Crystal Structure - Wikipedia, The Free Encyclopedia

8/10/2019 Crystal Structure - Wikipedia, The Free Encyclopedia

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9/23/2014 Crystal structure - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Crystal_structure 1

Insulin crystals

Crystal structureFrom Wikipedia, the free encyclopedia

In mineralogy and crystallography, a crystal structureis a unique arrangement of atoms or molecules in a

crystallineliquid or solid.[1]A crystal structure describes a highly ordered structure, occurring due to the

intrinsic nature of molecules to form symmetric patterns. A crystal structure can be thought of as an

infinitely repeating array of 3D 'boxes', known as unit cells. The unit cell is calculated from the simplest

possible representation of molecules, known as the asymmetric unit. The asymmetric unit is translated to

the unit cell through symmetry operations, and the resultant crystal lattice is constructed through repetition

of the unitcell infinitely in 3-dimensions. Patterns are located upon the points of a lattice, which is an array

of points repeating periodically in three dimensions. The lengths of the edges of a unit cell and the angles

between them are called the lattice parameters.The symmetry properties of the crystal are embodied in its

space group.[1]

A crystal's structure and symmetryplay a role in determining many of its physical properties, such as

cleavage, electronic band structure, and optical transparency.

Contents

1 Unit cell

1.1 Miller indices

1.2 Planes and directions

1.2.1 Cubic structures

2 Classification

2.1Lattice systems

2.2 Atomic coordination

2.2.1 Close packing

2.3 Bravais lattices

2.4 Point groups

2.5 Space groups

3 Grainboundaries

4 Defects and impurities5 Prediction of structure

6 Polymorphism

7 Physical properties

8 See also

9 References

10 External links
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Unit cell

The crystal structure of a material (the arrangement of atoms within a given type of crystal) can be

described in terms of its unit cell. The unit cell is a small box containing one or more atoms arranged in 3-

dimensions. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of

the crystal. The unit cell is represented in terms of its lattice parameters, which are the lengths of the cell

edges (a,b and c) and the angles between them (alpha, beta and gamma), while the positions of the atoms

inside the unit cell are described by the set of atomic positions (xi , yi , zi) measured from a lattice point.Commonly, atomic positions are represented in terms of fractional coordinates, relative to the unit cell

lengths.

Simple cubic (P)

Body-centered cubic (I)

Face-centered cubic (F)

The atom positions within the unit cell can be calculated through application of symmetry operations to the

asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit

cell. This does not, however imply that the entirety of the asymmetric unit must lie within the boundaries o

the unit cell. Symmetric transformations of atom positions are calculated from the space group of the cryst

structure, and this is usually a black box operation performed by computer programs. However, manualcalculation of the atomic positions within the unit cell can be performed from the asymmetric unit, through

the application of the symmetry operators described within the 'International Tables for Crystallography:

Volume A'[2]

Miller indices

Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (mn

The , m, and ndirectional indices are separated by 90, and are thus orthogonal.[3]

By definition, (mn) denotes a plane that intercepts the three points a1/, a2/m, and a3/n, or some multiple

thereof. That is, the Miller indices are proportional to the inversesof the intercepts of the plane with the un

cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not

intersect that axis (i.e., the intercept is "at infinity"). A plane containing a co-ordinate axis is translated so

that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane

are integers with no common factors. Negative indices are indicated with horizontal bars, as in (123). In an

orthogonal co-ordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components o

a vector normal to the plane.
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Planes with different Miller indices incubic crystals

Considering only (mn) planes intersecting one or more lattice points (the lattice planes), the perpendicula

distance dbetween adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to

the planes by the formula:

Planes and directions

The crystallographic directions are geometric lines linking nodes (atoms, ions or molecules) of a crystal.

Likewise, the crystallographic planes are geometricplaneslinking nodes. Some directions and planes have

a higher density of nodes. These high density planes have an influence on the behavior of the crystal as

follows:[1]

Optical properties: Refractive index is directly related to

density (or periodic density fluctuations).

Adsorption and reactivity: Physical adsorption and

chemical reactions occur at or near surface atoms or

molecules. These phenomena are thus sensitive to the

density of nodes.

Surface tension: The condensation of a material means

that the atoms, ions or molecules are more stable if they

are surrounded by other similar species. The surface

tension of an interface thus varies according to the

density on the surface.

Microstructural defects: Pores and crystallites tend to

have straight grain boundaries following higher density planes.

Cleavage: This typically occurs preferentially parallel to higher density planes.

Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The

perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one

node in a more dense direction requires a lesser distortion of the crystal lattice.

Some directions and planes are defined by symmetry of the crystal system. In monoclinic, rombohedral,

tetragonal, and trigonal/hexagonal systems there is one unique axis (sometimes called the principal axis)

which has higher rotational symmetry than the other two axes. The basal planeis the plane perpendicular

to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis

designation is arbitrary and there is no principal axis.

Cubic structures
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Dense crystallographic planes

For the special case of simple cubic crystals, the lattice

vectors are orthogonal and of equal length (usually

denoted a); similarly for the reciprocal lattice. So, in

this common case, the Miller indices (mn) and [mn]

both simply denote normals/directions in Cartesian

coordinates. For cubic crystals with lattice constant a,

the spacing dbetween adjacent (mn) lattice planes is

(from above):

Because of the symmetry of cubic crystals, it is possible

to change the place and sign of the integers and have

equivalent directions and planes:

Coordinates in angle bracketssuch as

denote afamilyof directions that are equivalent

due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.

Coordinates in curly bracketsor bracessuch as {100} denote a family of plane normals that are

equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not

orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice

vectors of the cubic supercell and hence are again simply the Cartesian directions.

Classification

The defining property of a crystal is its inherent symmetry, by which we mean that under certain

'operations' the crystal remains unchanged. All crystals have translational symmetry in three directions, bu

some have other symmetry elements as well. For example, rotating the crystal 180 about a certain axis ma

result in an atomic configuration that is identical to the original configuration. The crystal is then said to

have a twofold rotational symmetry about this axis. In addition to rotational symmetries like this, a crystal

may have symmetries in the form of mirror planes and translational symmetries, and also the so-called

"compound symmetries," which are a combination of translation and rotation/mirror symmetries. A full

classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.[4]

Lattice systems

These lattice systems are a grouping of crystal structures according to the axial system used to describe

their lattice. Each lattice system consists of a set of three axes in a particular geometric arrangement. There

are seven lattice systems. They are similar to but not quite the same as the seven crystal systems and the six

crystal families.

The 7 lattice systems
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(From least to most symmetric) The 14 Bravais Lattices

1. triclinic(none)

2. monoclinic(1 diad)

simple base-centered

3. orthorhombic(3 perpendicular diads)

simple base-centered body-centered face-centered

4. rhombohedral(1 triad)

5. tetragonal(1 tetrad)

simple body-centered

6. hexagonal(1 hexad)

7. cubic(4 triads)

simple (SC) body-centered (bcc) face-centered (fcc)
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HCP lattice (left) and the fcc lattice (right)

The simplest and most symmetric, the cubic (or isometric) system, has the symmetry of a cube, that is, it

exhibits four threefold rotational axes oriented at 109.5 (the tetrahedral angle) with respect to each other.

These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal

tetragonal, rhombohedral (often confused with the trigonal crystal system), orthorhombic, monoclinic and

triclinic.

Atomic coordination

By considering the arrangement of atoms relative to each other, their coordination numbers (or number of

nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the

structures and alternative ways of visualizing them.[5]

Close packing

The principles involved can be understood by

considering the most efficient way of packing

together equal-sized spheres and stacking close-

packed atomic planes in three dimensions. For

example, if plane A lies beneath plane B, there

are two possible ways of placing an additional

atom on top of layer B. If an additional layer was

placed directly over plane A, this would give riseto the following series :

...ABABABAB....

This arrangement of atoms in a crystal structure

is known as hexagonal close packing (hcp).

If, however, all three planes are staggered relative to each other and it is not until the fourth layer is

positioned directly over plane A that the sequence is repeated, then the following sequence arises:

...ABCABCABC...

This type of structural arrangement is known as cubic close packing (ccp).

The unit cell of a ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not

immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There

are four different orientations of the close-packed layers.

The packing efficiencycan be worked out by calculating the total volume of the spheres and dividing by

the volume of the cell as follows:
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The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only

one size. Most crystalline forms of metallic elements are hcp, fcc, or bcc (body-centered cubic). The

coordination number of atoms in hcp and fcc structures is 12 and its atomic packing factor (APF) is the

number mentioned above, 0.74. This can be compared to the APF of a bcc structure, which is 0.68.

Bravais lattices

When the crystal systems are combined with the various possible lattice centerings, we arrive at the Brava

lattices.[3]They describe the geometric arrangement of the lattice points, and thereby the translational

symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices that are distinct from one

another in the translational symmetry they contain. All crystalline materials recognized until now (not

including quasicrystals) fit in one of these arrangements. The fourteen three-dimensional lattices, classified

by crystal system, are shown above. The Bravais lattices are sometimes referred to asspace lattices.

The crystal structure consists of the same group of atoms, the basis, positioned around each and everylattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the

arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the

group of atoms, or unit cell, is described by its crystallographic point group.

Point groups

The crystallographic point group or crystal classis the mathematical group comprising the symmetry

operations that leave at least one point unmoved and that leave the appearance of the crystal structure

unchanged. These symmetry operations include

Reflection, which reflects the structure across a reflection plane

Rotation, which rotates the structure a specified portion of a circle about a rotation axis

Inversion, which changes the sign of the coordinate of each point with respect to a center of symmet

or inversion point

Improper rotation, which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called

ymmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven

crystal systems.

Space groups

In addition to the operations of the point group, the space group of the crystal structure contains

translational symmetry operations. These include:

Pure translations, which move a point along a vector

Screw axes, which rotate a point around an axis while translating parallel to the axis.[6]
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Glide planes, which reflect a point through a plane while translating it parallel to the plane.[6]

There are 230 distinct space groups.

Grain boundaries

Grain boundaries are interfaces where crystals of different orientations meet.[3]A grain boundary is a

single-phase interface, with crystals on each side of the boundary being identical except in orientation. The

term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms

that have been perturbed from their original lattice sites, dislocations, and impurities that have migrated to

the lower energy grain boundary.

Treating a grain boundary geometrically as an interface of a single crystal cut into two parts, one of which

is rotated, we see that there are five variables required to define a grain boundary. The first two numbers

come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation o

the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to

this plane).[5]

Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a

common way to improve strength, as described by the HallPetch relationship. Since grain boundaries are

defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material.

The high interfacial energy and relatively weak bonding in most grain boundaries often makes them

preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are

also important to many of the mechanisms of creep.[5]

Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large

enough that grain boundaries account for a small fraction of the material. However, very small grain sizes

are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of thematerial, with profound effects on such properties as diffusion and plasticity. In the limit of small

crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any

crystalline character, and thus becomes an amorphous solid.[5]

efects and impurities

Real crystals feature defects or irregularities in the ideal arrangements described above and it is these

defects that critically determine many of the electrical and mechanical properties of real materials. When

one atom substitutes for one of the principal atomic components within the crystal structure, alteration inthe electrical and thermal properties of the material may ensue.[7]Impurities may also manifest as spin

impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of

certain properties such as specific heat may be affected by small concentrations of an impurity, as for

example impurities in semiconducting ferromagnetic alloys may lead to different properties as first

predicted in the late 1960s.[8][9]Dislocations in the crystal lattice allow shear at lower stress than that

needed for a perfect crystal structure.[10]

Prediction of structure
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Quartz is one of the several thermodynamically

stable crystalline forms of silica, SiO2. The most

important forms of silica include: -quartz, -

quartz, tridymite, cristobalite, coesite, and

stishovite.

spontaneously convert from a metastable form (or thermodynamically unstable form) to the stable form at

particular temperature. They also exhibit different melting points, solubilities, and X-ray diffraction

patterns.

One good example of this is the quartz form of silicon dioxide, or SiO2. In the vast majority of silicates, th

Si atom shows tetrahedral coordination by 4 oxygens.

All but one of the crystalline forms involve tetrahedral

SiO4units linked together by shared vertices in

different arrangements. In different minerals the

tetrahedra show different degrees of networking and

polymerization. For example, they occur singly, joined

together in pairs, in larger finite clusters including

rings, in chains, double chains, sheets, and three-

dimensional frameworks. The minerals are classified

into groups based on these structures. In each of its 7

thermodynamically stable crystalline forms or

polymorphs of crystalline quartz, only 2 out of 4 of

each the edges of the SiO4tetrahedra are shared withothers, yielding the net chemical formula for silica:

SiO2.

Another example is elemental tin (Sn), which is

malleable near ambient temperatures but is brittle when

cooled. This change in mechanical properties due to

existence of its two major allotropes, - and -tin. The

two allotropes that are encountered at normal pressure

and temperature, -tin and -tin, are more commonly

known asgray tinand white tinrespectively. Two more

allotropes, and , exist at temperatures above 161 C and pressures above several GPa.[17]White tin is

metallic, and is the stable crystalline form at or above room temperature. Below 13.2 C, tin exists in the

gray form, which has a diamond cubic crystal structure, similar to diamond, silicon or germanium. Gray tin

has no metallic properties at all, is a dull-gray powdery material, and has few uses, other than a few

specialized semiconductor applications.[18]Although the - transformation temperature of tin is nominall

13.2 C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 C, and upon addition

of Sb or Bi the transformation may not occur at all.[19]

Physical properties

Twenty of the 32 crystal classes are piezoelectric, and crystals belonging to one of these classes (point

groups) display piezoelectricity. All piezoelectric classes lack a centre of symmetry. Any material develop

a dielectric polarization when an electric field is applied, but a substance that has such a natural charge

separation even in the absence of a field is called a polar material. Whether or not a material is polar is

determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are

pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.
http://en.wikipedia.org/wiki/Pyroelectrichttp://en.wikipedia.org/wiki/Polar_point_grouphttp://en.wikipedia.org/wiki/Dielectrichttp://en.wikipedia.org/wiki/Piezoelectricityhttp://en.wikipedia.org/wiki/Piezoelectrichttp://en.wikipedia.org/wiki/Semiconductorhttp://en.wikipedia.org/wiki/Germaniumhttp://en.wikipedia.org/wiki/Siliconhttp://en.wikipedia.org/wiki/Diamondhttp://en.wikipedia.org/wiki/Diamond_cubichttp://en.wikipedia.org/wiki/Allotropehttp://en.wikipedia.org/wiki/Allotropehttp://en.wikipedia.org/wiki/Brittlehttp://en.wikipedia.org/wiki/Tinhttp://en.wikipedia.org/wiki/Silicateshttp://en.wikipedia.org/wiki/Silicon_dioxidehttp://en.wikipedia.org/wiki/Quartzhttp://en.wikipedia.org/wiki/X-ray_diffractionhttp://en.wikipedia.org/wiki/Melting_pointshttp://en.wikipedia.org/wiki/Stablehttp://en.wikipedia.org/wiki/Stishovitehttp://en.wikipedia.org/wiki/Coesitehttp://en.wikipedia.org/wiki/Cristobalitehttp://en.wikipedia.org/wiki/Tridymitehttp://en.wikipedia.org/wiki/Quartzhttp://en.wikipedia.org/wiki/Quartzhttp://en.wikipedia.org/wiki/Silicahttp://en.wikipedia.org/wiki/Crystallinehttp://en.wikipedia.org/wiki/Quartzhttp://en.wikipedia.org/wiki/File:Quartz,_Tibet.jpg


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There areafew crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior.

This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the

ferroelectriccrystal does not exhibit a polarization. Upon the application of an electric field of sufficient

magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently

large counter-charge, in the same way that a ferromagnet can be reversed. However, although they are

called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).

See also

References

Brillouin zone

Crystal engineering

Crystal growth

Crystallographic database

Fractional coordinates

HermannMauguin notation

Laser-heated pedestal growthLiquid crystal

Patterson function

Periodic table (crystal structure)

Primitive cell

Schoenflies notation

Seedcrystal

WignerSeitz cell

1. ^ abcSolid State Physics (2nd Edition), J.R. Hook, H.E. Hall, Manchester Physics Series, John Wiley & Sons,

2010, ISBN 978-0-471-92804-1

2. ^International Tables for Crystallography (2006). Volume A, Space-group symmetry.

3. ^ abcEncyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN

(Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3

4. ^Ashcroft, N.; Mermin, D. (1976) Solid State Physics, Brooks/Cole (Thomson Learning, Inc.), Chapter 7, ISB

0-03-049346-3

5. ^ abcdMcGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3

6. ^ abDonald E. Sands (1994). "4-2 Screw axes and 4-3 Glide planes" (http://books.google.com/books?

id=h_A5u5sczJoC&pg=PA71).Introduction to Crystallography(Reprint of WA Benjamin corrected 1975 ed.)

Courier-Dover. pp. 7071. ISBN 0486678393.

7. ^Nikola Kallay (2000)Interfacial Dynamics(http://books.google.com/books?

id=ZXsBk20WO1sC&printsec=frontcover), CRC Press, ISBN 0-8247-0006-6
http://en.wikipedia.org/wiki/Special:BookSources/0824700066http://books.google.com/books?id=ZXsBk20WO1sC&printsec=frontcoverhttp://en.wikipedia.org/wiki/Special:BookSources/0486678393http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=h_A5u5sczJoC&pg=PA71http://en.wikipedia.org/wiki/Special:BookSources/0070514003http://en.wikipedia.org/wiki/Special:BookSources/0030493463http://en.wikipedia.org/wiki/Special:BookSources/9780471928041http://en.wikipedia.org/wiki/Wigner%E2%80%93Seitz_cellhttp://en.wikipedia.org/wiki/Seed_crystalhttp://en.wikipedia.org/wiki/Schoenflies_notationhttp://en.wikipedia.org/wiki/Primitive_cellhttp://en.wikipedia.org/wiki/Periodic_table_(crystal_structure)http://en.wikipedia.org/wiki/Patterson_functionhttp://en.wikipedia.org/wiki/Liquid_crystalhttp://en.wikipedia.org/wiki/Laser-heated_pedestal_growthhttp://en.wikipedia.org/wiki/Hermann%E2%80%93Mauguin_notationhttp://en.wikipedia.org/wiki/Fractional_coordinateshttp://en.wikipedia.org/wiki/Crystallographic_databasehttp://en.wikipedia.org/wiki/Crystal_growthhttp://en.wikipedia.org/wiki/Crystal_engineeringhttp://en.wikipedia.org/wiki/Brillouin_zonehttp://en.wikipedia.org/wiki/Ferromagnetismhttp://en.wikipedia.org/wiki/Ferroelectrichttp://en.wikipedia.org/wiki/Perovskite_(structure)


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External links

The internal structure of crystals... Crystallography for beginners

(http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html)

8. ^Hogan, C. M. (1969). "Density of States of an Insulating Ferromagnetic Alloy".Physical Review188(2): 870

Bibcode:1969PhRv..188..870H (http://adsabs.harvard.edu/abs/1969PhRv..188..870H).

doi:10.1103/PhysRev.188.870 (http://dx.doi.org/10.1103%2FPhysRev.188.870).

9. ^Zhang, X. Y.; Suhl, H (1985). "Spin-wave-related period doublings and chaos under transverse pumping".

Physical Review A32(4): 25302533. Bibcode:1985PhRvA..32.2530Z

(http://adsabs.harvard.edu/abs/1985PhRvA..32.2530Z). doi:10.1103/PhysRevA.32.2530

(http://dx.doi.org/10.1103%2FPhysRevA.32.2530). PMID 9896377

(https://www.ncbi.nlm.nih.gov/pubmed/9896377).

10. ^Courtney, Thomas (2000).Mechanical Behavior of Materials. Long Grove, IL: Waveland Press. p. 85.

ISBN1-57766-425-6.

11. ^L. Pauling (1929). "The principles determining the structure of complex ionic crystals".J. Am. Chem. Soc.51

(4):10101026. doi:10.1021/ja01379a006 (http://dx.doi.org/10.1021%2Fja01379a006).

12. ^Pauling, Linus (1938). "The Nature of the Interatomic Forces in Metals".Physical Review54(11): 899.

Bibcode:1938PhRv...54..899P (http://adsabs.harvard.edu/abs/1938PhRv...54..899P).

doi:10.1103/PhysRev.54.899 (http://dx.doi.org/10.1103%2FPhysRev.54.899).

13. ^Pauling, Linus (1947).Journal of the American Chemical Society69(3): 542. doi:10.1021/ja01195a024(http://dx.doi.org/10.1021%2Fja01195a024).

14. ^Pauling, L. (1949). "A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds".

Proceedings of the Royal Society A196(1046): 343. Bibcode:1949RSPSA.196..343P

(http://adsabs.harvard.edu/abs/1949RSPSA.196..343P). doi:10.1098/rspa.1949.0032

(http://dx.doi.org/10.1098%2Frspa.1949.0032).

15. ^Hume-rothery, W.; Irving, H. M.; Williams, R. J. P. (1951). "The Valencies of the Transition Elements in the

Metallic State".Proceedings of the Royal Society A208(1095): 431. Bibcode:1951RSPSA.208..431H

(http://adsabs.harvard.edu/abs/1951RSPSA.208..431H). doi:10.1098/rspa.1951.0172


16. ^Altmann, S. L.; Coulson, C. A.; Hume-Rothery, W. (1957). "On the Relation between Bond Hybrids and the

Metallic Structures".Proceedings of the Royal Society A240(1221): 145. Bibcode:1957RSPSA.240..145A

(http://adsabs.harvard.edu/abs/1957RSPSA.240..145A). doi:10.1098/rspa.1957.0073


17. ^Molodets, A. M.; Nabatov, S. S. (2000). "Thermodynamic Potentials, Diagram of State, and Phase Transition

of Tin on Shock Compression".High Temperature38(5): 715721. doi:10.1007/BF02755923

(http://dx.doi.org/10.1007%2FBF02755923).

18. ^Holleman, Arnold F.; Wiberg, Egon; Wiberg, Nils; (1985). "Tin".Lehrbuch der Anorganischen Chemie(in

German) (91100 ed.). Walter de Gruyter. pp. 793800. ISBN 3-11-007511-3.

19. ^Schwartz, Mel (2002). "Tin and Alloys, Properties".Encyclopedia of Materials, Parts and Finishes(2nd ed.)

CRCPress. ISBN 1-56676-661-3.
http://en.wikipedia.org/wiki/Special:BookSources/1-56676-661-3http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/3-11-007511-3http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://dx.doi.org/10.1007%2FBF02755923http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1098%2Frspa.1957.0073http://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1957RSPSA.240..145Ahttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Proceedings_of_the_Royal_Society_Ahttp://dx.doi.org/10.1098%2Frspa.1951.0172http://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1951RSPSA.208..431Hhttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Proceedings_of_the_Royal_Society_Ahttp://dx.doi.org/10.1098%2Frspa.1949.0032http://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1949RSPSA.196..343Phttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Proceedings_of_the_Royal_Society_Ahttp://dx.doi.org/10.1021%2Fja01195a024http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1103%2FPhysRev.54.899http://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1938PhRv...54..899Phttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1021%2Fja01379a006http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/J._Am._Chem._Soc.http://en.wikipedia.org/wiki/Linus_Paulinghttp://en.wikipedia.org/wiki/Special:BookSources/1-57766-425-6http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.ncbi.nlm.nih.gov/pubmed/9896377http://en.wikipedia.org/wiki/PubMed_Identifierhttp://dx.doi.org/10.1103%2FPhysRevA.32.2530http://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1985PhRvA..32.2530Zhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1103%2FPhysRev.188.870http://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1969PhRv..188..870Hhttp://en.wikipedia.org/wiki/Bibcodehttp://www.xtal.iqfr.csic.es/Cristalografia/index-en.html


13/13


Appendix A from the manual for Atoms, software for XAFS (http://iffwww.iff.kfa-

juelich.de/icp/atoms/atoms.sgml-7.html)

Intro to Minerals: Crystal Class and System (http://dave.ucsc.edu/myrtreia/crystal.html)

Introduction to Crystallography and Mineral Crystal Systems

(http://www.rockhounds.com/rockshop/xtal/index.html)

Crystal planes and Miller indices (http://www.ece.byu.edu/cleanroom/EW_orientation.phtml)

Interactive 3D Crystal models (http://www.ibiblio.org/e-notes/Cryst/Cryst.htm)

Specific Crystal 3D models (http://chemannex.weebly.com/crystal-structures.html)

Crystallography Open Database (with more than 140.000 crystal structures)

Crystal Lattice Structures: Other Crystal Structure Web Sites (http://cst-

www.nrl.navy.mil/lattice/others.html)

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Categories: Chemical properties Condensed matter physics Crystallography Materials science

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