) \, \mathbf{a}_{1} + \left(x_{9}+y_{9}\right) \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Si} \\ \mathbf{B}_{28} & = & \left(-x_{9}+y_{9}\right) \, \mathbf{a}_{1} + \left(-x_{9}-y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Si} \\ \mathbf{B}_{29} & = & \left(x_{9}+y_{9}\right) \, \mathbf{a}_{1} + \left(x_{9}-y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Si} \\ \mathbf{B}_{30} & = & \left(-x_{9}-y_{9}\right) \, \mathbf{a}_{1} + \left(-x_{9}+y_{9}\right) \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Si} \\ \end{array} \]

References

  • R. M. Hazen and A. Y. Au, High–pressure crystal chemistry of phenakite (Be2SiO4) and bertrandite (Be4Si2O7(OH)2), Phys. Chem. Miner. 13, 69–78 (1986), doi:10.1007/BF00311896.
  • T. Ito and J. West, The Structure of Bertrandite (H2Be4Si2O9), Zeitschrift für Kristallographie – Crystalline Materials 83, 384–393 (1932), doi:10.1524/zkri.1932.83.1.384.
  • L. P. Solov'eva and N. V. Belov, Precise Determination of the Crystal Structure of Bertrandite Be4[Si207](OH)2, Sov. Phys. Crystallogr. 9, 458–460 (1965).

Found in

  • B. Lafuente, R. T. Downs, H. Yang, and N. Stone, The power of databases: the RRUFF project, in Highlights in Mineralogical Crystallography, edited by T. Armbruster and R. M. Danisi (De Gruyter, Berlin, 2015), chap. 1, 1–30.

Geometry files


Prototype Generator

aflow --proto=A4B7C2D2_oC60_36_2b_a3b_2a_b --params=

Species:

Running:

Output: