Lattice Systems

Abstract

One possible way of categorizing crystals is by the symmetry of their lattice. Where a crystal system is one of seven ways to categorize crystals based on the symmetry of the crystal (cubic, hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, triclinic), a lattice system is one of seven ways to categorize crystals based on the symmetry of their lattice (cubic, hexagonal, rhombohedral, tetragonal, orthorhombic, monoclinic, triclinic).

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.

–Chen Ning Yang

Review Questions

• 1. Define the terms Bravais lattice, lattice system, and crystal system.

• 2. According to the classification system used in the International Tables for Crystallography, what is the difference between a trigonal crystal and a rhombohedral one? Why is it wrong to define trigonal and rhombohedral crystal systems interchangeably? What crystal systems would exist if crystals were categorized by the symmetry of their lattices rather than the symmetry of the crystal itself?

• 3. The lattice structure of NaCl is usually described as face-centred cubic. The ionic positions are:

  Na+	0,0,0	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,0	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,0,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	0,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$
  Cl−	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,0,0	0, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,0	0,0,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

  Sketch one unit cell of this structure. How many ions comprise the basis? Annotate your drawing to show these ions.

  A body-centred unit cell can be inscribed within the face-centred one. Sketch this new unit cell within a single face-centred cubic NaCl one and indicate its basis. What is the new Bravais lattice which is formed?

  It is always possible to draw a primitive unit cell of any crystalline structure. Inscribe a primitive NaCl unit cell within a single face-centred cubic one and indicate its basis. What is the Bravais lattice now?

• 4. Use the metric tensor to calculate the angle between [111] and [123] in an orthorhombic unit cell with lattice constants:

  a = 4 Å, b = 6 Å, c = 11 Å.

• 5. Use the metric tensor to calculate the angle between [123] and [321] in a monoclinic unit cell with lattice constants:

  a = 1 Å, b = 2 Å, c = 3 Å, β = 95°.

• 6. Consider the diamond cubic C allotrope with a = 3.56 Å. Atomic positions are at 000, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$0, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$0$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, 0$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$. Calculate its theoretical density, the C-C bond length, and the C-C-C tetrahedral bond angle.

• 7. Use the metric tensor to calculate the distance between atoms at 000 and $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ in a monoclinic unit cell with lattice constants a = 1 Å, b = 2 Å, c = 3 Å, β = 95°.