Transcript Lecture 4 X-Ray Diffraction and Reciprocal Lattice

Lecture 4
X-Ray Diffraction and Reciprocal
Lattice
Diffraction
Diffraction occurs as waves interact with a regular structure whose
repeat distance is about the same as the wavelength. The phenomenon
is common in the natural world, and occurs across a broad range of
scales. For example, light can be diffracted by a grating having scribed
lines spaced on the order of a few thousand angstroms, about the
wavelength of light.
It happens that X-rays have wavelengths on the order of a few
angstroms, the same as typical interatomic distances in crystalline
solids. That means X-rays can be diffracted from minerals which, by
definition, are crystalline and have regularly repeating atomic structures.
X-ray Scattering
When certain geometric requirements are met, X-rays scattered from a
crystalline solid can constructively interfere, producing a diffracted
beam. In 1912, W. L. Bragg recognized a predictable relationship
among several factors.
1. The distance between similar atomic planes in a mineral (the
interatomic spacing) which we call the d-spacing and measure in
angstroms.
2. The angle of diffraction which we call the theta angle and measure in
degrees. For practical reasons the diffractometer measures an angle
twice that of the theta angle. We call the measured angle '2-theta'.
3. The wavelength of the incident X-radiation, symbolized by the Greek
letter lambda and, in ordinary cases, equal to 1.54 angstroms.
Bragg’s Law
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Most methods of determining the atomic structure of crystals are based on the
idea of scattering of radiation.
X-ray is one of the types of radiation used.
The wavelength of the radiation should have awavelength comparable to a
typical interatomic distance which in solids is of a few angstroms (10-8 cm).
The x-ray wavelength, l can be estimated as follows:
Therefore, x-rays of energy 2-10 keV are suitable for studying the crystal
structure.
X-rays interact with electronic shells of atoms in a solid. Electrons absorb and
re-radiate x-rays which can then be detected. Nuclei are too heavy to respond.
The reflectivity of x-rays is of the order of 10-3 – 10-5, so that the penetration in
the solid is deep. Therefore x-rays serve as a bulk probe.,
Bragg’s Law
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In 1913, Bragg found that crystalline solids have remarkably
characteristic patterns of reflected x-ray radiation.
In crystalline materials, for certain wavelength and incident
directions, intense peaks of scattered radiation were observed.
Bragg accounted for this by regarding a crystal as made out of
parallel planes of atoms spaced d apart.
The conditions for a sharp peak in the intensity of the scattered
radiation were:
i.
The x-rays should be specularly (angle of
incidence = angle of reflection) reflected in the
atomic plane.
ii. The reflected rays from successive planes must
interfere constructively.
X-ray Diffraction Techniques
The quantities measured:
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The scattering angle 2q between the diffracted and incident
beams. By substituting sinq into Bragg’s Law, the interplanar
spacing and the orientaion of the plane responsible for the
diffraction can be determined.
The intensity of the diffracted beam, This quantity determines the
cell-structure factor and therefore gives information concerning
arrangement of atoms in the unit cell.
Laue camera
SiGe
forward reflexion
photo (spots on
ellipses)
Unfiltered X-radiation
through collimator
back reflexion
crystal held
on goniometer
head
photo (spots on
hyperbolas)
• continuous band of wavelength provided
• suitable for determination of crystal orientation & symmetry
• each spot corresponds to planes of atoms of spacing d satisfying 2dsinq=l
• diffraction pattern exhibits the symmetry of the crystal in the direction of k0
• spots corresponding to reciprocal vectors [h,k,l] and [h,k,l] are symmetric
with respect to the centre of the diffraction pattern
Rotating crystal
axis of rotation=crystal axis
unwrapped film
g2
monochromatic
X-ray
r
beam
trap
d2
d1
l 2
l 1
l 0
l1
l2
film held against
inside wall of
cylinder
Let axis of rotation be the C-axis so that Laue eqn C(cosg-cosg0)=ll
Since g0p/2, C cosg=ll. For l=0 “cone” is a plane ^ C. For l=1,2
dl=rcosgl. Therefore
ll r 2  d l2
ll
C

cos g l
dl
Powder method
powder sample: many crystallites
in random orientations
wrapped film
r
2q cone of x-rays
satisfying the
Laue eqn
unwrapped film
monochromatic
X-ray
D  4qr
From 2d sin q  l d 
l
2 sin q
with q 
Lack of orientation make crystal structure determination difficult
D
4r
Electron and Neutron
diffraction
Electron diffraction
• Suitable for study only thin films
and crystals, surfaces because
of strong interaction of the electrons
with atoms and electrons in
solids
• LEED –Low Energy Elastic Diffraction:
probes structure of 2D surface of solids.
Surface should be free of absorbed atoms
• Wavelength of the electron l(Å)=
=12/Ee(eV), l=1Å for Ee=150eV
Neutron diffraction
• Weak interaction with
charged particles, but strong
interaction with magnetic nuclei
and electronic magnetisation
• Wavelength of the neutron l(Å)=
=0.28/Ee(eV), l=1Å for Ee=0.08eV
(lh/p, Ee=p2/2M)
Fe
(lh/p, Ee=p2/2m)
o
30
o
50
o
70
Bragg’s Law
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The figure above shows x-rays which are specularly reflected from
adjacent planes.
The path difference between the two x-rays is equal to 2dsinq.
For constructive inteference this difference must be an integer
number of wavelengths.
This leads to the the Bragg condition:
The number m is known as the order of the corresponding reflection
(or order of inteference).
Deriving Bragg’s Law
Bragg's Law can easily be derived by
considering the conditions necessary
to make the phases of the beams
coincide when the incident angle is
equal to the 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 (Fig). 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
traveling adjacent and parallel. This
extra distance must be an integral (n)
multiple of the wavelength (l) for the
phases of the two beams to be the
same:
nl = AB +BC (2).
Deriving Bragg’s Law
Recognizing d as the hypotenuse of
the right triangle AbZ, we can use
trigonometry to relate d and q to the
distance (AB + BC). The distance AB is
opposite q so,
AB = d sinq (3).
Because AB = BC and nl = AB +BC
becomes,
nl = 2AB (4)
Substituting eq. (3) in eq. (4) we have,
nl = 2 d sinq, (1)
and Bragg's Law has been derived.
The location of the surface does not
change the derivation of Bragg's Law.
Bragg’s Law
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There are a number of various setups for studying crystal structure
using x-ray diffraction. In most cases, the wavelength is fixed, and
the angle is varied to observe diffraction peaks corresponding to
reflections from different crystallographic planes.
Using Bragg law, the distance between planes can be determined.
Sharp x-ray diffraction peaks are observed only in the directions and
wavelengths for which the x-rays scattered from all lattice points interferes
constructively.
Diffraction Condition and Reciprocal Lattice
•To find condition of constructive interference
two

scatterers separated by a lattice vector T are
considered.
•X-rays are incident from infinity, along direction
with wavelength and wavevector 2pk̂ l .
•Assume the scattering is elastic i.e the x-rays are
scattered in direction with same wavelength , so
that the wavevector k̂ '  2pk̂ ' l
.
•The path difference between the x-ray scattered from the two atoms should
be an integer number wavelengths.
Therefore, the condition for constructive

interference is (k̂ 'k̂ )  T  ml .
•Where m is an integer. Multiplying both sides of the above equation by leads
to a condition on the incident and scattered wave vectors
  
(k'k)  T  2pm
Diffraction Condition and Reciprocal
Lattice
  
Defining the scattering wave vector k  k 'k
,
the diffraction condition can be written as
.
 
k  G

Where G is, a vector where
 
G  T  2pm

G satisfying this condition form a reciprocal lattice.
A set of vectors

Vectors G are called reciprocal lattice vectors.
A reciprocal lattice is defined with reference to a particular
Bravais

lattice which is determined by a set of lattice vectors T .
The Bravais lattice that determines a particular reciprocal lattice is
referred to as the direct lattice, when viewed in relation to its
reciprocal.
  
(k'k)  T  2pm
Algorithm for determining reciprocal of direct
lattice
Let a1, a 2 and a 3 be a set of primitive vectors of the direct lattice.
The reciprocal lattice that can be generated from the primitive
vectors are
.

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2p  
2p  
2p  
b1 
a 2  a3
b2 
a 3  a1
b3 
a1  a 2
V
V
V
  
Where V  a1  (a 2  a 3  is the volume of a unit cell so that
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G  m1b1  m 2b 2  m3b3
To proof the relation between direct and
reciprocal lattice is true
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To show that G  m1b1  m 2b 2  m3b3 satisfies the
equation
1D and 2-D: Direct to Reciprocal Lattice
For 1-D and 2-D cases use equation
For 3-D case use
3-D Direct Lattice to Reciprocal Lattice
From the condition:
it can be said that constructive
interference occurs provided that the scattering wave vector is a vector of
the reciprocal lattice.
Bragg’s Law using Reciprocal Lattice Vector
In elastic scattering the photon energy is conserved, so the
magnitudes of k and k’ are equal, therefore k2 =k’2. There from
This another way of stating Bragg’s Law.
To Show: Reciprocal Lattice Vector is
Orthogonal to the plane (hkl)
Definition of Miller Indices
If G is normal to the plane (hkl)
Therefore G is orthogonal to the plane (hkl).
To show: The distance between two adjacent
parallel planes of direct lattice is d = 2p/G
Note: The nearest plane which is parallel
to (hkl) goes through the origin of the
Cartesian coordinates.
Therefore the interplanar distance is
given by the projection of one of the
vectors xa1, ya2, za3 to the direction
normal to the (hkl) plane.
The direction is given by unit vector G/G, since G is normal to the
plane. Therefore
To show:
is equivalent to
Bragg Law
From :
This angle is opposite to the angle between k and G.
Therefore,
k – same direction
as X-ray beam
q
Plane
G