Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: A5B40CD12_cP116_224_bd_e3k_a_k-001

This structure originally had the label A5B40CD12_cP116_224_cd_e3k_a_k. Calls to that address will be redirected here.

If you are using this page, please cite:
D. Hicks, M.J. Mehl, M. Esters, C. Oses, O. Levy, G.L.W. Hart, C. Toher, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 3, Comp. Mat. Sci. 199, 110450 (2021). (doi=10.1016/j.commatsci.2021.110450)

Links to this page

https://aflow.org/p/E8XB
or https://aflow.org/p/A5B40CD12_cP116_224_bd_e3k_a_k-001
or PDF Version

12-phosphotungstic acid [H$_{3}$PW$_{12}$O$_{40}\cdot$5H$_{2}$O] ($H4_{16}$) Structure: A5B40CD12_cP116_224_bd_e3k_a_k-001

Picture of Structure; Click for Big Picture
Prototype (H$_{2}$O)$_{5}$O$_{40}$PW$_{12}$
AFLOW prototype label A5B40CD12_cP116_224_bd_e3k_a_k-001
Strukturbericht designation $H4_{16}$
Mineral name 12-phosphotungstic acid
ICSD 31897
Pearson symbol cP116
Space group number 224
Space group symbol $Pn\overline{3}m$
AFLOW prototype command aflow --proto=A5B40CD12_cP116_224_bd_e3k_a_k-001
--params=$a, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak z_{8}$

  • This structure is a bit problematic. (Brown, 1977) offers evidence that the true structure of the hydrated phosphotungstic acid is H$_{3}$PW$_{12}$O$_{40}$ · 6H$_{2}$O, rather than 5H$_{2}$O as deduced by (Keggin, 1934). The presence of the extra water molecule does change the structure somewhat, and Keggin does not give the positions of the hydrogen atoms not attached to water molecules. Ordinarily we would mark this structure obsolete, but it is possible to remove water molecules from this structure and still have a molecule that is recognizably related to the acid, as in H$_{3}$PW$_{12}$O$_{40}$ · 3H$_{2}$O. In addition, the lattice constant reported by Keggin is 12.141Å compared to Brown's 12.506Å, a change consistent with the loss of a water molecule. Given this, we will not depreciate this five Water molecule structure.
  • (Gottfried, 1937) does not give the positions of the water molecules, and they reverse the $x$ and $z$ coordinates for O-IV and W.
  • This structure is a partially dehydrated form of H$_{3}$PW$_{12}$O$_{40}$ · 29H$_{2}$O ($H4_{21}$). Further dehydration produces the H$_{3}$PW$_{12}$O$_{40}$ · 3H$_{2}$O structure.
  • The positions of the three hydrogen atoms are unknown. If we follow (Marosi, 2000), then the 'free' hydrogens are likely to be bound to some of the water molecules, giving the unusual stoichiometry in the prototype.
  • The ICSD entry for (Keggin, 1934) does not list the positions of the water molecules.

\[ \begin{array}{ccc} \mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&a \,\mathbf{\hat{z}} \end{array}\]

Basis vectors

Lattice coordinates Cartesian coordinates Wyckoff position Atom type
$\mathbf{B_{1}}$ = $\frac{1}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ = $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ (2a) P I
$\mathbf{B_{2}}$ = $\frac{3}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ = $\frac{3}{4}a \,\mathbf{\hat{x}}+\frac{3}{4}a \,\mathbf{\hat{y}}+\frac{3}{4}a \,\mathbf{\hat{z}}$ (2a) P I
$\mathbf{B_{3}}$ = $0$ = $0$ (4b) H I
$\mathbf{B_{4}}$ = $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}$ = $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}$ (4b) H I
$\mathbf{B_{5}}$ = $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{3}$ = $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ (4b) H I
$\mathbf{B_{6}}$ = $\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ = $\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ (4b) H I
$\mathbf{B_{7}}$ = $\frac{1}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ = $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{3}{4}a \,\mathbf{\hat{y}}+\frac{3}{4}a \,\mathbf{\hat{z}}$ (6d) H II
$\mathbf{B_{8}}$ = $\frac{3}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ = $\frac{3}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{3}{4}a \,\mathbf{\hat{z}}$ (6d) H II
$\mathbf{B_{9}}$ = $\frac{3}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ = $\frac{3}{4}a \,\mathbf{\hat{x}}+\frac{3}{4}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ (6d) H II
$\mathbf{B_{10}}$ = $\frac{1}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ = $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{3}{4}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ (6d) H II
$\mathbf{B_{11}}$ = $\frac{3}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ = $\frac{3}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ (6d) H II
$\mathbf{B_{12}}$ = $\frac{1}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ = $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{3}{4}a \,\mathbf{\hat{z}}$ (6d) H II
$\mathbf{B_{13}}$ = $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ = $a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{14}}$ = $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ = $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{15}}$ = $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{16}}$ = $x_{4} \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{17}}$ = $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ = $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{18}}$ = $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ = $- a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{19}}$ = $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}+a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{20}}$ = $- x_{4} \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (8e) O I
$\mathbf{B_{21}}$ = $x_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ = $a x_{5} \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{22}}$ = $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ = $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{23}}$ = $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}- a \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{24}}$ = $x_{5} \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{5} \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{25}}$ = $z_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ = $a z_{5} \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{26}}$ = $z_{5} \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a z_{5} \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{27}}$ = $- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ = $- a \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{28}}$ = $- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{29}}$ = $x_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ = $a x_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{30}}$ = $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{31}}$ = $x_{5} \, \mathbf{a}_{1}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{5} \,\mathbf{\hat{x}}- a \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{32}}$ = $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ = $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{33}}$ = $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ = $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{34}}$ = $- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ = $- a x_{5} \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{35}}$ = $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}+a \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{36}}$ = $- x_{5} \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{5} \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{37}}$ = $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ = $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{38}}$ = $- x_{5} \, \mathbf{a}_{1}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{5} \,\mathbf{\hat{x}}+a \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{39}}$ = $- x_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ = $- a x_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{40}}$ = $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{41}}$ = $\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ = $a \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{42}}$ = $\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{43}}$ = $- z_{5} \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a z_{5} \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{44}}$ = $- z_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ = $- a z_{5} \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ (24k) O II
$\mathbf{B_{45}}$ = $x_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ = $a x_{6} \,\mathbf{\hat{x}}+a x_{6} \,\mathbf{\hat{y}}+a z_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{46}}$ = $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ = $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a z_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{47}}$ = $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{6} \,\mathbf{\hat{y}}- a \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{48}}$ = $x_{6} \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{6} \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{49}}$ = $z_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ = $a z_{6} \,\mathbf{\hat{x}}+a x_{6} \,\mathbf{\hat{y}}+a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{50}}$ = $z_{6} \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a z_{6} \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{51}}$ = $- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ = $- a \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{52}}$ = $- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{6} \,\mathbf{\hat{y}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{53}}$ = $x_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ = $a x_{6} \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{y}}+a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{54}}$ = $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a z_{6} \,\mathbf{\hat{y}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{55}}$ = $x_{6} \, \mathbf{a}_{1}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{6} \,\mathbf{\hat{x}}- a \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{56}}$ = $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ = $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{57}}$ = $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ = $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a z_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{58}}$ = $- x_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ = $- a x_{6} \,\mathbf{\hat{x}}- a x_{6} \,\mathbf{\hat{y}}- a z_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{59}}$ = $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{6} \,\mathbf{\hat{y}}+a \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{60}}$ = $- x_{6} \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{6} \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{61}}$ = $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ = $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{62}}$ = $- x_{6} \, \mathbf{a}_{1}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{6} \,\mathbf{\hat{x}}+a \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{63}}$ = $- x_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ = $- a x_{6} \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{y}}- a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{64}}$ = $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a z_{6} \,\mathbf{\hat{y}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{65}}$ = $\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ = $a \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{66}}$ = $\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{6} \,\mathbf{\hat{y}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{67}}$ = $- z_{6} \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a z_{6} \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{68}}$ = $- z_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ = $- a z_{6} \,\mathbf{\hat{x}}- a x_{6} \,\mathbf{\hat{y}}- a x_{6} \,\mathbf{\hat{z}}$ (24k) O III
$\mathbf{B_{69}}$ = $x_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ = $a x_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{70}}$ = $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ = $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{71}}$ = $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{72}}$ = $x_{7} \, \mathbf{a}_{1}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{7} \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{73}}$ = $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ = $a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{74}}$ = $z_{7} \, \mathbf{a}_{1}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a z_{7} \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{75}}$ = $- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ = $- a \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{76}}$ = $- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{77}}$ = $x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ = $a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{78}}$ = $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{79}}$ = $x_{7} \, \mathbf{a}_{1}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{7} \,\mathbf{\hat{x}}- a \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{80}}$ = $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ = $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{81}}$ = $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ = $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{82}}$ = $- x_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ = $- a x_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{83}}$ = $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{84}}$ = $- x_{7} \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{7} \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{85}}$ = $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ = $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{86}}$ = $- x_{7} \, \mathbf{a}_{1}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{7} \,\mathbf{\hat{x}}+a \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{87}}$ = $- x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ = $- a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{88}}$ = $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{89}}$ = $\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ = $a \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{90}}$ = $\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{91}}$ = $- z_{7} \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a z_{7} \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{92}}$ = $- z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ = $- a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ (24k) O IV
$\mathbf{B_{93}}$ = $x_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ = $a x_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{94}}$ = $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ = $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{95}}$ = $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{96}}$ = $x_{8} \, \mathbf{a}_{1}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{8} \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{97}}$ = $z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ = $a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{98}}$ = $z_{8} \, \mathbf{a}_{1}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a z_{8} \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{99}}$ = $- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ = $- a \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{100}}$ = $- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{101}}$ = $x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ = $a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{102}}$ = $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{103}}$ = $x_{8} \, \mathbf{a}_{1}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a x_{8} \,\mathbf{\hat{x}}- a \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{104}}$ = $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ = $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{105}}$ = $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ = $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{106}}$ = $- x_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ = $- a x_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{107}}$ = $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{108}}$ = $- x_{8} \, \mathbf{a}_{1}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{8} \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{109}}$ = $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ = $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{110}}$ = $- x_{8} \, \mathbf{a}_{1}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a x_{8} \,\mathbf{\hat{x}}+a \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{111}}$ = $- x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ = $- a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{112}}$ = $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{113}}$ = $\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ = $a \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{114}}$ = $\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $a \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{115}}$ = $- z_{8} \, \mathbf{a}_{1}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ = $- a z_{8} \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ (24k) W I
$\mathbf{B_{116}}$ = $- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ = $- a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ (24k) W I

References

  • J. F. Keggin, The structure and formula of 12-phosphotungstic acid, Proc. Royal Soc. London A 144, 75–100 (1934), doi:10.1098/rspa.1934.0035.
  • C. Gottfried and F. Schossberger, eds., Strukturbericht Band III 1933-1935 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).
  • L. Marosi, E. E. Platero, J. Cifre, and C. O. Areána, Thermal dehydration of H$_{3+x}$PV$_{x}$M${12−x}$O$_{40}$$\cdot$yH$_{2}$O Keggin type heteropolyacids; formation, thermal stability and structure of the anhydrous acids H$_{3}$PM$_{12}$O$_{40}$, of the corresponding anhydrides PM$_{12}$O$_{38.5}$ and of a novel trihydrate H$_{3}$PW$_{12}$O$_{40}$$\cdot$3H$_{2}$O, Journal of Materials Chemistry 10, 1949–1955 (2000), doi:10.1039/b001476l.

Found in

  • G. M. Brown, M.-R. Noe-Spirlet, W. R. Busing, and H. A. Levy, Dodecatungstophosphoric acid hexahydrate, (H$_{5}$O$_{2}^+$)$_{3}$(PW$_{12}$O$_{40}^{3-}$). The true structure of Keggin's 'pentahydrate' from single-crystal X-ray and neutron diffraction data, Acta Crystallogr. Sect. B 33, 1038–1046 (1977), doi:10.1107/S0567740877005330.

Prototype Generator

aflow --proto=A5B40CD12_cP116_224_bd_e3k_a_k --params=$a,x_{4},x_{5},z_{5},x_{6},z_{6},x_{7},z_{7},x_{8},z_{8}$

Species:

Running:

Output: