60
J. Appl. Cryst. (1979). 12, 60-65
FN: A Criterion for Rating Powder Diffraction Patterns and Evaluating the R e l i a b i l i t y
of Powder-Pattern Indexing*
BY GORDON S. SMITH AND ROBERT L. SNYDERt
Lawrence Livermore Laboratory, University of Californ
Structure Factor
Example
Wurtzite
Crystal Data
Formula:
ZnS
Z:
2
Space Group:
P63mc (186)
a:
3.861
c:
6.316
Structure
Both atoms occupy Wycoff position 2b:
1/3 , 2/3 , z
2/3 , 1/3 , z+
Zn:
S:
If z is not fixed, why is it
exactly zero for S?
z = 0.381
z =
Plane Point Groups
Point Groups
Symmetries created by symmetry operators passing
through a single point.
Actually, point group symmetries include such operators
as 5-fold symmetry, however, since our interest is in
objects that are periodic, only certain
Unit Cell Volume
To use space group information to build probable models of crystalline
materials, we need some physical property data such as density and
often the volume of the unit cell.
Vuc a b c
b c a
c a b
Example
Coesite
Monoclinic
a = 7.135,
Plane Group Identification
Determining Plane Groups
While the flow chart is certainly a logical approach to determining the
plane group, a more fundamental approach is also possible.
Locate the motif present in the pattern. This can be a molecule,
molecul
Structure Prediction
Tungsten
Space Group: Im3m (#229)
a = 3.16
Density: 19.3 g/cm3
Formula Weight: 183.85 g/g-mole
What is radius of a Tungsten atom?
Density
Density of Tungsten
Where Do The 2 W's Go?
In Space Group Im3m
Wyckoff Position 2a
1. 0, 0, 0
2
Plane Lattices
Plane Lattices
We have seen the types of symmetries that a motif can
possess in two dimensions. This gives rise to the 10 plane
point groups.
When periodicity is added to the motif there is one important
criterion that must be met (weve act
Symmetry
Basic 2-D Symmetry
Weve already seen that periodicity (translational
symmetry) is very important in the study of arrays of
objects. There are two other basic operators that move
objects in the plane:
Mirrors
Rotations
Mirror Symmetry
2-fold Rot
Before continuing with scattering, consider the various ways
wave phenomena are represented
A cos(t )
2
c
is angular frequency and is
related to wavelength
Periodic functions (like waves) can also be described with
exponentials.
i
e e
i 2 k
k integer
We have seen that the calculation of the magnitudes of lattice vectors
and of the angles between lattice vectors by vector methods is very
straightforward. We can now show that the reciprocal lattice concept
provides analogous and equally straightforward
More Space Groups
Replacing regular rotation axes with screw axes creates some of the
nonsymmorphic space groups, for instance P21.
General Position:
x, y , z
x , 12 y, z
A mirror h can be added to P21 to produce an interesting result.
General Position:
x
Space Groups
Tiling the Point Groups
Periodic groups can be built from the crystallographic
point groups by adding translations to the symmetry
operators.
Recall:
t n n1a n2b n3c
Seitz Operator
R | t r Rr t
The point operation R (proper rotation, imprope
The Structure Factor
Addressing Points in a Macroscopic Crystal
Lattice: Rm m1a m2b m3c
Atom in Unit Cell: n xna ynb znc
Combined: Rnm Rm n
Scatter From a Crystal
Since a scattered beam passing through the origin has zero
phase shift well use it as a refe
Classification of Point Groups
and Space Groups
into Crystal Systems
A set of coordinate axes must be attached to the origin in
a point group so that points in objects can be described
mathematically.
In crystallography we use
a right-handed coordinate
sy
Displaced Operators
Taking a closer look shows that the symmetry operators in this space
group (which are mostly glides and screws) do not intersect at the origin,
but are displaced. This is true of many of the nonsymmorphic groups.
Lets begin by investig
Physical Properties
and
Symmetry
(elementary introduction)
Piezoelectricity
The piezoelectric effect is understood as the interaction between the
mechanical and the electrical state in crystalline materials with no
inversion symmetry. The piezoelectric ef
Glide Planes
Glide Planes
Where screw axes can be constructed from any
proper rotation axis and a fractional translation
parallel to that axis, glide planes are only possible
with mirror, m, operations.
Glide planes have a much richer combination of
possi
Transformations
Recall the construction of the I centered monoclinic space group I2
I-centered
Monoclinic
New basis axes (a and c) could be chosen to give a C centered
cell. b remained unchanged.
aC a I c I
bC b I
cC a I
1 0 1
T 0 1 0
1 0 0
Th2Fe17 has
Plane-Direction Relationships
Planes (hkl) and Directions [UVW]
are not Generally Perpendicular to Each Other
Only in Cubic System
We can at least answer the following:
What Direction is Parallel to a Plane?
What Direction is Parallel to a Plane?
Use AB a
Absorption
Thompson Scattering is coherent. Photons interact with atoms
and are scattered but do not change wavelength.
Compton Scattering is incoherent. The incident photon
wavelength changes.
The Photoelectric effect is similar to normal X-ray productio
Discovery of X-rays
The science of x-ray crystallography can be traced back
to two major discoveries.
The discovery of x-rays is one of these (1895).
The other, Laue's discovery that crystalline materials
can act as diffraction gratings when interacting
The Laue Experiment
The Geometrical Condition
Wave Interference
When a set of waves sum they interfere constructively if they have the
same phase. If waves have differing starting points (phase) some
positive amplitudes combine with negative amplitudes an
Classification of Point Groups
and Space Groups
into Crystal Systems
A set of coordinate axes must be attached to the origin in
a point group so that points in objects can be described
mathematically.
In crystallography we use
a right-handed coordinate
sy
32 Crystallographic Point Groups
Point Groups
The 32 crystallographic point groups (point groups consistent
with translational symmetry) can be constructed in one of two
ways:
1. From 11 initial pure rotational point groups, inversion
centers can be added
Three Dimensional Symmetry
Symmetry Elements
Rotation Axis
Operation: Rotation about an axis
International Notation: n
Schoenflies Notation: Cn
4 (C4)
Mirror Plane
Operation: Reflection across a plane
International notation: m
Schoenflies notation:
m ()
Point Groups
Group Theory
Point Group
A collection of symmetry elements obeying the properties of
a mathematical group and having one point in common,
which remains fixed through all symmetry operations.
Point groups describe the allowable collection of s
Plane Lattices
Plane Lattices
We have seen the types of symmetries that a motif can
possess in two dimensions. This gives rise to the 10 plane
point groups.
When periodicity is added to the motif there is one important
criterion that must be met:
The latt
Lattices
Crystal
A periodic arrangement of atoms in three dimensions
Periodicity
Characteristic repetition distance (t) of a motif.
The collection of repetition distances defines a unit cell
Point Lattice
Replacement of repeated motif with an array of poi
Symmetry
Basic 2-D Symmetry
Weve already seen that periodicity (translational
symmetry) is very important in the study of arrays of
objects. There are two other basic operators that move
objects in the plane:
Mirrors
Rotations
Mirror Symmetry
2-fold Rot
Plane Point Groups
Point Groups
Symmetries created by symmetry operators passing
through a single point.
Actually, point group symmetries include such operators
as 5-fold symmetry, however, since our interest is in
objects that are periodic, only certain
Structure Determination
In a single crystal X-ray experiment we measure
I hkl
After a few corrections (polarization, absorption), we
can recover magnitude of structure factor from
F hkl I hkl
Problem!
Phase angle is missing from the Fs.
The goal of stru
X-Ray Production
Energy Wavelength Equivalency
hc
eV
V
6.626 1034 J sec 2.998 1010 cm sec
108 cm 1.602 1019 J V
1.24 103 V
12.4
kV
Producing X-Rays
Producing X-Rays
Allowed: n > 0, l = 1, and j = 0 or 1
Cu atom
Satellite Transitions
A double vacancy
Morphology
of
Crystalline Materials
Well formed crystals are characterized by flat faces and
sharp angles .
PbSO4
D2 h
(mmm)
Crystals
Crystal Morphology
Nicholas Steno (1669): Law of Constancy of
Interfacial Angles
120o
120o
120o
Quartz
120o
120o
120o
120
Screw Axes
Lets have another look at C2. Some new symmetry has appeared!
The hurricane symbol denotes a special type of axis that combines
rotation (2-fold) with translation ( b). This is called a screw axis and
a little math will show that it is consiste
Space Group Notation
From a space group entry in "The International Tables for
Crystallography" Vol. A, one can ascertain the following
information:
Herman-Mauguin (HM) Symbol (Long, Short)
Point Group (HM, Schoenflies)
Location and identification of symm