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    Metric, Myth & Quasicrystals
    Metric, Myth & Quasicrystals
    Metric, Myth & Quasicrystals
    Ebook137 pages1 hour

    Metric, Myth & Quasicrystals

    By Antony J. Bourdillon

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    Metric, Myth & Quasicrystallography describes the first measurement of the metric in quasicrystals and the first measurement at atomic scale. Quasicrystals are ordinary as window glass, but they have been mistified owing to their sharp diffraction patterns with 5-fold symmetries, impossible in crystals. Out of the fog, the patterns are not in Bragg order; the series is not properly Fibonacci: simplified indexation of the pattern is used to simulate a structure due to a single, aligned, edge-sharing unit-cell that is consistent with all data. Since it is unlikely that the sharp diffraction pattern is due to unmeasured poly polyhedra, does the International Union of Crystallography have to redefine crystals yet again?
    In modern physics, the metric relates the covariant components of invariant vectors with corresponding contravariant components. In crystallography it relates dimensions in momentum space to atomic locations in real space. In quasicrystals, the pattern in momentum space is logarithmic. Theory and simulation show why this has to be. Consequences follow. In particular, we show not only where the atoms are but also why they are there.
    A debate is reported so that the reader will be encouraged to make his own mind. When logarithmic periodicity is discovered and explained in one branch of physics, it should be expected in others.
    LanguageEnglish
    PublisherAuthorHouse
    Release dateAug 30, 2012
    ISBN9781477247853
    Metric, Myth & Quasicrystals
    Author

    Antony J. Bourdillon

    Antony Bourdillon M.A., D.Phil. (Oxon) Ph.D. (Cantab) took his degrees at the Clarendon Laboratory, University of Oxford. He did post doctoral work at the Cavendish Laboratory, Cambridge University before joining its Department of Materials Science & Metallurgy as permanent faculty. Subsequently, he was Director of the Electron Microscope Unit at the University of New South Wales and then Professor at the State University of New York. In Singapore he became Professor of Physics; Professor of Materials Science; Principal Fellow at the Institute of Materials Research and Engineering; and proposer, builder and Director of the Singapore Synchrotron Light Source. More recently, he has conducted independent research, consulting for various university institutes and companies, including an X-ray Lithography Consortium that he championed. He has published 6 monographs and 200 journal articles.

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      Metric, Myth & Quasicrystals - Antony J. Bourdillon

      © 2012 by Antony J. Bourdillon. All rights reserved

      No part of this book may be reproduced, stored in a retrieval system, or transmitted by any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission from the author.

      Published by AuthorHouse 07/16/2012

      ISBN: 978-1-4772-4786-0 (sc)

      ISBN: 978-1-4772-4785-3 (e)

      And by UHRL

      ISBN: 978-0-9789-8393-2;

      Any people depicted in stock imagery provided by Thinkstock are models, and such images are being used for illustrative purposes only. Certain stock images © Thinkstock

      Because of the dynamic nature of the Internet, any web addresses or links contained in this book may have changed since publication and may no longer be valid. The views expressed in this work are solely those of the author and do not necessarily reflect the views of the publisher, and the publisher hereby disclaims any responsibility for them.

      Front cover shows an icosahedron at right with an embedded golden triad at left, defining Cartesian axes.

      Contents

      Chapter 1      Introduction

      Chapter 2      The metric for the quasi-Bragg law measured for the first time

      Chapter 3       Evaluation

      Reference

      ENDNOTES

      Logarithmically periodic solids bibliography

      Logarithmically Periodic Solids, Nova Science, 2011, ISBN 978-1-61122-977-6 Quasicrystals’ 2D tiles in 3D superclusters, UHRL, 2010, ISBN: 978-0-9789839-2-5 Quasicrystals-and quasi drivers, UHRL, 2009, ISBN 978-0-9789-8391-8 Logarithmically periodic solids-properties, evidence and uncertainties, in Quasicrystals: Types, Systems, and Techniques, Nova Science, 2010, ISBN 978-1-61761-123-0

      A.J.Bourdillon, Nearly free-electron energy-bands in a logarithmically periodic solid, Solid State Comm. 149 1221-5 (2009).

      A.J.Bourdillon, Indexed scattering powers in a logarithmically periodic solid, International Journal of Condensed Matter, Advanced Materials, and Superconductivity Research (2010)

      A.J.Bourdillon, Fine Line Structure in Convergent Beam Electron Diffraction in Icosahedral Al6Mn, Phil. Mag. Lett. 55 2l-26 (1987)

      http://www.quasicrystalmetric.us The metric for the quasi-Bragg law measured for the first timehttp://www.quasicrystal.us The Quasi-Bragg law and metric-from geometric scatterers in three dimensional space.

      http://www.youtube.com/watch?v=0LDS0sQQvpk : the Quasi Bragg law and metric http://www.youtube.com : quasicrystals, logarithmically periodic http://www.youtube.com : quasicrystals, here are the atoms http://www.UHRL.net, the structure of quasicrystalshttp://www.quasicrystaltiling.us two-dimensional tiles in three-dimensional space-edge sharing tiles in logarithmically periodic solids.

      Chapter 1

      Introduction

      Quasicrystals have intrigued a few ‘scientists’ for many years. After 80 summers of practiced crystallography, these materials were discovered to have 5-fold symmetries. This is impossible in crystals, where space filling of unit cells restricts them to 14 types of Bravais lattice (cubic, tetrahedral, orthorhombic etc.), none of them having that symmetry. The purpose of this book is to unconfuse after 30 subsequent winters. It shows that quasicrystals are about as ordinary as window glass. They require no special understanding.

      Structure is important in the study of materials of all types. It determines most of the physical properties that we examine in science. As always, observation and measurement are critical. In this book we place special emphasis on the metric, which has had a fundamental role in modern science. Since Einstein, it has related covariant to contravariant components of invariant vectors. In crystallography, it relates measurements in the diffraction pattern to locations of atoms in real space. In this book, the first measurement of the metric in quasicrystals is described.

      The book is arranged firstly to describe experimental and computational results and secondly to aid the reader in assessment by including debates with journal referees. The debate has to occur, though inconsistncies between anonymous referees, destroys their credibility. Starting with the particular, what matters when a paper is refused by some while the theory is true? In a scientific revolution the reviewing system does not work because most referees are competitors. This is not to say that all referees have rejected our theory. It is accepted by a reviewer in Acta Crystallographica A (chapter 3) and in Solid State

      Communications it has been embraced enthusiastically, This is new to me and interesting. Two good reviews outweigh a thousand that are illogical. All have been analyzed, some in this book, others previously. The reader may judge for himself. He will find that no argument against the theory has been given that is not either false in fact or fallacious in logic. However the reader will surely wonder whether such a reviewing process is not wasteful of voluntary effort. When neither scientists nor editors follow the demands of logic, medieval orthodoxy will continue to reign.

      The discovery of quasicrystals is sometimes called a revolution in science. The International Union of Crystallography (IUCr) even chose to redefine crystals:

      "A material is a crystal if it has essentially a sharp diffraction pattern. The word essentially means that most of the intensity of the diffraction is concentrated in relatively sharp Bragg peaks, besides the always present diffuse scattering. In all cases, the positions of the diffraction peaks can be expressed by

      Image425.JPG

      Here Image434.JPG and /?, are the basis vectors of the reciprocal lattice and integer coefficients respectively and the number n is the minimum for which the positions of the peaks can be described with integer coefficient hi.

      "The conventional crystals are a special class, though very large, for which n = 3." i

      The first of many myths is that diffraction in quasicrystals is Bragg diffraction. In the weakest possible sense, perhaps it is, but not in any sense that Bragg described. The weak sense is that thediffraction is due to a ragged three dimensional grating where diffracting planes of atoms are variably spaced. In this very weak sense, the diffracting planes are normal to the scattering vector, a feature that we use in our simplified indexation, or naming, of the pattern. In the sense of Bragg’s law, the diffraction from quasi crystals is not Bragg diffraction. All orders are forbidden except the second, and that only with modification. This law describes diffraction due to a lattice of atoms onto a lattice in reciprocal space. In quasicrystal diffraction, the pattern in reciprocal space is not a regular lattice. The angular spacing between diffracted beams is not even a Fibonacci series as commonly claimed, because the series ratio, Fn+1/Fn is not a varying number tending to the golden section Image442.JPG at large n; but is the constant Image449.JPG . It is therefore a geometric series; far from the arithmetic series prescribed by Bragg’s law. (Reader can find an example of the quasicrystalline pattern in chapter 2 ii.) In quasicrystals, the lattice is logarithmiciii. It is contradictoryiv to claim the diffraction is both Fibonacci and Bragg. There are other differences too: among them the quasi-Bragg law has a special metric that is described in this book. This metric is measured and explained for the first time. If IUCr were to need to include quasicrystals in their definition, then it would have to redefine crystals again. It is no use saying we must do as they do-not while we use a valid optional convention of indexation that is consistent with all measurement. With the proper law it is possible to measure atomic positions from the diffraction pattern as is typical in crystallography, and we find these are consistent with transmission electron microscope (TEM) imaging exactly, and also with known atomic sizes, known that is for the first time.

      This book is written so that the reader may independently evaluate the great divide. X-ray cry stall ographers and electron microscopists use indexation differently. Their methods of indexation are correspondingly different but equally valid according to Bragg’s law. The X-rays have typical wavelengths about 0.1 nm, short of some of the atomic interplanar spacings. In transmission electron microscopy (TEM), electron wavelengths are 30 times shorter. Scattering angles are correspondingly smaller. The restriction, imposed by the Ewald sphere construction in X-ray diffraction (XRD), is relaxed in TEM where the deviation parameter takes complementary significance. In a vertical mount TEM all Bragg planes contain the vertical axis, and all their normals are horizontal. TEM is content to name the beams on a plane, without concern for full circle diffractometry. The whole pattern is indexed from a few beams around each axisv This is the method used here and we show that three dimensions are sufficient and easily manageable. It is unnecessary to index every axis separately because of symmetry. The icosahedral structure has few axes. The three main axes are,

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