KJM-MENA 4010 – Module 13 Powder X-ray Diffraction Theoretical Basis

KJM-MENA 4010 – Module 13 Powder X-ray Diffraction Theoretical Basis
Slides borrowed from: KJM-MENA 4010 – Module 13 Powder X-ray Diffraction Theoretical Basis David Wragg Fredrik Lundvall, Georgios Kalantzapoulos, Marion Duparc

Literature Powder Diffraction Theory and Practice
R. E. Dinnebier and S. J. L. Billinge (Eds.) Cambridge, UK, RSC Publishing 2008, 604 pp. (HB) ISBN

Literature Fundamentals of Powder Diffraction and
Structural Characterization of Materials by Vitalij Pecharsky, Peter Zavalij Paperback: 744 pages Publisher: Springer; 2nd ed. edition (November 26, 2008) ISBN-10:

What is X-ray Diffraction?
X-rays are diffracted by crystalline mateials Diffraction pattern is unique to crystal lattice Crystal Structure- Contains all the geometric information about the material! Crystallography 30+ Nobel prizes This is VITAL information!

Example 1 Phase ID and quantification from powder

Example 2 Structural information from powder

Example 3: in situ powder XRD
Structure and behaviour of methanol in SAPO-34 at room temperature Wragg et al. Micro. Meso. Mater. 2010 3% change in c-axis caused by filling of cages during reaction Wragg et al, J. Catal. 2009 These results, two published in journals and one presented at an international conference and soon to be submitted for publication were all obtained using the inGAP gas mixing system and combined PXRD/MS/Raman at SNBL. In situ XRD-MS-Raman study of the behaviuour of SAPO-34. first observation of the expansion of the catalyst during reaction driven by intermediates in cages. Direct observation of intermediates in cages. Correlation of intermediate growth (XRD and Raman) with expansion. Gas mixing system, MS, Raman. In situ and Ex situ studies of methanol and water adsorption at RT. Crystal structure of SAPO-34-Methanol. Behaviour from in situ XRD. Gas mixing system. In situ studies of SAPO-18. very different behaviour to SAPO-34. Gas mixing system, MS. In situ comparison of SAPO-18 and SAPO-34 with PXRD/MS Wragg et al. Micro. Meso. Mater. 2011 Direct Observation of intermediates in SAPO-34 by PXRD. Growth from simultaneous PXRD and Raman

Objectives Know the basics of how X-rays interact with matter
Understand the basic description diffraction works (Bragg’s Law) Know how the safety hazards in the X-ray lab Collect the appropriate data for what you want to do Know what kind of data you can extract from a powder pattern Know a bit about crystal symmetry

Part 1: X-rays and Matter

X-rays Electromagnetic waves with wavelengths similar to the length of chemical bonds Produced in the lab by bombarding a metal target with electrons in a vacuum tube X-rays are produced by decay of electrons excited to higher energy levels in the target Several wavelengths from different transitions

X-rays continued The multiple wavelengths generated in the tube are selected using optics Simplest is a nickel foil to remove Kβ In our routine instrument a single wavelength (Kα1) is selected using a monochromator Single peak for each diffraction peak

X-ray diffraction When X-rays are incident with the crystal lattice of a solid they are diffracted by the planes of atoms in the lattice The diffracted beams form a diffraction pattern due to interference This phenomenon is described by the famous Bragg equation: nλ = 2dsinθ Where d is the lattice spacing, λ is the wavelength of the diffracted beam and θ is the diffraction angle

Powder diffraction A diffractogram is a plot of the intensities of the diffracted beams vs their diffraction angles There are several ways of measuring the intensities Random distribution of particle orientations is essential! Sample spinning helps with this

Diffractometer Set Ups
Simplest setup is Debye- Scherrer gemometry X-rays are fired at the sample and are diffracted as they pass through Diffracted beams are detected on the other side Commonly used in old film based diffractometers

Bragg Brentano Geometry
Most common geometry for modern powder diffraction with position sensitive detectors “Parafocussing” geometry, orientation of the focus circle changes with detector position This type of geometry is used on DIFF5 Videos?

Safety X-ray exposure can cause serious burns and radiation related diseases All the instruments in the lab have interlocked shielding to prevent you from being exposed to X-rays The routine instrument has a robotic sample changer with very powerful motors Do not get in the way of the sample changer or goniometer! Always alert the operators of the instrument to any serious chemical hazards from your samples Remember your risk assessments!

Part 2: Describing Crystal structures
How do we go from the basic unit to the crystal?

Symmetry!

The Unit Cell Smallest repeat unit of the crystal lattice
Not the smallest repeated structural motif (asymmetric unit)! Unit cells are classified according to their cell dimensions (a, b and c) and angles (α, β, γ) and lattice centring.

7 crystal systems: b ≠ 90o All crystalline compounds with long range order have a structure whose unit cell fits one of these 7 crystal systems

Lattices Identical mathematical points
A 3D unit cell describes their repetition in space The most simple lattice correspond to the 7 unit cells shown for the 7 crystal systems, with one lattice point in origin Are there more lattices (according to their definition of identical surroundings for all lattice points)? Yes… 14 altogether

Primitive and non-primitive unit cells
A primitive lattice contains one lattice point A non-primitive contains more than one lattice point: 2 or 4

Crystal systems and Bravais lattices
b ≠ 90o

Effects of Lattice centring
In a centred lattice some reflection classes systematically have zero intensity Systematic absences This is used in indexing to determine the lattice type

Origin of Systematic Absences
Caused by interference in the same way as diffraction Two sets of lattice planes oriented so that, although the Bragg condition is satisfied, they produce reflections which are 180° out of phase! 1 2 Path difference 1-3 = 2d sin (θ) 3 d 4 d/2 Path difference 1-2 or 3-4 = 2(d/2) sin (θ) e.g. BCC (1 0 0) reflection

Counting of atoms in 3D A corner-atom is shared between 8 cells Þ 1/8 atom per cell An edge-atom is shared between 4 cells Þ 1/4 atom per cell A face-atom is shared between 2 cells Þ 1/2 atom per cell An atom inside 1 cell Þ 1 atom per cell

How to describe a crystal structure?
Periodic lattice + Atoms (including symmetry operations) that follow the lattice

Symmetry operations Mirror plane m Rotation axis n (2,3,4,6)
Inversion axis n (1,2…) Centrosymmetry 1 Glide plane n, d, a, b, c Screw axis 21, 31 Point group symmetry Special symmetry operations involving translations Remember: Mirroring is a left – right hand operation

Mirror plane m

Rotation axis 4-fold rotation axis  = 360/n )

Rotation inversion axis n

Screw axis, Xy Translation: y/x Rotation: 360º/x
regular 6-fold rotation axis screw axis 65

Glide plane )

Space groups All 3-dimensional symmetry can be described by 230 Space groups Symmetry elements in each group and relationships between the groups are described in volume A of the International Tables for Crystallography

Part 2: Powder Diffraction

Powder diffraction vs Single Crystal
A diffractogram is a plot of the intensities of the diffracted beams vs their diffraction angles There are several ways of measuring the intensities Random distribution of particle orientations is essential! Sample spinning helps with this

Extracting data from the pattern
Peak positions: unit cell parameters, Miller indices, d-spacings of layers, fingerprint for phases, indexing, spacegroup determination Intensities: Atom types present in phases, site occupancies, thermal parameters, instrumental factors(!) Peak broadening: Instrumental factors, Crystallite size and strain, shape, structural defects

“Fingerprint” phase identification
Bragg’s law shows us that the diffraction pattern is very characteristic of the crystal lattice for a given phase We can use the diffraction pattern for phase identification Visual (if you know the pattern of your phase) Databases (COD, PDF2, ICDD FindIt) When we know the phase we can study it further…

Quantitative Analysis
If the sample contains more than one phase then the pattern corresponds to the weight percentage of each phase Can fit using intensities of single characteristic peaks Now easy to determine from full profile using software (TOPAS is excellent for this) Common application of XRD in industry Be aware that preferred orientation (see later) can cause problems

Peak positions: Miller Indices
These are labels for the planes of atoms in the lattice Each set of Miller planes gives rise to a peak in the diffraction pattern whose diffraction angle is related to the d-spacing by the Bragg equation Take the reciprocal of the fractional intercepts on each axis and write in round brackets i.e. - 1/2a, 1/2b, 1/2c - (222) plane (100), (010), (001) define a, b and c axes

Peak Positions: Miller Indices

Peak positions: Indexing and cell refinement
Assignment of Miller indices to a pattern according to a specific unit cell Can be done by hand for high symmetry examples Use of software is more common (TREOR, TOPAS, DICVOL etc) If we know the cell (from indexing or phase identification) we can refine the lattice parameters against the data

Peak positions: Lattice Parameter refinement
Changes in the lattice parameters can be used to study phase modification, e.g.: Substitution of atoms in a phase- the substituent usually has a different atomic radius and different bond lengths to the original atom Lattice parameters change with substitution Can also be used to look at vacancies Changes in peak intensities may also be observed (see later…) Least squares refinement methods are used this can be done with simple scripts and spreadsheets or complex software packages like GSAS and TOPAS

Peak Positions: Space Group Determination
Crystal symmetry is described by space groups These group lattices by the symmetry elements present Lattice centering Familiar mirror planes, inversion centres and rotation axes from molecular symmetry Translational symmetry (crystal lattices only): Screw axes and glide planes Centring and translational symmetry elements lead to further interference phenomena in the diffraction pattern: Systematic absences Systematic absences are used to determine the spacegroup from the diffraction pattern Difficult, limited number of observations in powder data (single crystal is easier) Usually done with software

Peak positions: Space Groups
Space group (Hermann-Maugin) names are symbols describing the symmetry elements E.g. P21/m (number 11): Primitive lattice with a 21 screw axis and a mirror plane The 230 3D space groups are listed in volume A of the International Tables for Crystallography which are available to us at UiO through an online site license

Peak Intensities More complex than peak positions
Many variables, some instrumental, some sample Instrument Radiation choice (λ in the Bragg equation) Instrument geometry and optics Lorentz polarisation Sample Atom type and position in the structure (scattering and structure factors) Thermal displacement parameters Absorption Site multiplicity (from space group) Sample preparation Poor prep gives poor data Preferred orientation

Peak Intensities: Atomic Scattering Factors
Each atom has a characteristic scattering curve for the radiation used, therefore the scattering is usually plotted against sinθ/λ The scattering factor decreases with θ Scattering increases with atomic number because X-rays are scattered by electrons Peak intensities are characteristic of the atoms present!

Peak Intensities: Site Occupancy
If you know the type of atom or atoms on a particular site you can study the occupancy If the site is occupied by two atoms with very different scattering factors (atomic number) you can work out the amount of each It may be possible to work out the level of vacancies on a site from the intensity

Peak Intensities: Thermal Displacement Parameters
AKA “temperature factors” and “Debye-Waller factors” Describe the thermal motion of the atoms Difficult to determine due to correlation with background (especially at high diffraction angles), site occupancy, adsorption and scattering factor

Peak Intensity: Adsorption
Intensity loss due to sample absorbing the x-ray beam Sample and geometry dependent For flat plate absorption is a linear function For cylindrical sample/Debye-Scherrer there is some angle dependence

Peak Intensity: Preferred Orientation
Random distribution of crystallites is assumed for powder diffraction But real crystallites can have shapes which do not lend themselves to this! E.g. Mica flakes- tend to lie flat on plate sample holders

Peak Intensity: Preferred Orientation
Preferred orientation can lead to changes in peak intensity as certain orientations are under represented in the sample Strong P.O. effects can lead to peaks being completely absent from the diffractogram Can be avoided by correct sample preparation Capillaries Roughened sample holders Can also be modelled in profile fits

Peak Shape: Size and Strain
Peak shapes are made up of two basic contributions: Instrumental (radiation, optics and geometry) Includes the bulk of asymmetric broadening Can be fitted with fundamental parameters of analytical peak shapes Sample (crystallite size and strain, defects, shape) Analysis can be complex Defect and shape analysis are beyond the scope of this course….

Peak Shape: Crystallite Size
Scherrer equation describes the relationship between peak broadening and crystallite size K: Constant (0.89) λ: Wavelength (Å) β: Broadening (rad) θ: Diffraction angle (rad) Beware! The Scherrer equation is not a panacea: Weight averaged size of the crystallites may not represent the real distribution of sizes (Fourier transform methods can help) Assumes spherical crystallites- real crystals have many shapes (this can be fitted with more complex methods) Strain can also cause similar broadening effects (this can in some cases be deconvoluted) Strongly dependent on the quality of peak fit

Peak Shape: Strain Strain (simplified) εstr = β/{4 tan θ}
Slightly different broadening effect to crystallite size

Sample Preparation Essential for reliable PXRD data!
Bad sample preparation can lead to: Incorrect peak positions Bad peak shapes Incorrect intensities 5 minutes of sample preparation can save hours of work identifying and fitting phases from bad data!

Sample Preparation Correct preparation “Powder mountain”
Sample holder must be full, with powder homogeneously packed Top of sample must be flat and level with sample holder Correct preparation Correct peak positions Correct peak intensities Sharp peaks “Powder mountain” Wrong peak positions Wrong peak intensities Wrong peak shapes “Powder Valley” Wrong peak positions Wrong peak intensities Broad/double peaks

Sample preparation Correct Powder valley Powder mountain

Preparing a Sample Correctly
well-type sample holders Fill slightly above the top press down with a glass slide to give a flat even surface at the level of the sample holder surface Plate type sample holders Suspend your powder in isopropanol Carefully spread your sample/solvent suspension on the plate to give a flat surface Always make sure your sample is flat!

Choosing your data collection Strategy
Using the correct data collection strategy to get the data you need from your sample saves time for you and on the instrument Increasing the beam size on the sample means greater intensity and better signal to noise (count statistics) you may get some extra background if your sample does not cover the whole sample holder V10, V20 in program name = size of beam on sample in mm Step size Number of points where the detector stops and counts to form the pattern You should try to have more than 5 points across the top half of your peaks If you have nanoparticle samples the peaks will be broader, so you can use a larger step and count for longer in the same instrument time! If you are uncertain or think you need a new program for your sample ask! David-RECX

Setting the parameters in FP
Anette

Data Collection Phase identification
What have I made? Is it crystalline? Don’t need amazing data for this! Collect your data with a short timescale (1-3), large beam size on the sample Measurements with a timescale of 1 can be measured as “priority” samples for a fast turnaround

Data Collection Looking for impurities and mixtures
Does my sample contain other minor phases? Need to be able to see the weaker peaks! Longer timescale (3-7), large beam size on the sample

Data Collection Refinement of structural parameters
I need to get structural parameters from my sample. Need high quality data with a large angular range Longer timescale (6-10; longer measurements), Choose beam size to match sample, measure to high angles (at least 80 ° 2θ)

Data Collection Nano samples (broad peaks)
I want to get the best possible intensities from poorly crystalline samples, preferably without counting forever! Need good intensity but not high peak resolution Choose a program with a larger step size! “Nanosample_programs” folder

Normal, sharp diffraction peak
Nano samples Step size gives 4 data points above FWHM FWHM 0.075° Normal, sharp diffraction peak

Nano Samples Step size of 0.02 ° gives many points above FWHM
High noise

Nano Samples Larger step (0.1°) allows longer counting with same overall measurement time Much lower noise! And better fit!

Data Collection Film samples
I want to look for small peaks from a thin film Need to count for a relatively long time to see the weak peaks, BUT remember to think about what you intend to use your data for! Start with a quick scan to find where your peaks are If you don’t need to get real structural information from your pattern don’t count for too long! Get some idea of what you have made and use a program which measures the range you expect to find your peaks in

Fluorescence – example
Highly increased background Fluorescence by Co compared with Ni when using CuKa1 radiation Fluorescence reduces the signal-to-noise ratio

Fluorescence Fluorescence occurs when electrons are excited to high energy states by incoming X-rays As the electrons return to their normal energy levels energy is given out as fluorescence Lower energy than the incident X-rays Co Kα transition (6.9 KeV) is just below Cu Kα (8 KeV) Easily excited by Cu radiation Fe Kα (6.4 KeV) is also excited by Cu radiation

Dealing with Fluorescence
DIFF5 (Routine XRD) has electronic fluorescence suppression for Fe Measurement programs in “fluorescence supress” folder RECX2 : Mo radiation Fluorescence from Co is no longer excited. RECX1: Full electronic fluorescence suppression for Co and Fe. You will see how Fluorescence is supressed in XRD lab on a Rigaku instrument in FP

Summary What are X-rays? X-ray safety Basics of Crystal Symmetry
Correct sample preparation Information found in the powder pattern Some common problems Collecting the right data

KJM-MENA 4010 – Module 13 Powder X-ray Diffraction Theoretical Basis

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