α-N ($P2_{1}3$) Structure: A_cP8_198_2a-001
| Prototype | N |
| AFLOW prototype label | A_cP8_198_2a-001 |
| ICSD | 27249 |
| Pearson symbol | cP8 |
| Space group number | 198 |
| Space group symbol | $P2_13$ |
| AFLOW prototype command |
aflow --proto=A_cP8_198_2a-001
--params=$a, \allowbreak x_{1}, \allowbreak x_{2}$ |
- Solid nitrogen is found in three forms (Mills, 1969; Donohue, 1974):
- The ground state $\alpha$–N structure, stable below 35.6K, found either in a centrosymmetric or a non-centrosymmetric cubic structure.
- The hexagonal $\beta$-phase, which has freely rotating N$_{2}$ molecules and is stable up to the melting point, and
- High-pressure $\gamma$–N, stable above 355 MPa.
- There is considerable controversy about the crystal structure of $\alpha$–N, as outlined in (Donohue, 1982, 280-285). This page assumes the non-centrosymmetric $P2_{1}3$ #198 structure. The other possibility is the $Pa\overline{3}$ #205 structure, where the N$_{2}$ dimers are centered on an inversion site. (Venables, 1974) makes a convincing case that the ground state is $Pa\overline{3}$, but we present both structures. Density Functional Theory calculations show no appreciable difference in energy between the $Pa_{3}$ and $P2_{1}3$ structures. (Mehl, 2015)
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&a \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}+a x_{1} \,\mathbf{\hat{y}}+a x_{1} \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{2}}$ | = | $- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{1} \,\mathbf{\hat{y}}+a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}+a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{4}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{1} \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+a x_{2} \,\mathbf{\hat{y}}+a x_{2} \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{6}}$ | = | $- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{2} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{2} \,\mathbf{\hat{y}}+a \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{7}}$ | = | $- x_{2} \, \mathbf{a}_{1}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+a \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{2} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{8}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $a \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{2} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{2} \,\mathbf{\hat{z}}$ | (4a) | N II |
References
- S. J. L. Placa and W. C. Hamilton, Refinement of the crystal structure of α-N$_2$, Acta Crystallogr. Sect. B 28, 984–985 (1972), doi:10.1107/S0567740872003541.
- R. L. Mills and A. F. Schuch, Crystal Structure of Gamma Nitrogen, Phys. Rev. Lett. 23, 1154–1156 (1969), doi:10.1103/PhysRevLett.23.1154.
- J. Donohue, The Structures of the Elements (Robert E. Krieger Publishing Company, New York, 1974).
- J. A. Venables and C. A. English, Electron diffraction and the structure of α-N$_2$, Acta Crystallogr. Sect. B 30, 929–935 (1974), doi:10.1107/S0567740874004067.
- M. J. Mehl, D. Finkenstadt, C. Dane, G. L. W. Hart, and S. Curtarolo, Finding the stable structures of N$_{1-x}$W$_x$ with an {\em ab initio} high-throughput approach, Phys. Rev. B 91, 184110 (2015), doi:10.1103/PhysRevB.91.184110.
Found in
- J. Donohue, The Structures of the Elements (Robert E. Krieger Publishing Company, New York, 1974).
Prototype Generator
aflow --proto=A_cP8_198_2a --params=$a,x_{1},x_{2}$