Kurdistan Iraqi Region Ministry of Higher Education Sulaimani University College of Science Physics Department

 

Reciprocal Lattices

Simulation using Matlab

Prepared by

Bnar Jamal Hsaen

Hanar Kamal Rashed

Kizhan Nury Hama Sur

Supervised by

Dr. Omed Gh. Abdullah

2008 – 2009

2  

{…But say: oh My Lord! Advance me in knowledge}

(Surat Taha:14)

We dedicate this research to:

- Those who helped us during the preparation of this research,

- Our Department.

- Those who reading this research.

3  

Acknowledgements

We would like to express our gratitude and thankfulness to our

supervisor Dr. Omed Gh. Abdullah, for continues help and guidance

throughout this work. We are also indebted to Mr. Yadgar Abdullah for

providing us with sources and his encouragement during writing this research

paper.

True appreciation for Department of Physics in the College of Science at

the University of Sulaimani, for giving us an opportunity to carry out this

work. We wish to extend my sincere thanks to all teachers’ staff who taught

us along our study.

Also we express our thankfulness to the library of our department for

providing us with references.

Finally thanks and love to our family for their patience and supporting

during our study.

 

Bnar - Hanar - Kizhan

2009

 

   

4  

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

5  

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-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.

6  

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.

7  

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

8 

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

9  

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

10 

lattices in

ais Lattices:

base-cente

base-cente

body-cent

body-cent

three dime

ered

ered

tered

tered

ension.

body-centered

face-centered

face-c

centered

11  

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

12 

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 13 

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

14 

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

15 

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

16 

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

17 

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

calle

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

ed the Oct

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)

tahedral (

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

or Oh) ho18 

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

le, see fig

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

gure (1.16)

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

). A guest

osest packe

nes in a

riangular

toms touch

of the tetra

will just

2247r, see

ms in the B

matic poly

dron is a

t sphere w

ed.

closest

groove

h along

ahedron

fill this

e figure

B layer

yhedron

a cavity

will just

 

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

19 

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

Fig (1.

(1.18): The

The sodiu

(Z = 4), w

9).

9): The Cr

e - The

ed as a CC

the Td ho

which is ha

.20): The C20 

e Crystal S

um chlorid

with smalle

rystal Stru

structure

CP lattice

les (Z = 8

alf the dim

Crystal St

Structure

de structur

er cations

ucture of H

of the m

of cations

8), see figu

mension of

tructure of

of NaCl.

re may als

occupyin

Halite latte

mineral f

s (Z = 4),

ure (1.20)

f the CCP

f Fluorite.

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

21 

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

22  

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.

23  

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

24  

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.

25  

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

26  

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.

27  

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.

28  

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

29 

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

30  

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)

31  

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

kS =

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

ij

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

. ji G rer r

32 

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)

33  

where jf is atomic scattering factor (form factor). The usual form of this result

follows on writing, the lattice vector as:

j j j jr u a v b w c= + +rr r r

(2.11)

Then, for the reflection labeled by u , v , w (i.e. Reciprocal lattice

vector), we have:

. 2j j j jG r u h v w lπ ⎡ ⎤= + +⎣ ⎦r r

(2.12)

( )2

1

j j j

G

ni u h v k w l

ij

S f e π + +

=

= ∑ (2.13)

Where n is a number of atom in unit cell, and j j ju v w 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

34  

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

35  

(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

36  

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.

37  

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

38  

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:

39  

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.

40  

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

41  

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.

42  

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

43  

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.

44  

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

45  

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-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.

46  

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

47  

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

48  

% 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

49  

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

50  

%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