Transcript Elementary Crystallography for X

Elementary Crystallography
for X-ray Diffraction
A simplified introduction
for EPS 400-002
assembled by Jim Connolly
What is crystallography?

Originated as the study of macroscopic crystal forms

“Crystal” has been traditionally defined in terms of the structure
and symmetry of these forms.

Modern crystallography has been redefined by x-ray diffraction.
Its primary concern is with the study of atomic arrangements in
crystalline materials

The definition of a crystal has become that of Buerger (1956): “a
region of matter within which the atoms are arranged in a threedimensional translationally periodic pattern.”

This orderly arrangement in a crystalline material is known as
the crystal structure.

X-ray crystallography is concerned with discovering and
describing this structure (using diffraction as a tool).

Automated processing of diffraction data often effectively
distances the analyst from the crystallographic underpinnings
What Crystallographic Principles are
most important to understand?

conventions of lattice description, unit
cells, lattice planes, d-spacing and
Miller indices,

crystal structure and symmetry
elements,

the reciprocal lattice (covered in session
on diffraction)
Description of Crystal Structure

Three-dimensional motif (groups of atoms or
molecules) is the “core” repeated unit

The motif is repeated in space by movement
operations – translation, rotation and
reflection

Crystal structures are “created” in a two-step
process:
– Point-group operations create the motif
– Translation operation produce the crystal structure

“Infinite” repetition is necessary for XRD to
determine the structure
The Lattice

Lattice is “an imaginary pattern of points (or nodes) in
which every point (node) has an environment that is
identical to that of any other point (node) in the
pattern. A lattice has no specific origin, as it can be
shifted parallel to itself.” (Klein, 2002)

The lattice must be described in terms of 3dimensional coordinates related to the translation
directions. Lattice points, Miller indices, Lattice
planes (and the “d-spacings” between them) are
conventions that facilitate description of the lattice.

Although it is an imaginary construct, the lattice is
used to describe the structure of real materials.
1
Symmetry Operations

Crystal structure is “created” by replicating a 3-d motif
with a variety of replication (or movement) operations:
– Rotation (symbols: 1,2,3,4,6 = # of times form is
repeated in a 360º rotation)
– Reflection (symbol: m or ♦, Form replicated across
a mirror plane)
– Inversion (symbol: i. Form replicated by projection
through a point of inversion) 1
– Rotation-inversion (symbol:
ī for single rotation
with inversion.  for 3-fold rotation w. inversion at
each rotation)
Klein Mineralogy Tutorial – Sect II, Symmetry Operations
1
Symmetry Operations

Translation of form along in 3-d space creates the
repeating structure. This occurs as:
– Simple Linear Translation of the motif
– Linear Translation combined with a mirror
operation (Glide Plane)
– Linear Translation combined with a rotational
operation (Screw Axis)

These operations on the basic lattice shapes can
produce a large number of distinct lattice structures
Klein Mineralogy Tutorial – Sect III, One-dimensional order
Lattice Notation

Origin chosen as 000

Axis directions a, b, c
and unit measurements
defined by particular
crystal system

Axes are shown with
brackets, i.e., [001]

Lattice points are
defined in 3-dimesions
as units along the axes,
without brackets (i.e.,
111, 101, 021, etc.)

Lattice planes are defined using Miller indices, calculated as
the reciprocals of the intercepts of the planes on the coordinate
axes (the plane above containing 100, 010, and 002).
Klein Mineralogy Tutorial – Section II, Miller Indices; Miller Index Handout
Spacing of Lattice Planes






Miller indices define a
family of parallel planes
The distance between
these planes is defined as
“d” and referred to as the
“d-spacing”.
Calculations can be very
simple or exceedingly
complex (next slide)
Determination of the unit
cell parameters from
diffraction data is called
indexing the unit cell
These calculations are
usually done by computer
MDI’s Jade can do unit cell
indexing
“Crysfire” is a widely used
free system for indexing
• h, k, l are
Miller indices
• a, b, c are
unit cell
distances
• , ,  are
angles
between the
lattice
directions
Complexity of
calculations is
dependent on
the symmetry
of the crystal
system.
Lattices and Crystal Systems
There are 5 possible planar (2-d) lattices in the different
coordinate systems:
(Mineralogy Tutorial III, 2d Order, Generation of 2d Nets)
Translating these lattices into the 3rd dimension generates the
14 unique Bravais Lattices
(Mineralogy Tutorial III, 3d Order, Generation of 10 Bravais Lattices)
The Bravais Lattices (pt 1)
The Bravais Lattices (pt 2)
System
Type
Edge - Angle Relations
Symmetry
P
abc

Monoclinic
P (b = twofold axis)
C
abc
 =  = 90  
P (c = twofold axis)
C
abc
   = 90  
P
C (or A, B)
I
F
abc
 =  =  = 90
mmm
Tetragonal
P
I
a1 = a 2  c
 =  =  = 90
4/mmm
Hexagonal
R
P
a1 = a 2  c
 =  = 90,  = 120
m
6/mmm
Cubic
P
I
F
a1 = a 2 = a 3
 =  =  = 90
m3m
Orthorhombic
Ī
2/m
P=Primitive F=Face Centered I=Body Centered C=Centered on opposing faces
The Six Crystal Systems
Triclinic
The 32 Point Groups

Operation of translation-free symmetry
operations on the 14 Bravais lattices
produces the 32 Point Groups

These are also known as the “Crystal
Classes”

These are shown schematically on the next
slides

Mineralogy Tutorials II, Crystal Classes
demonstrates the generation of these Point
Groups

Mineralogy Tutorial available for class use on EPSCI Network.
Login and connect to \\eps1\mintutor3
Double-click on “Mineralogy_Tutorials.exe”
The Point
Groups
(Part 1)
The Point
Groups
(Part 2)
The 230 Space Groups

Operation of the translation operations on the 32
point groups produces the (somewhat intimidating)
space groups (listed on following slides).

Screw Axes combine rotation about an axis with
translation parallel to it. Rotations can be 180º, 120º,
90º or 60º defining 2-, 3-, 4- and 6-fold axes
respectively.

Glide Planes combine reflection across a plane
combined with translation parallel to it. Glides are
expressed as a direction (a,b,c) with a subscript
indicating how many glides occur in one unit
distance.
(See Mineralogy Tutorials – III – 3d Order on Screw Axes and Glide
Planes; Space Group Elements in Structures explores space group
symmetry in several common minerals)
The 230 Space Groups (pt. 1)
Crystal Class
Space Group
The 230 Space Groups (pt. 2)
Crystal Class
Space Group
And now for something completely different . . .