This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.
 

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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.

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	m-3	23, m-3
			m-3m	432, -43m, m-3m

For tetragonal structures, the lattice spacing (d-spacing) 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 transition-metal (TM) oxide crystal, the divalent TM ion is in a site that has octahedral O_h symmetry and the d-levels 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_3z2_-r2 and d_x2_-y2 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 Jahn-Teller ion is Mn³⁺ as shown in Figure 3551c. The ion in the high-spin 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.

Table 3551c. Other characteristics of tetragonal structures.