Fundamentals of X-ray diffraction and scattering

Don SavageStaff member: In the Nanoscale Imaging and

Analysis Center (NIAC)

dsavage@wisc.edu1231 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 = hν =hc/λ, V (volts) =1239.8/λ(nm)

Copper is a common anode choice

8.048 EV for Cu Kα

Cu Kα2 λ = 0.1544390 nmCu Kα1 λ = 0.1540562 nm

Lab sources

Point source – Useful with area detector

Bruker d-8 – Conditioning Opticscrossed multilayer mirrors and point collimator yields a Cu Kα parallel point beam with diameters from 0.1 to 2.0 mm

Line source – Useful when you have a large uniform sample (e.g., for a perfect crystal or uniform smooth film)

Panalytical – Conditioning optics• Multilayer mirror and channel cut crystal - Cu Kα1• Multilayer mirror only – Cu Kα• Slits only

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 scatteringdepends 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: nλ=2 d sin(θ)λ=2 dhkl sin(θ)

d sin(θ)

What does a lab diffractometer measure?

• Angles and X-ray intensities (counts)

additional degrees of rotational freedom

“theta- 2theta” diffraction geometry

ω

X-ray detectors

Would like to count single x-ray photons with high dynamic range as quickly as possible

0-dTraditional: Scintillation counter- serial detector (slow)- x-ray photon generates electron pulse

1-dlinear photo diode array –can now count in parallel

2-dphoto plate (first x-ray detectors)- not quantitativewire arraycharged coupled device (CCD) array2-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

tens

ity (c

ount

s)

2 θ (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

RIR – reference intensity ratio

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 2θ range

B(2θ) = K λ/[t cos(θ)],

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 λ the wavelength

Texture: Best observed with an area detectorThousands 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 2θ angle hkl peak position at different angles ψ with respect to the surface normal

ψ

Residual stress using the sin2ψ 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, φ, and chi

Pixcel detector for fast mapping

Channel-cut analyzer crystal with12 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)

thicknessSiGe (004)

63 period InGaAs/InAlAs deposited on InP (001)

4.47 nm In79Ga19As3.91 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 = 2πn/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, Bragg’s law is satisfied

• k0 is in the direction of incoming x-ray• k1 is the direction of the diffracted beam

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 2θ valuesIn parallel during an ω−2θ 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: ALD of Alumina on Sapphire

As deposited:density= 2.95 gm/cm2

thickness = 114 nm

After 1050 C anneal for 2 hrsdensity= 3.4 gm/cm2

thickness = 96 nm

Smaller critical angle means lower density

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 detector10 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 Kαmicro 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

Inte

nsity

(cps

)

q (inverse Angstroms)

Log

Inte

nsity

(cps

)

q (inverse Angstroms)

Rigaku Saxq ~ 0.08 to 1.2 nm -1d ~ 80 nm to 5 nm

Bruker d8q ~ 0.4 to 7.2 nm -1d ~ 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

Nayomi Plaza will present recent work using SAX on:

Moisture-induced changes in the cellulose crystalline structure of wood cell walls

X-ray diffraction summary

Diffraction is ideally suited for looking at order in materialsPolycrystalline 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