X-ray diffraction

X-ray scattering is

a non-destructive analytical technique

X-ray scattering gives information

about crystallographic structure, chemical composition, physical properties of materials and thin films

When x-rays interact with matter, they are

scattered
reflected
scattered incoherently
absorbed
refracted
transmitted

XRD is based on

scattering of x-rays on electron cloud of atoms

Diffraction occurs when

each object on a periodic array scatters radiation coherently, producing constructive interference at specific angles

Requirements for XRD

wavelength of incident radiation is approximately equal to lattice parameters in the crystal (100 pm or 1 Angstrom)

radiation must not destroy the sample

radiation absorption must be very small

X-rays production at the interaction between an electron beam and the matter

bremsstrahlung

characteristic x-rays

Bremsstrahlung

radiation which is emitted when electrons are decelerated or “braked” when they are fired at a metal target (continuous spectrum)

Characteristic x-rays

produced when electrons from higher energy levels fill core holes

Bragg’s Law describes

the conditions for constructive interference of X-rays diffracted by a crystal lattice, stating that 2dsin(θ) = nλ, where d is the interplanar spacing, θ is the angle of incidence, n is an integer, and λ is the X-ray wavelength

The space between diffracting planes of atoms determines

peak positions

The peak intensity is determined by

what atoms are in the diffracting plane

Laue method is mainly used

to determine the orientation of large single crystals

Laue method

white radiation (all wavelength, fixed angle) is reflected from, or transmitted through, a fixed crystal

(Laue method) reflection only occurs for

specific combinations of wavelength and angle

(Laue method) crystal orientation is determined from

position of the spots

Incident angle

angle between x-ray source and the sample

Diffracted angle

angle between the incident beam and detector angle

The incident angle is always … of the detector angle

half

Bregg-Brentano geometry

both X-ray tube and the detector can move

Grazing-incidence X-ray diffraction

X-ray tube has a fixed position

(Grazing-incidence X-ray diffraction) x-rays are focused only at surface of the sample by

fixing the incident angle at a very small value (less than 5 degrees) which decreases penetration depth

Grazing-incidence X-ray diffraction can

perform many analysis possible with XRD + can resolve info as a function of depth (depth-profiling) by collecting successive diffraction patterns with varying incident angles

Grazing-incidence X-ray diffraction examples of measurements

orientation of thin film with respect to substrate

lattice mismatch between film and substrate

epitaxy/texture

macro- and microstains

reciprocal space map

X-ray reflectivity (XRR)

a varying and small (grazing) incident angle, combined with a matching detector angle collects the X-rays reflected from the sample surface

(XRR) Interference fringes in the reflected signal can be used to determine

thickness of film layers

density and composition of thin film layers

roughness of films and interfaces

XRD can be used for

phrase identification (quantitative phase analysis)

crystalline symmetry group and unit cell lattice parameters

residual strain (macrostrain)

epitaxy

texture

orientation

crystallite size and macrostain

Phase Identification

measures the relative amounts of phases in a mixture by analyzing the relative peak intensities

Diffraction pattern for every phase is

unique

Same chemical composition can show … diffraction patterns due to … phases

different

(phase identification) position and relative intensity of a series of peaks is

compared with the reference pattern from a database

(phase identification) a single crystal shows

one family of peaks in the diffraction patterns
(for other angles planes are not properly aligned to diffract)

(phase identification) A polycrystalline materials shows

all possible peaks due to diffraction on various crystallites

(for every set of planes, a small percentage of crystallites are properly oriented to diffract)

Basic assumptions for powder XRD

for every set of planes there is an equal number of crystallites that will diffract

there is a statistically relevant number of crystallites

XRD database

Powder Diffraction File

Quantitative Phase Analysis

With a high signal to noise ratio, the amount of specific phase present can be accurately determined

Quantitative Phase Analysis relationship

linear relationship between ratio of peak intensities and weight fractions of any two phases in a mixture

Reference Intensity Ratio (RIR) method

gives fast semi-quantitative results

Whole pattern fitting (Rietveld refinement)

possible, more accurate, more complicated analysis

Crystallite Size and Microstrain

broadening of diffraction peaks indicates crystallites smaller than 120 nm and microstrains

Broadening is

used to identify the average crystallite size of nanoparticles using Scherrer equation

Calibration curve shows

contribution of the instrument to the broadening

High resolution XRD of a single peak

allows separating microstrain and crystallite size by using a Williamson-Hull plot

Preferred orientation (texture)

Systematic variation in diffraction peak intensities can be caused by preferred orientation of crystallites

Texture quantification

a pole figure maps the intensity of a single peak as a function of tilt and rotation of the sample

X-ray absorption

X-rays are absorbed by all matter through the photoelectric effect

X-ray absorption steps

An X-ray is absorbed by an atom, promoting a core-level electron out of the atom

The atom is left in an ionized state with an empty electronic level (a core hole)

The electron ejected from the atom is called the photoelectron

X-ray fluorescense (XRF)

An x-ray photon with energy equal to the difference in the core levels is emitted

(XRF) The photon energy is

characteristic to the atom

XRF can be used for

quantitative composition analysis of materials

X-ray Absorption Fine-Structure (XAFS) or X-ray absorption spectroscopy (XAS)

modulation of x-ray absorption coefficient at energies near and above an x-ray absorption edge

XAS is split into 2 regimes

XANES X-ray absorption Near-Edge Spectroscopy

EXAFS Extended X-ray Absorption Fine-structure (info about local coordination and chemical state of element)

XAFS scharacteristics

local atomic coordination

chemical/oxidation state

applies to any element

works at low concentrations

minimal sample requirements

(XANES) Edge position shift is used

for detecting oxidation states

(XANES) Coordination chemistry

for ions with partially filled d shells, the p-d hybridization changes significantly due to distortion of regular octahedra

very large change for tetrahedral coordination

leads to a dramatic pre-edge peak due to absorption to a localised electronic state

Synchrotron

a circular particle accelerator

Synchrotron mechanism

electrons are accelerated in very strong electric fields

trajectories are bent into a circle using dipole bend magnets

electron beams are focused with quadrupole and sextupole magnets

electrons circulate very close to speed of light

wherever the path bends, velocity vector changes

this acceleration causes electrons to produce EM radiation

Synchrotron X-ray production

array of magnets of alternating polarity between which the beam travels

alternating magnetic field causes the path of electrons to wiggle back and forth

acceleration causes emission of radiation at each pole (50-100 poles)

unlike bend magnets, ID properties can be chosen to optimise beam specifically for experiments

wavelength range 0.01 nm to 1 micrometer

Synchrotron radiation is

far more intense that normal lab sourxces

(Synchrotron radiation) brilliance is

much greater than in other sources

(Synchrotron radiation) light comes

in rapid pulses (useful for time resolution)

Synchrotron radiation is

naturally collimated in vertical plane

well-matched to crystal monochromators