Crystallography and Diffraction. Theory and Modern Methods of Analysis Lectures 9-10 Powder Diffraction Dr. I. Abrahams Queen Mary University of London Lectures co-financed by the European Union in scope of the European Social Fund

Powder Diffraction In order to observe a diffracted X-ray beam from a single crystal, the crystal must be rotated with respect to the incident X-ray beam so that a particular set of hkl planes makes the correct angle  hkl with the incident beam so as to satisfy Bragg’s law. Crystalline powders contain large numbers of small crystallites, each of which is randomly orientated. Therefore at any one time all the allowed hkl planes will be correctly orientated in the X-ray beam. i.e for every possible hkl some crystals will make the correct  hkl for diffraction to occur Crystal 1 shows the (002) sets of planes but is not correctly aligned with the incident beam 2 shows the same crystal with the parallel set of (001) planes which are correctly aligned, so diffraction occurs 3 shows another crystal where the (002) planes are correctly aligned so diffraction occurs.

Lectures co-financed by the European Union in scope of the European Social Fund Powder Diffractometer In the laboratory X-ray powder method a single wavelength of collimated X-ray beam is incident to the powder. The diffracted beams are subsequently observed by radiation detectors. The resulting diffracted beams can be measured and the corresponding d- spacings calculated. The final results are essentially intensity versus d- spacings (or 2  ). Modern diffractometers are computer controlled and allow for fast and efficient data collection and subsequent analysis.

Lectures co-financed by the European Union in scope of the European Social Fund Powder Diffractometer

Lectures co-financed by the European Union in scope of the European Social Fund Modern PanAlytical X’Pert Pro diffractometer with Bragg-Brentano  /  focusing geometry.

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Powder Neutron Diffraction Neutrons with wavelengths of around 1 Å can be used for diffraction purposes. Neutrons are scattered by nuclei and this has important consequences for the nature of the diffraction data obtained. Some physical properties of thermal neutrons The de Broglie equation relates to the neutron mass and velocities to wavelength. Therefore, the neutrons generated from reactors or pulsed sources at certain speeds can be in useful ranges for diffraction experiments.

Lectures co-financed by the European Union in scope of the European Social Fund The scattering power (scattering length b for neutrons or scattering factor f for X-rays) of an atom towards neutrons is different than it is towards X-rays. This is mainly because neutrons are scattered through interaction with atomic nuclei rather than atomic electrons as occurs with X-rays, making the relationship between the neutron scattering length of an atom and its atomic number weak

Lectures co-financed by the European Union in scope of the European Social Fund In systems containing light and heavy atoms X-ray scattering will be dominated by scattering from the heavy atoms, while neutron scattering does not show this correlation and it is often easier to locate light atoms in the presence of heavier ones. Able to distinguish between neighbouring elements in the periodic table such as manganese and iron or cobalt and nickel. Isotopic substitution experiments possible as scattering lengths of isotopes differ. Less dependence of scattering on Bragg angle, , leading to greater intensity at higher angles. Advantages of neutron diffraction

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Ref: Neutron Diffraction, Bacon Clarendon Press 1975.

Lectures co-financed by the European Union in scope of the European Social Fund Neutron diffraction is also a powerful technique for the study of magnetic structure. Neutrons possess a spin of ½ and therefore, have a magnetic dipole moment that can interact with unpaired electrons (mostly in d or f orbitals) giving rise to an additional scattering effect. In diffraction experiments, where magnetic ordering may occur over different length scales to the crystallographic ordering, this gives rise to magnetic superlattice peaks in the diffraction pattern.

Lectures co-financed by the European Union in scope of the European Social Fund e.g. Antiferromagnetic transition in MnO Ref: Solid State Chemistry and its applications A.R. West, Wiley 1984.

Lectures co-financed by the European Union in scope of the European Social Fund

Neutron sources Two main types of neutron sources. 1. Reactor Sources e.g. ILL High Flux reactor source. Neutrons produced as a product of the fission of 235 U. Neutrons can then be moderated to give a range of energies and then conducted to different instruments via guide tubes. Diffraction experiments are normally at a constant wavelength, but can also be energy dispersive. 2. Pulsed Sources e.g. ISIS Proton spallation source. Pulses of protons are accelerated in a synchrotron and then fired at a heavy metal target (tantalum in this case) to generate neutrons. These can be moderated or used directly. A Maxwellian distribution of energies is produced. Diffraction experiments normally use a large range of energies (time of flight method).

Lectures co-financed by the European Union in scope of the European Social Fund The time of flight method Schematic diagram of the time-of-flight powder diffractometer The distance between the moderator and the sample is L 0, with the detector located at a distance L 1 from the sample. A variable wavelength/fixed angle scan is used. The variation of wavelength arises due to the time distribution of neutrons arriving at the detector following the initial pulse and hence this is known as the time-of-flight method. For high resolution L 0 is large  greater distribution of energies.

Lectures co-financed by the European Union in scope of the European Social Fund The neutron velocity is given by: Combining the de Broglie and Bragg equations: Thus

Lectures co-financed by the European Union in scope of the European Social Fund t.o.f d-spacing

Lectures co-financed by the European Union in scope of the European Social Fund Multiplicities In powder diffraction because three-dimensional space is reduced to one dimension in the powder pattern then if two sets of planes have the same d-spacing they will both contribute to the same peak. e.g. For cubic crystals Therefore multiplicity = 6 Two different reflections (not hkl permutations) may also have the same d-spacing e.g. In cubic crystals

Lectures co-financed by the European Union in scope of the European Social Fund Characterisation of Materials by Powder Diffraction X-ray powder diffraction is a fundamental tool for materials characterisation. e.g. (1) Qualitative analysis (2) Unit cell refinement (3) Structure refinement (4) Structure determination (5) Quantitative analysis (6) Phase transition studies (7) Superlattice identification (8) Kinetic studies (9) Crystallite size determination (10) Strain analysis (11) Preferred orientation (Texture)

Lectures co-financed by the European Union in scope of the European Social Fund Qualitative Analysis The primary role of the majority of X-ray powder diffractometers is phase identification. This relies on the fact that every unique crystal structure has its own unique powder pattern, i.e. its own fingerprint. PDF The Powder Diffraction File (PDF-2, PDF-3 or PDF-4) represents the most complete database of powder diffraction data. The database can be searched in a number of ways by a variety of commercial search engines. A typical search would involve the loading of a powder pattern and automatic measurement of the diffraction peaks. These would then be compared to those in the database and matches ranked according to how good the match is. To narrow the search it is advisable to restrict the search to compounds containing only those elements you know were in the reaction mixture.

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Crystallographic Databases ICSD ICSDInorganic Crystal Structure Database ca. 50,000 Inorganic compounds CSD CSD Cambridge Structural Database, Organic/Organometallic Structures > 100,000 compounds CDIF CDIFCrystal class and unit cell data PDB PDB Protein database MDF MDFMetals Data file

Lectures co-financed by the European Union in scope of the European Social Fund ICSD on the Web

Lectures co-financed by the European Union in scope of the European Social Fund

Using crystallographic data, powder diffraction patterns may be calculated using a variety of software: e.g. LAZY-PULVERIX POWDER-CELL

Lectures co-financed by the European Union in scope of the European Social Fund Unit cell refinement Assuming that you have been able to identify phases and index the powder pattern, it is often useful to refine unit cell dimensions. This is particularly important if you want to identify solid-solutions. (1) Measure the peaks accurately (2) Index the peaks (3) Refine the unit cell dimensions and zero-point correction. There are a number of programs available to do this. However they all rely on accurate peak measurement. Some incorporate automatic peak measurement Often where the structure is known you are better off carrying out a Rietveld refinement. Alternatively a LeBail type fit can be carried out.

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e.g. Thermal variation of cubic lattice parameter in Bi 3 YO 6

Lectures co-financed by the European Union in scope of the European Social Fund Structure Refinement The Rietveld method is a structure refinement technique for powder diffraction data which fits the whole powder pattern including peak shapes and background. This will be dealt with in the next set of lectures.

Lectures co-financed by the European Union in scope of the European Social Fund Structure Determination For structure refinement an initial structural model is required before refinement can proceed. In structure determination the initial model is found using an ab-initio approach. There are a number of stages in structure determination (1) Unit cell determination and indexing (2) Space Group identification (3) Intensity extraction (4) Initial model determination (5) Rietveld refinement (6) Difference Fourier maps generated to locate remaining atoms (7) Final Rietveld Refinement

Lectures co-financed by the European Union in scope of the European Social Fund Unit cell determination and indexing There are a number of methods for indexing. All require accurate measurement of diffraction peaks. Consider a cubic unit cell. You will recall: Let n = h 2 + k 2 + l 2 In a cubic system only certain values of n are possible

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We may express Bragg’s law in the form: Since for a cubic cell: Substituting: Now sinceIs a constant for a particular cell then on dividing sin 2  by the allowed values of n the value of the constant should be evident and hence the value of a derived. Examples to be given in workshops

Lectures co-financed by the European Union in scope of the European Social Fund The same rules can be applied for tetragonal and orthorhombic cells, but it gets more complicated. For tetragonal cells we have two constants: Allowed values of h 2 +k 2 are 1,2,4,5,8… To can obtain C we can use the differences and look for Cl 2 values in the correct ratios.

Lectures co-financed by the European Union in scope of the European Social Fund Today, this type of indexing is done automatically Typically 20 peaks or more are used. These should include all the high d-spacing peaks. The most popular programs are Visser ITO TREOR DICVOL Typically these programs will offer a number of possible solutions.

Lectures co-financed by the European Union in scope of the European Social Fund Most automatic indexing programs work with reciprocal space Q values where: Thus in the general triclinic case there are 6 unknowns in reciprocal space: Two basic approaches are used to solve this equation. The Shirley method works through the different crystal systems starting with cubic and increments the lattice parameters until a solution is found. The ITO method starts with triclinic symmetry and tries to find a solution and work out the higher symmetry from this.

Lectures co-financed by the European Union in scope of the European Social Fund TREOR output for bismuth zirconium vanadate

Lectures co-financed by the European Union in scope of the European Social Fund Once a unit cell is found one must decide whether it is the correct cell. In order to do this one must determine the reduced cell (the simplest primitive unit cell) and then see if there are any other higher symmetry cells. For every normal unit cell there may be many alternative cells, but for all of these there is only one reduced cell. This is done routinely in single crystal diffraction but often forgotten in powder diffraction. Today there automatic procedures for calculating the reduced cell. Le-Page is a freely available software package that will calculate the reduced cell and examine the parameters for higher metric symmetry.

Lectures co-financed by the European Union in scope of the European Social Fund Le-Page screen for MgO showing reduced cell for cubic F-centred cell.

Lectures co-financed by the European Union in scope of the European Social Fund Space Group Assignment Space group assignment is carried out using systematic absences in the reflection data. Intensity Extraction Intensity extraction is usually carried using either the method of Pawley or Le-Bail. These will be dealt with in the next lectures. Both methods use a Rietveld-like approach in that they fit the whole pattern, but with no structural model. These methods rely on accurate high resolution data since peak overlap needs to be minimised. The result is a set of quasi-single crystal data which can be analysed using standard single crystal packages.

Lectures co-financed by the European Union in scope of the European Social Fund Determination of Initial Model Three methods are now commonly used. (1) Patterson vector density methods (2) Direct methods (3) Probabilistic methods The first two are identical to the single crystal methods already discussed.

Lectures co-financed by the European Union in scope of the European Social Fund Quantitative Analysis For mixtures of phases X-ray powder diffraction can be used to quantify the different fractions present. In some cases it may be the only method possible for example a mixture of different phases of silica. There are a number of well established methods. 1. Standard Additions (Lennox, 1957) 2. Absorption diffraction (Alexander and Klug, 1948, Smith et al 1979) 3. Internal standard (Klug and Alexander, 1974) Multiphase refinement using the Rietveld method now offers a quick and accurate way to determine phase fractions through multi-phase refinement. However: Accuracy is a problem. Remember the sensitivity of X-ray diffraction methods is estimated to be  5% error.

Lectures co-financed by the European Union in scope of the European Social Fund In a multiphase Rietveld analysis individual scale factors S i are refined for each component i. The weight fraction W i of each component is given by: Where  is the density and V is the unit cell volume.

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Phase Transition Studies X-ray powder diffraction is ideally suited for examining crystallographic phase transitions Both compositional and temperature dependence can be examined.    BIMEVOX

Lectures co-financed by the European Union in scope of the European Social Fund Superlattices Powder diffraction is often superior to single crystal diffraction in identifying superlattice reflections. Superlattices occur when part of a structure orders on a larger length scale than the original subcell. This ordering may be 1-, 2- or 3- dimensional and may be commensurate or incommensurate. In diffraction data a superlattice manifests itself in the form of additional peaks.

Lectures co-financed by the European Union in scope of the European Social Fund Superlattice peaks in  -BiMgVOX a  = 6 a m b  = b am c  = c am In commensurate supercells, the supercell has a periodicity that is a simple integral multiple of the subcell. e.g.  -BIMEVOXes

Lectures co-financed by the European Union in scope of the European Social Fund 3-D incommensurate modulation in Bi 3 TaO 7 Incommensurate supercells show a periodicity that is not a simple integral multiple of the subcell. In these cases structural analysis from powders is very difficult.   0.38 Ref: I. Abrahams, F. Krok, M. Struzik and J. R. Dygas, Solid State Ionics, 179 (2008) 1013.

Lectures co-financed by the European Union in scope of the European Social Fund Kinetic Studies Depending on the reaction rate kinetic studies can be carried out using diffraction methods. For fast reactions high flux instruments are required. e.g. Kinetic studies using time-resolved X-ray diffraction on ID11 at ESRF H. Spiers et al. J. Mater. Chem. 14 (2004) 1104 Formation of Mg 0.5 Zn 0.5 Fe 2 O scans of 0.06 s acquisition time Reaction occurs over a period of ca. 0.7 s

Lectures co-financed by the European Union in scope of the European Social Fund Kinetic Studies e.g. Ageing effects in BIMgVOX.07 using conventional X-ray diffraction Annealing of a sample of Bi 2 Mg 0.07 V 0.93 O at 470  C. Change from  to . ref: M.Malys, W.Wrobel, F.Krok, I.Abrahams, M.Hołdyński, J.R. Dygas in prepraration

Lectures co-financed by the European Union in scope of the European Social Fund Crystallite size The peak width of a diffraction peak is related to the coherence length of the lattice. i.e. the crystallite size. Essentially this means that the smaller the crystallite size the broader the peaks. The Scherrer equation can be used to estimate crystallite size. Where is the wavelength,  B is the Bragg angle of the peak being measured and B is the line broadening in radians. The value of B can be estimated by measuring the peak width at half height of the sample and a well crystallised reference.

Lectures co-financed by the European Union in scope of the European Social Fund e.g. the peak width at half height of sample peak at  B = 21.5  measured using Cu K  radiation ( = Å) was radians. The peak width at half height of a well crystallised sample of KCl at a similar value of  was radians. Å radians

Lectures co-financed by the European Union in scope of the European Social Fund Strain Analysis Strain can manifest itself in two types of ways in diffraction patterns 1. Macrostress Uniform compressive or tensile stress. Lattice parameters get larger or smaller resulting in peak shift 2.Microstress Distributions of compressive and or tensile stresses. Results in a distribution of d-values about the normal i.e. peak broadening. Residual microstresses result mainly from dislocations, but also from vacancies, defects, shear planes, thermal expansion and contraction etc.

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Like particle size broadening strain broadening   has a Bragg angle dependence. However since particle size shows a cos  dependence they can be separated. Where  is the residual strain. The line broadening  is therefore the sum of the particle size broadening and the strain broadening terms. Lorentzian broadening Gaussian broadening  and  can be separated using a Williamson-Hall plot. i.e. a plot of  cos  vs 4  sin  gives a straight line of slope  and y-intercept of / 

Lectures co-financed by the European Union in scope of the European Social Fund Preferred orientation: texture analysis At the beginning of these lectures we described the powder diffraction experiment as occurring on a powder containing many randomly orientated crystallites. In some cases this distribution of crystallites is quasi- or non-random. For example where the crystallites show a needle like morphology, it is likely that the needles will lie flat on the specimen holder. So if the needle axis was coincident with the c-axis for example, 00 l peaks would be enhanced or suppressed with respect to the other hkl peaks depending on whether reflection or transmission geometry was used.

Lectures co-financed by the European Union in scope of the European Social Fund Sample of Fe 2 (CO) 9 showing preferred orientation. Upper reflection mode, lower transmission mode.

Lectures co-financed by the European Union in scope of the European Social Fund Preferred orientation can be modelled in Powder diffraction patterns by a number of methods. For neutron diffraction where the sample is contained in a cylindrical can, the March-Dollase model is relatively effective and assumes that crystals are either rod or disc shaped. The correction O h Where, R 0 is a refineable coefficient, M is the number of reflections equivalent to vector h and  j is the angle between the preferred orientation direction and the reflection vector h. R 0 = 1 no preferred orientation R 0 1 preferred orientation present

Lectures co-financed by the European Union in scope of the European Social Fund Preferred orientation is particularly important in metals (sheets and wires) and also in thin film technology. Analysis of texture involves the use of a texture cradle, which allows for rotation of the sample about its axes. This allows the intensity of a reflection as a function of sample orientation. The detector is set to a particular 2  value so as to correspond to the particular reflection to be studied, and the sample rotated through the Debye ring. 111 Pole figure of Brass Ref: Elements of X-ray Diffraction, B.D. Cullity,Addison Wessley Publishing Co 1956.

Lectures co-financed by the European Union in scope of the European Social Fund Consider a sheet of a cubic metal containing only 10 grains. If we plot the 100 intensity as a function of the sample rotation angle we would get a random distribution of spots if the grains were randomly orientated, or would see them clumped together if there was preferred orientation. Note: because the crystal is cubic we see 3 x 10 spots for 100, 010 and 001

Lectures co-financed by the European Union in scope of the European Social Fund Eulerian texture cradle (Bruker AXS) Pole Figure (Univ. of Cambridge)