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8/10/2019 Crystal Structure - Wikipedia, The Free Encyclopedia
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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.
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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)8/10/2019 Crystal Structure - Wikipedia, The Free Encyclopedia
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http://en.wikipedia.org/wiki/Crystal_structure 12
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
(http://dx.doi.org/10.1098%2Frspa.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
(http://dx.doi.org/10.1098%2Frspa.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.html8/10/2019 Crystal Structure - Wikipedia, The Free Encyclopedia
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9/23/2014 Crystal structure - Wikipedia, the free encyclopedia
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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