1. CHAPTER 2 : CRYSTAL DIFFRACTION AND PG Govt College for Girls
RECIPROCAL LATTICE
Prof. Harvinder Kaur
PG Govt College for Girls
Sector-11, Chandigarh

2. Outline Reciprocal Lattice Reciprocal Lattice of various crystals
Simple cubic lattice
Face-centered cubic lattice
Body-centered cubic lattice
Bragg law of Diffraction
Experimental Diffraction methods
Structure and Form factor

3. Reciprocal Lattice
The reciprocal lattice of a Bravais Lattice is the set of all vectors K such that
eiK.R = 1
for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.
For an infinite three dimensional lattice, defined by its primitive vectors, its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formulae

4. Properties Of Reciprocal Lattice
Direct lattice is a lattice in ordinary space whereas the reciprocal lattice is a lattice in the Fourier space.
The primitive vectors in reciprocal lattice has the dimensions of (length)-1 whereas the primitive vectors of the direct lattice have the dimensions of length
A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal whereas a microscopic image is a map of direct lattice
When we rotate a crystal, both direct and reciprocal lattice rotates
Each point in the reciprocal lattice represents a set of parallel planes of the crystal lattice
If the coordinates of reciprocal vector G have no common factor, then G is inversely proportional to the spacing of the lattice planes normal to G
The volume of unit cell of the reciprocal lattice is inversely proportional to the volume of unit cell of the direct lattice
The direct lattice is the reciprocal of its own reciprocal lattice
The unit cell of the reciprocal lattice need not be parallelopiped

5. Reciprocal Lattice In Two Dimensions

6. Reciprocal Lattice of Simple Cubic Lattice
The simple cubic Bravais, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side ( in the crystallographer's definition).
The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space

7. Reciprocal lattice to Face-centered cubic
The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice.
FCC in real space
BCC in fourier space

8. Primitive Translation Vectors:
Reciprocal lattice of Body-centered cubic
The reciprocal lattice to an BCC lattice is the face-centered cubic (FCC) lattice.
Primitive Translation Vectors:
FCC in real space

9. Real: FCC Reciprocal: BCC Real: BCC Reciprocal: FCC
Brillouin zones for FCC,BCC, HCP
Real: FCC Reciprocal: BCC
Real: BCC Reciprocal: FCC
HCP

10. Bragg law of Diffraction
Prof. W.L.Bragg observed that X-rays can be reflected by the cleavage planes of the crystal as if these parallel planes were acting like mirrors to the light beam. The reflected beam lead to well defined diffraction patterns on a photographic plane. The cleavage planes are atomic planes of a crystal that are systematically stacked one over the other as a parallel set of planes. X-rays from the incident plane can penetrate deeper into a target crystal and get reflected. Thus, several beams of X-ray reflected from various planes are obtained. The final diffracted beam is the result of the superposition of these beams.
Bragg law, dsin = n 
where, d is the interplanar spacing,  is the wavelength of incident X rays and  is the angle of reflection

11. Bragg law of Diffraction
Significance of Bragg’s law
Bragg’s law is a consequence of periodicity of the lattice
The law does not refer to the arrangement of atoms in the basis associated with each lattice point
The composition of the basis determine the relative intensity of the various orders n of diffraction
Since  ~ 1Ao is inevitable, we can’t use visible light for such studies

12. Bragg law in Reciprocal Lattice
(Ewald Construction)
Chose a point according to the orientation of the specimen with respect to the incident beam.
Draw a vector AO in the incident direction of length 2p/l terminating at the origin

13. Draw a vector AB to the point of the
(Ewald Construction)
Construct a circle of radius 2p/l with center at A. Note whether this circle passes through any point of the reciprocal lattice; if it does....
Draw a vector AB to the point of the
intersection

14. Draw a vector OB to the point of the intersection

15. Draw a line AE perpendicular to OB
Complete the construction to all the intersection points in the same fashion
Draw a line AE perpendicular to OB

16. By means of the Ewald construction we can write the Bragg law in
vector form: Let G = OB and k = AO. For diffraction, it is necessary
that the vector k + G, that is, the vector AB, be equal in magnitude to
the vector k or
(k + G)2 = k2 or
2k . G + G2 = 0 (1)
If we call the scattered wave vector k*, than
k* = k + G (2)
so we can write
k*2 = k2 (3)
and k* - k = G, showing (2) that the scattering changes only the direction
on k, and (3) that the scattered wave differs from the incident wave by a
reciprocal lattice vector G.
Equations (2) and (3) are the momentum and energy conservation law for
x-ray diffraction, which is an example of elastic scattering.
Whether or not the k circle intersects a lattice vector, and hence reflects,
depends on its magnitude and orientation. Using (1) we can construct in the
reciprocal lattice the locus of all those waves that can produce Bragg
reflection. This locus represents a set of planes in three dimensions.
The volume terminated by those planes is called Brillouin zone.

17. Experimental Diffraction Methods

18. Experimental Diffraction Methods
Laue’s Method : In this method a single crystal is held stationary in the path of a beam of e-m radiation (X-rays) or the Neutron radiation of continuous wavelengths. While  is kept constant, the wavelength ,  is varied so that the Bragg law is satisfied. A plane film receives the diffracted beams. A developed film after its exposure shows a diffraction pattern that consists of series of spots.
Laue spots in a diffraction pattern are actually map of the reciprocal lattice of the crystal under experiment.

19. Experimental Diffraction Methods
Rotating Crystal Method : In this method , a single crystal is rotated about the fixed axis in a beam of monochromatic X-rays or neutrons. The angle  is variable while the wavelength is kept constant. The variation of angle  due to rotation of the crystal brings different atomic planes in the crystal into position for which Bragg’s reflection holds good. To record such reflections a film is mounted on a cylindrical holder that is concentric with a rotating spindle.

20. Experimental Diffraction Methods
Powder Method : If a powdered specimen is used, instead of a single crystal, then there is no need to rotate the specimen, because there will always be some crystals at an orientation for which diffraction is permitted. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample.
This method is useful for samples that are difficult to obtain in single crystal form.
The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal.
For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appearance of a continuous line.

21. The Powder Method
If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result.

22. The Powder Method
If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards.

23. The Powder Method
A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film.

24. Structure and Form Factor
A crystal is a periodic arrangement of atoms in a particular pattern. Each of the atoms may scatter incident radiation such as X-rays, electrons and neutrons. Because of the periodic arrangement of the atoms, the interference of waves scattered from different atoms may cause a distinct pattern of constructive and destructive interference to form. This is the diffraction pattern caused by the crystal.
In the kinematical approximation for diffraction, the intensity of a diffracted beam is given by:
                            
where       is the wave function of a beam scattered a vector     , and       is the so called structure factor which is given by:
                         
Here, rj is the position of an atom j in the unit cell, and fj is the scattering power of the atom, also called the atomic form factor. The sum is over all atoms in the unit cell. It can be shown that in the ideal case, diffraction only occurs if the scattering vector is equal to a reciprocal lattice vector   .

25. Structure Factor For Specific Lattice Types
Body-centered cubic (BCC) : As a convention, the body-centered cubic system is described in terms of a simple cubic lattice with primitive vectors ax,ay,az with the basis consisting of ro = 0 and r1 = (a/2) (x+y+z). In a monoatomic crystal, all the form factors f are the same. The intensity of a diffracted beam scattered with a vector
(2/a) (hx+ky+lz) by a crystal plane with Miller indices given by (h,k,l)

26. Structure Factor For Face Centered Cubic
The face-centered cubic system is described as r0 =0, r1 = a/2(x+y), r2 = (a/2)(y+z)
r3 = (a/2)(x+z) with indices given by (1/2,1/2,0), (0,1/2,1/2) and (1/2,0,1/2)