Tetragonal bipyramidal systems

B. Reference system E>4h(5Aig), stronger axial field Case 4 Compressed tetragonal bipyramid... 

C. Reference system E>4h(5Eig), weaker axial field Case 5 Elongated tetragonal bipyramid... 

For a compressed tetragonal bipyramid (axial strengthening) the separation Ai3 = s(r7) - (/, ) of the lowest multiplets r6 (ground) and r7 increases rapidly when the system deviates from the regular octahedron. 

Each crystal system contains several classes that exhibit only a partial symmetry for instance, only one-half or one-quarter of the maximum number of faces permitted by the symmetry may have been developed. The holohedral class is that which has the maximum number of similar faces, i.e. possesses the highest degree of symmetry. In the hemihedral class only half this number of faces have been developed, and in the tetrahedral class only one-quarter have been developed. For example, the regular tetrahedron (4 faces) is the hemihedral form of the holohedral octahedron (8 faces) and the wedge-shaped sphenoid is the hemihedral form of the tetragonal bipyramid Figure 1.9). 

Figure 1.10. Simple combination forms in the tetragonal system a) tetragonal prism b) tetragonal bipyramid (c) combination of prism and bipyramid d) combination of two bipyramids...

The crystal has a four-fold axis and a horizontal plane of symmetry (perpendicular to the four-fold axis). It also has four two-fold axes perpendicular to the four-fold axis and four mirror planes along those axes. The crystal therefore belongs to the tetragonal system. The form can be described as ditetragonal bipyramidal and the slanted faces are referred to as pyramid faces. 

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