Transcript Slide 1

X RAY DIFFRACTION- XRD
SOLID MATTERAMORPHOUS:
Atoms arranged in a random manner , like in liquids- eg: Glass
CRYSTALLINE:
Atoms arranged in a regular pattern. Smallest volume element
repeats in three dimensions describing the crystal. The
smallest volume element is UNIT CELL. Dimensions of the unit
cell described by the edges a,b, and c and the angles between
them alpha, beta and gamma.
X - RAYS
• German scientist Rontgen discovered X-rays
in 1895 accidentally when working with
discharge tube.
• Barium platinocyanide screen placed near
the tube began to glow, Glow continued
even when a wooden screen was placed
between them.
• As cause was not known, called as X-rays.
• It could pass through opaque bodies. Wave
length shorter than that of ultraviolet light.
Essential elements of a coolidge X- ray vacuum tube:
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Cathode- tungsten filament heated to incandescence by a low voltage AC
from a step down transformer/ storage battery.
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Emits large number of electrons known as thermions focused on a target
using cylindrical shields (molybdenum)
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Shield maintained at a negative potential surrounding the cathode.
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Electrons accelerated to very high speeds by DC potential difference about
50kV to 100kV applied between cathode and anode (anticathode). The high
DC from a step up transformer.
Electrons
Tungsten filament
Shield
Cooling water
X rays
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The Coolidge tube (1913) is also called hot cathode tube
It works with a very good quality vacuum (about 10-4 Pa, or 10-6 Torr).
The filament is the cathode of the tube. The high voltage potential is between the
cathode and the anode, the electrons are accelerated and then hit the anode.
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There are two designs: end-window tubes and side-window tubes.
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In the end-window tubes, the filament is around the anode, the electrons have a
curved path.
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Special about side-window tubes is:
An Electrostatic lens focuses the beam onto a very small spot on the anode
The anode is specially designed to dissipate the heat and wear resulting from this
intense focused barrage of electrons:
– Mechanically spun to increase the area heated by the beam.
– Cooled by circulating coolant.
The anode is precisely angled at 1-20 degrees off perpendicular to the electron
current so as to allow escape of some of the X-ray photons which are emitted
essentially perpendicular to the direction of the electron current.
The anode is usually made out of tungsten or molybdenum.
The tube has a window designed for escape of the generated X-ray photons.
The power of a Coolidge tube usually ranges from 1 to 4 kW
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Introduction to X-ray Diffraction
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References:
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Elements of Modern X-ray Physics, Jens Als-Nielsen and Des McMorrow, John Wiley &
Sons, Ltd., 2001
(Modern x-ray physics & new developments)
• X-ray Diffraction, by B.E. Warren, General Publishing Company, 1969, 1990
(Classic x-ray physics book)
• Elements of X-ray Diffraction,2nd Ed., by B.D. Cullity, Addison-Wesley, 1978
(Covers most techniques used in traditional material characterization)
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High Resolution X-ray Diffractometry and Topography, by D. Keith Bowen and Brian K.
Tanner, Taylor & Francis, Ltd., 1998
(Semiconductors and thin film analysis)
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Modern Aspects of Small-Angle Scattering, by H. Brumberger, Editor, Kluwer Academic
Publishers, 1993
(SAXS techniques)
• Principles of Protein X-ray Crystallography, by Jan Drenth, Springer, 1994
(Crystallography)
The incoming beam (coming from upper left) causes each
scatterer to re-radiate a small portion of its energy as a
spherical wave.
If scatterers are arranged symmetrically with a separation
d, these spherical waves will be in synch only in
directions where their path-length difference 2 d sin θ
equals an integer multiple of the wavelength λ.
In that case, part of the incoming beam is deflected by an
angle 2θ, producing a reflection spot in the diffraction
pattern
An intuitive understanding of XRD can be
obtained from the Bragg Model of Diffraction.
• In this model, a given reflection is associated with a set of
evenly spaced sheets running through the crystal, usually
passing through the centers of the atoms of the crystal
lattice.
• The orientation of a particular set of sheets is identified by
its three MILLER INDICES (h, k, l), and let their spacing
be noted by d.
• WILLIAM LAWARENCE BRAGG proposed a model in
which the incoming X-rays are scattered specularly
(mirror-like) from each plane; from that assumption, Xrays scattered from adjacent planes will combine
constructively when the angle θ between the plane and
the X-ray results in a path-length difference that is an
integer multiple n of the X-ray wave length λ.
• A reflection is said to be indexed when its
Miller indices have been identified from the
known wavelength and the scattering angle
2θ. Such indexing gives the unit cell
parameters, the lengths and angles of the
unit-cell, as well as its space group. Since
BRAGG’S LAW does not interpret the
relative intensities of the reflections,
however, it is generally inadequate to solve
for the arrangement of atoms within the
unit-cell; for that, a Fourier transform
method must be carried out.
BRAGG’S LAW
Theoretical Considerations
An X-ray diffraction pattern formed when
X-rays are focused on a crystalline
material, (a protein).
Each dot, called a reflection, forms from
the coherent interference of scattered Xrays passing through the crystal.
X-ray scattering techniques are a
family of non-destructive analytical
techniques which reveal information
about the crystallographic structure,
chemical composition, and physical
properties of materials and thin films.
These techniques are based on
observing the scattered intensity of an XRAY beam hitting a sample as a function
of incident and scattered angle,
polarization, and wavelength or energy.
X-ray diffraction techniques
• X-ray diffraction finds the geometry or shape of a molecule
using x-rays. X-ray diffraction techniques are based on the
elastic scattering of x-rays from structures that have long
range order.
• Single-crystal X-ray diffraction is a technique used to solve the
complete structure of crystalline materials, ranging from simple
inorganic solids to complex macromolecules, such as proteins.
• Powder diffraction (XRD) is a technique used to characterize
the crystallographic structure, crystallite size (grain size), and
preferred orientation in polycrystalline or powdered solid
samples. Powder diffraction is commonly used to identify
unknown substances, by comparing diffraction data against a
database maintained by the International Centre for Diffraction
Data. It may also be used to characterize heterogeneous solid
mixtures to determine relative abundance of crystalline
compounds and, when coupled with lattice refinement
techniques, such as Rietveld refinement, can provide
structural information on unknown materials. Powder
diffraction is also a common method for determining strains in
• Thin film diffraction and grazing incidence x-ray
diffraction may be used to characterize the
crystallographic structure and preferred orientation
of substrate-anchored thin films.
• High-resolution x-ray diffraction is used to
characterize thickness, crystallographic structure,
and strain in thin epitaxial films. It employs
parallel-beam optics.
• X-ray pole figure analysis enables one to analyze
and determine the distribution of crystalline
orientations within a crystalline thin-film sample.
• X-ray rocking curve analysis is used to quantify
grain
Scattering techniques
Elastic scattering
• Materials that do not have long range order may also be studied by
scattering methods that rely on elastic scattering of monochromatic x-rays.
• Small angle X-ray scattering (SAXS) probes structure in the nanometer to
micrometer range by measuring scattering intensity at scattering angles 2θ
close to 0°.
• X-ray reflectivity is an analytical technique for determining thickness,
roughness, and density of single layer and multilayer thin films.
• Wide angle X-ray scattering (WAXS), a technique concentrating on
scattering angles 2θ larger than 5°.
Inelastic scattering
• When the energy and angle of the inelastically scattered x-rays are
monitored scattering techniques can be used to probe the electronic band
structure of materials.
• Compton scattering
• Resonant inelastic x-ray scattering (RIXS)
• X-ray Raman scattering
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X-ray Generation & Properties
Lattice Planes and Bragg's Law
Powder Diffraction
Thin Film Diffraction
Texture Measurement (Pole Figures)
Residual Stress Measurements
Small Angle X-ray Scattering (SAXS)
X-ray Crystallography
1. X-ray Generation & Properties
• X-rays are electromagnetic radiation with typical photon energies in the
range of 100 eV - 100 keV. For diffraction applications, only short
wavelength x-rays (hard x-rays) in the range of a few angstroms to 0.1
angstrom (1 keV - 120 keV) are used.
• Because the wavelength of x-rays is comparable to the size of atoms,
they are ideally suited for probing the structural arrangement of atoms
and molecules in a wide range of materials. The energetic x-rays can
penetrate deep into the materials and provide information about the bulk
structure.
• X-rays are produced generally by either x-ray tubes or synchrotron
radiation. In a x-ray tube, which is the primary x-ray source used in
laboratory x-ray instruments, x-rays are generated when a focused
electron beam accelerated across a high voltage field bombards a
stationary or rotating solid target. As electrons collide with atoms in the
target and slow down, a continuous spectrum of x-rays are emitted,
which are termed Bremsstrahlung radiation. The high energy electrons
also eject inner shell electrons in atoms through the ionization process.
When a free electron fills the shell, a x-ray photon with energy
characteristic of the target material is emitted.
• Common targets used in x-ray tubes include Cu and Mo,
which emit 8 keV and 14 keV x-rays with corresponding
wavelengths of 1.54 Å and 0.8 Å, respectively. (The
energy E of a x-ray photon and it's wavelength is related
by the equation E = hc/l, where h is Planck's constant
and c the speed of light)
• In recent years synchrotron facilities have become
widely used as preferred sources for x-ray diffraction
measurements. Synchrotron radiation is emitted by
electrons or positrons travelling at near light speed in a
circular storage ring. These powerful sources, which are
thousands to millions of times more intense than
laboratory x-ray tubes, have become indispensable tools
for a wide range of structural investigations and brought
advances in numerous fields of science and technology.
2. Lattice Planes and Bragg's Law
• X-rays primarily interact with electrons in atoms. When xray photons collide with electrons, some photons from the
incident beam will be deflected away from the direction
where they original travel, much like billiard balls bouncing
off one anther. If the wavelength of these scattered x-rays
did not change (meaning that x-ray photons did not lose
any energy), the process is called elastic scattering
(Thompson Scattering) in that only momentum has been
transferred in the scattering process. These are the x-rays
that we measure in diffraction experiments, as the
scattered x-rays carry information about the electron
distribution in materials. On the other hand, In the inelastic
scattering process (Compton Scattering), x-rays transfer
some of their energy to the electrons and the scattered xrays will have different wavelength than the incident xrays.
• Diffracted waves from different atoms can
interfere with each other and the resultant
intensity distribution is strongly modulated
by this interaction. If the atoms are
arranged in a periodic fashion, as in
crystals, the diffracted waves will consist of
sharp interference maxima (peaks) with the
same symmetry as in the distribution of
atoms. Measuring the diffraction pattern
therefore allows us to deduce the
distribution of atoms in a material.
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The peaks in a x-ray diffraction pattern are directly related to the atomic distances.
Consider an incident x-ray beam interacting with the atoms arranged in a periodic
manner as shown in 2 dimensions
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The atoms, represented as green spheres in the graph, can be viewed as forming
different sets of planes in the crystal (colored lines). For a given set of lattice plane
with an inter-plane distance of d, the condition for a diffraction (peak) to occur can be
written as
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known as the Bragg's law, after W.L. Bragg, who first proposed it.
n is an integer representing the order of the diffraction peak. The Bragg's Law is one
of most important laws used for interpreting x-ray diffraction data.
Here, atoms are used as scattering points in this example, Bragg's Law applies to
scattering centers consisting of any periodic distribution of electron density. Ie., the
law holds true if the atoms are replaced by molecules or collections of molecules,
such as colloids, polymers, proteins and virus particles
3. Powder Diffraction
• Powder XRD (X-ray Diffraction) is perhaps the
most widely used x-ray diffraction technique for
characterizing materials. As the name suggests,
the sample is usually in a powdery form,
consisting of fine grains of single crystalline
material to be studied. The technique is used
also widely for studying particles in liquid
suspensions or polycrystalline solids (bulk or
thin film materials).
• The term 'powder' really means that the crytalline
domains are randomly oriented in the sample.
Therefore when the 2-D diffraction pattern is
recorded, it shows concentric rings of scattering
peaks corresponding to the various d spacings in
the crystal lattice. The positions and the intensities
of the peaks are used for identifying the underlying
structure (or phase) of the material. For example,
the diffraction lines of graphite would be different
from diamond even though they both are made of
carbon atoms. This phase identification is
important because the material properties are
highly dependent on structure (just think of
graphite and diamond).
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Powder diffraction data can be collected using either transmission
or reflection geometry, as shown below.
Because the particles in the powder sample are randomly oriented,
these two methods will yield the same data.
In the MRL x-ray facility, powder diffraction data are measured
using the Philips XPERT MPD diffractometer, which measures data
in reflection mode and is used mostly with solid samples, or the
custom built 4-circle diffractometer, which operates in transmission
mode and is more suitable for liquid phase samples
MOUNTING THE CRYSTAL
DIFFRACTOMETER
A powder XRD scan from a
K2Ta2O6 sample is as
shown -as a plot of
scattering intensity v/s. the
scattering angle 2theta or
the corresponding dspacing.
The peak positions,
intensities, widths and
shapes all provide important
information about the
structure of the material.
4. Thin Film Diffraction
• Thin film diffraction refers not to a specific technique but rather a
collection of XRD techniques used to characterize thin film samples
grown on substrates. These materials have important technological
applications in microelectronic and optoelectronic devices, where high
quality epitaxial films are critical for device performance. Thin film
diffraction methods are used as important process development and
control tools, as hard x-rays can penetrate through the epitaxial layers
and measure the properties of both the film and the substrate.
• There are several special considerations for using XRD to characterize
thin film samples. (i) reflection geometry is used for these measurements
as the substrates are generally too thick for transmission. (ii) high angular
resolution is required because the peaks from semiconductor materials
are sharp due to very low defect densities in the material. Multiple
bounce crystal monochromators are used to provide a highly collimated
x-ray beam for these measurements.
Eg: in the Philips MRD used in the x-ray facility, a 4-crystal monochromator
made from Ge is used to produce an incident beam with less than 5 arc
seconds of angular divergence.
Basic XRD measurements made on thin film samples include:
• Precise lattice constants measurements derived from 2q - q scans, which
provide information about lattice mismatch between the film and the
substrate and therefore is indicative of strain & stress
• Rocking curve measurements made by doing a q scan at a fixed 2q
angle, the width of which is inversely proportionally to the dislocation
density in the film and is therefore used as a gauge of the quality of the
film.
• Superlattice measurements in multilayered heteroepitaxial structures,
which manifest as satellite peaks surrounding the main diffraction peak
from the film. Film thickness and quality can be deduced from the data.
• Glancing incidence x-ray reflectivity measurements, which can determine
the thickness, roughness, and density of the film. This technique does
not require crystalline film and works even with amorphous materials.
• Texture measurements-(discussed separately)
• The graph shows
the high resolution
XRD data of the
superlattice peaks
on the GaN (002)
reflections.
Red line denotes results of
computer simulation of
the structure.
5. Texture Measurement (Pole Figure)
• Texture measurements are used to determine the
orientation distribution of crystalline grains in a
polycrystalline sample. A material is termed textured if the
grains are aligned in a preferred orientation along certain
lattice planes. One can view the textured state of a
material (typically in the form of thin films) as an
intermediate state in between a completely randomly
oriented polycrystalline powder and a completely oriented
single crystal. The texture is usually introduced in the
fabrication process (e.g. rolling of thin sheet metal,
deposition,etc.) and affect the material properties by
introducing structural anisotropy.
• A texture measurement is also referred to as
a pole figure as it is often plotted in polar
coordinates consisting of the tilt and rotation
angles with respect to a given crytallographic
orientation. A pole figure is measured at a
fixed scattering angle (constant d spacing)
and consists of a series of f -scans (in- plane
rotation around the center of the sample) at
different tilt or Y -(azimuth) angles, as
illustrated below.
• The pole figure data are
displayed as contour plots
or elevation graphs with
zero angle in the center.
Below we show two pole
figure plots using the same
data set. An orientation
distribution function (ODF)
can be calculated using
the pole figure data.
6. Residual Stress Measurement
• Structural and residual stress in materials can be
determined from precision lattice constants
measurements. For polycrystalline samples high
resolution powder diffraction measurements
generally will provide adequate accuracy for stress
evaluation. For textured (oriented) and single
crystalline materials, 4-circle diffractometry is
needed in which the sample is rotated so that
measurements on multiple diffraction peaks can be
carried out. The interpretation of stress
measurement data is complicated and model
dependent. Consult the reference literature for more
details
7. Small Angle X-ray Scattering (SAXS)
• SAXS measurements typically are concerned with
scattering angles < 1o. As dictated by Bragg's Law, the
diffraction information about structures with large dspacings lies in the region. Therefore the SAXS technique
is commonly used for probing large length scale
structures such as high molecular weight polymers,
biological macromolecules (proteins, nucleic acids, etc.),
and self-assembled superstructures (e.g. surfactant
templated mesoporous materials).
• SAXS measurements are technically challenging because
of the small angular separation of the direct beam (which
is very intense) and the scattered beam. Large specimento-detector distances (0.5 m - 10 m) and high quality
collimating optics are used to achieve good signal-tonoise ratio in the SAXS measurement.
• The MRL x-ray facility has cutting edge
capabilities for SAXS measurements with
three custom-built SAXS instruments
including one 3.5-meter long ultra-small
angle SAXS instrument with state-of-theart optics and area detector for low
scattering density samples (see
instrumentation section for more details)
8. X-ray Crystallography
• X-ray crystallography is a standard technique for solving
crystal structures. Its basic theory was developed soon
after x-rays were first discovered more than a century
ago. However, over the years it has gone through
continual development in data collection instrumentation
and data reduction methods. In recent years, the advent
of synchrotron radiation sources, area detector based
data collection instruments, and high speed computers
has dramatically enhanced the efficiency of
crystallographic structural determination. Today x-ray
crystallography is widely used in materials and biological
research. Structures of very large biological machinery
(e.g. protein and DNA complexes, virus particles) have
been solved using this method.
• In x-ray crystallography, integrated intensities of
the diffraction peaks are used to reconstruct the
electron density map within the unit cell in the
crystal. To achieve high accuracy in the
reconstruction, which is done by Fourier
transforming the diffraction intensities with
appropriate phase assignment, a high degree of
completeness as well as redundancy in diffraction
data is necessary, meaning that all possible
reflections are measured multiple times to reduce
systematic and statistical error. The most efficient
way to do this is by using an area detector which
can collect diffraction data in a large solid angle.
The use of high intensity x-ray sources, such as
synchrotron radiation, is an effective way to
reduce data collection time.
• One of the central difficulties in structural
determination using x-ray crystallography is
referred to as the "phase problem", which arises
from the fact that the diffraction data contains
information only on the amplitude but not the
phase of the structure factor. Over the years
many methods have been developed to deduce
the phases for reflections, including
computationally based direct methods,
isomorphous replacement, and multi-wavelength
anormalous diffraction (MAD) methods.
X-ray crystallography
Procedure
• The technique of single-crystal X-ray
crystallography has three basic steps. The first
— and often most difficult — step is to obtain an
adequate crystal of the material under study.
The crystal should be sufficiently large (typically
larger than 100 micrometres in all dimensions),
pure in composition and regular in structure, with
no significant internal imperfections such as
cracks or twinning. A small or irregular crystal
will give fewer and less reliable data, from which
it may be impossible to determine the atomic
arrangement.
• In the second step, the crystal is placed in an
intense beam of X-rays, usually of a single
wavelength (monochromatic X-rays), producing
the regular pattern of reflections. As the crystal is
gradually rotated, previous reflections disappear
and new ones appear; the intensity of every spot
is recorded at every orientation of the crystal.
Multiple data sets may have to be collected, with
each set covering slightly more than half a full
rotation of the crystal and typically containing
tens of thousands of reflection intensities.
• In the third step, these data are combined
computationally with complementary
chemical information to produce and refine
a model of the arrangement of atoms
within the crystal. The final, refined model
of the atomic arrangement — now called a
crystal structure — is usually stored in a
public database.
• A real 3-dimensional
crystal contains many sets
of planes. For diffraction,
crystal must have the
correct orientation with
respect to the incoming
beam.
• Perfect, infinite crystal and
perfectly collimated beam:
diffraction condition must
be satisfied ``exactly.''
• Strains, defects, finite size
effects, instrumental
resolution: diffraction
peaks are broadened.
More formally, the scattered
intensity is proportional to
the square of the Fourier
transform of the charge
density:
where
is the charge density.
For perfect crystals, I(q) consists of
delta functions (perfectly sharp
scattering). For imperfect crystals,
the peaks are broadened. For
liquids and glasses, it is a
continuous, slowly varying function
Features of Electron, X-ray, or
Neutron Diffraction
• For a known structure, pattern can be calculated exactly.
• Symmetry of the diffraction pattern given by symmetry of
the lattice.
• Intensities of spots determined by basis of atoms at each
lattice point.
• Sharpness and shape of spots determined by perfection
of crystal.
• Liquids, glasses, and other disordered materials produce
broad fuzzy rings instead of sharp spots.
• Defects and disorder in crystals also result in diffuse
scattering.
The ``Ultimate'' (Technically
Challenging) Experiment
• Sample is tiny (micron-sized).
• The effect is weak (light elements, small modulations,
subtle modifications of the long-range order).
• Instrumental resolution (angle and energy) is ``perfect''
allowing detailed measure- ments of structural disorder.
• Measurement is time-resolved (nanosecond time scale).
• To achieve all of the above, will need lots of intensity in
the primary beam together with sensitive detection
systems.
Powder vs. Single Crystal X-ray Diffraction
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SINGLE CRYSTAL
Put a crystal in the beam, observe what reflections
come out at what angles for what orientations of the
crystal with what intensities.
Advantages
In principle you can learn everything there is to
know about the structure.
Disadvantages
You may not have a single crystal. It is timeconsuming and difficult to orient the crystal. If more
than one phase is present, you will not necessarily
realize that there is more than one set of reflections.
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POWDER
Samples consists of a collection of many small
crystallites with random orientations. Average over
crystal orientations and measure the scattered
intensity as a function of outgoing angle.
Disadvantage
Inversion of the measured intensities to find the
structure is more difficult and less reliable.
Advantages
It is usually much easier to prepare a powder
sample. You are guaranteed to see all reflections.
The best way to follow phase changes as a
function of temperature, pressure, or some other
variable.
Overview of single-crystal X-ray
diffraction
• The oldest and most precise method of X-ray
crystallography is single-crystal X-ray diffraction, in which a
beam of X-rays are reflected from evenly spaced planes of a
single crystal, producing a diffraction pattern of spots called
reflections.[1] Each reflection corresponds to one set of
evenly spaced planes within the crystal. The density of
electrons within the crystal is determined from the position
and brightness of the various reflections observed as the
crystal is gradually rotated in the X-ray beam; this density,
together with supplementary data, allows the atomic
positions to be inferred. For single crystals of sufficient
purity and regularity, X-ray diffraction data can determine the
mean chemical bond lengths and angles to within a few
thousandths of an Ångström and to within a few tenths of a
degree, respectively. The data also allow the static and
dynamic disorder in the atomic positions to be estimated,
which is usually less than a few tenths of an Ångström.
Limitations
As the crystal's repeating unit, its unit cell, becomes larger and
more complex, the atomic-level picture provided by X-ray
crystallography becomes less well-resolved (more "fuzzy") for a
given number of observed reflections. Two limiting cases of X-ray
crystallography are often discerned, "small-molecule" and
"macromolecular" crystallography. Small-molecule
crystallography typically involves crystals with fewer than 100
atoms in their asymmetric unit; such crystal structures are
usually so well resolved that its atoms can be discerned as
isolated "blobs" of electron density.
By contrast, macromolecular crystallography often involves tens of
thousands of atoms in the unit cell. Such crystal structures are
generally less well-resolved (more "smeared out"); the atoms and
chemical bonds appear as tubes of electron density, rather than
as isolated atoms. In general, small molecules are also easier to
crystallize than macromolecules; however, X-ray crystallography
has proven possible even for viruses with hundreds of thousands
of atoms.
• The three-dimensional
structure of penicillin,
for which Dorothy
Crowfoot Hodgkin was
awarded the Nobel
Prize in Chemistry in
1964. The green, white,
red and blue spheres
represent atoms of
carbon, hydrogen,
oxygen and nitrogen,
respectively.
The diffraction imaging layout
at beamline 9.0.1: from left,
coherent x-rays illuminate the
sample (center), which is
mounted on a silicon nitride
window just 50 nanometers
thick in a movable frame
• Aerogels, sometimes
called "frozen smoke, "
can be made from
different materials. This
silicon aerogel is an
efficient insulator.
• A 500-nanometer cube
of aerogel from the
interior of the 3-D
volume, reconstructed
by X-ray diffraction.
The foam structure
shows globular nodes
that are
interconnected by thin
beam-like struts.
Published on 31st July 2008