R$_{1}$ Au$_{3}$Zn Structure: A3B_tI64_142_def_d-001
| Prototype | Au$_{3}$Zn |
| AFLOW prototype label | A3B_tI64_142_def_d-001 |
| ICSD | 58628 |
| Pearson symbol | tI64 |
| Space group number | 142 |
| Space group symbol | $I4_1/acd$ |
| AFLOW prototype command |
aflow --proto=A3B_tI64_142_def_d-001
--params=$a, \allowbreak c/a, \allowbreak z_{1}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak x_{4}$ |
- Au$_{3}$Zn is known to exist in three forms, depending upon the exact composition and temperature (Hisatsune, 1998):
- The tetragonal $R_{1}$ phase (this structure) is stable below $≈ 475$K with a composition very nearly stoichiometric.
- The orthorhombic $R_{2}$ phase is stable below $≈ 550$K with a composition range somewhat wider than the $R_{1}$ phase.
- The tetragonal $H$ phase has the $D0_{23}$ structure and is stable at temperatures up to $≈ 700$K over a considerably wider range of stoichiometries than either the $R_{1}$ or $R_{2}$ phases.
- (Iwaskai, 1962) gave structure of the $R_{1}$ phase in setting 1 of space group $I4_{1}/acd$ #142. We used FINDSYM to transform this to the standard setting 2.
- (Iwaskai, 1962) gave the lattice constants in kX units. We used the conversion factor 1 kX = 1.00202Å. (Wood, 1947)
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}- \frac{1}{2}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\left(z_{1} + \frac{1}{4}\right) \, \mathbf{a}_{1}+z_{1} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+c z_{1} \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{2}}$ | = | $z_{1} \, \mathbf{a}_{1}+\left(z_{1} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+c \left(z_{1} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{3}}$ | = | $- \left(z_{1} - \frac{1}{4}\right) \, \mathbf{a}_{1}- \left(z_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}- c z_{1} \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{4}}$ | = | $- \left(z_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{1} - \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}- c \left(z_{1} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{5}}$ | = | $- \left(z_{1} - \frac{3}{4}\right) \, \mathbf{a}_{1}- z_{1} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{4}a \,\mathbf{\hat{y}}- c z_{1} \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{6}}$ | = | $- z_{1} \, \mathbf{a}_{1}- \left(z_{1} - \frac{3}{4}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{4}a \,\mathbf{\hat{y}}- c \left(z_{1} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{7}}$ | = | $\left(z_{1} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+c \left(z_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{8}}$ | = | $\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{1} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+c \left(z_{1} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Au I |
| $\mathbf{B_{9}}$ | = | $\left(z_{2} + \frac{1}{4}\right) \, \mathbf{a}_{1}+z_{2} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{10}}$ | = | $z_{2} \, \mathbf{a}_{1}+\left(z_{2} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+c \left(z_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{11}}$ | = | $- \left(z_{2} - \frac{1}{4}\right) \, \mathbf{a}_{1}- \left(z_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}- c z_{2} \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{12}}$ | = | $- \left(z_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{2} - \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}- c \left(z_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{13}}$ | = | $- \left(z_{2} - \frac{3}{4}\right) \, \mathbf{a}_{1}- z_{2} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{4}a \,\mathbf{\hat{y}}- c z_{2} \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{14}}$ | = | $- z_{2} \, \mathbf{a}_{1}- \left(z_{2} - \frac{3}{4}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{4}a \,\mathbf{\hat{y}}- c \left(z_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{15}}$ | = | $\left(z_{2} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{16}}$ | = | $\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{2} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (16d) | Zn I |
| $\mathbf{B_{17}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{4}\right) \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{18}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- \left(x_{3} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{19}}$ | = | $\left(x_{3} + \frac{1}{4}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{20}}$ | = | $- \left(x_{3} - \frac{1}{4}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}$ | (16e) | Au II |
| $\mathbf{B_{21}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- \left(x_{3} - \frac{3}{4}\right) \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+\frac{3}{4}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{22}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\left(x_{3} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{23}}$ | = | $- \left(x_{3} - \frac{3}{4}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{4}a \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{24}}$ | = | $\left(x_{3} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (16e) | Au II |
| $\mathbf{B_{25}}$ | = | $\left(x_{4} + \frac{3}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\left(2 x_{4} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{26}}$ | = | $- \left(x_{4} - \frac{3}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{8}\right) \, \mathbf{a}_{2}- \left(2 x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{27}}$ | = | $\left(x_{4} + \frac{1}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{3}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{28}}$ | = | $- \left(x_{4} - \frac{1}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{3}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{29}}$ | = | $- \left(x_{4} - \frac{5}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{7}{8}\right) \, \mathbf{a}_{2}- \left(2 x_{4} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{30}}$ | = | $\left(x_{4} + \frac{5}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{7}{8}\right) \, \mathbf{a}_{2}+\left(2 x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{31}}$ | = | $- \left(x_{4} - \frac{7}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{5}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
| $\mathbf{B_{32}}$ | = | $\left(x_{4} + \frac{7}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{5}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (16f) | Au III |
References
- H. Iwaskai, Study on the Ordered Phases with Long Period in the Gold-Zinc Alloy System II. Structure Analysis of Au$_{3}$Zn[R$_{1}$], Au$_{3}$Zn[R$_{2}$] and Au$_{3}+$Zn, J. Phys. Soc. Jpn. 17, 1620–1633 (1962), doi:10.1143/JPSJ.17.1620.
- E. A. Wood, The Conversion Factor for kX Units to Angström Units, J. Appl. Phys. 18, 929–930 (1947), doi:10.1063/1.1697570.
Found in
- K. Hisatsune, Y. Takuma, Y. Tanaka, K. Udoh, T. Morimura, and M. Hasaka, Martensite transformation in Au$_{3}$Zn alloy, Solid State Commun. 106, 509–512 (1998), doi:10.1016/S0038-1098(98)00077-5.
Prototype Generator
aflow --proto=A3B_tI64_142_def_d --params=$a,c/a,z_{1},z_{2},x_{3},x_{4}$