Describing condensed phase structures
Describing the structure of an isolated small
molecule is easy to do
– Just specify the bond distances and angles
How do we describe the structure of a condensed
phase ?
– we have ~ Avogadro’s number of atoms to locate
– we should either give up on specifying the position of
every atom or find a trick to help us out
The structure of liquids and glasses
We can use pair distribution functions to describe
the structure of such systems
Page 1
The structure of crystalline materials
We can use the symmetry of a crystal to reduce the
number of unique atom positions we have to specify
The most important type of symmetry is translational
– this can be described by a lattice
The structure associated with the lattice can be carved
up into boxes (unit cells) that pack together to
reproduce the whole crystal structure
The lattice and unit cell in 1D
Page 2
Lattices and unit cells 2 D
Unit cell choice
There is always more than possible choice of unit cell
By convention the unit cell is chosen so that it is as
small as possible while reflecting the full symmetry
of the lattice
Page 3
Unit cell choice in 2D
Picking a unit cell for NaCl
Page 4
Other types of symmetry
Crystallographers
make use of all the
symmetry in a crystal to minimize the
number of independent coordinates
Lattice symmetry
Point symmetry
Other translational symmetry elements
– screw axes and glide planes
Point symmetry elements
A
point symmetry operation does not alter
at least one point that it operates on
– rotation axes
– mirror planes
– rotation-inversion axes
Screw
axes and glide planes are not point
symmetry elements !!!
Page 5
Benzene
A two fold rotation
Page 6
A mirror plane
An inversion center
Page 7
A rotation inversion axis
Point symmetry elements
compatible with 3D translations
Symmetry element
Symbol
Mirror plane
m
Rotation axis
n = 2,3,4,6
Inversion axis
n (= 1,2,3,4,6)
Center of symmetry 1
υνδερσχορε
Page 8
Point symmetry and packing
Unit cells in 3D
Page 9
The seven crystal systems
System
Unit Cell
Minimum Symmetry
Triclinic
α ≠ β ≠ γ ≠ 90º
a≠b≠c
None
Monoclinic
α = γ = 90º
β ≠ 90º
a≠b≠c
One two-fold axis or one
symmetry plane
Orthorhombic
α = β = γ = 90º
a≠b≠c
Any combination of three
mutually perpendicular two-fold
axes or planes of symmetry
Trigonal
α = β = γ ≠ 90º
a=b=c
One three-fold axis
Hexagonal
α = β = 90º
γ = 120º
a=b≠c
One six-fold axis or one six-fold
improper axis
Tetragonal
α = β = γ = 90º
a=b≠c
One four-fold axis or one fourfold improper-axis
Cubic
α = β = γ = 90º
a=b=c
Four three-fold axes at 109º 23’ to
each other
The symmetry elements of a cube
Page 10
Centering
Bravais Lattices
Page 11
Screw axes and glide planes
Crystalline
solids often posses symmetry
that can be described as a combination of a
rotation and a translation “a screw axis” or
a combination of a reflection and a
translation “a glide plane”
A two fold screw
Page 12
An “a glide”
Lattice planes
It
is possible to describe certain directions and planes
with respect to the crystal lattice using a set of three
integers referred to as Miller Indices
Page 13
Miller indices (hkl)
Miller Indices are the
reciprocal intercepts of the
plane on the unit cell axes
Identify plane adjacent to
origin
– can not determine for plane
passing through origin
Find intersection of plane on
all three axes
Take reciprocal of intercepts
If plane runs parallel to axis,
intercept is at ∞, so Miller
index is 0
Examples of Miller indices
Page 14
Families of planes
Miller
indices describe the orientation a
spacing of a family of planes
– The spacing between adjacent planes in a
family is referred to as a “d-spacing”
Three different
families of planes
d-spacing between
(300) planes is one
third of the (100)
spacing
d-spacing formulae
For a unit cell with orthogonal axes
– (1 / d2hkl) = (h2/a2) + (k2/b2) + (l2/c2)
Hexagonal unit cells
– (1 / d2hkl) = (4/3)([h2 + k2 + hk]/ a2) + (l2/c2)
Page 15
Unit cells and dhkl
Bragg’s law
2d sinθ = nλ
Consider crystal to contain repeating ‘reflecting’
planes (lattice planes)
Page 16
Fractional coordinates
The
positions of atoms inside a unit cell are
specified using fractional coordinates
(x,y,z)
– These coordinates specify the position as
fractions of the unit cell edge lengths
Specifying orientation and direction
Miller
indices (hkl) are used to specify the orientation
and spacing of a family of planes. {hkl} are used to
specify all symmetry equivalent sets of planes
Miller indices [hkl] are used to specify a direction in
space with respect to the unit cell axes and <hkl> are
used to specify a set of symmetry equivalent directions
– To specify a direction parallel to a line joining the origin and
a point with coordinates x,y,z in the unit cell multiply x,y,z
by the smallest number that will result in three integers
» these are the Miller indices specifying the direction
» Passes through 0.3333,0.6667,1 so Miller indices are [123]
Page 17
Density
Density
measurement and calculation can
be used to
– determine number of formula units in unit cell
– check that your supposed formula is correct
– establish defect mechanism
Density = (FW x Z) /(V x N) = (FW x Z x 1.66) /V
V – unit cell volume
Z – formula units in cell
FW – formula weight
Page 18