1 / 60

What is X-ray Diffraction Properties and generation of X-ray Bragg’s Law Basics of Crystallography

Attention Please!. From: LAM, Mandi [mailto:suetmlam5@cityu.edu.hk] Sent: Thursday, October 10, 2013 3:46 PM To: Email List Subject: Reminder: Completion of Teaching and Learning Questionnaire (TLQ) - AP 5301 Dear Students,

uta
Download Presentation

What is X-ray Diffraction Properties and generation of X-ray Bragg’s Law Basics of Crystallography

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Attention Please! From: LAM, Mandi [mailto:suetmlam5@cityu.edu.hk] Sent: Thursday, October 10, 2013 3:46 PMTo: Email ListSubject: 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 LamAP 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

  2. 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 http://www.matter.org.uk/diffraction/Default.htm

  3. http://www.youtube.com/watch?v=vYztZlLJ3ds ~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 Photograph of the hand of an old man using X-ray. http://www.youtube.com/watch?v=IRBKN4h7u80

  4. Properties and Generation of X-ray • X-rays are electromagneticradiation 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

  5. http://www.youtube.com/watch?v=lwV5WCBh9a0 to~1:08 A Modern Automated X-ray Diffractometer X-ray Tube Detector Sample stage Cost: $560K to 1.6M

  6. 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 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. http://www.youtube.com/watch?v=UjyHK7jy1Vw X-ray tube

  7. http://www.youtube.com/watch?v=Bc0eOjWkxpU at~1:10- 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 spectrumcaused 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

  8. 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. V – applied voltage

  9. 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

  10. Characteristic x-ray Spectra Z

  11. 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

  12. 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 ,  x x

  13. 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

  14. 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.

  15. 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.”

  16. Filtering of X-ray K absorption edge of Ni • 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. 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.

  17. What Is Diffraction? A wave interacts with A single particle The particle scatters the incident beam uniformly in alldirections. 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

  18. 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 B X-ray of  I d http://www.youtube.com/watch?v=1FwM1oF5e6o to~1:17 diffraction & interference

  19. Constructive and Destructive Interference of Waves Constructiveinterference 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 Constructive Interference Destructive Interference In Phase Out Phase http://www.youtube.com/watch?v=kSc_7XBng8w

  20. http://www.youtube.com/watch?v=hQUsnMzTdpU Bragg’s Law and X-ray DiffractionHow waves reveal the atomic structure of crystals n-integer Diffraction occurs only when Bragg’s Law is satisfied Condition for constructive interference (X-rays 1 & 2) from planes with spacing d nl = 2dsin() X-ray1 X-ray2 l =3Å =30o Atomic plane d=3Å 2-diffraction angle http://www.youtube.com/watch?v=UfDW0-kghmI at~3:00-5.50

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

  22. http://www.youtube.com/watch?v=wJ1s-Ztxuzg crystal lattice Basics of Crystallography 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. http://www.youtube.com/watch?v=Rm-i1c7zr6Q&list=TLyPTUJ62VYE4wC1snHSChDl0NGo9IK-Nl

  23. Seven crystal Systems SystemAxial lengths Unit cell and angles Rhombohedral a=b=c ==90o a 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 a c abc ===90o Triclinic abc 90o c a a b b

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

  25. 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 4 2 1 h k l 4Å 8Å 3Å  8Å  /4 1 /3 0 1 0 h k l http://www.youtube.com/watch?v=C9h1gLQmUto Miller indices of a given plane

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

  27. Indexing of Planes and Directions (111) c c [111] (110) b b [110] a a 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.

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

  29. XRD PatternSignificance of Peak Shape in XRD • Peak position • Peak width • Peak intensity http://www.youtube.com/watch?v=nstYtUFELVQ

  30. Peak PositionDetermined-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 Note: Most accurate d-spacings are those calculated from high-angle peaks.

  31. Determine crystal structure and atomic arrangement in a unit cell Peak Intensity 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/ 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 jthatom in the unit cell

  32. 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, .

  33. 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

  34. Cubic Structuresa = 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: 0,0,00,0,0,½, 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

  35. 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] axis [010] 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.

  36. 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. 1 1’ 2 2’ 3 3’ d001  d002 z Structure factor and intensity of diffraction (001) (002) FCC

  37. 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. 1 1’ 2 2’ 3 3’ d001 d002 Structure factor and intensity of diffraction /4 /4 /2 /2

  38. 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) Structure factor and intensity of diffraction for FCC z D B y A C x

  39. 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 Structure factor and intensity of diffraction for FCC 2i Ihkl lFhkll2 A(0,0,0),B(½,0,½), C(½,½,0) & D(0,½,½)

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

  41. Diffractions Possibly Present for Cubic Structures

  42. Peak Width - Full Width at Half Maximum (FWHM) Determine • Particle or grain size 2. Residual strain

  43. 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

  44. Effect of Lattice Strain on Diffraction Peak Position and Width No Strain Uniform Strain (d1-do)/do Peak moves, no shape changes Non-uniform Strain d1constant Peak broadens

  45. 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 orderbroad maxima Monatomic gas Atoms are arranged Scattering I perfectly at random decreases with 

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

  47. Diffraction of X-rays by Polycrystals 2 2 Powder Diffraction (most widely used) A powder sample is in fact an assemblage of small crystallites, oriented at random in space. d3 d1 d2 Powder sample d1 crystallite d2 d3 Polycrystalline sample http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:08-1:46

  48. X-ray detector Sample holder Detection of Diffracted X-ray by A Diffractometer X-ray tube • 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  2 http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:44-1:56 and 15:44-16:16

  49. 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).

  50. Quality of data JCPDS Card 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.

More Related