Crystal structure

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.

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.

Ice Ih Crystal Lattice
The (3-D) crystal structure of H2O ice Ih (c) consists of bases of H2O ice molecules (b) located on lattice points within the (2-D) hexagonal space lattice (a). The values for the H–O–H angle and O–H distance have come from Physics of Ice[1] with uncertainties of ±1.5° and ±0.005 Å, respectively. The white box in (c) is the unit cell defined by Bernal and Fowler[2]

Unit cell

Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.[4] The geometry of the unit cell is defined as a parallelepiped, providing six lattice parameters taken as the lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates (xi, yi, zi) along the cell edges, measured from a reference point. It is only necessary to report the coordinates of a smallest asymmetric subset of particles. This group of particles may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the space group of the crystal structure.[5]

Lattic simple cubic

Simple cubic (P)

Lattice body centered cubic

Body-centered cubic (I)

Lattice face centered cubic

Face-centered cubic (F)

Miller indices

Miller Indices Cubes
Planes with different Miller indices in cubic crystals

Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices , m, and n as directional orthogonal parameters, which are separated by 90°.[6]

By definition, the syntax (ℓmn) 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 inverses of the intercepts of the plane with the unit 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 coordinate 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 coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane.

Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes), the distance d between 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 geometric planes linking 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:[3]

  • 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.
Cristal densite surface
Dense crystallographic planes
  • 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, rhombohedral, 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 plane is 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

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 (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) 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 brackets such as ⟨100⟩ denote a family of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.
  • Coordinates in curly brackets or braces such 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.

Interplanar spacing

The spacing d between adjacent (hkl) lattice planes is given by:[7]

  • Cubic:
  • Tetragonal:
  • Hexagonal:
  • Rhombohedral:
  • Orthorhombic:
  • Monoclinic:
  • Triclinic:

Classification by symmetry

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, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may 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.[8]

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.

Crystal family Lattice system Schönflies 14 Bravais Lattices
Primitive Base-centered Body-centered Face-centered
triclinic Ci Triclinic
monoclinic C2h Monoclinic, simple Monoclinic, centered
orthorhombic D2h Orthorhombic, simple Orthorhombic, base-centered Orthorhombic, body-centered Orthorhombic, face-centered
tetragonal D4h Tetragonal, simple Tetragonal, body-centered
hexagonal rhombohedral D3d Rhombohedral
hexagonal D6h Hexagonal
cubic Oh Cubic, simple Cubic, body-centered Cubic, face-centered

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.

Bravais lattices

Bravais lattices, also referred to as space lattices, describe the geometric arrangement of the lattice points,[6] and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above.

The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group.

Crystal systems

A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.

Crystal family Crystal system Point group / Crystal class Schönflies Point symmetry Order Abstract group
triclinic pedial C1 enantiomorphic polar 1 trivial
pinacoidal Ci (S2) centrosymmetric 2 cyclic
monoclinic sphenoidal C2 enantiomorphic polar 2 cyclic
domatic Cs (C1h) polar 2 cyclic
prismatic C2h centrosymmetric 4 Klein four
orthorhombic rhombic-disphenoidal D2 (V) enantiomorphic 4 Klein four
rhombic-pyramidal C2v polar 4 Klein four
rhombic-dipyramidal D2h (Vh) centrosymmetric 8
tetragonal tetragonal-pyramidal C4 enantiomorphic polar 4 cyclic
tetragonal-disphenoidal S4 non-centrosymmetric 4 cyclic
tetragonal-dipyramidal C4h centrosymmetric 8
tetragonal-trapezohedral D4 enantiomorphic 8 dihedral
ditetragonal-pyramidal C4v polar 8 dihedral
tetragonal-scalenohedral D2d (Vd) non-centrosymmetric 8 dihedral
ditetragonal-dipyramidal D4h centrosymmetric 16
hexagonal trigonal trigonal-pyramidal C3 enantiomorphic polar 3 cyclic
rhombohedral C3i (S6) centrosymmetric 6 cyclic
trigonal-trapezohedral D3 enantiomorphic 6 dihedral
ditrigonal-pyramidal C3v polar 6 dihedral
ditrigonal-scalenohedral D3d centrosymmetric 12 dihedral
hexagonal hexagonal-pyramidal C6 enantiomorphic polar 6 cyclic
trigonal-dipyramidal C3h non-centrosymmetric 6 cyclic
hexagonal-dipyramidal C6h centrosymmetric 12
hexagonal-trapezohedral D6 enantiomorphic 12 dihedral
dihexagonal-pyramidal C6v polar 12 dihedral
ditrigonal-dipyramidal D3h non-centrosymmetric 12 dihedral
dihexagonal-dipyramidal D6h centrosymmetric 24
cubic tetartoidal T enantiomorphic 12 alternating
diploidal Th centrosymmetric 24
gyroidal O enantiomorphic 24 symmetric
hextetrahedral Td non-centrosymmetric 24 symmetric
hexoctahedral Oh centrosymmetric 48

In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

Point groups

The crystallographic point group or crystal class is 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 symmetry 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 symmetry 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.[9]
  • Glide planes, which reflect a point through a plane while translating it parallel to the plane.[9]

There are 230 distinct space groups.

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.[10]

Close packing

Close packing
The hcp lattice (left) and the fcc lattice (right)

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 rise to 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 efficiency can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows:

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.

Grain boundaries

Grain boundaries are interfaces where crystals of different orientations meet.[6] 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 of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).[10]

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 Hall–Petch 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.[10]

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 the material, 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.[10]

Defects 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 in the electrical and thermal properties of the material may ensue.[11] 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.[12][13] Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure.[14]

Prediction of structure

Sodium-chloride-3D-ionic
Crystal structure of sodium chloride (table salt)

The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as evolutionary algorithms, random sampling, or metadynamics.

The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of Pauling's rules, first set out in 1929 by Linus Pauling, referred to by many since as the "father of the chemical bond".[15] Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures.[16]

In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": ​12, ​13, ​23, ​14, ​34, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.[17][18]

After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number.[19] The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The "d-weight" calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.[20]

In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as unlimited big in all directions. Starting from a triclinic structure with no further symmetry property assumed, the system may be driven to show some additional symmetry properties by applying Newton's Second Law on particles in the unit cell and a recently developed dynamical equation for the system period vectors [21] (lattice parameters including angles), even if the system is subject to external stress.

Polymorphism

Quartz, Tibet
Quartz is one of the several crystalline forms of silica, SiO2. The most important forms of silica include: α-quartz, β-quartz, tridymite, cristobalite, coesite, and stishovite.

Polymorphism is the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials including polymers, minerals, and metals. According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature. Polymorphism is related to allotropy, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit, amorphous fraction or crystallographic defects. Polymorphs have different stabilities and may spontaneously convert from a metastable form (or thermodynamically unstable form) to the stable form at a 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, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral {SiO4} units 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 the 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the {SiO4} tetrahedra are shared with others, 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, α- and β-tin. The two allotropes that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as gray tin and white tin respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa.[22] 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.[23] Although the α–β transformation temperature of tin is nominally 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.[24]

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 center of symmetry. Any material develops 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.

There are a few 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 ferroelectric crystal 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

  1. ^ Petrenko, V. F.; Whitworth, R. W. (1999). Physics of Ice. Oxford University Press. ISBN 9780198518945.
  2. ^ Bernal, J. D.; Fowler, R. H. (1933). "A Theory of Water and Ionic Solution, with Particular Reference to Hydrogen and Hydroxyl Ions". The Journal of Chemical Physics. 1 (8): 515. Bibcode:1933JChPh...1..515B. doi:10.1063/1.1749327.
  3. ^ a b c Hook, J.R.; Hall, H.E. (2010). Solid State Physics. Manchester Physics Series (2nd ed.). John Wiley & Sons. ISBN 9780471928041.
  4. ^ West, Anthony R. (1999). Basic Solid State Chemistry (2nd ed.). Wiley. p. 1. ISBN 978-0-471-98756-7.
  5. ^ International Tables for Crystallography (2006). Volume A, Space-group symmetry.
  6. ^ a b c Encyclopaedia 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
  7. ^ "4. Direct and reciprocal lattices". CSIC Dept de Cristalografia y Biologia Estructural. 6 Apr 2017. Retrieved 18 May 2017.
  8. ^ Ashcroft, N.; Mermin, D. (1976). "Chapter 7". Solid State Physics. Brooks/Cole (Thomson Learning, Inc.). ISBN 978-0030493461.
  9. ^ a b Donald E. Sands (1994). "§4-2 Screw axes and §4-3 Glide planes". Introduction to Crystallography (Reprint of WA Benjamin corrected 1975 ed.). Courier-Dover. pp. 70–71. ISBN 978-0486678399.
  10. ^ a b c d Parker, C.B., ed. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). ISBN 978-0070514003.
  11. ^ Kallay, Nikola (2000). Interfacial Dynamics. CRC Press. ISBN 978-0824700065.
  12. ^ Hogan, C. M. (1969). "Density of States of an Insulating Ferromagnetic Alloy". Physical Review. 188 (2): 870–874. Bibcode:1969PhRv..188..870H. doi:10.1103/PhysRev.188.870.
  13. ^ Zhang, X. Y.; Suhl, H (1985). "Spin-wave-related period doublings and chaos under transverse pumping". Physical Review A. 32 (4): 2530–2533. Bibcode:1985PhRvA..32.2530Z. doi:10.1103/PhysRevA.32.2530. PMID 9896377.
  14. ^ Courtney, Thomas (2000). Mechanical Behavior of Materials. Long Grove, IL: Waveland Press. p. 85. ISBN 978-1-57766-425-3.
  15. ^ L. Pauling (1929). "The principles determining the structure of complex ionic crystals". J. Am. Chem. Soc. 51 (4): 1010–1026. doi:10.1021/ja01379a006.
  16. ^ Pauling, Linus (1938). "The Nature of the Interatomic Forces in Metals". Physical Review. 54 (11): 899–904. Bibcode:1938PhRv...54..899P. doi:10.1103/PhysRev.54.899.
  17. ^ Pauling, Linus (1947). "Atomic Radii and Interatomic Distances in Metals". Journal of the American Chemical Society. 69 (3): 542–553. doi:10.1021/ja01195a024.
  18. ^ Pauling, L. (1949). "A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds". Proceedings of the Royal Society A. 196 (1046): 343–362. Bibcode:1949RSPSA.196..343P. doi:10.1098/rspa.1949.0032.
  19. ^ 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 A. 208 (1095): 431. Bibcode:1951RSPSA.208..431H. doi:10.1098/rspa.1951.0172.
  20. ^ 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 A. 240 (1221): 145. Bibcode:1957RSPSA.240..145A. doi:10.1098/rspa.1957.0073.
  21. ^ Liu, Gang (2015). "Dynamical equations for the period vectors in a periodic system under constant external stress". Can. J. Phys. 93 (9): 974–978. arXiv:cond-mat/0209372. Bibcode:2015CaJPh..93..974L. doi:10.1139/cjp-2014-0518.
  22. ^ Molodets, A. M.; Nabatov, S. S. (2000). "Thermodynamic Potentials, Diagram of State, and Phase Transitions of Tin on Shock Compression". High Temperature. 38 (5): 715–721. doi:10.1007/BF02755923.
  23. ^ Holleman, Arnold F.; Wiberg, Egon; Wiberg, Nils (1985). "Tin". Lehrbuch der Anorganischen Chemie (in German) (91–100 ed.). Walter de Gruyter. pp. 793–800. ISBN 978-3-11-007511-3.
  24. ^ Schwartz, Mel (2002). "Tin and Alloys, Properties". Encyclopedia of Materials, Parts and Finishes (2nd ed.). CRC Press. ISBN 978-1-56676-661-6.

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Cubic crystal system

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

There are three main varieties of these crystals:

Primitive cubic (abbreviated cP and alternatively called simple cubic)

Body-centered cubic (abbreviated cI or bcc),

Face-centered cubic (abbreviated cF or fcc, and alternatively called cubic close-packed or ccp)Each is subdivided into other variants listed below. Note that although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell often is not.

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Factor X, also known by the eponym Stuart–Prower factor, is an enzyme (EC 3.4.21.6) of the coagulation cascade. It is a serine endopeptidase (protease group S1, PA clan). Factor X is synthesized in the liver and requires vitamin K for its synthesis.

Factor X is activated, by hydrolysis, into factor Xa by both factor IX (with its cofactor, factor VIII in a complex known as intrinsic Xase) and factor VII with its cofactor, tissue factor (a complex known as extrinsic Xase). It is therefore the first member of the final common pathway or thrombin pathway.

It acts by cleaving prothrombin in two places (an arg-thr and then an arg-ile bond), which yields the active thrombin. This process is optimized when factor Xa is complexed with activated co-factor V in the prothrombinase complex.

Factor Xa is inactivated by protein Z-dependent protease inhibitor (ZPI), a serine protease inhibitor (serpin). The affinity of this protein for factor Xa is increased 1000-fold by the presence of protein Z, while it does not require protein Z for inactivation of factor XI. Defects in protein Z lead to increased factor Xa activity and a propensity for thrombosis.

The half life of factor X is 40–45 hours.

Fracture (mineralogy)

In the field of mineralogy, fracture is the texture and shape of a rock's surface formed when a mineral is fractured. Minerals often have a highly distinctive fracture, making it a principal feature used in their identification.

Fracture differs from cleavage in that the latter involves clean splitting along the cleavage planes of the mineral's crystal structure, as opposed to more general breakage. All minerals exhibit fracture, but when very strong cleavage is present, it can be difficult to see.

HRAS

GTPase HRas also known as transforming protein p21 is an enzyme that in humans is encoded by the HRAS gene. The HRAS gene is located on the short (p) arm of chromosome 11 at position 15.5, from base pair 522,241 to base pair 525,549. HRas is a small G protein in the Ras subfamily of the Ras superfamily of small GTPases. Once bound to Guanosine triphosphate, H-Ras will activate a Raf kinase like c-Raf, the next step in the MAPK/ERK pathway.

Intercalation (chemistry)

In chemistry, intercalation is the reversible inclusion or insertion of a molecule (or ion) into materials with layered structures. Examples are found in graphite and transition metal dichalcogenides.

Lysozyme

Lysozyme, also known as muramidase or N-acetylmuramide glycanhydrolase is an antimicrobial enzyme produced by animals that forms part of the innate immune system. Lysozyme is a glycoside hydrolase that catalyzes the hydrolysis of 1,4-beta-linkages between N-acetylmuramic acid and N-acetyl-D-glucosamine residues in peptidoglycan, which is the major component of gram-positive bacterial cell wall. This hydrolysis in turn compromises the integrity of bacterial cell walls causing lysis of the bacteria.

Lysozyme is abundant in secretions including tears, saliva, human milk, and mucus. It is also present in cytoplasmic granules of the macrophages and the polymorphonuclear neutrophils (PMNs). Large amounts of lysozyme can be found in egg white. C-type lysozymes are closely related to alpha-lactalbumin in sequence and structure, making them part of the same family. In humans, the lysozyme enzyme is encoded by the LYZ gene.Lysozyme is thermally stable, with a melting point reaching up to 72 ℃ at pH 5.0. However, in human milk it loses activity very quickly at that temperature. Its isoelectric point is 11.35. In a large range of pH (6-9) lysozyme can survive.

Mineral

A mineral is, broadly speaking, a solid chemical compound that occurs naturally in pure form. A rock may consist of a single mineral, or may be an aggregate of two or more different minerals, spacially segregated into distinct phases. Compounds that occur only in living beings are usually excluded, but some minerals are often biogenic (such as calcite) and/or are organic compounds in the sense of chemistry (such as mellite). Moreover, living beings often syntesize inorganic minerals (like hydroxylapatite) that also occur in rocks.

In geology and mineralogy, the term "mineral" is usually reserved for mineral species: crystalline compounds with a fairly well-defined chemical composition and a specific crystal structure. Minerals without a definite crystalline structure, such as opal, obsidian, and anthracite, are then more properly called mineraloids. If a chemical compound may occur naturally with different crystal structures, each structure is considered different mineral species. Thus, for example, quartz and stishovite are two different minerals consisting of the same compound, silicon dioxide.

The International Mineralogical Association (IMA) is the world's premier standard body for the definition and nomenclature of mineral species. As of November 2018, the IMA recognizes 5,413 official mineral species. out of more than 5,500 proposed or traditional ones.The chemical composition of a named mineral species may vary somewhat by the inclusion of small amounts of impurities. Specific varieties of a species sometimes have conventional or official names of their own. For example, amethyst is a purple variety of the mineral species quartz. Some mineral species can have variable proportions of two or more chemical elements that occupy equivalent positions in the mineral's structure; for example, the formula of mackinawite is given as (Fe,Ni)9S8, meaning FexNi9-xS8, where x is a variable number between 0 and 9. Sometimes a mineral with variable composition is split into separate species, more or less arbitrarily, forming a mineral group; that is the case of the silicates CaxMgyFe2-x-ySiO4, the olivine group.

Besides the essential chemical composition and crystal structure, the description of a mineral species usually includes its common physical properties such as habit, hardness, lustre, diaphaneity, colour, streak, tenacity, cleavage, fracture, parting, specific gravity, magnetism, fluorescence, radioactivity, as well as its taste or smell and its reaction to acid.

Minerals are classified by key chemical constituents; the two dominant systems are the Dana classification and the Strunz classification. Silicate minerals comprise approximately 90% of the Earth's crust. Other important mineral groups include the native elements, sulfides, oxides, halides, carbonates, sulfates, and phosphates.

Mineralogy

Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization.

Monoclinic crystal system

In crystallography, the monoclinic crystal system is one of the 7 crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base. Hence two vectors are perpendicular (meet at right angles), while the third vector meets the other two at an angle other than 90°.

NEU2

Sialidase-2 is an enzyme that in humans is encoded by the NEU2 gene.This gene belongs to a family of glycohydrolytic enzymes which remove sialic acid residues from glycoproteins and glycolipids. Expression studies in COS7 cells confirmed that this gene encodes a functional sialidase. Its cytosolic localization was demonstrated by cell fractionation experiments.

Orthorhombic crystal system

In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.

Periodic table (crystal structure)

For elements that are solid at standard temperature and pressure the table gives the crystalline structure of the most thermodynamically stable form(s) in those conditions. In all other cases the structure given is for the element at its melting point. Data is presented only for the first 114 elements as well as the 118th (hydrogen through flerovium and oganesson), and predictions are given for elements that have never been produced in bulk (astatine, francium, and elements 100–114 and 118).

Proto-oncogene tyrosine-protein kinase Src

Proto-oncogene tyrosine-protein kinase Src, also known as proto-oncogene c-Src or simply c-Src , is a non-receptor tyrosine kinase protein that in humans is encoded by the SRC gene. This protein phosphorylates specific tyrosine residues in other proteins.

c-Src stands for "cellular Src kinase" and should not be confused with "C-terminal Src kinase" (CSK) which is an enzyme which phosphorylates c-Src at its C-terminus and provides negative regulation of Src's enzymatic activity. Similarly, c-Src should not be mistaken for v-Src, a viral (hence the prefix v-) gene that shares similarity with c-Src and is also an oncogene, which can be found in Rous sarcoma virus.

An elevated level of activity of c-Src tyrosine kinase is suggested to be linked to cancer progression by promoting other signals. Mutations in this gene could be involved in the malignant progression of colon cancer.

This proto-oncogene may play a role in the regulation of embryonic development and cell growth.

c-Src includes an SH2 domain, an SH3 domain, and a tyrosine kinase domain.

Src (pronounced "sarc" as it is short for sarcoma) was originally discovered by J. Michael Bishop and Harold E. Varmus, for which they were awarded the 1989 Nobel Prize in Physiology or Medicine. It belongs to a family of Src family kinases.This gene is similar to the v-Src gene of Rous sarcoma virus.

Two transcript variants encoding the same protein have been found for this gene.

Thrombin

Thrombin (EC 3.4.21.5, fibrinogenase, thrombase, thrombofort, topical, thrombin-C, tropostasin, activated blood-coagulation factor II, blood-coagulation factor IIa, factor IIa, E thrombin, beta-thrombin, gamma-thrombin) is a serine protease, an enzyme that, in humans, is encoded by the F2 gene. Prothrombin (coagulation factor II) is proteolytically cleaved to form thrombin in the clotting process. Thrombin in turn acts as a serine protease that converts soluble fibrinogen into insoluble strands of fibrin, as well as catalyzing many other coagulation-related reactions.

Transthyretin

Transthyretin (TTR or TBPA) is a transport protein in the serum and cerebrospinal fluid that carries the thyroid hormone thyroxine (T4) and retinol-binding protein bound to retinol. This is how transthyretin gained its name: transports thyroxine and retinol. The liver secretes transthyretin into the blood, and the choroid plexus secretes TTR into the cerebrospinal fluid.

TTR was originally called prealbumin (or thyroxine-binding prealbumin) because it ran faster than albumin on electrophoresis gels.

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