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A crystalline solid: atomic resolution image of strontium titanate. Brighter atoms are Sr and darker ones are Ti.

Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals. 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 = write.

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. Wulff 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 depend 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 distribution 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. When neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute hydrogen for deuterium. Because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies.

Contents

Theory

Condensed matter physics
Linbo3 Unit Cell.png
Phases · Phase transition

An image of a small object is usually generated by using a lens to focus the illuminating radiation, as is done with the rays of the visible spectrum in light microscopy. However, the wavelength of visible light (about 4000 to 7000 angstroms) is three orders of magnitude longer then the length of typical atomic bonds and atoms themselves (about 1 to 2 angstroms). Therefore, obtaining information about the spatial arrangement of atoms requires the use of radiation with shorter wavelengths, such as X-rays. Employing shorter wavelengths implied 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, and by critical-angle reflection inside long tapered capillaries.)[1] Diffracted x-ray beams cannot be focused to produce images, so the sample structure must be reconstructed from the diffraction pattern. Sharp features in the diffraction pattern arise from periodic, repeating structure in the sample, which are often very strong due to coherent reflection of many photons from many regularly spaced instances of similar structure, while non-periodic components of the structure result in diffuse (and usually weak) diffraction features.

Because of their highly ordered and repetitive structure, crystals give diffraction patterns of sharp Bragg reflection spots, and are ideal for analyzing the structure of solids.

Notation

  • Coordinates in square brackets such as [100] denote a direction vector (in real space).
  • Coordinates in angle 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 [hkl].
  • Indices in curly brackets or braces such as {100} denote a family of planes and their normals which are equivalent in cubic materials due to symmetry operations, much the way angle brackets denote a family of directions. In non-cubic materials, <hkl> is not necessarily perpendicular to {hkl}.

Technique

Some materials studied using crystallography, proteins 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 crystallize 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).

Crystallography in materials engineering

An example of a closed packed lattice

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. Mostly, materials do not occur in a single crystalline, but poly-crystalline form, such that the powder diffraction method plays a most important role in structural determination.

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. Such mechanisms can be studied by crystallographic texture measurements.

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.

Material Properties
Specific heat c=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)
Compressibility \beta=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)
Thermal expansion \alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)

Crystallography is useful in phase identification. When performing any process on a material, it may be 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) is a freely accessible repository for the structures of proteins and other biological macromolecules. Computer programs like RasMol or Pymol 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.

Scientists of note

See also

References

  1. ^ A. Snigirev et al. (2007). "Two-step hard X-ray focusing combining Fresnel zone plate and single-bounce ellipsoidal capillary". Journal of Synchrotron Radiation 14 (Pt 4): 326–330. doi:10.1107/S0909049507025174. PMID 17587657. 

Further reading

  • Burns, G.; Glazer, A.M. (1990). Space Groups for Scientists and Engineers (2nd ed.). Boston: Academic Press, Inc. ISBN 0-12-145761-3. 
  • Clegg, W (1998). Crystal Structure Determination (Oxford Chemistry Primer). Oxford: Oxford University Press. ISBN 0-19-855-901-1. 
  • Drenth, J (1999). Principles of Protein X-Ray Crystallography. New York: Springer-Verlag. ISBN 0-387-98587-5. 
  • Giacovazzo, C; Monaco HL, Viterbo D, Scordari F, Gilli G, Zanotti G, and Catti M (1992). Fundamentals of Crystallography. Oxford: Oxford University Press. ISBN 0-19-855578-4. 
  • Glusker, JP; Lewis M, Rossi M (1994). Crystal Structure Analysis for Chemists and Biologists. New York: VCH Publishers. ISBN 0-471-18543-4. 
  • O'Keeffe, M.; Hyde, B.G. (1996). Crystal Structures; I. Patterns and Symmetry. Washington, DC: Mineralogical Society of America, Monograph Series. ISBN 0-939950-40-5. 

Applied Computational Powder Diffraction Data Analysis

  • Edited by R. A. Young (1993). Young, R.A.. ed. The Rietveld Method. Oxford: Oxford University Press & International Union of Crystallography. ISBN 0-19-855577-6. 

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Wikibooks

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From Wikibooks, the open-content textbooks collection

Introduction

Crystallography is a branch of geometry that deals with indefinitely repeating patterns. Two-dimensional crystallography can be used, for example, to describe the way tiles cover a floor. Extending the field into three dimensions allows a general description of the way atoms or molecules arrange themselves into crystals. The three-dimensional crystallography was proven to be complete over a century ago. The fact that the mathematics itself cannot be advanced without some change of its axioms has meant that it is studied less often as pure mathematics, than as a means of understanding the details of complex structures in matter. Two fields are particularly reliant on it: materials scientist use it to describe the structure of engineering materials, often with particular attention to crystallographic defects; biochemists use it to describe the structure of biopolymers (see proteomics for an example), which usually must be processed laboriously before they form crystals. In mathematical terms, a crystal is an object with translational symmetry, i.e. it can be moved some distance and remain the same. This type of symmetry is fundamentally different from the more familiar mirror symmetry (the human face) or rotational symmetry (an airplane propeller), in that objects we imagine to represent translational symmetry must be larger than ourselves. We can imagine passing Alice through the looking glass, or spinning a propeller by one blade's fraction of a rotation, while we stand still. For an experience of translational symmetry, however, we must move ourselves, and not notice the difference. This can happen in an ocean, a desert, or a large suburb, if every wave, or dune, or tract home looks exactly like the next. Just as no eye is the exact mirror of its opposite, and no propeller is perfectly balanced, no physical crystal is perfect. There will always be a boundary that gets nearer or farther after a unit of translation. Strictly speaking, any true crystal must fill the entire universe.

Abstraction & categorization

If we imagine a "perfect" housing development (dystopian though it may be) which covers a two-dimensional plane with an infinitely-repeating pattern of homes, and want to apply crystallography to it, we can save a lot of work by eliminating all the geometric complexity of garages and sprinkler heads and such. To make things as simple as possible, we could abstract every house down to a single point, although we need to keep track of each house's orientation.

The set of all these points is conventionally known as a lattice, and it contains all information about repetition. It is defined as a set of points, each with an identical environment. To keep the description complete, we could also create a blueprint for a generic home in this development. This second set of information is called the motif, and it describes what is repeated. As long as each point on the lattice is taken to be the same point on the blueprint for every house, it doesn't matter where that point is. It could be the top of the gable, the doorknob, the house's cener of gravity, or one particular corner of the lot. It could even be a point several miles from the house, but since all lattice points are the same by definition, it's more convenient to choose a point that's closer. In any case, the combination of lattice and motif will still produce the same result.

The definition of a lattice imposes some restrictions which may not be obvious at first. While there are many types of symmetry, only a few can be applied to a true lattice. For instance, while rotational symmetry exists for any fraction of a revolution, only 2-, 3-, 4-, and 6-fold rotations will allow all the points of a lattice to align with one another.

Notation

Throughout this wikibook, we will use standard crystallographic notation to describe points, directions, and planes according to the crystallographic lattice. All of these geometric concepts are associated with vectors, but the special properties of an infinite lattice mean that standard Cartesian coordinates are not always the most helpful. The origin should be a point on the lattice, and the basis vectors should be chosen to connect lattice points. Furthermore, the space we create should be modular, to reflect the fact that translation to a new lattice point is congruent to no translation at all. See High School Mathematics Extensions/Primes/Modular Arithmetic for a review of the concept of modularity.

Constructing a modular space also frees us to consider only a small area (or volume) the size of the separation between lattice points, rather than an infinite space. Conceptually, this small space has sharp boundaries where passing out one side is the same as passing in from the opposite side, like a game of Asteroids.

The basis vectors are chosen to connect lattice points. The only requirement is that enough vectors are chosen that the set of vectors spans the space; in three dimensions, any three vectors that don't lie in the same plane are enough.

The Cartesian coordinate system is convenient because it is orthornormal; it saves a lot of work if we construct our space using vectors that are orthogonal, or normalized, or both, but the highest priority is that the basis vectors connect lattice points. Fortunately, many common crystals (silicon, iron, copper, diamond, table salt) are cubic, which allows the use of an orthonormal basis set. For non-cubic crystals, we must construct a metric tensor in order to make use of the full range of vector operations. Fortunately, the list of all possible metric tensors for crystal lattices is relatively short and easy to generate.

Having defined the space in which a crystal lattice sits, we can then represent vectors by ordered sets of numbers. These numbers are presented differently depending on what the vector means:

Position                (u,v,w)
Direction               [u v w]
Family of directions    <u v w>
Plane                   (u v w)
Family of planes        {u v w}

In all these cases, a negative value is represented by adding a bar over the term. For instance, [1 2 0] is the direction found by adding \vec{a} -2\vec{b}.


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