PHY1039
Properties of Matter
Crystallography, 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
Crystals of silica (SiO2)
Single crystals of Si
High resolution electron
microscope image of
Mg2Al4Si5O18
A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 - 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.
Primitive Unit Cells
2D crystal
A primitive unit cell contains
only one lattice point, e.g. four
shared between four cells.
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
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 is the angle between b and c
b is the angle between a and c
g is the angle between a and b
Describing Directions in a Crystal
A direction can be described by a vector t:
t=Ua+Vb+Wc
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:
𝒕 = 𝑈𝑉𝑊
c
b
a
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
One red and one
blue circle are
attached to each
lattice point.
Basis is also called
the “motif”
http://www.doitpoms.ac.uk/tlplib/crystallography3/structure.php
N-Fold Rotational Symmetry
Atoms in a crystal have rotational symmetry.
It is not possible to fill 2-D or 3-D space using 5-fold symmetry!
(But see the final slide!)
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.
Symmetry Elements
Triclinic
a≠b≠c;
α≠β≠γ
Monoclinic
a≠b≠c; α=γ=
90; β>90
Orthorhombic
See Bravais lattices in 3D:
a≠b≠c;
α=β=γ= 90
Tetragonal
a=b≠c;
α=β=γ=90°
Trigonal
a=b≠c; α=β=
90; γ= 120
Cubic
a=b=c;
α=β=γ= 90
Hexagonal
a=b≠c;
α=β=90°;
γ=120°
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 <111> )
1 hexad (parallel to [001])
http://www.doitpoms.ac.uk/tlplib/crystallography3/systems.php
Primitive
=Simple
Triclinic
a≠b≠c;
α≠β≠γ
Monoclinic
a≠b≠c; α=γ=
90; β>90
Orthorhombic
See Bravais lattices in 3D:
a≠b≠c;
α=β=γ= 90
Tetragonal
a=b≠c;
α=β=γ=90°
Trigonal
a=b≠c; α=β=
90; γ= 120
Cubic
a=b=c;
α=β=γ= 90
Hexagonal
a=b≠c;
α=β=90°;
γ=120°
http://www.doitpoms.ac.uk/tlplib/crystallography3/systems.php
Bodycentred
Basecentred
Facecentred
(Primitive)
Only Po has a simple cubic structure!
Basis with more than one atom
NaCl crystals are described by a facecentred 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
c
a, b, and c
b
a
a
b
c
http://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_index.php
Bracket Notation in Miller Indices
• 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), (100) (010) and (001).
• These planes all "look" the same and are related to each
other by the symmetry elements present in a cube.
{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
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:
𝑑ℎ𝑘𝑙 =
𝑎
ℎ2 + 𝑘 2 + 𝑙 2
Shown here: 213
planes in a cubic unit
cell.
Other equations are used for other crystal systems.
Penrose Tiling and Quasicrystals
Penrose titles have five-fold symmetry, and
can fill two-dimensional areas. But, the
pattern is non-periodic.
In 1982, rapidly cooled AlMg alloy produced a
diffraction pattern with 5fold rotational symmetry.
Now called “quasicrystals”
Images from: http://www.ams.org/samplings/feature-column/fcarc-penrose