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Crystallography (from
the
Greek words crystallon
= cold drop / frozen drop, with
its meaning extending to all
solids with some degree of
transparency, and graphein
= write) is the experimental
science of determining the
arrangement of
atoms in
solids. In older usage, it is
the scientific study of
crystals.
Before the development of X-ray
diffraction crystallography (see
below), the study of crystals was
based on the geometry of the
crystals. This involves measuring
the angles of crystal faces
relative to theoretical reference
axes (crystallographic
axes), and establishing the
symmetry of the crystal in
question. The former is carried
out using a
goniometer. The position in 3D
space of each crystal face is
plotted on a stereographic net,
e.g.
Wolff net or
Lambert net. In fact, the
pole to each face is plotted
on the net. Each point is labelled
with its
Miller Index. The final plot
allows the symmetry of the crystal
to be established.
Crystallographic methods now
rely on the analysis of the
diffraction patterns that
emerge from a sample that is
targeted by a beam of some type.
The beam is not always
electromagnetic radiation,
even though
X-rays are the most common
choice. For some purposes
electrons or
neutrons are used, which is
possible due to the wave
properties of the particles.
Crystallographers often explicitly
state the type of illumination
used when referring to a method,
as with the terms
X-ray diffraction,
neutron diffraction and
electron diffraction.
These three types of radiation
interact with the specimen in
different ways.
X-rays interact with the
spatial distribution of the
valence electrons, while
electrons are
charged particles and
therefore feel the total charge
distrubution of both the
atomic nuclei and the
surrounding electrons.
Neutrons are scattered by the
atomic nuclei through the
strong nuclear forces, but in
addition, the
magnetic moment of neutrons is
non-zero. They are therefore also
scattered by
magnetic fields. Because of
these different forms of
interaction, the three types of
radiation are suitable for
different crystallographic
studies.
Theory
In many cases, an image of a
microscopic object is generated by
focusing the rays of the
visible spectrum using a
lens as in light
microscopy. However, because
the wavelength of visible light is
long compared to
atomic bond lengths and
atoms themselves, it is
necessary to use radiation with
shorter wavelengths, such as
X-rays. Employing shorter
wavelengths implies abandoning
microscopy and true imaging,
however, because there exists no
material from which a lens capable
of focusing this type of radiation
can be created. (That said,
scientists have had some success
focusing X-rays with microscopic
Fresnel zone plates made from
gold). Generally, in
diffraction-based imaging, the
only wavelengths used are those
that are too short to be focused.
This difficulty is the reason that
crystals must be used.
Because of their highly ordered
and repetitive structure, crystals
are an ideal material for
analyzing the structure of solids.
To use X-ray diffraction as an
example, a single X-ray photon
diffracting off of one electron
cloud will not generate a strong
enough signal for the equipment to
detect. However, many X-rays
diffracting off many electron
clouds in approximately the same
relative position and orientation
throughout the crystal will result
in constructive
interference and hence a
detectable signal.
Notation
See
Miller index for a full
treatment of this topic.
- Coordinates in square
brackets such as
[100] denote a direction (in
real space).
- Coordinates in angle
brackets or chevrons
such as <100> denote a
family of directions which
are equivalent due to symmetry
operations. If it refers to a
cubic system, this example could
mean [100], [010], [001] or the
negative of any of those
directions.
- Coordinates in
parentheses such as (100)
denote the direction of a plane
normal.
- Coordinates in curly
brackets or braces
such as {100} denote a
family of plane normals which
are equivalent due to symmetry
operations, much the way angle
brackets denote a family of
directions.
Technique
Some materials studied using
crystallography, DNA for example,
do not occur naturally as
crystals. Typically, such
molecules are placed in solution
and allowed to crystallize over
days, weeks, or months through
vapor
diffusion. A drop of solution
containing the molecule, buffer,
and precipitants is sealed in a
container with a reservoir
containing a
hygroscopic solution. Water in
the drop diffuses to the
reservoir, slowly increasing the
concentration and allowing a
crystal to form. If the
concentration were to rise more
quickly, the molecule would simply
precipitate out of solution,
resulting in disorderly granules
rather than an orderly and hence
usable crystal.
Once a crystal is obtained,
data can be collected using a beam
of radiation. Although many
universities that engage in
crystallographic research have
their own X-ray producing
equipment,
Synchrotrons are often used as
X-ray sources, because of the
purer and more complete patterns
such sources can generate.
Synchrotron sources also have a
much higher intensity of x-ray
beams, so data collection takes a
fraction of the time normally
necessary at weaker sources.
Producing an image from a
diffraction pattern requires
sophisticated
mathematics and often an
iterative process of modelling
and refinement. In this
process, the mathematically
predicted diffraction patterns of
an hypothesized or "model"
structure are compared to the
actual pattern generated by the
crystalline sample. Ideally,
researchers make several initial
guesses, which through refinement
all converge on the same answer.
Models are refined until their
predicted patterns match to as
great a degree as can be achieved
without radical revision of the
model. This is a painstaking
process, made much easier today by
computers.
The mathematical methods for
the analysis of diffraction data
only apply to patterns,
which in turn result only when
waves diffract from orderly
arrays. Hence crystallography
applies for the most part only to
crystals, or to molecules which
can be coaxed to crystalize for
the sake of measurement. In spite
of this, a certain amount of
molecular information can be
deduced from the patterns that are
generated by fibers and powders,
which while not as perfect as a
solid crystal, may exhibit a
degree of order. This level of
order can be sufficient to deduce
the structure of simple molecules,
or to determine the coarse
features of more complicated
molecules (the double-helical
structure of
DNA, for example, was deduced
from an X-ray diffraction pattern
that had been generated by a
fibrous sample).
Materials science
Crystallography is a tool that
is often employed by materials
scientists. In single crystals,
the effects of the crystalline
arrangement of atoms is often easy
to see macroscopically, because
the natural shapes of crystals
reflect the atomic structure. In
addition, physical properties are
often controlled by crystalline
defects. The understanding of
crystal structures is an important
prerequisite for understanding
crystallographic defects.
A number of other physical
properties are linked to
crystallography. For example, the
minerals in
clay form small, flat,
platelike structures. Clay can be
easily deformed because the
platelike particles can slip along
each other in the plane of the
plates, yet remain strongly
connected in the direction
perpendicular to the plates.
In another example,
iron transforms from a
body-centered cubic (bcc)
structure to a
face-centered cubic (fcc)
structure called austenite when it
is heated. The fcc structure is a
close-packed structure, and the
bcc structure is not, which
explains why the volume of the
iron decreases when this
transformation occurs.
Crystallography is useful in
phase identification: That is,
when performing some kind of
processing on a material, it is
often desired to find out what
compounds and what phases are
present in the material. Each
phase has a characteristic
arrangement of atoms. Techniques
like X-ray diffraction can be used
to identify which patterns are
present in the material, and thus
which compounds are present (note:
the determination of the "phases"
within a material should not be
confused with the more general
problem of "phase determination,"
which refers to the phase of waves
as they diffract from planes
within a crystal, and which is a
necessary step in the
interpretation of complicated
diffraction patterns).
Crystallography covers the
enumeration of the symmetry
patterns which can be formed by
atoms in a crystal and for this
reason has a relation to group
theory and geometry. See
Symmetry group.
Biology
X-ray crystallography is the
primary method for determining the
molecular conformations of
biological
macromolecules, particularly
protein and
nucleic acids such as
DNA and
RNA. In fact, the
double-helical structure of DNA
was deduced from crystallographic
data. The first crystal structure
of a macromolecule was solved in
1958 (Kendrew, J.C. et al. (1958)
A three-dimensional model of the
myoglobin molecule obtained by
X-ray analysis (Nature 181,
662-666). The Protein Data Bank (PDB)
at
http://www.rcsb.org is a
freely accessible repository for
the structures of
proteins and other biological
macromolecules.
RasMol can be used to
visualize biological molecular
structures.
Electron crystallography has
been used to determine some
protein structures, most notably
membrane proteins and
viral capsids.
