DOCSLIB.ORG

Chapter 36 Diffraction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: August 15, 2020)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 36 Diffraction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: August 15, 2020) Chapter 36 Diffraction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: August 15, 2020) Interference is the more general concept: it refers to the phenomenon of waves interacting. Waves will add constructively or destructively according to their phase difference. Diffraction usually refers to the spreading wave pattern from a finite-width aperture. 1. Diffraction by a single slit Fig . Single slit diffraction. Imagine the slit divided into many narrow zones, width y (= = a/N). Treat each as a secondary source of light contributing electric field amplitude E to the field at P. We consider a linear array of N coherent point oscillators, which are each identical, even to their polarization. For the moment, we consider the oscillators to have no intrinsic phase difference. The rays shown are all almost parallel, meeting at some very distant point P. If the spatial extent of the array is comparatively small, the separate wave amplitudes arriving at P will be essentially equal, having traveled nearly equal distances, that is E E ()r E (r ) ... E (r ) E ()r 0 0 1 0 2 0 N 0 N The sum of the interfering spherical wavelets yields an electric field at P, given by the real part of i( kr1t ) i( kr2 t ) i( krN t ) E Re[ E0 ()r e E0 ()r e ... E0 ()r e ] i( kr1t ) ik( r2 r 1 ) ik( r3 r 1 ) ik( rN r1 )) Re[ E0 ()r e 1[ e e ... e ]] (( Note )) When the distances r1 and r 2 from sources 1 and 2 to the field point P are large compared with the separation , then these two rays from the sources to the point P are nearly parallel . The path difference r2 – r1 is essentially equal to sin . Here we note that the phase difference between adjacent zone is a k( r r ) k sin k( sin ) 2 1 N k( r3 r2 ) k( r4 r3 ) k( rN rN 1) 2 where k is the wavenumber, k . It follows that k( r2 r1) k( r3 r1) 2 k( r4 r1) 3 (rN r1) (N )1 Thus the field at the point P may be written as i( kr1 t) i i2 i( N )1 E Re[E0()r e 1[ e e ...e ]] We now calculate the complex number given by Z 1 ei ei2 ... ei( N )1 1 eiN 1 ei eiN 2/ (eiN 2/ eiN 2/ ) ei 2/ (ei 2/ ei 2/ ) N sin( ) ei( N 1) / 2 2 sin( ) 2 If we define D to be the distance from the center of the line of oscillators to the point P, that is 1 1 kD (N )1 k sin kr (N )1 kr 2 1 2 1 1 k(D r ) (N )1 1 2 Then we have the form for E as N sin( ) ~ E Re[ E ()r ei( kD t ) 2 ] Re[ Eei t ] 0 sin( ) 2 The intensity distribution within the diffraction pattern due to N coherent, identical, distant point sources in a linear array is equal to c ~ 2 I S 0 E 2 N sin2 ( ) sin2 ( ) sin2 ( ) 2 2 2 I I0 I0 Im 2 2 2 sin ( ) sin ( ) 2 2N 2 in the limit of N→∞, where 2 sin 2 ( ) 2N 2N c I 0 [E (r)]2 0 2 0 c c I I N 2 0 [NE (r)]2 0 E 2 m 0 2 0 2 0 is the phase difference N Nk sin ka sin With k sin where a = N. We make a plot of the relative intensity I/Im as a function of . sin 2 ( ) I 2 2 I m 2 Note that 2 sin ( ) 2 d 2 2 2 IêIm 0.06 0.05 0.04 0.03 0.02 0.01 bê2p -4 -2 0 2 4 The numerator undergoes rapid fluctuations, while the denominator varies relatively slowly. The combined expression gives rise to a series of sharp principal peaks separated by small subsidiary maxima. The principal minimum occurs in directions in direction m such that ka 1 sin m , asin 2 m 2 mm 2 2 m k 2 or m m a 2. Shape of the intensity Here we examine the function form of the relative intensity sin 2 ( ) I 2 , I 2 0 ( ) 2 where 2 2 a sin a . 2 in the small limit of . The function I/I0 is equal to 4/ = 0.405 for = , and is ½ for = 2.78311 = 0.8859 . The main feature of the intensity I/I0 is that the intensity is large only in 1 1 , or sin 2 a 2 a or (sin ) a (diffraction limit) a For the diffraction pattern of a circular aperture of the diameter d, we have the condition of 22.1 d Fig. Fraunhofer diffraction pattern for a circular aperture (Mathematica). We use the FFT program of the Mathematica 3. Phasor diagram In order to discuss the intensity I, we need to calculate the sum defined by i i2 (Ni )1 Z E0 r 1)[( e e ... e ] E 0 1[ ei ei2 ... e (Ni )1 ] N in the complex plane, where k sin . For simplicity, here we solve this problem N geometrically using the phasor diagram (in the real x-y plane). There are N isosceles triangular lattices (see the Figs. below). The first one is of length E0/N and it has a phase equal to zero. The next one is of length E0/N and it has a phase equal to φ. The next one is of length E0/N and it has a phase equal to 2 φ, and soon. So we get an equiangular polygon with N sides. Fig. The resultant amplitude of N = 6 equally spaced sources with net successive phase difference φ. = N φ = 6 φ. ______________________________________________________________________ Fig. The resultant amplitude of N = 36 equally spaced sources with net successive phase difference φ. We now consider the system with a very large N. We may imagine dividing the slit into N narrow strips. In the limit of large N, there is an infinite number of infinitesimally narrow strips. Then the curve trail of phasors become an arc of a circle, with arc length equal to the length E0. The center C of this arc is found by constructing perpendiculars at O and T. The radius of arc is given by E0 E0 R R(N) , or R N in the limit of large N, where R is the side of the isosceles triangular lattice with the vertex angle φ, and the phase difference is given by N ka sin with the value being kept constant. Note that is the change of phase for two rays a separated by , N ka k sin , or sin N Then the amplitude Ep of the resultant electric field at P is equal to the chord OT , which is equal to sin E E 2R sin 2 0 sin E 2 . P 2 2 0 2 Then the intensity I for the single slits with finite width a is given by sin I I ( 2 )2 . m 2 where Im is the intensity in the straight-ahead direction where a The phase difference is given by ka sin 2 sin 2p sin . We make a plot of I/Im as a function of , where p = a/ is changed as a parameter. Fig. The relative intensity in single-slit diffraction for various values of the ratio p = a/. The wider the slit is the narrower is the central diffraction maximum. 4. Diffraction patterns 4.1 Young’s double slit experiment (two slits with finite width) We consider the Young’s double slits (the slits are separated by d). Each slit has a finite width a. Fig. Geometric construction for describing the Young’s double-slit experiment (not to scale). The intensity is given by sin 2 2 2 I I0 cos ( ) , 2 2 where 2d 2a kd sin sin ka sin sin We note that the peak due to the double slit diffraction occurs at 2 cos 1 , 2 m m 2 d The intensity becomes zero when 2 sin 0 , 2n n 2 a Fig. Diffraction pattern with the double slit with the distance d and the single slit with the distance a. d a . 1.0 0.8 0.6 0.4 0.2 -20 -10 0 10 20 1.000 0.500 0.100 0.050 0.010 0.005 0.001 -20 -10 0 10 20 Fig. Intensity ratio I/I0 vs angle (degrees) when p = d/ = 48 and q = a/ = 4. Fig. Fraunhofer diffraction pattern for the Young’s double slits. See my article on the Fraunhofer diffraction in http://physics.binghamton.edu/Sei_Suzuki/suzuki.html Fig. The intensity I as a function of y (x = 300) in the above Fraunhofer diffraction pattern. 4.2 DC SQUID Analogy of the diffraction with double slits and single slit See my article on the Josephson junction and DC SQUID for the detail; Fig. Diffraction effect of Josephson junction. A magnetic field B along the z direction, which is penetrated into the junction (in the normal phase). We consider a junction (1) of rectangular cross section with magnetic field B applied in the plane of the junction, normal to an edge of width w. q 2 J Jsin[ A d l ] , 0 1 ℏ c 1 with q = -2e. We use the vector potential A given by 1 By Bx A( B r )( , ,0) , 2 2 2 A' A (By ,0,0) , where Bxy . 2 Then we have q xb qB J J sin[ (By)dx] J sin[ yW], 0 1 ℏc 0 1 ℏc xa qB dI JLdy J Lsin[ yW ]dy , 1 0 1 ℏc Or t 2/ qB I J L sin[ yW ]dy .
Load more

Recommended publications
  • The Reciprocal Lattice
    Appendix A THE RECIPROCAL LATTICE When the translations of a primitive space lattice are denoted by a, b and c, the vector p to any lattice point is given p = ua + vb + we. The definition of the reciprocal lattice is that the translations a*, b* and c*, which define the reciprocal lattice fulfil the following relationships: (A. I) a*.b = b*.c = c*.a = a.b* = b.c* =c. a*= 0 (A.2) It can then be easily shown that: (1) Ic* I = 1/c-spacing of primitive lattice and similarly forb* and a*. (2) a*= (b 1\ c)ja.(b 1\ c) b* = (c 1\ a)jb.(c 1\ a) c* =(a 1\ b)/c-(a 1\ b) These last three relations are often used as a definition of the reciprocal lattice. Two properties of the reciprocal lattice are particularly important: (a) The vector g* defined by g* = ha* + kb* + lc* (where h, k and I are integers to the point hkl in the reciprocal lattice) is normal to the plane of Miller indices (hkl) in the primary lattice. (b) The magnitude Ig* I of this vector is the reciprocal of the spacing of (hkl) in the primary lattice. AI ALLOWED REFLECTIONS The kinematical structure factor for reflections is given by Fhkl = I;J;exp[- 2ni(hu; + kv; + lw)] (A.3) where ui' v; and W; are the coordinates of the atoms and hkl the Miller indices of the reflection g. If there is only one atom at 0, 0, 0 in the unit cell then the structure factor will be independent of hkl since for all values of h, k and I we have Fhkl =f.
    [Show full text]
  • The 4 Reciprocal Lattice
    4 The Reciprocal Lattice by Andr~ Authier This electronic edition may be freely copied and redistributed for educational or research purposes only. It may not be sold for profit nor incorporated in any product sold for profit without the express pernfission of The Executive Secretary, International Union of Crystallography, 2 Abbey Square, Chester CIII 211[;, [;K Copyright in this electronic ectition (<.)2001 International l.Jnion of Crystallography Published for the International Union of Crystallography by University College Cardiff Press Cardiff, Wales © 1981 by the International Union of Crystallography. All rights reserved. Published by the University College Cardiff Press for the International Union of Crystallography with the financial assistance of Unesco Contract No. SC/RP 250.271 This pamphlet is one of a series prepared by the Commission on Crystallographic Teaching of the International Union of Crystallography, under the General Editorship of Professor C. A. Taylor. Copies of this pamphlet and other pamphlets in the series may be ordered direct from the University College Cardiff Press, P.O. Box 78, Cardiff CF1 1XL, U.K. ISBN 0 906449 08 I Printed in Wales by University College, Cardiff. Series Preface The long term aim of the Commission on Crystallographic Teaching in establishing this pamphlet programme is to produce a large collection of short statements each dealing with a specific topic at a specific level. The emphasis is on a particular teaching approach and there may well, in time, be pamphlets giving alternative teaching approaches to the same topic. It is not the function of the Commission to decide on the 'best' approach but to make all available so that teachers can make their own selection.
    [Show full text]
  • Chapter 2 X-Ray Diffraction and Reciprocal Lattice
    Chapter 2 X-ray diffraction and reciprocal lattice I. Waves 1. A plane wave is described as Ψ(x,t) = A ei(k⋅x-ωt) A is the amplitude, k is the wave vector, and ω=2πf is the angular frequency. 2. The wave is traveling along the k direction with a velocity c given by ω=c|k|. Wavelength along the traveling direction is given by |k|=2π/λ. 3. When a wave interacts with the crystal, the plane wave will be scattered by the atoms in a crystal and each atom will act like a point source (Huygens’ principle). 4. This formulation can be applied to any waves, like electromagnetic waves and crystal vibration waves; this also includes particles like electrons, photons, and neutrons. A particular case is X-ray. For this reason, what we learn in X-ray diffraction can be applied in a similar manner to other cases. II. X-ray diffraction in real space – Bragg’s Law 1. A crystal structure has lattice and a basis. X-ray diffraction is a convolution of two: diffraction by the lattice points and diffraction by the basis. We will consider diffraction by the lattice points first. The basis serves as a modification to the fact that the lattice point is not a perfect point source (because of the basis). 2. If each lattice point acts like a coherent point source, each lattice plane will act like a mirror. θ θ θ d d sin θ (hkl) -1- 2. The diffraction is elastic. In other words, the X-rays have the same frequency (hence wavelength and |k|) before and after the reflection.
    [Show full text]
  • Lecture Notes
    Solid State Physics PHYS 40352 by Mike Godfrey Spring 2012 Last changed on May 22, 2017 ii Contents Preface v 1 Crystal structure 1 1.1 Lattice and basis . .1 1.1.1 Unit cells . .2 1.1.2 Crystal symmetry . .3 1.1.3 Two-dimensional lattices . .4 1.1.4 Three-dimensional lattices . .7 1.1.5 Some cubic crystal structures ................................ 10 1.2 X-ray crystallography . 11 1.2.1 Diffraction by a crystal . 11 1.2.2 The reciprocal lattice . 12 1.2.3 Reciprocal lattice vectors and lattice planes . 13 1.2.4 The Bragg construction . 14 1.2.5 Structure factor . 15 1.2.6 Further geometry of diffraction . 17 2 Electrons in crystals 19 2.1 Summary of free-electron theory, etc. 19 2.2 Electrons in a periodic potential . 19 2.2.1 Bloch’s theorem . 19 2.2.2 Brillouin zones . 21 2.2.3 Schrodinger’s¨ equation in k-space . 22 2.2.4 Weak periodic potential: Nearly-free electrons . 23 2.2.5 Metals and insulators . 25 2.2.6 Band overlap in a nearly-free-electron divalent metal . 26 2.2.7 Tight-binding method . 29 2.3 Semiclassical dynamics of Bloch electrons . 32 2.3.1 Electron velocities . 33 2.3.2 Motion in an applied field . 33 2.3.3 Effective mass of an electron . 34 2.4 Free-electron bands and crystal structure . 35 2.4.1 Construction of the reciprocal lattice for FCC . 35 2.4.2 Group IV elements: Jones theory . 36 2.4.3 Binding energy of metals .
    [Show full text]
  • Introduction to Higher Dimensional Description of Quasicrystal Structures
    ISQCS, June 23-27, 2019, Sendai, Tohoku University Introduction to higher dimensional description of quasicrystal structures Hiroyuki Takakura Division of Applied Physics, Faculty of Engineering, Hokkaido University ISQCS, June 23-27, 2019, Sendai, Tohoku University Outline • Diffraction symmetries & Space groups of iQCs • Section method • Fibonacci structure • Icosahedral lattices • Simple models of iQCs • Real iQC structures • Cluster based model of iQCs • Summary ISQCS, June 23-27, 2019, Sendai, Tohoku University Crystal Amorphous Their diffraction patterns ISQCS, June 23-27, 2019, Sendai, Tohoku University Diffraction symmetries and space groups of iQCs ISQCS, June 23-27, 2019, Sendai, Tohoku University X-ray transmission Laue patterns of iQC 2-fold 3-fold 5-fold i-Zn-Mg-Ho F-type ISQCS, June 23-27, 2019, Sendai, Tohoku University Electron diffraction pattern of iQC i-AlMn 1 The arrangement of the diffraction spots is not periodic but quasi-periodic. D.Shechtman et al., Phys.Rev.Lett., 53,1951(1984). ISQCS, June 23-27, 2019, Sendai, Tohoku University Symmetry of iQC 2 Point group 31.72º 5 Order : 120 37.38º 2 5 3 20.90º 3 2 2 Asymmetric region: 6 +10 +15 + m + center ISQCS, June 23-27, 2019, Sendai, Tohoku University X-ray diffraction patterns of iQCs P-type i-Zn-Mg-Ho F-type i-Zn-Mg-Ho 2fy 2fy 5f 5f 3f 3f 2fx 2fx Liner plots ISQCS, June 23-27, 2019, Sendai, Tohoku University X-ray diffraction patterns of iQCs P-type i-Zn-Mg-Ho F-type i-Zn-Mg-Ho 2fy 2fy 5f 5f 3f 3f 2fx All even or all odd for 2fx No reflection condition Log plots ISQCS, June 23-27, 2019, Sendai, Tohoku University Vectors used for indexing 6 Any vectors can be used if all the reflections can be indexed correctly.
    [Show full text]
  • Quasicrystals
    Volume 106, Number 6, November–December 2001 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 106, 975–982 (2001)] Quasicrystals Volume 106 Number 6 November–December 2001 John W. Cahn The discretely diffracting aperiodic crystals Key words: aperiodic crystals; new termed quasicrystals, discovered at NBS branch of crystallography; quasicrystals. National Institute of Standards and in the early 1980s, have led to much inter- Technology, disciplinary activity involving mainly Gaithersburg, MD 20899-8555 materials science, physics, mathematics, and crystallography. It led to a new un- Accepted: August 22, 2001 derstanding of how atoms can arrange [email protected] themselves, the role of periodicity in na- ture, and has created a new branch of crys- tallography. Available online: http://www.nist.gov/jres 1. Introduction The discovery of quasicrystals at NBS in the early Crystal periodicity has been an enormously important 1980s was a surprise [1]. By rapid solidification we had concept in the development of crystallography. Hau¨y’s made a solid that was discretely diffracting like a peri- hypothesis that crystals were periodic structures led to odic crystal, but with icosahedral symmetry. It had long great advances in mathematical and experimental crys- been known that icosahedral symmetry is not allowed tallography in the 19th century. The foundation of crys- for a periodic object [2]. tallography in the early nineteenth century was based on Periodic solids give discrete diffraction, but we did the restrictions that periodicity imposes. Periodic struc- not know then that certain kinds of aperiodic objects can tures in two or three dimensions can only have 1,2,3,4, also give discrete diffraction; these objects conform to a and 6 fold symmetry axes.
    [Show full text]
  • Phys 446: Solid State Physics / Optical Properties
    Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 2 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 2 (Ch. 2.1-2.3, 2.6-2.7) Last week: • Crystals, • Crystal Lattice, • Reciprocal Lattice Today: • Types of bonds in crystals Diffraction from crystals • Importance of the reciprocal lattice concept Lecture 2 Andrei Sirenko, NJIT 2 1 (3) The Hexagonal Closed-packed (HCP) structure Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl • The HCP structure is made up of stacking spheres in a ABABAB… configuration • The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atoms • Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are not all perpendicular) • The number of nearest-neighbours is 12 • The ideal ratio of c/a for Rotated this packing is (8/3)1/2 = 1.633 three times . Lecture 2 Andrei Sirenko, NJITConventional HCP unit 3 cell Crystal Lattice http://www.matter.org.uk/diffraction/geometry/reciprocal_lattice_exercises.htm Lecture 2 Andrei Sirenko, NJIT 4 2 Reciprocal Lattice Lecture 2 Andrei Sirenko, NJIT 5 Some examples of reciprocal lattices 1. Reciprocal lattice to simple cubic lattice 3 a1 = ax, a2 = ay, a3 = az V = a1·(a2a3) = a b1 = (2/a)x, b2 = (2/a)y, b3 = (2/a)z reciprocal lattice is also cubic with lattice constant 2/a 2. Reciprocal lattice to bcc lattice 1 1 a a x y z a2 ax y z 1 2 2 1 1 a ax y z V a a a a3 3 2 1 2 3 2 2 2 2 b y z b x z b x y 1 a 2 a 3 a Lecture 2 Andrei Sirenko, NJIT 6 3 2 got b y z 1 a 2 b x z 2 a 2 b x y 3 a but these are primitive vectors of fcc lattice So, the reciprocal lattice to bcc is fcc.
    [Show full text]
  • Development of Rotation Electron Diffraction As a Fully Automated And
    Bin Wang Development of rotation electron Development of rotation electron diffraction as a fully automated and accurate method for structure determination and accurate automated as a fully diffraction electron of rotation Development diffraction as a fully automated and accurate method for structure determination Bin Wang Bin Wang was born in Shanghai, China. He received his B.Sc in chemistry from Fudan University in China in 2013, and M.Sc in material chemistry from Cornell University in the US in 2015. His research mainly focused on method development for TEM. ISBN 978-91-7797-646-2 Department of Materials and Environmental Chemistry Doctoral Thesis in Inorganic Chemistry at Stockholm University, Sweden 2019 Development of rotation electron diffraction as a fully automated and accurate method for structure determination Bin Wang Academic dissertation for the Degree of Doctor of Philosophy in Inorganic Chemistry at Stockholm University to be publicly defended on Monday 10 June 2019 at 13.00 in Magnélisalen, Kemiska övningslaboratoriet, Svante Arrhenius väg 16 B. Abstract Over the past decade, electron diffraction methods have aroused more and more interest for micro-crystal structure determination. Compared to traditional X-ray diffraction, electron diffraction breaks the size limitation of the crystals studied, but at the same time it also suffers from much stronger dynamical effects. While X-ray crystallography has been almost thoroughly developed, electron crystallography is still under active development. To be able to perform electron diffraction experiments, adequate skills for using a TEM are usually required, which makes ED experiments less accessible to average users than X-ray diffraction. Moreover, the relatively poor data statistics from ED data prevented electron crystallography from being widely accepted in the crystallography community.
    [Show full text]
  • Reciprocal Lattice
    Reciprocal Lattice From Quantum Mechanics we know that symmetries have kinematic im- plications. Each symmetry predetermines its specific quantum numbers and leads to certain constraints (conservation laws/selection rules) expressed in terms of these quantum numbers. In this chapter, we consider kinematic consequences of the discrete translation symmetry. The shortest way to in- troduce the relevant quantities|the vectors of reciprocal lattice|is to con- sider the Fourier analysis for crystal functions. Fourier analysis for crystal functions Let A(r) be some function featuring discrete translation symmetry: A(r + T) = A(r) ; (1) where T is any translation vector of a certain translation group GT . We want to expand A(r) in a Fourier series of functions respecting translation symmetry (i.e. obeying translation symmetry with the same group GT ). Without loss of generality, we confine the coordinate r to a unit cell of the Bravais lattice of the group GT . Hence, we work with the vector space of functions defined within the unit cell and satisfying the periodicity condition (1). Within this vector space, we want to construct the orthogonal basis of functions that would be eigenfunctions of continuous translations. These functions are the plane waves eiG·r with the wavevector G constrained by the condition (1). Clearly, the constraint reduces to G · T = 2π × integer (for any G and T) : (2) From (2) we immediately see that the set of vectors G is a group, GG, with the vector addition as a group operation. The group GG can be interpreted as a group of translations and visualized with corresponding Bravais lattice referred to as reciprocal lattice.
    [Show full text]
  • 02 Geometry, Reclattice.Pdf
    Geometry of Crystals Crystal is a solid composed of atoms, ions or molecules that demonstrate long range periodic order in three dimensions The Crystalline State State of Fixed Fixed Order Properties Matter Volume Shape Gas No No No Isotropic Liquid Yes No Short-range Isotropic Solid Yes Yes Short-range Isotropic (amorphous) Solid Yes Yes Long-range Anisotropic (crystalline) Atoms Lattice constants A Crystal Lattice a, b B C Not only atom, ion or molecule b positions are repetitious – there are a certain symmetry relationships in their arrangement. Crystalline Lattice structure = Basis + Crystal Lattice a a r ua One-dimensional lattice with lattice parameter a a a r ua b b b Two-dimensional lattice with lattice parameters a, b and Crystal Lattice r ua b wc Crystal Lattice Lattice vectors, lattice parameters and interaxial angles c Lattice vector a b c c Lattice parameter a b c b Interaxial angle b a a A lattice is an array of points in space in which the environment of each point is identical Crystal Lattice Lattice Not a lattice Crystal Lattice Unit cell content Coordinates of all atoms Types of atoms Site occupancy Individual displacement parameters y1 y2 y3 0 x1 x2 x3 Crystal Lattice Usually unit cell has more than one molecule or group of atoms They can be represented by symmetry operators rotation Symmetry Symmetry is a property of a crystal which is used to describe repetitions of a pattern within that crystal. Description is done using symmetry operators Rotation (about axis O) = 360°/n where n is the fold of the axis Translation n = 1, 2, 3, 4 or 6) m O i Mirror reflection Inversion Two-dimensional Symmetry Elements 1.
    [Show full text]
  • Chapter 3 X-Ray Diffraction • Bragg's Law • Laue's Condition
    Chapter 3 X-ray diffraction • Bragg’s law • Laue’s condition • Equivalence of Bragg’s law and Laue’s condition • Ewald construction • geometrical structure factor 1 Bragg’s law Consider a crystal as made out of parallel planes of ions, spaced a distance d apart. The conditions for a sharp peak in the intensity of the scattered radiation are 1. That the x-rays should be specularly reflected by the ions in any one plane and 2. That the reflected rays from successive planes should interfere constructively Path difference between two rays reflected from adjoining planes: 2d sin θ For the rays to interfere constructively, this path difference must be an integral number of wavelength λ nλ = 2d sin θ Bragg’s condition. 2 Bragg angle θ is just the half of the total angle 2 θ by which the incident beam is deflected. There are different ways of sectioning the crystal into planes, each of which will it self produce further reflection. The same portion of Bravais lattice shown in the previous page, with a different way of sectioning the crystal planes. The incident ray is the same. But both the direction and wavelength (determined by Bragg condition with d replaced by d’) of the reflected ray are different from the previous page. 3 Von Laue formulation of X-ray diffraction by a crystal • No particular sectioning of crystal planes r • Regard the crystal as composed of identical microscopic objects placed at Bravais lattice site R • Each of the object at lattice site reradiate the incident radiation in all directions.
    [Show full text]
  • Reciprocal Lattice-1.Pdf
    R. I. Badran Solid State Physics The Reciprocal Lattice Two types of lattice are of a great importance: 1. Reciprocal lattice 2. Direct lattice (which is the Bravais lattice that determines a given reciprocal lattice). What is a reciprocal lattice? A reciprocal lattice is regarded as a geometrical abstraction. It is essentially identical to a "wave vector" k-space. Definition: Since we know that R may construct a set of points of a Bravais lattice, thus a reciprocal lattice can be defined as: - The collection of all wave vectors that yield plane waves with a period of the Bravais lattice.[Note: any vector is a possible period of the Bravais lattice] - A collection of vectors G satisfying eiGR 1orG R 2n , where n is an integer and is defined as: k1n1 k2n2 k3n3 . Here , is a reciprocal lattice vector which can be defined as:G k b k b k b , where k , k and k are integers. 1 1 2 2 3 3 1 2 3 [Note: In some text books you may find thatG K ]. - The reciprocal lattice vector which generates the reciprocal lattice is constructed from the linear combination a2 a3 of the primitive vectorsb ,b , andb , where b1 2 1 2 3 V cell and b2 and can be obtained from cyclic permutation of 1 2 3. Notes: 1. Since , this implies thatbi a j 2 ij , where ij 1if i=j and ij 0 if ij. 2. The two lattices (reciprocal and direct) are related by the above definitions in 1. 3. Rotating a crystal means rotating both the direct and reciprocal lattices.
    [Show full text]