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Table 3551a and Figure 3551a show the tetragonal crystal systems and the schematic illustrations of the tetragonal lattices, respectively. Furthermore, Table 3551c shows the cell edges and angles of tetragonal crystals.
Table 3551a. Tetragonal crystal systems.
Crystal
family 
Crystal
system 
Required
symmetries
of point group 
Point
group 
Space
group 
Bravais
lattices 
Lattice
system 
Tetragonal 
1 fourfold axis of rotation 
7 
68 
2 
Tetragonal 
Figure 3551a. Schematic illustrations of the Bravais
lattices of tetragonal crystals.
Table 3551b. Relationship between Laue classes and point groups.
System 
Essential symmetry 
Lattice symmetry 
Laue class (Diffraction symmetry) 
Point Groups (Hermann–Mauguin notation) 
Triclinic 
None 


1, 1 
Monoclinic 

2/m 
2/m 
2, m, 2/m 
Orthorhombic 
222 or 2mm

mmm

mmm 
222, mm2, mmm 
Tetragonal 

4/mmm

4/m

4, 4, 4/m 
4/mmm 
422, 42m, 4mm, 4/mmm 
Trigonal 


3 
3, 3 
3m1 
321, 3m1, 3m1 
31m 
312, 31m, 31m 
Hexagonal 

6/mmm

6/m 
6, 6, 6/m 
6/mmm 
622, 62m, 6mm, 6/mmm 
Cubic 
23 
m3m

m3 
23, m3 
m3m 
432, 43m, m3m 
For tetragonal structures, the lattice spacing (dspacing) can be given by, (You can download the excel file for your own calculations)
 [3551] where,
a and c  The lattice constants.
h, k, and l  The Miller indices.
As shown in Figure 3551b, in a cubic transitionmetal (TM) oxide crystal, the divalent TM ion is in a site that has octahedral O_{h} symmetry and the dlevels split into threefold degenerate (lower energy) t_{2g} states and twofold degenerate (higher energy) e_{g} states that can accommodate six and four electrons, respectively (including spin states). The tetragonal symmetry splits the levels further. The t_{2g} states split into a singlet, d_{xy}, and a doublet d_{xz} and d_{yz}. The e_{g} states split into d_{3z}2_{r}2 and d_{x}2_{y}2 levels.
Figure 3551b. Ligand field splitting of d orbitals in an octahedral ligand field.
In some TM (transition metal) cases, the filling of orbitals with electrons may affect the local structure and thus induce geometrical distortion around the TM ion. The Jahn–Teller effect, also called Jahn–Teller distortion, describes this type of distortions. A typical JahnTeller ion is Mn^{3+} as shown in Figure 3551c. The ion in the highspin configuration contains a single electron in the upper e_{g} state when it is placed in an octahedral LF (ligand field). A tetragonal distortion can lower the energy of the system. The lowering in total energy is due to the lowering of one of the e_{g} orbitals by lengthening the bond along the z axis. Note that the overall energy of the system is not further lowered by splitting the t_{2g} state because the center of gravity is retained.
Figure 3551c. JohnTeller effect for Mn^{3+} (3d^{4}).
Table 3551c. Other characteristics of tetragonal structures.
Contents 
Page 
Angles in unit cells 
page3555 
Volume of unit cells 
page3033 
Bravais lattices 
page4546 
Relationship between threedimensional crystal families, crystal systems, space group, point group, lattice systems and symmetries 
page4549 
