Mn$_{2}$B ($D1_{f}$) Structure: AB2_oF48_70_e_ef-001
| Prototype | BMn$_{2}$ |
| AFLOW prototype label | AB2_oF48_70_e_ef-001 |
| Strukturbericht designation | $D1_{f}$ |
| ICSD | 86398 |
| Pearson symbol | oF48 |
| Space group number | 70 |
| Space group symbol | $Fddd$ |
| AFLOW prototype command |
aflow --proto=AB2_oF48_70_e_ef-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak y_{3}$ |
Other compounds with this structure
Cr$_{2}$B
- Early works, e.g. (Pearson, 1958), referred to this structure as Mn$_{4}$B, with the same space group and Wyckoff positions. The stoichiometry was fixed by assuming that the (16e) boron positions were only half-occupied. The (Tergenius, 1981) refinement of the structure showed that the (16e) sites were totally filled, fixing the stoichiometry to Mn$_{2}$B. A similar reanalysis showed that the similar structure known as Cr$_{4}$B also had composition Cr$_{2}$B.
- Tergenius gives the atomic positions using the first setting of space group $Fddd$ #70. We have translated this into the second setting, where the origin is on an inversion site. As a part of this process the primitive axes were also rotated compared to Tergenius.
- Mn$_{2}$B ($D1_{f}$) and CuMg$_{2}$ ($C_{b}$) have the same AFLOW prototype label, AB2_oF48_70_e_ef. They are generated by the same symmetry operations with different sets of parameters (
--params) specified in their corresponding CIF files.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}b \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $- \left(x_{1} - \frac{1}{4}\right) \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}+\frac{1}{8}b \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16e) | B I |
| $\mathbf{B_{2}}$ | = | $x_{1} \, \mathbf{a}_{1}- \left(x_{1} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{8}b \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16e) | B I |
| $\mathbf{B_{3}}$ | = | $\left(x_{1} + \frac{3}{4}\right) \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}+\frac{3}{8}b \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16e) | B I |
| $\mathbf{B_{4}}$ | = | $- x_{1} \, \mathbf{a}_{1}+\left(x_{1} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(x_{1} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{3}{4}\right) \,\mathbf{\hat{x}}+\frac{3}{8}b \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16e) | B I |
| $\mathbf{B_{5}}$ | = | $- \left(x_{2} - \frac{1}{4}\right) \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+\frac{1}{8}b \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16e) | Mn I |
| $\mathbf{B_{6}}$ | = | $x_{2} \, \mathbf{a}_{1}- \left(x_{2} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{8}b \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16e) | Mn I |
| $\mathbf{B_{7}}$ | = | $\left(x_{2} + \frac{3}{4}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+\frac{3}{8}b \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16e) | Mn I |
| $\mathbf{B_{8}}$ | = | $- x_{2} \, \mathbf{a}_{1}+\left(x_{2} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(x_{2} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{2} + \frac{3}{4}\right) \,\mathbf{\hat{x}}+\frac{3}{8}b \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16e) | Mn I |
| $\mathbf{B_{9}}$ | = | $y_{3} \, \mathbf{a}_{1}- \left(y_{3} - \frac{1}{4}\right) \, \mathbf{a}_{2}+y_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Mn II |
| $\mathbf{B_{10}}$ | = | $- \left(y_{3} - \frac{1}{4}\right) \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}- \left(y_{3} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}- b \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Mn II |
| $\mathbf{B_{11}}$ | = | $- y_{3} \, \mathbf{a}_{1}+\left(y_{3} + \frac{3}{4}\right) \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | Mn II |
| $\mathbf{B_{12}}$ | = | $\left(y_{3} + \frac{3}{4}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+\left(y_{3} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+b \left(y_{3} + \frac{3}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | Mn II |
References
- L.-E. Tergenius, Refinement of the crystal structure of orthorhombic Mn$_{2}$B (formerly denoted Mn$_{4}$B), J. Less-Common Met. 82, 335–340 (1981), doi:10.1016/0022-5088(81)90236-8.
- W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, no. N.R.C. No. 4303 in International Series of Monographs on Metal Physics and Physical Metallurgy (Pergamon Press, Oxford, London, Edinburgh, New York, Paris, Frankfort, 1958), 1964 reprint with corrections edn.
Prototype Generator
aflow --proto=AB2_oF48_70_e_ef --params=$a,b/a,c/a,x_{1},x_{2},y_{3}$