γ-brass (Cu$_{9}$Al$_{4}$, $D8_{3}$) Structure: A4B9_cP52_215_ei_3efgi-001
| Prototype | Al$_{4}$Cu$_{9}$ |
| AFLOW prototype label | A4B9_cP52_215_ei_3efgi-001 |
| Strukturbericht designation | $D8_{3}$ |
| Mineral name | γ-brass |
| ICSD | 1625 |
| Pearson symbol | cP52 |
| Space group number | 215 |
| Space group symbol | $P\overline{4}3m$ |
| AFLOW prototype command |
aflow --proto=A4B9_cP52_215_ei_3efgi-001
--params=$a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}, \allowbreak x_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak z_{8}$ |
Other compounds with this structure
Cu$_{9}$Ga$_{4}$, InMn$_{3}$
- (Arnberg, 1978) give the Wyckoff positions of the Cu-IV and Cu-V atoms as (6g) $(x,1/2,1/2)$, but give the coordinates in the form $(x,0,0)$ corresponding to the (6f) site.
- (Stokhuyzen, 1974) used (6f) for both types of atoms in the isostructural system Ga$_{9}$Al$_{4}$.
- (Pearson, 1958) places the Cu-IV atoms on a (6f) site and Cu-V on (6g), but does not give explicit coordinates.
- Placing the Cu-V atoms on (6f) sites yields an interatomic distance of 1.8Å. This contradicts (Arnberg, 1978), who say that the minimum interatomic distance between the Cu-IV and Cu-V atoms is 2.48Å . Placing the Cu-V atoms on (6g) sites gives this distance, in agreement with (Pearson, 1958), so we make this choice for the crystal structure.
- This is a variety of $\gamma$-brass comparable to the $D8_{2}$ structure. In fact, if we
- Replace the Al and Cu-III atoms by Zn, while setting $x_{4}=x_{1}+1/2$,
- Replace the Al II and Cu-VI atoms by Zn, with $x_{8} = x_{7} + 1/2$ and $z_{8} = z_{7} + 1/2$,
- Set $x_{3} = x_{2} + 1/2$ and
- Set $x_{6} = x_{5} + 1/2$,
- (Pearson, 1958), 252, gives a list of compounds which can take on the $D8_{1}$, $D8_{2}$, or $D8_{3}$ structure, depending on the exact composition.
- (Mizutani, 2010) classifies this as a
P-cell
$\gamma$-brass.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&a \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}+a x_{1} \,\mathbf{\hat{y}}+a x_{1} \,\mathbf{\hat{z}}$ | (4e) | Al I |
| $\mathbf{B_{2}}$ | = | $- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}- a x_{1} \,\mathbf{\hat{y}}+a x_{1} \,\mathbf{\hat{z}}$ | (4e) | Al I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}+a x_{1} \,\mathbf{\hat{y}}- a x_{1} \,\mathbf{\hat{z}}$ | (4e) | Al I |
| $\mathbf{B_{4}}$ | = | $x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}- a x_{1} \,\mathbf{\hat{y}}- a x_{1} \,\mathbf{\hat{z}}$ | (4e) | Al I |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+a x_{2} \,\mathbf{\hat{y}}+a x_{2} \,\mathbf{\hat{z}}$ | (4e) | Cu I |
| $\mathbf{B_{6}}$ | = | $- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}- a x_{2} \,\mathbf{\hat{y}}+a x_{2} \,\mathbf{\hat{z}}$ | (4e) | Cu I |
| $\mathbf{B_{7}}$ | = | $- x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+a x_{2} \,\mathbf{\hat{y}}- a x_{2} \,\mathbf{\hat{z}}$ | (4e) | Cu I |
| $\mathbf{B_{8}}$ | = | $x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}- a x_{2} \,\mathbf{\hat{y}}- a x_{2} \,\mathbf{\hat{z}}$ | (4e) | Cu I |
| $\mathbf{B_{9}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}+a x_{3} \,\mathbf{\hat{z}}$ | (4e) | Cu II |
| $\mathbf{B_{10}}$ | = | $- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}+a x_{3} \,\mathbf{\hat{z}}$ | (4e) | Cu II |
| $\mathbf{B_{11}}$ | = | $- x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}- a x_{3} \,\mathbf{\hat{z}}$ | (4e) | Cu II |
| $\mathbf{B_{12}}$ | = | $x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}- a x_{3} \,\mathbf{\hat{z}}$ | (4e) | Cu II |
| $\mathbf{B_{13}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (4e) | Cu III |
| $\mathbf{B_{14}}$ | = | $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (4e) | Cu III |
| $\mathbf{B_{15}}$ | = | $- x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (4e) | Cu III |
| $\mathbf{B_{16}}$ | = | $x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (4e) | Cu III |
| $\mathbf{B_{17}}$ | = | $x_{5} \, \mathbf{a}_{1}$ | = | $a x_{5} \,\mathbf{\hat{x}}$ | (6f) | Cu IV |
| $\mathbf{B_{18}}$ | = | $- x_{5} \, \mathbf{a}_{1}$ | = | $- a x_{5} \,\mathbf{\hat{x}}$ | (6f) | Cu IV |
| $\mathbf{B_{19}}$ | = | $x_{5} \, \mathbf{a}_{2}$ | = | $a x_{5} \,\mathbf{\hat{y}}$ | (6f) | Cu IV |
| $\mathbf{B_{20}}$ | = | $- x_{5} \, \mathbf{a}_{2}$ | = | $- a x_{5} \,\mathbf{\hat{y}}$ | (6f) | Cu IV |
| $\mathbf{B_{21}}$ | = | $x_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{z}}$ | (6f) | Cu IV |
| $\mathbf{B_{22}}$ | = | $- x_{5} \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{z}}$ | (6f) | Cu IV |
| $\mathbf{B_{23}}$ | = | $x_{6} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (6g) | Cu V |
| $\mathbf{B_{24}}$ | = | $- x_{6} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (6g) | Cu V |
| $\mathbf{B_{25}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+a x_{6} \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (6g) | Cu V |
| $\mathbf{B_{26}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- a x_{6} \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (6g) | Cu V |
| $\mathbf{B_{27}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+a x_{6} \,\mathbf{\hat{z}}$ | (6g) | Cu V |
| $\mathbf{B_{28}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}- a x_{6} \,\mathbf{\hat{z}}$ | (6g) | Cu V |
| $\mathbf{B_{29}}$ | = | $x_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{30}}$ | = | $- x_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a z_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{31}}$ | = | $- x_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{32}}$ | = | $x_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a z_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{33}}$ | = | $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{34}}$ | = | $z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{35}}$ | = | $- z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{36}}$ | = | $- z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a z_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{37}}$ | = | $x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{38}}$ | = | $- x_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{39}}$ | = | $x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}- a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{40}}$ | = | $- x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a z_{7} \,\mathbf{\hat{y}}+a x_{7} \,\mathbf{\hat{z}}$ | (12i) | Al II |
| $\mathbf{B_{41}}$ | = | $x_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{42}}$ | = | $- x_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a z_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{43}}$ | = | $- x_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{44}}$ | = | $x_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a z_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{45}}$ | = | $z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{46}}$ | = | $z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{47}}$ | = | $- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{48}}$ | = | $- z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a z_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{49}}$ | = | $x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{50}}$ | = | $- x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{51}}$ | = | $x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}- a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
| $\mathbf{B_{52}}$ | = | $- x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a z_{8} \,\mathbf{\hat{y}}+a x_{8} \,\mathbf{\hat{z}}$ | (12i) | Cu VI |
References
- L. Arnberg and S. Westman, Crystal perfection in a noncentrosymmetric alloy. Refinement and test of twinning of the γCu$_{9}$Al$_{4}$ structure 34, 399–404 (1978), doi:10.1107/S0567739478000807.
- R. Stokhuyzen, J. K. Brandon, P. C. Chieh, and W. B. Pearson, Copper-Gallium, $\gamma_1$Cu$_{9}$Ga$_{4}$ 30, 2910–2911 (1974), doi:10.1107/S0567740874008478.
- 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. \bibitem{Mizutani_Hume-Rothery_2010U. Mizutani, Hume-Rothery Rules for Structurally Complex Alloy Phases} (CRC Press, Boca Raton, London, New York, 2010).\bibAnnoteFile{Mizutani_Hume-Rothery_2010
Found in
- P. Villars and K. Cenuzal, eds., Structure Types (Springer, Berlin, Heidelberg, 2005), Landolt-Börnstein - Group III Condensed Matter (Numerical Data and Functional Relationships in Science and Technology)}, vol. 43A2, chap. Cu$_{9}Al$_{4 in Part 2: Space Groups (218) P-43n - (195) P23.
Prototype Generator
aflow --proto=A4B9_cP52_215_ei_3efgi --params=$a,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},z_{7},x_{8},z_{8}$