Table 11.
Quaternary lattice metric singularity. The four lattices yield the same set of unique calculated d-spacings. For each lattice, the table gives the conventional cell along with the corresponding reduced cell and normalized reduced form. The normalized reduced forms reveal extra specialization in forms F2–F4
| Lattice I | Lattice II | Lattice III | Lattice IV | |
|---|---|---|---|---|
| Cubic I | Tetragonal P | Orthorhombic F | Orthorhombic P | |
| Conventional cells | ||||
|
| ||||
| Cell | Cell 1 | Cell 2a | Cell 3b | Cell 4c |
| a(Å) | 10.0000 | 7.0711 | 4.7140 | 3.5355 |
| b(Å) | 10.0000 | 7.0711 | 10.0000 | 5.0000 |
| c(Å) | 10.0000 | 5.0000 | 14.1421 | 7.0711 |
| α (°) | 90.0 | 90.0 | 90.0 | 90.0 |
| β (°) | 90.0 | 90.0 | 90.0 | 90.0 |
| γ (°) | 90.0 | 90.0 | 90.0 | 90.0 |
| V(Å3) | 1000.0 | 250.0 | 666.67 | 125.0 |
|
| ||||
| Reduced cells | ||||
|
| ||||
| Cell | R1 | R2d | R3e | R4f |
| a(Å) | 8.6603 | 5.0000 | 4.7140 | 3.5355 |
| b(Å) | 8.6603 | 7.0711 | 5.5277 | 5.0000 |
| c(Å) | 8.6603 | 7.0711 | 7.4536 | 7.0711 |
| α (°) | 109.471 | 90.0 | 82.251 | 90.0 |
| β (°) | 109.471 | 90.0 | 71.565 | 90.0 |
| γ (°) | 109.471 | 90.0 | 64.761 | 90.0 |
| V(Å3) | 500.0 | 250.0 | 166.67 | 125.0 |
|
| ||||
| Normalized reduced forms | ||||
|
| ||||
| Form | F1 | F2 | F3 | F4 |
| a·a | 3 | 1 | 4 | 1 |
| b·b | 3 | 2 | 5.5 | 2 |
| c·c | 3 | 2 | 10 | 4 |
| b·c | −1 | 0 | 1 | 0 |
| a·c | −1 | 0 | 2 | 0 |
| a·b | −1 | 0 | 2 | 0 |
| Form No. | 5 | 21 | 26 | 32 |
Transformations
a
Cell 2 → Cell 1 [ 0 0 2 / 1− 1 0 / 1 1 0 ]∆ = 4.
b
Cell 3 → Cell 1 [1/2−2/3−1/2 / 2 1/3 0 / 1/2 −2/3 1/2]∆ = 3/2.
c
Cell 4 → Cell 1 [ 0 2 0 / 2 0 1 / 2 0− 1 ]∆ = 8.
d
R2 → R1 [ 1 −1 0 / −1 0 1 / −1 0 −1 ]∆ = 2.
e
R3 → R1 [ 1 1 0 / −2 1 0 / 0 −1 1 ]∆ = 3.
F
R4 → R1 [ 0 −1 −1 / 2 1 0 / 0 −1 1 ]∆ = 4.