X-ray Diffraction (XRD) What is X-ray Diffraction Properties and generation of X-ray Bragg’s Law Basics of Crystallography XRD Pattern Powder Diffraction Applications of XRD http://www.matter.org.uk/diffraction/x-ray/default.htm

X-ray and X-ray Diffraction http://www.youtube.com/watch?v=vYztZlLJ3ds at~0:40-3:10 X-ray and X-ray Diffraction X-ray was first discovered by W. C. Roentgen in 1895. Diffraction of X-ray was discovered by W.H. Bragg and W.L. Bragg in 1912 Bragg’s law: n=2dsin X-rays were discovered in 1895 by the German physicist Roentgen and were so named because their nature was unknown at the time. Unlike ordinary light, these rays were invisible, but they traveled in straight lines and affected photographic film in the same way as light. On the other hand, they were much more penetrating than light and could easily pass through the human body, wood, quite thick pieces of metal and other “opaque” objects. X-rays are a form of electromagnetic radiation (like light); they are of higher energy, however, and can penetrate the body to form an image on film. Photograph of the hand of an old man using X-ray.

Properties and Generation of X-ray X-rays are electromagnetic radiation with very short wavelength ( 10-8 -10-12 m) The energy of the x- ray can be calculated with the equation E = h = hc/ e.g. the x-ray photon with wavelength 1Å has energy 12.5 keV h=4.136x10-15 eV sec.

A Modern Automated X-ray Diffractometer X-ray Tube Detector http://www.youtube.com/watch?v=lwV5WCBh9a0 at~6:00 how to operate XRD equipment Sample stage Cost: $560K to 1.6M

Production of X-rays - + http://www.youtube.com/watch?v=Bc0eOjWkxpU to~1:10 Production of X-rays Production of X-rays Cross section of sealed-off filament X-ray tube W filament + - target X-rays Vacuum http://www.youtube.com/watch?v=wbbsbE2mQuA How X-ray works The target is always water-cooled to prevent its heating since most of the kinetic energy of the electrons is converted into heat in the target, less than 1 percent being transformed into x-rays. The filament is heated by a filament current of ~3 amp and emits electrons which are rapidly drawn to the target by the high negative voltage between the cathode and anode. A small metal cup maintained at the same negative voltage as the filament repels the electrons and tends to focus them into a narrow region of the target. X-rays are emitted from the focal spot in all directions and escape from the tube through windows in the tube housing. Since the windows must be vacuum tight and yet highly transparent to x-rays, they are usually made of beryllium. X-rays are produced whenever high-speed electrons collide with a metal target. A source of electrons – hot W filament, a high accelerating voltage (30-50kV) between the cathode (W) and the anode, which is a water-cooled block of Cu or Mo containing desired target metal. https://www.youtube.com/watch?v=3_bZCA7tlFQ How does X-ray tube work

http://www.youtube.com/watch?v=Bc0eOjWkxpU at~1:06-3:10 X-ray Spectrum A spectrum of x-ray is produced as a result of the interaction between the incoming electrons and the nucleus or inner shell electrons of the target element. Two components of the spectrum can be identified, namely, the continuous spectrum caused by bremsstrahlung (German word: braking radiation) and the characteristic spectrum. I Mo k characteristic radiation continuous radiation k  SWL - short-wavelength limit http://www.youtube.com/watch?v=3fe6rHnhkuY Bremsstrahlung http://www.youtube.com/watch?v=n9FkLBaktEY characteristic X-ray

Short-wavelength Limit The short-wavelength limit (SWL or SWL) corresponds to those x-ray photons generated when an incoming electron yield all its energy in one impact. Planck constant h=6.63x10-34 J.s and speed of light c=3.0x108 m/s Electron charge e=1.6x10-19 C. V – applied voltage

Characteristic x-ray Spectra Sharp peaks in the spectrum can be seen if the accelerating voltage is high (e.g. 25 kV for molybdenum target). These peaks fall into sets which are given the names, K, L, M…. lines with increasing wavelength. Mo

Excitation of K, L, M and N shells and Formation of K to M Characteristic X-rays If an incoming electron has sufficient kinetic energy for knocking out an electron of the K shell (the inner-most shell), it may excite the atom to an high-energy state (K state). One of the outer electron falls into the K-shell vacancy, emitting the excess energy as a x-ray photon. Characteristic x-ray energy: Ex-ray=Efinal-Einitial L K K2 K1 K I II III K L M N L subshells K state (shell) Energy K K Since the energy of the emitted x-ray photon is related to the difference in energy between the sharply defined levels of the atom, it is referred as a characteristic x-ray. L state K excitation L M state EK>EL>EM EK>EK M L excitation N state 4/23/2017 ground state

Characteristic x-ray Spectra Z

Characteristic X-ray Lines K K and K2 will cause Extra peaks in XRD pattern, but can be eliminated by adding filters. ----- is the mass absorption coefficient of Zr. I K1 <0.001Å K2 K =2dsin  (Å) Spectrum of Mo at 35kV

Absorption of x-ray All x-rays are absorbed to some extent in passing through matter due to electron ejection or scattering. The absorption follows the equation where I is the transmitted intensity; I0 is the incident intensity x is the thickness of the matter; is the linear absorption coefficient (element dependent);  is the density of the matter; (/) is the mass absorption coefficient (cm2/gm). I0 I ,  I  is dependent on the density,  and thus / is a constant of the materials. Mass absorption is also wavelength dependent. x x

Effect of , / (Z) and t on Intensity of Diffracted X-ray incident beam crystal diffracted beam film http://www.matter.org.uk/diffraction/x-ray/x_ray_diffraction.htm

Absorption of x-ray /  The mass absorption coefficient is also wavelength dependent. Discontinuities or “Absorption edges” can be seen on the absorption coefficient vs. wavelength plot. These absorption edges mark the point on the wavelength scale where the x-rays possess sufficient energy to eject an electron from one of the shells. Absorption edges /  Absorption coefficients of Pb, showing K and L absorption edges.

Filtering of X-ray The absorption behavior of x-ray by matter can be used as a means for producing quasi- monochromatic x-ray which is essential for XRD experiments. The rule: “Choose for the filter an element whose K absorption edge is just to the short- wavelength side of the K line of the target material.”

Filtering of X-ray Choose for the filter an element whose K absorption edge is just to the short-wavelength side of the K line of the target material. A common example is the use of nickel to cut down the K peak in the copper x-ray spectrum. The thickness of the filter to achieve the desired intensity ratio of the peaks can be calculated with the absorption equation shown in the last section. K absorption edge of Ni Cu K  1.5405Å 1.4881Å   No filter Ni filter Comparison of the spectra of Cu radiation (a) before and (b) after passage through a Ni filter. The dashed line is the mass absorption coefficient of Ni.

What Is Diffraction? A wave interacts with A single particle The particle scatters the incident beam uniformly in all directions. A crystalline material The scattered beam may add together in a few directions and reinforce each other to give diffracted beams. http://www.matter.org.uk/diffraction/introduction/what_is_diffraction.htm

What is X-ray Diffraction? The atomic planes of a crystal cause an incident beam of x-rays (if wavelength is approximately the magnitude of the interatomic distance) to interfere with one another as they leave the crystal. The phenomenon is called x-ray diffraction.                                                                                                           Bragg’s Law: n= 2dsin()  ~ d 2B atomic plane In XRD, X-rays with ~0.5-2Å, is incident on a specimen and is diffracted by crystalline phase in specimen according to Bragg’s Law. The intensity of the diffracted X-ray is measured as a function of the diffraction angle and the specimen orientation. XRD patterns will provide a lot of information about materials structure and properties. X-ray of  B I d http://www.youtube.com/watch?v=1FwM1oF5e6o to~1:17 diffraction & interference

Constructive and Destructive Interference of Waves Constructive interference occurs only when the path difference of the scattered wave from consecutive layers of atoms is a multiple of the wavelength of the x-ray. /2 http://www.matter.org.uk/diffraction/geometry/superposition_of_waves_exercises.htm For demonstration of constructive and destructive interference of waves. Constructive Interference Destructive Interference In Phase Out Phase http://www.youtube.com/watch?v=kSc_7XBng8w http://micro.magnet.fsu.edu/primer/java/interference/waveinteractions/index.html

http://www.youtube.com/watch?v=UfDW0-kghmI at~3:00-6:00 Bragg’s Law and X-ray Diffraction How waves reveal the atomic structure of crystals http://www.eserc.stonybrook.edu/ProjectJava/Bragg/index.html nl = 2dsin() n-integer Diffraction occurs only when Bragg’s Law is satisfied Condition for constructive interference (X-rays 1 & 2) from planes with spacing d X-ray1 X-ray2 l =3Å =30o Atomic plane d=3 Å 2-diffraction angle http://www.youtube.com/watch?v=hQUsnMzTdpU When Lambda=0.15nm, path difference between X-rays 1 and 2 equals to nxLambda. Braggs were awarded the Nobel Prize in physics in 1915 for their work in determining Crystal structures beginning with NaCl, ZnS and diamond. Bragg’s law can also be used to explain the interference pattern any beam, e.g., electrons, neutrons, ions and protons with a wavelength similar to the distance between the atomic or molecular structures of interest. http://www.youtube.com/watch?v=UfDW0-kghmI at~3:00-6:00

Deriving Bragg’s Law - nl = 2dsin Constructive interference occurs only when nl = AB + BC X-ray 2 X-ray 1 AB=BC nl = 2AB Sin=AB/d AB=dsin nl =2dsin What is hkl, basics of crystallography l=2dhklsinhkl n – integer, called the order of diffraction

Basics of Crystallography http://www.youtube.com/watch?v=Mm-jqk1TeRY crystal packing in lattices to~2:25 smallest building block c Single crystal d3 CsCl   b a  Unit cell (Å) z [001] d1 y [010] Lattice d2 x [100] crystallographic axes A crystal consists of a periodic arrangement of the unit cell into a lattice. The unit cell can contain a single atom or atoms in a fixed arrangement. Crystals consist of planes of atoms that are spaced a distance d apart, but can be resolved into many atomic planes, each with a different d-spacing. a,b and c (length) and ,  and  (angles between a,b and c) are lattice constants or parameters which can be determined by XRD. The size and shape of the unit cell can be described by the three vectors, a, b and c. also called the crystallographic axes of the cell. How to define atomic planes? http://www.youtube.com/watch?v=Rm-i1c7zr6Q&list=TLyPTUJ62VYE4wC1snHSChDl0NGo9IK-Nl Lattice structures

Seven crystal Systems Rhombohedral a a=b=c ==90o Cubic a=b=c System Axial lengths Unit cell and angles Rhombohedral a a=b=c ==90o Cubic a=b=c ===90o a Hexagonal a=bc ==90o =120o c Tetragonal a=bc ===90o c a Monoclinic a abc ==90o c b Orthorhombic c a abc ===90o Triclinic abc 90o c a a b b

Plane Spacings for Seven Crystal Systems hkl 1 hkl hkl hkl hkl hkl hkl

Miller Indices - hkl Miller indices-the reciprocals of the Miller indices form a notation system in crystallography for planes in crystal lattices. Miller indices-the reciprocals of the fractional intercepts which the plane makes with crystallographic axes (010) a b c http://en.wikipedia.org/wiki/Miller_index a b c Axial length 4Å 8Å 3Å Intercept lengths 1Å 4Å 3Å Fractional intercepts ¼ ½ 1 Miller indices 4 2 1 h k l 4Å 8Å 3Å  8Å  /4 1 /3 0 1 0 h k l https://www.youtube.com/watch?v=enVpDwFCl68 Miller indices example crystallography for everyone

Planes and Spacings - a http://www.matter.org.uk/diffraction/geometry/planes_in_crystals.htm

Indexing of Planes and Directions http://www.youtube.com/watch?v=9Rjp9i0H7GQ Directions in crystals Indexing of Planes and Directions (111) c c [111] (110) b b [110] a a [] square bracket, <> angular bracket, () parenthesis, {} brace a direction [uvw] a set of equivalent directions <uvw> <100>:[100],[010],[001] [100],[010] and [001] a plane (hkl) a set of equivalent planes {hkl} {110}:(101),(011),(110) (101),(101),(101),etc.

X-ray Diffraction Pattern BaTiO3 at T>130oC (hkl) Simple Cubic I 20o 40o 2 60o dhkl Bragg’s Law: l=2dhklsinhkl l(Cu K)=1.5418Å

XRD Pattern Significance of Peak Shape in XRD Peak position Peak width Peak intensity I 2 http://www.youtube.com/watch?v=MU2jpHg2vX8 XRD peak analysis

Peak Position Determine d-spacings and lattice parameters Fix l (Cu k)=1.54Å dhkl = 1.54Å/2sinhkl For a simple cubic (a=b=c=a0) 2 a0 = dhkl (h2+k2+l2)½ e.g., for BaTiO3, 2220=65.9o, 220=32.95o, d220 =1.4156Å, a0=4.0039Å Because the value of sin changes very slows with  in the neighborhood of 90o. For this reason, a very accurate value of sin can be obtained from a measurement of  which is itself not particular precise, provided that  is near 90o or diffraction angle 2 is near 180o. Note: Most accurate d-spacings are those calculated from high-angle peaks.

Peak Intensity X-ray intensity: Ihkl  lFhkll2 Fhkl - Structure Factor Determine crystal structure and atomic arrangement in a unit cell X-ray intensity: Ihkl  lFhkll2 Fhkl - Structure Factor N Fhkl =  fjexp[2i(huj+kvj+lwj)] j=1 fj – atomic scattering factor fj  Z, sin/ In order to describe the intensity of the diffraction peaks of various crystal plane of different unit cells, the Structure Factor is introduced. Table of fj values, as a function of (sin/), for the elements and some ionic states of the elements can be found from references. The summation is over the contents of the unit cell, i.e. the structure factor describes the collective scattering effect of all atoms in a unit cell. F depends on atomic arrangement in a unit cell and orientation of specimen. To calculate F, one has to know the number of atoms present in the unit cell and their locations. Low Z elements may be difficult to detect by XRD N – number of atoms in the unit cell, uj,vj,wj - fractional coordinates of the jth atom in the unit cell

Cubic Structures a = b = c = a Simple Cubic Body-centered Cubic Face-centered Cubic BCC FCC [001] z axis a a [010] y a 1 atom 2 atoms 4 atoms [100] x 8 x 1/8 =1 8 x 1/8 + 1 = 2 8 x 1/8 + 6 x 1/2 = 4 Location: 0,0,0 0,0,0, ½, ½, ½, 0,0,0, ½, ½, 0, ½, 0, ½, 0, ½, ½, - corner atom, shared with 8 unit cells - atom at face-center, shared with 2 unit cells 8 unit cells

Structures of Some Common Metals [001] axis l = 2dhklsinhkl (001) plane d010 Mo Cu a d001 (010) plane (002) a d002 = ½ a [010] [010] axis a BCC FCC [100] h,k,l – integers, Miller indices, (hkl) planes (001) plane intercept [001] axis with a length of a, l = 1 (002) plane intercept [001] axis with a length of ½ a, l = 2 (010) plane intercept [010] axis with a length of a, k = 1, etc.

Structure factor and intensity of diffraction z Structure factor and intensity of diffraction (001) (002) Sometimes, even though the Bragg’s condition is satisfied, a strong diffraction peak is not observed at the expected angle. Consider the diffraction peak of (001) plane of a FCC crystal. Owing to the existence of the (002) plane in between, complications occur. FCC d001 d002  1 2 3 1’ 2’ 3’

Structure factor and intensity of diffraction 1 2 3 1’ 2’ 3’ ray 1 and ray 3 have path difference of  but ray 1 and ray 2 have path difference of /2. So do ray 2 and ray 3. It turns out that it is in fact a destructive condition, i.e. having an intensity of 0. the diffraction peak of a (001) plane in a FCC crystal can never be observed. /4 /4 /2 /2

l=2dhklsinhkl d002 d001 d001sin001=d002sin002 since d001=2d002 http://emalwww.engin.umich.edu/education_materials/microscopy.html l=2dhklsinhkl d001sin001=d002sin002 since d001=2d002 If sin002=2sin001 i.e., 002>001 Bragg’s law holds and (002) diffraction peak appears 1 1’ 1 1’ 2 2’ 2’ 2 001 002 d002 /2 3 3’ 3 3’ /4 d001 002 001 When =001 no diffraction occurs, while  increases to 002, diffraction occurs.

Structure factor and intensity of diffraction for FCC z e.g., Aluminium (FCC), all atoms are the same in the unit cell four atoms at positions, (uvw): A(0,0,0), B(½,0,½), C(½,½,0) & D(0,½,½) D B y A C x

Structure factor and intensity of diffraction for FCC Ihkl  lFhkll2 For a certain set of plane, (hkl) F =  f () exp[2i(hu+kv+lw)] = f ()  exp[2i(hu+kv+lw)] = f (){exp[2i(0)] + exp[2i(h/2 + l/2)] + exp[2i(h/2 + k/2)] + exp[2i(k/2 + l/2)]} = f (){1 + ei(h+k) + ei(k+l) + ei(l+h)} Since e2ni = 1 and e(2n+1)i = -1, if h, k & l are all odd or all even, then (h+k), (k+l), and (l+h) are all even and F = 4f; otherwise, F = 0 A(0,0,0), B(½,0,½), C(½,½,0) & D(0,½,½) e2ni = con(2n) + isin(2n) = 1

XRD Patterns of Simple Cubic and FCC 2 FCC Diffraction angle 2 (degree)

Diffractions Possibly Present for Cubic Structures

Peak Width - Full Width at Half Maximum (FWHM) Determine Particle or grain size 2. Residual strain The smaller FWHM, the better the peak location and phase identification.

Effect of Particle (Grain) Size As rolled 300oC As rolled t Grain size 200oC I K1 B K2 (FWHM) 250oC Grain size 450oC 300oC 0.9 Peak broadening B = t cos 450oC As grain size decreases hardness increases and peak become broader 2 (331) Peak of cold-rolled and annealed 70Cu-30Zn brass

Effect of Lattice Strain on Diffraction Peak Position and Width No Strain Uniform Strain (d1-do)/do Peak moves, no shape changes Measure residual strains or stresses in samples. Non-uniform Strain d1constant Peak broadens

XRD patterns from other states of matter Crystal Constructive interference Structural periodicity Diffraction Sharp maxima 2 Liquid or amorphous solid Lack of periodicity One or two Short range order broad maxima Monatomic gas Atoms are arranged Scattering I perfectly at random decreases with 

X-ray Diffraction (XRD) What is X-ray Diffraction Properties and generation of X-ray Bragg’s Law Basics of Crystallography XRD Pattern Powder Diffraction Applications of XRD http://www.matter.org.uk/diffraction/x-ray/laue_method.htm 45

Diffraction of X-rays by Crystals-Laue Method Back-reflection Laue crystal Film X-ray [001] Transmission Laue crystal Film http://www.youtube.com/watch?v=UfDW0-kghmI at~1:20-3:00 http://www.youtube.com/watch?v=2JwpHmT6ntU

Powder Diffraction (most widely used) Diffraction of X-rays by Polycrystals Powder Diffraction (most widely used) A powder sample is in fact an assemblage of small crystallites, oriented at random in space. d3 d1 d2 2 Powder sample d1 crystallite d2 d3 Polycrystalline sample http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:20-1:56

Detection of Diffracted X-ray by A Diffractometer detector Sample holder x-ray detectors (e.g. Geiger counters) is used instead of the film to record both the position and intensity of the x-ray peaks The sample holder and the x-ray detector are mechanically linked If the sample holder turns , the detector turns 2, so that the detector is always ready to detect the Bragg diffracted x-ray X-ray tube  The focusing circle – X-ray source, sample and detector on same circle, detector moving through 2 theta and sample moving through theta. 2 http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:44-1:56 and 15:44-16:16

Phase Identification One of the most important uses of XRD Obtain XRD pattern Measure d-spacings Obtain integrated intensities Compare data with known standards in the JCPDS file, which are for random orientations (there are more than 50,000 JCPDS cards of inorganic materials).

JCPDS Card Quality of data 1.file number 2.three strongest lines Intensities are expressed as percentages of I1, the intensity of the strongest line on the pattern. There are about 50,000 JCPDS cards of inorganic materials 1.file number 2.three strongest lines 3.lowest-angle line 4.chemical formula and name 5.data on dif- fraction method used 6.crystallographic data 7.optical and other data 8.data on specimen 9.data on diffraction pattern.

Other Applications of XRD To identify crystalline phases To determine structural properties: Lattice parameters (10-4Å), strain, grain size, expitaxy, phase composition, preferred orientation order-disorder transformation, thermal expansion To measure thickness of thin films and multilayers To determine atomic arrangement To image and characterize defects Detection limits: ~3% in a two phase mixture; can be ~0.1% with synchrotron radiation. Lateral resolution: normally none XRD is a nondestructive technique The Pacific Northwest National Laboratory is developing a range of technologies to broaden the field of explosives detection. Phased contrast X-ray imaging, which uses silicon gratings to detect distortions in the X-ray wave front, may be applicable to mail or luggage scanning for explosives; it can also be used in detecting other contraband, small-parts inspection, or materials characterization. - Uploaded on Feb 9, 2010 https://www.youtube.com/watch?v=CpJZfeJ4poE phased contrast x-ray imaging https://www.youtube.com/watch?v=6POi6h4dfVs Determining strain pole figures from diffraction experiments

Phase Identification -Effect of Symmetry on XRD Pattern a b c Phase Identification -Effect of Symmetry on XRD Pattern Cubic a=b=c, (a) b. Tetragonal a=bc (a and c) c. Orthorhombic abc (a, b and c) 2 Number of reflection Peak position Peak splitting

Finding mass fraction of components in mixtures The intensity of diffraction peaks depends on the amount of the substance By comparing the peak intensities of various components in a mixture, the relative amount of each components in the mixture can be worked out ZnO + M23C6 + 

Preferred Orientation (Texture) In common polycrystalline materials, the grains may not be oriented randomly. (We are not talking about the grain shape, but the orientation of the unit cell of each grain, ) This kind of ‘texture’ arises from all sorts of treatments, e.g. casting, cold working, annealing, etc. If the crystallites (or grains) are not oriented randomly, the diffraction cone will not be a complete cone Grain Random orientation Preferred orientation https://www.youtube.com/watch?v=UfDW0-kghmI at~1:20

Preferred Orientation (Texture) (110) Random orientation

Preferred Orientation (Texture) Simple cubic Random orientation Texture 20 30 40 50 60 70 PbTiO3 (001)  MgO (001) highly c-axis oriented 2 I I (110) PbTiO3 (PT) simple tetragonal (111) Preferred orientation  Figure 1. X-ray diffraction -2 scan profile of a PbTiO3 thin film grown on MgO (001) at 600°C. Figure 2. X-ray diffraction  scan patterns from (a) PbTiO3 (101) and (b) MgO (202) reflections.

Preferred Orientation (Texture) https://www.youtube.com/watch?v=R9o39StS5ik Goniometer Rotations for X-Ray Crystallography By rotating the specimen about three major axes as shown, these spatial variations in diffraction intensity can be measured. 4-Circle Goniometer For pole-figure measurement  A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials science. https://en.wikipedia.org/wiki/Pole_figure Pole figures displaying crystallographic texture of -TiAl in an 2-gamma alloy, as measured by high energy X-rays.[

In Situ XRD Studies Temperature Electric Field Pressure

High Temperature XRD Patterns of Decomposition of YBa2Cu3O7- Collecting diffraction patterns every few minutes. High temperature reactions can be mapped over hundreds of different temperatures in an overnight run. BaCuO2 start to form at ~950C. T 2

In Situ X-ray Diffraction Study of an Electric Field Induced Phase Transition (330) Single Crystal Ferroelectric 92%Pb(Zn1/3Nb2/3)O3 -8%PbTiO3 E=6kV/cm (330) peak splitting is due to Presence of <111> domains Rhombohedral phase No (330) peak splitting Tetragonal phase K1 K2 E=10kV/cm K1 O-experimental data and --- fit data K2

Specimen Preparation Powders: 0.1m < particle size <40 m Peak broadening less diffraction occurring Double sided tape Glass slide Bulks: smooth surface after polishing, specimens should be thermal annealed to eliminate any surface deformation induced during polishing. http://www.youtube.com/watch?v=lwV5WCBh9a0 at~2:00-5:10

a b Do review problems for XRD Next Lecture Transmission Electron Microscopy a b 62