Roger Johnson Structure and Dynamics: X-ray Diffraction Lecture 6
|
|
- Ann Wilkerson
- 5 years ago
- Views:
Transcription
1 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 be comparable to, or smaller than, interatomic distances. In a crystal, atoms are typically spaced at distances of the order m (the Angstrom, Å) and X-rays have a wavelength in the range 0.1 to 6 Å. We will learn how x-ray diffraction can be employed to determine a crystal s lattice parameters, space group and atomic fractional coordinates. Furthermore, the energy levels of bound electrons are such that the elements have characteristic absorption lines in the x-ray region of the electromagnetic spectrum, which can also be exploited Elastic X-ray scattering The elementary scatterer of x-rays is the electron. Here we consider an elastic scattering process in which the energy of the incident x-ray is the same as that scattered, and momentum is conserved through momentum transfer to the scattering medium. It is sufficient to consider a classical model in which the electric field of the incident radiation exerts a force on the electron, causing it to accelerate, oscillate, and hence emit the scattered x-ray. In a typical experiment an x-ray beam quantified by its flux, Φ 0, (rate of photons passing through a unit area) is incident on the scatterer. The intensity of the scattered x-rays, I, (rate of scattered photons) is measured by a detector that subtends a solid angle, Ω, positioned some distance, R, away from the scatterer. The efficiency of the scattering process is quantified by the differential scattering cross-section defined as: = I dω Φ 0 Ω The differential scattering cross-section for one electron can be shown to be ( ) ( dσ e 2 ) 2 = dω 4πɛ 0 mc 2 P = r0p 2 where r 0 is the Thomson scattering length, or the classical radius of the electron (r 0 = 2.82x10 15 m). Accordingly, the scattering process takes the name Thomson scattering. P = ˆɛ ˆɛ 2 is a polarisation factor, where ˆɛ and ˆɛ are the polarisation of the incident and scattered x- rays, respectively. We define a scattering amplitude, A, such that = AA dω and A electron = r 0 P 1 2. Page 1 of 6
2 Figure 1: The atomic form factors for hydrogen and neon. The scattering of x-rays by an atom is calculated by considering scattering from the electron charge density of the atom, ρ(r), in units of the electron charge. We assume that the incident and scattered x-ray beams are plane waves with wave vectors k and k, respectively. X-rays scattered from a volume element of the charge density at some vector r from the origin will acquire a phase difference of φ(r) = (k k ) r = Q r, where Q is the scattering vector. Integrating over the full charge density we have f 0 (Q) = ρ(r)e iq r dr where f 0 (Q) is the atomic form factor. At Q = 0, all electrons scatter in-phase, and f 0 (Q) = Z, the total number of electrons. As Q tends to infinity, f 0 (Q) tends to zero. The hydrogen and neon form factors are plotted in Figure 1 Itinerant electrons contribute weakly to the scattering of x-rays, as the atomic form factor decays rapidly with increasing Q. The majority of x-ray scattering occurs through interaction with tightly bound core electrons, and it is important to consider the quantisation of energy levels below the itinerant continuum. When the incident x-ray energy is near to an atomic resonance (excitation of a core electron into the continuum followed by instantaneous decay back to the core state) we see a departure from energy-independent Thomson scattering. Such resonant effects on the scattering process can be accounted for by including energy dependent, complex terms in the atomic form factor, known as anomalous terms: f(q) = f 0 (Q) + f ( hω) + if ( hω) The anomalous terms are independent of Q, but strongly dependent on the x-ray energy hω, and become large near atomic resonances. Below an atomic resonance the core electrons are tightly bound, whereas above the resonance energy they become essentially free. Their ability to respond to the driving force of Page 2 of 6
3 Figure 2: Anomalous terms in the neon atomic form factor. the incident x-ray s electric field is therefore reduced below the resonance, and increased above. This is captured by f ( hω). Atomic resonances coincide with x-ray absorption events, and the respective dispersion is contained in the complex term if ( hω). Both terms are plotted as a function of energy for neon in Figure 2. Finally, the scattering amplitude for an atom is the atomic form factor times the electron scattering amplitude. Hence, the differential scattering cross-section for the atom is = A atom (Q)A dω atom(q) = f(q) 2 r0p X-ray scattering from a perfect crystal The scattering amplitude of an atom is equal to the Fourier transform of the charge density times the electron scattering length times a polarisation factor. The same is true for a crystal, but now the charge density is periodic by the translational symmetry of the lattice, and we might have multiple atoms in the basis. We saw in Lecture 4 that the Fourier transform of a real space lattice is the reciprocal space lattice. Therefore, the scattering amplitude of a crystal is proportional to the reciprocal lattice of the crystal weighted by the structure factor: A crystal = r 0 P 1 2 F (Q) where the summation is over all reciprocal lattice points, R, and F (Q) = j f j(q)e Q r j is the structure factor summed over all j atoms in a single unit cell. The differential cross-section is then: = A crystal (Q)A crystal(q) = r 20P N (2π)3 δ(q τ ) F (Q) 2 dω V In summary, we find that the differential cross-section for a crystal is proportional to the squared modulus of the structure factor and to the number of unit cells in the sample. It is only non-zero if Q is equal to a reciprocal lattice vector, τ. R e iq R The delta function δ(q τ ) is traditionally expressed as the Laue equation, Q = τ which describes the condition for diffraction, where Q = ha + kb + lc. τ Page 3 of 6
4 Figure 3: The scattering triangle Figure 4: The diffraction condition (expanded scattering triangle). By geometry (see Figure 3) we find that the length of the scattering vector, Q = k k, is related to the total scattering angle, 2θ, between k and k by Q = 4π λ sin(θ) There exists a family of crystal planes perpendicular to Q that we also label with the Miller indices (h, k, l). The spacing between the planes is d = 2π Q Combining the above two equations we obtain Bragg s Law, λ = 2d sin(θ) which is a real space analogy to the Laue equation. The Laue equation is more attractive as it allows us to use integer values of h, k, and l. However, the Bragg equation is perhaps more intuitive, and helps us understand the experimental condition for diffraction. We must orient our crystal planes of spacing d with respect to the incident x-ray beam at the scattering angle, θ, such that the diffraction condition is met. We must then position our detector to measure the peak diffraction intensity at an angle 2θ with respect to the incident beam. We can see this geometry by expanding the scattering triangle, shown in Figure 4. Page 4 of 6
5 6.4. Determining the lattice parameters By measuring the scattering angle of diffraction peaks with known h, k, and l, and with non-parallel scattering vectors, we can determine the lattice parameters of the crystal (using monochromatic x-rays): sin hkl (θ) = λ h [h, k, l] 2 G k l where the reciprocal metric tensor, G is a symmetric 3x3 tensor, written in terms of the lattice parameters with 6 unknowns. Aside: It is often the case that one cannot identify the Miller indices of a reflection without prior knowledge of the lattice parameters. In a typical experiment on an unknown crystal one has to measure a very large number of diffraction peaks, and use computer algorithms to fit the lattice parameters Determining the space group The space group is determined as follows: a) Having determined the lattice parameters, label every measured diffraction peak with Miller indices (hkl). b) Identify if any peaks are systematically absent, and determine whether or not the absences are consistent with a lattice centring extinction condition or roto-translation extinction conditions, or both. c) Determine the relative value of F (hkl) 2 for each reflection. The measured diffraction intensity for a given peak is related to F (hkl) 2 by I = Ψ 0 L(θ)P A(θ, λ)r 2 0N F (hkl) 2 Here, L(θ) is a correction, known as the Lorentz correction, which is applied in order to transform intensities measured in terms of angles of the experiment, into peaks in reciprocal space (it is a Jacobian), and A(θ, λ) is an absorption correction. The correction terms L(θ), P, and A(θ, λ) are included here for completeness, but they are non-examinable. You would be given F (Q) 2 directly. d) Identify any symmetry in the values of F (hkl) 2 in reciprocal space in order to determine the Laue Class. By combining the Laue Class with the observed extinction conditions we can determine the space group up to the presence of inversion symmetry Determining the atomic fractional coordinates We have shown that the structure factor is proportional to the Fourier transform of the charge density, ρ(r), integrated over the unit cell. Therefore, by taking the inverse Fourier transform of the structure factor we can find an expression for the charge density, which is dependent upon the element types and their fractional coordinates. ρ(r) = 1 F (Q)e iq r V Q Page 5 of 6
6 To determine the charge density exactly one needs in principle to know the structure factor at every point of the reciprocal lattice. However, for a given wavelength we can only measure up to a value of Q max = 2 k i, giving our structure solution a real space resolution of 2π/ Q max. A direct reconstruction of the charge density from diffraction data is impossible. In a diffraction experiment we measure intensities, which are proportional to the modulus square of the structure factor. Therefore, all phase information in the scattering amplitude, Q r, is lost. This is known as the phase problem. Mathematical tools have been developed, known as direct methods, that attempt to reconstruct the charge density from diffraction data based upon physical constraints on the inverse Fourier transform. For example, the charge density must be positive (for x-rays), and it must resemble a periodic array of atoms. In many cases one has a good guess at the crystal structure, owing largely to years of work by crystallographers who have assembled an extensive database. In these cases it is sufficient to refine a good guess directly against F (Q) 2 values using non-linear least squares methods, so long as the crystal structure can be quantified by a modest number of free parameters. For example, consider a 1D diatomic chain: We want to measure the position of atom B with respect to atom A, which we define with the fractional coordinate, x. F (h) = f A (h) + f B (h)e 2πihx I(h) F (h) 2 = f 2 A(h) + f 2 B(h) + 2f A (h)f B (h)cos(2πhx) So if we measure the series of diffraction peaks h = 1, 2, 3, 4..., we will find that their relative intensity is modulated with a periodicity dependent upon the fractional coordinate, x. Page 6 of 6
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
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 informationScattering by two Electrons
Scattering by two Electrons p = -r k in k in p r e 2 q k in /λ θ θ k out /λ S q = r k out p + q = r (k out - k in ) e 1 Phase difference of wave 2 with respect to wave 1: 2π λ (k out - k in ) r= 2π S r
More informationResolution: 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 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 informationPHYS Introduction to Synchrotron Radiation
PHYS 570 - Introduction to Synchrotron Radiation Term: Spring 2015 Meetings: Tuesday & Thursday 17:00-18:15 Location: 204 Stuart Building Instructor: Carlo Segre Office: 166A Life Sciences Phone: 312.567.3498
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 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 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 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 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 informationAn Introduction to Diffraction and Scattering. School of Chemistry The University of Sydney
An Introduction to Diffraction and Scattering Brendan J. Kennedy School of Chemistry The University of Sydney 1) Strong forces 2) Weak forces Types of Forces 3) Electromagnetic forces 4) Gravity Types
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 informationStructural characterization. Part 1
Structural characterization Part 1 Experimental methods X-ray diffraction Electron diffraction Neutron diffraction Light diffraction EXAFS-Extended X- ray absorption fine structure XANES-X-ray absorption
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 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 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 informationCrystal Structure SOLID STATE PHYSICS. Lecture 5. A.H. Harker. thelecture thenextlecture. Physics and Astronomy UCL
Crystal Structure thelecture thenextlecture SOLID STATE PHYSICS Lecture 5 A.H. Harker Physics and Astronomy UCL Structure & Diffraction Crystal Diffraction (continued) 2.4 Experimental Methods Notes: examples
More informationScattering of Electromagnetic Radiation. References:
Scattering of Electromagnetic Radiation References: Plasma Diagnostics: Chapter by Kunze Methods of experimental physics, 9a, chapter by Alan Desilva and George Goldenbaum, Edited by Loveberg and Griem.
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 informationInteraction X-rays - Matter
Interaction X-rays - Matter Pair production hν > M ev Photoelectric absorption hν MATTER hν Transmission X-rays hν' < hν Scattering hν Decay processes hν f Compton Thomson Fluorescence Auger electrons
More informationSOLID STATE 9. Determination of Crystal Structures
SOLID STATE 9 Determination of Crystal Structures In the diffraction experiment, we measure intensities as a function of d hkl. Intensities are the sum of the x-rays scattered by all the atoms in a crystal.
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 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 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 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 informationScattering and Diffraction
Scattering and Diffraction Adventures in k-space, part 1 Lenson Pellouchoud SLAC / SSL XSD summer school 7/16/018 lenson@slac.stanford.edu Outline Elastic Scattering eview / overview and terminology Form
More informationDIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY. Regents' Professor enzeritus Arizona State University
DIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY Regents' Professor enzeritus Arizona State University 1995 ELSEVIER Amsterdam Lausanne New York Oxford Shannon Tokyo CONTENTS Preface to the first
More informationWhich of the following can be used to calculate the resistive force acting on the brick? D (Total for Question = 1 mark)
1 A brick of mass 5.0 kg falls through water with an acceleration of 0.90 m s 2. Which of the following can be used to calculate the resistive force acting on the brick? A 5.0 (0.90 9.81) B 5.0 (0.90 +
More informationSetting The motor that rotates the sample about an axis normal to the diffraction plane is called (or ).
X-Ray Diffraction X-ray diffraction geometry A simple X-ray diffraction (XRD) experiment might be set up as shown below. We need a parallel X-ray source, which is usually an X-ray tube in a fixed position
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 information1.4 The Compton Effect
1.4 The Compton Effect The Nobel Prize in Physics, 1927: jointly-awarded to Arthur Holly Compton (figure 9), for his discovery of the effect named after him. Figure 9: Arthur Holly Compton (1892 1962):
More informationSolid State Spectroscopy Problem Set 7
Solid State Spectroscopy Problem Set 7 Due date: June 29th, 2015 Problem 5.1 EXAFS Study of Mn/Fe substitution in Y(Mn 1-x Fe x ) 2 O 5 From article «EXAFS, XANES, and DFT study of the mixed-valence compound
More informationProtein crystallography. Garry Taylor
Protein crystallography Garry Taylor X-ray Crystallography - the Basics Grow crystals Collect X-ray data Determine phases Calculate ρ-map Interpret map Refine coordinates Do the biology. Nitrogen at -180
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 informationPROBING CRYSTAL STRUCTURE
PROBING CRYSTAL STRUCTURE Andrew Baczewski PHY 491, October 10th, 2011 OVERVIEW First - we ll briefly discuss Friday s quiz. Today, we will answer the following questions: How do we experimentally probe
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 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 informationCrystal Structure and Dynamics
Crystal Structure and Dynamics Paolo G. Radaelli, Michaelmas Term 2012 Part 2: Scattering theory and experiments Lectures 5-7 Web Site: http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option
More informationX-ray diffraction is a non-invasive method for determining many types of
Chapter X-ray Diffraction.1 Introduction X-ray diffraction is a non-invasive method for determining many types of structural features in both crystalline and amorphous materials. In the case of single
More informationPHYS Introduction to Synchrotron Radiation
C. Segre (IIT) PHYS 570 - Spring 2018 January 09, 2018 1 / 20 PHYS 570 - Introduction to Synchrotron Radiation Term: Spring 2018 Meetings: Tuesday & Thursday 13:50-15:05 Location: 213 Stuart Building Instructor:
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 informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle 5.7 Probability,
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 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 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 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 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 information2 X-ray and Neutron Scattering
X-ray and Neutron Scattering F. Fiori, F. Spinozzi Theoretical aspects of X-ray and neutron scattering are presented and discussed in this chapter. Theoretical concepts are linked to the most actual and
More informationAPEX CARE INSTITUTE FOR PG - TRB, SLET AND NET IN PHYSICS
Page 1 1. Within the nucleus, the charge distribution A) Is constant, but falls to zero sharply at the nuclear radius B) Increases linearly from the centre, but falls off exponentially at the surface C)
More informationX-Ray Scattering Studies of Thin Polymer Films
X-Ray Scattering Studies of Thin Polymer Films Introduction to Neutron and X-Ray Scattering Sunil K. Sinha UCSD/LANL Acknowledgements: Prof. R.Pynn( Indiana U.) Prof. M.Tolan (U. Dortmund) Wilhelm Conrad
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 informationStructure and Dynamics : An Atomic View of Materials
Structure and Dynamics : An Atomic View of Materials MARTIN T. DOVE Department ofearth Sciences University of Cambridge OXFORD UNIVERSITY PRESS Contents 1 Introduction 1 1.1 Observations 1 1.1.1 Microscopic
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 information6. X-ray Crystallography and Fourier Series
6. X-ray Crystallography and Fourier Series Most of the information that we have on protein structure comes from x-ray crystallography. The basic steps in finding a protein structure using this method
More informationNeutron and x-ray spectroscopy
Neutron and x-ray spectroscopy B. Keimer Max-Planck-Institute for Solid State Research outline 1. self-contained introduction neutron scattering and spectroscopy x-ray scattering and spectroscopy 2. application
More informationCHEM-E5225 :Electron Microscopy. Diffraction 1
CHEM-E5225 :Electron Microscopy Diffraction 1 2018-10-15 Yanling Ge Text book: Transmission electron microscopy by David B Williams & C. Barry Carter. 2009, Springer Outline Diffraction in TEM Thinking
More informationMP464: Solid State Physics Problem Sheet
MP464: Solid State Physics Problem Sheet 1) Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred
More informationPhysics 504, Lecture 22 April 19, Frequency and Angular Distribution
Last Latexed: April 16, 010 at 11:56 1 Physics 504, Lecture April 19, 010 Copyright c 009 by Joel A Shapiro 1 Freuency and Angular Distribution We have found the expression for the power radiated in a
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 informationParticles and Waves Particles Waves
Particles and Waves Particles Discrete and occupy space Exist in only one location at a time Position and velocity can be determined with infinite accuracy Interact by collisions, scattering. Waves Extended,
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 informationWave properties of matter & Quantum mechanics I. Chapter 5
Wave properties of matter & Quantum mechanics I Chapter 5 X-ray diffraction Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed. Crystals
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 1 (2/3/04) Overview -- Interactions, Distributions, Cross Sections, Applications
.54 Neutron Interactions and Applications (Spring 004) Chapter 1 (/3/04) Overview -- Interactions, Distributions, Cross Sections, Applications There are many references in the vast literature on nuclear
More information(10%) (c) What other peaks can appear in the pulse-height spectrum if the detector were not small? Give a sketch and explain briefly.
Sample questions for Quiz 3, 22.101 (Fall 2006) Following questions were taken from quizzes given in previous years by S. Yip. They are meant to give you an idea of the kind of questions (what was expected
More informationChapter 27. Quantum Physics
Chapter 27 Quantum Physics Need for Quantum Physics Problems remained from classical mechanics that relativity didn t explain Blackbody Radiation The electromagnetic radiation emitted by a heated object
More informationMP464: Solid State Physics Problem Sheet
MP464: Solid State Physics Problem Sheet 1 Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred rectangular
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 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 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 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 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 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 informationCS273: Algorithms for Structure Handout # 13 and Motion in Biology Stanford University Tuesday, 11 May 2003
CS273: Algorithms for Structure Handout # 13 and Motion in Biology Stanford University Tuesday, 11 May 2003 Lecture #13: 11 May 2004 Topics: Protein Structure Determination Scribe: Minli Zhu We acknowledge
More informationX-rays and their interaction with matter
1 X-rays and their interaction with matter X-rays were discovered by Wilhelm Conrad Röntgen in 1895. Since that time they have become established as an invaluable probe of the structure of matter. The
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 informationLecture 21 Reminder/Introduction to Wave Optics
Lecture 1 Reminder/Introduction to Wave Optics Program: 1. Maxwell s Equations.. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave
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 informationLecture 5 Scattering geometries.
Lecture 5 Scattering geometries. 1 Introduction In this section, we will employ what we learned about the interaction between radiation and matter, as well as the methods of production of electromagnetic
More informationLecture 5 Wave and particle beams.
Lecture 5 Wave and particle beams. 1 What can we learn from scattering experiments The crystal structure, i.e., the position of the atoms in the crystal averaged over a large number of unit cells and over
More informationThe Larmor Formula (Chapters 18-19)
2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 1 The Larmor Formula (Chapters 18-19) T. Johnson Outline Brief repetition of emission formula The emission from a single free particle - the Larmor
More informationApplied Nuclear Physics (Fall 2006) Lecture 19 (11/22/06) Gamma Interactions: Compton Scattering
.101 Applied Nuclear Physics (Fall 006) Lecture 19 (11//06) Gamma Interactions: Compton Scattering References: R. D. Evans, Atomic Nucleus (McGraw-Hill New York, 1955), Chaps 3 5.. W. E. Meyerhof, Elements
More informationis the minimum stopping potential for which the current between the plates reduces to zero.
Module 1 :Quantum Mechanics Chapter 2 : Introduction to Quantum ideas Introduction to Quantum ideas We will now consider some experiments and their implications, which introduce us to quantum ideas. The
More informationExercise 1 Atomic line spectra 1/9
Exercise 1 Atomic line spectra 1/9 The energy-level scheme for the hypothetical one-electron element Juliettium is shown in the figure on the left. The potential energy is taken to be zero for an electron
More informationSmall Angle Neutron Scattering in Different Fields of Research. Henrich Frielinghaus
Small Angle Neutron Scattering in Different Fields of Research Henrich Frielinghaus Jülich Centre for Neutron Science Forschungszentrum Jülich GmbH Lichtenbergstrasse 1 85747 Garching (München) h.frielinghaus@fz-juelich.de
More informationLattice Vibrations. Chris J. Pickard. ω (cm -1 ) 200 W L Γ X W K K W
Lattice Vibrations Chris J. Pickard 500 400 300 ω (cm -1 ) 200 100 L K W X 0 W L Γ X W K The Breakdown of the Static Lattice Model The free electron model was refined by introducing a crystalline external
More informationPhonons I - Crystal Vibrations (Kittel Ch. 4)
Phonons I - Crystal Vibrations (Kittel Ch. 4) Displacements of Atoms Positions of atoms in their perfect lattice positions are given by: R 0 (n 1, n 2, n 3 ) = n 10 x + n 20 y + n 30 z For simplicity here
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 informationHigh-Resolution. Transmission. Electron Microscopy
Part 4 High-Resolution Transmission Electron Microscopy 186 Significance high-resolution transmission electron microscopy (HRTEM): resolve object details smaller than 1nm (10 9 m) image the interior of
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 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 informationDetermining Protein Structure BIBC 100
Determining Protein Structure BIBC 100 Determining Protein Structure X-Ray Diffraction Interactions of x-rays with electrons in molecules in a crystal NMR- Nuclear Magnetic Resonance Interactions of magnetic
More informationIntroduction to Crystallography and Electron Diffraction
Introduction to Crystallography and Electron Diffraction Marc De Graef Carnegie Mellon University Sunday July 24, 2016 M&M Conference, July 24-28, 2016, Columbus, OH Overview Introductory remarks Basic
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 informationNeutron Instruments I & II. Ken Andersen ESS Instruments Division
Neutron Instruments I & II ESS Instruments Division Neutron Instruments I & II Overview of source characteristics Bragg s Law Elastic scattering: diffractometers Continuous sources Pulsed sources Inelastic
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More information2018 Quantum Physics
2018 Quantum Physics Text: Sears & Zemansky, University Physics www.masteringphysics.com Lecture notes at www.tcd.ie/physics/study/current/undergraduate/lecture-notes/py1p20 TCD JF PY1P20 2018 J.B.Pethica
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 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 informationMain Notation Used in This Book
Main Notation Used in This Book z Direction normal to the surface x,y Directions in the plane of the surface Used to describe a component parallel to the interface plane xoz Plane of incidence j Label
More information