What use is Reciprocal Space? An Introduction
|
|
- Shawn Francis
- 6 years ago
- Views:
Transcription
1 What use is Reciprocal Space? An Introduction a* b* x You are here John Bargar 5th Annual SSRL Workshop on Synchrotron X-ray Scattering Techniques in Materials and Environmental Sciences June 1-3, 2010
2 OUTLINE I. What is the reciprocal lattice? 1. Bragg s law. 2. Reciprocal Lattice. II. How do you use it? 1. Ewald sphere. 2. Types of scans: 3. Etc.
3 Starting from Braggs law Bragg s Law: nλ = 2d sinθ Good phenomenologically Good enough for a Nobel prize (1915) A B θ 2θ θ A B d d BUT There are a gabillion planes in a crystal. How do we keep track of them? How do we know where in space a crystal lattice will diffract? What are their diffraction intensities?
4 Better approach Make a map of the diffraction conditions of the crystal. For example, define a map spot for each diffraction condition. Each spot represents kajillions of parallel atomic planes. Such a map could provide a convenient way to describe the relationships between planes in a crystal a considerable simplification of a messy and redundant problem. Objective of this talk: show that the reciprocal lattice provides such a map
5 To show this, start again from diffracting planes Define unit vectors s 0, s Notice that s-s 0 = 2Sinθ Substitute in Bragg s law 1/d = 2Sinθ/λ A B θ s 0 s s 0 s θ A B Diffraction occurs when s-s 0 /λ =1/d s 0 2θ d d (Note, for those familiar with q q = 2π s-s 0 Bragg s law: q = 2π/d = 4πSinθ/ λ
6 Now, add the points A B Define a map point at the end of the scattering vector at Bragg condition θ s s 0 λ θ Map point A B Diffraction occurs when scattering vector connects to map point. Scattering vectors (s-s 0 /λ or q) have reciprocal lengths (1/λ). Diffraction points define a reciprocal lattice. 2θ d d Reciprocal cuz units are in inverse distance Vector representation carries Bragg s law into 3D.
7 Families of planes become points! Single point now represents all planes in all unit cells of the crystal that are parallel to the crystal plane of interest and have same d value. A s s 0 λ A B θ s 0 /λ s/λ B d d
8 Families of planes become points on a line! Parallel planes with different d-spacings have reciprocal lattice points on a line. A A Diffraction occurs when s-s 0 /λ = 1/d = 2Sinθ/λ B θ s 0 /λ s/λ B Larger d = smaller s-s 0 d 2d d
9 Families of planes become points on a line! Parallel planes with different d-spacings have reciprocal lattice points on a line. A A Diffraction occurs when s-s 0 /λ = 1/d = 2Sinθ/λ B θ s 0 /λ s/λ B Larger d = smaller s-s 0 d d 2d Smaller d = larger s-s 0 ½ d
10 Crystals have families of planes in real space
11 Differently oriented planes project into different directions in reciprocal space. Thus, the RECIPROCAL LATTICE is obtained s 0 s s 0 λ Origin 1/d b* (010) (110) a* (200) s Families of planes become points! Distances between origin and RL points give 1/d. Reciprocal Lattice Axes: a* normal to b-c plane b* normal to a-c plane c* normal to a-b plane Index RL points based upon axes Diffraction occurs when s-s 0 /λ = 1/d = 2Sinθ/λ
12 Symmetry and peaks The symmetry of the unit cell is reflected in the reciprocal lattice, which means s 0 s s 0 λ Origin 1/d a* b* (010) (110) s the locations and number of diffraction peaks is determined by unit cell symmetry (200) But, peak intensities are determined by atomic number and atomic position in unit cell (more on this from Apurva, Joanne, and Misra/Joanna)
13 Reciprocal Lattice of γ-lialo 2 (008) (600) (400) (004) (200) (110) a* b* c* a* a* c* Projection along c: hk0 layer Note 4-fold symmetry Projection along b: h0l layer a = b = 5.17 Å; c = 6.27 Å; P (tetragonal) a* = b* = 0.19 Å -1 ; c* = 0.16 Å -1 general systematic absences (00ln; l 4), ([2n-1]00)
14 OUTLINE I. What is the reciprocal lattice? 1. Bragg s law. 2. Reciprocal Lattice. II. How do you use it? 1. Ewald sphere. 2. Types of scans: Longitudinal or θ-2θ, Rocking curve scan Arbitrary reciprocal space scan
15 Graphical Representation of Bragg s Law Bragg s law is obeyed for any triangle inscribed within the circle: Sinθ = (1/d)/(2/λ) A A θ s 0 θ 2/λ s s s 0 = 1/d s 0
16 The Ewald Sphere An elegant way to understand / rationalize diffraction phenomena A Circumscribe circle with radius 2/λ around scattering vectors A s 0 /λ s/λ s s 0 λ =1/d s-s 0 /λ = 1/d = 2Sinθ/λ Diffraction occurs only when map point intersects circle.
17 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s 0 s
18 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ Reciprocal lattice rotates by θ during scan
19 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 2θ
20 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 2θ
21 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 2θ
22 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 2θ
23 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 2θ The periodicity of peaks is linear in units of Sinθ/λ - not θ! q-scanning is more efficient than θ scanning
24 2. Rocking Curve scan Sample moves on θ, Detector fixed Provides information on sample mosaicity & quality of orientation First crystallite s-s 0 /λ Second crystallite 2θ Third crystallite
25 2. Rocking Curve scan Sample moves on θ, Detector fixed Provides information on sample mosaicity & quality of orientation Reciprocal lattice rotates by θ during scan s-s 0 /λ 2θ
26 A related point: in a powder, orientational averaging produces rings instead of spots s 0 /λ s/λ
27 3. Arbitrary Reciprocal Lattice scans Choose path through RL to satisfy experimental need, (Arturas will give examples) s-s 0 /λ 2θ
28 A note about q In practice q is used instead of s-s 0 q = k -k 0 = 2π * s-s 0 q = 4πSinθ/λ A q A B θ k 0 k θ B d 2θ d
29 What we haven t talked about: Intensities of peaks (Mehta/Stubbs) Peak width & shape (Stubbs) Scattering from thin films (PM session)
30 QUIZ (summary): 1. Symmetry of the reciprocal lattice = symmetry of the unit cell. 2. Dimensions that are large in direct space are small in reciprocal space. 3. Which is a more useful parameter: q or 2θ? Answer: q is! 4. Why? Cuz q contains info about θ and λ! 5. Peak shape has information about what? (orientation, size, strain, etc Mehta talk).
31 The End The Beginning
Resolution: maximum limit of diffraction (asymmetric)
Resolution: maximum limit of diffraction (asymmetric) crystal Y X-ray source 2θ X direct beam tan 2θ = Y X d = resolution 2d sinθ = λ detector 1 Unit Cell: two vectors in plane of image c* Observe: b*
More informationThe Reciprocal Lattice
59-553 The Reciprocal Lattice 61 Because of the reciprocal nature of d spacings and θ from Bragg s Law, the pattern of the diffraction we observe can be related to the crystal lattice by a mathematical
More informationHandout 7 Reciprocal Space
Handout 7 Reciprocal Space Useful concepts for the analysis of diffraction data http://homepages.utoledo.edu/clind/ Concepts versus reality Reflection from lattice planes is just a concept that helps us
More informationX-ray, Neutron and e-beam scattering
X-ray, Neutron and e-beam scattering Introduction Why scattering? Diffraction basics Neutrons and x-rays Techniques Direct and reciprocal space Single crystals Powders CaFe 2 As 2 an example What is the
More informationSOLID STATE 18. Reciprocal Space
SOLID STATE 8 Reciprocal Space Wave vectors and the concept of K-space can simplify the explanation of several properties of the solid state. They will be introduced to provide more information on diffraction
More informationX-ray Diffraction. Diffraction. X-ray Generation. X-ray Generation. X-ray Generation. X-ray Spectrum from Tube
X-ray Diffraction Mineral identification Mode analysis Structure Studies X-ray Generation X-ray tube (sealed) Pure metal target (Cu) Electrons remover inner-shell electrons from target. Other electrons
More informationFROM DIFFRACTION TO STRUCTURE
3.012 Fund of Mat Sci: Structure Lecture 19 FROM DIFFRACTION TO STRUCTURE Images removed for copyright reasons. 3-fold symmetry in silicon along the [111] direction. Forward (left) and backward (right)
More information3.012 Fund of Mat Sci: Structure Lecture 18
3.012 Fund of Mat Sci: Structure Lecture 18 X-RAYS AT WORK An X-ray diffraction image for the protein myoglobin. Source: Wikipedia. Model of helical domains in myoglobin. Image courtesy of Magnus Manske
More informationData collection Strategy. Apurva Mehta
Data collection Strategy Apurva Mehta Outline Before.. Resolution, Aberrations and detectors During.. What is the scientific question? How will probing the structure help? Is there an alternative method?
More informationClass 29: Reciprocal Space 3: Ewald sphere, Simple Cubic, FCC and BCC in Reciprocal Space
Class 29: Reciprocal Space 3: Ewald sphere, Simple Cubic, FCC and BCC in Reciprocal Space We have seen that diffraction occurs when, in reciprocal space, Let us now plot this information. Let us designate
More information3.012 Structure An Introduction to X-ray Diffraction
3.012 Structure An Introduction to X-ray Diffraction This handout summarizes some topics that are important for understanding x-ray diffraction. The following references provide a thorough explanation
More informationBasic Crystallography Part 1. Theory and Practice of X-ray Crystal Structure Determination
Basic Crystallography Part 1 Theory and Practice of X-ray Crystal Structure Determination We have a crystal How do we get there? we want a structure! The Unit Cell Concept Ralph Krätzner Unit Cell Description
More informationScattering Techniques and Geometries How to choose a beamline. Christopher J. Tassone
Scattering Techniques and Geometries How to choose a beamline Christopher J. Tassone Why Care About Geometries? How do you decide which beamline you want to use? Questions you should be asking Do I want
More informationThe ideal fiber pattern exhibits 4-quadrant symmetry. In the ideal pattern the fiber axis is called the meridian, the perpendicular direction is
Fiber diffraction is a method used to determine the structural information of a molecule by using scattering data from X-rays. Rosalind Franklin used this technique in discovering structural information
More informationChapter 2. X-ray X. Diffraction and Reciprocal Lattice. Scattering from Lattices
Chapter. X-ray X Diffraction and Reciprocal Lattice Diffraction of waves by crystals Reciprocal Lattice Diffraction of X-rays Powder diffraction Single crystal X-ray diffraction Scattering from Lattices
More informationData processing and reduction
Data processing and reduction Leopoldo Suescun International School on Fundamental Crystallography 2014 May 1st, 2014 Reciprocal lattice c* b* b * dh' k' l' 1 dh' k' l' * dhkl 1 dhkl a a* 0 d hkl c bc
More informationThe structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures
Describing condensed phase structures Describing the structure of an isolated small molecule is easy to do Just specify the bond distances and angles How do we describe the structure of a condensed phase?
More informationCrystal Structure and Electron Diffraction
Crystal Structure and Electron Diffraction References: Kittel C.: Introduction to Solid State Physics, 8 th ed. Wiley 005 University of Michigan, PHY441-44 (Advanced Physics Laboratory Experiments, Electron
More informationAnalytical Methods for Materials
Analytical Methods for Materials Lesson 15 Reciprocal Lattices and Their Roles in Diffraction Studies Suggested Reading Chs. 2 and 6 in Tilley, Crystals and Crystal Structures, Wiley (2006) Ch. 6 M. DeGraef
More informationProtein Crystallography
Protein Crystallography Part II Tim Grüne Dept. of Structural Chemistry Prof. G. Sheldrick University of Göttingen http://shelx.uni-ac.gwdg.de tg@shelx.uni-ac.gwdg.de Overview The Reciprocal Lattice The
More informationFundamentals of X-ray diffraction
Fundamentals of X-ray diffraction Elena Willinger Lecture series: Modern Methods in Heterogeneous Catalysis Research Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals
More informationdisordered, ordered and coherent with the substrate, and ordered but incoherent with the substrate.
5. Nomenclature of overlayer structures Thus far, we have been discussing an ideal surface, which is in effect the structure of the topmost substrate layer. The surface (selvedge) layers of the solid however
More information3.012 PS Issued: Fall 2003 Graded problems due:
3.012 PS 4 3.012 Issued: 10.07.03 Fall 2003 Graded problems due: 10.15.03 Graded problems: 1. Planes and directions. Consider a 2-dimensional lattice defined by translations T 1 and T 2. a. Is the direction
More informationPSD '18 -- Xray lecture 4. Laue conditions Fourier Transform The reciprocal lattice data collection
PSD '18 -- Xray lecture 4 Laue conditions Fourier Transform The reciprocal lattice data collection 1 Fourier Transform The Fourier Transform is a conversion of one space into another space with reciprocal
More informationExperimental Determination of Crystal Structure
Experimental Determination of Crystal Structure Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 624: Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
More informationRoad map (Where are we headed?)
Road map (Where are we headed?) oal: Fairly high level understanding of carrier transport and optical transitions in semiconductors Necessary Ingredients Crystal Structure Lattice Vibrations Free Electron
More informationSolid State Physics 460- Lecture 5 Diffraction and the Reciprocal Lattice Continued (Kittel Ch. 2)
Solid State Physics 460- Lecture 5 Diffraction and the Reciprocal Lattice Continued (Kittel Ch. 2) Ewald Construction 2θ k out k in G Physics 460 F 2006 Lect 5 1 Recall from previous lectures Definition
More informationX-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity
X-ray Diraction Interaction o Waves Reciprocal Lattice and Diraction X-ray Scattering by Atoms The Integrated Intensity Basic Principles o Interaction o Waves Periodic waves characteristic: Frequency :
More informationScattering and Diffraction
Scattering and Diffraction Andreas Kreyssig, Alan Goldman, Rob McQueeney Ames Laboratory Iowa State University All rights reserved, 2018. Atomic scale structure - crystals Crystalline materials... atoms
More informationHomework 4 Due 25 October 2018 The numbers following each question give the approximate percentage of marks allocated to that question.
Name: Homework 4 Due 25 October 218 The numbers following each question give the approximate percentage of marks allocated to that question. 1. Use the reciprocal metric tensor again to calculate the angle
More informationV 11: Electron Diffraction
Martin-Luther-University Halle-Wittenberg Institute of Physics Advanced Practical Lab Course V 11: Electron Diffraction An electron beam conditioned by an electron optical system is diffracted by a polycrystalline,
More informationDiffraction Geometry
Diffraction Geometry Diffraction from a crystal - Laue equations Reciprocal lattice Ewald construction Data collection strategy Phil Evans LMB May 2013 MRC Laboratory of Molecular Biology Cambridge UK
More informationDiffraction Experiments, Data processing and reduction
Diffraction Experiments, Data processing and reduction Leopoldo Suescun Laboratorio de Cristalografía Química del Estado Sólido y Materiales, Facultad de Química, Universidad de la República, Montevideo,
More informationSwanning about in Reciprocal Space. Kenneth, what is the wavevector?
Swanning about in Reciprocal Space or, Kenneth, what is the wavevector? Stanford Synchrotron Radiation Laboratory Principles The relationship between the reciprocal lattice vector and the wave vector is
More informationSolid State Physics Lecture 3 Diffraction and the Reciprocal Lattice (Kittel Ch. 2)
Solid State Physics 460 - Lecture 3 Diffraction and the Reciprocal Lattice (Kittel Ch. 2) Diffraction (Bragg Scattering) from a powder of crystallites - real example of image at right from http://www.uni-wuerzburg.de/mineralogie/crystal/teaching/pow.html
More informationPhys 412 Solid State Physics. Lecturer: Réka Albert
Phys 412 Solid State Physics Lecturer: Réka Albert What is a solid? A material that keeps its shape Can be deformed by stress Returns to original shape if it is not strained too much Solid structure
More informationCrystals, X-rays and Proteins
Crystals, X-rays and Proteins Comprehensive Protein Crystallography Dennis Sherwood MA (Hons), MPhil, PhD Jon Cooper BA (Hons), PhD OXFORD UNIVERSITY PRESS Contents List of symbols xiv PART I FUNDAMENTALS
More informationde Broglie Waves h p de Broglie argued Light exhibits both wave and particle properties
de Broglie argued de Broglie Waves Light exhibits both wave and particle properties Wave interference, diffraction Particle photoelectric effect, Compton effect Then matter (particles) should exhibit both
More informationDiffraction. X-ray diffraction
Diffraction Definition (from Cambridge Advanced Learner s Dictionary ): - diffraction noun [U] SPECIALIZED (a pattern caused by) a change in the direction of light, water or sound waves - diffract verb
More informationSchool on Pulsed Neutrons: Characterization of Materials October Neurton Sources & Scattering Techniques (1-2)
1866-6 School on Pulsed Neutrons: Characterization of Materials 15-26 October 2007 Neurton Sources & Scattering Techniques (1-2) Guenter Bauer Forschungzentrum Julich GmbH Julich Germany The Abdus Salam
More information4. Other diffraction techniques
4. Other diffraction techniques 4.1 Reflection High Energy Electron Diffraction (RHEED) Setup: - Grazing-incidence high energy electron beam (3-5 kev: MEED,
More informationX-ray diffraction geometry
X-ray diffraction geometry Setting controls sample orientation in the diffraction plane. most important for single-crystal diffraction For any poly- (or nano-) crystalline specimen, we usually set: 1 X-ray
More informationData Collection. Overview. Methods. Counter Methods. Crystal Quality with -Scans
Data Collection Overview with a unit cell, possible space group and computer reference frame (orientation matrix); the location of diffracted x-rays can be calculated (h k l) and intercepted by something
More informationSurface Sensitivity & Surface Specificity
Surface Sensitivity & Surface Specificity The problems of sensitivity and detection limits are common to all forms of spectroscopy. In its simplest form, the question of sensitivity boils down to whether
More information2. Diffraction as a means to determine crystal structure
Page 1 of 22 2. Diffraction as a means to determine crystal structure Recall de Broglie matter waves: 2 p h E = where p = 2m λ h 1 E = ( ) 2m λ hc E = hυ = ( photons) λ ( matter wave) He atoms: [E (ev)]
More informationCrystal planes. Neutrons: magnetic moment - interacts with magnetic materials or nuclei of non-magnetic materials. (in Å)
Crystallography: neutron, electron, and X-ray scattering from periodic lattice, scattering of waves by periodic structures, Miller indices, reciprocal space, Ewald construction. Diffraction: Specular,
More informationIntroduction to Materials Science Graduate students (Applied Physics)
Introduction to Materials Science Graduate students (Applied Physics) Prof. Michael Roth Chapter Reciprocal Lattice and X-ray Diffraction Reciprocal Lattice - 1 The crystal can be viewed as made up of
More informationWave diffraction and the reciprocal lattice
Wave diffraction and the reciprocal lattice Dept of Phys M.C. Chang Braggs theory of diffraction Reciprocal lattice von Laue s theory of diffraction Braggs view of the diffraction (1912, father and son)
More informationWhat is a Quasicrystal?
July 23, 2013 Rotational symmetry An object with rotational symmetry is an object that looks the same after a certain amount of rotation. Rotational symmetry An object with rotational symmetry is an object
More informationX-ray Data Collection. Bio5325 Spring 2006
X-ray Data Collection Bio535 Spring 006 Obtaining I hkl and α (Ihkl) from Frame Images Braggs Law -predicts conditions for in-phase scattering by equivalent atoms lying in planes that transect a crystal.
More informationMaterials Science and Engineering 102 Structure and Bonding. Prof. Stephen L. Sass. Midterm Examination Duration: 1 hour 20 minutes
October 9, 008 MSE 0: Structure and Bonding Midterm Exam SOLUTIONS SID: Signature: Materials Science and Engineering 0 Structure and Bonding Prof. Stephen L. Sass Midterm Examination Duration: hour 0 minutes
More informationGeometry of Crystal Lattice
0 Geometry of Crystal Lattice 0.1 Translational Symmetry The crystalline state of substances is different from other states (gaseous, liquid, amorphous) in that the atoms are in an ordered and symmetrical
More information3.012 PS Issued: Fall 2003 Graded problems due:
3.012 PS 4 3.012 Issued: 10.07.03 Fall 2003 Graded problems due: 10.15.03 Graded problems: 1. Planes and directions. Consider a 2-dimensional lattice defined by translations T 1 and T 2. a. Is the direction
More informationCrystallography Reading: Warren, Chapters 2.1, 2.2, 2.6, 8 Surface symmetry: Can be a clue to underlying structure. Examples:
Crystallography Reading: Warren, Chapters 2.1, 2.2, 2.6, 8 Surface symmetry: Can be a clue to underlying structure. Examples: Snow (SnowCrystals.com) Bismuth (Bao, Kavanagh, APL 98 66103 (2005) Hexagonal,
More informationCrystal Structure Determination II
Crystal Structure Determination II Dr. Falak Sher Pakistan Institute of Engineering and Applied Sciences 09/10/2010 Diffraction Intensities The integrated intensity, I (hkl) (peak area) of each powder
More informationStructure of Surfaces
Structure of Surfaces C Stepped surface Interference of two waves Bragg s law Path difference = AB+BC =2dsin ( =glancing angle) If, n =2dsin, constructive interference Ex) in a cubic lattice of unit cell
More informationGeneral theory of diffraction
General theory of diffraction X-rays scatter off the charge density (r), neutrons scatter off the spin density. Coherent scattering (diffraction) creates the Fourier transform of (r) from real to reciprocal
More informationNew algoritms for electron diffraction of 3D protein crystals. JP Abrahams, D Georgieva, L Jiang, I Sikhuralidze, NS Pannu
New algoritms for electron diffraction of 3D protein crystals JP Abrahams, D Georgieva, L Jiang, I Sikhuralidze, NS Pannu Why new algorithms? New research questions New experimental techniques Better insight
More informationDetermining the distance between the planes of a model crystal lattice using Bragg s Law. Abstract
1 2 3 4 Determining the distance between the planes of a model crystal lattice using Bragg s Law Meena Sharma University of the Fraser Valley, Department of Physics, Abbotsford, V2S 7M8 5 6 7 Esther Campbell
More information2. Diffraction as a means to determine crystal structure
2. Diffraction as a means to determine crystal structure Recall de Broglie matter waves: He atoms: [E (ev)] 1/2 = 0.14 / (Å) E 1Å = 0.0196 ev Neutrons: [E (ev)] 1/2 = 0.28 / (Å) E 1Å = 0.0784 ev Electrons:
More informationPhysical Chemistry Analyzing a Crystal Structure and the Diffraction Pattern Virginia B. Pett The College of Wooster
Physical Chemistry Analyzing a Crystal Structure and the Diffraction Pattern Virginia B. Pett The College of Wooster L. W. Haynes and his Senior Independent Study students conducted the 2 + 2 photo addition
More informationX-ray analysis. 1. Basic crystallography 2. Basic diffraction physics 3. Experimental methods
X-ray analysis 1. Basic crystallography 2. Basic diffraction physics 3. Experimental methods Introduction Noble prizes associated with X-ray diffraction 1901 W. C. Roentgen (Physics) for the discovery
More informationGood Diffraction Practice Webinar Series
Good Diffraction Practice Webinar Series High Resolution X-ray Diffractometry (1) Mar 24, 2011 www.bruker-webinars.com Welcome Heiko Ress Global Marketing Manager Bruker AXS Inc. Madison, Wisconsin, USA
More information... 3, , = a (1) 3 3 a 2 = a (2) The reciprocal lattice vectors are defined by the condition a b = 2πδ ij, which gives
PHZ646: Fall 013 Problem set # 4: Crystal Structure due Monday, 10/14 at the time of the class Instructor: D. L. Maslov maslov@phys.ufl.edu 39-0513 Rm. 114 Office hours: TR 3 pm-4 pm Please help your instructor
More informationFormation of the diffraction pattern in the transmision electron microscope
Formation of the diffraction pattern in the transmision electron microscope based on: J-P. Morniroli: Large-angle convergent-beam diffraction (LACBED), 2002 Société Française des Microscopies, Paris. Selected
More informationStrain-induced single-domain growth of epitaxial SrRuO 3 layers on SrTiO 3 : a high-temperature x-ray diffraction study
Strain-induced single-domain growth of epitaxial SrRuO 3 layers on SrTiO 3 : a high-temperature x-ray diffraction study Arturas Vailionis 1, Wolter Siemons 1,2, Gertjan Koster 1 1 Geballe Laboratory for
More informationIntroduction to X-ray and neutron scattering
UNESCO/IUPAC Postgraduate Course in Polymer Science Lecture: Introduction to X-ray and neutron scattering Zhigunov Alexander Institute of Macromolecular Chemistry ASCR, Heyrovsky sq., Prague -16 06 http://www.imc.cas.cz/unesco/index.html
More informationSurface crystallography
Modern Methods in Heterogeneous Catalysis Research Surface crystallography Dirk Rosenthal Department of Inorganic Chemistry Fritz-Haber-Institut der MPG Faradayweg 4-6, DE 14195 Berlin Part of the lecture
More informationQuantum Condensed Matter Physics Lecture 5
Quantum Condensed Matter Physics Lecture 5 detector sample X-ray source monochromator David Ritchie http://www.sp.phy.cam.ac.uk/drp2/home QCMP Lent/Easter 2019 5.1 Quantum Condensed Matter Physics 1. Classical
More informationX-ray Crystallography. Kalyan Das
X-ray Crystallography Kalyan Das Electromagnetic Spectrum NMR 10 um - 10 mm 700 to 10 4 nm 400 to 700 nm 10 to 400 nm 10-1 to 10 nm 10-4 to 10-1 nm X-ray radiation was discovered by Roentgen in 1895. X-rays
More informationPrentice Hall Mathematics, Geometry 2009 Correlated to: Maine Learning Results 2007 Mathematics Grades 9-Diploma
A. NUMBER: Students use numbers in everyday and mathematical contexts to quantify or describe phenomena, develop concepts of operations with different types of numbers, use the structure and properties
More informationClass 27: Reciprocal Space 1: Introduction to Reciprocal Space
Class 27: Reciprocal Space 1: Introduction to Reciprocal Space Many properties of solid materials stem from the fact that they have periodic internal structures. Electronic properties are no exception.
More informationSupplementary Information
Supplementary Information Supplementary Table 1. Atomic details for the crystal structures of silver closo-boranes. See Table 1 for further details. α Ag 2 B 10 H 10 Wyckoff x y z U / Å 2 Occ. Ag 4d 0.250
More informationRoger Johnson Structure and Dynamics: X-ray Diffraction Lecture 6
6.1. Summary In this Lecture we cover the theory of x-ray diffraction, which gives direct information about the atomic structure of crystals. In these experiments, the wavelength of the incident beam must
More informationMATHEMATICS AS/M/P1 AS PAPER 1
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS AS PAPER 1 Bronze Set B (Edexcel Version) CM Time allowed: 2 hours Instructions to candidates:
More informationChap 3 Scattering and structures
Chap 3 Scattering and structures Dept of Phys M.C. Chang Von Laue was struck in 1912 by the intuition that X-ray might scatter off crystals in the way that ordinary light scatters off a diffraction grating.
More informationPSD '17 -- Xray Lecture 5, 6. Patterson Space, Molecular Replacement and Heavy Atom Isomorphous Replacement
PSD '17 -- Xray Lecture 5, 6 Patterson Space, Molecular Replacement and Heavy Atom Isomorphous Replacement The Phase Problem We can t measure the phases! X-ray detectors (film, photomultiplier tubes, CCDs,
More informationRajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi
TEQIP WORKSHOP ON HIGH RESOLUTION X-RAY AND ELECTRON DIFFRACTION, FEB 01, 2016, IIT-K. Introduction to x-ray diffraction Peak Positions and Intensities Rajesh Prasad Department of Applied Mechanics Indian
More informationCRYSTAL STRUCTURE, PHASE CHANGES, AND PHASE DIAGRAMS
CRYSTAL STRUCTURE, PHASE CHANGES, AND PHASE DIAGRAMS CRYSTAL STRUCTURE CRYSTALLINE AND AMORPHOUS SOLIDS Crystalline solids have an ordered arrangement. The long range order comes about from an underlying
More informationBasics of XRD part III
Basics of XRD part III Dr. Peter G. Weidler Institute of Functional Interfaces IFG 1 10/31/17 KIT The Research University of the Helmholtz Association Name of Institute, Faculty, Department www.kit.edu
More informationLecture 23 X-Ray & UV Techniques
Lecture 23 X-Ray & UV Techniques Schroder: Chapter 11.3 1/50 Announcements Homework 6/6: Will be online on later today. Due Wednesday June 6th at 10:00am. I will return it at the final exam (14 th June).
More informationTEP Examination of the structure of NaCl monocrystals with different orientations
Examination of the structure of NaCl TEP Related topics Characteristic X-radiation, energy levels, crystal structures, reciprocal lattices, Miller indices, atomic form factor, structure factor, and Bragg
More informationIntroduction to crystallography The unitcell The resiprocal space and unitcell Braggs law Structure factor F hkl and atomic scattering factor f zθ
Introduction to crystallography The unitcell The resiprocal space and unitcell Braggs law Structure factor F hkl and atomic scattering factor f zθ Introduction to crystallography We divide materials into
More informationChapter 1 X-ray Absorption Fine Structure (EXAFS)
1 Chapter 1 X-ray Absorption Fine Structure (EXAFS) 1.1 What is EXAFS? X-ray absorption fine structure (EXAFS, XAFS) is an oscillatory modulation in the X-ray absorption coefficient on the high-energy
More informationChapter 20: Convergent-beam diffraction Selected-area diffraction: Influence of thickness Selected-area vs. convergent-beam diffraction
1 Chapter 0: Convergent-beam diffraction Selected-area diffraction: Influence of thickness Selected-area diffraction patterns don t generally get much better when the specimen gets thicker. Sometimes a
More informationKeble College - Hilary 2012 Section VI: Condensed matter physics Tutorial 2 - Lattices and scattering
Tomi Johnson Keble College - Hilary 2012 Section VI: Condensed matter physics Tutorial 2 - Lattices and scattering Please leave your work in the Clarendon laboratory s J pigeon hole by 5pm on Monday of
More informationSummary Chapter 2: Wave diffraction and the reciprocal lattice.
Summary Chapter : Wave diffraction and the reciprocal lattice. In chapter we discussed crystal diffraction and introduced the reciprocal lattice. Since crystal have a translation symmetry as discussed
More informationRöntgenpraktikum. M. Oehzelt. (based on the diploma thesis of T. Haber [1])
Röntgenpraktikum M. Oehzelt (based on the diploma thesis of T. Haber [1]) October 21, 2004 Contents 1 Fundamentals 2 1.1 X-Ray Radiation......................... 2 1.1.1 Bremsstrahlung......................
More informationTHE FIVE TYPES OF PLANAR 2-D LATTICES. (d) (e)
THE FIVE TYPES OF PLANAR 2-D LATTICES (a) (d) (b) (d) and (e) are the same (e) (c) (f) (a) OBLIQUE LATTICE - NO RESTRICTIONS ON ANGLES BETWEEN THE UNIT CELL EDGES (b) RECTANGULAR LATTICE - ANGLE BETWEEN
More informationPhysics with Neutrons I, WS 2015/2016. Lecture 11, MLZ is a cooperation between:
Physics with Neutrons I, WS 2015/2016 Lecture 11, 11.1.2016 MLZ is a cooperation between: Organization Exam (after winter term) Registration: via TUM-Online between 16.11.2015 15.1.2015 Email: sebastian.muehlbauer@frm2.tum.de
More informationThe Solid State. Phase diagrams Crystals and symmetry Unit cells and packing Types of solid
The Solid State Phase diagrams Crystals and symmetry Unit cells and packing Types of solid Learning objectives Apply phase diagrams to prediction of phase behaviour Describe distinguishing features of
More informationLattice (Sieć) A collection of nodes, i.e. points with integral coordinates. In crystallography, a lattice is an
Prof. dr hab. Mariusz Jaskólski GLOSSARYUSZ TERMINÓW KRYSTALOGRAFICZNYCH (dla osób nie znających jeszcze krystalografii, ale znających język angielski) Symmetry (Symetria) Property of physical and mathematical
More informationHelpful resources for all X ray lectures Crystallization http://www.hamptonresearch.com under tech support: crystal growth 101 literature Spacegroup tables http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Crystallography
More informationIntroduction to Triple Axis Neutron Spectroscopy
Introduction to Triple Axis Neutron Spectroscopy Bruce D Gaulin McMaster University The triple axis spectrometer Constant-Q and constant E Practical concerns Resolution and Spurions Neutron interactions
More informationStructure Factors. How to get more than unit cell sizes from your diffraction data.
Structure Factors How to get more than unit cell sizes from your diffraction data http://homepages.utoledo.edu/clind/ Yet again expanding convenient concepts First concept introduced: Reflection from lattice
More informationSchematic representation of relation between disorder and scattering
Crystal lattice Reciprocal lattice FT Schematic representation of relation between disorder and scattering ρ = Δρ + Occupational disorder Diffuse scattering Bragg scattering ρ = Δρ + Positional
More information- A general combined symmetry operation, can be symbolized by β t. (SEITZ operator)
SPACE GROUP THEORY (cont) It is possible to represent combined rotational and translational symmetry operations in a single matrix, for example the C6z operation and translation by a in D 6h is represented
More informationScattering Lecture. February 24, 2014
Scattering Lecture February 24, 2014 Structure Determination by Scattering Waves of radiation scattered by different objects interfere to give rise to an observable pattern! The wavelength needs to close
More informationSynchrotron X-ray surface scattering techniques. Joanne E. Stubbs Center for Advanced Radiation Sources, GeoSoilEnviroCARS, University of Chicago
Synchrotron X-ray surface scattering techniques Joanne E. Stubbs Center for Advanced Radiation Sources, GeoSoilEnviroCARS, University of Chicago Mineral Surfaces Dynamic Geochemical Microenvironments Dissolution
More informationGEOL. 40 ELEMENTARY MINERALOGY
CRYSTAL DESCRIPTION AND CALCULATION A. INTRODUCTION This exercise develops the framework necessary for describing a crystal. In essence we shall discuss how we fix the position of any crystallographic
More information