# Attention Please! What is X-ray Diffraction

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Attention Please! What is X-ray Diffraction
From: LAM, Mandi Sent: Thursday, October 10, :46 PM To: List Subject: Reminder: Completion of Teaching and Learning Questionnaire (TLQ) - AP 5301 Dear Students, To enable us to have a better understanding of your valuable comment on our teaching and learning activities, you are invited to complete the TLQ evaluation for AP 5301 as soon as possible and no later than 27 October 2013. You may access the TLQ system via one of the followings: (1)    Blackboard Portal (http://www.cityu.edu.hk/cityu/logon/eportal.htm) or (2)    TLQ Website for Students (http://onlinesurvey.cityu.edu.hk/) Thank you very much for your cooperation in advance. Regards, Mandi Lam AP General Office What is X-ray Diffraction Properties and generation of X-ray Bragg’s Law Basics of Crystallography XRD Pattern Powder Diffraction Applications of XRD

Lecture-5 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

X-ray and X-ray Diffraction
~3:10 X-ray and X-ray Diffraction X-ray was first discovered by W. C. Roentgen in 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 1985 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 ( 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 at~6:00 how to operate XRD equipment Sample stage Cost: \$560K to 1.6M

Production of X-rays - +
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 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. X-ray tube

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 Bremsstrahlung 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. 4/15/2017 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

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

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 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 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 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.

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 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 For demonstration of constructive and destructive interference of waves. Constructive Interference Destructive Interference In Phase Out Phase

Bragg’s Law and X-ray Diffraction How waves reveal the atomic structure of crystals 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 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. at~3:

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
crystal lattice 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?

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
fractional intercepts which the plane makes with crystallographic axes (010) a b c a b c Axial length 4Å 8Å 3Å Intercept lengths 1Å 4Å 3Å Fractional intercepts ¼ ½ 1 Miller indices h k l 4Å 8Å 3Å  8Å  / /3 h k l Miller indices of a given plane

Planes and Spacings - a

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

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) 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

Scattering of x-ray by an atom
x-ray also interact with the electrons in an atom through scattering, which may be understood as the redistribution of the x-ray energy spatially. The Atomic Scattering Factor, f is defined to describe this distribution of intensity with respect to the scattered angle, .

Atomic Scattering Factor - f
f is element-dependent and also dependent on the bonding state of the atoms. This parameter influence directly the diffraction intensity. Table of f values, as a function of (sin/), for the elements and some ionic states of the elements can be found from references. I  f Direction of incident beam atom

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 atoms [100] x 8 x 1/8 =1 8 x 1/8 + 1 = x 1/8 + 6 x 1/2 = 4 Location: 0,0,0 0,0,0, ½, ½, ½, ,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

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 

Diffraction of X-rays by Crystals Laue Method
at~1:20-3:00 Back-reflection Laue crystal Film X-ray [001] Transmission Laue crystal Film

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 Usually, there are two ways to collect diffracted X-rays. d1 crystallite d2 d3 Polycrystalline sample at~1:08-1:46

Detection of Diffracted X-ray by A Diffractometer
Sample holder X-ray detector 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 at~1:44-1: 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

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

Preferred Orientation (Texture)
Simple cubic Random orientation Texture 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)
By rotating the specimen about the three major axes as shown, these spatial variations in diffraction intensity can be measured. 4-Circle Goniometer For pole-figure measurement

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. at~2:00-5:10