CRYSTALLOGRAPHY -GAYATHRI K

1. CRYSTALLOGRAPHY
• Crystallography is the experimental science of determining the
arrangement of atoms in crystalline solids (see crystal structure). The
word "crystallography" is derived from the Greek words crystallon "cold
drop, frozen drop", with its meaning extending to all solids with some
degree of transparency, and graphein "to write". In July 2012, the United
Nations recognised the importance of the science of crystallography by
proclaiming that 2014 would be the International Year of
Crystallography.[1]
• Before the development of X-ray diffraction crystallography (see
below), the study of crystals was based on physical measurements of
their geometry using using a goniometer [2]. This involved measuring
the angles of crystal faces relative to each other and to
theoretical reference axes (crystallographic axes), and
establishing the symmetry of the crystal in question. The
position in 3D space of each crystal face is plotted on a
stereographic net such as a Wulff net or Lambert net. 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.

2. • X-rays interact with the spatial distribution of electrons in the sample.
• Electrons are charged particles and therefore interact with the total charge distribution of
both the atomic nuclei and the electrons of the sample.
• Neutrons are scattered by the atomic nucleiproduce diffraction patterns with high noise
levels. However, the material can sometimes be treated to substitute deuterium for hydrogen.
• through the strong nuclear forces, but in addition, the magnetic moment of neutrons is non-
zero. They are therefore also scattered by magnetic fields. When neutrons are scattered
from hydrogen-containing materials, they

3. Coordinates in square brackets such as [100] denote a direction vecctor.
Coordinates in a ngle brackets or chevrons such as <100> denote a family of directions
which are related by symmetry operations. In the cubic crystal system for example, <100>
would mean [100], [010], [001] or the negative of any of those directions.
Miller indices in parentheses such as (100) denote a plane of the crystal structure, and
regular repetitions of that plane with a particular spacing. In the cubic system, the normal to the
(hkl)
plane is the direction [hkl], but in lower-symmetry cases, the normal to (hkl) is not parallel
to Indices in curly brackets or braces such as {100} denote a family of planes and their normals.
In cubic materials the symmetry makes them equivalent, just as the way angle brackets denote
a family of directions. In non-cubic materials, <hkl> is not necessarily perpendicular to {hkl}.

4. Some materials that have been analyzed crystallographically, such as proteins, do not occur
naturally as crystals. Typically, such molecules are placed in solution and allowed to slowly
crystallize 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.

5. Complementary neutron crystallography
techniques are used to identify the positions
of hydrogen atoms, since X-rays only interact
very weakly with light elements such as
hydrogen.
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.

6. 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 crystallize for the sake
of measurement. In spite of this, a certain amount of molecular information can be deduced
from 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. For example, the double-helical structure of DNA was deduced from an X-ray
diffraction pattern that had been generated by a fibrous sample.

7. X-Ray Crystallography
• X-ray crystallography (XRC) is the
experimental science determining the atomic
and molecular structure of a crystal, in which
the crystalline structure causes a beam of
incident X-rays to diffract into many specific
directions. By measuring the angles and
intensities of these diffracted beams, a
crystallographer can produce a three-
dimensional picture of the density of electrons
within the crystal. From this electron density,
the mean positions of the atoms in the crystal
can be determined, as well as their chemical
bonds, their crystallographic disorder, and
various other information.

8. X-ray diffraction is the elastic
scattering of x-ray photons by atoms
in a periodic lattice. The scattered
monochromatic x-rays that are in
phase give constructive interference.
Figure 1 illustrates how diffraction of
x-rays by crystal planes allows one to
derive lattice spacings by using the
Bragg's law.

9. becasue electrons have wave properties they can be different by crystals. electrons will be
diffracted when the angle of incidence, θ on a crystal plane satisfies the bragg equation.
nλ=2 d sin θ
where λ is the wavelength of the electrons, d is the spacing of the crystal planes and is an
interger.
A simple way to derive the bragg equation is as follow. the path difference between electrons
scattered from adjucent crystal planes is 2d sin θ. for constructive interference between the two
scattered beams the difference must be an interger multiple of electron wavelengths, which
gives the bragg equation.

10. • Dual lattice" redirects here. For duals of order-theoretic lattices, see order dual.
• The computer-generated reciprocal lattice of a fictional monoclinic 3D crystal.
• A two-dimensional crystal and its reciprocal lattice
• In physics, the reciprocal lattice represents the Fourier transform of another lattice
(usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented
by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known
as the direct lattice. While the direct lattice exists in real-space and is what one would
commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space
(also known as momentum space or less commonly as K-space, due to the relationship
between the Pontryagin duals momentum and position). The reciprocal of a reciprocal lattice
is the original direct lattice, since the two are Fourier transforms of each other.

11. • The reciprocal lattice plays a very
fundamental role in most analytic
studies of periodic structures,
particularly in the theory of
diffraction. In neutron and X-ray
diffraction, due to the Laue
conditions, the momentum
difference between incoming and
diffracted X-rays of a crystal is a
reciprocal lattice vector. The
diffraction pattern of a crystal can
be used to determine the reciprocal
vectors of the lattice. Using this
process, one can infer the atomic
arrangement of a crystal.

12. • Miller indices form a notation system in crystallography for planes in crystal (Bravais)
lattices.
• In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller
indices. They are written (hkℓ), and denote the family of planes orthogonal to {displaystyle
hmathbf {b_{1}} +kmathbf {b_{2}} +ell mathbf {b_{3}} }h{mathbf {b_{1}}}+k{mathbf
{b_{2}}}+ell {mathbf {b_{3}}}, where {displaystyle mathbf {b_{i}} }{mathbf {b_{i}}} are the
basis of the reciprocal lattice vectors (note that the plane is not always orthogonal to the
linear combination of direct lattice vectors {displaystyle hmathbf {a_{1}} +kmathbf {a_{2}}
+ell mathbf {a_{3}} }h{mathbf {a_{1}}}+k{mathbf {a_{2}}}+ell {mathbf {a_{3}}} because
the reciprocal lattice vectors need not be mutually orthogonal). By convention, negative
integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms,
i.e. their greatest common divisor should be 1. Miller indices are also used to designate
reflexions in X-ray crystallography.

13. • In this case the integers are not necessarily in lowest terms, and can be thought of as
corresponding to planes spaced such that the reflexions from adjacent planes would have a
phase difference of exactly one wavelength (2π), regardless of whether there are atoms on
all these planes or not.
• There are also several related notations:[1]
• the notation {hkℓ} denotes the set of all planes that are equivalent to (hkℓ) by the symmetry of
the lattice.
• In the context of crystal directions (not planes), the corresponding notations are:
• [hkℓ], with square instead of round brackets, denotes a direction in the basis of the direct
lattice vectors instead of the reciprocal lattice;

14. • similarly, the notation <hkℓ> denotes the set of
all directions that are equivalent to [hkℓ] by
symmetry.
• Miller indices were introduced in 1839 by the
British mineralogist William Hallowes Miller,
although an almost identical system (Weiss
parameters) had already been used by German
mineralogist Christian Samuel Weiss since
1817.[2] The method was also historically
known as the Millerian system, and the indices
as Millerian,[3] although this is now rare.
• The Miller indices are defined with respect to
any choice of unit cell and not only with respect
to primitive basis vectors, as is sometimes
state.

15. • In crystallography, crystal structure is a
description of the ordered arrangement of
atoms, ions or molecules in a crystalline
material.[3] Ordered structures occur from the
intrinsic nature of the constituent particles to
form symmetric patterns that repeat along the
principal directions of three-dimensional space
in matter.
• The smallest group of particles in the material
that constitutes this repeating pattern is the unit
cell of the structure. The unit cell completely
reflects the symmetry and structure of the entire
crystal, which is built up by repetitive translation
of the unit cell along its principal axes. The
translation vectors define the nodes of the
Bravais lattice.

16. • The lengths of the principal axes, or edges, of the unit cell and the
angles between them are the lattice constants, also called lattice
parameters or cell parameters. The symmetry properties of the
crystal are described by the concept of space groups.[3] All
possible symmetric arrangements of particles in three-
dimensional space may be described by the 230 space groups.
• The crystal structure and symmetry play a critical role in
determining many physical properties, such as cleavage,
electronic band structure, and optical transparency.

17. • Finally, we would like to emphasize that despite the
profusion of increasingly sophisticated methods of
crystallographic analysis, the simple optical microscope
remains an extraordinarily discerning and comparatively
inexpensive device for the study of crystalline matter and
its imperfections.