Fundamentals of X-ray diffraction and scattering Don Savage [email protected] 1231 Engineering Research Building (608) 263-0831
Fundamentals of X-ray
diffraction and scattering
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