Introduction to X-Ray Powder Diffraction Data Analysis Mohammad Aminul Islam PhD Student Solar Energy Research Institute (SERI),UKM Supervisors Prof. Dr. Nowshad Amin Dr. Mohamad Yusof Sulaiman

M.A.Islam, SERI, UKM Röntgen publishes the discovery of X-rays Laue observes diffraction of X-rays from a crystal A Brief History of XRD

An X-ray diffraction pattern is a plot of the intensity of X-rays scattered at different angles by a sample The detector moves in a circle around the sample – The detector position is recorded as the angle 2theta (2θ) – The detector records the number of X- rays observed at each angle 2θ – The X-ray intensity is usually recorded as “counts” or as “counts per second” To keep the X-ray beam properly focused, the sample will also rotate. – On some instruments, the X-ray tube may rotate instead of the sample. M.A.Islam, SERI, UKM3   X-ray tube Detector sample Position [°2Theta] (Cu K-alpha) Intensity (Counts)

Each “phase” produces a unique diffraction pattern A phase is a specific chemistry and atomic arrangement. Quartz, cristobalite, and glass are all different phases of SiO 2 – They are chemically identical, but the atoms are arranged differently. – As shown, the X-ray diffraction pattern is distinct for each different phase. – Amorphous materials, like glass, do not produce sharp diffraction peaks. M.A.Islam, SERI, UKM4 Position [°2Theta] (Cu K-alpha) Quartz Cristobalite Glass The X-ray diffraction pattern is a fingerprint that lets you figure out what is in your sample.

The diffraction pattern of a mixture is a simple sum of the diffraction patterns of each individual phase. From the XRD pattern you can determine: – What crystalline phases are in a mixture – How much of each crystalline phase is in the mixture (quantitative phase analysis, QPA, is covered in another tutorial) – If any amorphous material is present in the mixture M.A.Islam, SERI, UKM Position [°2Theta] (Cu K-alpha) Quartz Cristobalite Glass Position [°2Theta] (Copper (Cu)) Mixture

Qualitative Analysis of XRD Data M.A.Islam, SERI, UKM6

Experimental XRD data are compared to reference patterns to determine what phases are present The reference patterns are represented by sticks The position and intensity of the reference sticks should match the data – A small amount of mismatch in peak position and intensity is acceptable experimental error M.A.Islam, SERI, UKM7

Specimen Displacement Error will cause a small amount of error in peak positions M.A.Islam, SERI, UKM8 Peaks that are close together should be shifted the same direction and by the same amount The peak shift follows a cosθ behavior, so peak shift might change direction over a large angular range

Most diffraction data contain K-alpha1 and K- alpha2 peak doublets rather than just single peaks The k-alpha1 and k-alpha2 peak doublets are further apart at higher angles 2theta The k-alpha1 peaks always as twice the intensity of the k-alpha2 At low angles 2theta, you might not observe a distinct second peak M.A.Islam, SERI, UKM9 K-alpha1 K-alpha2 K-alpha1 K-alpha2 K-alpha1 K-alpha2

Kicker Box The X-ray diffraction pattern is a sum of the diffraction patterns produced by each phase in a mixture M.A.Islam, SERI, UKM10 Each different phase produces a different combination of peaks. The pattern shown above contains equal amounts of TiO 2 and Al 2 O 3 The TiO 2 pattern is more intense because TiO 2 diffracts X-rays more efficiently

Kicker Box Diffraction peak broadening may contain information about the sample microstructure Peak broadening may indicate: – Smaller crystallite size in nanocrystalline materials – More stacking faults, microstrain, and other defects in the crystal structure – An inhomogeneous composition in a solid solution or alloy However, different instrument configurations can change the peak width, too M.A.Islam, SERI, UKM11 These patterns show the difference between bulk compound (blue) and nanocrystalline compound (red) These patterns show the difference between the exact same sample run on two different instruments. When evaluating peak broadening, the instrument profile must be considered.

M.A.Islam, SERI, UKM12 Crystallite Shape Though the shape of crystallites is usually irregular, we can often approximate them as: sphere, cube, tetrahedra, or octahedra parallelepipeds such as needles or plates prisms or cylinders Most applications of Scherrer analysis assume spherical crystallite shapes If we know the average crystallite shape from another analysis, we can select the proper value for the Scherrer constant K Anistropic peak shapes can be identified by anistropic peak broadening if the dimensions of a crystallite are 2x * 2y * 200z, then (h00) and (0k0) peaks will be more broadened then (00l) peaks.

M.A.Islam, SERI, UKM13 Anistropic Size Broadening The broadening of a single diffraction peak is the product of the crystallite dimensions in the direction perpendicular to the planes that produced the diffraction peak.

Quantitative Analysis of XRD Data M.A.Islam, SERI, UKM14

Diffraction peak positions can be used to calculated unit cell dimensions The unit cell dimensions can be correlated to interatomic distances Anything the changes interatomic distances- temperature, subsitutional doping, stress- will be reflected by a change in peak positions M.A.Islam, SERI, UKM deg d= Å deg d= Å Brag’s Scattering Law

M.A.Islam, SERI, UKM16 To calculate unit cell lattice parameters from the diffraction peak positions

Crystallite Size Broadening M.A.Islam, SERI, UKM The Scherrer Equation was published in 1918 Peak width (B, FWHM) is inversely proportional to crystallite size (D), K is the Scherrer Constant. Peak Width due to crystallite size varies inversely with crystallite size – as the crystallite size gets smaller, the peak gets broader The peak width varies with 2  as cos  – The crystallite size broadening is most pronounced at large angles 2Theta However, the instrumental profile width and microstrain broadening are also largest at large angles 2theta peak intensity is usually weakest at larger angles 2theta – If using a single peak, often get better results from using diffraction peaks between 30 and 50 deg 2theta below 30deg 2theta, peak asymmetry compromises profile analysis

The Scherrer Constant, K The constant of proportionality, K (the Scherrer constant) depends on the how the width is determined, the shape of the crystal, and the size distribution – the most common values for K are: 0.94 for FWHM of spherical crystals with cubic symmetry 0.89 for integral breadth of spherical crystals w/o cubic symmetry 1, because 0.94 and 0.89 both round up to 1 – K actually varies from 0.62 to 2.08 For an excellent discussion of K, refer to JI Langford and AJC Wilson, “Scherrer after sixty years: A survey and some new results in the determination of crystallite size,” J. Appl. Cryst. 11 (1978) p M.A.Islam, SERI, UKM

Remember, Crystallite Size is Different than Particle Size A particle may be made up of several different crystallites Crystallite size often matches grain size, but there are exceptions

Microstrain Broadening lattice strains from displacements of the unit cells about their normal positions often produced by dislocations, domain boundaries, surfaces etc. microstrains are very common in nanocrystalline materials the peak broadening due to microstrain will vary as: M.A.Islam, SERI, UKM

Contributions to Microstrain Broadening M.A.Islam, SERI, UKM21 Non-uniform Lattice Distortions Dislocations Antiphase Domain Boundaries Grain Surface Relaxation Other contributions to broadening – faulting – solid solution inhomogeneity – temperature factors

M.A.Islam, SERI, UKM22 Williamson Hull Plot y-intercept slope FW(S)*Cos(Theta) Sin(Theta) Williamson-Hull plot are used for Size & Strain Analysis

M.A.Islam, SERI, UKM23 Williamson Hull Plot Strain type -Tensile strain -Compressive strain Tensile strain Compressive strain

Dislocations Broadening M.A.Islam, SERI, UKM24 Line broadening due to dislocations has a strong hkl dependence. The dislocation densities of thin films are calculated by the Williamson and Smallman’s relation. δ = n / D 2  Sometimes growth mechanism leads to the dislocation.  Dislocation indicating the imperfection in a crystal associated with miss registry of the lattice in one part of the crystal with respect to another part.

Texture Coefficient Broadening  Quantitative information related to the preferential crystallite orientation of the crystallites along a crystal plane (hkl) in a thin film can be described by the term texture coefficient given in references. TC i (hkl) = N[{I i (hkl)/ I o (hkl)}/∑ N i=1 { I i (hkl)/ I o (hkl)}] Where, TC i is the texture coefficient of the plane i, I i is the measured integral intensity, I 0 is the integral intensity of the JCPDS powder diffraction pattern of the corresponding peak i and N is the number of reflections considered for the analysis.  The value TC(hkl) ≤ 1 represents the films with randomly oriented crystallites,  while values TC(hkl) ≥ 1 indicates preferred orientation of the crystallites in that particular direction. M.A.Islam, SERI, UKM

 The degree of preferred orientation of the sample as a whole was measured by the standard deviation ‘σ’ for the samples. Standard Deviation Broadening σ (hkl) = √{∑ N i=1 [TC i (hkl)- TC io (hkl)] 2 /N} where TC i0, is the texture coefficient of the powder sample which is always in unity.  The value of σ is an indicator of the degree of orientation of a sample.  The decreasing trend of σ values indicate the increase of randomization of the sample.  A value of σ = 1 indicate the completely randomly orientated sample.

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