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Aperiodic Crystal
Aperiodic crystals lack three-dimensional translational symmetry but still exhibit three-dimensional long-range order. In other words, they are periodic crystals in higher dimensions. They are classified into three different categories: incommensurate modulated structures, incommensurate composite structures, and quasicrystals. The diffraction patterns of aperiodic crystals contain two sets of peaks, which include "main reflections" and "satellite reflections". Main reflections are usually stronger in intensity and span a lattice defined by three-dimensional reciprocal lattice vectors(a^*, b^*, c^*). Satellite reflections are weaker in intensity and are known as "lattice ghosts". These reflections do not correspond to any lattice points in physical space and cannot be indexed with the original three vectors. To understand aperiodic crystal structures, one must use the superspace approach. In materials science, "superspace" or higher-dimensional space refers to the concept of desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Space
In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the ''direct lattice''. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice (e.g., a lattice of a crystal), the reciprocal lattice exists in reciprocal space (also known as ''momentum space'' or less commonly as ''K-space'', due to the relationship between the Pontryagin duals momentum and position). The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The reciprocal lattice is the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasicrystal
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five-fold. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the field of crystallography. In crystallography the quasicrystals were predicted in 1981 by a five-fold symmetry study of Alan Lindsay Mackay,—that also brought in 1982, with the crystallographic Fourier transform of a Penrose tiling,Alan L. Mackay, "Crystall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bragg Peak
The Bragg peak is a pronounced peak on the Bragg curve which plots the energy loss of ionizing radiation during its travel through matter. For protons, α-rays, and other ion rays, the peak occurs immediately before the particles come to rest. It is named after William Henry Bragg, who discovered it in 1903. When a fast charged particle moves through matter, it ionizes atoms of the material and deposits a dose along its path. A peak occurs because the interaction cross section increases as the charged particle's energy decreases. Energy lost by charged particles is inversely proportional to the square of their velocity, which explains the peak occurring just before the particle comes to a complete stop. In the upper figure, it is the peak for alpha particles of 5.49 MeV moving through air. In the lower figure, it is the narrow peak of the "native" proton beam curve which is produced by a particle accelerator of 250 MeV. The figure also shows the absorption of a beam of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biphenyl Molecule Planar Ball
Biphenyl (also known as diphenyl, phenylbenzene, 1,1′-biphenyl, lemonene or BP) is an organic compound that forms colorless crystals. Particularly in older literature, compounds containing the functional group consisting of biphenyl less one hydrogen (the site at which it is attached) may use the prefixes xenyl or diphenylyl. It has a distinctively pleasant smell. Biphenyl is an aromatic hydrocarbon with a molecular formula (C6H5)2. It is notable as a starting material for the production of polychlorinated biphenyls (PCBs), which were once widely used as dielectric fluids and heat transfer agents. Biphenyl is also an intermediate for the production of a host of other organic compounds such as emulsifiers, optical brighteners, crop protection products, and plastics. Biphenyl is insoluble in water, but soluble in typical organic solvents. The biphenyl molecule consists of two connected phenyl rings. Properties and occurrence Biphenyl occurs naturally in coal tar, crude oil, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biphenyl
Biphenyl (also known as diphenyl, phenylbenzene, 1,1′-biphenyl, lemonene or BP) is an organic compound that forms colorless crystals. Particularly in older literature, compounds containing the functional group consisting of biphenyl less one hydrogen (the site at which it is attached) may use the prefixes xenyl or diphenylyl. It has a distinctively pleasant smell. Biphenyl is an aromatic hydrocarbon with a molecular formula (C6H5)2. It is notable as a starting material for the production of polychlorinated biphenyls (PCBs), which were once widely used as dielectric fluids and heat transfer agents. Biphenyl is also an intermediate for the production of a host of other organic compounds such as emulsifiers, optical brighteners, crop protection products, and plastics. Biphenyl is insoluble in water, but soluble in typical organic solvents. The biphenyl molecule consists of two connected phenyl rings. Properties and occurrence Biphenyl occurs naturally in coal ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steric Effects
Steric effects arise from the spatial arrangement of atoms. When atoms come close together there is a rise in the energy of the molecule. Steric effects are nonbonding interactions that influence the shape ( conformation) and reactivity of ions and molecules. Steric effects complement electronic effects, which dictate the shape and reactivity of molecules. Steric repulsive forces between overlapping electron clouds result in structured groupings of molecules stabilized by the way that opposites attract and like charges repel. Steric hindrance Steric hindrance is a consequence of steric effects. Steric hindrance is the slowing of chemical reactions due to steric bulk. It is usually manifested in ''intermolecular reactions'', whereas discussion of steric effects often focus on ''intramolecular interactions''. Steric hindrance is often exploited to control selectivity, such as slowing unwanted side-reactions. Steric hindrance between adjacent groups can also affect torsiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Structure Types
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification. The word ''crystal'' derives from the Ancient Greek word (), meaning both "ice" and "rock crystal", from (), "icy cold, frost". Examples of large crystals include snowflakes, diamonds, and table salt. Most inorganic solids are not crystals but polycrystals, i.e. many microscopic crystals fused together into a single solid. Polycrystals include most metals, rocks, ceramics, and ice. A third category of sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
