1. science
2. mathematics
3. geometry

THE PRELIMINARY ANALYSIS OF AN ORGANIC COMPOUND BY
USING X-RAY DIFFRACTION TECHNIQUES
A
PROJECT REPORT
Submitted to School of Physics
In partial fulfilment for the award of degree of
Masters of Sciences
in PHYSICS
Supervised By:
Submitted By:
Dr.Kamni Pathania
RAHUL PURI - 2012SMY01
HITESH KHARKA -2012SMY19
SHRI MATA VAISHNO DEVI UNIVERSITY, KATRA, (J&K)
May, 2014
DECLERATION
We, Rahul Puri & Hitesh Kharka declare that the work reported in this project report has
entirely been done by us under the supervision of Dr. Kamni Pathania, Assistant Professor,
School Of Physics, Shri Mata Vaishno Devi University, Katra. No part of this work has been
submitted in part or full for a degree in any other university.
May, 2014
RAHUL PURI (2012SMY01)
HITESH KHARKA (2012SMY19)
CERTIFICATE
This is to be certified that the project entitled “THE PRELIMINARY ANALYSIS OF AN
ORGANIC COMPOUND BY USING X-RAY DIFFRACTION TECHNIQUES ” undertaken
by below mentioned students of M.Sc. Physics (semester IV), School Of Physics, Shri Mata
Vaishno Devi University, katra, has been submitted to the supervisior and to the department for
the fulfilment of the requirement for Masters Degree in Physics. They have completed their
project under my supervision.
Teacher-In-Charge
Dr. Kamni Pathania
Submitted By
RAHUL PURI (2012SMY01)
HITESH KHARKA (2012SMY19)
Head of the Department
Dr. Jitendra Sharma
ACKNOWLEDGEMENT
We epress our gratitude to “Almighty” for bestowing upon us the potentialities of working
under his noble care.
It is a pleasing privilege for us to express our profound gratefulness to our esteemed
supervisor, Dr.Kamni, Assistant Professor, School of Physics, Shri Mata Vaishno Devi
University, Katra. For her keen interest, encouragement, continued inspiration, constructive
criticism and all round help throughout the course of this project work. Her crystallized
wisdom becomes our lamp posts which provided us right directions to steer through this
project.
We are grateful to Dr. Jitendra Sharma, Director, School of Physics, Shri Mata Vaishno Devi
University, Katra who provided us readily all facilities in the School of Physics, as also much
needed encouragement in pursuing our project.
We are deeply obliged to Dr. S.K.Wanchoo, Dr. Yugal Khajuria , Dr. Vivek Kumar Singh, Dr.
Ram Parkash, Dr. Vinay Kumar Dhiman and Mr. Pankaj Biswas, faculty members School of
Physics, Shri Mata Vaishno Devi University, Katra for their help in this project.
It’s our great pleasure to express profound reverences and deep sense of gratitude to Dr.
Rajnikant, HOD, Department of Physics, University of Jammu for his suggestions, guidance
and never ending optimism throughout the project report.
We wish to record our special thanks to Dr. Vivek Gupta, Lecturer, Department of Physics,
University of Jammu and Mr. Kuldeep Singh, PHD scholar at Department of Physics,
University of Jammu for the continuous encouragement and timely suggestions at every stage
of our work.
We cannot express in words the depth of gratitude to our beloved Parents, who through the
force of their unsurpassable love, affection and encouragement have been the greatest source of
inspiration behind this study.
RAHUL PURI (2012smy01)
HITESH KHARKA (2012smy19)
INDEX
UNIT 1:- BASICS OF CRYSTALLOGRAPHY & XRAYS
CONTENTS
1.1
CLASSIFICATION OF SOLIDS
1.2
CRYSTALLOGRAPHY
1.3
CRYSTAL STRUCTURE
1.4
SYMMETERY ELEMENTS IN A CRYSTAL
1.5
POINT GROUPS
1.6
SPACE GROUPS
1.7
UNIT CELL
1.8
LATTICE SYSTEMS (BRAVAIS’ LATTICES)
1.9
MILLER INDICES
1.10 CUBIC CELLS AND THEIR CHARACTERISTICS
1.11 CLOSED PACKED STRUCTURES
1.12 X-RAYS AND THEIR CHARACTERISTICS
1.13 HISTORY OF X-RAYS
1.14 PRODUCTION OF X-RAYS
PAGE NO.
UNIT 2:- X-RAY DIFFRACTION METHODS
&PRELIMINARY X-RAY STUDY OF AN
ORGANIC COMPOUND
CONTENTS
2.1
DIFFRACTION & X-RAY DIFFRACTION
2.2
BRAGG’S LAW
2.3
RECIPROCAL LATTICE
2.4
X-RAY DIFFRACTION TECHNIQUES
2.5
SINGLE & DOUBLE OSCILLATION METHOD
2.6
ROTATION METHOD
2.7
WEISSENBERG METHOD
2.8
STRUCTURE OF THE CRYSTAL
2.9
MOUNTING OF THE CRYSTAL
2.10 SINGLE OSCILLATION PHOTOGRAPH
2.11 ROTATION PHOTOGRAPH
2.12 WEISSENBERG PHOTOGRAPH
2.13 INDEXING OF WEISSENBERG PHOTOGRAPH
2.14 EVALUATION OF LATTICE PARAMETERS
2.15 DETERMINATION OF INTERFACIAL ANGLES
2.16 CONCLUSION
PAGE NO.
FIGURE CAPTION :
1.1
CLASSIFICATION OF SOLIDS
1.2
LATTICE
1.3
BASIS
1.4
COMBINATION OF LATTICE AND BASIS
1.5
2-D CRYSTAL
1.6
REFLECTION IN APLANE
1.7
INVERSION THROUGH A POINT
1.8
ROTATION SYMMETRY
1.9
ROTATION INVERSION SYMMETRY
PAGE NO.
1.10 SCREW AXES
1.11 GLIDE PLANES
1.12 WEIGNER-SEITZ UNIT CELL
1.13 DIFFERENT CUBIC UNIT CELLS
1.14 TRIGONAL HOLES
1.15 TETRAHEDRAL HOLES
1.16 OCTAHEDRAL HOLES
1.17 HEXAGONAL CLOSED PACKED STRUCTURE
1.18 CUBIC CLOSED PACKED STRUCTURE
1.19 PRODUCTION OF X-RAYS
2.1
DIFFRACTION
2.2
ILLUSTRATION OF BRAGG’S LAW
2.3
GEOMETRICAL REPRESENTATION OF
RECIPROCAL LATTICE
2.4
REPRESENTATION OF TILT AND BOW ERRORS
2.5
A ROTATION PHOTOGRAPH
2.6
COMPONENTS OF WEISSENBERG APPARATUS
2.7
SCHEMATIC DIAGRAM OF WEISSENBERG
APPARATUS
2.8
STRUCTURE OF THE CRYSTAL
2.9
SINGLE OSCILLATION PHOTOGRAPH OF TITLE
2.10 ROTATION PHOTOGRAPH OF TITLE COMPOUND
2.11 INDEXED PICTURE OF ROTATION PHOTOGRAPH
2.12 ZERO LAYER WEISSENBERG PHOTOGRAPH
OF TITLE COMPOUND ( about b axis )
2.13 INDEXED PICTURE OF ZERO LAYER
WEISSENBERG PHOTOGRAPH
TABLE CAPTION
1.1
POSSIBLE GLIDE PLANES
1.2
POSSIBLE TYPES OF SYMMETRY
1.3
TYPES OF LATTICE SYSTEMS
1.4
BRAVAIS’ LATTICES IN 3-D
1.5
CALCULATION OF CELL PARAMETER (say b)
USING ROTATION PHOTOGRAPH
1.6
CALCULATION OF CELL PARAMETER (say a)
USING WEISSENBERG PHOTOGRAPH
1.7
CALCULATION OF CELL PARAMETER (say c)
USING WEISSENBERG PHOTOGRAPH
PAGE NO.
UNIT -1
BASICS OF CRYSTALLOGRAPHY & X-RAYS
1.1 CLASSIFICATION OF SOLIDS
SOLIDS
NON-CRYSTALLINE
POLYCRYSTALLINE
CRYSTALLINE
AMORPHOUS
FIGURE 1.1 : CLASSIFICATION OF SOLIDS
All of us are familiar with the three states of matter viz. Solids, Liquids and Gases. Solids are
further classified into two categories, CRYSTALLINE and NON-CRYSTALLINE SOLIDS.
CRYSTALLINE SOLIDS :- The solids which possess a regular and periodic arrangement of
atoms of three dimensional pattern are called crystalline solids. Crystalline solids possess long
range order of atoms. These solids have sharp melting points. Crystalline solids are anisotropic
in nature i.e. the physical properties have different values in different directions. These solids
are also sometimes called as True solids. Examples:- Quartz, Sugar , NaCl, Carbon etc.
NON-CRYSTALLINE SOLIDS:- The solids which do not possess a regular and periodic
arrangement of atoms are called non-crystalline solids. Non-crystalline solids possess short
range order of atoms. These solids do not have sharp melting points. Non-crystalline solids are
isotropic in nature i.e. they have same values of physical properties in different directions.
These are further classified in two categories: Amorphous and Polycrystalline. Examples:Glass, Rubber, Sulphur etc.
1.2 CRYSTALLOGRAPHY
Crystallography is the science that examines the arrangement of atoms in solids. In other
words, the study of geometric form and physical properties of crystalline solids by using Xrays, electron beams & neutron beams etc. constitutes the science of crystallography. Since in
crystalline solids, the atoms and molecules are arranged in regular and periodic manner, so
most of the solids are crystalline in nature because energy released during the formation of
ordered structure is more than that released during the formation of disordered structure.Thus,
crystalline is the low energy state and most of the solids preferred this state.
Modern crystallography is largely based on the analysis of the DIFFRACTION of X-RAYS by
crystals acting as optical gratings. Using X-ray crystallography, chemists are able to determine
the internal structures and bonding arrangements of minerals and molecules, including the
structures of large complex molecules such as proteins and DNA.
1.3 CRYSTAL STUCTURE
CRYSTAL STRUCTURE = Space Lattice + Basis.
What is a lattice?
A lattice is a hypothetical regular and periodic arrangement of points in space. It is used to
describe the structure of a crystal. Let’s see how a two-dimensional lattice may look.
FIGURE 1.2: LATTICE
A basis is a collection of atoms in particular fixed arrangement in space. We could have a basis
of a single atom as well as a basis of a complicated but fixed arrangement of hundreds of
atoms. Below we see a basis of two atoms inclined at a fixed angle in a plane.
FIGURE 1.3 : BASIS
Let us now attach the above basis to each lattice point (in black) as follows.
FIGURE 1.4 : COMBINATION OF LATTICE & BASIS
Next remove the lattice points in black (remember that the lattice is an abstract entity). Let’s
see what we have got?
FIGURE 1.5 : 2-D CRYSTAL
We have got the actual two-dimensional crystal in real space. So we may write:
Lattice + basis = crystal.
Mathematically,
If r be the coordinates of a lattice point from any origin, then if we apply a lattice translation
thought the lattice translation vector
Then, we would arrive at any other lattice point r around which the environment would look
exactly the same as around r.
a, b and c are the primitive translation vectors or basis vectors which form the primitive cell of
the lattice and u, v, w are integers.
1.4 SYMMETRY ELEMENTS IN CRYSTALS
A symmetry operation is a transformation on the body which leaves it invariant. In other
words, after performing an operation on the body, if the body becomes indistinguishable from
initial configuration, the body is said to possess a symmetry element corresponding to that
particular operation. Thus symmetry is a property by means of which an object is brought into
self-coincidence by certain operation.
Suppose that the water molecule is rotated by 180̊ about an axis (dotted lines) in such a way
that the hydrogen atoms swap their places. This leaves the chemical environment as identical
as it was before the rotation makes the molecule indistinguishable from its original position.
Here the rotation about the bisector H-O-H bond is a symmetry operation & the rotation axis is
the symmetry element.
A symmetry element may be thought of as a geometric entity, which generates symmetry
operation, i.e. a point, a line or plane with respect to which symmetry operation may be
performed.
Symmetry elements are of two types:1. Non-Translational Symmetry Elements:These symmetry elements are further described as:
a) Reflection in a Plane
b) Inversion
c) Rotation Axis
d) Rotation Inversion Axis
2. Translational Symmetry Elements:a) Screw Axis
b) Glide Plane
Non-Translational Symmetry Elements
1.Reflection in a plane
In a reflection symmetry, the crystal is brought into self-coincidence by reflection along a
plane is called mirror plane. Its symbol is ‘m’. Objects which are mirror images are known as
enantiomers .
FIGURE 1.6: REFLECTION IN A PLANE
2. Inversion
A crystal is said to possess an inversion symmetry if for every lattice point represented by
position vector ‘-r’. So, the operation of inversion is the combined operation of rotation
through 180̊ followed by reflection in a plane perpendicular to the rotation axis and symbolizes
as ‘I’. It can also be described as the transformation of the coordinates i.e. xˊ=x, yˊ=y, zˊ=z.
The standard symbol of inversion is I . Cyclohexane is a very simple example to explain the
inversion symmetry.
FIGURE 1.7 :INVERSION THROUGH A POINT
3. Rotation Axis
If a body remains invariant after a rotation through an angle ψ ( known as throw of an axis ),
the body is said to possess n-fold rotational symmetry. The value of n determines the degree of
rotation & n takes the value 1,2,3,4 and 6.Where as 5 & 7 fold symmetry do not exist in crystal
structure because structures corresponding to these do not fill the space completely & leave
voids in stacked arrangement. Depending upon the value of n, the rotation axes are named as
one fold (monad), two fold (diad), three fold (triad) ,four fold (tetrad) and six fold (hexad).
FIGURE 1.8 :ROTATION SYMMETRY
1.Rotation Inversion Axis
Rotation with subsequent inversion can be obtained to give a new symmetry element-the
rotation inversion axes. It is represented as 2̅,3̅,4̅,or 6̅.
FIGURE 1.9: ROTATION INVERSION SYMMETRY
Note:-Improper rotation axes
An improper rotation is the combination of rotation with either an inversion or a reflection in a
plane perpendicular to the rotation axis. The former is known as Rotoinversion and the later is
called as Rotoreflection.
Translational Symmetry Elements
1.Screw axes
These combine the rotation of an ordinary symmetry axes with a translation parallel to it and is
equal to the fraction of unit distance in this direction. Screw rotation is the combination of a
rotation and a translation parallel to the axis of rotation. A screw axis is defined as an axis
about which a proper rotation (θ = 2π/n) is combined with a non-primitive translation parallel
to the axis of rotation, transforms an array of atoms into self coincidence. In such a motion
where the rotation of an angle θ ( takes the object from e to f ) is combined with the
translation t to take the object from f to g. This is equivalent to a single hybrid operation called
“ screw operation” which takes the object directly from e to g. This is equivalent to a single
hybrid operation called “screw operation” which takes the object directly from e to g.
FIGURE 1.10: SCREW AXES
The cumulative translation distance after n screw operations must be an integral multiple (say
m) of lattice translation (say a) along the screw axis. i.e.
nT = ma
 T = ma/n
where m and n both are integers with m < n and n = 1,2,3,4 & 6 where each digit represent a
proper rotation. For m=0 and m=n both correspond to the pure rotations. For m =1, 2, 3,…..(n1) there exist only 11 screw axes. They are symbolically represented as nₐ and are as follows :
21, 31, 32, 41, 42, 43, 61, 62, 63, 64 & 65 .
2.Glide planes
When a mirror plane is combined with a simultaneous translation operation in a crystal, it
results in a glide plane. A glide mirror plane combine the mirror image with a translation
parallel to the mirror plane over a distance which is half the distance in a glide direction. Thus,
the combination of a mirror plane and a translation parallel to the reflecting plane produces a
glide plane. This is an illustrated in the figure below
FIGURE 1.11: GLIDE PLANES
Table 1.1: Possible Glide Planes
Type of glide
Symbol
Translation component (T)
Axial Glide
A
a/2
Axial Glide
B
b/2
Axial Glide
C
c/2
Diagonal Glide
N
(a+b)/2, (b+c)/2 or (c+a)/2
Diamond Glide
D
(a+b)/4, (b+c)/4or (c+a)/4
1.5 POINT GROUP
A point group is the group of the symmetry operations all of which leave one point unmoved,
the translation thus being excluded. In one dimension there can be only be reflection across the
point. In two dimensions, there can be inversion about a point, rotation about a line and
reflection across a plane. The possible types of point group symmetry are presented below in
the table.
Table 1.2 : Possible Types of Symmetry
ONE-DIMENSIONAL
TWO-DIMENSIONAL
THREE-DIMENSIONAL
 Reflection(point)
 Rotation(point)
 Inversion(point)
 Reflection(line)
 Rotation(line)
 Reflection(plane)
The concept of point group is convenient for the analysis of symmetry of a finite body. If we
confine ourselves to the crystallographic point groups, we have the limitation that only those
symmetry elements are permitted which can operate on lattices, i.e. which can give rise to
parallelpipeds which can be stacked together in the same orientation to fill space. This
condition limits us to axis of 1, 2, 3, 4 and 6-fold symmetry, with or without inversion in the
three dimensional case.
Thus, a point group in the lattice is defined as the collection of symmetry operations which
when applied about a lattice point, leave the lattice invariant. In other words, all possible
symmetry elements in point group must pass through point. Suitable combinations of a various
symmetry elements give rise to 32 allowed point groups. Since crystals belonging to different
crystal systems show different point group symmetries, therefore, the classification of crystal
structures can be made easily on the basis of point groups
.
1.6 SPACE GROUP
There are infinitely extended arrays of symmetry elements disposed on a space lattice. A space
group acts as three dimensional kaleidoscope: an object submitted to its symmetry operation is
multiplied and periodically repeated in such a way that it generates a number of
interpenetrating identical space lattices. In other words, to understand the symmetry of a crystal
which is three dimensional in nature, certain point group operations enables one to describe the
crystal in an unambiguous manner in space. Thus the groups so formed by the combination of
32 point groups and 14 Bravais lattices are generally known as space groups.
Therefore, a space group is described as a set of symmetry elements, the operation of any of
which brings the infinite array of points to which they belong into self confidence. A simple
practical approach, however, is to consider that the translation in a lattice introduces two new
kinds of symmetry operations, screw axes and glide planes.
1.7 UNIT AND PRIMITIVE CELL
A primitive cell (sometimes called a primitive unit cell) is the smallest area cell (for a 2-dim
lattice) or smallest volume cell (for a 3-dim lattice) which serves as the basic building block of
the lattice. If we keep stacking the primitive cells without leaving any gap we would have
obtained the lattice over all space. The primitive cell is mathematically obtained by the
primitive translation vectors a, b and c, and for a three dimensional lattice its volume is given
by
|a . b x c|
In some situations symmetry dictates that we use another unit cell which is bigger than the
primitive unit cell but which is simpler to work with. Such a unit cell is called a conventional
unit cell. If we keep stacking the conventional unit cells without leaving any gap we would
have obtained the lattice over all space, just as before. Unlike a conventional unit cell, one can
associate no more than one lattice point with a primitive unit cell.
WEIGNER-SEITZ PRIMITIVE CELL
A primitive unit cell may also be constructed as follows:
(i) We start with an array of points in the lattice.
(ii) We connect any one lattice point to all the neighbouring lattice points with lines.
(iii) At the mid point of these lines we draw normals (if we started out with a two dimensional
lattice) or normal-planes (if we started out with a three dimensional lattice). The smallest area
(or volume) enclosed in this way is called the Wigner-Seitz primitive cell of the direct lattice.
All space may be filled up without leaving any gap by joining these Wigner-Seitz primitive
cells.
The following figure shows the Weigner-Seitz primitive cell.
FIGURE 1.12: WEIGNER-SEITZ UNIT CELL
LATTICE PARAMETERS OF A UNIT CELL
1) CRYSTALLOGRAPHIC AXES: There are two lattice parameters of a unit cell. The lines
drawn parallel to the lines of intersection of a unit cell, which do not lie in the same plane are
called crystallographic axes.
2) INTERFACIAL ANGLES: The angles between the three crystallographic axes are called
interfacial angles.
3) PRIMITIVES: The three sides of the unit cell are known as Primitives.
1.8 LATTICE SYSTEMS
These lattice systems are a grouping of crystal structures according to the axial system used to
describe their lattice. Each lattice system consists of a set of three axes in a particular
geometric arrangement. There are seven lattice systems as follows.
Table 1.3 : Types of Lattice Systems
BRAVAIS’ LATTICES
In geometry and crystallography, a Bravais lattice, studied by Auguste Bravais (1850), is an
infinite array of discrete points generated by a set of discrete translation operations described
by:
R = n1a1 + n2a2 + n3a3
Where ni are any integers and ai are known as the primitive vectors which lie in different
directions and span the lattice. This discrete set of vectors must be closed under vector addition
and subtraction. For any choice of position vector R, the lattice looks exactly the same.
A crystal is made up of a periodic arrangement of one or more atoms (the basis) repeated at
each lattice point. Consequently, the crystal looks the same when viewed from any equivalent
lattice point.
The 14 three-dimensional lattices, classified by crystal system are shown below in table 1.4.
Table 1.4 : The 14 Bravais’ Lattices
The 7 lattice systems
The 14 Bravais lattices
P
Triclinic
P
C
P
C
P
I
Monoclinic
Orthorhombic
Tetragonal
P
Trigonal
I
F
P
Hexagonal
P (pcc)
I (bcc)
F (fcc)
Cubic
1.9 MILLER INDICES
A crystal lattice may be considered as an assembly of a number of equidistant parallel planes
passing through the lattice points and are called lattice planes. The orientation of a plane in a
lattice is specified by giving its miller indices. Therefore, miller indices are the symbolic
representation for the orientation of an atomic plane in a crystal lattice and are defined as the
three smallest possible integers required to represent a plane with respect to the three
crystallographic axis having unit cell translation a, b, c respectively. Once the choice of the
unit cell is made, it is possible to express the intercepts of all other planes as ma, nb and qc
where m, n and q are small integers or infinity. These three numbers m, n, q might be used as
indices to denote a given face. However, because sometimes an index would have the value
infinity which is troublesome in some mathematical calculations, so it is more convenient to
use indices number proportional to the reciprocals of m, n, q. If h=1/m, k=1/n, and l=1/q then
h, k, l are known as Miller indices of the face and it can be designated as (hkl).
To obtain the Miller indices of a plane the following step-wise procedure may be applied:
1) Determine the intercept of plane along x, y and z-axis in terms of lattice parameters.
2) Divide these intercepts by the appropriate unit translations.
3) Determine the reciprocals of these numbers.
4) Reduce these reciprocal to the smallest set of integral number and enclose them in brackets.
1.10 CUBIC CELLS AND THIER CHARACTERISTICS
In crystallography, the cubic crystal system is a crystal system where the unit cell is in the
shape of a cube. This is one of the most common and simplest shapes found in crystals and
minerals.
There are three main varieties of these crystals:
1) Primitive cubic (alternatively called simple cubic)
2) Body - centered cubic
3) Face - Centered cubic
Although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit
cell often is not. This is related to the fact that in most cubic crystal systems, there is more than
one atom per cubic unit cell.
SIMPLE CUBIC CRYSTAL(SCC)
BASE CENTERED CUBIC(BCC)
FACE CENTE RED CUBIC(FCC)
FIGURE 1.13: DIFFERENT CUBIC UNIT CELLS
SCC :- The primitive cubic system consists of one lattice point on each corner of the cube.
Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit
cell therefore contains in total one atom (1⁄8 × 8). Co-ordination number for SCC is 6 and
packing fraction is 0.52.
Atomic radius(r) in terms of cube edge (a) is given by :r = a/2
BCC :- The body-centred cubic system has one lattice point in the center of the unit cell in
addition to the eight corner points. It has a net total of 2 lattice points per unit cell (1⁄8 × 8 + 1).
Co-ordination number for BCC is 8 and packing fraction is 0.68.
Atomic radius(r) in terms of cube edge (a) is given by :r = ( 1.71a)/4
FCC:- The face-centered cubic system has lattice points on the faces of the cube, that each
gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4
lattice points per unit cell (1⁄8 × 8 from the corners plus 1⁄2 × 6 from the faces). Co-ordination
number for FCC is 12and packing fraction is 0.74.
Atomic radius(r) in terms of cube edge (a) is given by :r = (1.41a)/4
1.11 CLOSED PACKED STRUCTURES
The term "closed packed structures" refers to the most tightly packed or space-efficient
composition of crystal structures (lattices). Imagine an atom in a crystal lattice as a sphere.
While cubes may easily be stacked to fill up all empty space, unfilled space will always exist in
the packing of spheres. To maximize the efficiency of packing and minimize the volume of
unfilled space, the spheres must be arranged as close as possible to each other. These
arrangements are called closed packed structures.
Introduction
The packing of spheres can describe the solid structures of crystals. In a crystal structure, the
centers of atoms, ions, or molecules lie on the lattice points. Atoms are assumed to be
spherical to explain the bonding and structures of metallic crystals. These spherical particles
can be packed into different arrangements. In closest packed structures, the arrangement of the
spheres are densely packed in order to take up the greatest amount of space possible.
Types of Holes From Close-Packing of Spheres
When a single layer of spheres is arranged into the shape of a hexagon, gaps are left uncovered.
The hole formed between three spheres is called a trigonal hole because it resembles a triangle.
In the example below, two out of the six trigonal holes have been highlighted green.
Trigonal holes:
FIGURE 1.14: TRIGONAL HOLES
Once the first layer of spheres is laid down, a second layer may be placed on top of it. The
second layer of spheres may be placed to cover the trigonal holes from the first layer. Holes
now exist between the first layer (the orange spheres) and the second (the lime spheres), but
this time the holes are different. The triangular-shaped hole created over a orange sphere from
the first layer is known as a tetrahedral hole. A hole from the second layer that also falls
directly over a hole in the first layer is called an octahedral hole.
FIGURE 1.15: TETRAHEDRAL HOLES
FIGURE 1.16: OCTAHEDERAL HOLES
Closed Pack Crystal Structures
Hexagonal Closed Packed (HCP)
In a hexagonal closest packed structure, the third layer has the same arrangement of spheres as
the first layer and covers all the tetrahedral holes. Since the structure repeats itself after every
two layers, the stacking for HCP may be described as "a-b-a-b-a-b." The atoms in a hexagonal
closest packed structure efficiently occupy 74% of space while 26% is empty space.
FIGURE 1.17: HEXAGONAL CLOSED PACKED STRUCTURE
2.Cubic Closed Packed (CCP)
The arrangement in a cubic closest packing also efficiently fills up 74% of space. Similar to
hexagonal closest packing, the second layer of spheres is placed on to of half of the depressions
of the first layer. The third layer is completely different than that first two layers and is stacked
in the depressions of the second layer, thus covering all of the octahedral holes. The spheres in
the third layer are not in line with those in layer A, and the structure does not repeat until a
fourth layer is added. The fourth layer is the same as the first layer, so the arrangement of
layers is "a-b-c-a-b-c."
FIGURE 1.18: CUBIC CLOSED PACKED STRUCTURE
1.12 X-RAYS AND THIER CHARACTERISTICS
X-ray is a form of electromagnetic radiation. Most X-rays have a wavelength in the range of
0.01 to 10 nm, corresponding to frequencies in the range 3×1016 Hz to 3×1019 Hz and
energies in the range 100 eV to 100 KeV. X-ray wavelengths are shorter than those of UV
rays and typically longer than those of gamma rays. In many languages, X-radiation is
referred to with terms meaning Rontgen radiation, after Wilhelm Rontgen, who is usually
credited as its discoverer, and who had named it X-radiation to signify an unknown type of
radiation.
Characteristics of X-Rays
 X-rays are invisible.
 X-rays are electrically neutral. They have neither a positive nor a negative charge. They
cannot be accelerated or made to change direction by a magnet or electrical field.
 X-rays have no mass.
 X-rays travel at the speed of light in a vacuum.
 X-rays cannot be optically focused.
 X-rays form a polyenergetic or heterogenous beam.
 The x-ray beam used in diagnostic radiography comprises many photons that have many
different energies.
 X-rays travel in straight lines.
 X-rays can cause some substances to fluoresce.
 X-rays cause chemical changes to occur in radiographic and photographic film.
 X-rays can be absorbed or scattered by tissues in the human body.
 X-rays can produce secondary radiation.
 X-rays can cause chemical and biologic damage to living tissue.
1.13 HISTORY OF X-RAYS
X-rays were invented in 1895 by a German physicist named Wilhelm Conrad Rontgen. He
discovered x-rays at the University of Wurzburg while experimenting with electron beams
in a gas discharge tube. He
noticed that a fluorescent screen in his laboratory began to
glow when the tube was turned on. This surprised him because he thought that the heavy
cardboard surrounding the tube would catch most of the radiation. This shows that x-rays
penetrate most materials. Rontgen began to place different things between the tube and the
screen, but none of them stopped the screen from glowing.
Finally, he placed his hand in between the tube and the screen and the silhouette of his
bones was shown on the screen. He had discovered the most useful application for x-rays.
His first four photographs included the hand of his wife, a set of weights, a compass, and a
piece of metal, which he included in his paper, "On a New Kind of Rays," published
December 28, 1895. He published a total of three papers between 1895 and 1897 and none
of his conclusions have yet been proven false. In 1901, he won the first Nobel Prize for
Physics. Rontgen's discovery was one of the most remarkable discoveries in medical
history. X-rays allow doctors to look directly through tissues and see broken bones,
cavities, swallowed objects etc. X-rays can also be used to examine soft tissues.
Rontgen's had Twelve Discoveries that are still relevant today. X-rays:
 Are highly penetrating, invisible rays which are a form of electromagnetic radiation.
 Are electrically neutral and therefore not affected by either electric or magnetic fields.
 Can be produced over a wide variety of energies and wavelengths (polyenergetic and
heterogeneous).
 Release very small amounts of heat upon passing through matter.
 Travel in straight lines.
 Travel at the speed of light, 3x108 m/s in a vacuum.
 Can ionize matter.
 Cause fluorescence (the emission of light) of certain crystals.
 Cannot be focused by a lens.
 Affect photographic film.
 Produce chemical and biological changes in matter through ionization and excitation.
 Produce secondary and scatter radiation.
1.14 PRODUCTION OF X-RAYS
The production of x-rays requires a rapidly moving stream of electrons that are suddenly
decelerated or stopped. The source of electrons is the cathode, or negative electrode.
Electrons
are stopped or decelerated by the anode, or positive electrode. Electrons move
between the cathode and the anode because there is a potential difference in charge between
electrodes.
FIGURE 1.19: PRODUCTION OF X-RAYS
CATHODE

Negatively charged electrode.

Consists of a filament and a focusing cup. Filament is a coiled tungsten wire that serves
as the source of electrons during x-ray production.

Most x-ray tubes are referred to as dual-focus tubes because they use two filaments; a
large and a small. Only one filament is energized at any one time during x-ray
production.

Focusing cup is made of nickel and mostly surrounds the filament to focus the stream of
electrons before they strike the anode.
ANODE

Positively charged electrode

Consists of a target and in rotating anode tubes, a stator and rotor.

The target stops the electrons and creates the opportunity for the production of x-rays.

The target of rotating anode tubes is made of tungsten and rhenium alloy.

Tungsten generally makes up 90% of the composition of the rotating target, with
rhenium making up the other 10%.

Rotating anodes generally have a target angle ranging from 6 to 20 degrees.

Tungsten is used as the material of choice for the rotating targets because of its high
atomic number of 74 and a high melting point of 3370 degrees F.

Anode rotates from 3,300 rpm to 10,000 rpm.
The components of the X-ray tube include a glass envelope containing a high vacuum. A
cathode or negative electrode which contains a tungsten filament, which when heated emits
electrons in a process called 'thermionic emission'. The cathode also has a focussing cup to
better direct the emitted electrons across the vacuum to hit the target. The anode or positive
electrode is a thick copper rod with a small tungsten target at the end. Tungsten is required
as it has a high atomic number to improve the efficiency of bremsstrahlung X-ray
production , and a high melting point. There is a good deal of heat generated and hence the
need for high melting points and the copper anode is able to conduct heat away effectively.
Some machines have an oil based cooling system ported through the anode, while others
have a spinning electrode to effectively increase the surface area. Either way, considerable
heat needs to be dissipated.
A potential difference or voltage is applied between the cathode and anode. The tungsten
filament (cathode) is heated by an independent current and the thermionically emitted
electrons are accelerated across the potential difference to a high velocity before striking the
tungsten target. The high vacuum is needed to reduce the electron/atom collisions which
waste accelerating energy. The electrons that hit the tungsten target undergo sudden
deflection because of the interactions with the tungsten nucleus. The tungsten target is
usually angled to direct the resultant x-rays towards a consistent portion or window in the
tube-wall.
Some additions to this basic set up include the anode hood made of copper and tungsten
that act like blinkers to prevent stray electrons from striking the walls of the tube. The
copper catches the electrons and the tungsten attenuates the photons produced in the copper.
The window is thin and made of beryllium.
UNIT-2
X-RAY DIFFRACTION METHODS
&PRELIMINARY X-RAY STUDY OF AN
ORGANIC COMPOUND
2.1 DIFFRACTION & X-RAY DIFFRACTION
DIFFRACTION:
Diffraction refers to various phenomena which occur when a wave encounters an obstacle or a
slit. In classical physics, the diffraction phenomenon is described as the apparent bending of
waves around small obstacles and the spreading out of waves past small openings. These
characteristic behaviours are exhibited when a wave encounters an obstacle or a slit that is
comparable in size to its wavelength.
FIGURE 2.1 : DIFFRACTION
X-RAY DIFFRACTION
The atomic planes of a crystal cause an incident beam of x-rays to interfere with one another as
they leave the crystal. This phenomenon is known as X-ray diffraction.
2.2 BRAGG’S LAW
Introduction
The structures of crystals and molecules are often being identified using x-ray diffraction
studies, which are explained by Bragg’s Law. The law explains the relationship between an xray light shooting into and its reflection off from crystal surface. Bragg’s Law was introduced
by Sir W.H. Bragg and his son Sir W.L. Bragg. The law states that when the x-ray is incident
onto a crystal surface, its angle of incidence, θ, will reflect back with a same angle of
scattering, θ. And, when the path difference, d is equal to a whole number, n, of wavelength, a
constructive interference will occur. The condition for Bragg’s diffraction to be occurred is that
wavelength is always less than or equal to twice of interplanar spacing of the crystal.
DERIVATION OF BRAGG’S LAW
Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with
wavelength comparable to atomic spacings, are incident upon a crystalline sample, scattered by
the atoms in the system and undergo constructive interference in accordance to Bragg's law.
FIGURE 2.2: ILLUSTRATION OF BRAGG’S LAW
Consider conditions necessary to make the phases of the beams coincide when the incident
angle equals and reflecting angle. The rays of the incident beam are always in phase and
parallel up to the point at which the top beam strikes the top layer at atom z. The second beam
continues to the next layer where it is scattered by atom B. The second beam must travel the
extra distance AB + BC if the two beams are to continue travelling adjacent and parallel. This
extra distance must be an integral (n) multiple of the wavelength (λ) for the phases of the two
beams to be the same:
nλ = AB+BC
(1)
Recognizing d as the hypotenuse of the right triangle ABz, we can use trigonometry to relate d
and θ to the distance (AB + BC). The distance AB is opposite θ so,
AB = dsinθ
(2)
Because AB = BC eq. (1) becomes,
nλ=2AB
(3)
Substituting eq. (2) in eq. (3) we have,
nλ=2d⋅sinθ
Equation (4) is required expression of Bragg’s Law.
(4)
1.3
RECIPROCAL LATTICE
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the
spatial wavefunction of the original lattice (or direct lattice) is represented. This space is also
known as momentum space or less commonly k-space. The reciprocal lattice of a reciprocal
lattice is the original lattice.
This projection is based on the fact that a family of parallel planes is repeated by their common
normal & if we relate the length of normal to the normal to inter-planar spacing, we have
displayed both orientation & spacing of planes, this in fact devises the new projection called
the Reciprocal Lattice. Thus with reciprocal lattice we are able to know not only the shape but
also the size of unit cell. Thus the reciprocal lattice is a lattice whose repeat distance along a
row of points is reciprocally related to inter-planar spacing of direct lattice, such that the rows
of points in the reciprocal lattice are normal to the planes of direct lattice. A representation of
reciprocal lattice is shown in figure below.
FIGURE 2.3 : GEOMETRICAL REPRESENTATION OF A RECIPROCAL LATTICE
GEOMETRICAL CONSTRUCTION: Fix up some point in the direct lattice as a common origin.
 From the origin, draw normals to each and every set of parallel planes in direct lattice.
 Fix the length of the each normal equal to the reciprocal of inter-planar spacing.
 Put a point at the end of each normal.
The collection of points is called the reciprocal lattice.
PROPERTIES OF RECIPROCAL LATTICE
 Every crystal may be imagined to be made of two lattices, direct lattice and reciprocal
lattice.
 A diffraction pattern is a map of its reciprocal lattice.
 It has dimensions of length.
 The volume of the unit cell of reciprocal lattice is inversely proportional to the volume
of unit cell of crystal lattice.
 The unit cell of reciprocal lattice need not to be parallelepiped.
 Reciprocal lattice of a SC lattice is also SC, BCC is FCC and FCC is BCC.
RELATIONSHIP BETWEEN DTRECT AND RECIPROCAL LATTICE
Reciprocal lattice vector for the three axes of a unit cell can be calculated to be equal to
a* = 2π .(b× c)/[a.b×c]
b* = 2π.(c× a)/[a.b×c]
c* = 2π .(a× b)/[a.b×c]
Here some of the important mathematical properties of reciprocal lattice vectors
a*.a = 1;
b*.b = 1;
c*.c = 1
(a*)* = a;
(b*)* = b;
(c*)* = c
2.4 X-RAY DIFFRACTION TECHNIQUES :AN EXPERIMENTAL OVERVIEW
In order to examine a material by X-ray diffraction methods, the crystal under examination
must fulfil the following requirements.
1. It should have uniform internal structure and good morphology including shape and
size.
2. It must be a single crystal.
3. The crystal should not be bent or otherwise physically disturbed or fractured.
The crystals are examined under an optical polarising microscope before subjecting them
to X-radiation so as to ensure that they do not have any morphological or internal
imperfections. Crystals with imperfections give dark bright regions at the same time.
Following are the techniques which are often employed to carry out the preliminary
examination.
1) SINGLE OSCILLATION/ DOUBLE OSCILLATION METHOD
2) ROTATION METHOD
3) WEISSENBERG METHOD
2.5 SINGLE OSCILLATION/ DOUBLE OSCILLATION METHOD
Oscillation technique is applied to carry out preliminary X-ray analysis of single crystals.
In this method, monochromatic x-rays are used. The oscillation range lies between 10° to
20° and it can be achieved by the mechanism of two pins provided in the camera
arrangement. Due to this arrangement, the probability of the plane overlap is reduced.
In an oscillation photograph of a misaligned crystal, the layer line is tilted w.r.t. the crystal
axes of the film. Any error due to this misalignment of the crystal can be thought of as the
combination of tilt and bow error. If the setting error is in horizontal plane, the trace in the
layer line on the film is tilted away from the central line and it is called bow error. If the
error is in vertical plane, the trace of the layer line on the film is then tilted w.r.t. the
central axes of the film and is called as tilt error.
In order to correct these errors, a double oscillation photographic technique is employed.
After first exposure to the sample for one hour, the crystal is rotated through 180° and the
film cassette is given a shift of 2 mm. Exposure to the film is given for the same range of
oscillation but for the difference of time duration of exposure time.
After developing the film, the distance between the two images of the zero layer line is
measured in mm at points 90°(45mm) to the right(∆ŕ́́́') and 90°(45mm) to the left(∆́́l') of
the direct beam point. The difference between the parameters should be positive if the line
has the same relative position as at the centre of the film and it will be negative if they
have crossed .This is shown in figure 2.4
Figure 2.4 : REPRESENTATION OF TILT AND BOW ERRORS
The true error is calculated by subtracting the known film shift (∆f) from both the
quantities.
∆r = ∆r'-∆f and ∆l = ∆l'-∆f
Following relations are used to calculate the corrections (degrees)
Tilt error =│∆r-∆l│/2 and Bow error =│∆r+∆l│/2
And accordingly the corrections are applied to the arcs of the goniometer for the correct
alignment of the crystal.
2.6 ROTATION METHOD
Rotation method records the diffraction pattern of a single crystal as it rotates completely
about a crystallographic axes. Reflections are produced according to Bragg’s condition by
varying the angle θ for a wavelength λ..The variation of angle θ is brought about by
rotating the crystal along one of the three crystallographic axes. Since the crystal is
rotating; its planes pass through the glancing angles to give rise to the diffracted rays,
recorded in photographic plate coaxial with the rotation axes. All the reflections are
concentrated in a small region in the form of a number of parallel straight lines i.e. these
lines are highly symmetrical to the central line called the Zero Layer Line. The zero layer
line gives us the reflections corresponding to the Laue cone of order n=1 and so on. These
layer lines help to calculate the repeat distance along the axes of rotation axes. A typical
rotation photograph after unrolling and developing looks like as presented in figure 2.5
below.
Figure 2.5 : A ROTATION PHOTOGRAPH
The formula used for the calculations of repeat distance can be written as
T=nλ/sin[𝐭𝐚𝐧−𝟏 (
𝒀𝒏
𝑹
)]
where Yn is the height of the nth layer from the zero line. R is the radius of the cylindrical
film and λ is the wavelength of the radiation used. By rotating the crystal in X-ray beam
about each of the crystallographic axes and measuring the repeat distance along the axes,
the length of the cell edges can be obtained.
2.7 WEISSENBERG METHOD
PRINCIPLE :Like Rotation and Oscillation method, Weissenberg method is based upon rotating a
single crystal in a beam of monochromatic x-rays. In this method the rotation of the crystal
is accompanied by the translation of the film holder by maintaining a definite instrumental
relationship shown afterwards. Here it is customary to have rotation axes of the crystal
horizontal, which facilitates the equi-inclination setting by varying µ,the angle between the
rotation axes and the x-ray beam.
CONSTRUCTION OF WEISSENBERG APPARATUS :The Weissenberg apparatus consists of following parts:1) TRIPLE BASE
2) CAMERA
3) GONIOMETER
4) LAYER LINE SCREEN
CAMERA
GONIOMETER
Figure 2.6: COMPONENTS OF WEISSENBERG APPARATUS
1) TRIPLE BASE : The Weissenberg apparatus consists of three bases. The lower base,
the middle base and the upper base. Lower base has three levelling screws which adjust
the height of the base and rest of the instrument is translated on it. Middle base can be
translated to either side by means of crank and screw arrangement. Upper base consists
of motor gear train, camera guide and crystal mount.
2) CAMERA : The Weissenberg camera consists of a steel cylinder having a horizontal
slot parallel to the axes of rotation called collimator slot. It carries a groove in its inner
edge. The inner diameter of the camera is 57.3mm and its length is 145mm.The camera
rests on a carrier which is so designed to carry as it physically possible over the shaft
track without any vibrations and friction.
3) GONIOMETER : Goniometer consists of two mutually perpendicular arcs and two
sledges. By moving these arcs and sledges the crystal mounted on its head can be set in
a way that its edges coincides with axis of rotation and set in the path of
coming X-ray beam.
4) LAYER LINE SCREEN :- These are used to get different layer line photographs.
Layer line is split into two slots. The diameter of screen is 48.5mm and width is 4-5
inch which slips over the crystal and the beam catcher. They help to record the
diffraction pattern of any desired line.
BRIEF EXPLANATION OF WEISSENBERG METHOD
In this method the translation of film is coupled with rotation of crystal so that the crystal
rotates and the film shifts or translates parallel to the axis of rotation and reflections
occuring
at a particular layer at different lines are recorded on the same film.The
coupling is such that 1mm of translation of film is equal to the 2°rotation of crystal. The
crystal is rotated through a range which covers the entire film located in the camera or
cassette, usually this rotation range is below 220°.
Figure 2.7: SCHEMATIC REPRESENTATION OF WEISSENBERG APPARATUS
2.8 STRUCTURE OF THE CRYSTAL
Chalcones are open chain flavonoids. These are abundant in edible plants and are
considered to be precursors of flavonoids and isoflavonoids. Chalcones and its
derivatives have attracted particular interest during the last few decades due to the
use of such rin system as the core structure in many drug substances covering wide
range of pharmacological applications. Chalcone moiety is the backbone of several
antiulcer, cardiovascular and antidepressant drugs. Chalcone has the chemical
formula C15H12O and is a light yellow powder. Chalcone is extracted from a
natural source preferably from a Glycyrrhiza species. Some of the general physical
and chemical properties of chalcones are:
(i)
The chemical formula for chalcone is C15H12O and has a molecular
weight of 208.26gmol-1.
(ii)
It is a light yellow powder and slightly soluble in cold water and show
positive solubility in organic solvents like ethyl alcohol, benzene, acetone
and methanol.
(iii)
Its boiling point is 345-348oC and melting point is 55-57oC.
(iv)
Its density is 1.071g/cm3.
The compound undertaken for the present sudy was synthesized by ClaisenSchmidt condensation of 4-(methyl sulfanyl) benzaldehyde (0.01mol) with 3, 4dimethoxy acetophenone (0.01mol) in methanol (60ml) in the presence of catalytic
amount of sodium hydroxide solution (5ml, 30%). After stirring for three hours
the contents of the flask were poured into ice cold water (500ml) and left to stand
for twelve hours. The resulting crude solid was filtered and dried. Crystal
suitable for single crystal X-ray diffraction was grown by slow evaporation from
acetone at room temperature. The chemical structure of the compound is given
below:
Figure 2.8 : CHEMICAL STRUCTURE OF THE CHALCONE DERIVATIVE
2.9 SELECTION MOUNTING AND CENTERING OF THE CRYSTAL
SELECTION OF THE CRYSTAL
The crystal chosen for the collection of the X-ray diffraction data must be a single crystal
i.e it should not be fractured, bent or otherwise physically distorted.
1) It must be of proper size.
2) The dimension of the crystal should be small so as to minimize absorption of X-rays.
3) Crystals should be reviewed in microscope under polarized light before subjecting to
ensure that they do not possess any flaws like cracks, twinning etc.
MOUNTING OF THE CRYSTAL
To obtain good diffraction pattern, the crystal should be mounted properly. After having
examined the crystal under polarizing microscope, a good crystal is selected and
morphological information is obtained. One of the axes of the crystal is chosen as the
axes of rotation. The crystal is then stuck to one end of the thin borosilicate glass fibre,
which is chosen because it has minimum scattering and absorption of X-rays. The other
end of the glass fibre is set within the groove of a small spindle on the top of the
goniometer head with the help of plasticine.
CENTERING OF THE CRYSTAL
For the crystal to be accurately centered in the X-ray beam, the Weissenberg goniometer
is placed in front of the X-ray window. The collimator of the apparatus is adjusted in such
a way that parallel monochromatic X-ray beam passes through it. The position of the
collimator is checked by the microscope attached to the moveable base of the
Weissenberg apparatus. The most intense beam is allowed to fall on the crystal is done
roughly with the help of microscope fitted to the moveable base of the goniometer. After
that the crystal fixed to goniometer head is allowed to expose to X-ray radiations and the
different X-ray diffraction techniques are followed for the preliminary studies of a single
crystal.
2.10 SINGLE/ DOUBLE OSCILLATION PHOTOGRAPH
In the present study all X-ray photographs have been taken using CuK-alpha radiation.
The generator is operated at 10 mA and 30 KV. Fig 2.9 shows a single oscillation
photograph of the title crystal. The crystal has been exposed for one hour to X-ray
radiation. The oscillation range of the crystal is set at 17°. If the photograph is slightly
mis-aligned, the exact centering of the crystal is finally achieved by double oscillation
photograph mechanism proposed by Stout and Jensen (1968). In this technique, firstly the
crystal is oscillated within a fixed range of 17° and an exposure of one hour is given to
the film. On completion of the exposure the crystal is rotated through 180° and the film
cassette is given a shift of 2mm. Exposure to the same film for two hours is given for the
same range of oscillation. Exposure time is changed in order to differentiate between the
layer lines recorded on the same film from the previous ones. The distance between the
two images of zero layer linenon the photographs is measured in terms in mm at points
90° (45mm) to the right(∆r) and to the left (∆l) of the direct beam point. The true error is
calculated by the known film shift(∆f) from both the quantities:
∆r = ∆ŕ́ - ∆f and ∆l = ∆ĺ́ - ∆f
The correction in degrees is then calculated by makng use of the following equations :
Tilt error =│∆r-∆l│/2 and Bow error =│∆r+∆l│/2
Accordingly the correction is applied to the arcs of the goniometer for the correct
alignment of the crystal (stout et. al. , 1968). Since the sample get aligned in the first go
itself therefore it is desired not to take a double oscillation picture. From the photograph
it is found that the crystal is perfectly aligned.
Figure 2.9 : SINGLE OSCILLATION PHOTOGRAPH
2.11 ROTATION PHOTOGRAPH
Figure 2.10 represents the rotation photograph of the title compound. When the crystal is
rotated about b-axis, the crystal is exposed to the X-ray for four hours and the range of
the crystal rotation is fixed at 210°. The layer line are found to be perfectly parallel and
straight. This confirms the perfect centering of the crystal.
Figure 2.10 : ROTATION PHOTOGRAPH
Figure 2.11 : INDEXED PICTURE OF ROTATION PHOTOGRAPH
The value of the identity period has been calculated and given in the following table :Table 2.1: Calculation of cell parameter (say b) using rotation photograph.
λ = 0.5418 nm
S.No. Layer
Line 2 Yn
No.
ξn=sin[tan(Yn/R)]
Yn
(in mm) (in
Identity
(in r.l.u.)
period
along Y-axis
b=(nλ/ξn)å
m
m
)
1
1
15.5
7.75
0.2611
5.90
2
2
35
17.5
0.52125
5.91
Mean value of ‘b’= 5.905 å
2.12 WEISSENBERG PHOTOGRAPH
In oscillation and rotation methods the information contained in a two dimensional
reciprocal lattice plane is considered into one-dimensional layer line. The indexing of
reflections become difficult and to overcome this difficulty Weissenberg introduced a
very useful method called Wiessenberg method. In this method, single layer line is
selected by slotted screen which allows only those diffracted beams from the particular
layer chosen during the rotation of the crystal. The crystal is rotated through a range
which covers the entire film held in the cassette. This angular range is usually set below
220° and in the present case, the range fixed for rotation of the crystal is 215°. In order to
check the zero layer setting first a test exposure photograph for one hour was taken. The
final photograph for the five hours exposure using CuK-α radiation is showm in the
photograph of figure 2.12
Figure 2.12 : ZERO LAYER WEISSENBERG PHOTOGRAPH
2.13 INDEXING OF ZERO LAYER LINE PHOTOGRAPH
Indexing of a two dimensional X-ray photograph simply means the reconstruction of
reciprocal lattice points from a set of Weissenberg photographs. The indexing of zero
level Weissenberg photograph is done by tracing the spots carefully on a trace paper. It is
apparent that some reflections lie on the prominent slanting layer lines, running across the
fil, but other spots are found to form festoons in nature. These prominent lines are taken
to be axes of the crystal imprinted on film. After marking the spots above and below the
central region along the prominent axial lines, a central line is drawn which divide the
photograph into two halves, the upper half and the lower half. To draw festoons, a
template devised by Burger (1966) is used. The traced copy of the photograph is placed
over the template in such a way that the two axes which are 90 mm apart coincide with
the engraved axes of the template. When the festoons are drawn, the spots are assigned
the proper indices depending upon the nature of the axes chosen. The smae procedure is
followed for both the halves of the traced picture and this is how ythe indexing of whole
photograph takes place. Figure 2.13 represents the sketch of festoons drawn in case of
zero level Weissenberg photograph.
Figure 2.13 :
INDEXED PICTURE OF ZERO LAYER WEISSENBERG PHOTOGRAPH
2.14 EVALUATION OF LATTICE PARAMETERS
CELL DIMENSIONS ALONG A & C AXES
As we know, Weissenberg photograph is a map of reciprocal lattice points, the cell
constants are measured by measuring the reciprocal lattice constants and then to convert
these into direct lattice ones. The calculation are generally made on the zero level film.
The distance of each spot on its corresponding lattice axis on the zero layer Weisenberg
photograph is measured from the central line of the film. For more accuracy, we take the
mean of the distance of the corresponding points, lying on the same axial line and on the
both the halves of the photograph, from the central line. Let 2Yn be the perpendicular
distance between the corresponding reflections on the axial lines, then
ξn = 2Sin Yn.
The dimensions of the cell along the given axis is :
a (say)= nλ/ 2sinYn
where, ‘n’ represents the order of reflection and ‘λ’ the wavelength of X-rays used.the
values of the cell parameters, a and c calculated for different sets of corresponding points
are given in the tables below.
Table 2.2: Calculation of cell parameter (say a) using weissenberg photograph.
λ = 0.5418 nm
Spot No.
Prep.
Distance Yn=ln/2
between
(in mm)
ξn= 2sinYn
Cell
(in r.l.u.)
along a-axis
dimension
a=(nλ/ξn) Ǻ
corresponding
spots on axis(ln)
(in mm)
1
14
7
0.2437
6.33
2
19
9.5
0.3301
9.34
Mean value of ‘a’ = 7.83 Ǻ
Table 2.3: Calculation of cell parameter (say c) using weissenberg photograph.
λ = 0.5418 nm
Spot No.
Prep. Distance between Yn=ln/2
corresponding
(in mm)
ξn= 2sinYn
(in r.l.u.)
spots on axis(ln)
Cell dimension
along
c-
axis
(in mm)
c=(nλ/ξn) Ǻ
1
14.5
7.25
0.2524
6.11
2
23
11.5
0.3987
7.73
Mean value of ‘c’ = 6.92 Ǻ
2.15 DETERMINATION OF INTERFACIAL ANGLES
The angle β̽ between the reciprocal axes (a̽ and c̽) is given by :
β̽ = (Xβ̽ /X180̽)× 180º
where Xβ̽ is the distance between two axes and X180̽ is the distance between two traces of
single axis along the central line of film.
For the title crystal, Xβ̽ =41mm and X180̽ = 90 mm
Therefore , β̽ = (41/90) × 180º
= 82º
Now, angle between a and c is :
β = 180º-82º = 98º
The other two interfacial angles are determined by using mathematical expression:
a:b:c =sin α/a̽ : sin β/b̽ : sin γ/c̽
where α, β and γ are the interaxial angles between the axes a , b and c respectively.
Thus,
a : b = sin α/a̽ : sin β/b̽
Or
sin α = a̽×a / (b̽ × b × sin β)
Similarly,
Sin γ = c̽× c/ (b̽ × b × sin β)
By doing the calculations we found that α = γ = 90º and β = 98º.
Thus from the above calculations of the unit cell of title crystal we find that a ≠ b ≠ c
and α = γ = 90º ≠ β which indicates that the crystal system is MONOCLINIC.
Now , As we know, the density of molecules is given by the formula:
Density = (Molecular weight × No. of molecules)/ (Volume of unit cell × Avagadro’s
No.)
Where molecular weight of
is
, the assumed density
g/cm^3 and the volume of unit cell is :
of compound is
Volume = a×b×c
Here β̽= 82º and β= 98º
Thus, Volume =
cm^3.
Therefore ,No. of molecules per unit cell =(Density× Volume of unit cell × Avagadro’s
No.)/ Molecular Weight
=
Hence the no. of molecules per unit cell =
2.16 CONCLUSION
The crystal symmetry and unit cell parameter of the compound <name> was determined
by Oscillation, Rotation and Weissenberg photographs. The cell parameters obtained
from these photographs show that the crystal under study belongs to
SYSTEM.
MONOCLINIC
REFERENCES
 Solid State Physics by M.A.Wahab.
 Applied Solid State Physics by Dr. Rajnikant.
 Buerger ,M.J.(1942), X-ray Crystallography , John Wiley and sons.
 Kuldeep Singh , Ph. D. Thesis, University of Jammu.
 Stout, G.H. Jensen, L.H. Crystal Structure Determination, macmillan
Co., New York(1968)
 Bragg, W.L. Proc.Royal Society London, (A)89,245(1913)
 Weissenberg K.,Z.Physik,23,339(1924)
 Internet.