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The concept of Glazer is derived from crystallographic assumptions and was introduced mainly to facilitate the indexation of XRD spectra of distorted perovskites. The papers cited here describe a relatively simple technique to derive the type of distortion starting from the primitive perovskite unit cell.
The idea relies simply on the relation between the resulting lattice constants and a given distortion of an octahedron. Since the perovskite structure is a network of corner-sharing octahedra, there are several possibilities for tilts:
The table below is derived by accounting for the crystallographic / symmetrical changes due to tilting. Tilt axes, amplitudes and sequences are given by the symbols in row two applying the following notation:
Third and following rows list the derived lattice centring, the multiple cell used for the indexation technique described in the paper, the relative deviation of the pseudocubic cell parameters and the space group symbol. The last row will list examples if there are any.
Octahedral distortion is most often observed, when the Goldschmidt number t is too low, i.e. the A-cations are not large enough. A large number of perovskite-type mixed metal oxides do not meet the stability criterion for the ideal perovskite structure 0.9 < t < 1; most of these obey the GdFeO3-structure which is case (10) in the table below.
Refer to the original literature for a complete survey:
The table from the second paper:
Complete list of possible simple tilt systems
| Serial number |
Symbol | Lattice centring |
Multiple cell |
Relative pseudocubic subcell parameters |
Space group | Known Examples |
|---|
| (1) | a+ b+ c+ | I | 2ap x 2bp x 2cp | ap Ç bp Ç cp | Immm (No. 71 ) |
|
| (2) | a+ b+ b+ | I | ap Ç bp = cp | Immm (No. 71 ) |
||
| (3) | a+ a+ a+ | I | ap = bp = cp | Im3 (No. 204) |
||
| (4) | a+ b+ c‚ | P | ap Ç bp Ç cp | Pmmn (No. 59) |
||
| (5) | a+ a+ c‚ | P | ap = bp = cp | Pmmn (No. 59) |
||
| (6) | a+ b+ b‚ | P | ap = bp Ç cp | Pmmn (No. 59) |
||
| (7) | a+ a+ a‚ | P | ap = bp = cp | Pmmn (No. 59) |
||
| (8) | a+ b‚ c‚ | A | ap Ç bp Ç cp a Ç 90ƒ |
A21/m11 (No. 11) |
||
| (9) | a+ a‚ c‚ | A | ap = bp Ç cp a Ç 90ƒ |
A21/m11 (No. 11) |
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| (10) | a+ b‚ b‚ | A | ap Ç bp = cp a Ç 90ƒ |
Pmnb (No. 62)* |
GdFeO3-type perovskites | |
| (11) | a+ a‚ a‚ | A | ap = bp = cp a Ç 90ƒ |
Pmnb (No. 62)* |
||
| (12) | a‚ b‚ c‚ | F | ap Ç bp Ç cp a Ç b Ç g Ç 90ƒ |
F-l (No. 2) |
||
| (13) | a+ b‚ b‚ | F | ap Ç bp = cp a Ç b Ç g Ç 90ƒ |
I2/a (No. 15)* |
||
| (14) | a‚ a‚ a‚ | F | ap = bp = cp a Ç b Ç g Ç 90ƒ |
R-3c (No. 167) |
(Bi Fe O3) |
| (15) | a0 b+ c+ | I | 2ap x 2bp x 2cp | ap < bp Ç cp | Immm (No. 71 ) |
|
| (16) | a0 b+ b+ | I | ap < bp = cp | I4/mmm (No. 139) |
||
| (17) | a0 b+ c‚ | B | ap < bp Ç cp | Bmmb (No. 63) |
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| (18) | a0 b+ b‚ | B | ap < bp = cp | Bmmb (No. 63) |
||
| (19) | a0 b‚ c‚ | F | ap < bp Ç cp a Ç 90ƒ |
F2/m11 (No. 12) |
||
| (20) | a0 b‚ b‚ | F | ap < bp = cp a Ç 90ƒ |
Imcm (No. 74)* |
| (21) | a0 a0 c+ | C | 2ap x 2bp x cp | ap = bp < cp | C4/mmb (No. 127 ) |
|
| (22) | a0 a0 c‚ | F | 2ap x 2bp x 2cp | ap = bp < cp | F4/mmc (No. 140) |
| (23) | a0 b0 c0 | P | ap x bp x cp | ap = bp = cp | Pm3m (No. 221 ) |
simple perovskite |
*These space-group symbols refer to axes chosen according to the matrix transformation
| 1 | 0 | 0 | |
| 0 | 1/2 | ‚1/2 | |
| 0 | 1/2 | 1/2 |
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