β-Pu Structure: A_mC34_12_ah3i2j-001
| Prototype | Pu |
| AFLOW prototype label | A_mC34_12_ah3i2j-001 |
| ICSD | 43336 |
| Pearson symbol | mC34 |
| Space group number | 12 |
| Space group symbol | $C2/m$ |
| AFLOW prototype command |
aflow --proto=A_mC34_12_ah3i2j-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak \beta, \allowbreak y_{2}, \allowbreak x_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}$ |
- Plutonium has been found in a variety of structures (Donohue, 1982):
- $\alpha$–Pu
- $\beta$–Pu (this structure)
- $\gamma$–Pu
- $\delta$–Pu is in the face-centered cubic $A1$ structure
- $\delta$'-Pu is in the body-centered tetragonal $A6$ (In) structure
- $\epsilon$–Pu is in the body-centered cubic $A2$ structure
- (Zachariasen, 1963) give the coordinates of this structure in the $I2/m$ setting of space group #12. We used FINDSYM to transform this to the standard $C2/m$ setting.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \cos{\beta} \,\mathbf{\hat{x}}+c \sin{\beta} \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (2a) | Pu I |
| $\mathbf{B_{2}}$ | = | $- y_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}c \cos{\beta} \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+\frac{1}{2}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4h) | Pu II |
| $\mathbf{B_{3}}$ | = | $y_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}c \cos{\beta} \,\mathbf{\hat{x}}- b y_{2} \,\mathbf{\hat{y}}+\frac{1}{2}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4h) | Pu II |
| $\mathbf{B_{4}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\left(a x_{3} + c z_{3} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{3} \sin{\beta} \,\mathbf{\hat{z}}$ | (4i) | Pu III |
| $\mathbf{B_{5}}$ | = | $- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $- \left(a x_{3} + c z_{3} \cos{\beta}\right) \,\mathbf{\hat{x}}- c z_{3} \sin{\beta} \,\mathbf{\hat{z}}$ | (4i) | Pu III |
| $\mathbf{B_{6}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\left(a x_{4} + c z_{4} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{4} \sin{\beta} \,\mathbf{\hat{z}}$ | (4i) | Pu IV |
| $\mathbf{B_{7}}$ | = | $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $- \left(a x_{4} + c z_{4} \cos{\beta}\right) \,\mathbf{\hat{x}}- c z_{4} \sin{\beta} \,\mathbf{\hat{z}}$ | (4i) | Pu IV |
| $\mathbf{B_{8}}$ | = | $x_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (4i) | Pu V |
| $\mathbf{B_{9}}$ | = | $- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}- c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (4i) | Pu V |
| $\mathbf{B_{10}}$ | = | $\left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} + y_{6}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VI |
| $\mathbf{B_{11}}$ | = | $- \left(x_{6} + y_{6}\right) \, \mathbf{a}_{1}- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}- c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VI |
| $\mathbf{B_{12}}$ | = | $- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}- \left(x_{6} + y_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}- c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VI |
| $\mathbf{B_{13}}$ | = | $\left(x_{6} + y_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VI |
| $\mathbf{B_{14}}$ | = | $\left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} + y_{7}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VII |
| $\mathbf{B_{15}}$ | = | $- \left(x_{7} + y_{7}\right) \, \mathbf{a}_{1}- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}- c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VII |
| $\mathbf{B_{16}}$ | = | $- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}- \left(x_{7} + y_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}- c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VII |
| $\mathbf{B_{17}}$ | = | $\left(x_{7} + y_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (8j) | Pu VII |
References
- W. H. Zachariasen and F. H. Ellinger, The Crystal Structure of Beta Plutonium Metal, Acta Cryst. 16, 369–375 (1963), doi:10.1107/S0365110X63000992.
Found in
- J. Donohue, The Structures of the Elements (Robert E. Krieger Publishing Company, New York, 1974).
Prototype Generator
aflow --proto=A_mC34_12_ah3i2j --params=$a,b/a,c/a,\beta,y_{2},x_{3},z_{3},x_{4},z_{4},x_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7}$