INTRODUCTION TO XRD

The parameters that define a unit cell are:

a, b, c = unit cell dimensions along x, y, z respectively

, , = angles between b,c (); a,c (); a,b ()

z

y

x

Unit cell: the building block of crystalline solids

90;cba

Shapes of unit cells

All the possible shapes of a unit cell are defined by 7 crystal systems, which are based on the relationship among a,b,c and , , ,

Cubic system

90;cba Tetragonal system

90;cba Orthorhombic system

90;cba Triclinic system

12090 ;;cba Hexagonal system

90;cba Rhombohedral system

90;cba Monoclinic system

Primitive lattice: the unit cell has a lattice point at each corner only (P)

Body centred lattice: the unit cell has a lattice point at each corner and one in the centre (I)

Types of Unit Cells

Face-centred lattice: the unit cell has a lattice point at each corner and one in the centre of one pair of opposite faces (A), (B), (C)

All-face-centred lattice: the unit cell has a lattice point at each corner and one in the centre of each face (F)

CHARACTERIZATION OF THE STRUCTURE OF SOLIDS

Three main techniques:

X-ray diffraction Electron diffraction Neutron diffraction

Principles of x-ray diffraction

Single crystalPowder

X-rays are passed through a crystalline material and the patterns produced give information of size and shape of the unit cell

X-rays passing through a crystal will be bent at various angles: this process is calleddiffraction

X-rays interact with electrons in matter, i.e. are scattered by the electron clouds of

atoms

WHAT IS DIFFRACTION?

Diffraction – the spreading out of waves as they encounter a barrier.

What is a Diffraction pattern?

- an interference pattern that results from the superposition of waves.

- Mathematically, this process can be described by Fourier transform, if the diffraction is kinematic (electron or X-ray has been scatted only once inside the object).

Laser diffraction pattern of a thin grating films, where the size of holes is closed to the wavelength of the laser (Ruby red light 594 um) .

Fourier transform of regular lattices:

Reciprocal spaceReal space

INTERACTION BETWEEN X-RAY AND MATTER

d

wavelength Pr

intensity Io

incoherent scattering

Co (Compton-Scattering)

coherent scattering

Pr(Bragg´s-scattering)

absorbtionBeer´s law I = I0*e-µd

fluorescense

> Pr

photoelectrons

The angles at which x-rays are diffracted depends on the distance between adjacent layers of atoms or ions. X-rays that hit adjacent layers can add their energies constructively when they are “in phase”. This produces dark dots on a detector plate

Scattering of x-rays by crystallographic planes

We need to consider how x-rays are diffracted by parallel crystallographic planes

diffracted x-rays

lattice planes

atoms on lattice planes

d

incident x-rays

X-rays diffracted in phase will give a signal. “In phase” means that the peak of one wave matches the peak of the following wave

d

D

E

F

C

The two x-ray beams travel at different distances. This difference is related to the distance between parallel planes

We connect the two beams with perpendicular lines (CD and CF) and obtain twoequivalent right triangles. CE = d (interplanar distance)

d

DE sin DEd sin EFDE EFd sin

length path in difference sin2 DEEFd

The angle of incidence of the x-rays is is

The angle of diffraction is the sum of these two angles, 2

The angle at which the x-rays are diffracted is equal to the angle of incidence,

These conditions are met when the difference in path length equals an integral numberof wavelengths, n. The final equation is the BRAGG’S LAW

sin2dn Data are collected by using x-rays of a known wavelength. The position of the sampleis varied so that the angle of diffraction changes

When the angle is correct for diffraction a signal is recorded

With modern x-ray diffractometers the signals are converted into peaks

Inte

nsity

(a.

u.)

2 degrees

(200) (110) (400)

(310)(301)

(600) (411) (002)(611)(321)

Reflection (signal) only occurs when conditions for constructive interference between the beams are met

TEST

NaCl is used to test diffractometers. The distance between a set of planes inNaCl is 564.02 pm. Using an x-ray source of 75 pm, at what diffraction angle (2) should peaks be recorded for the first order of diffraction (n = 1) ?

Hint: To calculate the angle from sin , the sin-1 function on the calculator must be used

7.62 2 ; 3.81

0.066 pm 564.02 2

pm 75 sin

sin pm 564.022pm 751

dn

sin2

Lattice Planes and Miller Indices

Atoms or ions in lattices can be thought of as being connected by lattice planes.Each plane is a representative member of a parallel set of equally spaced planes.

A family of crystallographic planes is always uniquely defined by three indices, h, k, l, (Miller indices) usually written (h, k, l)

The Miller indices are defined byZ

l,Y

k,X

h111

Note - plane // to axis,intercept = ∞ and 1/∞ = 0

X, Y, Z are the intersections of one plane with on a, b, c respectively

)hk(

)lh(

)kl(

0

0

0family of lattice planes parallel to

z

y

x

How to Determine Miller Indices

EXAMPLES OF CRYSTALLOGRAPHIC PLANES

(111)

(212)

a

b

c

(100)

a

b

c

0.5

a

b

c

Inter-Planar Spacing, dhkl, and Miller Indices

The inter-planar spacing (dhkl) between crystallographic planes belonging to the same family (h,k,l) is denoted (dhkl)

Distances between planes defined by the same set of Miller indices are unique for each material

2D

d'h’k’l’

dhkl

Inter-planar spacings can be measured by x-ray diffraction (Bragg’s Law)

The lattice parameters a, b, c of a unit cell can then be calculated

The relationship between d and the lattice parameters can be determined geometrically and depends on the crystal system

Crystal system dhkl, lattice parameters and Miller indices

Cubic

Tetragonal

Orthorhombic

2

22

a

l k h

d

2

2

1

2

2

2

1

c

l

a

k h

d

2

2

2

22

2

2

1

c

l

b

k

a

h

d

22

2

The expressions for the remaining crystal systems are more complex

THE POWDER TECHNIQUE

An x-ray beam diffracted from a lattice plane can be detected when the x-ray source, the sample and the detector are correctly oriented to give Bragg diffraction

A powder or polycrystalline sample contains an enormous number of small crystallites, which will adopt all possible orientations randomly

Thus for each possible diffraction angle there are crystals oriented correctly for Bragg diffraction

Each set of planes in a crystalwill give rise to a cone of diffraction

Each cone consists of a set of closely spaced dots each one of which represents a diffraction from a single crystallite

FORMATION OF A POWDER PATTERN

Single set of planes

Powder sample

Experimental Methods

To obtain x-ray diffraction data, the diffraction angles of the various cones, 2, must be determined

The main techniques are: Debye-Scherrer camera (photographic film) or powder diffractometer

Debye Scherrer Camera

Powder Diffractometer

The detector records the angles at which the families of lattice planes scatter (diffract) the x-ray beams and the intensities of the diffracted x-ray beams

The detector is scanned around the sample along a circle, in order to collect all the diffracted x-ray beams

The angular positions (2) and intensities of the diffracted peaks of radiation (reflections or peaks) produce a two dimensional pattern

This pattern is characteristic of the material analysed (fingerprint)

Each reflection represents the x-ray beam diffracted by a family of lattice planes (hkl)

Inte

nsity

2 degrees

(200) (110) (400)

(310)(301)

(600) (411) (002)(611) (321)

APPLICATIONS AND INTERPRETATION OF X-RAY POWDER DIFFRACTION DATA

Number and positions (2) of peaks

crystal class

lattice type

cell parameters

Intensity of peakstypes of atoms

position of atoms

Information is gained from:

Identification of unknown phases

Determination of phase purity

Determination and refinement of lattice parameters

Investigation of phase changes

Structure refinement

Determination of crystallite size

Powder diffraction data from known compounds have been compiled into a database (PDF) by the Joint Committee on Powder Diffraction Standard, (JCPDS)

This technique can be used in a variety of ways

The powder diffractogram of a compound is its ‘fingerprint’ and can be used to identify the compound

‘Search-match’ programs are used to compare experimental diffractograms with patterns of known compounds included in the database

Identification of compounds

PDF - Powder Diffraction File

A collection of patterns of inorganic and organic compounds

Data are added annually (2008 database contains 211,107 entries)

Example of Search-Match Routine

Outcomes of solid state reactions

Product: SrCuO2?Pattern for SrCuO2from database

Product: Sr2CuO3?

Pattern for Sr2CuO3from database

CuO2SrCO3 2SrCuO

3CuOSr2?

When a sample consists of a mixture of different compounds, the resultant diffractogram shows reflections from all compounds (multiphase pattern)

Phase purity

Sr2CuO2F2+

Sr2CuO2F2+ + impurity

*

http://www.talmaterials.com/technew.htm

ZrO2 (monoclinic)

3 mol % Y2O3 in ZrO2 (tetragonal)

8 mol % Y2O3 in ZrO2 (cubic)

Effect of defects

Determination of crystal class and lattice parameters

X-ray powder diffraction provides information on the crystal class of the unit cell (cubic, tetragonal, etc) and its parameters (a, b, c) for unknown compounds

Indexing Assigning Miller indices to peaks

1

Determination of lattice parameters

Bragg equation and lattice parameters

2

2222

22

4lkh

asin

Cubic system

Crystal class comparison of the diffractogram of the unknowncompound with diffractograms of known compounds(PDF database, calculated patterns)

3

PROBLEM

NaCl shows a cubic structure. Determine a (Å) and the missing Miller indices( = 1.54056 Å).

2 () h,k,l

27.47 111

31.82 ?

45.62 ?

56.47 222

Selected data from the NaCl diffractogram

? ?

2222

22

4lkh

asin

638.5

2473.56

sin4

12541.1

sin4 2

2

2

2222

lkh

a

Use at least two reflections and then average the results

(222)Å

a (Å)

Miller Indices

2222

22

4lkh

asin

A

2222 lkhAsin

018670

63854

540561

4 2

2

2

2

..

.

aA

2222

lkhA

sin

82312 .40264

0186702

82312

..

.sin 2004222 lkh

62452 . 80528018670

262452

..

.sin 2208222 lkh

Systematic Absences

Conditions for reflection

number)(even 2,, 222222 nlklhkh Fi.e indices are all odd or all even

I nlkh 2222

P No conditions

For body centred (I) and all-face centred (F) lattices restriction on reflections fromcertain families of planes, (h,k,l) occur. This means that certain reflections do not appear in diffractograms due to ‘out-of-phase” diffraction

This phenomenon is known as systematic absences and it is used to identifythe type of unit cell of the analysed solid. There are no systematic absences forprimitive lattices (P)

Considering systematic absences, assign the following sets of Miller indices to either the correct lattice(s).

Lattice Type

Miller Indices P I F

1 0 0 Y N N

1 1 0 Y Y N

1 1 1 Y N Y

2 0 0 Y Y Y

2 1 0 Y N N

2 1 1 Y Y N

2 2 0 Y Y Y

3 1 0 Y Y N

3 1 1 y N Y

Autoindexing

Generally indexing is achieved using a computer program.This process is called ‘autoindexing’

Input: •Peak positions (ideally 20-30 peaks)•Wavelength (usually =1.54056 Å)•The uncertainty in the peak positions•Maximum allowable unit cell volume

Problems: •Impurities•Sample displacement•Peak overlap

Derivation of 2222

22

4lkh

asin

sin2d2

22

a

l k h

d

2

2

1

222

2

22

22

2222

2

l k h4a

l k h2a

l k h

1a

l k h

1ad

l k h

ad

2

2

2

222

2

;

2sin

sin

sin