RECIPROCAL LATTICE

Revision
 Lattice Plane Spacing

For cubic a=b=c

Reciprocal la3ce
 Reciprocal Lattice Vectors
q To proceed further with the Fourier analysis of the electron
concentra1on, we must find the vectors G of the Fourier sum
.
q There is a powerful, somewhat abstract procedure for doing this.
The procedure forms the theore1cal basis for much of solid
state physics, where Fourier analysis is the order of the day.

q We construct the axis vectors b1, b2, b3 of the reciprocal la8ce:

q The factors are not used by crystallographers but are

convenient in solid state physics.
 Fourier Series

n(r)=
 Reciprocal Lattice Vectors
q If a1, a2, a3 are primi@ve vectors of the crystal laAce, then b1, b2,
b3 are primi1ve vectors of the reciprocal la8ce.
q Each vector defined by the rela@on of b and a is orthogonal to
two axis vectors of the crystal laAce. Thus b1, b2, b3 have the
property

q Points in the reciprocal lattice are mapped by the set of vectors

How to draw of reciprocal la/ce?
Prob.: (i) Show that the SC laAce is reciprocal to the SC la8ce. (ii)
Show that the BCC la8ce is reciprocal to the FCC la8ce and vice-
versa.

Solution:
(i) The primitive translation vectors of a simple cubic lattice may be
taken as

The primitive translation vectors of the reciprocal lattice are found

from the standard prescription equation.
Primi@ve transla@on vectors of the BCC and FCC laAce.
Prob.: Write the reciprocal lattice vectors (G) for SC, BCC and FCC
and their volumes in both lattice spaces i.e direct and reciprocal
lattices.
Solu1on:
The primitive translation vectors of the bcc lattice,

The primi@ve transla@on vectors of the reciprocal laAce are found

from the standard prescrip@on equa@on.

The volume on DL.

The volume on RL.

The reciprocal lattice.

 Reciprocal Lattice Vectors
q The vectors G in the Fourier series are just the reciprocal laAce
vectors G, for then the Fourier series representa1on of the
electron density has the desired invariance under any crystal
transla1on T=u1a1 + u2a2 + u3a3. From

q The is given as;

q The argument of the exponen@al has the form @mes an

integer, because v1u1+v2u2+v3u3 is an integer, being the sum of
products of integers.
q The argument of the exponen@al has the form @mes an
integer, because v1u1+v2u2+v3u3 is an integer, being the sum of
products of integers. Thus by the Fourier analysis to periodic
func@ons n (r), we have the desired invariance:

q Every crystal structure has two lattices associated with it, the
crystal lattice (direct lattice) and the reciprocal lattice.
q A diffraction pattern of a crystal is, as we shall show, a map of the
reciprocal lattice of the crystal.
q A microscope image, if it could be resolved on a fine enough
scale, is a map of the crystal structure in real space. The two
lattices are related by the definitions reciprocal lattice.
q Thus, when we rotate a crystal in a holder, we rotate both the
direct lattice and the reciprocal lattice.
q Vectors in the direct lattice have the dimensions of [length];
vectors in the reciprocal lattice have the dimensions of [l/length].
q The reciprocal lattice is a lattice in the Fourier space associated
with the crystal.
q Wave vectors are always drawn in Fourier space, so that every
position in Fourier space may have a meaning as a description of
a wave, but there is a special significance to the points defined
by the set of G’s associated with a crystal structure.
 Diffrac4on Condi4ons
q The set of reciprocal lattice vectors G determines the possible x-
ray reflections.
 Diffraction Conditions
The set the difference in phase factors is between
beams scattered from volume elements ‘r’ apart. The wave vectors
of the incoming and outgoing beams are k and k’.
q The amplitude of the wave scattered from a volume element is
proportional to the local electron concentration n(r). The total
amplitude of the scattered wave in the direction of k’ is
proportional to the integral over the crystal of n(r)dV times the
phase factor .
q In other words, the amplitude of the electric or magnetic field
vectors in the scattered electromagnetic wave is proportional to
the following integral which defines the quantity F that we call
the scattering amplitude:
q We introduce into (F equa@on) the Fourier components of n(r) to
obtain for the sca;ering amplitude F;

q If , then the argument of the

exponen@al vanishes and F =VnG .
 Scattering amplitude in diffraction condition

q When the scattering vector is equal to a particular reciprocal

lattice vector, the scattering amplitude is maximum. The F is
negligibly small when differs significantly from any reciprocal
lattice vector.
q
q This is the central result of the theory of elastic scattering of
waves in a periodic lattice. If G is a reciprocal lattice vector, so is
-G, and with this substitution we can write as;

q This particular expression is often used as the condition for

diffraction.
Prob. 1: The spacing d(hkl) between parallel laAce planes that are
normal to the direc@on , then prove
that .

Prob. 2: Using , find the Bragg’s diffraction

equation where the angle between the incident beam and the
crystal plane.
 Laue Equa(ons
q The original result of diffraction theory, namely that
may be expressed in another way to give what are called the Laue
equations. These are valuable because of their geometrical
representation.
q The Laue equations can be represented by;
T=
 Ewald construction
A beautiful construction, the Ewald construction, is exhibited in Fig.
below. This helps us visualize the nature of the accident that must
occur in order to satisfy the diffraction condition in three
dimensions.
Wave Diffraction of waves by
crystals
 DIFFRACTION OF WAVES BY CRYSTALS

q The diffrac1on depends on the crystal structure and on the

wavelength.
q At op@cal wavelengths such as 5000Å, the superposi@on of the
waves sca\ered elas@cally by the individual atoms of a crystal
results in ordinary op1cal refrac1on.
q When the wavelength of the radia@on is comparable with or
smaller than the la8ce constant, the beams are diffracted in
direc@ons quite different from the incident direc@on.
q W. L. Bragg came with a simple explana@on of the diffracted
beams from a crystal that convince only because it reproduces
the correct result with experiment.
 DIFFRACTION OF WAVES BY CRYSTALS

Fig.: The Bragg equa/on ; here d is the spacing of parallel atomic

planes and is the difference in phase between reflec6ons from successive planes. The
reflec/ng planes have nothing to do with the surface planes bounding the par/cular
specimen.
Fig.: Wavelength versus par@cle energy, for photons (X-Ray
Diffrac1on), neutrons (Neutron Diffrac1on), and electrons (TEM,
SEM, EDX etc.).
qC.

OR

q Bragg reflection can occur only when

wavelength is lesser or equal to 2d. Ques.: Why we cannot use
visible light in such diffraction?
q Although the reflec@on from each plane is specular, for only
certain values of angle (theta) will the reflec@ons from all
periodic parallel planes add up in phase to give a strong
reflected beam.
q If each plane were perfectly reflecting, only the first plane of a
parallel set would see the radiation, and any wavelength would
be reflected.
q But each plane reflects 10-3 to 10-5 of the incident radiation, so
that 103 to 105 planes may contribute to the formation of the
Bragg-reflected beam in a perfect crystal.
q The Bragg law is a consequence of the periodicity of the lattice.
q Notice that the law does not refer to the composition of the
basis of atoms associated with every lattice point.
q The composition of the basis determines the relative intensity of
the various orders of diffraction (denoted by n above) from a
given set of parallel planes.
q Bragg reflec@on from a single crystal is shown in Fig.

q Bragg reflec@on from a powder in Fig.

Scattered wave amplitude
 SCATTERED WAVE AMPLITUDE
q The Bragg derivation of the diffraction condition (equation) gives
a neat statement of the condition for the constructive
interference of waves scattered from the lattice points.
q It needs a deeper analysis to determine the scattering intensity
from the basis of atoms, which means from the spatial
distribution of electrons within each cell by Fourier Analysis .
q A crystal is invariant under any translation of the form
T=u1a1+u2a2+u3a3, where u1, u2, u3 are integers and a1, a2, a3 are
the crystal axes.
q Any local physical property of the crystal, such as the charge
concentration, electron number density, or magnetic moment
density is invariant under T.
q The most important to us is that the electron number density
n(r) is a periodic function of r, with periods a1, a2, a3 in the
directions of the three crystal axes, respectively.
 Fourier Analysis
q Inside the the charge concentration, electron number density,
or magnetic moment density is invariant under T.
q The electron number density n(r) is a periodic function of r,
which is related with.

q Such periodicity creates an ideal situa@on for Fourier analysis.

The most interes@ng proper@es of crystals are directly related to
the Fourier components of the electron density.

q We consider first a func@on n(x) in one dimension with period a

in the direc@on x. We expand n(x) in a Fourier series of Sines and
Cosines:
A periodic func@on n(x) of period a, and the terms that may
appear in the Fourier transform n(x) .
 q For n(x) has the period a, the Fourier series can be expanded with
n(x+a)

q We say that is a point in the reciprocal lattice or Fourier

space of the crystal.
q In one dimension these points lie on a line. The reciprocal lattice
points tell us the allowed terms in the Fourier series. A term is
allowed if it is consistent with the periodicity of the crystal.
q A term is allowed if it is consistent with the periodicity of the crystal;
other points in the reciprocal space are not allowed in the Fourier
expansion of a periodic function n(x) i.e. similar to the T vector in the
crystals or

q It is convenient to write the Fourier series in the compact form:

where the sum is over all integers p: posi@ve, nega@ve, and zero.
The coefficients np now are complex numbers.

q Thus the number density n(x) is a real func@on, as desired. It

means that the
Ques.: Show that number
density n(x) is a real function?
q Inversion of the FS of n(x) will be

q The reciprocal lattice points tell us the allowed terms in the

Fourier series. A term is allowed if it is consistent with the
periodicity of the crystal.

q A term is allowed if it is consistent with the periodicity of the

crystal; other points in the reciprocal space are not allowed in
the Fourier expansion of a periodic function n(r) =n(r+T).
q The extension of the Fourier analysis to periodic func1ons n(r) in
three dimensions is straighforward. We must find a set of vectors
G such that

q n(r) is invariant under all crystal transla@ons T that leave the crystal
invariant. It will be shown below that the set of Fourier coefficients
nG determines the x-ray scaXering amplitude.

q Using inversion of Fourier Series, the nG can be determined by

following relation:

which is related with Vc is the volume of a cell of the crystal.

FOURIER ANALYSIS OF THE BASIS FOR XRD ANALYSIS

qThe electron number density n(r) is a periodic function of r

q The inversion of Fourier Series, the nG is given as;

q The scattering amplitude for a crystal of N cells

FG=N*[nG*volume of the cell (Vc)]

q When the diffrac@on condi@on is sa@sfied, the sca;ering
amplitude, as discussed earlier, for a crystal of N cells may be
wri\en as ;

The quan@ty SG is called the structure factor and is defined as an

integral over a single cell, with r = 0 at one corner.

q If rj is the vector to the center of atom j, then the function nj(r =rj)
defines the contribution of that atom to the electron
concentration at r.
q The total electron concentra@on at r due to all
atoms in the single cell is the sum

over the all ‘s’ atoms of the basis.

q The structure factor may now be written as integrals over the s
atoms of a cell:

q We now define the atomic form factor as

q So, the structure factor of the basis in the form

q The usual form of this result follows on wri@ng for atom j:

q Then, for the reflection labelled in the reciprocal space lattice by

v1, v2, v3, we have

q Then the structure factor of the basis in the form for atom j:

q The structure factor S need not be real because the scattered

intensity will involve SG*SG, where SG* is the complex conjugate of
SG so that SG*SG is real.
q Now we take the values of xj, yj and zj (direct la8ce space) for SC,
BCC and FCC and try to find the value of v1, v2 and v3(reciprocal
la8ce space) for SC, BCC and FCC having either maximum or
minimum values of the SG. This structure factor can be used to
calculate the scaXered intensity of the unit cell.
 Structure Factor of the bcc Lattice
q The bcc basis referred to the cubic cell has
iden@cal atoms at x1 =y1 =z1= 0 and at
x2=y2=z2=1/2 i.e. Basic points are (0 0 0)
and (1/2 ½ ½).

q The structure factor for bcc lattice becomes

where f is the form factor of an atom

q The value of S is zero whenever the exponential has the value -1,
which is whenever the argument is (odd integer). Thus
we have
BCC Selection rule BCC: (h+k+l) even allowed
In BCC 100, 111, 210, etc. go missing

002 x 022
BCC crystal x x
202 x 222
x 011 x
Important note:
§ The 100, 111, 210, etc. points in the 101
reciprocal lattice exist (as the corresponding x x
real lattice planes exist), however the
intensity decorating these points is zero. 000 x 020
x 110 x
100 missing reflection (F = 0) 200 x 220

Weighing factor for each point motif

Reciprocal Crystal = FCC

2 2
F =4f FCC lattice with Intensities as the motif
Prob. 6: Metallic sodium has a bcc structure. The diffrac@on pa\ern
does not contain lines such as (100), (300), (111), or (221), but lines
such as (200), (110), and (222) will be present; here the indices
(v1v2v3) are referred to a cubic cell. What is the physical
interpreta@on of the result that the (100) reflec@on vanishes?

Solution:
 Structure Factor of the fcc La0ce
q The basis of the fcc structure referred to the cubic cell has
identical atoms at (000); (011); (101); (110).

q The structure factor for fcc laAce becomes

where f is the form factor of an atom

 Structure Factor of the fcc Lattice

q If all indices are even integers, S = 4f; similarly if all indices

are odd integers.
q But if only one of the integers is even, two of the exponents
will be odd multiples of and S will vanish i.e. S=0.
q If only one of the integers is odd, the same argument applies
and S will also vanish i.e. S=0.
q Thus in the fcc lattice no reflections can occur for which the
indices are partly even and partly odd. The reflections are
occurred when all indices are either even or odd integers.
 002 2/a 022
FCC
a 202
222

111

000

Lattice = FCC 020

×
200
220
100 missing reflection (F = 0)
110 missing reflection (F = 0)

Weighing factor for each point motif

Reciprocal Crystal = BCC
BCC lattice with Intensities as the motif
2 2
F = 16 f
 Structure Factor of the sc Lattice
SC Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)

Selection rule: All (hkl) allowed

In simple cubic crystals there are No missing
reflections
+ 001 011
Single
sphere motif
101
Lattice = SC

000 010
=

100 1/a 110

a
Reciprocal Crystal = SC
SC crystal SC lattice with Intensities as the motif at each
reciprocal lattice point
Diffrac3on peaks occur in Cubic Crystals
Bragg’s Equations
Prob. 7: Identify the structure
of the KCl and KBr.

Solution:
 Atomic Form Factor
q The structure factor, there occurs the quan@ty fj, which is a
measure of the scaXering power of the jth atom in the unit cell.
The value of f involves the number and distribu1on of atomic
electrons, and the wavelength and angle of scaXering of the
radia1on. We now give a classical calcula@on of the sca\ering
factor.

q The sca\ered radia@on from a single atom takes account of

interference effects within the atom. We defined the form factor,

with the integral extended over the electron concentra1on

associated with a single atom.
the number of atomic electrons.

q Therefore f is the ratio of the radiation amplitude scattered by

the actual electron distribution in an atom to that scattered by
one electron localized at a point. In the forward direction G = 0,
and f reduces again to the value Z.
q The overall electron distribution in a solid as seen in x-ray
diffraction is fairly close to that of the appropriate free atoms.
This statement does not mean that the outermost or valence
electrons are not redistributed somewhat in forming the solid; it
means only that the x-ray reflection intensities are represented
well by the free atom values of the form factors and are not very
sensitive to small redistributions of the electrons.
XRD pattern study
Bragg’s Equations
Processes for indexing of XRD pattern
Information in a diffraction pattern

q Peak positions: The peak positions tells us about translational symmetry

namely size and shape of the unit cell.
q Peak intensity: The peak intensity tell us about the electron density inside
the unit cell, namely where the atoms are located.
q Peak shapes and widths: The peak widths and shapes can give information
on deviation from the perfect crystals and learn about extended defects
and micro strains.
 Systematic peaks
Permitted Reflections

SC (100), (110), (111),

(200), (210), (211),
(220), (300), (221)
………
BCC (110), (200), (211),
(220), (310), (222)….
FCC (111), (200), (220),
(311)…..

n lCu Ka = 2 d ( hkl ) Sinq hkl

Bragg
q In SC, all possible planes will diffract having an h2 + k2 + l2 pattern of 1,
2, 3, 4, 5, 6, 8, . . . .
q In BCC, diffraction occurs only from planes having an even h2 + k2 + l2
sum of 2, 4, 6, 8, 10, 12, 14, 16, . . . .
q For FCC, more destructive interference occurs, and planes having h2 +
k2 + l2 sums of 3, 4, 8, 11, 12, 16, . . . will diffract.

Note: By calculating the values of sin2𝜃 and then finding the appropriate
pattern, the crystal structure can be determined for metals having one of
these simple structures.
Prob.: The results of an x-ray diffraction experiment using x-
rays with 𝜆= 0.7107 Å (radiation obtained from a molybdenum
(Mo) target) show that diffracted peaks occur at the following
2𝜃 angles:

Determine the crystal structure, the indices of the plane

producing each peak, and the lattice parameter of the material.
Prob.: For BCC iron, compute (a) the interplanar spacing, and
(b) the diffraction angle for the (220) set of planes. The lattice
parameter for Fe is 0.2866nm. Also, assume that monochromatic
radiation having a wavelength of 0.1790nm is used, and the
order of reflection is 1.
Prob.: Figure shows an x-ray diffraction pattern for lead taken
using a diffractometer and monochromatic x-radiation having a
wave- length of 0.1542 nm; each diffraction peak on the pattern
has been indexed. Determine the crystal structure, compute the
interplanar spacing for each set of planes indexed; also
determine the lattice parameter of Pb for each of the peaks.
Prob.: The first five peaks of the x-ray diffraction pattern for tungsten,
which has a crystal structure; monochromatic x-radiation having a
wavelength of 0.1542nm was used.
(a) Index (i.e., give h, k, and l indices) for each of these peaks.
(b) Determine the inter planar spacing for each of the peaks.
Problem: Shows the first five peaks of the x-ray diffraction pattern for
tungsten, which has a crystal structure; monochromatic x-radiation having
a wavelength of 0.1542 nm was used.
(a) Index (i.e., give h, k, and l indices) for each of these peaks.
(b) Determine the inter planar spacing for each of the peaks.
Problem: If x-rays with a wave- length of 0.07107nm are used, determine
(a) the crystal structure of the metal;
(b) the indices of the planes that produce each of the peaks; and
(c) the lattice parameter of the metal.