Condensed Matter Physics

Prof. G.Rangarajan
Department of Physics
Indian Institute of Technology Madras

Module - 01
Lecture -4
Diffraction Methods for Crystal Structures

In the last lecture we saw how the symmetry of a crystal structure plays a crucial role in
determining the behavior of solids in particular crystalline solids and condensed matter
in general. So, the hallucination of the crystal structure namely the determination of the
special position of the various atoms and molecules, which go to make up a given solid is
the first important step in understanding phenomena in the solid state.

Until the discovery of x-ray diffraction by crystalline solids this was not possible, but the
advent of x-ray diffraction by a crystalline solid which serves as a three dimensional
grating for the x-rays. Revolutionized the situation and enabled a detailed mapping of the
atomic and molecular positions inside a solid. Most of the phenomena in condense
matter take place under the influence of applied electric and magnetic fields or through
the scattering of particles such as electrons neutrons protons etcetera.

As well as elementary excitations such as photons, phonons, magnons, polarons,

plasmons, etcetera by the atoms and molecules in the medium. So, these are all scattering
events, which can also be used to reveal the details regarding crystal structure, spin
arrangement, etcetera in condense matter, as well as to find, to determine the normal
modes of these excitations. This is the case for example, in x-ray, electron or neutron
diffraction or in roman brilone are inelastic neutron scattering. In this lecture, we
describe the basic concepts which are common to these phenomena and examine some of
their applications to the study of solids and liquids.

In order to do this, we have to formulate the theory of what happens, when x-rays are
diffracted by a crystalline solid. We consider a perfect crystal which consists of a regular
lattice of discrete atoms and molecules, when an atom is excited by an electromagnetic
wave it reradiates this wave in all directions in face with the initial excitations at that
atom. This is just a consequence of what Huygens postulated in wave optic much earlier.
The amplitude of the beam, x-ray beam in any direction is obtained by summing up the
contribution in that direction from each atom taking into account the face differences
between these atoms.

(Refer Slide Time: 04:25)

This figure shows the direction of the incident and diffracted waves for an arbitrary angle
of incidence on a lattice of discrete atoms. So, we have a lattice, which is shown by the
heavy dots. So, those are the atoms and you have a pair of a set of parallel crystal planes
and then the incident x-ray beam is scattered by an atom for example, at B the incident
wave from A is scattered by the atom at B into a direction marked as C in this figure. So,
that is the basic scattering or diffraction event.

We all know from elementary discussions, which I do not propose to go into this is based
on finding this amplitude and finding the condition for constructive interference of the
scattered or diffracted amplitudes. So, diffraction is known to occur when the waves
reinforce each other that is what I meant by constructive interference. The condition for a
diffraction maximum is given by the so, called famous Bragg law of x-ray diffraction.
(Refer Slide Time: 06:02)

Which is given as 2 d sin theta equals n lambda, here d is the inter planar separation
distance, theta is the angle of incidence let me say angle of scattering and lambda is the
wave length of the incident x-rays, n is any integer, corresponding to different values of
n, diffraction maxima will arise. We will now try to see how this comes about by
calculating the amplitude of the diffracted waves from all the atoms at a point, when a
wave is incident at one of the atoms as shown in this figure.

(Refer Slide Time: 07:33)

So, you have an x-ray of wave vector k 0 incident x-ray. It defines a direction the
magnitude is 2 pi by lambda, where lambda is the wave length and the direction is along
the direction of propagation of the wave.

(Refer Slide Time: 08:26)

And now, k dash is the scattered wave vector, wave vector of the scattered x-rays. So, we
are considering a small volume element at a point or prime. Scattered by atoms in a small
volume around at position vector r prime. Now, if I consider the point at which we
measure the scattered amplitude is if it is taken if the point at which scattered x-ray
amplitude is calculated is given by the position vector r and if this is sufficiently far
away from the arrays of atom, which do the scattering.

Then we can assume that the secondary wavelengths of Huygens theory, which originate
from the scattering atoms may this spherical wave fronts of the secondary wavelets will
become plane waves at sufficiently large distances and these plane waves can be written
as exponential i k dash dot r minus omega t. The amplitude of the wave as it leaves the
atom at r prime the incident amplitude is proportional to exponential i k 0 dot r prime
minus omega t, this is the incident amplitude at the atom located in r.
(Refer Slide Time: 11:14)

So, we have a situation, where we can easily calculate the amplitude at the observation
point.

(Refer Slide Time: 11:26)

The amplitude the observation point of the scattered wave is proportional to or we can
write as sum A exponential i k not dot r prime minus omega t.
(Refer Slide Time: 11:55)

This we will write later, times exponential i k dash dot r minus r prime that is the
distance between r and r prime times exponential minus i omega t. So, that would be the
amplitude. So, this can be rewritten as A exponential i k dot r times exponential i k
naught minus k prime dot r prime this is simply neglecting omitting the time dependent
term.

(Refer Slide Time: 13:17)

So, this is simply rearranged in the form. So, this is the amplitude when we drop to the
time dependent the total amplitude is got by summing over all the atoms. So, the atomic
terms are here. So, this is a summation over all atoms that is the total amplitude. So, that
can also be rewritten as A e to the power i k dot r integral all atoms since we are having
atoms, which are practically continuously distributed. So, we can write a scattering
amplitude times exponential i delta k dot r prime, where delta k is just this quantity this
is i write it as delta k.

(Refer Slide Time: 14:23)

So, this is a, this f of r prime is a periodic function depending on the distribution of

atoms. So, it has a maximum value at the site of a lattice point it is unity and it is zero
everywhere else where ever there is an atom there is scattering amplitude. So, it has this
has which has the same periodicity is the same as that of the crystal lattice. So, it has the
same thing therefore, this amplitude will be nonzero.
(Refer Slide Time: 15:32)

Only when this exponential term does not vanish, in other words the condition for a
diffraction maximum is got as delta k dot r equal to n into 2 pi, where n is an integer this
is known as Laue condition, this was first written by max von Laue. Well I will write you
just keep the notation and keep the r prime. So, this is the condition for constructive
interference or reinforcement of the scattered amplitudes. So, that there is a Bragg
reflection so, strictly this lead to a Bragg reflection. We can write this condition in
component form because this is a vector equation.

So, we can write it in component form. So, if we do this we get three conditions in terms
of integers, which are 2 pi by a multiples of 2 pi by a, 2 pi by b, 2 pi by c for a lattice
with periodicity of a b and c in the three mutually orthogonal crystallographic directions.
So, it is now, useful to construct a lattice with unit cell with lengths of this and such a
lattice is known as a lattice with unit cell of dimensions 2 pi by a, 2 pi by b, 2 pi by c is
known as the reciprocal lattice in crystallography, this is a very useful concept .The
concept of a reciprocal lattice it is called reciprocal lattice because it involves a unit cell
whose lengths are the reciprocals of the three lattice parameters.
(Refer Slide Time: 19:03)

So, we can write the basis vectors of such a reciprocal lattice. If a b and c are the lattice
vectors of the original lattice, reciprocal lattice vectors are given by a star equals 2 pi
times b cross c by a dot b cross c, b star is 2 pi times c cross a by a dot b cross c.

(Refer Slide Time: 19:28)

And c star is 2 pi times a cross b by a dot b cross c. So, we define the reciprocal lattice
vectors in this way and a vector, which connects two points in the reciprocal lattice, is
known as a reciprocal lattice vector.
(Refer Slide Time: 20:01)

A reciprocal lattice vector is usually denoted by the symbol G.

(Refer Slide Time: 20:14)

So, the condition for maximum then becomes delta k equals n dot G, since diffraction
takes place diffraction means it is elastic scattering. What is elastic scattering elastic?
Scattering is a scattering event in which the energy is conserved apart from momentum
conservation. Energy is also conserved in the sense that the incident and scattered wave
vector will have equal magnitudes and differ only in direction. So, delta k will have a
magnitude equal to two times k naught sin theta. So, if you plug this in where theta is the
scattering angle. So, we get two times sin theta equal to n into 2 pi.

(Refer Slide Time: 21:54)

And we can show that the inter planar spacing is just 1 by G therefore; we arrive at the
Bragg condition 2 d sine theta equal to n lambda. So, we see that the Laue condition and
Bragg condition for diffraction maximum are really one and the same. Because, the inter
planar spacing in all crystals is a the order of d is the order of a few angstroms therefore,
x-rays, which have the wavelength of the same order or smaller are commonly used to
observe the diffraction of waves in solids. The technique of determining the structures of
crystals using x-ray diffraction is known as x-ray crystallography. X-ray crystallography
gives you the basic principles governing how to determine crystal structures using x-ray
diffraction. There are several methods we will discuss a few of them here.
(Refer Slide Time: 23:12)

The first one is known as the Debye Scherrer method. This is the most popular and most
commonly used method. So, I am discussing that first.

(Refer Slide Time: 23:26)

So, the first method is known as Debye Scherrer it is also known as the powder method,
since it uses samples in the form of powder, powder method. So, you have a sample in
the form of a powder or poly crystalline sample that is why this is very popular that you
do not require a crystal you can take any solid in the form of a poly crystalline or powder
sample. So, you do not have to take special efforts to grow a single crystal. So, you have
a large number of tiny crystals oriented in all possible directions randomly with respect
to the incident x-ray direction wherever you apply the incident x-ray direction the crystal
grains will be oriented in all possible directions isotropically because, there is no way of
saying, which way it will be preferably preferentially oriented.

So, all grains will be oriented randomly around the incident direction whatever be that
direction. So, when a monochromatic x-rays we use a poly crystalline sample and
monochromatic x-rays that is x-rays of a definite wave length. So, these are the two
ingredients in the Debye Scherrer method when you have a monochromatic x-ray beam
which is incident on such a poly crystalline sample, since the grains are oriented in all
possible directions Bragg reflection takes place, almost all the planes because you have a
large number of grains oriented in all possible directions.

So, some plane or other will satisfy the Bragg law condition. So for, every set of h k l
planes containing different h k l planes, h k l or the miller indices of the crystallography
planes. So, for every set of there are different planes with different values of h k and l the
miller indices. So, there will be several grains in the powder sample, which are oriented
in such a direction as to satisfy the Bragg condition for the given incident x-ray.

So, you take the sample in the form of very fine powder, which is usually put inside a
thin glass capillary tube or is attached to a glass fiber by coating the glass fiber with
grease and then rolled in the powder then this is mounted at the center of a cylindrical
camera, which is shown that is known as the Debye Scherrer camera it is shown in the
next figure.
(Refer Slide Time: 26:46)

So, that is the Debye Scherrer camera and you see the sample mounted at the center. A
strip of photographic plate is positioned inside the camera around the inner wall. So, you
have a strip positioned around the inner wall of this camera. So, there is a photographic
film. In the middle such that the films surrounds the sample, the camera has a hole for
the incident x-ray and a diametrically opposite hole for the outgoing x-ray. So, there is an
entrance for the incident x-rays and an exit.

So, that is the basic details of the Debye Scherrer camera when the monochromatic x-ray
beam falls on the specimen the incident beam undergoes Bragg reflection from the
various crystallographic planes. So, the rays reflected from one set of h k l planes making
an angle theta with the incident beam will all lie on the surface of a cone, whose apex is
at the point of contact of the incident radiation with the specimen this is shown in the
next figure. where how the x-ray diffraction pattern is formed is shown. So, the incident
x-ray beam is coming and then all the points situated on that cone will correspond to the
same Bragg reflection condition. So, the diffraction beam from two sets of plane is
shown in this figure.
(Refer Slide Time: 28:39)

So, from this figure you can see that the solid angle subtended by the cone is 4 theta. So,
each set of planes for a given specific h k l value gives rise to a such a cone of reflected
rays and therefore, these cones intersect the cylindrical photographic plate in circles as
illustrated in the next figure.

(Refer Slide Time: 29:09)

So, these cones intersect the photographic plate. So, you have the diffracted x-ray beam
from the powder sample intersects the photographic plate. So, you get each cone
generates a pair of arcs on the photographic plate. The arcs are positioned symmetrically
with respect to the central hole there will be several such pairs. So, those will be different
pairs of arcs each corresponding to one set of h k l planes.

(Refer Slide Time: 30:01)

So, this is how a typical record will show powder diffraction pattern will be formed on
the photographic plate as shown in the figure.

(Refer Slide Time: 30:11)

So, this is quite simple, this picture that is shown in this figure is a very simple one
because, it corresponds to a simple a monatomic cubic crystal. If the crystal structure is
complex this pattern also will become more complicated with closely spaced arcs of
varying intensities. So, how to interpret the x-ray pattern, it is easier for simple cubic
structures. So, this such a simple description of how the crystal structure can be
determine using the diffraction pattern generated by a simple monatomic cubic lattice is
shown in this figure is described now.

(Refer Slide Time: 31:00)

So, if you have a cubic crystal then the condition is we already saw this mentioned this in
an earlier lecture.

(Refer Slide Time: 31:08)

Where a is the lattice parameter of a cubic solid crystal, d h k l is the inter planar spacing
corresponding to a set of h k l planes, where h k l are the miller indices. So, substituting
these values in the Braggs law, we get sin square theta and rearranging the terms you get
n square lambda square h square plus k square plus l square by four a square. So, that is
the basic equation.

(Refer Slide Time: 32:17)

So, the lambda is fixed in a given x-ray experiment. The unit cell parameter is fixed for
the given solid, n is the order of the diffraction so, which takes integer values. So, we can
see that sin square theta is proportional to h square plus k square plus l square everything
else is fixed. Now, the Bragg angle is related to the distance between the arcs, this
distance is what is measured and this is given by the solid angle is 4 theta times R, which
is the radius of the camera gives you this distance. So, you can measure this distance and
knowing the angle of scattering and knowing the radius of the Debye Scherrer camera.
(Refer Slide Time: 33:18)

The distance from the distance which is accurately measured from the diffracted pattern
we can calculate the theta values using this equation. So, the sin square theta is
calculated then if it is tabulated the common factor between the different values of sin
square theta gives you lambda square by 4 a square. So, what do you do? You write
down the calculate the sin square theta and then find the common factor for example,
table shows the different h k l planes and the corresponding h square plus k square plus l
square. So, we can calculate the common factor lambda square by a square and hence
find the lattice parameter a.

(Refer Slide Time: 34:18)

Now, we will also see that for a given lattice type Bragg reflection is possible only from
certain h k l planes this is known as the extinction roll.

(Refer Slide Time: 34:35)

This simplifies the analysis of the x-ray diffraction pattern, not all values of h k l give
rise to a Bragg reflection. So, depending on the given lattice type Bragg reflection occurs
only from those crystal planes which satisfy certain conditions on the h k and l values.

(Refer Slide Time: 35:02)

This is to understand this; we have to write down what is known as the structure factor.
Which is really determined for a given h k l planes as sigma over j f j. f j is the scattering
amplitude times exponential i 2 pi into h x j plus k y j plus l z j . So, that defines the
where k j y j z j are the coordinates of the jth atom, which lies on a crystal lattice plane
whose miller indices are h k and l. So, once you have this, the summation over j is over
all such identical atoms in the unit cell, f j is known as the atomic scattering factor. Let
us now evaluate the structure factor; the structure factor goes into the intensity of the
diffracted beam. So, if you take a simple cubic structure.

(Refer Slide Time: 36:49)

A simple cubic solid it has atoms at the corners of this cube, cubic unit cell which is
primitive. So, there is only one atom per unit cell you may wonder there are 8 atoms why
do I say there is only one atom. You must remember this unit cell is surrounded by other
unit cells in all the three directions on either side therefore, all these eight atoms are
shared by these adjacent atom unit cells as well.
(Refer Slide Time: 37:28)

So, there is only one eighth of an atom which contributes to a given unit cell and since
there are eight atoms at the eight vertices of a cube eight times one eight gives you one.
So, that is the really the only one atom in the unit cell, which is at 0, 0, 0 the origin, we
can we take this atom at the origin. So, s h k l substituting in that expression is just f
because it is summation is just over one term. So, it is nonzero, the structure factor is
nonzero for any sets of h k l. So, all the crystal planes give rise to Bragg diffraction. So,
there is no extinction roll really, but come to a body centered cubic lattice

(Refer Slide Time: 38:27)

A body centered cubic lattice has two atoms per unit cell one atom at 0, 0, 0 and another
one atom in the body center of the cube, which is half, half, half inside the distance of
half the lattice separation the distance between consecutive atoms. So, substituting this
you get s h k l equals one plus f times one plus exponential i pi the 2 pi cancels with the
two factor 2 cancels the 2 in the denominator.

So, I get h plus k plus l. So, that will be the scattering amplitude. So, this will be nonzero
only if h plus k plus l is even. Only when this is an even number this leads to exponential
i pi times 2 or 4, otherwise it becomes 0. So, only even values of h plus k plus l, the sum
of h k l should be an even number. That is the extinction rule for a body centered cubic
lattice. Going on to a face centered cubical lattice.

(Refer Slide Time: 40:14)

We see that there are four atoms in the unit cell, cubic unit cell. one at 0, 0, 0 then one at
0, half, half at the face center, one at half, 0, half and another at half, half, 0 these are the
coordinates of the centers of the faces. So, the s h k l in this case is going to be f times 1
plus exponential i pi into h plus k plus exponential i pi into k plus l plus exponential i pi
into l plus h.
(Refer Slide Time: 41:25)

So, the structure factor will be nonzero in this case only when this will be nonzero only
when h k and l are all odd or are all even. So, that is the extinction rule for a face
centered lattice.

(Refer Slide Time: 42:22)

So, these three extinction rules are summarized here. In the next table and the next table
as I showed you already. List the different h plus k plus l values and the corresponding h
square plus k square plus l square values for these combinations, which enter into the
Bragg condition.
(Refer Slide Time: 42:43)

From the above and the extinction rule it can be concluded that the possible ratios of h
square plus k square plus l square values for the simple cubic, body centered cubic and
face centered cubic structure these are shown in the next table. So, the ratios of the h
square plus k square plus l square corresponding to different h k l values for which Bragg
reflections are allowed for the case of a simple cubic ratio is 1 is to 2 is to 3 is to 4
everything all the numbers. For a body centered cubic lattice it is 2 is to 4 is to 6 is to 8 is
to 10, which can be reduced to 1 is to 2 is to 3 is to 4 etcetera. And the face centered
cubic lattice it is only 3 is to 4 is to 8 is to 11 is to 12 is to 16 etcetera. So, you see how
there are systematic extinctions.

So, the experimentally calculated ratios knowing these h square plus k square plus l
square values you can calculate the ratios of sin square theta values. And so, we can
calculate, compare the h square plus k square plus l square values and the sin square
theta values the lattice type is immediately found by matching these ratios. So, that is
how the analysis is done for the Debye Scherrer method. The next method that we talk
about is the Laue method in this technique.
(Refer Slide Time: 44:23)

This is the second method. The first methods were the Debye Scherrer method. In this in
the Debye Scherrer method we use monochromatic x-rays, here we use continuous or
white x-rays not monochromatic x-rays and in the Debye Scherrer method we use a
polycrystalline sample, but here this is specimen is in the form of a single crystal
specimen.

So, the diffraction pattern will be a set of spots, where there is maximum of diffracted
intensity the arrangement of these spots indicates the symmetry of the crystal with
respect to the incident x-ray, if the x-ray direction beam is directed along a n fold
rotation axis of symmetry. The diffraction pattern will then display the symmetry of the n
fold axis. So, the crystallographic directions in a crystal can be identified using the Laue
method.
(Refer Slide Time: 45:41)

So, the next figure shows a Laue pattern for a single crystal of ammonium chloride. So,
you see how the diffraction spots are distributed.

(Refer Slide Time: 45:50)

The third method goes by the name the Bragg Diffractometer this is the bread and butter
of every x-ray crystallography. This is used for the determination of crystal structure of
materials.
(Refer Slide Time: 46:18)

And the next figure shows the experimental arrangement in a Diffractometer. You have
the schematic diagram is shown with a monochromatic x- ray beam incident on a single
crystal and then the diffracted beam is seen by a interms of using an x-ray detector
mounted on rotatable arm the diffraction . So, the x-rays get Bragg reflected from
various crystal planes in different directions the x-ray detector shows a peak at those
angles at which the Braggs law is satisfied by a set of crystal planes.

(Refer Slide Time: 47:06)

So, such a diffraction pattern for iron is shown in figure can see the diffraction maxima
the Bragg diffraction maxima corresponding to different h k l planes. When the intensity
of the scattered x-ray is plotted as a function of the diffraction angle. So, we have now,
considered all the important techniques of x-ray diffraction.

(Refer Slide Time: 47:57)

We in addition to x-rays we can also use a beam of electrons or a beam of neutrons these
are particles which can again get scattered and we can have electron diffraction or
neutron diffraction.