Powder X-ray Diffraction Brendan J. Kennedy School of Chemistry The University of Sydney Crystallite Size Distribution Microstrain Analysis Extended Defect Concentration Three Unique Features of Synchrotron Radiation •Intensity •Energy Range •Enables Rapid Data •Enables Spectroscopy –Elemental Identification Collection −Bonding Studies Kinetics Unstable Compounds -Speciation Environmental Cells •Enables Optimal Conditions -Enables Focussing –Environmental Cells Small Samples –Selected Elements Small areas/volumes •Low Divergence Enables High Resolution •Micro Beams •Small Volumes •Complex Materials What is special about a crystal?
Crystal Structure
• CRYSTAL: Contains a periodical array of atoms/ions. This can be represented by a simple lattice of points. • A group of atoms is associated with each lattice points. •  LATTICE: An infinite array of points in space, in which each point has identical surroundings to all others. • CRYSTAL STRUCTURE: The periodic arrangement of atoms in the crystal. The unit cell is a basic parallelopiped shaped block from which the whole volume of the crystal may be built by repetition in 3 dimensions. Any point in the unit cell may be specified with respect to the origin by parameters x, y, z measured parallel to the unit cell axes and expressed as fractions. Example of 2D symmetry in a wallpaper pattern To show symmetry: .1. Pick a point .2. Find all equivalent points .These points form a 2D lattice form a parallelogram gives a possible unit cell (http://ww.clarku.odu/-djoyca/wallpaper/) Example of 2D symmetry in a
wallpaper pattern . Connecting 4 lattice points to form a parallelogram gives a possible unit cell . Unit cel’ – the basic unit that repeats in every direction . Different unit cells can be chosen .But some unit cells are preferable for higher symmetry
Hexagonal
P C l F PCC Lattice BCC Lattice Sodium Chloride
(NaCl) Na is much
smaller than Cl Face Centered
Cubic Rocksal! structure Lattice type F Also NaF, KBr, 
MgO.... α-Po  is primitive-
Centered Cubic Identical atoms at corners but nothing
at the and body or face centers. Lattice type P α-Iron  is Body-
Centered Cubic Identical atoms at corners and body center (nothing at face centers) Lattice type I Also Nb, Ta, Ba, Mo...
Diffraction of X-ray Waves
•  Diffraction:  When light passes sharp edges or goes through narrow slits the rays are  deflected and produce fringes of light and dark bands. But not all planes result in diffraction !!!
Lattice Planes
•  It is possible to describe certain directions and planes with respect to the crystal lattice using a set of integers referred to as Miller Indicies Miller Indices: 1. Find the intercepts on the axes in terms of the lattice constant a, b, c 2. Take the reciprocals of these numbers, reduce to the three integers having the same ratio (hkl) Set of symmetry-related planes: (hkl)
Families of Planes
• Miller indices describe the orientation of a family of planes – the spacing between adjacent planes in a family is referred to as a "d-spacing" • different families of planes – d-spacing between (400) planes is 1/4 that of the (100) spacing. The (300) plane does not contain atoms and so is not observed === (300)
Single Crystal vs Powder
Powder – A Polycrystalline Mass
All orientations of crystallites possible Single crystal reciprocal lattice - smeared into spherical shells (220) . By varying the angle θ, the Bragg's Law conditions are
 satisfied by different d-spacings in polycrystalline materials. . Plotting the angular positions and intensities of the resultant diffracted peaks produces a pattern which is characteristic of the sample. Powder Diffraction
Information Contained in a
Centering and Absences
The positions of the atoms in a unit cell determine the intensities of the reflections Consider diffraction from (100) planes in If the pathiength between rays 1 and 2 differs by λ, the path length between rays 1 and 3 will differ by λ/2 and destructive interference in (b) will lead to no diffracted intensity
Centering and Absences
•  We can extend these types of calculation to include other modes of lattice centering. They all lead to systematic absences

Multiplicity
For high symmetry materials the Bragg angles and d-spacings for different reflections may be equivalent to one another For example (100), (010), (001) etc are equivalent in a cubic material In a powder, all planes with the same d-spacing contribute to the scattered intensity at a given Bragg angle 
The number of planes that are symmetry equivalent is referred to as
the multiplicity and its appears as a multiplicative term in powder
diffraction intensity calculations 
The multiplicity of a reflection depends upon the symmetry of the crystal only be 4 as (100) and (001) are not equivalent
Diffraction Patterns
Spacing of peaks depends on size of unit cell and the space group. The bigger the unit cell and/or the lower the symmetry the more diffraction peaks are observed. Intensity of peaks depends on (amongst other things) the arrangement of the atoms in the unit cell. For two materials that had identical unit cells, the peak positions would be IDENTICAL, however their intensities would be DIFFERENT.
Need for High Resolutio
Peak Overlap
•  Powder Diffraction patterns are a one dimensional representation of a three dimensional structure. •  Often peaks due to individual Bragg reflections overlap
The Solution - Rietveld
Yicak Y ic the net intensity calculated at point i in the pattern, Yiback is the background intensity, • G ik is a normalised peak profile function, •  I , is the intensity of the k th Bragg reflection, k, ... k, are the reflections contributing intensity to point i, the superscript p corresponds to the possible phases present in the sample.
The Answers
• The Profile  R The Goodness of fit
Phase Analysis
•  Where a mixture of different phases is present, the resultant diffraction pattern is formed by addition of the individual patterns. •  The intensity of the peaks is proportional to the amount of the phase present. Quantitative Phase Analysis Bragg scattering is proportional to N/V where N is the number of unit cells and V the unit cell volume. There for the weight of a phase in the beam is: III ( SZMV ) (SMPV) S - the scale factor Z the number of formula unites per unit cell M the molecular weight of the formula unit l is the index running over all phases . Hence SZVM is proportional to the weight of the diffracting sample An Example
The background
• Fluorescent radiation from the sample •  Diffraction from the continuous spectrum •  Diffuse scattering – Incoherent – Temperature diffuse •  Other materials – Specium holder – air etc •  Here the capillary is a feature. G •  Beam Divergence can results in asymmetric peaks at low angles. • Results from not integrating over the entire Debye cone.
A Common Problem
•  If the structural model is wrong then the most common response of Rietveld programs is to: – broaden the peaks, – Increase the displacement parameters, • The former is  most noticeable at high angles where intensity is lowest. •  Due to absorption of the X-rays powder X-ray diffraction often yields poor displacement parameters
An Alternate Model
• The high angle splitting is well modeled by a tetragonal model - but this overestimates some intensities. • The Truth lies somewhere in the middle • The sample contains a mixture of both phases! Strengths and Limitations of Powder X-ray Diffraction Strengths Non-destructive – small
amount of sample Relatively rapid
Identification of compounds / phases – not just elements
Quantification of concentration of phases – (sometimes)
Classically for powders, but
solids possible too Gives information regarding crystallinity, strain, crystallite
size, and orientation Limitations

Experiment Design Issues
What Wavelength? • Absorption is your enemy! • Short Wavelengths are best! BUT.... • Consider required resolution.  And •  Avoid Absorption Edges. What Size Capillary? •  Small capillaries reduce absorption AND improve resolution. •  BUT reduce amount of material.