Reciprocal Lattices Simulation Using Matlab

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

{…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.

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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

   

 

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Contents   

Chapter 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 Bragg’s 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

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Abstract 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 Facecentered 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.

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Chapter 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.

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1.2 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 α,β,γ. 7   

Fig (1.1) 1): The uniite cell of the crystaal structuree. 1.3 Classifica C ation of crrystals byy symmetrry: The definning propperty of a crystal c is its i inherennt symmettry, by wh hich we meaan that unnder certaain operaations' thee crystal remains unchangeed. For exam mple, rotaating the crrystal 1800 degrees about a a ceertain axis may resu ult in an atom mic configguration which w is identical to the orriginal configuratio on. The crysstal is thenn said to have h a twofold rotaational sym mmetry abbout this axis. a In addiition to rootational syymmetries like thiss, a crystal may havve symmeetries in the form f of mirror m plannes and traanslational symmetrries, and aalso the so o-called com mpound symmetrie s es which are a combinattion of translatio on and rotattion/mirroor symmettries. A fuull classificcation of a crystal iis achieved d when all of o these inhherent sym mmetries of o the crysstal are ideentified. The crystal system ms are a grouping g of o crystal structures s according g to the axiaal system used u to describe thheir latticee. Each crrystal systtem consissts of a set of o three axes a in a particularr geometrrical arranngement. There aree seven uniqque crystaal systemss. The simplest an nd most symmetricc, the cub bic (or isom metric) sysstem, the other o six systems, s in n order off decreasinng symmeetry, are hexaagonal, tettragonal, rhombohe r edral (also o known as a trigonall), orthorh hombic, monnoclinic and a triclinnic. Somee crystallo ographerss considerr the hex xagonal crysstal system m not to be its ow wn crystal system, but insteaad a part of the trigoonal crystaal system.

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1.4 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.

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Th he 7 Crystal syystems

The 14 Bravaais Lattices:

tricclinic

simple

base-centeered

simple

base-centeered

simple

body-centtered

simple

body-centtered

moonoclinic

body-centered

ortthorhombic

hexxagonal

rhoombohedral

tetrragonal

face-centered

cub bic

Fig.((1.2): The 14 1 Bravais lattices in three dimeension. 10   

face-ccentered

There 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

P or sc, I or bcc,F or

a=b=c

fcc

α =β =γ=90°

Cubic

3

Tetragonal

2

P, I

Orthorhombic

4

P, C, I, F

Monoclinic

2

F, C

Triclinic

1

P

Trigonal

1

R

a=b≠c α=β =γ=90° a≠b≠ c α=β =γ=90° a≠b≠ c α=β=90 °≠β a≠b≠ c α≠β≠γ a=b=c α=β =γ <120° ,≠90° a=b≠c

Hexagonal

1

P

α =β =90° γ=120°

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1.5 Three T Dim mension Crystal C L Lattice Im mages: 1.5.11 Simple lattices an nd their unit u cell: Simple Cubic C (SC C) - There is one hosst atom (laattice poinnt) at each h corner of a cubic uniit cell. Thee unit cell is describ bed by threee edge lenngths a = b = c = 2r (rr is the host atom raadius), andd the anglees betweenn the edgees, alpha = beta = gam mma = 90 degrees as shownn in figuree (1.3). There T is oone atom wholly insidde the cubbe (Z = 1). Unit ceells in wh hich there are host atoms (orr lattice poinnts) only at a the eightt corners are a called primitive..

F (1.3): The Fig T Crystaal structurre of Simpple Cubic ((SC). Body Ceentered Cuubic (BCC C) - Theree is one hoost atom aat each co orner of the cubic unitt cell and one atom m in the ceell center.. Each atoom touchees eight otheer host atooms along the body diagonal of the cubbe (a = 2..3094r, Z = 2) as show wn in figuure (1.4).

Fig (11.4): The Crystal strructure off Body Cennter Cubicc (BCC). Face Cenntered Cubbic (FCC)) - There is i one hosst atom at each corn ner, one hostt atom in each e face,, and the host h atomss touch aloong the faace diagon nal (a = 12   

2.82284r, Z = 4) as shown in fiigure (1.5 5). This laattice is ""closest paacked", becaause spherres of equual size occcupy the maximum m amountt of space in this arranngment (774.05%); since s this closest paacking is based b on a cubic array, it is calleed "cubic closest paacking": CCP C = FCC C.

Fig (1..5): The Crystal C struucture of Face F Centtered Cubbic (FCC). FCC Priimitive - It is also possible to choose a primiitive unit cell to desccribe the FCC F lattice. The celll is a rhom mbohedron, with a = b = c = 2r, and alphha = beta = gammaa = 60 deggrees, as shown s in figure (1..6). [A cu ube is a rhom mbohedronn with alppha = beta = gammaa = 90 deggrees!]

Fig (1.6): Permitivve Face Centered C C Cubic (FCC C). Simple Hexagonal H l (SH) - Spheres S off equal sizee are most densely packed (witth the leasst amount of emptyy space) in n a plane when w eachh sphere touches t six other sphheres arrannged in thhe form of o a reguular hexaggon. When n these hexaagonally closest packed plaanes (the plane thhrough thee centers of all spheeres) are stacked s dirrectly on top of onee another,, a simple hexagonaal array 13   

resuults; this iss not, how wever, a thhree-dimen nsional cloosest packked arrang gement. The unit cell, outlined in black, is compossed of onee atom at each corn ner of a prim mitive unitt cell (Z = 1), the eddges of which w are: a = b = c = 2r, wheere cell edgees a and b lie in the t hexaggonal plan ne with anngle a-b = gamma = 120 degrrees, and edge e c is the verticaal stacking g distance, as shownn in figure (1.7).

Fig (1.7): Thee Crystal structure s of o Simple Hexagona H al (SH).

1.5.22 Closest Packing: Hexagonnal Closest Packing (HCP) - To T form a three-dim mensional closest packked structuure, the hexagonal h closest packed plaanes must be stackeed such that atoms inn successiive planes nestle in i the triangular ""grooves" of the precceeding plane. Notee that theree are six of o these "ggrooves" ssurroundin ng each atom m in the heexagonal plane, p butt only threee of them m can be ccovered by y atoms in thhe adjacennt plane. The T first plane p is laabeled "A A" and thee second plane p is labeeled "B", and a the peerpendicuular interpllanar spaccing betweeen plane A and planne B is 1..633r (com mpared too 2.000r for f simplee hexagonnal). If th he third planne is againn in the "A A" orientattion and succeeding s g planes aare stacked d in the repeeating patttern ABABA... = (A AB), the resulting r c closest paacked struccture is HCP P, see figuure (1.8).

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F (1.8): The Crysstal Structuure of Hexxagonal Closest Fig C Paccking (HC CP). HCP Coordinationn - Each host h atom in an HC CP lattice iis surroun nded by and touches 12 nearest neighborss, each at a distance of 2r: sixx are in thee planar hexaagonal arrray (B layeer), and siix (three in n the A laayer abovee and threee in the A laayer below w) form a trigonal t prrism aroun nd the cenntral atom,, see figure (1.9).

F (1.9): The nearrest neighb Fig bors in HC CP Structuure. Cubic Cllosest Paccking (CCP P) - If the atoms in the third llayer lie over o the threee groovess in the A layer whhich were not coverred by thee atoms in n the B layeer, then thee third layyer is diffeerent from m either A or B and is labeled "C". If a foourth layeer then reppeats the A layer orientatioon, and suucceeding g layers repeeat the paattern AB BCABCA... = (AB BC), the resulting closest packed struccture is CCP C = FCC C, as show wn in figu ure (1.10). Again, thhe perpen ndicular spaccing betweeen any tw wo successsive layerss is 1.633rr.

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Fig (1.110): The Crystal C Struucture of Cubic Cloosest Packking (CCP) P). CCP Coordinationn - Each host h atom m in a CCP P lattice iis surroun nded by and touches 12 nearest neighborss, each at a distance of 2r: sixx are in thee planar hexaagonal (B) plane, and a six (thhree in thee C layer above annd three in n the A layeer below) form f a triggonal anti--prism (also knownn as a distoorted octah hedron) arouund the cenntral atom m, see figurre (1.11).

F (1.11): The nearrest neighbors in CC Fig CP Structture. Rhombohedral (R R) lattice - If, in n the (A ABC) layeered latticce, the interrplanar sppacing is not n the closest packeed value (1.633r), thhen the prrimitive (Z = 1) unit cell c is a rhhomboheddron with a = b = c <> 2r annd alpha = beta = gam mma <> 60 degrees, as shhown in figure (11.12). Thhe non-prrimitive hexaagonal uniit cell (Z = 3).may also a be cho osen.

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Fig (11.12): Thee Crystal Structure S of o Rhomboohedral (R R) lattice. 2- & 3-laayer repeaats - Theree is only one o way to producee a repeat pattern (cryystal latticee) in two layers off hexagonally closeest packedd planes: (AB) ( = HCP P. Likewise, there is only onne way to o producee a repeat pattern in three layeers of hexaagonally closest paccked planees: (ABC) = CCP, seee figure (1.13). (

Fiig (1.13): The repeaat pattern of 2- & 3--layer hexxagonally cclosest pa acked. 4-layer repeats r - Howeverr, there arre two ways w to prroduce a closest packked latticee in four laayers: (AB BAC) and (ABCB), as shownn in figure (1.14). By extension, e , there are increasinng numberrs of ways to producce closest packed latticces in fivve layers, six layeers, etc., up u to andd includinng non-rep peating randdom stacking. Thus, there aree many clo osest (andd pseudo-cclosest) paackings in naatural andd artificial materials.

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Fig (1.144): The reppeat patterrn of 4-layyer hexagoonally cloosest packeed. 1.5.33 Holes (IInterstices) in Clossest Packeed Arrayss Tetraheddral Hole - Considder any tw wo successive plannes in a closest packked latticee. One attom in thhe A layeer nestles in the trriangular groove form med by thrree adjacennt atoms in i the B laayer, and the t four attoms touch h along the edges e (of length 2r)) of a reguular tetrah hedron; thee center oof the tetraahedron is a cavity callled the Teetrahedrall (or Td) hole; h a gueest sphere will just fill this caviity (and toouch the four host spheres) if its raddius is 0.22247r, seee figure (1.15).

Fig (11.15): Scheematics off Tetraheddral Hole. Octahedrral Hole - Adjacennt to the Td hole, thhree atom ms in the B layer matic poly yhedron toucch three attoms in thhe A layer such that a trigonall antiprism (a regular octahedron) is formeed; the ceenter of thhe octahedron is a cavity calleed the Octtahedral (or Oh) hole, see fig gure (1.16)). A guestt sphere will w just 18   

fill this t cavityy (and toucch the six host spherres) if its radius r is 00.4142r. Itt can be show wn that thhere are twice t as many m Td as Oh holles in anyy closest packed bilayyer.

Fig (1.116): The Scchematicss of Octaheedral Holee.

1.5.44 Simple Crystal Structures S s: CsCl Strructure - Each ionn resides on a sepaarate, inteerpenetratiing SC latticce such thhat the catiion is in thhe center of o the anioon unit cell and visaa versa, see figure f (1.117). The tw wo lattices have thee same uniit cell dim mension.

Fig (1.17): ( Thee Crystal Structure S of CsCl. NaCl Strructure - Each ion resides on o a separrate, interppenetratin ng FCC latticce. The tw wo latticess have the same unit cell dimension, ass shown in n figure (1.18).

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Fig (1.18): (1 Thee Crystal Structure S of NaCl. Halite Sttructure - The sodiuum chlorid de structurre may alsso be view wed as a CCP P lattice of o anions (Z ( = 4), with w smalleer cations occupyinng all Oh cavities c (Z = 4), see fiigure (1.199).

Fig (1.199): The Crrystal Stru ucture of Halite H lattees. Fluorite Structuree - The structure of the mineral m ffluorite (ccalcium fluoride) mayy be vieweed as a CC CP lattice of cationss (Z = 4), with the smaller anioon occupyiing all of the Td holes (Z = 8), 8 see figuure (1.20)). The Td cavities c residde on a SC C lattice which w is haalf the dim mension off the CCP lattice.

Fig (1..20): The Crystal C Sttructure off Fluorite.. 20   

Zinc Bleende Struccture - Thee structuree of cubicc ZnS (minneral nam me "zinc blennde") mayy be vieweed as a CC CP lattice of anionss (Z = 4), with the smaller catioons occuppying everry other Td hole (Z Z = 4), as a shown in figure (1.21). [Notte: the othher ZnS mineral, m w wurtzite, caan be desccribed as a HCP laattice of anioons with caations in every e otheer Td hole.]

Fig (1.211): The Crrystal Stru ucture of Zinc Z Blendde. Zinc Bleende latticces - Thee lattice of o cations in zinc bblende is a FCC latticce of the same dim mension as a the aniion latticee, so the structure can be desccribed as interpenetr i rating FCC lattices of the sam me unit ceell dimenssion, as show wn in figuure (1.22). Note thaat the only y differennce betweeen the hallite and zincc blende structures s is a simpple shift in n relative position of the tw wo FCC latticces.

Fig (1.222): The interpenetr i rating FC CC latticess.

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Chapter Two X-ray Diffraction and Crystal Structure 2.1 Introduction: Wilhelm Röntgen discovered X-rays in 1895. Seventeen years later, Max von Laue suggested that they might be diffracted when passed through a crystal, for by then he had realized that their wavelengths are comparable to the separation of lattice planes. This suggestion was confirmed almost immediately by Walter Friedric and Paul Knipping and has grown since then into a technique of extraordinary power. The bulk of this section will deal with the determination of structures using X-ray diffraction. The mathematical procedures necessary for the determination of structure from Xray diffraction data are enormously complex, but such is the degree of integration of computers into the experimental apparatus that the technique is almost fully automated, even for large molecules and complex solids. The analysis is aided by molecular modelling techniques, which can guide the investigation towards a plausible structure. X-rays are typically generated by bombarding a metal with high-energy electrons. The electrons decelerate as they plunge into the metal and generate radiation with a continuous range of wavelengths called Bremsstrahlung. Superimposed on the continuum are a few high-intensity, sharp peaks. These peaks arise from collisions of the incoming electrons with the electrons in the inner shells of the atoms. A collision expels an electron from an inner shell, and an electron of higher energy drops into the vacancy, emitting the excess energy as an X-ray photon. If the electron falls into a K shell (a shell with n = 1), the X-rays are classified as K-radiation, and similarly for transitions into the L (n = 2) and M (n = 3) shells. Strong, distinct lines are labelled Kα, Kβ, and so on.

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Von Laue’s original method consisted of passing a broad-band beam of X-rays into a single crystal, and recording the diffraction pattern photographically. The idea behind the approach was that a crystal might not be suitably orientated to act as a diffraction grating for a single wavelength but, whatever its orientation, diffraction would be achieved for at least one of the wavelengths if a range of wavelengths was used.

2.2 Bragg’s diffraction law: The Bragg's law is the result of experiments into the diffraction of Xrays or neutrons off crystal surfaces at certain angles, derived by physicist Sir William Lawrence Bragg in 1912 and first presented on 1912-11-11 to the Cambridge Philosophical Society. Although simple, Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond. When X-rays hit an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as the Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible [see Figure(2.1)]. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Both neutron and

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X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.

Fig.(2.1): Rayleigh or X-ray scattering. The interference is constructive when the phase shift is a multiple to 2π, as shown in Figure (2.2); this condition can be expressed by Bragg's law: 2 sin

(2.1)

where •

n is an integer determined by the order given,

•

λ is the wavelength of x-rays, and moving electrons, protons and

neutrons, •

d is the spacing between the planes in the atomic lattice, and

•

θ is the angle between the incident ray and the scattering planes

Fig.(2.2): The conventional derivation of Bragg’s law treats each lattice plane as a reflecting the incident radiation. Constructive interference (a ‘reflection’) occurs when difference in phase is equal to an integer number of wavelengths. According to the 2θ deviation, the phase shift causes constructive (left figure) or destructive (right figure) interferences. Note that moving particles, including electrons, protons and neutrons, have an associated De Broglie wavelength. 24   

2.3 Experimentation diffraction method: A large range of laboratory equipment is available for X-ray diffraction and spectroscopy, and the International Union of Crystallography has published a useful, which shows what apparatus and supplies are available and where to find them. A laboratory manual by Azaroff and Donahue describes twenty-one experiments in X-ray crystallography. This section deals with some experimental methods to find crystal structure by X-ray diffraction.

2.3.1 The Laue Method: Diffraction patterns from a single crystal are produced using a beam of white, X-ray radiation. The range of wavelengths in the white X-ray radiation assures that diffracting conditions will be met. The Laue method is mainly used to determine the orientation of large single crystals. White radiation is reflected from, or transmitted through, a fixed crystal. The diffracted beams form arrays of spots, that lie on curves on the film. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg’s law for the values of d and

involved.

Each curve therefore corresponds to a different wavelength. The spots lying on any one curve are reflections from planes belonging to one zone. Laue reflections from planes of the same zone all lie on the surface of an imaginary cone whose axis is the zone axis. There are two practical variants of the Laue method, the back-reflection and the transmission Laue method:

1- Back-reflection Laue: In the back-reflection method, the film is placed between the X-ray source and the crystal. The beams which are diffracted in a backward direction are recorded. One side of the cone of Laue reflections is defined by 25   

the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an hyperbola, as shown in Figure (2.3).

Fig.(2.3): Back-Reflection Laue Method.

2- Transmission Laue: In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal. One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse, as illustrated in Figure (2.4).

Fig.(2.4): Transmission Laue Method. The crystal orientation is determined from the position of the spots. Each spot can be indexed, i.e. attributed to a particular plane, using special charts.

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The Greninger chart is used for back-reflection patterns and the Leonhardt chart for transmission patterns. The Laue technique can also be used to assess crystal perfection from the size and shape of the spots. If the crystal has been bent or twisted in anyway, the spots become distorted and smeared out.

2.3.2 The rotation method: In the rotation method the Bragg condition for each reflection is satisfied for monochromatic radiation by rotating the sample crystal. Each lattice plane is brought in turn into the diffraction condition for a short period of time as the crystal rotates. An equivalent description is to imagine reciprocal lattice points traversing the Ewald sphere as the lattice rotates. This may be visualised with the aid of Figure (2.5). The method is utilized in determining the structure of unknown materials and to provide an unequivocal determination of unit cell dimensions.

Fig.(2.5): 2-dimensional section through reciprocal space showing how the Ewald sphere sweeps through reciprocal lattice points bringing them into the diffraction condition.

27   

The Ewald sphere of radius 1/ is shown in two positions with respect to the reciprocal lattice after a rotation about the axis O which is normal to the paper. The shaded region represents that part of reciprocal space which cuts the sphere as it rotates. (In fact the Ewald sphere is fixed and the reciprocal lattice rotates but for simplicity the figure has been drawn in the opposite fashion; the two situations are however equivalent). It is seen that there is a specific region of reciprocal space which is missed by the Ewald sphere as it rotates. This is called the blind region. Any reflections which are not collected due to being in the blind region can be collected by re-orienting the sample crystal so that they enter the region of space traversed by the Ewald sphere. Alternatively the presence of symmetry elements in the crystal may imply that symmetry equivalent reflections of those lost in the blind region may be observed elsewhere in reciprocal space where they do cut the Ewald sphere.

2.3.3 X-ray Powder Diffracon: Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. A powder or polycrystalline sample is irradiated with a beam of X-rays and the resulting powder diffraction pattern is recorded with a detector - photographic film, image plate, etc. The powder method is the most widely applied technique in the field of X-ray diffraction analysis for the identification of phases or compounds and the measurement of lattice spacing. A Powder Diffraction File exists with over a hundred thousand characteristic diffraction patterns ("fingerprints") for elements, alloys, minerals and organic compounds. Ideally, every possible crystalline orientation is represented equally in a powdered sample. The resulting orientational averaging causes the three dimensional reciprocal space that is studied in single crystal diffraction to be projected onto a single dimension. The three dimensional space can be described with (reciprocal) axes x*, y* and z* or alternatively in spherical 28   

coorrdinates q,, φ*, χ*. Inn powder diffractio on intensityy is homoogeneous over o φ* and χ* and onnly q remains as ann importan nt measuraable quanttity, as sh hown in Figuure (2.6). In practiice, it is sometimees necesssary to rootate the sample orienntation to eliminatee the effectts of textu uring and achieve a truue random mness.

Figg.(2.6): Tw wo-dimenssional pow wder diffra action setuup with flaat plate deetector. When thhe scattereed radiatiion is colllected onn a flat pllate detecctor the rotattional aveeraging leads to sm mooth diffr fraction rinngs arounnd the beaam axis ratheer than thhe discretee Laue spoots as obsserved forr single crrystal diffrraction. The angle bettween thee beam axxis and thee ring is called c the scattering g angle and in X-rayy crystalloography always a deenoted ass 2θ. In aaccordancce with Braggg's law, each e ring corresponnds to a particular p r reciprocal l lattice veector G in thhe sample crystal. This T leads to t the defiinition of the scatterring vecto or as: 2 sin

4 sin

/

(2.2) (

Powder diffractioon data arre usually y presenteed as a ddiffractog gram in whicch the difffracted inntensity I is shown as functioon either of the scaattering anglle 2θ or ass a functioon of the scattering s vector q. The latterr variable has the advaantage thaat the difffractogram m no lon nger depeends on tthe value of the wavvelength λ. λ To faciilitate com mparabilitty of dataa obtained with diifferent wavvelengths the t use off q is thereefore recom mmended and gainiing accepttability. An instrumen i nt dedicateed to perfoorm powd der measurrements iss called a powder p diffrractometerr. Relative to other methods of analy ysis, powdder diffracction allo ows for rapidd, non-desstructive analysis a of multi-co omponent mixtures without th he need for extensive e sample preparation p n. Identification is performedd by comp parison 29   

of the diffraction pattern to a known standard or to a database such as the International Centre for Diffraction or the Cambridge Structural Database (CSD). Advances in hardware and software, particularly improved optics and fast detectors, have dramatically improved the analytical capability of the technique, especially relative to the speed of the analysis. The fundamental physics upon which the technique is based provides high precision and accuracy in the measurement of interplanar spacings, sometimes to fractions of an Ångström. The ability to analyze multiphase materials also allows analysis of how materials interact in a particular matrix.

3.3.4 Electrons or Neutrons Diffraction: Because it is relatively easy to use electrons or neutrons having wavelengths smaller than a nanometre, electrons and neutrons may be used to study crystal structure in a manner very similar to X-ray diffraction. Electrons do not penetrate as deeply into matter as X-rays, hence electron diffraction reveals structure near the surface; neutrons do penetrate easily and have an advantage that they possess an intrinsic magnetic moment that causes them to interact differently with atoms having different alignments of their magnetic moments.

2.4 Reciprocal lattice: In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that ·

1

(2.3)

for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. For an infinite three dimensional lattice, defined by its primitive vectors ( ,

,

), its reciprocal lattice can be determined by generating its

three reciprocal primitive vectors, through the formulae 2

(2.4)

·

30   

2

(2.5)

·

2

(2.6)

·

Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion: 2

(2.7)

This method appeals to the definition, and allows generalization to arbitrary dimensions. Curiously, the cross product formula dominates introductory materials on crystallography. The above definition is called the "physics" definition, as the factor of 2π comes naturally from the study of periodic structures. An equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice to be

·

1 which changes the definitions of the

reciprocal lattice vectors to be (2.8)

·

and so on for the other vectors. The crystallographer's definition has the advantage that the definition of direction of

is just the reciprocal magnitude of

in the

, dropping the factor of 2π. This can simplify certain

mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed. Each point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal to the real space planes, and the magnitude of the reciprocal lattice vector is equal to the reciprocal of the interplanar spacing of the real space planes. The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. For Bragg reflections in neutron and X-ray diffraction, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice 31   

vecttor. The diffraction d n pattern of a crysstal can be b used tto determiine the recipprocal vecctors of thhe lattice. Using thiis process, one can infer the atomic arranngement of o a crystaal. • Vector algebra a is very convvenient fo or describiing otherw wise comp plicated diffractioon problem ms • The recipprocal latttice offerss a simple approachh to handliing diffracction in terms of vectors • Use of thhe reciproocal latticee permits the t analyssis of diffr fraction prroblems that cannnot be acceessed by Bragg’s B Laaw. • The recipprocal latttice is impportant in all phasess of solid state physics, so an underrstanding of o this conncept is usseful in annd of itselff

2.5 Diffractio D on in recip procal space: Definingg that the diffractionn vector G, G where G = k-k0. The difffraction condditions in reciprocaal space are ∆

, this means thhat the su ufficient

conddition to diffraction d as shownn in Figuree (2.7) wass: |∆ | = 4p(sinq)/l 4 = |Ghkl| = 2p/d 2 hkl

(2.9) (

Figg.(2.7): Diffraction D in reciproocal spacee. More dirrectly, whhen the diiffraction conditionn ∆

is satisfiied, the

scatttering ampplitude (sttructure faactor) is deetermined by:

S

k

=

n

∑

j=1

fie

r r iG .r j

(2.10) ( 32 

 

where fj is atomic scattering factor (form factor). The usual form of this result follows on writing, the lattice vector as: r r r r r j = u j a + v jb + w jc

(2.11)

Then, for the reflection labeled by u , v , w (i.e. Reciprocal lattice vector), we have:

rr G.rj = 2π ⎡⎣u j h + v j + w j l ⎤⎦

SG =

n

∑

j =1

fie

(

2π i u j h + v j k + w jl

(2.12)

) (2.13)

Where n is a number of atom in unit cell, and ujvj wj was position of each atom in the unit cell.

2.6 Fourier analysis: In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions. The attempt to understand functions by breaking them into basic pieces that are easier to understand is one of the central themes in Fourier analysis. Fourier analysis is named after Joseph Fourier who showed that representing a function by a trigonometric series greatly simplified the study of heat propagation. Today the subject of Fourier analysis encompasses a vast spectrum of mathematics with parts that, at first glance, may appear quite different. In the sciences and engineering the process of decomposing a function into simpler pieces is often called an analysis. The corresponding operation of rebuilding the function from these pieces is known as synthesis. In this context the term Fourier synthesis describes the act of rebuilding and the term Fourier analysis describes the process of breaking the function into a sum of simpler pieces. In

33   

mathematics, the term Fourier analysis often refers to the study of both operations. In Fourier analysis, the term Fourier transform often refers to the process that decomposes a given function into the basic pieces. This process results in another function that describes how much of each basic piece are in the original function. It is common practice to also use the term Fourier transform to refer to this function. However, the transform is often given a more specific name depending upon the domain and other properties of the function being transformed, as elaborated below. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis has a corresponding inverse transform that can be used for synthesis. In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (17681830) for the purpose of solving the heat equation in a metal plate. It led to a revolution in mathematics, forcing mathematicians to reexamine the foundations of mathematics.

2.7 Fourier series: In this section, ƒ(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x; to write such a function as an infinite sum, or series of simpler 2π–periodic functions, it will be start by using an infinite sum of sine and cosine functions on the interval [−π, π], and then discuss different formulations and generalizations. Fourier's formula for 2π-periodic functions using sines and cosines For a 2π-periodic function ƒ(x) that is integrable on [−π, π], the numbers 34   

(2.14) and (2.15) are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by: (2.16) The partial sums for ƒ are trigonometric polynomials. One expects that the functions SN ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum

is called the Fourier series of ƒ. It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.

2.8 Exponential Fourier series: Using Euler's formula, (2.17) where i is the imaginary unit, to give a more concise formula: (2.18) The Fourier coefficients are then given by: (2.19) The Fourier coefficients an, bn, cn are related via and

35   

The notation cn is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of ƒ, such as F or

and functional notation often replaces subscripting.

Thus:

(2.20) In engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

36   

Chapter Three Reciprocal Lattice Simulation 3.1 Introduction: The concept of reciprocally has been introduced in the X-ray Diffraction within the Bragg’s equation. This inverse scaling between real and reciprocal space is based on Fourier transforms. Josiah Willard Gibbs first made the formalisation of reciprocal lattice vectors in 1881. The reciprocal vectors lie in “reciprocal space”, an imaginary space where planes of atoms are represented by reciprocal points, and all lengths are the inverse of their length in real space. In 1913, P. P. Ewald demonstrated the use of the Ewald sphere together with the reciprocal lattice to understand diffraction. It geometrically represents the conditions in reciprocal space where the Bragg equation is satisfied. Diffraction patterns from single crystals can provide a good deal of information about the atomic structure of the compound. Many compounds, however, can only be obtained as powders. Although a powder diffraction pattern yields much less information than that generated by a single crystal, it is unique to each substance, and is therefore highly useful for purposes of identification. A diffraction pattern is the 2-D picture obtained by shining shortwavelength radiation through a material. The incident radiation is scattered coherently by the atoms making up the material, and the resultant scattered radiation generates a pattern of interference that is dependent upon the relative positioning of the atoms. The first radiation ever used in crystal diffraction was white (broadband) X-ray radiation. If X-rays could be diffracted in the manner of light through an optical grating, it would be conclusive proof of their wave nature. At the 37   

same time as these first studies of X-rays were being conducted, early theories of crystal structure were being proposed in which crystals were postulated to be composed of regular sub-units. These theories led von Laue, in 1912, to suggest that a crystal could provide the "grating" needed for the X-ray experiment. Soon thereafter, the first X-ray diffraction photos were produced. The diffraction of either photons or electrons (sometimes neutrons) is one of the most powerful techniques for surface structure determination. Unfortunately, the diffraction pattern is not a direct representation of the realspace arrangement of the atoms in a solid or on a surface. The most convenient way to link the real structure of the material to it's diffraction pattern is through the reciprocal lattice. In order for measureable diffraction to occur, the wavelength of the interrogating wave-particle should be on the same order as the periodicity of the features. For atoms or molecules in a crystalline solid, this periodicity is a few Angstroms. This means that if we are using photons to examine the lattice spacing of a solid, their wavelength should be a few Angstroms (X-rays).

3.2 Reciprocal Lattice to SC Lattice: The primitive translation vectors of a simple cubic lattice may be taken as the set:

Here , ,

are orthogonal vectors of unit length. The volume of the cell

is: |

.

|

The primitive translation vectors of the reciprocal lattice are found from the standard prescription:

38   

Here the reciprocal lattice is itself a simple cubic lattice, now of lattice constant

.

The interpretation of X-ray diffraction pattern (the reciprocal crystal structure) was done by using FFT command from MATLAB. The process of sketching the crystal structure of simple cubic was done by generating a 500x500 zeros matrix, and defining the atoms as a circles of values one, the position of the circles are arranged to be separated by distance (d=2r) as shown in Figure (3.1). The diffraction pattern was obtained by using the Fast Fourier Transformation to this matrix, and then taking the inverse Fast Fourier Transformation for the real part of the result (see Appendix). The sketch in Figure (3.2) shows the diffraction pattern of Simple cubic crystal structure. One observes that it’s reciprocal is also a simple cubic lattice as it was expected. The cubic lattice is therefore said to be dual, having its reciprocal lattice being identical.

39   

50 100 150 200 250 300 350 400 450

500

50

100

150

200

250

300

350

400

450

500

Fig.(3.1): The two dimensional crystal structure of Simple Cube.

50 100 150 200 250 300 350 400 450

500

50

100

150

200

250

300

350

400

450

Fig.(3.2): Schematic of the diffraction pattern of SC.

40   

500

3.3 Reciprocal Lattice to BCC Lattice: The primitive translation vectors of a body center cubic lattice may be taken as the set:

Where

is the side of the conventional cube and , ,

are orthogonal

unit vectors parallel to the cube edges. The volume of the cell is: |

|

.

The primitive translation vectors of the reciprocal lattice are found from the standard prescription:

These are just the primitive vectors of an FCC lattice, so that an FCC lattice is the reciprocal lattice of the BCC lattice. An attempted has been made to describe the reciprocal crystal structure (i.e. the diffraction pattern) for Body Center Cube crystal structure by introducing the (500x500) zeros matrix, and the atoms are pointed as a centers of the values one, the center of the atoms are arranged to be separated by distance (

√ √

) in the x-direction, and the distance (

√

) in the y-

direction, as shown in the Figure (3.3). The sketch in Figure (3.4) shows the diffraction pattern of the previous configurations. One observes that the lattice is Face Center cube, as it was expected.

41   

50 100 150 200 250 300 350 400 450

500

50

100

150

200

250

300

350

400

450

500

Fig.(3.3): The two dimensional crystal structure of Body Centered Cube.

50 100 150 200 250 300 350 400 450

500

50

100

150

200

250

300

350

400

450

Fig.(3.4): Schematic of the diffraction pattern of BCC.

42   

500

3.4 Reciprocal Lattice to FCC Lattice The primitive translation vectors of a face center cubic lattice may be taken as the set:

Where

is the side of the conventional cube and , ,

are orthogonal

unit vectors parallel to the cube edges. The volume of the cell is: |

.

|

The primitive translation vectors of the reciprocal lattice of the FCC are found from the standard prescription:

These are primitive translation vectors of an BCC lattice, so that an BCC lattice is the reciprocal lattice of the FCC lattice. The interpretation of X-ray diffraction pattern (the reciprocal crystal structure) of the crystal structure of Face Center Cubic was also done by generation a 500x500 zeros matrix, and defining the atoms as a circles of values one, the center of the atoms are arranged to be separated by distance (

√

) in the x-direction, as shown in the Figure (3.5). The diffraction

pattern was obtained by using the Fast Fourier Transformation to this matrix, and then taking the inverse Fast Fourier Transformation for the real part of the result. The sketch in Figure (3.6) shows the diffraction pattern of the previous configurations. One observes that the reciprocal crystal structure of Face Center cube was Body Center Cube.

43   

50 100 150 200 250 300 350 400 450

500

50

100

150

200

250

300

350

400

450

500

Fig.(3.5): The two dimensional crystal structure of Face Centered Cube.

50 100 150 200 250 300 350 400 450

500

50

100

150

200

250

300

350

400

450

Fig.(3.6): Schematic of the diffraction pattern of FCC.

44   

500

3.5 Conclusion: The crystal structure can be studied through the diffraction of photons, neutrons, and electrons. The diffraction depends on the crystal structure and on the wavelength. A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal. Every crystal structure has two lattices associated with it, namely the crystal lattice and the reciprocal lattice. An attempted has been made to describe the reciprocal lattice for different crystal structures in a simple way, using the Fast Fourier Transformation command (FFT) from MATLAB. To test the accuracy of this method, the reciprocal lattices of well known: simple cubic, body center, and face center crystal structures were examined. The investigation shows that the simple cubic Bravais lattice, with cubic primitive cell of side (a), have a simple cubic reciprocal lattice with a cubic primitive cell of side ( ). The simple cubic lattice is therefore said to be dual, having its reciprocal lattice being identical. The reciprocal lattice for Facecentered cubic lattice is a Body-centered cubic lattice. The reciprocal lattice for Body-centered cubic lattice is a Face-centered cubic lattice. The result of this project shows that the FFT is a powerful technique to studies a reciprocal lattice. Thus, the suggestion could be made to use the FFT 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 of constant values, which was established in present work.

45   

References 1 Charles Kittel, “Introduction to Solid State Physics”, Sixth Edition, John Wiley & Sons, Inc. (1986). 2 William Clegg, Alexander J. Blake, Peter Main, and, Robert Gould, “Crystal Structure Analysis: Principles and Practice”, Contributor William Clegg, Oxford University, (2001). 3 James D. Patterson, and Bernard C. Bailey, “ Solid-State Physics: Introduction to the Theory”, Springer-Verlag Berlin Heidelberg, (2007). 4 Richard J. D. Tilley, “Crystals and Crystal Structures”, John Wiley & Sons Ltd, England, (2006). 5 Uri Shmueli, “Theories and Techniques of Crystal Structure Determination”, Oxford University Press Inc., New York, (2007). 6 http://www.chem.lsu.edu/htdocs/people/sfwatkins/ch4570/lattices/ lattice.html 7 http://en.wikipedia.org/wiki/Bravais_lattice 8 http://en.wikipedia org/wiki/Crystal_structure 9 http://en.wikipedia.org/wiki/Reciprocal_lattice 10 http://en.wikipedia.org/wiki/Fourier_analysis 11 http://www.gwyndafevans.co.uk/thesis-html/node33.html 12 http://www.matter.org.uk/diffraction/x-ray/laue_method.htm

46   

Appendix % Simple Cubic clc clear all %r=input('Enter redius of the atome :') for i=1:500 for j=1:500 g(i,j)=0; end end r=25; d=r*2; for x=r:d:500 for y=r:d:500 for i=1:500 for j=1:500 if sqrt((i-x)^2+(j-y)^2)<=r; g(i,j)=1; end end end end end %colormap('gray') imagesc(g) pause farray=fft2(g,500,500); psf=abs(farray); imagesc(psf); pause aaa=fftshift(psf); imagesc(aaa); pause farray1=fft2(psf,500,500); psf1=abs(farray1); imagesc(psf1); pause

47   

% Body Center Cubic clc clear all %r=input('Enter redius of circle :') for i=1:500 for j=1:500 g(i,j)=0; end end r=30; dx=r*4*sqrt(2)/sqrt(3); dy=r*4/sqrt(3); for x=r:dx:500 for y=r:dy:500 for i=1:500 for j=1:500 if sqrt((i-x)^2+(j-y)^2)<=r; g(i,j)=1; end end end end end for x=r:dx:500 for y=r:dy:500 for i=1:500 for j=1:500 if sqrt((i-x-dx/2)^2+(j-y-dy/2)^2)<=r; g(i,j)=1; end end end end end %colormap('gray') imagesc(g) pause farray=fft2(g,500,500); psf=abs(farray); imagesc(psf); pause 48   

aaa=fftshift(psf); imagesc(aaa); pause farray1=fft2(psf,500,500); psf1=abs(farray1); imagesc(psf1); pause

% Face Center Cubic clc clear all %r=input('Enter redius of circle :') for i=1:500 for j=1:500 g(i,j)=0; end end r=25; d=r*2*sqrt(2); for x=r:d:500 for y=r:d:500 for i=1:500 for j=1:500 if sqrt((i-x)^2+(j-y)^2)<=r; g(i,j)=1; end end end end end for x=r:d:500 for y=r:d:500 for i=1:500 for j=1:500 if sqrt((i-x-d/2)^2+(j-y-d/2)^2)<=r; g(i,j)=1; end end end end end 49   

%colormap('gray') imagesc(g) pause farray=fft2(g,500,500); psf=abs(farray); imagesc(psf); pause aaa=fftshift(psf); imagesc(aaa); pause farray1=fft2(psf,500,500); psf1=abs(farray1); imagesc(psf1); pause

50