Question

Discuss physically why an actual crystal cannot possess a fivefold axis of rotational symmetry. Draw sketches illustrating a (100) plane, a (110) plane, and a (111) plane in a cubic unit cell. How many equivalent {100}, {110}, and {111} planes are there in a cubic crystal? Regard planes (hkl) and (hkl) as identical. How many equivalent {123} planes are there in a cubic crystal? How many equivalent {111} planes are there in an orthorhombic crystal?

Answer 1

3)If a crystal had a fivefold symmetry then it would always be possible to obtain a lattice vector which is shorter than a mathematically constructed shortest lattice vector. This is unphysical.So it is impossible to obtain a crystal with five fold symmetry.

5)a)There are 48 sets of equivalent {123} planes in a cubic crystal

b)There are 0 sets of equivalent {111} planes in a cubic crystal