X-ray Diffraction & Crystal Structure

Basic Concepts

T. P. RadhakrishnanSchool of Chemistry, University of Hyderabad

Email: tprsc@uohyd.ernet.inWeb: http://chemistry.uohyd.ernet.in/~tpr/

http://chemistry.uohyd.ernet.in/~ch521/

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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

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

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)

Crystal and its structure

3-dimensions

Anthony, Raghavaiah, Radhakrishnan, Cryst. Growth Des. 2003, 3, 631

Plass, Kim, Matzger, J. Am. Chem. Soc. 2004, 126, 9042

STM image of 1,3-diheptadecylisophthalate on HOPG (with a model of two molecules)

2-dimensional square lattice

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

Translation

Translation

Translation

Translation

Rotation

Rotation

Rotation

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

Crystal Systems in 2-dimensions - 4

square

rectangular

oblique

hexagonal

Crystal Systems in 3-dimensions - 7

Cubic Tetragonal Orthorhombic

Trigonal HexagonalMonoclinic Triclinic

Bravais lattices in 2-dimensions - 5

square rectangular

oblique hexagonal

centred rectangular

Primitive cube (P)

Bravais Lattices in 3-dimensions(in cubic system)

Body centred cube (I)

Face centred cube (F)

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

Point groupoperations

Point groupoperations +translationsymmetries

7 Crystal systems

14 Bravais lattices

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

C4

C4

Spherical basis

Non-spherical basis

Lattice +Nonspherical Basis

Point groupoperations

Point groupoperations +translationsymmetries

7 Crystal systems 32 Crystallographic point groups

14 Bravais lattices 230 space groups

Lattice +Spherical Basis

(01)

(10)

Miller plane in 2-D

Distance between lines = a

a

a

x

y

(11)

Distance between lines = a/2= 0.7 a

Miller plane in 2-D

x

y

(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

x

y

z

(100)

Miller plane in 3-D

Distance between planes = a

a

Miller plane in 3-D

(010)

Distance between planes = a

x

y

z

Miller plane in 3-D

(110)

Distance between planes = a/2= 0.7 a

x

y

z

Miller plane in 3-D

(111)

Distance between planes = a/3= 0.58 a

x

y

z

ah2+k2+l2

dhkl =

Spacing between Miller planes

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 /2

PhaseDisplacement

A sin{2(x/ - t)}

sin (0) = sin (n) = 0sin ([n+1/2] = +1 n even

-1 n odd

= wavelength = frequencyA = amplitude

Superposition of Waves

amplitude = A amplitude = 2A

Constructive interference

Superposition of Waves

amplitude = A amplitude = 1.4A

/4

Superposition of Waves

amplitude = A amplitude = 0

/2

Destructive interference

x

x+ /2

x+

1

2

3

Waves 1 and 2 interfere destructivelyWaves 1 and 3 interfere constructively

Crystals

Waves

Bragg Law

Powder diffraction

Systematic absences, Structure factor

Single crystals - Solution and Refinement

Diffraction line width

Applications of powder diffraction

dhkl

hkl plane

2dhkl sin = n

Wavelength =

Crystals

Waves

Bragg Law

Powder diffraction

Systematic absences, Structure factor

Single crystals - Solution and Refinement

Diffraction line width

Applications of powder diffraction

Single crystal Collection of several small crystals

Cones intersecting a film

Detector

Sample

X-ray tube

Powder diffraction setup

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

NaCl - powder x-ray datasource Cu-K ( = 1.540598 Å)

a = d(h2+k2+l2)½

Indexing

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 )

(h00)

a/h

a

Equivalent to hth order scattering

2d.sin = n2(d/n).sin =

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)

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.

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

Waves can be represented as vectors in complex space

real

imaginary

A

The wave vector can be written as

A(cos + i.sin) = Aei

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

Shkl = fn e2ihx +ky +lz ) n n n

Relates toAtom type

Atom position

Intensity of x-ray scattered from an(hkl) plane

Ihkl Shkl2

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

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

Water

Anode

X-rays

X-ray tube

Cathode

Filament

Tungsten wire at 1200 – 1800oCHeating current ~ 35 mAVoltage ~ 40 kV (Cu), 45 kV (Mo)

Goniometer

3-circle goniometer with fixed

CCD based detector

http://www.sensorsmag.com/articles/0198/cc0198/main.shtml

Charge Coupled Device

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

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)

•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

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

B

Amax

½Amax

B (Bragg angle) B

Particle size small Particle size large

1 2

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

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'

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

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 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

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

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

Thank you

http://chemistry.uohyd.ernet.in/~ch521/

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