Reciprocal Lattices Simulation Using Matlab

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Four's Stage Project:Submitted to the Sulaimani University - College of Science - Department of Physics Supervised by Dr. Omed Ghareb Abdullah

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<p>Kurdistan Iraqi Region Ministry of Higher Education Sulaimani University College of Science Physics Department Reciprocal Lattices Simulation using Matlab Prepared by Bnar Jamal Hsaen Hanar Kamal Rashed Kizhan Nury Hama Sur Supervised by Dr. Omed Gh. Abdullah 2008 2009 2 {But say: oh My Lord! Advance me in knowledge} (Surat Taha:14) We dedicate this research to: - Those who helped us during the preparation of this research, - Our Department. - Those who reading this research. 3 Acknowledgements We would like to express our gratitude and thankfulness to our supervisor Dr. Omed Gh. Abdullah, for continues help and guidance throughout this work. We are also indebted to Mr. Yadgar Abdullah for providing us with sources and his encouragement during writing this research paper. True appreciation for Department of Physics in the College of Science at the University of Sulaimani, for giving us an opportunity to carry out this work. We wish to extend my sincere thanks to all teachers staff who taught us along our study. Also we express our thankfulness to the library of our department for providing us with references. Finally thanks and love to our family for their patience and supporting during our study. Bnar - Hanar - Kizhan 2009 4ContentsChapter One: Crystal Structure 1.1 Introduction 1.2 Crystal structure 1.3 Classification of crystal by symmetry 1.4 The bravais lattices 1.5 Three dimension crystal lattice image 1.5.1 Simple lattices and their unit cell 1.5.2 Closest packing 1.5.3 Holes (interstices)in closest packing arrays 1.5.4 Simple crystal structures Chapter Two: X-Ray Diffraction and Crystal Structure 2.1 Introduction 2.2 Braggs diffraction law 2.3 Experimentation diffraction method 2.3.1 The Laue method 2.3.2 The rotation method 2.3.3 X-Ray powder diffraction 2.3.4 Electrons or neutron diffraction 2.4 Reciprocal lattice 2.5 Diffraction in reciprocal space 2.6 Fourier analysis 2.7 Fourier series 2.8 Exponential Fourier series Chapter Three: Reciprocal Lattice Simulation 3.1 Introduction 3.2 Reciprocal lattice to SCC lattice 3.3 Reciprocal lattice to BCC lattice 3.4 Reciprocal lattice to FCC lattice 3.5 Conclusion References. Appendix 5Abstract The diffraction of X-ray is a method for structural analysis of an unknown crystal. These beams are diffracted by the unknown structure and can interfere with one another. If they are in phase, they amplify each other and cause an increased intensity. If they are out in phase, then on average they cancel each other out, and the intensity becomes zero. The reciprocal relationship seen in the Bragg equation, together with the associated geometrical conditions, leads to a mathematical construction called the reciprocal lattice, which provides an elegant and convenient basis for calculations involving diffraction geometry. From a particular lattice structure built up from given types of atoms the diffraction intensities can be calculated, by a combination of the Fourier series for the lattice and a Fourier transform of individual atoms. By this techniques the reciprocal lattices are produce, which gives the amplitude of each scattered intensity for the wave vector. In this project, the authors show how the Fast Fourier Transformation may be used to simulate the X-ray diffraction from different crystal structures, for this reason, the reciprocal lattices of well known: simple cubic, body center, and face center crystal structures were examined. The result shows that the reciprocal lattices of a simple cubic Bravais lattice have a cubic primitive cell, while the reciprocal lattice for a Face-centered cubic lattice is a Body-centered cubic lattice, and the reciprocal lattice for Body-centered cubic lattice is a Face-centered cubic lattice. A good agreements between the theoretical and present results indicate that this technique can be used to simulate the more complex crystal structures. For more reliability simulation the Gaussian function could be used to express the atoms instead of the circles which was established in present work. 6Chapter One Crystal Structure 1.1 Introduction: Solids can be classified in to three categories according to its structure; amorphous, crystal, and polycrystal. The first type an amorphous solid is a solid in which there is no long-range order of the positions of the atoms. Most classes of solid materials can be found or prepared in an amorphous form. For instance, common window glass is an amorphous ceramic, many polymers (such as polystyrene) are amorphous, and even foods such as cotton candy are amorphous solids. In materials science, a crystal may be defined as a solid composed of atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions; while the polycrystalline materials are solids that are composed of many crystallites of varying size and orientation. The variation in direction can be random (called random texture) or directed, possibly due to growth and processing conditions. Fiber texture is an example of the latter. Almost all common metals, and many ceramics are polycrystalline. The crystallites are often referred to as grains; however, powder grains are a different context. Powder grains can themselves be composed of smaller polycrystalline grains. Polycrystalline is the structure of a solid material that, when cooled, form crystallite grains at different points within it. Where these crystallite grains meet is known as grain boundaries. 71.2 Crystal structure: In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a particular way. Motifs are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The crystal structure of a material or the arrangement of atoms in a crystal structure can be described in terms of its unit cell. The unit cell is a tiny box containing one or more motifs, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three dimensional shape. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi,yi,zi) measured from a lattice point. Although there are an infinite number of ways to specify a unit cell, for each crystal structure there is a conventional unit cell, which is chosen to display the full symmetry of the crystal [see figure (1.1)]. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible volume one can construct with the arrangement of atoms in the crystal such that, when stacked, completely fills the space. This primitive unit cell will not always display all the symmetries inherent in the crystal. A Wigner-Seitz cell is a particular kind of primitive cell which has the same symmetry as the lattice. In a unit cell each atom has an identical environment when stacked in 3 dimensional space. In a primitive cell, each atom may not have the same environment. Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by ,,. 1.3 Cmeaexamatomcrysaddithe fcomrotatall oaxiaset ouniqisomhexamoncrystrigo ClassificaThe definan that unmple, rotamic configstal is thenition to roform of mmpound stion/mirroof these inh The cryal system uof three aque crystametric) sysagonal, tetnoclinic astal systemonal crystaFig (1.1)ation of crning propnder certaating the crguration wn said to hotational symirror plansymmetrieor symmetherent symstal systemused to daxes in a al systemsstem, the otragonal, rand triclinm not to al system.1): The unirystals byperty of a cain operarystal 180which is have a twymmetrienes and traes which tries. A fummetries oms are a gescribe thparticulars. The siother six srhombohenic. Somebe its ow 8ite cell of y symmetrcrystal is iations' the0 degrees aidentical ofold rotas like thisanslationaare a ull classificof the crysgrouping oheir latticer geometrmplest ansystems, inedral (alsoe crystallown crystalthe crystary: its inherene crystal about a ceto the orational syms, a crystal symmetrcombinatcation of astal are ideof crystal se. Each crrical arrannd most n order ofo known aographers system, al structurent symmetremains ertain axis riginal commetry abl may havries, and ation of a crystal ientified. structures rystal systngement. symmetricf decreasinas trigonals considerbut insteae. try, by whunchangemay resunfiguratiobout this ave symmealso the sotranslatiois achievedaccordingtem consisThere arec, the cubng symmel), orthorhr the hexad a part hich we ed. For ult in an on. The axis. In etries in o-called on and d when g to the sts of a e seven bic (or etry, are hombic, xagonal of the 91.4 The Bravais lattices: When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. They describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices which are distinct from one another in the translational symmetry they contain. All crystalline materials recognized until now fit in one of these arrangements. The fourteen three-dimensional lattices, classified by crystal system, are shown in figure (1.2). The Bravais lattices are sometimes referred to as space lattices. The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is described by its crystallographic point group. ThtricmoorthexrhotetrcubFig.( he 7 Crystal syclinic onoclinic thorhombic xagonal ombohedral ragonal bic (1.2): The 1ystems 14 Bravais The 14 Bravasimple simple simple simple 10lattices inais Lattices: base-cente base-cente body-centbody-cent three dimeered ered tered tered ension. body-centered face-centered face-c centered 11There are seven crystal systems: 1. Triclinic, all cases not satisfying the requirements of any other system. There is no necessary symmetry other than translational symmetry, although inversion is possible. 2. Monoclinic, requires either 1 twofold axis of rotation or 1 mirror plane. 3. Orthorhombic, requires either 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes. 4. Tetragonal, requires 1 fourfold axis of rotation. 5. Rhombohedral, also called trigonal, requires 1 threefold axis of rotation. 6. Hexagonal, requires 1 six fold axis of rotation. 7. Cubic or Isometric, requires 4 threefold axes of rotation. The table (1.1) gives a brief characterization of the various crystal systems, the seven crystal systems make up fourteen Bravais lattice types in three dimensions. Table(1.1): Characterization of the various crystal system. System Number of Lattices Lattice Symbol Restriction on crystal cell angle Cubic 3 P or sc, I or bcc,F or fcc a=b=c = ==90 Tetragonal 2 P, I a=bc = ==90 Orthorhombic 4 P, C, I, F ab c = ==90 Monoclinic 2 F, C ab c ==90 Triclinic 1 P ab c Trigonal 1 R a=b=c = = </p>