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

{…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.

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

 

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.

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 Face-centered cubic lattice is a Body-Face-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.

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.

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.

1.3 C mea exam atom crys addi the f com rotat all o axia set o uniq isom hexa mon crys trigo Classifica The defin an that un mple, rota mic config stal is then ition to ro form of m mpound s tion/mirro of these inh The cry al system u of three a que crysta metric) sys agonal, tet noclinic a stal system onal crysta Fig (1.1) ation of cr ning prop nder certa ating the cr guration w n said to h otational sy mirror plan symmetrie or symmet herent sym stal system used to d axes in a al systems stem, the o tragonal, r and triclin m not to al system. 1): The uni rystals by perty of a c ain opera rystal 180 which is have a tw ymmetrie nes and tra es which tries. A fu mmetries o ms are a g escribe th particular s. The si other six s rhombohe nic. Some be its ow ite cell of y symmetr crystal is i ations' the 0 degrees a identical ofold rota s like this anslationa are a ull classific of the crys grouping o heir lattice r geometr mplest an systems, in edral (also e crystallo wn crystal the crysta ry: its inheren e crystal about a ce to the or ational sym s, a crysta l symmetr combinat cation of a stal are ide of crystal s e. Each cr rical arran nd most n order of o known a ographers system, al structure nt symmet remains ertain axis riginal co mmetry ab l may hav ries, and a tion of a crystal i entified. structures rystal syst ngement. symmetric f decreasin as trigonal s consider but instea e. try, by wh unchange may resu nfiguratio bout this a ve symme also the so translatio is achieved according tem consis There are c, the cub ng symme l), orthorh r the hex ad 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

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.

Th tric mo ort hex rho tetr cub Fig.( he 7 Crystal sy clinic onoclinic thorhombic xagonal ombohedral ragonal bic (1.2): The 1 ystems 14 Bravais The 14 Brava simple simple simple simple lattices in ais Lattices: base-cente base-cente body-cent body-cent three dime ered ered tered tered ension. body-centered face-centered 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

Cubic 3 P or sc, I or bcc,F or fcc a=b=c α =β =γ=90° Tetragonal 2 P, I a=b≠c α=β =γ=90° Orthorhombic 4 P, C, I, F a≠b≠ c α=β =γ=90° Monoclinic 2 F, C a≠b≠ c α=β=90 °≠β Triclinic 1 P a≠b≠ c α≠β≠γ Trigonal 1 R a=b=c α=β =γ <120° ,≠90° Hexagonal 1 P a=b≠c α =β =90° γ=120°

1.5 T 1.5.1 of a 2r (r gam insid poin the othe show host Three Dim 1 Simple Simple C cubic uni r is the ho mma = 90 de the cub nts) only a F Body Ce cubic unit er host ato wn in figu Fig (1 Face Cen t atom in e mension C lattices an Cubic (SC it cell. The st atom ra degrees be (Z = 1 at the eight Fig (1.3): T entered Cu t cell and oms along ure (1.4). 1.4): The ntered Cub each face, Crystal L nd their u C) - There e unit cell adius), and as shown ). Unit ce t corners a The Crysta ubic (BCC one atom the body Crystal str bic (FCC) , and the h Lattice Im unit cell: is one hos is describ d the angle n in figure ells in wh are called al structur C) - There m in the ce diagonal ructure of ) - There i host atoms mages: st atom (la bed by thre es between e (1.3). T hich there primitive. re of Simp e is one ho ell center. of the cub f Body Cen is one hos s touch alo attice poin ee edge len n the edge There is o are host . ple Cubic ( ost atom a . Each ato be (a = 2. nter Cubic st atom at ong the fa nt) at each ngths a = es, alpha = one atom atoms (or (SC). at each co om touche .3094r, Z c (BCC). each corn ace diagon h corner b = c = = beta = wholly r lattice orner of es eight = 2) as ner, one nal (a =

2.82 beca arran calle desc alph rhom (wit six hexa sphe 284r, Z = ause spher ngment (7 ed "cubic Fig (1. FCC Pri cribe the F ha = beta mbohedron Simple H th the leas other sph agonally eres) are s 4) as sh res of equ 74.05%); s closest pa .5): The C imitive - FCC lattic = gamma n with alp Fig (1.6) Hexagonal st amount heres arran closest p stacked dir own in fi ual size oc since this acking": C Crystal stru It is also e. The cel a = 60 deg pha = beta : Permitiv l (SH) - S of empty nged in th acked pla rectly on igure (1.5 ccupy the closest pa CCP = FCC ucture of F possible ll is a rhom grees, as s = gamma ve Face C Spheres of y space) in he form o anes (the top of one 5). This la maximum acking is b C. Face Cent to choos mbohedro shown in a = 90 deg Centered C f equal size n a plane w of a regu plane th e another, attice is " m amount based on a tered Cub e a primi n, with a = figure (1. grees!] Cubic (FCC e are mos when each ular hexag hrough the , a simple "closest pa t of space a cubic arr bic (FCC). itive unit = b = c = .6). [A cu C). t densely h sphere t gon. When e centers hexagona acked", in this ay, it is cell to 2r, and ube is a packed touches n these of all al array

resu The prim edge degr 1.5.2 pack that prec atom in th labe plan plan repe HCP ults; this is unit cell, mitive unit es a and rees, and e Fig 2 Closest Hexagon ked structu atoms in ceeding pl m in the he he adjacen eled "B", a ne B is 1. ne is again eating patt P, see figu s not, how outlined t cell (Z = b lie in t edge c is t (1.7): The Packing: nal Closes ure, the h n successi ane. Note exagonal p nt plane. T and the pe .633r (com n in the "A tern ABA ure (1.8). wever, a th in black, = 1), the ed the hexag he vertica e Crystal s t Packing hexagonal ive plane e that there plane, but The first p erpendicu mpared to A" orientat BA... = (A hree-dimen is compos dges of w gonal plan al stacking structure o (HCP) - T closest p s nestle i e are six o t only thre plane is la ular interpl o 2.000r f tion and s AB), the r nsional clo sed of one which are: ne with an g distance, of Simple H To form a acked pla in the tri of these "g ee of them abeled "A lanar spac for simple succeeding resulting c osest pack e atom at a = b = c ngle a-b = as shown Hexagona a three-dim anes must angular " grooves" s m can be c A" and the cing betwe e hexagon g planes a closest pa ked arrang each corn = 2r, whe = gamma n in figure al (SH). mensional be stacke "grooves" surroundin covered by e second p een plane nal). If th are stacked acked struc gement. ner of a ere cell = 120 (1.7). closest ed such of the ng each y atoms plane is A and he third d in the cture is

F and hexa A la three laye a fo repe struc spac Fig (1.8): HCP Co touches 1 agonal arr ayer below F Cubic Cl e grooves er, then the ourth laye eat the pa cture is C cing betwe The Crys ordination 2 nearest ray (B laye w) form a t Fig (1.9): losest Pac s in the A e third lay er then rep attern AB CCP = FCC een any tw stal Structu n - Each h neighbors er), and si trigonal pr The near cking (CCP A layer wh yer is diffe peats the BCABCA C, as show wo success ure of Hex host atom s, each at a ix (three in rism aroun rest neighb P) - If the hich were erent from A layer ... = (AB wn in figu sive layers xagonal C in an HC a distance n the A la nd the cen bors in HC atoms in not cover m either A orientatio BC), the ure (1.10) s is 1.633r Closest Pac CP lattice i of 2r: six ayer above ntral atom, CP Structu the third l red by the or B and on, and su resulting . Again, th r. cking (HC is surroun x are in the e and three , see figur ure. layer lie o e atoms in is labeled ucceeding closest he perpen CP). nded by e planar e in the e (1.9). over the n the B "C". If g layers packed ndicular

and hexa laye arou inter (Z = gam hexa Fig (1.1 CCP Co touches 1 agonal (B er below) f und the cen

F Rhombo rplanar sp = 1) unit c mma <> agonal uni 10): The C ordination 2 nearest ) plane, a form a trig ntral atom Fig (1.11) hedral (R pacing is n cell is a rh 60 degre it cell (Z = Crystal Stru n - Each h neighbors and six (th gonal anti-m, see figur : The near R) lattice not the clo

hombohed es, as sh = 3).may a ucture of host atom s, each at a hree in the -prism (al re (1.11). rest neigh - If, in sest packe dron with hown in also be cho Cubic Clo m in a CCP a distance e C layer so known bors in CC n the (A ed value ( a = b = c figure (1 osen. osest Pack P lattice i of 2r: six above an n as a disto CP Struct ABC) laye 1.633r), th <> 2r an 1.12). Th king (CCP) is surroun x are in the nd three in orted octah ture. ered lattic hen the pr nd alpha = he non-pr P). nded by e planar n the A hedron) ce, the rimitive = beta = rimitive

(cry HCP laye Fi pack By e lattic rand in na Fig (1 2- & 3-la ystal lattice P. Likewi ers of hexa ig (1.13): 4-layer r ked lattice extension, ces in fiv dom stack atural and 1.12): The ayer repea e) in two se, there agonally c The repea repeats - e in four la , there are ve layers, ing. Thus d artificial e Crystal S ats - There layers of is only on losest pac at pattern However ayers: (AB increasin six laye , there are materials Structure o e is only o f hexagon ne way to cked plane of 2- & 3-r, there ar BAC) and ng number ers, etc., u e many clo . of Rhombo one way t ally close o produce es: (ABC) -layer hex re two w (ABCB), rs of ways up to and osest (and ohedral (R o produce est packed e a repeat = CCP, se xagonally c ways to pr as shown to produc d includin d pseudo-c R) lattice. e a repeat d planes: ( pattern i ee figure ( closest pa roduce a n in figure ce closest ng non-rep closest) pa pattern (AB) = n three (1.13). acked. closest (1.14). packed peating ackings

1.5.3 pack form the e is a cavi (1.1 touc (a r Fig (1.14 3 Holes (I Tetrahed ked lattice med by thr edges (of cavity cal ity (and to 5). Octahedr ch three at egular oc 4): The rep Interstice dral Hole e. One at ree adjacen length 2r) lled the Te ouch the Fig (1 ral Hole -toms in th tahedron) peat patter s) in Clos - Consid tom in th nt atoms i ) of a regu etrahedral four host 1.15): Sche - Adjacen he A layer is forme rn of 4-lay sest Packe der any tw he A laye in the B la ular tetrah l (or Td) h spheres) ematics of nt to the T such that ed; the ce yer hexago ed Arrays wo succe er nestles ayer, and t hedron; the hole; a gue if its rad f Tetrahed Td hole, th a trigonal enter of th onally clo s ssive plan in the tr the four at e center o est sphere dius is 0.2 dral Hole. hree atom l antiprism he octahe osest packe nes in a riangular toms touch of the tetra will just 2247r, see ms in the B matic poly dron is a ed. closest groove h along ahedron fill this e figure B layer yhedron a cavity

fill t show bilay 1.5.4 lattic see f lattic (1.1 this cavity wn that th yer. 4 Simple CsCl Str ce such th figure (1.1 NaCl Str ce. The tw 8). y (and touc here are t Fig (1.1 Crystal S ructure - hat the cati 17). The tw Fig ( ructure - wo lattices ch the six twice as m 16): The Sc Structures Each ion ion is in th wo lattice (1.17): The Each ion s have the host spher many Td chematics s: n resides he center o s have the e Crystal S resides o same uni res) if its r as Oh hol s of Octahe on a sepa of the anio e same uni Structure on a separ t cell dim radius is 0 les in any edral Hole arate, inte on unit ce it cell dim of CsCl. rate, interp ension, as 0.4142r. It y closest e. erpenetrati ll and visa mension. penetratin s shown in t can be packed ing SC a versa, ng FCC n figure

CCP (Z = fluo anio resid Halite St P lattice o = 4), see fi Fluorite ride) may on occupyi de on a SC Fig (1 tructure - of anions ( igure (1.19 Fig (1.19 Structure y be viewe ing all of C lattice w (1.18): The The sodiu (Z = 4), w 9). 9): The Cr e - The ed as a CC the Td ho which is ha e Crystal S um chlorid with smalle rystal Stru structure CP lattice les (Z = 8 alf the dim

Structure de structur er cations ucture of H of the m of cations 8), see figu mension of of NaCl. re may als occupyin Halite latte mineral f s (Z = 4), ure (1.20) f the CCP so be view ng all Oh c es. fluorite (c with the ). The Td c lattice. wed as a cavities calcium smaller cavities

blen catio [Not anio lattic desc show zinc lattic Zinc Ble nde") may ons occup

te: the oth ons with ca Zinc Ble ce of the cribed as i wn in figu c blende s ces. ende Struc y be viewe pying ever her ZnS m ations in e Fig (1.21 ende lattic same dim interpenetr ure (1.22) structures Fig (1.2 cture - The ed as a CC ry other T mineral, w every othe 1): The Cr ces - The mension a rating FC . Note tha is a simp 22): The i e structure CP lattice Td hole (Z wurtzite, ca er Td hole. rystal Stru e lattice o as the ani C lattices at the only ple shift in interpenetr e of cubic of anions Z = 4), a an be desc ] ucture of Z of cations ion lattice of the sam y differen n relative rating FC c ZnS (min s (Z = 4), as shown cribed as Zinc Blend in zinc b e, so the me unit ce nce betwee position CC lattices neral nam with the in figure a HCP la de. blende is structure ell dimens en the hal of the tw s. me "zinc smaller (1.21). attice of a FCC can be sion, as lite and wo FCC

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 X-ray 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.

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 X-rays 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

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.

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

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.

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.

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

coor and Figu orien Fig rotat rathe The and Brag in th whic angl adva wav wav An i diffr rapid for e rdinates q, χ* and on ure (2.6). ntation to g.(2.6): Tw When th tional ave er than th angle bet in X-ray gg's law, e he sample Powder ch the dif le 2θ or as antage tha velength λ velengths t instrumen ractometer Relative d, non-des extensive , φ*, χ*. In nly q rem In practi eliminate wo-dimens he scattere eraging le he discrete tween the y crystallo each ring crystal. T 2 sin diffractio ffracted in s a functio at the dif λ. To faci the use of nt dedicate r. to other structive a sample p n powder ains as an ice, it is e the effect sional pow ed radiati ads to sm e Laue spo e beam ax ography a correspon This leads t 4 s on data ar ntensity I on of the s ffractogram ilitate com f q is there ed to perfo methods analysis o preparation diffractio n importan sometime ts of textu wder diffra ion is col mooth diffr ots as obs xis and the

always de nds to a p to the defi in / re usually is shown scattering m no lon mparabilit efore recom orm powd of analy f multi-co n. Identifi on intensity nt measura es necess uring and a action setu llected on fraction rin served for e ring is c enoted as particular r inition of y presente as functio vector q. nger depe ty of data mmended der measur ysis, powd omponent cation is y is homo able quant sary to ro achieve tru up with fla n a flat pl ngs aroun r single cr called the s 2θ. In a reciprocal the scatter ed as a d on either The latter ends on t a obtaine and gaini rements is der diffrac mixtures performed ogeneous o tity, as sh otate the ue random at plate de late detec nd the bea rystal diffr scattering accordanc l lattice ve ring vecto ( diffractog of the sca r variable the value d with di ing accept s called a p ction allo without th d by comp over φ* hown in sample mness. etector. ctor the am axis raction. g angle ce with ector G or as: (2.2) gram in attering has the of the ifferent tability. powder ows for he need parison

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 _· (2.5) 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 is just the reciprocal magnitude of in the direction of , 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

vect recip arran • • • • 2.5 D cond cond scatt tor. The d procal vec ngement o Vector a diffractio The recip terms of Use of th that cann The recip an under Diffractio Defining ditions in dition to d |∆ | = 4 More dir tering amp k

S

=

diffraction ctors of th of a crysta algebra is on problem procal latt vectors he recipro not be acce procal latt rstanding o on in recip g that the reciproca diffraction 4p(sinq)/l = Fig rectly, wh plitude (st 1 n i j

f

=

∑

n pattern he lattice. al. very conv ms tice offers ocal lattice essed by B tice is imp of this con procal sp diffraction al space as shown = |Ghkl| = 2 g.(2.7): D hen the di tructure fa . j i G r

e

r r of a crys Using thi venient fo s a simple e permits t Bragg’s La portant in ncept is us ace: n vector G are ∆ n in Figure 2p/dhkl Diffraction iffraction actor) is de stal can b is process or describi approach the analys aw. all phases seful in an G, where , this e (2.7) was in recipro condition etermined be used t , one can ing otherw h to handli sis of diffr s of solid nd of itself G = k-k0 means th s: ocal space n ∆ by: to determi infer the wise comp ing diffrac fraction pr state phy f . The diff hat the su ( e. is satisfi ( ine the atomic plicated ction in roblems sics, so fraction ufficient (2.9) ied, the (2.10)

where f_j is atomic scattering factor (form factor). The usual form of this result follows on writing, the lattice vector as:

j j j j

rr = u ar + v br + w cr _(2.11)

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

.

_j

2

_{j j j}

G r

r r

=

π

⎡

_⎣

u h v

+ +

w l

⎤

_⎦

(2.12)

(

)

2 1 j j j G n i u h v k w l i j

S

f e

π + + =

=

∑

(2.13) Where n is a number of atom in unit cell, and u v wj j j 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

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 (1768-1830) 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

(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

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.

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 short-wavelength 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

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 real-space 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:

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.

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

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

50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500

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.

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

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

50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500

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.

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

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

50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500

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 Face-centered cubic lattice is a Body-Face-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.

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

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

% 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);

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

%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