PHY1039

Properties of MatterCrystallography, Lattice Planes, Miller Indices,

and X-ray Diffraction

(See on-line resource: http://www.doitpoms.ac.uk/ )

March 19, 2012

Lectures 13 and 14

Crystals of silica (SiO2) Single crystals of Si

Crystals

High resolution electron microscope image of

Mg2Al4Si5O18

A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 - frontispiece

Atomic Arrangements in CrystalsPeriodic 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.

http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php

Primitive Unit Cells2D crystal

A primitive unit cell contains only one lattice point, e.g. four shared between four cells.

a is the angle between b and cb is the angle between a and cg 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 cThis lattice vector would be written in the form:

The set of directions that are symmetrically related to the direction [UVW] are written <UVW>.

For instance, in a cubic system, as shown here: [110], [101], [011] can be represented as <110>.

http://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.php

Describing Directions in a Crystal

𝒕=[𝑈𝑉 𝑊 ]

ba

c

Basis is also called the “motif”

http://www.doitpoms.ac.uk/tlplib/crystallography3/structure.php

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 15th 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°

http://www.doitpoms.ac.uk/tlplib/crystallography3/systems.php

See Bravais lattices in 3D:

Symmetry ElementsTranslational 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 <111> )

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

http://www.doitpoms.ac.uk/tlplib/crystallography3/systems.php

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

http://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_index.php

a

bc a, b, and c

a

b

c

• When referring to a specific plane, "round" brackets are used:

(hkl)

• When referring to a set of planes related by symmetry, then "curly" brackets are used:

{hkl}

• For example, they might be the (100) type planes in a cubic system, which are (100), (010), (001), (00) (00) and (00).

• These planes all "look" the same and are related to each other by the symmetry elements present in a cube.

Bracket Notation in Miller Indices

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

Convenient unit: 1 Ångstrom, Å = 10-10 m

Animation: http://www.doitpoms.ac.uk/tlplib/xray-diffraction/bragg.php

Relationship between Lattice Spacings and Lattice Constants, a, in a Cubic Crystal

𝑑h𝑘𝑙=𝑎

√h2+𝑘2+𝑙2

Other equations are used for other crystal systems.

The spacing between planes with Miller indices (hkl) is designated as dhkl

If the lattice constant for a cubic crystal is a (where a = b = c), then dhkl is calculated as:

planes in a cubic unit cell.

Shown here: d 213

Penrose Tiling and Quasicrystals

Images from: http://www.ams.org/samplings/feature-column/fcarc-penrose

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”