1

X-Ray Powder Diffraction and

Rietveld Refinement : An

Overview

Akhilesh Tripathi, PhD

Rigaku Americas Corp.

The Woodlands

TEXAS

Fundamentals of X-Ray diffraction

Properties of X-rays

Crystals & X-ray diffraction

Powder X-ray diffraction methods

X-ray diffractometer

X-Rays and Crystals Why use X-Rays for crystal structure

determination

4.913

around 0.1nm 1around several hundred nm

With the naked eye,

we can see colors and shapes of

objects.

We can see

atoms with X-

rays.

Non-destructive

Accurate atomic distance determination

Simple instrumentation

Properties of X-rays

What are X-rays?

X-rays are short wavelength electromagnetic waves.

Wilhelm Conrad Roentgen discovered X-Rays in 1895

10-2 1 102 104 106 108 1010 1012

10-3 10-1 101 103 105 107 109 1011 nm

Micro waveShort wave

Long waveinfrared

visible lightUltraviolet

Soft X-Rays

Gamma

Rays

Hard X-Rays

Properties of X-rays

X-Rays are a type of electromagnetic wave.

-rays are generated by nuclear reactions.

X-Rays are generated by energy transitions that happen when electrons hit a target.

An X-ray has a dual wave-particle character.

As a wave motion: showing the diffraction phenomenon.

As a particle: behaving as a particle that has an energy inversely proportional to its wavelength.

E 12.4E energykeVwavelength

Properties of X-rays

Fluorescent behavior

Emitting visible light: scintillation counter( NaI.)

Ionization behavior

Generating positive and negative ions:proportional

counter, position sensitive proportional counter.

The refractive index is nearly equal to 1

Not focused through a lens by refraction.

Showing total reflection at very small incident angles.

High penetrating power

Going through many materials due to high energy.

Generation of X-rays

X-rays are generated when electrons accelerated by high voltage from the filament bombard a target.Cu is commonly used for the target material.

Almost all of the energy from the electrons is transformed into heat, and a small amount of the energy into X-rays.

High voltage

Filament TargetX-rays

Electrons

Generation efficiency 1.110-9ZV Cu 0.1%Zatomic number of a target material applied voltage

V

Principle of X-ray Generation

Characteristic X-rays are produced when an outer orbital electron falls into a vacancy in the K-shell created when high speed electrons eject an inner electron.

L1

K

K1

K2

L-shell

electron -shell electron

shell

shell shell

0

1

2

3

4

5

6

7

8

0. 3 0. 5 0. 7 0. 9

Mo

K K

Wavelength()

Absorption coefficient

When X-rays pass through a thickness X cm , the

intensity of the X-rays decrease from I0 to I.

Linear absorption coefficientcm-1/Mass absorption coefficient cm2/g Density (g/cm3)

II0 expx Beer-Lambert formula

I0 I

x cm

Mass absorption coefficient of compounds

The Mass absorption coefficient of a compound, mixture, or liquid that consists of more than two elements is calculated as below.

/WA/AWB/B WAWB AB weight percent of elementWAWB1

ex. :TiO2 CuKTi/200 atomic weight47.9O/11.5 atomic weight16.0

/TiO2 124.547.9200 16.0211.5

47.916.02 47.916.02

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X-ray & materials interaction

X-raysTransmission x-rays

(EXAFS, XANES)

Scattered x-rays

(XRD, XRR, SAXS)

Fluorescent x-rays

(XRF)

Electrons (XPS)

Heat

Analysis techniques using x-rays as the incident beam

1. Introduction

13

Diffraction condition

Incident x-ray

Scattered x-ray

dhkl

...)3,2,1(,sin2 == nnd hkl

Interference

In phase constructive

Out of phase destructive

Crystals & X-ray diffraction

What is a crystal

A crystal is defined as a solid composed of atoms

arranged in a periodic pattern in three

dimensions.

Unit cell

The ideal crystal has the atomic arrangement in a

pattern periodic in three dimensions.

The smallest repeating unit is called a unit cell.

A unit

cell

Primitive

unit cell

Non-primitive

unit cell

Lattice constants

c

a

b

cubic abc primitive 90 body-centered

face-centered

tetragonal abc primitive 90 body-centered

orthorhombic abc primitive 90 body-centered

base-centered face-centered

rhombohedral abc primitive trigonal or 90hexagonal abc primitive

90 120

monoclinic abc primitive 90 base-centered

triclinic abc primitive 90

Miller indices

The lattice plane is defined by the three integer

numbers (h k l), which are called miller indices.

The distance between two lattice planes is called a

lattice spacing.

Miller index (hkl)

Lattice plane

1/h

1/l

1/kb/k

a/h

c/l

Miller indices

c/l (2/3)

b/k (1/2)

a/h (2/3)a/h (0/0)

b/k (1/1)

c/l (0/0)

(343)

(010)

a

b

c

Examples of the Miller indices

a

b

c

(111) (110) (010)

(010)

Lattice planes and lattice spacing

Examples of the Miller indices

Powder X-ray diffraction method

The principle of the Powder X-ray

Diffraction

The PXD covers powder crystals or polycrystalline

aggregates.

2

2

Incident X-ray

Debye ring

Powder sample

26

What does a powder really means?

Single Oriented Random

Crystalline size & debye ring

very small

Less than 0.1m

Preferred orientation

small

10mbig

50mSingle crystal

Measured data

0

2040

6080

100120

140

15 25 35 45 55 65 75 85

2 ( )

29

Diffraction by crystalline materials

Small crystallitesLarge crystallites

Random powderAmorphous

Para-crystal (type 1) Para-crystal (type 2)

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Diffraction pattern

Diffraction angle 2

Inte

nsity

Gas

Liquid

Amorphous

Crystal

Diffraction angle 2

Inte

nsity

Diffraction angle 2

Inte

nsity

X-ray Diffractometer

Key Components

X-ray

Wavelength

Intensity

Optics

Geometry,

Monochromator, etc.

Sample

Crystal structure,

Crystal quality, etc.

Detector

Energy resolution

Efficiency

Instrument

Goniometer

sample

counter

X-ray Generator

Water Supply

X-ray Tube

High Voltage Generator

Display

Data output

electronic circuit

panel

Control/Data

Processing Unit

Computer

Detector

Interactions between X-rays and some materials are

used to detect diffracted x-rays.

Position Sensitive Proportional Counter

Solid State Detector

Scintillation Counter

Imaging Plate

Photosensitivity Photofilm

Scintillation effect

Ionization effect

Scintillation counter

This is the counter which uses a luminescent material called the scintillator.

X-ray

anode

BeNaI crystal

Photomultiplier tube

photocathode

pulse

lightphotomultipliers

electron

2sample

X-ray diffractionX-ray source

counter

Incident X-ray

Principle of Focusing method

Bragg-Brentano focusing method

Well-known optical system that shows high intensity and peak resolution.

- The X-ray source, the sample, and the receiving slit must

be on the focusing circle.

- The sample must be equidistant from the source and the

receiving slit.

- The 2- scanning mechanism is needed.

Diffraction Pattern

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Anatomy of diffraction pattern

- Sample -

Peak positions

d values > Phase ID

Lattice parameter

d shift > Residual stress

Solid solution

Peak positions

d values > Phase ID

Lattice parameter

d shift > Residual stress

Solid solution

Intensity vs. Orientation

Preferred orientation

Fiber structure

Pole figure

Intensity vs. Orientation

Preferred orientation

Fiber structure

Pole figureFWHM

Crystal quality

Crystallite size

Lattice distortion

FWHM

Crystal quality

Crystallite size

Lattice distortion

Scattering Angle [deg.]

Inte

nsity

Integrated Int.

of amorphous

Integrated Int.

of crystal

> Qualitative analysis

Integrated Int.

of amorphous

Integrated Int.

of crystal

> Qualitative analysis

Crystallinity

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Peak positions

Wavelength

Systematic errors

Peak positions

Wavelength

Systematic errors

Intensity vs. Orientation

Irradiati