PHY1039 Properties of Matter Crystallography, Lattice Planes, Miller Indices, and X-ray Diffraction (See on-line resource: ) March 19, 2012 Lectures 13 and 14

Crystals of silica (SiO 2 )Single crystals of Si Crystals High resolution electron microscope image of Mg 2 Al 4 Si 5 O 18 A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, frontispiece

Atomic Arrangements in Crystals Periodic arrays of atoms extending in three dimensions How can we describe and determine crystal structures? Each atom in a crystal is associated with points in a lattice. A lattice is made of regularly-space points that fill 3-D space.

The unit cell can be translated in space to repeat the pattern containing all of the lattice points in the crystal. Primitive Unit Cells 2D crystal A primitive unit cell contains only one lattice point, e.g. four shared between four cells.

 is the angle between b and c  is the angle between a and c  is the angle between a and b Lattice Parameters a, b and c are the lengths of the edges of the unit cell (called the lattice constants) a, b and c can also be used in vectors to define the points on a lattice in 3D.

A direction can be described by a vector t: t = U a + V b + W c In shorthand, lattice vectors are written in the form: t = [UVW] Negative values are not prefixed with a minus sign. Instead a bar is placed above the number to denote that the value is negative: t = −U a + V b − W c This lattice vector would be written in the form: The set of directions that are symmetrically related to the direction [UVW] are written. For instance, in a cubic system, as shown here: [110], [101], [011] can be represented as. Describing Directions in a Crystal b a c

Basis is also called the “motif” One red and one blue circle are attached to each lattice point.

N-Fold Rotational Symmetry It is not possible to fill 2-D or 3-D space using 5-fold symmetry! (But see the final slide!) Atoms in a crystal have rotational symmetry.

Mirror Symmetry Crystal structures also exhibit mirror symmetry across certain planes in the crystal. The view on the left side of the plane is the mirror image of what is on the right side.

The 14 unique lattices are referred to as “Bravais lattices”. No one has ever found a 15 th Bravais lattice.

Triclinic Monoclinic Cubic Orthorhombic Tetragonal Trigonal Hexagonal a≠b≠c; α≠β≠γ a≠b≠c; α=γ= 90  ; β>90  a≠b≠c; α=β=γ= 90  a=b≠c; α=β= 90  ; γ= 120  a=b=c; α=β=γ= 90  a=b≠c; α=β=90°; γ=120° a=b≠c; α=β=γ=90 ° See Bravais lattices in 3D: Symmetry Elements Translational symmetry only Only one diad axis (parallel to [010]) 3 diad axes One tetrad (parallel to the [001] vector) 1 triad (parallel to [001]) 4 triads (all parallel to ) 1 hexad (parallel to [001])

Triclinic Monoclinic Cubic Orthorhombic Tetragonal Trigonal Hexagonal a≠b≠c; α≠β≠γ a≠b≠c; α=γ= 90  ; β>90  a≠b≠c; α=β=γ= 90  a=b≠c; α=β= 90  ; γ= 120  a=b=c; α=β=γ= 90  a=b≠c; α=β=90°; γ=120° a=b≠c; α=β=γ=90 ° Body- centred Face- centred Base- centred Primitive =Simple See Bravais lattices in 3D:

(Primitive)

Only Po has a simple cubic structure!

Basis with more than one atom NaCl crystals are described by a face- centred cubic (FCC) Bravais lattice. The basis consists of a Cl- ion at (0,0,0) and a Na+ ion at (1/2, 0, 0)

Basis with more than one atom

a b c a, b, and c a b c

Bracket Notation in Miller Indices

{110} Planes in a Cubic Crystal – Related by Symmetry

Convenient unit: 1 Ångstrom, Å = m

Animation:

Relationship between Lattice Spacings and Lattice Constants, a, in a Cubic Crystal Other equations are used for other crystal systems. The spacing between planes with Miller indices (hkl) is designated as d hkl If the lattice constant for a cubic crystal is a (where a = b = c), then d hkl is calculated as: planes in a cubic unit cell.

Penrose Tiling and Quasicrystals Images from: Penrose titles have five-fold symmetry, and can fill two-dimensional areas. But, the pattern is non-periodic. In 1982, rapidly cooled Al- Mg alloy produced a diffraction pattern with 5- fold rotational symmetry. Now called “quasicrystals”