Crystal Systems

Abstract

The seven crystal systems include triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Each differs by the symmetries present. A crystal’s symmetry will imply certain constraints on its basis vectors (i.e., lattice constants), and it is these constraints that are often used to determine or even define the crystal system; however, it should always be remembered that it is the symmetry, not the lattice constants, that determines the crystal system of any given crystal.

The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.

–Paul Valéry

Review Questions

• 1. Why is it wrong to define an orthorhombic crystal as one in which a ≠ b ≠ c and α = β = γ = 90°?

• 2. Determine the Miller indices of the direction in an orthorhombic unit cell that is perpendicular to both [100] and [111]. Assume a = 1 Å, b = 2 Å, c = 4 Å.

• 3. What is the angle between the [111] and [223] directions in an orthorhombic unit cell with lattice constants a = 4 Å, b = 6 Å, c = 11 Å?

• 4. List the crystal requirement(s) for tetragonal symmetry.

• 5. List the crystal requirement(s) for orthorhombic symmetry.

• 6. List the crystal requirement(s) for cubic symmetry.

• 7. Use the dot product to calculate the angle between [134] and $\left[5\overline{1}1\right]$ directions in a cubic crystal.

• 8. Use the cross-product to calculate the direction which is perpendicular to both [112] and $\left[2\overline{1}0\right]$ in an orthorhombic crystal with a = 1, b = 2, c = 3 Å.