Zn$_{22}$Zr Structure: A22B_cF184_227_cdfg_a-001
| Prototype | Zn$_{22}$Zr |
| AFLOW prototype label | A22B_cF184_227_cdfg_a-001 |
| ICSD | 106238 |
| Pearson symbol | cF184 |
| Space group number | 227 |
| Space group symbol | $Fd\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A22B_cF184_227_cdfg_a-001
--params=$a, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak z_{5}$ |
Other compounds with this structure
Be$_{22}$Mo, Be$_{22}$Re, Be$_{22}$W, Al$_{18}$Cr$_{2}$Mg$_{3}$, Ce(TiAl$_{10}$)$_{2}$, Dy(RhZn$_{10}$)$_{2}$, U(VAl$_{10}$)$_{2}$, Y(RuZn$_{10}$)$_{2}$
- (Samson, 1961) gives the atomic coordinates in terms of setting 1 of space group $F4dm$ #227. We have shifted this to the standard setting 2, where the inversion site of the lattice is at the origin.
- Samson also suggests that the
Zn$_{14}$Zr
structure is created when zirconium atoms replace some of the zinc atoms on the (16c) site [the (16d) site in Samson's orientation]. - This structure can be derived from the Mg$_{3}$Cr$_{2}$Al$_{18}$ structure.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\frac{1}{8} \, \mathbf{a}_{1}+\frac{1}{8} \, \mathbf{a}_{2}+\frac{1}{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (8a) | Zr I |
| $\mathbf{B_{2}}$ | = | $\frac{7}{8} \, \mathbf{a}_{1}+\frac{7}{8} \, \mathbf{a}_{2}+\frac{7}{8} \, \mathbf{a}_{3}$ | = | $\frac{7}{8}a \,\mathbf{\hat{x}}+\frac{7}{8}a \,\mathbf{\hat{y}}+\frac{7}{8}a \,\mathbf{\hat{z}}$ | (8a) | Zr I |
| $\mathbf{B_{3}}$ | = | $0$ | = | $0$ | (16c) | Zn I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}$ | (16c) | Zn I |
| $\mathbf{B_{5}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ | (16c) | Zn I |
| $\mathbf{B_{6}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ | (16c) | Zn I |
| $\mathbf{B_{7}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (16d) | Zn II |
| $\mathbf{B_{8}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}$ | (16d) | Zn II |
| $\mathbf{B_{9}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ | (16d) | Zn II |
| $\mathbf{B_{10}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{4}a \,\mathbf{\hat{z}}$ | (16d) | Zn II |
| $\mathbf{B_{11}}$ | = | $- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{12}}$ | = | $x_{4} \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{13}}$ | = | $x_{4} \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{14}}$ | = | $- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{15}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+a x_{4} \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{16}}$ | = | $- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{17}}$ | = | $\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{3}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{18}}$ | = | $- x_{4} \, \mathbf{a}_{1}+\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{19}}$ | = | $- x_{4} \, \mathbf{a}_{1}+\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{3}{4}\right) \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{20}}$ | = | $\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{21}}$ | = | $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}- a x_{4} \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{22}}$ | = | $\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}+a \left(x_{4} + \frac{3}{4}\right) \,\mathbf{\hat{z}}$ | (48f) | Zn III |
| $\mathbf{B_{23}}$ | = | $z_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{24}}$ | = | $z_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a z_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{25}}$ | = | $\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{1}- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}- a \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{26}}$ | = | $- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{27}}$ | = | $\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{28}}$ | = | $- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a z_{5} \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{29}}$ | = | $z_{5} \, \mathbf{a}_{1}+\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{2}- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{30}}$ | = | $z_{5} \, \mathbf{a}_{1}- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{31}}$ | = | $z_{5} \, \mathbf{a}_{1}+\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{32}}$ | = | $z_{5} \, \mathbf{a}_{1}- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a z_{5} \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{33}}$ | = | $- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}- a \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{34}}$ | = | $\left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}- \left(2 x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{35}}$ | = | $- z_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}+\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{36}}$ | = | $- z_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}- a z_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{37}}$ | = | $- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{1}+\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}+a \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{38}}$ | = | $\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{39}}$ | = | $- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}+\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{40}}$ | = | $\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}+a \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{41}}$ | = | $- z_{5} \, \mathbf{a}_{1}- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{42}}$ | = | $- z_{5} \, \mathbf{a}_{1}+\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a z_{5} \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{43}}$ | = | $- z_{5} \, \mathbf{a}_{1}- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{2}+\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{44}}$ | = | $- z_{5} \, \mathbf{a}_{1}+\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{45}}$ | = | $\left(2 x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
| $\mathbf{B_{46}}$ | = | $- \left(2 x_{5} - z_{5}\right) \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- a z_{5} \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}- a x_{5} \,\mathbf{\hat{z}}$ | (96g) | Zn IV |
References
- S. Samson, The Crystal Structure of the Intermetallic Compound ZrZn$_{22}$, Acta Cryst. 14 (1961), doi:10.1107/S0365110X61003600.
Found in
- T. B. Massalski, H. Okamoto, P. R. Subramanian, and L. Kacprzak, eds., Binary Alloy Phase Diagrams}, vol. 3 (ASM International, Materials Park, Ohio, USA, 1990), 2$^{nd$ edn. Hf-Re to Zn-Zr.
Prototype Generator
aflow --proto=A22B_cF184_227_cdfg_a --params=$a,x_{4},x_{5},z_{5}$