X-Ray diffraction by single-crystal

3D registration of the reciprocal space associated with the structure of a crystalline material to
measure the intensities of hundreds or thousands of reflections produced by the interaction of
an incident beam (X-rays, neutrons, electrons) with the single-crystal under study.

• Used to determine:

-crystal structure

- orientation

– degree of crystalline perfection/imperfections

• Sample is illuminated with monochromatic radiation

– The sample axis, phi, and the goniometer axes omega and 2θ are rotated to capture
diffraction spots from at least one hemisphere

– Easier to index and solve the crystal structure because it diffraction peak is uniquely
resolved

• X-rays striking a single crystal will produce diffraction spots in a sphere around the crystal.

– Each diffraction spot corresponds to a single (hkl)

– The distribution of diffraction spots is dependent on the crystal structure and the
orientation of the crystal in the diffractometer

– The diffracting condition is illustrated with the Ewald sphere in reciprocal space

• The design challenge for single crystal diffractometers: how to determine the position and
intensity of these diffraction spots

– Reflection vs transmission • Transmission: small samples & organic crystals •

Reflection: large samples, epitaxial thin films

– Laue vs. SCD • Laue: stationary sample bathed with white radiation (i.e. many
wavelengths) • SCD: monochromatic radiation hits a sample as it is rotated and
manipulated to bring different planes into diffracting condition

Powder X-ray diffraction

One-dimensional registration of the reciprocal space associated with the structure of a

crystalline material to measure the intensities of a few maxima produced by the superposition
of the reflections that originate through the interaction of an incident beam (X-rays, neutrons,
electrons) with the polycrystalline sample under study. A real powder consists of so many grains
that the dots of the reciprocal lattice form into continuous lines.

XRPD uses information about the position, intensity, width, and shape of diffraction peaks in a
pattern from a polycrystalline sample

The atoms in a crystal are a periodic array of coherent scatters and thus can diffract light
• Diffraction occurs when each object in a periodic array scatters radiation coherently, producing
concerted constructive interference at specific angles.

• The electrons in an atom coherently scatter light. – The electrons interact with the oscillating
electric field of the light wave.

• Atoms in a crystal form a periodic array of coherent scatters.

– The wavelength of X rays are similar to the distance between atoms.

– Diffraction from different planes of atoms produces a diffraction pattern, which

contains information about the atomic arrangement within the crystal

• X Rays are also reflected, scattered incoherently, absorbed, refracted, and transmitted when
they interact with matter.

Bragg’s law is a simplistic model to understand what conditions are required for diffraction

For parallel planes of atoms, with a space dhkl between the planes, constructive interference
only occurs when Bragg’s law is satisfied. – In many diffractometers, the X-ray wavelength l is
fixed. – Consequently, a family of planes produces a diffraction peak only at a specific angle q. –
Additionally, the plane normal must be parallel to the diffraction vector • Plane normal: the
direction perpendicular to a plane of atoms • Diffraction vector: the vector that bisects the angle
between the incident and diffracted beam  The space between diffracting planes of atoms
determines peak positions.  The peak intensity is determined by what atoms are in the
diffracting plane.

A SC specimen in a Bragg-Brentano diffractometer would produce only one family of peaks in

the diffraction pattern

A powder sample should contain thousands of crystallites. Therefore, all possible diffraction
peaks should be observed.

For every set of planes, there will be a small percentage of crystallites that are properly
oriented to diffract. Basic assumptions of powder diffraction are that for every set of
planes there is an equal number of crystallites that will diffract and that there is a
statistically relevant number of crystallites, not just one or two.

Powder diffraction data consists of a record of photon intensity versus detector angle 2θ

Diffraction patterns are best reported using dhkl and relative intensity rather than 2θ and
absolute intensity.

• The peak position as 2θ depends on instrumental characteristics such as wavelength.

– The peak position as dhkl is an intrinsic, instrument-independent, material property.

• Bragg’s Law is used to convert observed 2θ positions to dhkl.

• The absolute intensity, i.e. the number of X rays observed in a given peak, can vary due to
instrumental and experimental parameters.

– The relative intensities of the diffraction peaks should be instrument independent.

• To calculate relative intensity, divide the absolute intensity of every peak by the absolute
intensity of the most intense peak, and then convert to a percentage. The most intense peak of
a phase is therefore always called the “100% peak”.

– Peak areas are much more reliable than peak heights as a measure of intensity

The diffraction pattern for every phase is as unique as your fingerprint

• Phases with the same chemical composition can have drastically different diffraction patterns.
• While every diffraction pattern is different, some are very similar.

Instrumentation necessary for a diffraction experiment

High-voltage generator.

X-ray sources

Filters, monochromators and collimators

Detectors (Photographic films, electronic systems

X-rays with true 2D detectors: imaging plates, CCD cameras, multi-wires etc.

• A true 2D detector can intercept complete cones of diffracted radiation and very efficiently
record the diffraction pattern

Fast data acquisition, but not very high resolution Maximum 2θ that is readily achievable is often
quite limited

Debye rings from the 2D detector are integrated and converted into a conventional powder
pattern using FIT 2D or similar software X-ray beam size, detector pixel size and sample thickness
combine to limit the effective resolution of the data

Why use 2D detectors?

• Rapid acquisition of data from normal sized samples for time resolved or parametric studies –
Seconds/minutes per pattern

• Reasonable signal to noise and sampling statistics can be achieved even with very small
samples such as those used in high pressure diamond anvil cell experiments

1D detector: Debye-Scherrer camera

• Can record sections on these cones on film or some other x-ray detector – Simplest way of
doing this is to surround a capillary sample with a strip of film – Can covert line positions on film
to angles and intensities by electronically scanning film or measuring positions using a ruler and
guessing the relative intensities using a “by eye” comparison

• 1D position sensitive detectors based on many different types of technology are available. –
Fast data collection, but not as efficient as a 2D detector – But access to high 2θ by curving the
detector

Características importantes de las muestras para XRPD

• A flat sample by XRDP could have a smooth flat surface – If the surface is not smooth and, the
absorption of the X-rays can reduce the intensity of the peaks at low angle – The geometric array
of parallel beam can be used to analyze samples with strange shapes or rough surface

• Randomly oriented grains / crystallites

• Thickly packaged

• Grain size less than 10 microns

• ‘Infinitely’ thick