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 three-dimensional 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 3-dimensional 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.

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)

– 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

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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

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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 • α, β, γ 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

Triclinic P a ≠ b ≠ c α ≠ β ≠ γ

Ī

Monoclinic P (b = twofold axis) C

a ≠ b ≠ c α = γ = 90° ≠ β

2/m

P (c = twofold axis) C

a ≠ b ≠ c α ≠ β = 90° ≠ γ

Orthorhombic P C (or A, B) I F

a ≠ b ≠ c α = β = γ = 90°

mmm

Tetragonal P I

a1 = a2 ≠ c α = β = γ = 90°

4/mmm

Hexagonal R P

a1 = a2 ≠ c α = β = 90°, γ = 120°

m 6/mmm

Cubic P I F

a1 = a2 = a3 α = β = γ = 90°

m3m

The Six C

rystal System

s

P=Primitive F=Face Centered I=Body Centered C=Centered on opposing faces

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 . . .