X-ray Diffraction & Crystal Structure
Basic Concepts
T. P. Radhakrishnan
School of Chemistry, University of Hyderabad
Email: tprsc@uohyd.ernet.in
Web: http://chemistry.uohyd.ernet.in/~tpr/
This powerpoint presentation is available
at the following website
http://chemistry.uohyd.ernet.in/~ch521/
Click on x-ray_powd.ppt
Outline
 Crystals
symmetry
classification of lattices
Miller planes
 Waves
phase, amplitude
superposition of waves
 Bragg law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of Powder diffraction
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
Molecular Structure
Optical spectroscopy – IR, UV-Vis
Magnetic resonance – NMR, ESR
Mass spectrometry
X-ray diffraction
High resolution microscopy
Molecular Structure Resolved by
Atomic Force Microscopy
A
5Å
C
5Å
B
5Å
D
20 Å
Pentacene on Cu(111)
A. Molecular model of pentacene
B. STM image
C, D. AFM images (tip modified with CO molecule)
Gross, Mohn, Moll, Liljeroth, Meyer, Science 2009, 325, 1110
Crystal and its structure
3-dimensions
Anthony, Raghavaiah, Radhakrishnan, Cryst. Growth Des. 2003, 3, 631
STM image of 1,3-diheptadecylisophthalate on HOPG
(with a model of two molecules)
Plass, Kim, Matzger, J. Am. Chem. Soc. 2004, 126, 9042
2-dimensional square lattice
Point group symmetries :
Identity (E)
Reflection (s)
Rotation (Rn)
Rotation-reflection (Sn)
Inversion (i)
In periodic crystal lattice :
(i) Additional symmetry - Translation
(ii) Rotations – limited values of n
Translation
Translation
Translation
Translation
Rotation
Rotation
Rotation
Restriction on n-fold rotation symmetry
in a periodic lattice
a
a
q
q
a
na
(n-1)a/2
cos (180-q) = - cos q = (n-1)/2
n
qo
Rotation
3
180
2
2
120
3
1
90
4
0
60
6
-1
0
1
Crystal Systems in 2-dimensions - 4
square
oblique
rectangular
hexagonal
Crystal Systems in 3-dimensions - 7
Cubic
Monoclinic
Tetragonal
Triclinic
Trigonal
Orthorhombic
Hexagonal
Bravais lattices in 2-dimensions - 5
square
oblique
rectangular
centred rectangular
hexagonal
Bravais Lattices in 3-dimensions
(in cubic system)
Primitive cube (P)
Body centred cube (I)
Face centred cube (F)
Bravais Lattices in 3-dimensions - 14
Cubic
Tetragonal
Orthorhombic
Monoclinic
Triclinic
Trigonal
Hexagonal/Trigonal
-
P, F (fcc), I (bcc)
P, I
P, C, I, F
P, C
P
R
P
Point group 7 Crystal systems
operations
Point group
operations + 14 Bravais lattices
translation
symmetries
Lattice (o)
+ basis (x) = crystal structure
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Spherical
basis
C4
Non-spherical
basis
C4
Lattice +
Lattice +
Spherical Basis Nonspherical Basis
Point group 7 Crystal systems 32 Crystallographic
operations
point groups
Point group
operations + 14 Bravais lattices 230 space groups
translation
symmetries
Miller plane in 2-D
a
a
Distance
between
lines = a
y
(01)
x
(10)
Miller plane in 2-D
Distance
between
lines = a/2
= 0.7 a
y
x
(11)
Miller plane in 2-D
Distance between
lines = a/(2)2+(3)2
= 0.27 a
(2, 3, 0)
y
(23)
x
Take inverses
In 3-D: intercepts = 1/2, 1/3, 
Miller plane in 3-D
(100)
Distance
between
planes = a
z
y
x
a
Miller plane in 3-D
(010)
Distance
between
planes = a
z
y
x
Miller plane in 3-D
(110)
Distance
between
planes = a/2
= 0.7 a
z
y
x
Miller plane in 3-D
(111)
Distance
between
planes = a/3
= 0.58 a
z
y
x
Spacing between Miller planes
dhkl =
a
h2+k2+l2
for cubic crystal system
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
0
0
p
l/2
2p
l
A sin{2p(x/l - ut)}
Phase
Displacement
l = wavelength
u = frequency
A = amplitude
sin (0) = sin (np) = 0
sin ([n+1/2]p) = +1 n even
-1 n odd
Superposition of Waves
amplitude = A
amplitude = 2A
Constructive interference
Superposition of Waves
l/4
amplitude = A
amplitude = 1.4A
Superposition of Waves
l/2
amplitude = A
amplitude = 0
Destructive interference
x
1
x+ l/2
2
x+ l
Waves 1 and 2 interfere destructively
Waves 1 and 3 interfere constructively
3
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
Wavelength = l
q
q
dhkl
hkl plane
2dhkl sinq = nl
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
Single crystal
Cones intersecting a film
Collection of several small
crystals
Powder diffraction setup
q
q
Sample
Detector
X-ray tube
Powder x-ray diffractogram
(sodium chloride)
1296
220
200
Nacl
Counts
0
10
15
20
25
30
35
40
45
50
511
331
311
111
400
222
422
420
648
55
60
65
70
2q (degree)
75
80
85
90
95
100
NaCl - powder x-ray data
source Cu-Ka (l = 1.540598 Å)
Indexing
2q (deg.)
d (Å)
h
k
l
(h2+k2+l2)½
a (Å)
27.367
3.256
1
1
1
1.732
5.639
31.704
2.820
2
0
0
2.000
5.640
45.448
1.994
2
2
0
2.828
5.639
53.869
1.700
3
1
1
3.317
5.639
56.473
1.628
2
2
2
3.464
5.639
66.227
1.410
4
0
0
4.000
5.640
73.071
1.294
3
3
1
4.359
5.641
75.293
1.261
4
2
0
4.472
5.639
83.992
1.151
4
2
2
4.899
5.639
90.416
1.085
5
1
1
5.196
5.638
90.416
1.085
3
3
3
5.196
5.638
a = d(h2+k2+l2)½
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
Primitive cube
b.c.c. (h+k+l = odd absent )
f.c.c. (h, k, l all even or all odd present )
Equivalent to hth
order scattering
(h00)
a/h
a
2d.sinq = nl
2(d/n).sinq = l
(h00)
2'
3'
1'
2dh00sinq = l
q
q
d/2
d/2
d/2
1
3
xa
d/2
dh00 = a/h
Path difference 2'1', d = l
Path difference 3'1', d = l xa/(a/h)
= lhx
2
a/h
a
Phase difference 3'1' = (2p/l) lhx = 2phx
In 3-D, the phase difference 3'1' = 2p(hx+ky+lz)
The two waves 1 and 3 scattered from different
atomic layers have different phases, f1 and f2.
They will have different amplitudes A1and A2
if the atoms in the two planes are not the same.
The scattered x-ray intensity is
the sum of the contributions from the
different scattered waves
Two waves having the same frequency, but different amplitude
and phase can be represented as :
E1 = A1sinf1 and E2 = A2sinf2
3
1
2
Waves can be represented as
vectors in complex space
imaginary
The wave vector can be written as
A
f
A(cosf + i.sinf) = Aeif
real
Structure Factor
Atomic scattering factor,
amplitude of wave scattered by an atom
f=
amplitude of wave scattered by one electron
Wave scattered with phase, 2p(hx+ky+lz)
from atoms having scattering factor, f contribute to the
Structure Factor for the Miller plane, (hkl) :
Shkl = S fn e2pi(hxn+kyn +lzn)
n represent the atoms in the basis
Shkl = S fn e2pi(hxn+kyn +lzn)
Atom position
Relates to
Atom type
Intensity of x-ray scattered from an
(hkl) plane
Ihkl  Shkl2
Systematic Absences
Shkl = fA + fB e2pi(hx+ky+lz)
For body centred cubic lattice (bcc)
x = 1/2, y = 1/2, z = 1/2
2pi(hx+ky+lz) = pi(h+k+l)
(h+k+l) is even 
Shkl = fA + fB epi(h+k+l)
(h+k+l) is odd

epi(h+k+l) = +1
epi(h+k+l) = -1
If fA = fB = f
Shkl
= 2f
when h+k+l is even
=0
when h+k+l is odd
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
Single Crystal X-ray Diffractometer
with CCD detector
X-ray tube
Filament
Cathode
X-rays
Water
Anode
Tungsten wire at 1200 – 1800oC
Heating current ~ 35 mA
Voltage ~ 40 kV (Cu), 45 kV (Mo)
Goniometer
3-circle goniometer with fixed c
CCD based detector
Charge Coupled Device
http://www.sensorsmag.com/articles/0198/cc0198/main.shtml
Fourier Synthesis
Shkl = S fn e2pi(hxn+kyn +lzn)
SK = f(r).eiK.r dr
by Fourier transformation,
 (r)  f(r) = SK.e-iK.r.dK
Structure Solution
•The Fourier map provides a structure solution
•Using the initial solution a structure factor is
calculated for each (hkl)  Shkl(calc)
•For each (hkl) there is also an experimental
structure factor  Shkl(exp)
Structure Refinement
•Least square method to carry out regression
of Shkl(calc) against Shkl(exp). Quality of
refinement represented by the r factor
•The final model used for the best Shkl(calc)
is the structure solution
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
Effect of particle size on diffraction lines
Amax
½Amax
B
2q1
2q2
2qB (Bragg angle)
2q
Particle size small
2qB
2q
Particle size large
Scherrer formula for particle size estimation
t=
0.9l
B cosqB
t = average particle size
l = wavelength of x-ray
B = width (in radians) at half-height
qB = Bragg angle
A
B'
A'
E' C'
D'
B
C E
D
0
q1
q2
dA'D' = l
qB
qB
1
Path difference,d
d
dA'M' = ml
dB'E' = l+x
dB'L' = m(l+x)
= (m+1)l
2
3
M
N
L
L'
t = md
(for m: mx = l)
M'
N'
dC'N' = (m-1)l
m
qB
A'D'
2d sinqB = l
A'M'
2(md)sinqB = ml
2d sinqB = l
i.e.
B'L'
2(md) sinq1 = (m+1)l
C'N'
2(md) sinq2 = (m-1)l
sinq1
sinqB =
m
m+1
When m
q1 = qB

finite m: destructive interference is incomplete for q1 to q2
 Crystals
 Waves
 Bragg Law
 Powder diffraction
 Systematic absences, Structure factor
 Single crystals - Solution and Refinement
 Diffraction line width
 Applications of powder diffraction
1. Finger printing
a) Qualitative/quantitative analysis of mixtures
Excedrin - composition of caffeine, aspirin,
acitaminphen
Fly ash - for cement industry
b) Monitoring asbestos, silica in paints
c) Degradation of drugs due to humidity
d) ‘Builders’ in detergents
Sodium and potassium phosphates
e) Phase analysis of cement
2. Polymorph characterisation
a) Paints and pigments
White pigment, TiO2 - rutile, anatase, brookite
Quinacridone paints
b) Pharmaceuticals
Sulfathiazole (antibacterial) - four polymorphs
Ranitidine (antiulcer) - active/inactive polymorphs
c) Food industry
Chocolate - 5 polymorphs stable at room temperature
3. Determination of degree of crystallinity and stress
- linebroadening
a) ‘Excipients’ in pharmaceutical formulations
cellulose - different derivatives have different
extents of crystallinity
b) Photography
Silver halide in gelatin- stress due to drying of
gelatin
c) Polymers - crystalline/amorphous phases
d) Preliminary characterisation of nanomaterials
Thank you
This powerpoint presentation is available
at the following website
http://chemistry.uohyd.ernet.in/~ch521/
Click on x-ray_powd.ppt