Base-Centered Monoclinic La$_{2}$CuO$_{4}$ Structure: AB2C4_mC28_8_2a_4a_4a2b-001
| Prototype | CuLa$_{2}$O$_{4}$ |
| AFLOW prototype label | AB2C4_mC28_8_2a_4a_4a2b-001 |
| ICSD | 155497 |
| Pearson symbol | mC28 |
| Space group number | 8 |
| Space group symbol | $Cm$ |
| AFLOW prototype command |
aflow --proto=AB2C4_mC28_8_2a_4a_4a2b-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak \beta, \allowbreak x_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}$ |
- We have found three possible structures for La$_{2}$CuO$_{4}$, the parent compound of the high-temperature cuprate superconductors. All are distortions of the tetragonal K$_{2}$NiF$_{4}$/0201 [(La,Ba)$_{2}$CuO$_{4}$] High-T$_{c}$ Structure, and reduce to that structure with doping of the lanthanum site:
- (Longo, 1973) proposed a face-centered orthorhombic structure.
- Later other workers, including (Reehuis, 2006) and references therein, proposed a base-centered orthorhombic structure.
- (Reehuis, 2006) found that this base-centered monoclinic structure was a better fit to their neutron diffraction data.
- As the two orthorhombic structures are both referenced in the literature, we present them, as well as the more recent monoclinic structure.
- (Reehuis, 2006) gave the structure of the monoclinic phase in the $Bm$, unique axis $a$, setting of space group #8. We used FINDSYM to transform it to the standard $Cm$, unique axis $b$ setting. Although they never give a value, we assume $\alpha = 90°$, as the other lattice parameters are the same as in the $Cmca$ phase.
- (Rehuis, 2006) made several mistakes in the labeling of the Wyckoff positions in the $Bm11$ section of Table I:
- The copper sites labeled (4a) are actually (2a).
- The oxygen sites labeled (4a) are actually (4b).
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \cos{\beta} \,\mathbf{\hat{x}}+c \sin{\beta} \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $\left(a x_{1} + c z_{1} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{1} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Cu I |
| $\mathbf{B_{2}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\left(a x_{2} + c z_{2} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{2} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | Cu II |
| $\mathbf{B_{3}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\left(a x_{3} + c z_{3} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{3} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | La I |
| $\mathbf{B_{4}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\left(a x_{4} + c z_{4} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{4} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | La II |
| $\mathbf{B_{5}}$ | = | $x_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | La III |
| $\mathbf{B_{6}}$ | = | $x_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | La IV |
| $\mathbf{B_{7}}$ | = | $x_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | O I |
| $\mathbf{B_{8}}$ | = | $x_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + c z_{8} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{8} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | O II |
| $\mathbf{B_{9}}$ | = | $x_{9} \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + c z_{9} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{9} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | O III |
| $\mathbf{B_{10}}$ | = | $x_{10} \, \mathbf{a}_{1}+x_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + c z_{10} \cos{\beta}\right) \,\mathbf{\hat{x}}+c z_{10} \sin{\beta} \,\mathbf{\hat{z}}$ | (2a) | O IV |
| $\mathbf{B_{11}}$ | = | $\left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}+\left(x_{11} + y_{11}\right) \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (4b) | O V |
| $\mathbf{B_{12}}$ | = | $\left(x_{11} + y_{11}\right) \, \mathbf{a}_{1}+\left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (4b) | O V |
| $\mathbf{B_{13}}$ | = | $\left(x_{12} - y_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} + y_{12}\right) \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (4b) | O VI |
| $\mathbf{B_{14}}$ | = | $\left(x_{12} + y_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} - y_{12}\right) \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (4b) | O VI |
References
- M. Reehuis, C. Ulrich, K. Prokeš, A. Gozar, G. Blumberg, S. Komiya, Y. Ando, P. Pattison, and B. Keimer, Crystal structure and high-field magnetism of La$_{2}$CuO$_{4}$, Phys. Rev. B 73, 144513 (2006), doi:10.1103/PhysRevB.73.144513.
- J. M. Longo and P. M. Raccah, The structure of La$_{2}$CuO$_{4}$ and LaSrVO$_{4}$, J. Solid State Chem. 6, 526–531 (1973), doi:10.1016/S0022-4596(73)80010-6.
Prototype Generator
aflow --proto=AB2C4_mC28_8_2a_4a_4a2b --params=$a,b/a,c/a,\beta,x_{1},z_{1},x_{2},z_{2},x_{3},z_{3},x_{4},z_{4},x_{5},z_{5},x_{6},z_{6},x_{7},z_{7},x_{8},z_{8},x_{9},z_{9},x_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12}$