1. X-RAY DIFFRACTION  X- Ray Sources  Diffraction: Bragg’s Law  Crystal Structure Determination Elements of X-Ray Diffraction B.D. Cullity & S.R. Stock Prentice Hall, Upper Saddle River (2001) Recommended websites:  http://www.matter.org.uk/diffraction/  http://www.ngsir.netfirms.com/englishhtm/Diffraction.htm MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide

2.  How to produce monochromatic X-rays?  How does a crystal scatter these X-rays to give a diffraction pattern?  Bragg’s equation  What determines the position of the XRD peaks?  Answer) the lattice.  What determines the intensity of the XRD peaks?  Answer) the motif.  How to analyze a powder pattern to get information about the lattice type? (Cubic crystal types).  What other uses can XRD be put to apart from crystal structure determination?  Grain size determination  Strain in the material… What will you learn in this ‘sub-chapter’?

3.  For electromagnetic radiation to be diffracted* the spacing in the grating (~a series of obstacles or a series of scatterers) should be of the same order as the wavelength.  In crystals the typical interatomic spacing ~ 2-3 Å**  so the suitable radiation for the diffraction study of crystals is X-rays.  Hence, X-rays are used for the investigation of crystal structures.  Neutrons and Electrons are also used for diffraction studies from materials.  Neutron diffraction is especially useful for studying the magnetic ordering in materials. Some Basics ** Lattice parameter of Cu (a Cu ) = 3.61 Å  d hkl is equal to a Cu or less than that (e.g. d 111 = a Cu /  3 = 2.08 Å) ** If the wavelength is of the order of the lattice spacing, then diffraction effects will be prominent. Click here to know more about this

4. Beam of electrons Target X-rays An accelerating (or decelerating) charge radiates electromagnetic radiation  X-rays can be generated by decelerating electrons.  Hence, X-rays are generated by bombarding a target (say Cu) with an electron beam.  The resultant spectrum of X-rays generated (i.e. X-rays versus Intensity plot) is shown in the next slide. The pattern shows intense peaks on a ‘broad’ background.  The intense peaks can be ‘thought of’ as monochromatic radiation and be used for X-ray diffraction studies. Generation of X-rays

5. Mo Target impacted by electrons accelerated by a 35 kV potential shows the emission spectrum as in the figure below (schematic) 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 X-ray sources with different for doing XRD studies Target Metal Of K  radiation (Å) Mo0.71 Cu1.54 Co1.79 Fe1.94 Cr2.29

7. Absorption (Heat) Incident X-rays SPECIMEN Transmitted beam Fluorescent X-rays Electrons Compton recoilPhotoelectrons Scattered X-rays Coherent From bound charges Coherent From bound charges X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles) When X-rays hit a specimen, the interaction can result in various signals/emissions/effects. The coherently scattered X-rays are the ones important from a XRD perspective. Incoherent (Compton modified) From loosely bound charges Click hereClick here to know more Click hereClick here to know more

8.  Now we shall consider the important topic as to how X-rays interact with a crystalline array (of atoms, ions etc.) to give rise to the phenomenon known as X- ray diffraction (XRD).  Let us consider a special case of diffraction → a case where we get ‘sharp [1] diffraction peaks’.  Diffraction (with sharp peaks) (with XRD being a specific case) requires three important conditions to be satisfied:  Coherent, monochromatic, parallel waves (with wavelength ).  Crystalline array of scatterers* with spacing of the order of (~).  Fraunhofer diffraction geometry [1] The intensity-  plot looks like a ‘  ’ function. * A quasicrystalline array will also lead to diffraction with sharp peaks (which we shall not consider in this text). ** Amorphous material will give diffuse peak. Diffraction Click here to “Understand Diffraction” Click here to “Understand Diffraction” Coherent, monochromatic, parallel wave Fraunhofer geometry Diffraction pattern with sharp peaks Crystalline*, ** Aspects related to the wave Aspects related to the material Aspects related to the diffraction set-up (diffraction geometry)

9.  The waves could be:  electromagnetic waves (light, X-rays…),  matter waves** (electrons, neutrons…) or  mechanical waves (sound, waves on water surface…).  Not all objects act like scatterers for all kinds of radiation.  If wavelength is not of the order of the spacing of the scatterers, then the number of peaks obtained may be highly restricted (i.e. we may even not even get a single diffraction peak!).  In short diffraction is coherent reinforced scattering (or reinforced scattering of coherent waves).  In a sense diffraction is nothing but a special case of constructive (& destructive) interference. To give an analogy  the results of Young’s double slit experiment is interpreted as interference, while the result of multiple slits (large number) is categorized under diffraction.  Fraunhofer diffraction geometry implies that parallel waves are impinging on the scatteres (the object), and the screen (to capture the diffraction pattern) is placed far away from the object. ** With a de Broglie wavelength Some comments and notes Click here to know more about Fraunhofer and Fresnel diffraction geometries

10.  A beam of X-rays directed at a crystal interacts with the electrons of the atoms in the crystal.  The electrons oscillate under the influence of the incoming X-Rays and become secondary sources of EM radiation.  The secondary radiation is in all directions.  The waves emitted by the electrons have the same frequency as the incoming X-rays  coherent.  The emission can undergo constructive or destructive interference. XRD  the first step Schematics

11.  We can get a better physical picture of diffraction by using Laue’s formalism (leading to the Laue’s equations).  However, a parallel approach to diffraction is via the method of Bragg, wherein diffraction can be visualized as ‘reflections’ from a set of planes.  As the approach of Bragg is easier to grasp we shall use that in this elementary text.  We shall do some intriguing mental experiments to utilize the Bragg’s equation (Bragg’s model) with caution.  Let us consider a coherent wave of X-rays impinging on a crystal with atomic planes at an angle  to the rays.  Incident and scattered waves are in phase if the: i) in-plane scattering is in phase and ii) scattering from across the planes is in phase. Incident and scattered waves are in phase if Scattering from across planes is in phase In plane scattering is in phase Some points to recon with

12. Extra path traveled by incoming waves  AY Extra path traveled by scattered waves  XB These can be in phase if   incident =  scattered But this is still reinforced scattering and NOT reflection Let us consider in-plane scattering There is more to this Click here to know more and get introduced to Laue equations describing diffraction Click here

13. BRAGG’s EQUATION  A portion of the crystal is shown for clarity- actually, for destructive interference to occur many planes are required (and the interaction volume of x-rays is large as compared to that shown in the schematic).  The scattering planes have a spacing ‘d’.  Ray-2 travels an extra path as compared to Ray-1 (= ABC). The path difference between Ray-1 and Ray-2 = ABC = (d Sin  + d Sin  ) = (2d.Sin  ).  For constructive interference, this path difference should be an integral multiple of : n = 2d Sin   the Bragg’s equation. (More about this sooner).  The path difference between Ray-1 and Ray-3 is = 2  (2d.Sin  ) = 2  n = 2n. This implies that if Ray-1 and Ray-2 constructively interfere Ray-1 and Ray-3 will also constructively interfere. (And so forth). Let us consider scattering across planes Click hereClick here to visualize constructive and destructive interference Click hereClick here to visualize constructive and destructive interference

14.  The previous page explained how constructive interference occurs. How about the rays just of Bragg angle? Obviously the path difference would be just off as in the figure below. How come these rays ‘go missing’? Click hereClick here to understand how destructive interference of just ‘of-Bragg rays’ occur Click hereClick here to understand how destructive interference of just ‘of-Bragg rays’ occur Interference of Ray-1 with Ray-2 Note that they ‘almost’ constructively interfere!

15. Reflection versus Diffraction ReflectionDiffraction Occurs from surfaceOccurs throughout the bulk Takes place at any angleTakes place only at Bragg angles ~100 % of the intensity may be reflectedSmall fraction of intensity is diffracted Note: X-rays can ALSO be reflected at very small angles of incidence  Though diffraction (according to Bragg’s picture) has been visualized as a reflection from a set of planes with interplanar spacing ‘d’  diffraction should not be confused with reflection (specular reflection).

16.  n = 2d Sin  The equation is written better with some descriptive subscripts:  n is an integer and is the order of the reflection (i.e. how many wavelengths of the X-ray go on to make the path difference between planes).  Bragg’s equation is a negative statement  If Bragg’s eq. is NOT satisfied  NO ‘reflection’ can occur  If Bragg’s eq. is satisfied  ‘reflection’ MAY occur (How?- we shall see this a little later).  The interplanar spacing appears in the Bragg’s equation, but not the interatomic spacing ‘a’ along the plane (which had forced  incident =  scattered ); but we are not free to move the atoms along the plane ‘randomly’  click here to know more.click here  For large interplanar spacing the angle of reflection tends towards zero → as d increases, Sin  decreases (and so does  ).  The smallest interplanar spacing from which Bragg diffraction can be obtained is /2 → maximum value of  is 90 , Sin  is 1  from Bragg equation d = /2. Understanding the Bragg’s equation

17.  For Cu K  radiation ( = 1.54 Å) and d 110 = 2.22 Å n Sin  = n /2d  10.3420.7º First order reflection from (110)  110 20.6943.92º Second order reflection from (110) planes  110 Also considered as first order reflection from (220) planes  220 Relation between d nh nk nl and d hkl e.g. Order of the reflection (n)

18. In XRD n th order reflection from (h k l) is considered as 1 st order reflection from (nh nk nl) Hence, (100) planes are a subset of (200) planes Important point to note: In a simple cubic crystal, 100, 200, 300… are all allowed ‘reflections’. But, there are no atoms in the planes lying within the unit cell! Though, first order reflection from 200 planes is equivalent (mathematically) to the second order reflection from 100 planes; for visualization purposes of scattering, this is better thought of as the later process (i.e. second order reflection from (100) planes).

19. How is it that we are able to get information about lattice parameters of the order of Angstroms (atoms which are so closely spaced) using XRD? Funda Check  Diffraction is a process in which ‘linear information’ (the d-spacing of the planes) is converted to ‘angular information’ (the angle of diffraction,  Bragg ).  If the detector is placed ‘far away’ from the sample (i.e. ‘R’ in the figure below is large) the distances along the arc of a circle (the detection circle) get amplified and hence we can make ‘easy’ measurements.

20. Forward and Back Diffraction Here a guide for quick visualization of forward and backward scattering (diffraction) is presented

21. Funda Check  What is  (theta) in the Bragg’s equation?   is the angle between the incident x-rays and the set of parallel atomic planes (which have a spacing d hkl ). Which is 10  in the above figure.  It is NOT the angle between the x-rays and the sample surface (note: specimens could be spherical or could have a rough surface).

22.  We had mentioned that Bragg’s equation is a negative statement: i.e. just because Bragg’s equation is satisfied a ‘reflection’ may not be observed.  Let us consider the case of Cu K  radiation ( = 1.54 Å) being diffracted from (100) planes of Mo (BCC, a = 3.15 Å = d 100 ). The missing ‘reflections’ But this reflection is absent in BCC Mo The missing reflection is due to the presence of additional atoms in the unit cell (which are positions at lattice points)  which we shall consider next The wave scattered from the middle plane is out of phase with the ones scattered from top and bottom planes. I.e. if the green rays are in phase (path difference of ) then the red ray will be exactly out of phase with the green rays (path difference of /2).

23. However, the second order reflection from (100) planes (which is equivalent to the first order reflection from the (200) planes is observed This is because if the green rays have a path difference of 2 then the red ray will have path difference of → which will still lead to constructive interference! Continuing with the case of BCC Mo…

24.  Presence of additional atoms/ions/molecules in the UC  at lattice points  or as a part of the motif can alter the intensities of some of the reflections  Some of the reflections may even go missing Important points  Position of the ‘reflections’/‘peaks’ tells us about the lattice type.  The Intensities tells us about the motif.

25. Intensity of the Scattered waves Electron Atom Unit cell (uc) Scattering by a crystal can be understood in three steps A B C Polarization factor Atomic scattering factor (f) Structure factor (F) To understand the scattering from a crystal leading to the ‘intensity of reflections’ (and why some reflections go missing), three levels of scattering have to be considered: 1) scattering from electrons 2) scattering from an atom 3) scattering from a unit cell Click here to know the details Click here to know the details Structure factor calculations & Intensity in powder patterns  Structure Factor (F): The resultant wave scattered by all atoms of the unit cell  The Structure Factor is independent of the shape and size of the unit cell; but is dependent on the position of the atoms/ions etc. within the cell Click here to know more about  Bragg’s equation tells us about the position of the intensity peaks (in terms of  )  but tells us nothing about the intensities. The intensities of the peaks depend on many factors as considered here.

26. The concept of a Reciprocal lattice and the Ewald Sphere construction:  Reciprocal lattice and Ewald sphere constructions are important tools towards understanding diffraction. (especially diffraction in a Transmission Electron Microscope (TEM))  A lattice in which planes in the real lattice become points in the reciprocal lattice is a very useful one in understanding diffraction.  click here to go to a detailed description of these topics.click here Reciprocal Lattice & Ewald Sphere construction Click here to know more about

27. Bravais LatticeReflections which may be presentReflections necessarily absent SimpleallNone Body centred(h + k + l) even(h + k + l) odd Face centredh, k and l unmixedh, k and l mixed End centred (C centred) h and k unmixedh and k mixed Bravais LatticeAllowed Reflections SCAll BCC(h + k + l) even FCCh, k and l unmixed DC Either,  h, k and l are all odd or  all are even & (h + k + l) divisible by 4 Selection / Extinction Rules  As we have noted before even if Bragg’s equation is satisfied, ‘reflections may go missing’  this is due to the presence of additional atoms in the unit cell.  The reflections present and the missing reflections due to additional atoms in the unit cell are listed in the table below. Click here to see the derivations Structure factor calculations

28. h 2 + k 2 + l 2 SCFCCBCCDC 1100 2110 3111 4200 5210 6211 7 8220 9300, 221 10310 11311 12222 13320 14321 15 16400 17410, 322 18411, 330 19331 Allowed reflections in SC*, FCC*, BCC* & DC crystals * lattice decorated with monoatomic/monoionic motif Cannot be expressed as (h 2 +k 2 +l 2 )

29. The ratio of (h 2 + k 2 + l 2 ) derived from extinction rules (previous page) As we shall see soon the ratios of (h 2 + k 2 + l 2 ) is proportional to Sin 2   which can be used in the determination of the lattice type SC1234568… BCC1234567… FCC3481112… DC381116…  Note that we have to consider the ratio of only two lines to distinguish FCC and DC. I.e. if the ratios are 3:4 then the lattice is FCC.  But, to distinguish between SC and BCC we have to go to 7 lines!

30. Crystal structure determination Monochromatic X-rays Panchromatic X-rays Monochromatic X-rays Many  s (orientations) Powder specimen POWDER METHOD Single  LAUE TECHNIQUE  Varied by rotation ROTATING CRYSTAL METHOD  As diffraction occurs only at specific Bragg angles, the chance that a reflection is observed when a crystal is irradiated with monochromatic X-rays at a particular angle is small (added to this the diffracted intensity is a small fraction of the beam used for irradiation).  The probability to get a diffracted beam (with sufficient intensity) is increased by either varying the wavelength ( ) or having many orientations (rotating the crystal or having multiple crystallites in many orientations).  The three methods used to achieve high probability of diffraction are shown below. Only the powder method (which is commonly used in materials science) will be considered in this text.

31. THE POWDER METHOD Cubic crystal  In the powder method the specimen has crystallites (or grains) in many orientations (usually random).  Monochromatic* X-rays are irradiated on the specimen and the intensity of the diffracted beams is measured as a function of the diffracted angle.  In this elementary text we shall consider cubic crystals. (1) (2) (2) in (1)   * In reality this is true only to an extent

32.  In the powder sample there are crystallites in different ‘random’ orientations (a polycrystalline sample too has grains in different orientations)  The coherent x-ray beam is diffracted by these crystallites at various angles to the incident direction  All the diffracted beams (called ‘reflections’) from a single plane, but from different crystallites lie on a cone.  Depending on the angle there are forward and back reflection cones.  A diffractometer can record the angle of these reflections along with the intensities of the reflection  The X-ray source and diffractometer move in arcs of a circle- maintaining the Bragg ‘reflection’ geometry as in the figure (right) POWDER METHOD Different cones for different reflections

33. How to visualize the occurrence of peaks at various angles It is ‘somewhat difficult’ to actually visualize a random assembly of crystallites giving peaks at various angels in a XRD scan. The figures below are expected to give a ‘visual feel’ for the same. [Hypothetical crystal with a = 4Å is assumed with =1.54Å. Only planes of the type xx0 (like (100,110)are considered]. Random assemblage of crystallites in a material The sample is not rotating only the source and detector move in arcs of a circle As the scan takes place at increasing angles, planes with suitable ‘d’, which diffract are ‘picked out’ from favourably oriented crystallites h2h2 hkld Sin(  )  11004.000.1911.10 21102.830.2715.80 31112.310.3319.48 42002.000.3922.64 52101.790.4325.50 62111.630.4728.13 82201.410.5432.99 93001.330.5835.27 103101.260.6137.50

34.  In the power diffraction method a 2  versus intensity (I) plot is obtained from the diffractometer (and associated instrumentation).  The ‘intensity’ is the area under the peak in such a plot (NOT the height of the peak).  The information of importance obtained from such a pattern is the ‘relative intensities’ and the absolute value of the intensities is of little importance (for now).  I is really diffracted energy (as Intensity is Energy/area/time).  A table is prepared as in the next slide to tabulate the data and make calculations to find the crystal structure (restricting ourselves to cubic crystals for the present). Determination of Crystal Structure from 2  versus Intensity Data in Powder Method Powder diffraction pattern from Al Radiation: Cu K , = 1.54 Å Increasing  Increasing d

35. n 2  →  Intensity Sin  Sin 2  ratio Determination of Crystal Structure from 2  versus Intensity Data The following table is made from the 2  versus Intensity data (obtained from a XRD experiment on a powder sample (empty starting table of columns is shown below- completed table shown later).

36. Powder diffraction pattern from Al Radiation: Cu K , = 1.54 Å Note:  This is a schematic pattern  In real patterns peaks or not idealized  peaks  broadened  Increasing splitting of peaks with  g  (  1 &  2 peaks get resolved in the high angle peaks)  Peaks are all not of same intensity  No brackets are used around the indexed numbers (the peaks correspond to planes in the real space)

37. Powder diffraction pattern from Al 111 200 220 311 222 400 K  1 & K  2 peaks resolved in high angle peaks (in 222 and 400 peaks this can be seen) Radiation: Cu K , = 1.54 Å Note:  Peaks or not idealized  peaks  broadened  Increasing splitting of peaks with  g   Peaks are all not of same intensity In low angle peaks K  1 & K  2 peaks merged

38. What is the maximum value of  possible (experimentally)? Funda Check How are real diffraction patterns different from the ‘ideal computed ones?  We have seen real and ideal diffraction patterns. In ideal patterns the peaks are ‘  ’ functions.  Real diffraction patterns are different from ideal ones in the following ways:  Peaks are broadened Could be due to instrumental, residual ‘non-uniform’ strain (microstrain), grain size etc. broadening.  Peaks could be shifted from their ideal positions Could be due to uniform strain→ macrostrain.  Relative intensities of the peaks could be altered Could be due to texture in the sample. Funda Check Ans: 90   At  = 90  the ‘reflected ray’ is opposite in direction to the incident ray.  Beyond this angle, it is as if the source and detector positions are switched.   2  max is 180 .

39. Funda Check What will determine how many peaks I will get?  1)  smaller the wavelength of the X-rays, more will be the number of peaks possible.  From Bragg’s equation: [ =2dSin  ], (Sin  ) max will correspond to d min. (Sin  ) max =1. Hence, d min = /2. Hence, if is small then planes with smaller d spacing (i.e. those which occur at higher 2  values) will also show up in a XRD patter (powder pattern). Given that experimentally  cannot be greater than 90 .  2) Lattice type  in SC we will get more peaks as compared to (say) FCC/DC. Other things being equal.  3) Lower the symmetry of the crystal, more the number of peaks (e.g., in tetragonal crystal the 100 peak will lie at a different 2  as compared to the 001 peak).

40. # 22 Sin  Sin 2  ratioIndexd 138.5219.260.330.1131112.34 244.7622.380.380.1442002.03 365.1432.570.540.2982201.43 478.2639.130.630.40113111.22 582.4741.2350.660.43122221.17 699.1149.5550.760.58164001.01 7112.0356.0150.830.69193310.93 8116.6058.30.850.72204200.91 9137.4768.7350.930.87244220.83 10163.7881.890.990.98273330.78 Determination of Crystal Structure (lattice type) from 2  versus Intensity Data From the ratios in column 6 we conclude that FCC Let us assume that we have the 2  versus intensity plot from a diffractometer  To know the lattice type we need only the position of the peaks (as tabulated below) Solved example Using We can get the lattice parameter  which correspond to that for Al 1 Note: Error in d spacing decreases with  → so we should use high angle lines for lattice parameter calculation Click here to know more Note that Sin  cannot be > 1 XRD_lattice_parameter_calculation.ppt Note

41. 2  →  Sin  Sin 2  Ratios of Sin 2  Dividing Sin 2  by 0.134/3 = 0.044667 Whole number ratios 121.50.3660.13413 2250.4220.1781.333.994 3370.600.3622.708.108 4450.7070.5003.7311.1911 5470.7310.535411.9812 6580.8480.7195.3716.1016 7680.9270.8596.4119.2319 FCC Another example Given the positions of the Bragg peaks we find the lattice type Solved example 2

42. More Solved Examples on XRD Click here Comparison of diffraction patterns of SC, BCC & B2 structures Click here

43. Funda Check What happens when we increase or decrease ? We had pointed out that ~ a is preferred for diffraction. Let us see what happens if we ‘drastically’ increase or decrease. (This is only a thought experiment!!) If we ~double → we get too few peaks If we make small→ all the peaks get crowded to small angles With Cu K  = 1.54 Å And the detector may not be able to resolve these peaks if they come too close!

44. Bravais lattice determination Lattice parameter determination Determination of solvus line in phase diagrams Long range order Applications of XRD Crystallite size and Strain Determine if the material is amorphous or crystalline We have already seen these applications Click here to know more Next slide

45. Diffraction angle (2  ) → Intensity → 90 180 0 Crystal 90 180 0 Diffraction angle (2  ) → Intensity → Liquid / Amorphous solid 90 180 0 Diffraction angle (2  ) → Intensity → Monoatomic gas Schematic of difference between the diffraction patterns of various phases Sharp peaks Diffuse Peak No peak

46. Diffuse peak from Cu-Zr-Ni-Al-Si Metallic glass (XRD patterns) courtesy: Dr. Kallol Mondal, MSE, IITK Actual diffraction pattern from an amorphous solid Note  Sharp peaks are missing  Broad diffuse peak survives → the peak corresponds to the average spacing between atoms which the diffraction experiment ‘picks out’

47. Funda Check  What is the minimum spacing between planes possible in a crystal?  How many diffraction peaks can we get from a powder pattern? Let us consider a cubic crystal (without loss in generality) As h,k, l increases, ‘d’ decreases  we could have planes with infinitesimal spacing With increasing indices the interplanar spacing decreases The number of peaks we obtain in a powder diffraction pattern depends on the wavelength of x-ray we are using. Planes with ‘d’ < /2 are not captured in the diffraction pattern. These peaks with small ‘d’ occur at high angles in diffraction pattern.