DOCSLIB.ORG

The Reciprocal Lattice

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Thus if b.c.c. and f.c.c. crystals are referred to their primitive cells then reflections from the simple metals such as niobium and copper, which have only one atom at each lattice point, will all have intensities given by f(0)2 • Since the three basic g vectors derived below, for the b.c.c. structure for 180 ELECTRON BEAM ANALYSIS OF MATERIALS Table A.l Necessary conditions for allowed reflections in terms of the type of unit cells in crystals. Possible Forbidden Unit cell reflections reflections Primitive All values of h, k and I none Body centred (h + k + I) even (h + k +I) odd Face centred h, k, I all odd or even h, k and l mixed Base centred h and k both odd or even hand k mixed example, are [11 OJ*, [0 11 ]* and [10 1]* it is clear that the allowed reflections are those derived by summing or subtracting these vectors, i.e. reflections such that (h + k + l) is always even. A similar conclusion is reached if the cubic structure cell, which contains atoms at 0, 0, 0 and ±. ±. ±. is used since Fhkt = f { exp(- 2ni0) + exp[- 2ni( ~ + ~ + ~) ]} = {f(l + 1) = 2f if (h + k +/)is even !(1-1)=0 if(h+k+/)isodd. The allowed reflections in the various possible unit cells are shown in Table A.1 If the number of atoms in a unit cell is large there is the possibility that some of the allowed reflections will have zero intensity, e.g. the 200 reflection in silicon. Nevertheless all the allowed reflections must conform with the above classifications. It is of course an important part of structure determination to recognize which reflections are absent (see Chapter 4). The absent reflections in metals such as zirconium (i.e. those for which (h + 2k) is a multiple of 3 and I is odd) show that the structure is h.c.p .. A.2 RECIPROCAL LATTICE FOR F. C. C. AND B.C. C. CRYSTALS Referring the primitive translations to cubic axis for a primitive f.c.c. cell we have where i, j and k are unit vectors along the cubic axes. Since a*= b 1\ c a.(b 1\ c) APPENDIX A 181 then * _ !a[j + k] A !a[i + k] a - !a[i + jJ.!a[j + k] A !a[i + k] and 1 z[· · k] I I 4a 1 + J- = -[i + j- k] =-[III] ia3 [i+jJ.[i+j-k] a a Correspondingly, b* =~[Ill] a with a b.c.c. structure and referring translations to cubic axes then a f] a _ a _ a=-[111 b=-[111] c=-[111] 2 2 2 so that * _ !a[- i + j + k] A !a[i-j + k] a -!a[i+j-kJ·!a[-i+j+k] A!a[i-j+k] which reduces to -!a 2 [i + j] I sa1 3['•+J- • k] . ['•+J '] = -[110]a Similarly, I and c* = -[101] a Note that the basic vectors in the reciprocal lattice for these non-primitive cells are twice those of the crystal of the same system. Thus, the basic translations are aj2 (Ill) for a b.c.c. crystal and Ija (Ill), rather than I j2a ( Ill ) , for the reciprocal lattice for an f.c.c. crystal. A.3 DOUBLE DIFFRACTION Because electrons are strongly scattered it is possible that rescattering of a diffracted beam can give rise to a strong diffracted beam where structure factor considerations suggest the beam should be of zero intensity. The conditions under which a forbidden diffraction spot may appear are most easily seen using the Ewald sphere construction (see Chapter 4). Thus, if the reciprocal lattice point corresponding to g 1 for which the structure factor is large, lies on the Ewald sphere then a strong diffracted beam will be produced in the direction A 1 , as shown on Fig. A.l. Similarly if g2 also 182 ELECTRON BEAM ANALYSIS OF MATERIALS Fig. A.l Ewald sphere construction showing the conditions which must be fulfilled for double diffraction to occur. See text. lies on the sphere a diffracted beam would be expected in the direction A2 . However, if the structure factor is zero for g2 the intensity of A2 would be zero but rediffraction of the beam A 1 by planes perpendicular to g2 - g 1 will give rise to a beam A3 which is parallel to and therefore indistinguishable from A2 . Diffraction maxima due to double diffraction can be distinguished by rotating the crystal about the direction which contains the spot in question. The intensity of this spot will be unchanged unless it is due to double diffrac­ tion when it will disappear when g 1 is no longer excited. A.4 SHAPES OF DIFFRACTION MAXIMA Significant diffracted intensity is observed from thin samples even when the Bragg condition is not precisely satisfied ( [1 ], [2] ). It can be shown that only for an infinite crystal will the diffraction maxima be points and that as the crystal dimensions get smaller so the diffraction maxima get larger. For a parallelepiped crystal the diffracted amplitude is given by F ¢ g = V fA IBIC exp [ - 2:rri(ux + vy + wz) J dxdydz (A.4) c 0 0 0 APPENDIX A 183 I -3/A -2/A -1/A 0 1/A 2/A 3/A Fig. A.2 Predicted variation in intensity of a diffracted beam from a crystal of edge length A as a function of deviation from the Bragg condition where the deviation is measured in units of 1/A. See text. = ~ sin(nAu) sin(nBv) sin(nCw) (A.5) Vc nu nv nw where A, Band Care the edge lengths of the parallelepiped along x, y and z, and u, v, w are the values of s (the deviation from the Bragg condition) along x, y and z. Fig. A.2 shows how the intensity (obtained by multiplying ¢g by¢;, its complex conjugate), varies along u for v and w = 0. The central maximum has a width at half maximum height of 1/A and successive minima occur at intervals of 1/A. The intensity of the successive maxima decreases very rapidly, as shown in Fig. A.2 but because electron diffraction patterns have such a large dynamic range it is possible to detect maxima out to at least the fiftieth maximum for typical thickness samples. For very thin crystals, such as small precipitates, the spikes in reciprocal space give rise to visible intensity for all beam directions. The dimension of the central diffraction maximum along a given direction in reciprocal space is given by 1/d where d is the parallel dimension of the specimen. Thus a thin plate precipitate gives rise to a long spike in reciprocal space normal to the plate and a needle precipitate gives rise to a disc of intensity. REFERENCES I. Hirsch, P.B., Howie, A., Nicholson, R.B., Pashley, D.W. and Whelan, M.J. (1965) Electron Microscopy of Thin Crystals, Butterworths, Sevenoaks. 2. James, R. W. ( 1958) Optical Principles of the Diffraction of X-rays, Bell, London. Appendix B INTERPLANAR DISTANCES AND ANGLES IN CRYSTALS. CELL VOLUMES. DIFFRACTION GROUP SYMMETRIES B.! THE SEVEN SYSTEMS The axial lengths and angles and the symmetries exhibited m the seven crystal systems are shown in Table B.l. Table B.l The axial relationships and symmetries of the seven crystal systems. Axial length Crystal and angles Minimum symmetry elements Cubic Three equal axes Four, threefold rotation axes at right angles a = b = c, IX = f3 = y = 90° Hexagonal Two coplanar axes at One, sixfold rotation (or 120°, third axis at rotation-inversion) axis right angles a= b =/= C, IX= {3 = 90°, y = 120° Trigonal (or Three equal axes One, threefold rotation rhombohedral) equally inclined (or rotation-inversion) axis a = b = c, IX = f3 = y =!= 90° Tetragonal Three axes at right One, fourfold rotation (or angles rotation-inversion) axis a = b =!= c, IX = f3 = y = 90° Orthorhombic Three orthogonal Three, perpendicular twofold unequal axes (or rotation-inversion) axis a =!= b =!= c, IX = f3 = y = 90° Monoclinic Three unequal axes One, twofold rotation (or one pair not orthogonal rotation-inversion) axis a=!= b =f=c, IX= y = 90o =!= f3 APPENDIX B 185 Table 8.1 (Contd.) Triclinic Three unequal axes None none at right angles a +b +c, a +fJ + y +90° 8.2 INTERPLANAR SPACING The value of d, the distance between adjacent planes in the set (hkl), may be found from the following equations.
Load more

Recommended publications
  • 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]
  • Search for X-Ray Channeling Radiation at Fermilab's Fast Facility
    SEARCH FOR X-RAY CHANNELING RADIATION AT FERMILAB'S FAST FACILITY by Patricia Kobak Hall A senior thesis submitted to the faculty of Brigham Young University - Idaho in partial fulfillment of the requirements for the degree of Bachelor of Science Department of Physics and Astronomy Brigham Young University - Idaho April 2017 Copyright c 2017 Patricia Kobak Hall All Rights Reserved BRIGHAM YOUNG UNIVERSITY DEPARTMENT APPROVAL of a senior thesis submitted by Patricia Kobak Hall This thesis has been reviewed by the research advisor, research coordinator, and department chair and has been found to be satisfactory. Date Matt Zacherson, Committee Leader Date Richard Hatt, Committee Member Date Jon Johnson, Committee Member Date Todd Lines, Thesis Advisor Date Stephen McNeil, Chair ABSTRACT SEARCH FOR X-RAY CHANNELING RADIATION AT FERMILAB'S FAST FACILITY Patricia Kobak Hall Department of Physics and Astronomy Bachelor of Science The Fermilab FAST facility has capability of producing a 50 GeV electron beam, which will be used to generate hard x-rays through a process called Channeling Radiation. This paper will investigate experimental solutions for high background during the beginning of the Channeling Radiation study, as well as propose ideas for modifying the beam line to decrease background. ACKNOWLEDGMENTS I would like to acknowledge: my advisers, Brother Zacherson, Brother Hatt, and Brother Johnson for reading through this thesis, Brother Lines for his patience with me during this class, the mentorship of Dr. Tanaji Sen at Fermilab, and my patient husband for putting up with me. Contents Table of Contents vi List of Figures vii 1 Introduction1 2 Background4 2.1 Background of Channeling Radiation..................4 2.2 Classical Description of Channeling Radiation.............5 2.3 Equations of Motion...........................7 2.4 Quantum Description of Channeling Radiation.............8 2.5 Multiple Scattering............................9 3 Experimental Setup 10 4 Experimental Problem-Solving 12 4.1 Bremsstrahlung.............................
    [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]
  • 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 .
    [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]
  • Study of Crystal Defects Using Ion Channelling and Channelling Radiation
    Bull. Mater. Sci., Vol. 10, Nos 1 & 2, March 1988, pp. 105 115. (((~3 Printed in India. Study of crystal defects using ion channelling and channelling radiation ANAND P PATHAK School of Physics, University of Hyderabad, Central University P.O., Hyderabad 500 134, India. Abstraet. The channelling technique to study crystal defects is described and its appli- cations to various kind of defects to study their atomistic nature have been reviewed.Special emphasis has been placed on the applications to extended defects like dislocations. Finally a related new technique being developed for the last few years, namely the channelling radiation technique has been discussed along with its applications to study the dislocations. Keywords. Defects;dislocations: channelling; channelling radiation. I. Introduction Unless special care is taken in the fabrication and growth of material crystals, defects of one kind or the other are always present in these crystals. These defects can be classified as point defects (like vacancies and interstitials), line defects (dislocations), planar defects (for example stacking faults, grain boundaries etc.) and volume defects (like voids, gas bubbles, precipitates, amorphous regions etc.). From the viewpoint of fundamentals of defects physics in solids as well as from the application point of view, it is very important to know the nature and concentration of the defects present in solids. For such investigations, apart from the well-established method of electron microscopy, several new techniques have recently been developed. These include channelling, Mossbauer effect, perturbed angular correlation, positron annihilation, infrared absorption, electron paramagnetic resonance etc. The main advantage of these new techniques is their capability to select a given type of defect and provide information on the atomistic nature of the chosen defect.
    [Show full text]