Low Temperature White
Phosphorous Structure: A_aP24_2_12i-001
| Prototype | P |
| AFLOW prototype label | A_aP24_2_12i-001 |
| ICSD | 68326 |
| Pearson symbol | aP24 |
| Space group number | 2 |
| Space group symbol | $P\overline{1}$ |
| AFLOW prototype command |
aflow --proto=A_aP24_2_12i-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak \alpha, \allowbreak \beta, \allowbreak \gamma, \allowbreak x_{1}, \allowbreak y_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}$ |
- Phosphorus is found in at least three forms:
- Black phosphorus, Strukturbericht $A17$,
- Monoclinic Hittorf's phosphorus, and
- Low temperature triclinic
white
phosphorus, stable below 197K. (this structure)
- We use the data taken at 162K.
- The P$_{4}$ molecules exhibit considerable rigid-body motion, both rotational and translational (Simon, 1987).
- There is considerable disagreement about the name of this structure. Most authors refer to it as $\beta$-white phosphorous, but (Donohue, 1982) refers to it as $\alpha$-white phosphorous. Other authors refer to this as yellow phosphorous. Simply referring to this as low temperature phosphorous seems the easiest way to avoid confusion.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&b \cos{\gamma} \,\mathbf{\hat{x}}+b \sin{\gamma} \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c_{x} \,\mathbf{\hat{x}}+c_{y} \,\mathbf{\hat{y}}+c_{z} \,\mathbf{\hat{z}}\\c_{x} & = & c \cos{\beta} \\ c_{y} & = & c (\cos{\alpha} - \cos{\beta}\cos{\gamma}) / {\sin{\gamma}} \\ c_{z} & = & \sqrt{c^2 - c_{x}^2- c_{y}^2}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $\left(a x_{1} + b y_{1} \cos{\gamma} + c_{x} z_{1}\right) \,\mathbf{\hat{x}}+\left(b y_{1} \sin{\gamma} + c_{y} z_{1}\right) \,\mathbf{\hat{y}}+c_{z} z_{1} \,\mathbf{\hat{z}}$ | (2i) | P I |
| $\mathbf{B_{2}}$ | = | $- x_{1} \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}$ | = | $- \left(a x_{1} + b y_{1} \cos{\gamma} + c_{x} z_{1}\right) \,\mathbf{\hat{x}}- \left(b y_{1} \sin{\gamma} + c_{y} z_{1}\right) \,\mathbf{\hat{y}}- c_{z} z_{1} \,\mathbf{\hat{z}}$ | (2i) | P I |
| $\mathbf{B_{3}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\left(a x_{2} + b y_{2} \cos{\gamma} + c_{x} z_{2}\right) \,\mathbf{\hat{x}}+\left(b y_{2} \sin{\gamma} + c_{y} z_{2}\right) \,\mathbf{\hat{y}}+c_{z} z_{2} \,\mathbf{\hat{z}}$ | (2i) | P II |
| $\mathbf{B_{4}}$ | = | $- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $- \left(a x_{2} + b y_{2} \cos{\gamma} + c_{x} z_{2}\right) \,\mathbf{\hat{x}}- \left(b y_{2} \sin{\gamma} + c_{y} z_{2}\right) \,\mathbf{\hat{y}}- c_{z} z_{2} \,\mathbf{\hat{z}}$ | (2i) | P II |
| $\mathbf{B_{5}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\left(a x_{3} + b y_{3} \cos{\gamma} + c_{x} z_{3}\right) \,\mathbf{\hat{x}}+\left(b y_{3} \sin{\gamma} + c_{y} z_{3}\right) \,\mathbf{\hat{y}}+c_{z} z_{3} \,\mathbf{\hat{z}}$ | (2i) | P III |
| $\mathbf{B_{6}}$ | = | $- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $- \left(a x_{3} + b y_{3} \cos{\gamma} + c_{x} z_{3}\right) \,\mathbf{\hat{x}}- \left(b y_{3} \sin{\gamma} + c_{y} z_{3}\right) \,\mathbf{\hat{y}}- c_{z} z_{3} \,\mathbf{\hat{z}}$ | (2i) | P III |
| $\mathbf{B_{7}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\left(a x_{4} + b y_{4} \cos{\gamma} + c_{x} z_{4}\right) \,\mathbf{\hat{x}}+\left(b y_{4} \sin{\gamma} + c_{y} z_{4}\right) \,\mathbf{\hat{y}}+c_{z} z_{4} \,\mathbf{\hat{z}}$ | (2i) | P IV |
| $\mathbf{B_{8}}$ | = | $- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $- \left(a x_{4} + b y_{4} \cos{\gamma} + c_{x} z_{4}\right) \,\mathbf{\hat{x}}- \left(b y_{4} \sin{\gamma} + c_{y} z_{4}\right) \,\mathbf{\hat{y}}- c_{z} z_{4} \,\mathbf{\hat{z}}$ | (2i) | P IV |
| $\mathbf{B_{9}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + b y_{5} \cos{\gamma} + c_{x} z_{5}\right) \,\mathbf{\hat{x}}+\left(b y_{5} \sin{\gamma} + c_{y} z_{5}\right) \,\mathbf{\hat{y}}+c_{z} z_{5} \,\mathbf{\hat{z}}$ | (2i) | P V |
| $\mathbf{B_{10}}$ | = | $- x_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \left(a x_{5} + b y_{5} \cos{\gamma} + c_{x} z_{5}\right) \,\mathbf{\hat{x}}- \left(b y_{5} \sin{\gamma} + c_{y} z_{5}\right) \,\mathbf{\hat{y}}- c_{z} z_{5} \,\mathbf{\hat{z}}$ | (2i) | P V |
| $\mathbf{B_{11}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + b y_{6} \cos{\gamma} + c_{x} z_{6}\right) \,\mathbf{\hat{x}}+\left(b y_{6} \sin{\gamma} + c_{y} z_{6}\right) \,\mathbf{\hat{y}}+c_{z} z_{6} \,\mathbf{\hat{z}}$ | (2i) | P VI |
| $\mathbf{B_{12}}$ | = | $- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \left(a x_{6} + b y_{6} \cos{\gamma} + c_{x} z_{6}\right) \,\mathbf{\hat{x}}- \left(b y_{6} \sin{\gamma} + c_{y} z_{6}\right) \,\mathbf{\hat{y}}- c_{z} z_{6} \,\mathbf{\hat{z}}$ | (2i) | P VI |
| $\mathbf{B_{13}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + b y_{7} \cos{\gamma} + c_{x} z_{7}\right) \,\mathbf{\hat{x}}+\left(b y_{7} \sin{\gamma} + c_{y} z_{7}\right) \,\mathbf{\hat{y}}+c_{z} z_{7} \,\mathbf{\hat{z}}$ | (2i) | P VII |
| $\mathbf{B_{14}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \left(a x_{7} + b y_{7} \cos{\gamma} + c_{x} z_{7}\right) \,\mathbf{\hat{x}}- \left(b y_{7} \sin{\gamma} + c_{y} z_{7}\right) \,\mathbf{\hat{y}}- c_{z} z_{7} \,\mathbf{\hat{z}}$ | (2i) | P VII |
| $\mathbf{B_{15}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + b y_{8} \cos{\gamma} + c_{x} z_{8}\right) \,\mathbf{\hat{x}}+\left(b y_{8} \sin{\gamma} + c_{y} z_{8}\right) \,\mathbf{\hat{y}}+c_{z} z_{8} \,\mathbf{\hat{z}}$ | (2i) | P VIII |
| $\mathbf{B_{16}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- \left(a x_{8} + b y_{8} \cos{\gamma} + c_{x} z_{8}\right) \,\mathbf{\hat{x}}- \left(b y_{8} \sin{\gamma} + c_{y} z_{8}\right) \,\mathbf{\hat{y}}- c_{z} z_{8} \,\mathbf{\hat{z}}$ | (2i) | P VIII |
| $\mathbf{B_{17}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + b y_{9} \cos{\gamma} + c_{x} z_{9}\right) \,\mathbf{\hat{x}}+\left(b y_{9} \sin{\gamma} + c_{y} z_{9}\right) \,\mathbf{\hat{y}}+c_{z} z_{9} \,\mathbf{\hat{z}}$ | (2i) | P IX |
| $\mathbf{B_{18}}$ | = | $- x_{9} \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $- \left(a x_{9} + b y_{9} \cos{\gamma} + c_{x} z_{9}\right) \,\mathbf{\hat{x}}- \left(b y_{9} \sin{\gamma} + c_{y} z_{9}\right) \,\mathbf{\hat{y}}- c_{z} z_{9} \,\mathbf{\hat{z}}$ | (2i) | P IX |
| $\mathbf{B_{19}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + b y_{10} \cos{\gamma} + c_{x} z_{10}\right) \,\mathbf{\hat{x}}+\left(b y_{10} \sin{\gamma} + c_{y} z_{10}\right) \,\mathbf{\hat{y}}+c_{z} z_{10} \,\mathbf{\hat{z}}$ | (2i) | P X |
| $\mathbf{B_{20}}$ | = | $- x_{10} \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $- \left(a x_{10} + b y_{10} \cos{\gamma} + c_{x} z_{10}\right) \,\mathbf{\hat{x}}- \left(b y_{10} \sin{\gamma} + c_{y} z_{10}\right) \,\mathbf{\hat{y}}- c_{z} z_{10} \,\mathbf{\hat{z}}$ | (2i) | P X |
| $\mathbf{B_{21}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + b y_{11} \cos{\gamma} + c_{x} z_{11}\right) \,\mathbf{\hat{x}}+\left(b y_{11} \sin{\gamma} + c_{y} z_{11}\right) \,\mathbf{\hat{y}}+c_{z} z_{11} \,\mathbf{\hat{z}}$ | (2i) | P XI |
| $\mathbf{B_{22}}$ | = | $- x_{11} \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $- \left(a x_{11} + b y_{11} \cos{\gamma} + c_{x} z_{11}\right) \,\mathbf{\hat{x}}- \left(b y_{11} \sin{\gamma} + c_{y} z_{11}\right) \,\mathbf{\hat{y}}- c_{z} z_{11} \,\mathbf{\hat{z}}$ | (2i) | P XI |
| $\mathbf{B_{23}}$ | = | $x_{12} \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + b y_{12} \cos{\gamma} + c_{x} z_{12}\right) \,\mathbf{\hat{x}}+\left(b y_{12} \sin{\gamma} + c_{y} z_{12}\right) \,\mathbf{\hat{y}}+c_{z} z_{12} \,\mathbf{\hat{z}}$ | (2i) | P XII |
| $\mathbf{B_{24}}$ | = | $- x_{12} \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $- \left(a x_{12} + b y_{12} \cos{\gamma} + c_{x} z_{12}\right) \,\mathbf{\hat{x}}- \left(b y_{12} \sin{\gamma} + c_{y} z_{12}\right) \,\mathbf{\hat{y}}- c_{z} z_{12} \,\mathbf{\hat{z}}$ | (2i) | P XII |
References
- A. Simon, H. Borrmann, and H. Craubner, Crystal Structure of Ordered White Phosphorus (β-P), Phosphorus and Sulfur 30, 507–510 (1987), doi:10.1080/03086648708080631.
- J. Donohue, The Structures of the Elements (Robert E. Krieger Publishing Company, Malabar, Florida, 1982). Reprint of the 1974 John Wilely & Sons edition.
Prototype Generator
aflow --proto=A_aP24_2_12i --params=$a,b/a,c/a,\alpha,\beta,\gamma,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12}$