3 Crystal Structure Analysis Crystal Structure Analysis Xray diffraction 2. Single crystal Diffraction 3. Xray Powder Diffraction Electron Diffraction Neutron diffraction Essence of diffraction: Bragg Diffraction Crystals. Powders, and Diffraction X-ray Diffraction X'ray Diffraction-When an X'ray beam pattern.This phenomenon, known as X ra a bombards a crystal, the atomic structure of th crystal causes the beam to scatter in a spe X ravs and, the ds tanees bet weee atoms in the crystal are of similar magnitude Unit Cell Bragg Ang Crystal (Crystallite X-ray Powder Pattern Laue equations crystal? periodic repetition of identical unit Diffraction cells? diffraction grid constructive interference of scattered waves only in particular directions a( COS O-cosa)=hλ
1 3 Crystal Structure Analysis 1. Bragg Equation 2. Single Crystal Diffraction 3. X-ray Powder Diffraction Crystal Structure Analysis X-ray diffraction Electron Diffraction Neutron Diffraction Essence of diffraction: Bragg Diffraction Intensity Bragg Angle Unit Cell Crystal (Crystallite) X-ray Powder Pattern Electron Diffraction Powder Pattern Crystals, Powders, and Diffraction 0 Polycrystalline Specimen X-ray Diffraction ¾ When an X-ray beam bombards a crystal, the atomic structure of the crystal causes the beam to scatter in a specific pattern. This phenomenon, known as X-ray diffraction, occurs when the wavelength of the X rays and the distances between atoms in the crystal are of similar magnitude. X-ray Diffraction Laue Equations crystal ? periodic repetition of identical unit cells ? diffraction grid ? constructive interference of scattered waves only in particular directions: von Laue Diffraction a(cosa0 - cos a) = hl
von laue diffraction Laue diffraction For different incidence directions. the Single stal N b(co$。-cosp)=k dcosyo -cosy)=D. Laues the Bragg Condition of Crystal Diffraction Experiments Einstein AStrong reflection of the incident wave will occur for regards this the set of incident angles that satisfy the bragg (a)show its four-fold axis (b)show its three-fold axis Geometry of Bragg Diffraction Braggs Law ni=2dh sine a When xrays are scattered from a crystal lattice eaks of tensity are observe correspond to the following condition a The angle of incidence angle of scattering e The pathlength difference is equal to an inte The conditions for maximum intensity contained in B saw to calculate details about the crystal structure, or if the crystal Path difference for diffraction of rays from structure is known, to determine the wavelength adjacent planes is 2dhksinO, which must of the x rays incident upon the crystal correspond to ni for constructive interference
2 von Laue Diffraction ß For different incidence directions, the diffraction patterns are different g - g = l b - b = l a - a = l c(cos cos ) l b(cos cos ) k a(cos cos ) h 0 0 0 Laue Diffraction Single crystal Laue¢s Experiments Laue Images of Zincblende ZnS (a)show its four-fold axis，(b)show its three-fold axis Einstein regards this experiment as the most beautiful one in physics. Nobel Prize winner of 1914 the Bragg Condition of Crystal Diffraction d q l Strong reflection of the incident wave will occur for the set of incident angles that satisfy the Bragg condition: 2dsinq =nl the index n defines the path difference between waves i & ii when the diffraction occurs … for a given value of n this path difference is nl. Geometry of Bragg Diffraction Path difference for diffraction of rays from adjacent planes is 2dhklsinq, which must correspond to nl for constructive interference. q q dhklsinq dhklsinq q q nl = 2dhkl sinq Bragg's Law When x-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the following conditions: The angle of incidence = angle of scattering The pathlength difference is equal to an integer number of wavelengths. The conditions for maximum intensity contained in Bragg's law above allow us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the x-rays incident upon the crystal
Braggs Law The braggs British physicists William Henry Bragg n2= 2dsine (1862-1942) and William Lawrence bragg where n= order of diffraction (1890-1971)won Nobel Phvsical Prize in 1915 2=Xray wavelength due to their achievements on the structure Analysis via X'ray d= spacing between layers of atom 0 angle of diffraction Braggs Law is the fundamental law of x crvst Generation of X-rays X-ray Emission Spectrum knock inner core electrons are produced by thermionic emission from a W filament and are accelerated by a large potential difference innermost shell (K) I. the high energy electrons (? 50 kev)bombard a metal these vacancies are rapidly filled by electronic transitions from target (usually Cu, but can also be Mo the other orbitals not all transitions are possibl X-rays are generated by eraction between electron Ithe wavelengths are characteristic of the nd targe -target element 一 Heated filament 15418A PD? 50kV M anode Copper anode igh intensity X-rays can be generated using a part on X-ray generation Using a Syne Synchrotron Radiation resolution much quicker data collection. emit radiation tangentially. A particular wavelength can be lected from the continuous spectrum of X-rays generated. Synchrotron radiation: tunable Storage r
3 nl = 2dsinq where n Þ order of diffraction l Þ X-ray wavelength d Þ spacing between layers of atom q Þ angle of diffraction Bragg's Law is the fundamental law of x-ray crystallography. Bragg's Law The Braggs British physicists William Henry Bragg (1862~1942) and William Lawrence Bragg (1890~1971) won Nobel Physical Prize in 1915 due to their achievements on the Structure Analysis via X-ray. Generation of X-rays Copper anode Heated tungsten filament electrons X-rays - + cathode anode PD ? 50 kV ß electrons are produced by thermionic emission from a W filament and are accelerated by a large potential difference ß the high energy electrons (? 50 keV) bombard a metal target (usually Cu, but can also be Mo) ß X-rays are generated by the interaction between electrons and target X-ray Emission Spectrum ßupon collisions the high energy electrons can knock inner core electrons from the target atoms, leaving vacancies in the innermost shell (K) ß these vacancies are rapidly filled by electronic transitions from the other orbitals not all transitions are possible the wavelengths are characteristic of the target element Intensity l Wavelength c Kb2 Kb1 Ka1 Ka2 Copper anode: Ka 1.5418 Å Kb 1.3922 Å K M L Ka2 Ka1 Kb1 Kb2 X-ray Generation Using a Synchrotron High intensity X-rays can be generated using a particle accelerator such as a synchrotron: charged particles (electron or positrons) are accelerated round a circle and emit radiation tangentially. A particular wavelength can be selected from the continuous spectrum of X-rays generated. Synchrotron radiation: ß tunable ß intense X-rays X-rays beam Synchrotron Radiation More intense X-rays at shorter wavelengths mean higher resolution & much quicker data collection
Xray generators-The Synchrotron Schematic llustration angement for using bragg of an X-ray Diffractometer amples is illustrated determined by the r energy and Since it is usually difficult rajectory. When the particle the electrons(or positrons)are detector is moved through an accelerated toward the center of Radiation Facility Grenoble, France ectromagnetic radiation, an to mitted includes high energy x' particular planes can be of Common Mechanical Schematic Illustration of Xray Diffraction Movement in powder Diffractometers Matched = characteristic xrays Matched filters ar beam to optimize the fraction of energy which is in the Ka line. 0-20 geometry The Bragg- Brentano diffractometer is the dominant laue stationary single crvstal tube moves (and the specimen is fixed), this is called 0 Monochromatic Powder: specimen is polycrystalline, etry. The essential characteristics are (1)The relationship between e (the angle pecimen surface and the incident xray beam) and 20 e Jong Bouman' single crystal rota tes the incident b m and the illates about chosen axis in path of iving slit detector)is maintained throughout I Precession; chosen axis of single crystal (2)r, and r, are fixed and equal and define a diffractometer circle in which the specimen is always at the center
4 X-ray Generators ¾ The Synchrotron European Synchrotron Radiation Facility Grenoble, France Electrons (or positrons) are released from a particle accelerator into a storage ring. The trajectory of the particles is determined by their energy and the local magnetic field. Magnets of various types are used to manipulate the particle trajectory. When the particle beam is “bent”by the magnets, the electrons (or positrons) are accelerated toward the center of the ring. Charged particles moving under the influence of an accelerating field emit electromagnetic radiation, and when they are moving at close to relativistic speeds, the radiation emitted includes high energy xray radiation.. Schematic Illustration of an X-ray Diffractometer X-RAY SOURCE CRYSTAL DETECTOR PATH OF DETECTOR An experimental arrangement for using Bragg diffraction to determine the structure of single-crystal samples is illustrated schematically left Since it is usually difficult to move the X-ray source the sample itself is rotated with respect to the source and when the sample is moved through an angle q the detector is moved through an angle 2q. The wavelength of the x-ray source is well known so by measuring the angles at which strong diffraction peaks (i.e. strong detector signal) occur the spacing of particular planes can be determined. Type Tube Specimen Receiving Slit r1 r2 Brag -Brentano q:2q Fixed Varies as q Varies as 2q Fixed =r1 Brag -Brentano q:q Varies as q Fixed Varies as q Fixed =r1 Seeman-Bohlin Fixed Fixed Varies as 2q Fixed variabl e Texture Sensitive (Ladel) Fixed Varies as q processes about a Varies as 2q Fixed variabl e * Generally fixed, but can rotate about a or rock about goniometer axis. Common Mechanical Movement in Powder Diffractometers Schematic Illustration of X-ray Diffraction To obtain nearly monochromatic x-rays, an x-ray tube is used to produce characteristic x-rays. Matched filters are used in the x-ray beam to optimize the fraction of the energy which is in the Ka line. q-2q geometry The Bragg-Brentano diffractometer is the dominant geometry found in most laboratories. In this system, if the tube is fixed, this is called q-2 q geometry. If the tube moves (and the specimen is fixed), this is called q : q geometry. The essential characteristics are: (1) The relationship between q (the angle between the specimen surface and the incident x-ray beam) and 2q (the angle between the incident beam and the receiving slit detector) is maintained throughout the analysis. (2) r1 and r2 are fixed and equal and define a diffractometer circle in which the specimen is always at the center. Radiation Method White Laue: stationary single crystal Monochromatic Powder: specimen is polycrystalline, and therefore all orientations are simultaneously presented to the beam Rotation, Weissenberg: oscillation De Jong-Bouman: single crystal rota tes or oscillates about chosen axis in path of beam Precession: chosen axis of single crystal precesses about beam direction
Diffraction can occur whenever Braggs law is tisfied. With monochromatic radiation. an XRD: "Rocking" Curve Scan arbitrary setting of a single crystal in an x'ray beams. There would therefore be very littl(ed am will not generally produce any diffrac mple normal information in a single crystal diffraction pattern from using monochromatic radiation This problem can be by continuously A or 0 over a range of values, to satisf Rock"Sample Braggs law. Practically this is done by e Vary orientation of Ak relative to sample normal while (Dusing a range of x ray wavelengths (i.e. white Rock"sample over a very small angular range. Dby rotating the crystal or, using a powder or b Resulting data of Intensity vs theta (0, sample angle) polycrystalline specimen. shows detailed structure of diffraction peak being XRD: Rocking Curve Example Rocking Curves assessing crystal quality 002) Reflection T through 0 with the counter set at a known bragg angle, 20. The resulting intensity versus 0 curve is known as a rocking curve. The width of the rocking curve is a direct measure of the range of orientation on mosaic s read present in the irradiated area of the crystal, as each sub grain of is rotated. For a film that isn't truely epitaxial the width of a rocking curve of the layer peak will b infraction peak showing its detailed structure ment of the quality of the layer Rotating Crystal Method Laue method In the rotating crystal method. The laue method is mainly used to determin reflected from or transmitted throt rotated about the chosen axis. As the crvstal rotates, sets of lattice point make the Iw wn correct Bragg angle for the radiation that satisfies the bragg law for the values of d monochromatic incident beam and be formed aThe reflected beams are located surface of an imaginary cone whose axis is the out flat Experimental two practical variants of the laue method, the horizontal lines back-reflection and the transmission Laue method
5 ß Diffraction can occur whenever Bragg's law is satisfied. With monochromatic radiation, an arbitrary setting of a single crystal in an x-ray beam will not generally produce any diffracted beams. There would therefore be very little information in a single crystal diffraction pattern from using monochromatic radiation. ß This problem can be overcome by continuously varying l or q over a range of values, to satisfy Bragg's law. Practically this is done by: (1)using a range of x-ray wavelengths (i.e. white radiation), or (2)by rotating the crystal or, using a powder or polycrystalline specimen. XRD: “Rocking”Curve Scan Vary orientation of Dk relative to sample normal while maintaining its magnitude. How? “Rock”sample over a very small angular range. Resulting data of Intensity vs. theta (q, sample angle) shows detailed structure of diffraction peak being investigated. i k f k Dk “Rock”Sample Dk Sample normal XRD: Rocking Curve Example Rocking curve of single crystal GaN around (002) diffraction peak showing its detailed structure. 16.995 17.195 17.395 17.595 17.795 0 8000 16000 GaN Thin Film (002) Reflection Intensity (Counts/s) theta (deg) Rocking Curves ¾ assessing crystal quality To estimate the crystal quality a crystal is rotated through q with the counter set at a known Bragg angle, 2q. The resulting intensity versus q curve is known as a rocking curve. The width of the rocking curve is a direct measure of the range of orientation on mosaic spread present in the irradiated area of the crystal, as each sub grain of the crystal will come into orientation as the crystal is rotated. For a film that isn't truely epitaxial, the width of a rocking curve of the layer peak will be a measurement of the quality of the layer. Rotating Crystal Method In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic xray beam. A cylindrical film is placed around it and the crystal is rotated about the chosen axis. As the crystal rotates, sets of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, and at that point a diffracted beam will be formed. The reflected beams are located on the surface of imaginary cones. When the film is laid out flat, the diffraction spots lie on horizontal lines. Laue Method ß The Laue method is mainly used to determine the orientation of large single crystals. White radiation is reflected from, or transmitted through, a fixed crystal. ß The diffracted beams form arrays of spots, that lie on curves on the film. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d and q involved. Each curve therefore corresponds to a different wavelength. The spots lying on any one curve are reflections from planes belonging to one zone. Laue reflections from planes of the same zone all lie on the surface of an imaginary cone whose axis is the zone axis. ß Experimental ß There are two practical variants of the Laue method, the back-reflection and the transmission Laue method
Back-reflection Laue Methods Transmission Laue methods In the back-reflection In the transmission Laue ethod, the film is placed etween the laced behind the crystal and the crystal. The beams which are diffracted in a are transmitted through backward direction are the crystal Laue reflections is defined by the transmitted beam emitted beam. The The film intersects the the diffraction spots spots generally lying on ar generally lying on an hyperbola Chemical Crystallography- Single Crystal X-ray Diffraction Single Crystal Analysis at what angles for what orientations of the crystal with .Advantage: You can learn everything to know about the necessarily realize that there is more than one set 0.5mn Stoe IPDs Image plate diffraction System Single Crystal X-ray Diffraction( Cont) X-rays Diffraction Primary application is to determine atomic group, ete mmetry, unit cell dimensions, space struct erfect crystals. I(2 0) of delta function ctly sharp scattering WIL For imperfect crystals, the Thite xray" beam (xrays of variable7) such peaks are broadened that Bragg's equation would be satisfied by For liquids and glasses, it is a numerous atomic planes. ontinuous, slowly varying Modern me ation, Weissenberg tio w slm,function. rotatingeryst setup to X-ray diffraction works on the principle that xrays form limitations of the stationary methods predictable diffraction patterns when interacting with a crystalline matrix of atoms
6 Back-reflection Laue Methods ß In the back-reflection method, the film is placed between the x-ray source and the crystal. The beams which are diffracted in a backward direction are recorded. ß One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an hyperbola. Transmission Laue Methods ß In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal. ß One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse. Stoe IPDS Image Plate Diffraction System single crystal size < 0.5 mm Chemical Crystallography¾¾ Single Crystal Analysis Single Crystal X-ray Diffraction Single crystal x-ray diffraction is a kind of method by put a crystal in the beam, observing what reflections come out at what angles for what orientations of the crystal with what intensities. ßAdvantage: You can learn everything to know about the structure. ßDisadvantages: You however may not have a single crystal. It is time-consuming and difficult to orient the crystal. If more than one phase is present, you will not necessarily realize that there is more than one set of reflections. Single Crystal X-ray Diffraction (Cont.) ß Primary application is to determine atomic structure (symmetry, unit cell dimensions, space group, etc.,). ß Older methods used a stationary crystal with "white x-ray" beam (x-rays of variablel) such that Bragg's equation would be satisfied by numerous atomic planes. ß Modern methods (rotation, Weissenberg, precession, 4-circle) utilize various combination of rotating-crystal and camera setup to overcome limitations of the stationary methods X-rays Diffraction For perfect crystals, I(2 q) consists of delta functions (perfectly sharp scattering). For imperfect crystals, the peaks are broadened. For liquids and glasses, it is a continuous, slowly varying function. X-ray diffraction works on the principle that x-rays form predictable diffraction patterns when interacting with a crystalline matrix of atoms
Chemical Crystallography Powder analysi STADI-P Stoe Powder diffractometer Powder X-ray Diffraction Information from Powder XrD AMeasuring samples consisting of a collection of many small crystallites with random orientations. i Phase purity crystallinity of materials both qualitative and quantitative E Crystallinity amorphous content, particle size and strain sured powder patterns can be compared to a n Unit cell size and shape from peak position .Advantages over Single Crystal Diffraction It is usually much easier to prepare a powder sample You are guaranteed to see all reflections The Deby s wn eau Sherrer Amorphous Polycrystalline Crystalline
7 STADI-P Stoe Powder diffractometer powder sample in glass capillary Chemical Crystallography ¾¾ Powder Analysis powder Powder X-ray Diffraction Measuring samples consisting of a collection of many small crystallites with random orientations. Powder XRD is used routinely to assess the purity and crystallinity of materials Each crystalline phase has a unique powder diffraction pattern Measured powder patterns can be compared to a database for identification ßAdvantages over Single Crystal Diffraction It is usually much easier to prepare a powder sample. You are guaranteed to see all reflections. Information from Powder XRD Phase purity –both qualitative and quantitative Crystallinity –amorphous content, particle size and strain Unit cell size and shape –from peak positions 20 30 40 5 0 60 70 80 0 10000 20000 30000 40000 50000 20 3 0 40 50 6 0 70 80 0 50 100 150 200 20 30 40 50 60 70 80 200 400 600 800 1000 1200 1400 1600 1800 2000 Amorphous Polycrystalline Crystalline The DebyeSherrer Camera
I The Debye- Sherrer Camera Debye Scherrer Camera A very small amount of powdered material is sealed into a fine capillary tube made from pecimen is placed in the Debye Scherrer camera and is accurately aligned center of the camera. X-rays enter the camera 三 through a collimator. The powder diffracts the xrays in accordance with Braggs law to produce cones of diffracted 16·R ,10 ams.These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs is diffraction angle, R is radii of camera, 2L is the distance of on the film Debye-Scherrer Camera Powder diffraction film e Can record sections on these cones on film or some When the film is removed from the camera flattened ay of doing holes for the incident and transmitted beams sample with a strip of film Can cov this causes the highest angle back "reflected ares to intensities by electronically scanning film or measuring positions using a ruler and guessing the hole is for the transmitted beam and which is for the relative intensities using a "by eye comparison incident beam in the film X-ray powders Npos Peter Josephus Wilhelmus Debye and diffraction ●制‖‖( centered cubic stru diffraction )●0 Debye-Scherrer powder camera photographs of Zircon (ZrSio Zircon is a fairly complex tetragonal The schematic shows the debve cones that intersect tructure and this complexity is reflected in the tIons asured on the film to determine the d-spacings for diffraction pattern. he reflections measured
8 The Debye-Sherrer Camera 2L 180 4 R = p q · · p q = · 180 4 R 2 L q is diffraction angle, R is radii of camera, 2L is the distance of every pair of arcs in the image Debye Scherrer Camera ß A very small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract x-rays. The specimen is placed in the Debye Scherrer camera and is accurately aligned to be in the center of the camera. X-rays enter the camera through a collimator. ß The powder diffracts the x-rays in accordance with Braggs law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film. Debye-Scherrer Camera Can record sections on these cones on film or some other x-ray detector –Simplest way of doing this is to surround a capillary sample with a strip of film –Can covert line positions on film to angles and intensities by electronically scanning film or measuring positions using a ruler and guessing the relative intensities using a “by eye”comparison 1916 X-ray powders diffraction Powder Diffraction Film ß When the film is removed from the camera, flattened and processed, it shows the diffraction lines and the holes for the incident and transmitted beams. ß There are always two arcs in the x-ray beams Ka and Kb , this causes the highest angle back-reflected arcs to be doubled. From noting this, it is always clear which hole is for the transmitted beam and which is for the incident beam in the film. Dutch post stamp, 1936, memorizing Peter Josephus Wilhelmus Debye and his Nobel prize. ß The schematic shows the Debye cones that intersect the film in the camera, and how diffractions are measured on the film to determine the d-spacings for the reflections measured. Debye-Scherrer powder camera photographs of gold (Au), a Face centered cubic structure that exhibits a fairly simple diffraction Debye-Scherrer powder camera photographs of Zircon (ZrSiO4 ). Zircon is a fairly complex tetragonal structure and this complexity is reflected in the diffraction pattern
Measurement of Debve- Scherrer Powder X- ray Diffraction Photographs The pattern of lines on a (·) photograph (left figure) The distance Sl corresponds to HIT represents possible values of the gle between the diffracted Film from satisfy Braggs equation ways 20. We know that the distance between the holes in nλ=2 du sin e responds to a gle of0=π.Sowe -e can find e from 0=2 We know Braggs Law: ni=2dsin0 and the Interpretation of Powder Photographs equation for interplanar spacing, d, for cubic crystals is given b First task is to familiarize ourselves with these pattern. A The three most common structures are called face-centere cubic(FCC), body-centered cubic (BCC) and hexagonal close where a is the lattice parameter this gives packed (HCP) √h2+k2+12 [。m patterns of three common types of simple structunes I((b(ar Face-centered cubic generate a table of S,, 0 and sinze a,xv (b)Body-centered cubic (h (们c( c hexagonal clos Cubic structures Tetragonal and Hexagonal Structures For a cubic structure only one quantity is involved, he cell edge or the lattice parameter, we have For tetragonal structures, we have (h2+k2+12) Not al es of b2+k2+12 which we shall For he structures referred to hexagonal axes values ofn. the values of h k and l are (h2+hk+k2+212)
9 Measurement of Debye-Scherrer Photographs Film from powder camera laid flat. The pattern of lines on a photograph (left figure) represents possible values of the Bragg angles which satisfy Bragg’s equation: hkl hkl nl = 2d sin q Powder X-ray Diffraction (Powder diffraction film) The distance S1 corresponds to a diffraction angle of 2q. The angle between the diffracted and the transmitted beams is always 2q. We know that the distance between the holes in the film, W, corresponds to a diffraction angle of q = p . So we can find q from: 2W S1 p q = ) W S (1 2 2 - p or q = We know Bragg's Law: nl = 2dsinq and the equation for inter-planar spacing, d, for cubic crystals is given by: where a is the lattice parameter this gives: From the measurements of each arc we can now generate a table of S1 , q and sin2q. 2 2 2 hkl h k l a d + + = (h k l ) 4a sin 2 2 2 2 2 + + l q = Interpretation of Powder Photographs First task is to familiarize ourselves with these patterns. The three most common structures are called face-centered cubic (FCC), body-centered cubic (BCC) and hexagonal closepacked (HCP). Powder patterns of three common types of simple crystal structures. (a) Face-centered cubic (b) Body-centered cubic (c) Hexagonal closepacked Cubic Structures ßFor a cubic structure only one quantity is involved, the cell edge or the lattice parameter , we have Not all values of h2 + k2 + l2 , which we shall call N, are possible. Numbers such as 7, 15, 23, 28, 60 are said to be forbidden. For small values of N, the values of h, k and l are easily deduced. (h k l ) 4a sin 2 2 2 2 2 + + l q = Tetragonal and Hexagonal Structures ß For tetragonal structures, we have For hexagonal structures, or trigonal structures referred to hexagonal axes l ) c a (h k 4a sin 2 2 2 2 2 hkl 2 2 + + l q = l ) c a (h hk k 4a sin 2 2 2 2 2 hkl 2 2 + + + l q =
Indexing a Diffraction Pattern Indexing a diffraction pattern means assigning Miller indices hkl to each value of d If we know the unit cell, we can assign hkl 2dsin0=λ values to each d value using 1h2k212 Similarly. if we know hkl values, we can calculate the unit cell The split of the XRD lines However, often we don' t know hklor the unit 0o1100 he symmetry of structure decreased. the lines increased How Many Lines Are Possible? Observable diffraction n a cubic material, the largest d-spacing that can be observed is 100=010=001. For a primitive cell, we count h2+k2+P2 +k:+1 +上+ g e nd oes of M Danetng Planes tor acc and ke :1,2,3,4 1,5,6,8,9, Scumble -i Bcc: z4.6.8.10, 12. FCC:3,4,8,11,12,16,24 1 15 impossible Note we start with the largest d-spacing and work down Note: not all lines are present in every case 5i 3+1: 1 19. 3to 2d sin 0=n7. Ex: An element, BCC or FCC, shows diffraction peaks at Systematic Abser d centering Determine (a) Crystal structure?(b) Lattice constant? The presence of a centered lattice leads to the (e) What is the element. systematic absence of certain types of peak in For i centered lattices +k+l=2n for a line to be For an f centered lattice 43404721 h+k=2n. k+l=2n and h+I For a C centered lattice h+k=2 a=318A,BCC,→
10 Indexing a diffraction pattern means assigning Miller indices hkl to each value of d If we know the unit cell, we can assign hkl values to each d value using: Similarly, if we know hkl values, we can calculate the unit cell However, often we don’t know hkl or the unit cell… . 2 2 2 2 2 2 2 c l b k a h d 1 = + + Indexing a Diffraction Pattern 2dsinq = l The split of the XRD lines: The symmetry of structure decreased, the lines increased. h k l h2 + k 2 + l 2 h k l h2 + k2 + l 2 1 0 0 1 2 2 1, 3 0 0 9 1 1 0 2 3 1 0 10 1 1 1 3 3 1 1 11 2 0 0 4 2 2 2 12 2 1 0 5 3 2 0 13 2 1 1 6 3 2 1 14 2 2 0 8 4 0 0 16 In a cubic material, the largest d-spacing that can be observed is 100=010=001. For a primitive cell, we count according to h2+k2+l2 Note: 7 and 15 impossible Note: we start with the largest d-spacing and work down Note: not all lines are present in every case How Many Lines Are Possible? Observable diffraction peaks 2 2 2 h + k + l Ratio Simple cubic SC: 1,2,3,4,5,6,8,9,10,11,12.. BCC: 2,4,6,8,10, 12… . FCC: 3,4,8,11,12,16,24…. 2 2 2 hkl h k l a d + + = 2dsin q = nl Systematic Absences and Centering The presence of a centered lattice leads to the systematic absence of certain types of peak in the diffraction pattern For I centered lattices: h + k + l = 2n for a line to be present For an F centered lattice: h + k =2n, k + l = 2n and h + l = 2n For a C centered lattice: h + k =2n Ex: An element, BCC or FCC, shows diffraction peaks at 2q: 40, 58, 73, 86.8,100.4 and 114.7. Determine:(a) Crystal structure? (b) Lattice constant? (c) What is the element? 114.7 57.35 0.7090 6 (222) 100.4 50.2 0.5903 5 (310) 86.8 43.4 0.4721 4 (220) 73 36.5 0.3538 3 (211) 58 29 0.235 2 (200) 40 20 0.117 1 (110) 2theta theta (hkl) q 2 sin 2 2 2 h + k + l a =3.18 Å , BCC, ‡ W
