![Indexing of Planes and Directions -
Review
a
b
c
a
b
c
(111)
[110]
a direction: [uvw]
<uvw>: a set of equivalent
directions
a plane: (hkl)
{hkl}: a set of equi-
valent planes](https://image.slidesharecdn.com/topic5a-xrd-240304191648-95addc2a/85/XRD-calculations-and-characterization-pdf-17-320.jpg)
![3. By Laue Method - 1st Method Ever Used
Today - To Determine the Orientation of Single Crystals
Back-reflection Laue
Film
X-ray
crystal
crystal
Film
Transmission Laue
[001]
pattern](https://image.slidesharecdn.com/topic5a-xrd-240304191648-95addc2a/85/XRD-calculations-and-characterization-pdf-47-320.jpg)
XRD-calculations and characterization.pdf
- 1. X-ray Diffraction (XRD) • 1.0 What is X-ray Diffraction • 2.0 Basics of Crystallography • 3.0 Production of X-rays • 4.0 Applications of XRD • 5.0 Instrumental Sources of Error • 6.0 Conclusions
- 2. English physicists Sir W.H. Bragg and his son Sir W.L. Bragg developed a relationship in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (theta, θ). The variable d is the distance between atomic layers in a crystal, and the variable lambda λ is the wavelength of the incident X-ray beam; n is an integer. This observation is an example of X-ray wave interference (Roentgenstrahlinterferenzen), commonly known as X-ray diffraction (XRD), and was direct evidence for the periodic atomic structure of crystals postulated for several centuries. n λ =2dsinθ Bragg’s Law
- 3. Although Bragg's law was used to explain the interference pattern of X-rays scattered by crystals, diffraction has been developed to study the structure of all states of matter with any beam, e.g., ions, electrons, neutrons, and protons, with a wavelength similar to the distance between the atomic or molecular structures of interest. n λ =2dsinθ Bragg’s Law The Braggs were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS and diamond.
- 4. Deriving Bragg’s Law: nλ = 2dsinθ X-ray 1 X-ray 2 Constructive interference occurs only when n λ = AB + BC AB=BC n λ = 2AB Sinθ=AB/d AB=dsinθ n λ =2dsinθ λ = 2dhklsinθhkl AB+BC = multiples of nλ
- 5. Constructive and Destructive Interference of Waves Constructive Interference In Phase Destructive Interference Out of Phase
- 6. 1.0 What is X-ray Diffraction ? I www.micro.magnet.fsu.edu/primer/java/interference/index.html
- 7. Why XRD? • Measure the average spacings between layers or rows of atoms • Determine the orientation of a single crystal or grain • Find the crystal structure of an unknown material • Measure the size, shape and internal stress of small crystalline regions
- 8. X-ray Diffraction (XRD) The atomic planes of a crystal cause an incident beam of X-rays to interfere with one another as they leave the crystal. The phenomenon is called X-ray diffraction. incident beam diffracted beam film crystal Effect of sample thickness on the absorption of X-rays http://www.matter.org.uk/diffraction/x-ray/default.htm
- 9. Detection of Diffracted X-rays by Photographic film A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. Debye - Scherrer Camera Film X-ray film sample 2θ = 0° 2θ = 180° Point where incident beam enters
- 10. Bragg’s Law and Diffraction: How waves reveal the atomic structure of crystals n λ = 2dsinθ Atomic plane d=3 Å λ=3Å θ=30o n-integer X-ray1 X-ray2 l 2θ-diffraction angle Diffraction occurs only when Bragg’s Law is satisfied Condition for constructive interference (X-rays 1 & 2) from planes with spacing d http://www.eserc.stonybrook.edu/ProjectJava/Bragg/
- 11. Planes in Crystals-2 dimension To satisfy Bragg’s Law, θ must change as d changes e.g., θ decreases as d increases. λ = 2dhklsinθhkl Different planes have different spacings
- 12. 2.0 Basics of Crystallography 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 α, β and γ angles between a,b and c are lattice constants or parameters which can be determined by XRD. Beryl crystals smallest building block Unit cell Lattice (cm) (Å) CsCl d1 d2 d3 a b c α β γ
- 13. Seven Crystal Systems - Review
- 14. Miller Indices: hkl - Review (010) Miller indices-the reciprocals of the fractional intercepts which the plane makes with crystallographic axes 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 ∞ 0 1 0 0 1 0 h k l 4/ ∞ =0 a b c a b c
- 15. Several Atomic Planes and Their d-spacings in a Simple Cubic - Review a b c 1 0 0 1 0 0 Cubic a=b=c=a0 a b c 1 1 0 1 1 0 a b c 1 1 1 1 1 1 a b c 0 1 ½ 0 1 2 d100 d012 (100) (110) (111) Black numbers-fractional intercepts, Blue numbers-Miller indices (012)
- 16. Planes and Spacings - Review
- 17. Indexing of Planes and Directions - Review a b c a b c (111) [110] a direction: [uvw] <uvw>: a set of equivalent directions a plane: (hkl) {hkl}: a set of equi- valent planes
- 18. 3.0 Production of X-rays Cross section of sealed-off filament X-ray tube target X-rays tungsten filament Vacuum X-rays are produced whenever high-speed electrons collide with a metal target. A source of electrons – hot W filament, a high accelerating voltage between the cathode (W) and the anode and a metal target, Cu, Al, Mo, Mg. The anode is a water-cooled block of Cu containing desired target metal. glass X-rays copper cooling water electrons vacuum metal focusing cap beryllium window to transformer
- 19. Characteristic X-ray Lines Spectrum of Mo at 35kV Kα1 Kα Kβ λ (Å) <0.001Å Kα2 Kβ and Kα2 will cause extra peaks in XRD pattern, and shape changes, but can be eliminated by adding filters. ----- is the mass absorption coefficient of Zr. Intensity
- 20. Specimen Preparation Double sided tape Glass slide Powders: 0.1µm < particle size <40 µm Peak broadening less diffraction occurring Bulks: smooth surface after polishing, specimens should be thermal annealed to eliminate any surface deformation induced during polishing.
- 21. JCPDS Card 1.file number 2.three strongest lines 3.lowest-angle line 4.chemical formula and name 5.data on diffraction method used 6.crystallographic data 7.optical and other data 8.data on specimen 9.data on diffraction pattern. Quality of data Joint Committee on Powder Diffraction Standards, JCPDS (1969) Replaced by International Centre for Diffraction Data, ICDF (1978)
- 22. A Modern Automated X-ray Diffractometer X-ray Tube Detector Sample stage Cost: $560K to 1.6M θ 2θ
- 23. Basic Features of Typical XRD Experiment X-ray tube 1) Production 2) Diffraction 3) Detection 4) Interpretation
- 24. Detection of Diffracted X-rays by a Diffractometer Photon counter Detector Amplifier C Circle of Diffractometer Recording Focalization Circle Bragg - Brentano Focus Geometry, Cullity
- 25. Peak Position d-spacings and lattice parameters λ = 2dhklsinθhkl Fix λ (Cu kα) = 1.54Å dhkl = 1.54Å/2sinθhkl For a simple cubic (a = b = c = a0) a0 = dhkl /(h2 +k2 +l2 )½ e.g., for NaCl, 2θ220=46o , θ220=23o , d220 =1.9707Å, a0=5.5739Å (Most accurate d-spacings are those calculated from high-angle peaks) 2 2 2 0 l k h a dhkl + + =
- 26. Bragg’s Law and Diffraction: How waves reveal the atomic structure of crystals n λ = 2dsinθ Atomic plane d=3 Å λ=3Å θ=30o n-integer X-ray1 X-ray2 l 2θ-diffraction angle Diffraction occurs only when Bragg’s Law is satisfied Condition for constructive interference (X-rays 1 & 2) from planes with spacing d http://www.eserc.stonybrook.edu/ProjectJava/Bragg/ a0 = dhkl /(h2 +k2 +l2 )½ e.g., for NaCl, 2θ220=46o , θ220=23o , d220 =1.9707Å, a0=5.5739Å
- 27. XRD Pattern of NaCl Powder I Diffraction angle 2θ (degrees) (Cu Kα) Miller indices: The peak is due to X- ray diffraction from the {220} planes.
- 28. Significance of Peak Shape in XRD 1. Peak position 2. Peak width 3. Peak intensity
- 29. Peak Width-Full Width at Half Maximum FWHM Important for: • Particle or grain size 2. Residual strain Bragg angle 2θ Intensity Background Peak position 2θ Imax 2 max I max I mode Can also be fit with Gaussian, Lerentzian, Gaussian-Lerentzian etc.
- 30. No Strain Uniform Strain (d1-do)/do Non-uniform Strain d1≠constant Peak moves, no shape changes Peak broadens Effect of Lattice Strain on Diffraction Peak Position and Width Diffraction Line do d1 Shifts to lower angles Exceeds d0 on top, smaller than d0 on the bottom RMS Strain
- 31. 4.0 Applications of XRD • XRD is a nondestructive technique • To identify crystalline phases and orientation • To determine structural properties: Lattice parameters (10-4Å), strain, grain size, expitaxy, phase composition, preferred orientation (Laue) order-disorder transformation, thermal expansion • To measure thickness of thin films and multi-layers* • To determine atomic arrangement • Detection limits: ~3% in a two phase mixture; can be ~0.1% with synchrotron radiation Spatial resolution: normally none
- 32. 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).
- 33. Mr. Hanawalt Powder diffraction files: The task of building up a collection of known patterns was initiated by Hanawalt, Rinn, and Fevel at the Dow Chemical Company (1930’s). They obtained and classified diffraction data on some 1000 substances. After this point several societies like ASTM (1941-1969) and the JCPS began to take part (1969-1978). In 1978 it was renamed the Int. Center for Diffraction Data (ICDD) with 300 scientists worldwide. In 1995 the powder diffraction file (PDF) contained nearly 62,000 different diffraction patterns with 200 new being added each year. Elements, alloys, inorganic compounds, minerals, organic compounds, organo-metallic compounds. Hanawalt: Hanawalt decided that since more than one substance can have the same or nearly the same d value, each substance should be characterized by it’s three strongest lines (d1, d2, d3). The values of d1- d3 are usually sufficient to characterize the pattern of an unknown and enable the corresponding pattern in the file to be located.
- 34. a b c 2θ a. Cubic a=b=c, (a) b. Tetragonal a=b≠c (a and c) c. Orthorhombic a≠b≠c (a, b and c) • Number of reflections • Peak position • Peak splitting Phase Identification - Effect of Symmetry on XRD Pattern
- 35. More Applications of XRD Diffraction patterns of three Superconducting thin films annealed for different times. a. Tl2CaBa2Cu2Ox (2122) b. Tl2CaBa2Cu2Ox (2122)+ Tl2Ca2Ba2Cu3Oy (2223) b = a + c c. Tl2Ca2Ba2Cu3Oy (2223) CuO was detected by comparison to standards a b c (004) (004) Intensity
- 36. XRD Studies • Temperature • Electric Field • Pressure • Deformation
- 37. Effect of Coherent Domain Size (331) Peak of cold-rolled and Annealed 70Cu-30Zn (brass) 2θ Kα1 Kα2 As rolled 200o C 250o C 300o C 450o C As rolled 300o C 450o C Increasing Grain size (t) Peak Broadening Scherrer Model As grain size decreases hardness increases and peaks become broader Intensity ANNEALING TEMPERATURE (°C) HARDNESS (Rockwell B) θ λ Cos t B ⋅ ⋅ = 9 . 0
- 38. High Temperature XRD Patterns of the Decomposition of YBa2Cu3O7-δ T 2θ I Intensity (cps)
- 39. In Situ X-ray Diffraction Study of an Electric Field Induced Phase Transition Single Crystal Ferroelectric 92%Pb(Zn1/3Nb2/3)O3 -8%PbTiO3 E=6kV/cm E=10kV/cm (330) Kα1 Kα2 Kα1 Kα2 (330) peak splitting is due to Presence of <111> domains Rhombohedral phase Intensity (cps) Intensity (cps) No (330) peak splitting Tetragonal phase
- 40. What Is A Synchrotron? A synchrotron is a particle acceleration device which, through the use of bending magnets, causes a charged particle beam to travel in a circular pattern. Advantages of using synchrotron radiation: •Detecting the presence and quantity of trace elements •Providing images that show the structure of materials •Producing X-rays with 108 more brightness than those from normal X-ray tube (tiny area of sample) •Having the right energies to interact with elements in light atoms such as carbon and oxygen •Producing X-rays with wavelengths (tunable) about the size of atom, molecule and chemical bonds
- 41. Synchrotron Light Source Cost: $Bi Diameter: 2/3 length of a football field
- 42. 5.0 Instrumental Sources of Error • Specimen displacement • Instrument misalignment • Error in zero 2θ position • Peak distortion due to Kα2 and Kβ wavelengths
- 43. 6.0 Conclusions • Non-destructive, fast, easy sample prep • High-accuracy for d-spacing calculations • Can be done in-situ • Single crystal, poly, and amorphous materials • Standards are available for thousands of material systems
- 44. XRF: X-Ray Fluorescence XRF is a ND technique used for chemical analysis of materials. An X- ray source is used to irradiate the specimen and to cause the elements in the specimen to emit (or fluoresce) their characteristic X-rays. A detection system (wavelength dispersive) is used to measure the peaks of the emitted X-rays for qual/quant measurements of the elements and their amounts. The techniques was extended in the 1970’s to to analyze thin films. XRF is routinely used for the simultaneous determination of elemental composition and film thickness. Analyzing Crystals used: LiF (200), (220), graphite (002), W/Si, W/C, V/C, Ni/C
- 45. XRF Setup 1) X-ray irradiates specimen 2) Specimen emits characteristic X-rays or XRF 3) Analyzing crystal rotates to accurately reflect each wavelength and satisfy Bragg’s Law 4) Detector measures position and intensity of XRF peaks XRF is diffracted by a crystal at different φ to separate X-ray λ and to identify elements I 2φ NiKα nλ=2dsinφ - Bragg’s Law 2) 1) 3) 4)
- 46. Preferred Orientation A condition in which the distribution of crystal orientations is non-random, a real problem with powder samples. It is noted that due to preferred orientation several blue peaks are completely missing and the intensity of other blue peaks is very misleading. Preferred orientation can substantially alter the appearance of the powder pattern. It is a serious problem in experimental powder diffraction. Intensity Random orientation ------ Preferred orientation ------
- 47. 3. By Laue Method - 1st Method Ever Used Today - To Determine the Orientation of Single Crystals Back-reflection Laue Film X-ray crystal crystal Film Transmission Laue [001] pattern