XRD meebehheheehhehehehehehehehthod.pptx

Xrd

1. XRD method mgn

2. X-ray diffraction and structural analysis In 1885, Wilhelm Conrad Rontgen, a German physicist, discovered X-rays. These are electromagnetic radiations with a wavelength ranging from 0.01 to 100 Å, which corresponds to the frequency in the range of 30 PHz - 300 EHz.  What does X in X-ray represents? X-stands for un-known.

3. Diffraction Methods •  Radiation sources other than X-Rays such as neutrons or electrons can also be used in crystal diffraction experiments.   • The physical basis for the diffraction of electrons or neutrons is the same as that for the diffraction of X-Rays. The only difference is in the mechanism of scattering. The typical wavelength and energy involved for various sources are given

4. Wavelength range of X-rays in Electromagnetic spectrum Figure depicts the spectral range of electromagnetic waves . The wavelength of  X -rays is longer than gamma rays but shorter than ultraviolet rays . These rays are widely used for diagnostic radiography and crystallography, covering an extremely wide area.

5. Generation of X-rays In general, X-rays are generated either by X-ray tube or by synchrotron radiation. The  X-ray tube is the primary X-ray source used in laboratories , whereas the synchrotron radiation needs a large-scale accelerator for obtaining high-energy radiation. Figure demonstrates the X-rays induced by colliding high-energy thermal electrons with a metallic target (water cooled) such as copper, molybdenum, etc., which can emit strong X-rays of 1.541 Å, 0.71 Å, respectively. 

6. The emission of different X-rays from the copper metal is shown schematically in Figure

7. X-ray sources with different λ for doing XRD studies Mo Target impacted by electrons accelerated by a 35 kV potential shows the emission spectrum as in the figure The high intensity nearly monochromatic K x-rays can be used as a radiation source for X-ray diffraction (XRD) studies  a monochromator can be used to further decrease the spread of wavelengths in the X-ray

8. Diffraction Basics Father Sir William Henry Bragg (left) and Son Sir William Lawrence Bragg (right)Father Sir William Henry Bragg and Son Sir William Lawrence Bragg proposed that Bragg's diffraction occurs when electromagnetic radiation or subatomic particle wave with a wavelength comparable to atomic spacing is incident upon a crystalline sample and scattered in a specular fashion by the atoms in the system and undergoes a constructive interference. For a crystalline solid, the waves are scattered from lattice planes separated by the inter-planar distance  d . When the scattered waves interfere constructively, they remain in phase. Since the path length of each wave is equal to an integer multiple of the wavelength. The path difference between the two waves undergoing constructive interference is given by 2 d sin θ , where θ  is the scattering angle . This leads to Bragg's law, which describes the condition for constructive interference from successive crystallographic planes ( hkl  ) of the crystalline lattice.

9. Diffraction Basics Constructive interference: only occurs for certain  θ correlating to a ( hkl   ) plane, specifically when the path difference is equal to  n  wavelengths, as schematically shown in Figure.

10. X-ray diffractometer The typical set up of X-ray diffractometer geometry and its components are shown in Figure

11. X-ray diffractometer

12. X-ray diffraction and structural analysis: Constructive Interface The waves scattered from two successive planes interfering constructively . Figure demonstrates the interference of two waves constructively, when the path difference matches with wavelength . Since  n  = 1, the path difference should match with one wavelength.

13. Schematic of path difference between more planes Assuming that path difference of λ gives constructive interference: Similar to the path difference of λ, path difference of 2λ, 3λ …  nλ  also constructively interfere, as displayed in Figure. All Constructive interference: The path difference between Ray-1 and Ray-2 is λ then the path difference between Ray-1 and Ray-3 is 2λ and Ray-1 and Ray-4 is 3λ, etc.

14. Schematic of path difference going across planes.

15. Destructive Interference Visualize the exact destructive interference (between two planes, with a path difference of λ/2)

16. Destructive Interference

17. Destructive Interference At a different angle  θ ’, the waves scattered from two successive planes interfere (nearly) destructively. The overall wave nature of the incident and reflected waves are shown

18. Powder diffraction • The identification of crystal structure of unknown materials has been performed usually with a powder XRD. Powders, containing aggregation of fine grains with a single crystal structure, but oriented randomly, when exposed to X-rays. The diffraction intensity is assumed to be the sum of the X-rays reflected from all the fine grains. Hence, we can observe the diffraction peak attributed to Miller indices of the powders.  For known materials, there is a database maintained by the International Center for Diffraction Data (ICDD) or Joint Committee on Powder Diffraction Standards (JCPDS) for the identification of inorganic and organic materials with a crystal structure. One can identify the unknown materials by comparing their diffraction patterns and the database. In addition, XRD patterns can be used for ( i ) identification of crystalline phases and their degree of crystallization, (ii) estimating crystallite size and (iii) residual strain.

19. Use of XRD • Phase Composition of a sample:  – Quantitative Phase Analysis can determine the relative amounts of phases in a mixture by referencing the relative peak intensities  • Unit cell lattice parameters and Bravais lattice symmetry  – Index peak positions – Lattice parameters can vary and thereby giving the information about, alloying, doping, solid solutions, strains, etc. • Residual Strain ( macrostrain ) • Crystal Structure – By Rietveld refinement of the entire diffraction pattern • Epitaxy (growth of crystal) /Texture/Orientation • Crystallite Size and Microstrain – Indicated by peak broadening  – Other defects (stacking faults, etc.) can be measured by analysis of peak shapes and peak width • The above factors can be also determined as a function of time, temperature, and gas environment in some of the advanced XRD having in-situ heating and controlled atmosphere facilities.

20. Phase identification For XRD pattern of a unknown sample, the process of comparing with the database and identifying the sample structure after matching with the database. This process is almost similar to the one done for finger print search. The diffraction pattern for every phase is as unique as our fingerprint Phases with the same chemical composition can have drastically different diffraction patterns. Use the position and relative intensity of a series of peaks to match experimental data to the reference patterns in the database

21. Phase identification

22. X-ray diffraction Patterns for different structures Single crystal A single crystal specimen in a Bragg-Brentano diffractometer would produce only one family of peaks in the diffraction pattern as shown in Figure

23. X-ray diffraction Patterns for different structures Powders / Polycrystals powders / polycrystalline samples contain thousands of crystallites with each one having a single crystal structure. However, each of the crystals are oriented randomly. Therefore, all possible diffraction peaks should be observed.

24. X-ray diffraction Patterns for different structures Amorphous/ Liquid: Amorphous/Liquid samples contain a large number of atoms without forming any major crystalline phases. Hence, there is no possibility to get a clear diffraction peak in such a specimen. As a result, the diffraction from such structure provides a broad peak as shown in Figure.

25. X-ray diffraction Patterns for different structures Gas molecules move randomly within a given volume resulting no significant diffraction. Hence the intensity of X-ray drops largely with increasing angle and attains zero.

26. The structural make up of solids Small disordered cells are referred to be as amorphous, resulting in a broad diffraction pattern.  Crystallites show a sufficient extent of repeated unit cells to display diffraction.  Random orientation of crystallites produces the typical diffraction pattern.  Preferred orientation of crystallites will produce a significant distortion of diffraction intensities from the ideal.

27. Calculation of average crystallite size The width of the diffraction peak increases as the size of the crystal decreases . This allows one to calculate the average crystallite size from the XRD peak broadening. Let us consider a large sized crystal as shown in Figure. This would exhibit a XRD peak occurring only at a particular angle 2 θ B .

28. Calculation of average crystallite size Consider a crystal having thickness  t  as shown in Figure measured in a direction perpendicular (⊥) to a set of reflecting planes. Let there be ( m  +1) planes in this set. The Bragg angle  θ  as a variable and call  θ B  satisfies exactly the Bragg law, resulting in the diffraction condition

29. Calculation of average crystallite size Intensity of the beams diffracted at and above 2 θ 1 , and at and below 2 θ 2  is zero. However, the intensity of the diffracted beam at angles close to 2 θ B  , but not greater than 2 θ 1  or less than 2 θ 2  is not zero, but has an intermediate value between zero and one.

30. Calculation of average crystallite size The width  B  as given in figure can roughly be measured from XRD at an intensity equal to half the maximum intensity, called as Full Width at Half Maximum (FWHM) and given as The path difference equations for these two angles are

31. Calculation of average crystallite size where  k  is a constant representing shape factor . k   = 0.89 for spherical , K= 0.94 for cubic K= 0.9 for unknown size particles. This equation is known as Scherrer's formula for calculating the average crystallite size using XRD peaks. This equation not applicable for grains larger than 0.1 - 0.2   μ m .

32. Calculation of  d  -spacing and lattice parameters The various sets of planes in a lattice have different values of inter-planar spacing. The planes of larger spacing have low indices and pass through a high density of lattice points . The inter-planar spacing  d hkl  measured at right angles to the planes is a function of both the plane indices ( hkl ) and the lattice constants ( a,b,c ,  α, β, γ) , as shown in Figure The exact relation depends on the crystal system involved. The value of d can be calculated using Bragg's law

33. Calculation of  d  -spacing and lattice parameters

34. Calculation of  d  -spacing and lattice parameters

35. Calculation of structural parameters using XRD patterns Problem 1 Calculate the average crystallite size   d -spacing and lattice constant for the given body centered cubic structured polycrystalline material.

36. Problem solution 1

37. Step 1: Procedure for finding peak positions and full width at Half-maxima  (FWHM) The peak position and FWHM can be directly obtained from  xc  and  FWHM The obtained values are (1) Peak 1 (110); 2 θ  = 44.68°; FWHM = 0.80901°. (2) Peak 2 (200); 2 θ  = 65.045°; FWHM = 1.50079°. (3) Peak 3 (211); 2 θ  = 82.39°; FWHM = 1.5103°.