In the tetragonal system, we let all the angles remain
at 90º. Again, thinking about the name will give us a clue to the axis
system. “Tetra” means four, so the materials in this crystal
system have 4-fold axes. Also, we can release the constraint that all the
axes have to be the same length—this time, we’ll let one of
them be different. So mathematically, we would write this as:
lengths a = b
≠ c, and
angles α =
β =
γ = 90º.
Recall that the stereographic projection is used to
show the angular relationships, but not the lengths of the axes, so in
Figure 8 we show both the stereographic projection and a perspective
sketch of the axis set. Crystals that form in the tetragonal system have a
3-D shape that looks like spaghetti boxes with square ends, hat boxes, and
pieces of 4x4" fence posts. Their unifying characteristic is that
they have 90º angles, and a 4-fold axes in only the γ direction.
Later we will
show why this occurs mathematically. Some shapes with tetragonal symmetry
include dipyramids (two pyramids stacked with their bases touching) and
disphenoids (like tetrahedra, where two upper faces alternate with two
lower faces). Minerals that crystallize with tetragonal symmetry include
members of the scapolite group, chalcopyrite, and zircon.
Figure 8. Representations of the tetragonal
crystal system. Note how the lengths of a and
b are the same and c
is different, but the interaxial angles are all still equal to 90°(left): A 3-D perspective sketch
showing the relationship between the three crystallographic axes. (center): A solid composed by
these three axes showing one possible morphology in this crystal system. (right): A stereographic
projection showing the angular relationships between the three crystallographic axes.
