x-ray diffraction & crystal structure basic concepts t. p. radhakrishnan school of chemistry,...
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![Page 1: X-ray Diffraction & Crystal Structure Basic Concepts T. P. Radhakrishnan School of Chemistry, University of Hyderabad Email: tprsc@uohyd.ernet.in Web:](https://reader033.vdocuments.site/reader033/viewer/2022051618/56649d045503460f949d756b/html5/thumbnails/1.jpg)
X-ray Diffraction & Crystal Structure
Basic Concepts
T. P. RadhakrishnanSchool of Chemistry, University of Hyderabad
Email: [email protected]: http://chemistry.uohyd.ernet.in/~tpr/
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http://chemistry.uohyd.ernet.in/~ch521/
Click on x-ray_powd.ppt
This powerpoint presentation is available at the following website
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Outline
Crystalssymmetryclassification of latticesMiller planes
Waves phase, amplitudesuperposition of waves
Bragg law Powder diffraction Systematic absences, Structure factor Single crystals - Solution and Refinement Diffraction line width Applications of Powder diffraction
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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Molecular Structure
Optical spectroscopy – IR, UV-Vis
Magnetic resonance – NMR, ESR
Mass spectrometry
X-ray diffraction
High resolution microscopy
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Molecular Structure Resolved by Atomic Force Microscopy
Gross, Mohn, Moll, Liljeroth, Meyer, Science 2009, 325, 1110
A. Molecular model of pentacene
A B
C D
5 Å 5 Å
20 Å5 Å
Pentacene on Cu(111)B. STM imageC, D. AFM images (tip modified with CO molecule)
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Crystal and its structure
3-dimensions
Anthony, Raghavaiah, Radhakrishnan, Cryst. Growth Des. 2003, 3, 631
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Plass, Kim, Matzger, J. Am. Chem. Soc. 2004, 126, 9042
STM image of 1,3-diheptadecylisophthalate on HOPG (with a model of two molecules)
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2-dimensional square lattice
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Point group symmetries :Identity (E)Reflection ()Rotation (Rn)Rotation-reflection (Sn)Inversion (i)
In periodic crystal lattice :(i) Additional symmetry - Translation
(ii) Rotations – limited values of n
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Translation
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Translation
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Translation
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Translation
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Rotation
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Rotation
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Rotation
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Restriction on n-fold rotation symmetryin a periodic lattice
cos (180-) = - cos = (n-1)/2
n 3 2 1 0 -1o 180 120 90 60 0Rotation 2 3 4 6 1
a
a a
na(n-1)a/2
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Crystal Systems in 2-dimensions - 4
square
rectangular
oblique
hexagonal
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Crystal Systems in 3-dimensions - 7
Cubic Tetragonal Orthorhombic
Trigonal HexagonalMonoclinic Triclinic
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Bravais lattices in 2-dimensions - 5
square rectangular
oblique hexagonal
centred rectangular
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Primitive cube (P)
Bravais Lattices in 3-dimensions(in cubic system)
Body centred cube (I)
Face centred cube (F)
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Bravais Lattices in 3-dimensions - 14
Cubic - P, F (fcc), I (bcc)Tetragonal - P, IOrthorhombic - P, C, I, FMonoclinic - P, CTriclinic - PTrigonal - RHexagonal/Trigonal - P
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Point groupoperations
Point groupoperations +translationsymmetries
7 Crystal systems
14 Bravais lattices
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Lattice (o)
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
+ basis (x) = crystal structure
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C4
C4
Spherical basis
Non-spherical basis
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Lattice +Nonspherical Basis
Point groupoperations
Point groupoperations +translationsymmetries
7 Crystal systems 32 Crystallographic point groups
14 Bravais lattices 230 space groups
Lattice +Spherical Basis
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(01)
(10)
Miller plane in 2-D
Distance between lines = a
a
a
x
y
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(11)
Distance between lines = a/2= 0.7 a
Miller plane in 2-D
x
y
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(23)
Distance between lines = a/(2)2+(3)2
= 0.27 a
Miller plane in 2-D
x
y (2, 3, 0)
In 3-D: intercepts = 1/2, 1/3,
Take inverses
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x
y
z
(100)
Miller plane in 3-D
Distance between planes = a
a
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Miller plane in 3-D
(010)
Distance between planes = a
x
y
z
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Miller plane in 3-D
(110)
Distance between planes = a/2= 0.7 a
x
y
z
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Miller plane in 3-D
(111)
Distance between planes = a/3= 0.58 a
x
y
z
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ah2+k2+l2
dhkl =
Spacing between Miller planes
for cubic crystal system
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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0 0 /2
PhaseDisplacement
A sin{2(x/ - t)}
sin (0) = sin (n) = 0sin ([n+1/2] = +1 n even
-1 n odd
= wavelength = frequencyA = amplitude
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Superposition of Waves
amplitude = A amplitude = 2A
Constructive interference
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Superposition of Waves
amplitude = A amplitude = 1.4A
/4
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Superposition of Waves
amplitude = A amplitude = 0
/2
Destructive interference
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x
x+ /2
x+
1
2
3
Waves 1 and 2 interfere destructivelyWaves 1 and 3 interfere constructively
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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dhkl
hkl plane
2dhkl sin = n
Wavelength =
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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Single crystal Collection of several small crystals
Cones intersecting a film
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Detector
Sample
X-ray tube
Powder diffraction setup
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10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
1296
648
0
P o w d e r C e l l 2 . 2
Nacl
111
200
220
311
222
400
331
420
422
511
Powder x-ray diffractogram(sodium chloride)
2 (degree)
Counts
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NaCl - powder x-ray datasource Cu-K ( = 1.540598 Å)
a = d(h2+k2+l2)½
Indexing
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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Primitive cube
b.c.c. (h+k+l = odd absent )
f.c.c. (h, k, l all even or all odd present )
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(h00)
a/h
a
Equivalent to hth order scattering
2d.sin = n2(d/n).sin =
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xa
(h00)
a/ha
123
3'1'
2'
2dh00sin =
dh00 = a/h
Path difference 2'1', =
Path difference 3'1', = xa/(a/h)
= hx
Phase difference 3'1' = (2/hx = 2hx
In 3-D, the phase difference 3'1' = 2hx+ky+lz)
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The two waves 1 and 3 scattered from different
atomic layers have different phases, 1 and 2.
They will have different amplitudes A1and A2
if the atoms in the two planes are not the same.
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Two waves having the same frequency, but different amplitudeand phase can be represented as :
E1 = A1sin1 and E2 = A2sin2
The scattered x-ray intensity is the sum of the contributions from the
different scattered waves
3
21
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Waves can be represented as vectors in complex space
real
imaginary
A
The wave vector can be written as
A(cos + i.sin) = Aei
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Structure Factor
Atomic scattering factor,
f =amplitude of wave scattered by an atom
amplitude of wave scattered by one electron
Wave scattered with phase, 2hx+ky+lz) from atoms having scattering factor, f contribute to theStructure Factor for the Miller plane, (hkl) :
Shkl = fn e2ihx +ky +lz ) n n n
n represent the atoms in the basis
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Shkl = fn e2ihx +ky +lz ) n n n
Relates toAtom type
Atom position
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Intensity of x-ray scattered from an(hkl) plane
Ihkl Shkl2
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Systematic Absences
Shkl = fA + fB e2i(hx+ky+lz)
For body centred cubic lattice (bcc)x = 1/2, y = 1/2, z = 1/2
2i(hx+ky+lz) = i(h+k+l)
Shkl = fA + fB ei(h+k+l)
(h+k+l) is even ei(h+k+l) = +1
(h+k+l) is odd ei(h+k+l) = -1
If fA = fB = f
Shkl = 2f when h+k+l is even
= 0 when h+k+l is odd
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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Single Crystal X-ray Diffractometer with CCD detector
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Water
Anode
X-rays
X-ray tube
Cathode
Filament
Tungsten wire at 1200 – 1800oCHeating current ~ 35 mAVoltage ~ 40 kV (Cu), 45 kV (Mo)
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Goniometer
3-circle goniometer with fixed
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CCD based detector
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http://www.sensorsmag.com/articles/0198/cc0198/main.shtml
Charge Coupled Device
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Shkl = fn e2ihx +ky +lz )
Fourier Synthesis
n n n
by Fourier transformation,
SK = f(r).eiK.r dr
(r) f(r) = SK.e-iK.r.dK
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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)
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•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
Structure Refinement
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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Effect of particle size on diffraction lines
B
Amax
½Amax
B (Bragg angle) B
Particle size small Particle size large
1 2
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Scherrer formula for particle size estimation
t = 0.9
B cosB
t = average particle size = wavelength of x-rayB = width (in radians) at half-heightB = Bragg angle
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0
1
2
3
m
d
t = md
A
D
M
A'
D'
M'
B
B
B
1 2
C C'
N N'
Path difference,
A'D' =
A'M' = m
B'L' = m(+x) = (m+1)for m: mx =
C'N' = (m-1)
B'E' = x
B
L
E
B'
L'
E'
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A'D' 2d sinB =
A'M' 2(md)sinB = mi.e.2d sinB =
B'L' 2(md) sin1 = (m+1)
C'N' 2(md) sin2 = (m-1)
sin1
sinB=
mm+1
When m 1 = B
finite m: destructive interference is incomplete for 1 to 2
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Crystals
Waves
Bragg Law
Powder diffraction
Systematic absences, Structure factor
Single crystals - Solution and Refinement
Diffraction line width
Applications of powder diffraction
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1. Finger printing
a) Qualitative/quantitative analysis of mixturesExcedrin - composition of caffeine, aspirin, acitaminphen
Fly ash - for cement industry
b) Monitoring asbestos, silica in paints c) Degradation of drugs due to humidityd) ‘Builders’ in detergents
Sodium and potassium phosphates
e) Phase analysis of cement
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2. Polymorph characterisation
a) Paints and pigmentsWhite pigment, TiO2 - rutile, anatase, brookite
Quinacridone paints
b) Pharmaceuticals Sulfathiazole (antibacterial) - four polymorphs
Ranitidine (antiulcer) - active/inactive polymorphsc) Food industry
Chocolate - 5 polymorphs stable at room temperature
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3. Determination of degree of crystallinity and stress - linebroadening
a) ‘Excipients’ in pharmaceutical formulationscellulose - different derivatives have different extents of crystallinity
b) PhotographySilver halide in gelatin- stress due to drying of gelatin
c) Polymers - crystalline/amorphous phasesd) Preliminary characterisation of nanomaterials
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Thank you
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http://chemistry.uohyd.ernet.in/~ch521/
Click on x-ray_powd.ppt
This powerpoint presentation is available at the following website