Fundamentals of X-ray

diffraction and scattering

Don Savagedsavage@wisc.edu

1231 Engineering Research Building

(608) 263-0831

X-ray diffraction and X-ray scattering

Involves the elastic scattering of X-rays

Diffraction is primarily used for structure determination.

How are atoms or molecules arranged?

What is the crystal structure?

Scattering uses differences in electron density and looks at larger

structures.

X-rays are part of the electromagnetic spectrum

Laboratory X-ray sources

• Electrons bombard

target, give off X-rays

• Water cooling can be

used to increase the

power to the target

• Optics can be used to

filter and focus the X-

rays produced

eV = hn =hc/l,

V (volts) =1239.8/l(nm)

Copper is a common anode choice

Lab sources

Point source – Useful with area

detector

Bruker d-8 – source has crossed multilayer

mirrors to make a parallel point beam

Line source – Useful when you have a

large uniform sample (e.g., for a perfect

crystal or uniform smooth film)

Panalytical – Source has a multilayer mirror

and a channel cut crystal to make a

monochromatic, parallel line source

Mirror only, used for reflectivity

Slits only for Bragg-Brentano method

X-ray interactions with matter

X-ray interactions with matter

X-ray scattering by an atom

X-rays are scattered by electrons in an atom into (approximately) all directions,

though peaked in the forward direction. Wave picture of light is useful here:

Strength of the

scattering

depends on the

number of electrons

~ Z2 (Z is the atomic

number)

X-ray scattering by two (or several) atoms

Constructive interference in some places.

Destructive interference in others.

Two atoms: Several atoms:

From: C. Barret and T. B. Massalski, Structure of Metals, (1980).

X-ray diffraction from periodic arrangements of atoms

• Important Concept :X-

rays reflect from crystal

planes (only those that

scatter in-phase from

multiple planes yield

peaks)

• All “Peaks” in

Diffraction Satisfy

Bragg’s Law:

nl=2 d sin()

l=2 dhkl sin()

d sin(θ)

What does a lab diffractometer measure?

• Angles and X-ray intensities (counts)

additional degrees

of rotational freedom

“theta-two theta”

diffraction geometry

w

X-ray detectors

Would like to count single x-ray photons with high

dynamic range as quickly as possible

0-d

Traditional: Scintillation counter

- serial detector (slow)

- x-ray photon generates electron pulse

1-d

linear photo diode array –

can now count in parallel

2-d

photo plate (first x-ray detectors)

- not quantitative

wire array

charged coupled device (CCD) array

2-d photo diode array

Bruker d8 Vantec detector 2048 x 2048 pixel 14cm

active area

Panalytical Empyrean 255 x 255 diode array

Powder diffraction

Widely used –

• Phase identification

• Amorphous to crystalline ratio

Common industrial use: Quality

control (do I have the same mix)

Other uses:

• Grain size

• Film texture

• Stress measurement

www.mater.org.uk

Example of powder diffraction dataIn

tensity (

counts

)

2 (degrees)

Corundum

• Bruker d8 using 0.5 mm collimator

• 3 minute acquisition time

Phase identification

The diffraction pattern for a

particular phase is unique

• Phases with the same

composition can have

drastically different diffraction

patterns

• The peak positions and

relative intensities are

compared with reference

patterns in a database

http://prism.mit.edu/xray/oldsite/tutorials.htm

The scattering from a mixture is a simple sum the scattering from each component

phase (reference to a standard, as different compositions scatter more or less strongly)

Note: The amorphous to crystalline ratio is determined from relative

intensities (each phase is SiO2)

Example: Mixture of SiO2 phases

Quantification: Phases with different compositions

RIR calcite

[CaCO3] = 3.45

RIR dolomite

[CaMg(CO3)2] = 2.51

Crystallite size determination

Crystallites smaller than ~100nm broaden diffraction peaks• Analyze peak width with the Scherrer equation

• Must include instrument broadening

Microstrain may also broaden peaks but can be separated out by measuring peak

width over a wide 2 range

B(2) = K l/[t cos()],

B is the peak full width at half

maximum (radians), K is a shape

factor (0.8-1.2), t is the crystallite

size and l the wavelength

Texture: Best observed with an area detector

Thousands of crystalline grains are sampled

• Intensity in preferred directions shows the

orientations are not random (from the

deposition process or cold working)

• 2d detector with point source shows texture

directly

Stress can be inferred by measuring strain

Macrostrain determination in a polycrystalline sample

Look a at a high 2 angle hkl peak position at

different angles y with respect to the surface

normal

y

Residual stress using the sin2y method

https://mrl.illinois.edu/sites/default/files/pdfs/Workshop08_X-ray_Handouts.pdf

Single-crystal diffraction: requires high-resolution

• Obtain crystal structure and

orientation

• Measure crystal symmetry, lattice

constants and defects

• In epitaxial film growth

– Determine strain (film relaxation),

crystal mosaic, and film thickness

Requires accurate control of the sample orientation.

To satisfy Braggs law, the incident beam and the

detector have to be located precisely.

Panalytical Empyrean for high-resolution measurements

Hybrid monochromator:

curved multilayer mirror

coupled with 4-bounce

Ge(220) crystal

Sample stage moves in x, f, and chi

Pixcel detector

for fast mapping

Channel-cut analyzer

crystal with

12 Arc-second

acceptance angle

High-resolution X-ray analysis

SiGe deposited on Si(001)

Thickness 79 nm

Alloy composition Si80.5 Ge19.5

Si(004)

thickness

SiGe (004)

63 period InGaAs/InAlAs

deposited on InP (001)

4.47 nm In79Ga19As

3.91 nm In24.3Al75.7As

SL periodFits assume 100% coherent growth

Introduction to reciprocal space and the Ewald construction

Reciprocal lattice vectors• perpendicular to crystal planes

• spaced = 2pn/d hkl

Ewald construction links the

experiment to the lattice with

q (the scattering vector)

When q (the scattering vector) is

centered on a reciprocal lattice

point, Braggs law is satisfied

Possible ways to navigate in reciprocal space

Q =kf - ki

Why use reciprocal space mapping?

The relative positions of Bragg

peaks allow one to determine the

degree of relaxation (coherency)

Maps can take a long time to

acquire

Reciprocal space maps of epitaxial SiGe

(-2-2 4)(-2-2 4)

Ultra-fast reciprocal space mapping

(-2-2 4) reciprocal space map of SiGe on Si

Acquired in 3 minutes

Uses 255 lines of diodes at different 2 values

In parallel during an w-2 scan

X-ray reflectivity

Near surface and interface information

Density

Porosity

Film thickness

Surface and interface roughness

Works for amorphous films as well as crystalline

X-ray reflectivity

Contrast mechanism is differing refractive indices

(electron densities)

Film thickness measurements from 2nm - 300nm

Simulation and fitting: Determine interface

roughness and film porosity

Log inte

nsity

X-ray reflectivity information content

X-ray reflectivity from a thin layer

X-ray reflectivity data fitting

SAX (small angle x-ray scattering)

To look at larger periodic

structures or particle sizes,

look close to the incident

beam.

• Use transmission

• Cu radiation

• Need a vacuum to

reduce air scatter

Rigaku SAX system

Fixed area detector

10 cm with 1024 pixel

diameter

PIN diode on beam

stop measures beam

transmission

Sample to detector

distance 2 meters

Sample heating to 350 C

Cu Ka

micro source

Bruker d8 in SAX mode

Use when higher angles are

needed

Sample to detector distance

from 15 to 33.6 cm

Beam stop to block direct

transmitted x-ray beam

Sample heating to 350 C

SAX measurements from silver behenate

Log I

nte

nsity (

cps)

q (inverse Angstroms)

Log I

nte

nsity (

cps)

q (inverse Angstroms)

Rigaku Sax

q ~ 0.08 to 1.2 nm -1

d ~ 80 nm to 5 nm

Bruker d8

q ~ 0.4 to 7.2 nm -1

d ~ 16 nm to 0.9 nm

Smaller d possible

by moving the

detector closer

Some SAX applications

• Block copolymer ordering

• Nanoparticle size and distribution

• DNA in solution

X-ray diffraction summary

Diffraction is ideally suited for looking at order

in materials

Polycrystalline samples: Phase determination,

stress, grain size, and texture

Single-crystal diffraction: Epitaxial coherency,

mosaic spread, film thickness, and strain

Bruker d8

X-ray reflection and SAX

Crystallinity not needed

XRR of thin films: Thickness, density, and

interface roughness

SAX: Particle size (average) and long-range

domain ordering

Panalytical Empyrean