1.
Bravais lattice
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This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes, similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox
2.
Crystallography
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Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The word crystallography derives from the Greek words crystallon cold drop, frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein to write. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography, X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on measurements of their geometry. This involved measuring the angles of crystal faces relative to other and to theoretical reference axes. This physical measurement is carried out using a goniometer, the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to face is plotted on the net. Each point is labelled with its Miller index, the final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the patterns of a sample targeted by a beam of some type. X-rays are most commonly used, other beams used include electrons or neutrons and this is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography and these three types of radiation interact with the specimen in different ways. X-rays interact with the distribution of electrons in the sample. Electrons are charged particles and therefore interact with the charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the nuclear forces, but in addition. They are therefore also scattered by magnetic fields, when neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen, because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of an object is made using a lens to focus the beam. However, the wavelength of light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves
3.
Atom
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An atom is the smallest constituent unit of ordinary matter that has the properties of a chemical element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms, Atoms are very small, typical sizes are around 100 picometers. Atoms are small enough that attempting to predict their behavior using classical physics - as if they were billiard balls, through the development of physics, atomic models have incorporated quantum principles to better explain and predict the behavior. Every atom is composed of a nucleus and one or more bound to the nucleus. The nucleus is made of one or more protons and typically a number of neutrons. Protons and neutrons are called nucleons, more than 99. 94% of an atoms mass is in the nucleus. The protons have an electric charge, the electrons have a negative electric charge. If the number of protons and electrons are equal, that atom is electrically neutral, if an atom has more or fewer electrons than protons, then it has an overall negative or positive charge, respectively, and it is called an ion. The electrons of an atom are attracted to the protons in a nucleus by this electromagnetic force. The number of protons in the nucleus defines to what chemical element the atom belongs, for example, the number of neutrons defines the isotope of the element. The number of influences the magnetic properties of an atom. Atoms can attach to one or more other atoms by chemical bonds to form compounds such as molecules. The ability of atoms to associate and dissociate is responsible for most of the changes observed in nature. The idea that matter is made up of units is a very old idea, appearing in many ancient cultures such as Greece. The word atom was coined by ancient Greek philosophers, however, these ideas were founded in philosophical and theological reasoning rather than evidence and experimentation. As a result, their views on what look like. They also could not convince everybody, so atomism was but one of a number of competing theories on the nature of matter. It was not until the 19th century that the idea was embraced and refined by scientists, in the early 1800s, John Dalton used the concept of atoms to explain why elements always react in ratios of small whole numbers
4.
Ion
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An ion is an atom or a molecule in which the total number of electrons is not equal to the total number of protons, giving the atom or molecule a net positive or negative electrical charge. Ions can be created, by chemical or physical means. In chemical terms, if an atom loses one or more electrons. If an atom gains electrons, it has a net charge and is known as an anion. Ions consisting of only a single atom are atomic or monatomic ions, because of their electric charges, cations and anions attract each other and readily form ionic compounds, such as salts. In the case of ionization of a medium, such as a gas, which are known as ion pairs are created by ion impact, and each pair consists of a free electron. The word ion comes from the Greek word ἰόν, ion, going and this term was introduced by English physicist and chemist Michael Faraday in 1834 for the then-unknown species that goes from one electrode to the other through an aqueous medium. Faraday also introduced the words anion for a charged ion. In Faradays nomenclature, cations were named because they were attracted to the cathode in a galvanic device, arrhenius explanation was that in forming a solution, the salt dissociates into Faradays ions. Arrhenius proposed that ions formed even in the absence of an electric current, ions in their gas-like state are highly reactive, and do not occur in large amounts on Earth, except in flames, lightning, electrical sparks, and other plasmas. These gas-like ions rapidly interact with ions of charge to give neutral molecules or ionic salts. These stabilized species are commonly found in the environment at low temperatures. A common example is the present in seawater, which are derived from the dissolved salts. Electrons, due to their mass and thus larger space-filling properties as matter waves, determine the size of atoms. Thus, anions are larger than the parent molecule or atom, as the excess electron repel each other, as such, in general, cations are smaller than the corresponding parent atom or molecule due to the smaller size of its electron cloud. One particular cation contains no electrons, and thus consists of a single proton - very much smaller than the parent hydrogen atom. Since the electric charge on a proton is equal in magnitude to the charge on an electron, an anion, from the Greek word ἄνω, meaning up, is an ion with more electrons than protons, giving it a net negative charge. A cation, from the Greek word κατά, meaning down, is an ion with fewer electrons than protons, there are additional names used for ions with multiple charges
5.
Molecule
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A molecule is an electrically neutral group of two or more atoms held together by chemical bonds. Molecules are distinguished from ions by their lack of electrical charge, however, in quantum physics, organic chemistry, and biochemistry, the term molecule is often used less strictly, also being applied to polyatomic ions. In the kinetic theory of gases, the molecule is often used for any gaseous particle regardless of its composition. According to this definition, noble gas atoms are considered molecules as they are in fact monoatomic molecules. A molecule may be homonuclear, that is, it consists of atoms of one element, as with oxygen, or it may be heteronuclear. Atoms and complexes connected by non-covalent interactions, such as hydrogen bonds or ionic bonds, are not considered single molecules. Molecules as components of matter are common in organic substances and they also make up most of the oceans and atmosphere. Also, no typical molecule can be defined for ionic crystals and covalent crystals, the theme of repeated unit-cellular-structure also holds for most condensed phases with metallic bonding, which means that solid metals are also not made of molecules. In glasses, atoms may also be together by chemical bonds with no presence of any definable molecule. The science of molecules is called molecular chemistry or molecular physics, in practice, however, this distinction is vague. In molecular sciences, a molecule consists of a system composed of two or more atoms. Polyatomic ions may sometimes be thought of as electrically charged molecules. The term unstable molecule is used for very reactive species, i. e, according to Merriam-Webster and the Online Etymology Dictionary, the word molecule derives from the Latin moles or small unit of mass. Molecule – extremely minute particle, from French molécule, from New Latin molecula, diminutive of Latin moles mass, a vague meaning at first, the vogue for the word can be traced to the philosophy of Descartes. The definition of the molecule has evolved as knowledge of the structure of molecules has increased, earlier definitions were less precise, defining molecules as the smallest particles of pure chemical substances that still retain their composition and chemical properties. Molecules are held together by covalent bonding or ionic bonding. Several types of non-metal elements exist only as molecules in the environment, for example, hydrogen only exists as hydrogen molecule. A molecule of a compound is made out of two or more elements, a covalent bond is a chemical bond that involves the sharing of electron pairs between atoms
6.
Crystal
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A crystal or crystalline solid is a solid material whose constituents 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, 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 κρύσταλλος, meaning both ice and rock crystal, from κρύος, 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. Examples of polycrystals include most metals, rocks, ceramics, a third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and many plastics, Crystals are often used in pseudoscientific practices such as crystal therapy, and, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of a crystal is based on the arrangement of atoms inside it. A crystal is a solid where the form a periodic arrangement. For example, when liquid water starts freezing, the change begins with small ice crystals that grow until they fuse. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic order, even microscopically, there are distinct differences between crystalline solids and amorphous solids, most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its cell, a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal, the symmetry of a crystal is constrained by the requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries, called space groups. These are grouped into 7 crystal systems, such as cubic crystal system or hexagonal crystal system, Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. Euhedral crystals are those with obvious, well-formed flat faces, anhedral crystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces of a crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal. This occurs because some surface orientations are more stable than others, as a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces
7.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
8.
Translation (geometry)
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
9.
Lattice constant
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The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c, however, in the special case of cubic crystal structures, all of the constants are equal and we only refer to a. Similarly, in crystal structures, the a and b constants are equal. A group of lattice constants could be referred to as lattice parameters, however, the full set of lattice parameters consist of the three lattice constants and the three angles between them. For example, the constant for diamond is a =3.57 Å at 300 K. The structure is equilateral although its actual shape cannot be determined from only the lattice constant, furthermore, in real applications, typically the average lattice constant is given. Near the crystals surface, lattice constant is affected by the reconstruction that results in a deviation from its mean value. This deviation is especially important in nanocrystals since surface-to-nanocrystal core ratio is large, as lattice constants have the dimension of length, their SI unit is the meter. Lattice constants are typically on the order of several ångströms, lattice constants can be determined using techniques such as X-ray diffraction or with an atomic force microscope. Lattice constant of a crystal can be used as a length standard of nanometer range. In epitaxial growth, the constant is a measure of the structural compatibility between different materials. The volume of the cell can be calculated from the lattice constant lengths. If the unit cell sides are represented as vectors, then the volume is the dot product of one vector with the product of the other two vectors. The volume is represented by the letter V, for the general unit cell V = a b c 1 +2 cos α cos β cos γ − cos 2 α − cos 2 β − cos 2 γ. For monoclinic lattices with α = 90°, γ = 90°, for orthorhombic, tetragonal and cubic lattices with β = 90° as well, then V = a b c. Matching of lattice structures between two different semiconductor materials allows a region of band gap change to be formed in a material without introducing a change in crystal structure and this allows construction of advanced light-emitting diodes and diode lasers. Typically, films of different materials grown on the film or substrate are chosen to match the lattice constant of the prior layer to minimize film stress. An alternative method is to grade the lattice constant from one value to another by a controlled altering of the alloy ratio during film growth
10.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
11.
Space group
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
12.
Space groups
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
13.
Cleavage (crystal)
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Cleavage, in mineralogy, is the tendency of crystalline materials to split along definite crystallographic structural planes. Cleavage forms parallel to planes, Basal or pinacoidal cleavage occurs when there is only one cleavage plane. Mica also has basal cleavage, this is why mica can be peeled into thin sheets, cubic cleavage occurs on when there are three cleavage planes intersecting at 90 degrees. Halite has cubic cleavage, and therefore, when halite crystals are broken, octahedral cleavage occurs when there are four cleavage planes in a crystal. Octahedral cleavage is common for semiconductors, rhombohedral cleavage occurs when there are three cleavage planes intersecting at angles that are not 90 degrees. Prismatic cleavage occurs when there are two planes in a crystal. Dodecahedral cleavage occurs when there are six cleavage planes in a crystal, crystal parting occurs when minerals break along planes of structural weakness due to external stress or along twin composition planes. Parting breaks are very similar in appearance to cleavage, but only due to stress. Examples include magnetite which shows octahedral parting, the parting of corundum. Cleavage is a property traditionally used in mineral identification, both in hand specimen and microscopic examination of rock and mineral studies. As an example, the angles between the cleavage planes for the pyroxenes and the amphiboles are diagnostic. Crystal cleavage is of importance in the electronics industry and in the cutting of gemstones. Precious stones are generally cleaved by impact, as in diamond cutting, synthetic single crystals of semiconductor materials are generally sold as thin wafers which are much easier to cleave. Elemental semiconductors are diamond cubic, a group for which octahedral cleavage is observed. This means that some orientations of wafer allow near-perfect rectangles to be cleaved, most other commercial semiconductors can be made in the related zinc blende structure, with similar cleavage planes. Cleavage Mineral galleries, Mineral properties – Cleavage
14.
Electronic band structure
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In solid-state physics, the electronic band structure of a solid describes the range of energies that an electron within the solid may have and ranges of energy that it may not have. Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, the electrons of a single, isolated atom occupy atomic orbitals each of which has a discrete energy level. When two atoms together to form into a molecule, their atomic orbitals overlap. The Pauli exclusion principle dictates that no two electrons can have the quantum numbers in a molecule. Similarly if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms atomic orbitals overlap. Since the number of atoms in a piece of solid is a very large number the number of orbitals is very large. The energy of adjacent levels is so close together that they can be considered as a continuum and this formation of bands is mostly a feature of the outermost electrons in the atom, which are the ones responsible for chemical bonding and electrical conductivity. The inner electron orbitals do not overlap to a significant degree, Band gaps are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the atomic orbitals from which they arise, two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the associated with core orbitals are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands, higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies. Band theory is only an approximation to the state of a solid. These are the necessary for band theory to be valid, Infinite-size system, For the bands to be continuous. Since a macroscopic piece of material contains on the order of 1022 atoms, with modifications, the concept of band structure can also be extended to systems which are only large along some dimensions, such as two-dimensional electron systems. Homogeneous system, Band structure is a property of a material. Practically, this means that the makeup of the material must be uniform throughout the piece. Non-interactivity, The band structure describes single electron states, the existence of these states assumes that the electrons travel in a static potential without dynamically interacting with lattice vibrations, other electrons, photons, etc. Not only are there local small-scale disruptions, but also local charge imbalances and these charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum
15.
Parallelepiped
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In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle, in Euclidean geometry, its definition encompasses all four concepts. In this context of geometry, in which angles are not differentiated, its definition admits only parallelograms. The rectangular cuboid, cube, and the rhombohedron are all specific cases of parallelepiped, parallelepipeds are a subclass of the prismatoids. Any of the three pairs of faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four edges, the edges within each set are of equal length. Parallelepipeds result from linear transformations of a cube, since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci, each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not, a space-filling tessellation is possible with congruent copies of any parallelepiped. The volume of a parallelepiped is the product of the area of its base A, the base is any of the six faces of the parallelepiped. The height is the distance between the base and the opposite face. An alternative method defines the vectors a =, b = and c = to represent three edges that meet at one vertex, from the figure, we can deduce that the magnitude of α is limited to 0° ≤ α < 90°. On the contrary, the vector b × c may form with a an internal angle β larger than 90°, namely, since b × c is parallel to h, the value of β is either β = α or β = 180° − α. So cos α = ± cos β = | cos β |, and h = | a | | cos β |. We conclude that V = A h = | a | | b × c | | cos β |, which is, by definition of the scalar product, equivalent to the absolute value of a ·, Q. E. D. The latter expression is equivalent to the absolute value of the determinant of a three dimensional matrix built using a, b and c as rows, V = | det |. This is found using Cramers Rule on three reduced two dimensional matrices found from the original, the volume of any tetrahedron that shares three converging edges of a parallelepiped has a volume equal to one sixth of the volume of that parallelepiped. For parallelepipeds with a symmetry plane there are two cases, it has four rectangular faces it has two faces, while of the other faces, two adjacent ones are equal and the other two also
16.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
17.
Reciprocal lattice
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In physics, the reciprocal lattice represents the Fourier transform of another lattice. In normal usage, this first lattice is usually a periodic spatial function in real-space and is known as the direct lattice. The reciprocal lattice plays a role in most analytic studies of periodic structures. In neutron and X-ray diffraction due to the Laue conditions the momentum difference between incoming and diffracted X-rays of a crystal is a lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice, using this process, one can infer the atomic arrangement of a crystal. The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice, assuming an ideal Bravais lattice R n = n 1 ⋅ a 1 + n 2 ⋅ a 2 where n 1, n 2 ∈ Z. Any quantity, e. g. the electronic density in a crystal can be written as a periodic function f = f Due to the periodicity it is useful to write it in Fourier expansions. Mathematically, we can describe the lattice as the set of all vectors G m that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the lattice is the original lattice. Using column vector representation of vectors, the formulae above can be rewritten using matrix inversion. This method appeals to the definition, and allows generalization to arbitrary dimensions, the cross product formula dominates introductory materials on crystallography. The above definition is called the physics definition, as the factor of 2 π comes naturally from the study of periodic structures. The crystallographers definition has the advantage that the definition of b 1 is just the reciprocal magnitude of a 1 in the direction of a 2 × a 3 and this can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, each point in the reciprocal lattice corresponds to a set of lattice planes in the real space lattice. The direction of the lattice vector corresponds to the normal to the real space planes. The magnitude of the lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes. Reciprocal lattices for the crystal system are as follows. The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a primitive cell of side 2 π a
18.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
19.
Optics
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Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light, because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for using the classical description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice, practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines, physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that waves were in fact electromagnetic radiation. Some phenomena depend on the fact that light has both wave-like and particle-like properties, explanation of these effects requires quantum mechanics. When considering lights particle-like properties, the light is modelled as a collection of particles called photons, quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy, various engineering fields, photography, practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, lasers, and fibre optics. Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, the earliest known lenses, made from polished crystal, often quartz, date from as early as 700 BC for Assyrian lenses such as the Layard/Nimrud lens. The ancient Romans and Greeks filled glass spheres with water to make lenses, the word optics comes from the ancient Greek word ὀπτική, meaning appearance, look. Greek philosophy on optics broke down into two opposing theories on how vision worked, the theory and the emission theory. The intro-mission approach saw vision as coming from objects casting off copies of themselves that were captured by the eye, plato first articulated the emission theory, the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus, some hundred years later, Euclid wrote a treatise entitled Optics where he linked vision to geometry, creating geometrical optics. Ptolemy, in his treatise Optics, held a theory of vision, the rays from the eye formed a cone, the vertex being within the eye. The rays were sensitive, and conveyed back to the observer’s intellect about the distance. He summarised much of Euclid and went on to describe a way to measure the angle of refraction, during the Middle Ages, Greek ideas about optics were resurrected and extended by writers in the Muslim world
20.
Refractive index
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In optics, the refractive index or index of refraction n of a material is a dimensionless number that describes how light propagates through that medium. It is defined as n = c v, where c is the speed of light in vacuum, for example, the refractive index of water is 1.333, meaning that light travels 1.333 times faster in a vacuum than it does in water. The refractive index determines how light is bent, or refracted. The refractive indices also determine the amount of light that is reflected when reaching the interface, as well as the angle for total internal reflection. This implies that vacuum has a index of 1. The refractive index varies with the wavelength of light and this is called dispersion and causes the splitting of white light into its constituent colors in prisms and rainbows, and chromatic aberration in lenses. Light propagation in absorbing materials can be described using a refractive index. The imaginary part then handles the attenuation, while the real part accounts for refraction, the concept of refractive index is widely used within the full electromagnetic spectrum, from X-rays to radio waves. It can also be used with wave phenomena such as sound, in this case the speed of sound is used instead of that of light and a reference medium other than vacuum must be chosen. Thomas Young was presumably the person who first used, and invented, at the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances, newton, who called it the proportion of the sines of incidence and refraction, wrote it as a ratio of two numbers, like 529 to 396. Hauksbee, who called it the ratio of refraction, wrote it as a ratio with a fixed numerator, hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1. Young did not use a symbol for the index of refraction, in the next years, others started using different symbols, n, m, and µ. For visible light most transparent media have refractive indices between 1 and 2, a few examples are given in the adjacent table. These values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265. Moissanite lies at the end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, for infrared light refractive indices can be considerably higher
21.
Adsorption
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Adsorption is the adhesion of atoms, ions, or molecules from a gas, liquid, or dissolved solid to a surface. This process creates a film of the adsorbate on the surface of the adsorbent and this process differs from absorption, in which a fluid is dissolved by or permeates a liquid or solid, respectively. Adsorption is a process while absorption involves the whole volume of the material. The term sorption encompasses both processes, while desorption is the reverse of it, similar to surface tension, adsorption is a consequence of surface energy. In a bulk material, all the requirements of the constituent atoms of the material are filled by other atoms in the material. However, atoms on the surface of the adsorbent are not wholly surrounded by other adsorbent atoms, the exact nature of the bonding depends on the details of the species involved, but the adsorption process is generally classified as physisorption or chemisorption. It may also occur due to electrostatic attraction, pharmaceutical industry applications, which use adsorption as a means to prolong neurological exposure to specific drugs or parts thereof, are lesser known. The word adsorption was coined in 1881 by German physicist Heinrich Kayser, adsorption is usually described through isotherms, that is, the amount of adsorbate on the adsorbent as a function of its pressure or concentration at constant temperature. The quantity adsorbed is nearly always normalized by the mass of the adsorbent to allow comparison of different materials, to date,15 different isothem models were developed. The function is not adequate at very high pressure because in reality x / m has a maximum as pressure increases without bound. As the temperature increases, the k and n change to reflect the empirical observation that the quantity adsorbed rises more slowly. Irving Langmuir was the first to derive a scientifically based adsorption isotherm in 1918, the model applies to gases adsorbed on solid surfaces. It is a semi-empirical isotherm with a basis and was derived based on statistical thermodynamics. It is the most common isotherm equation to use due to its simplicity and it is based on four assumptions, All of the adsorption sites are equivalent and each site can only accommodate one molecule. The surface is energetically homogeneous and adsorbed molecules do not interact, at the maximum adsorption, only a monolayer is formed. Adsorption only occurs on localized sites on the surface, not with other adsorbates, the fourth condition is the most troublesome, as frequently more molecules will adsorb to the monolayer, this problem is addressed by the BET isotherm for relatively flat surfaces. The Langmuir isotherm is nonetheless the first choice for most models of adsorption, Langmuir suggested that adsorption takes place through this mechanism, A g + S ⇌ A S, where A is a gas molecule and S is an adsorption site. The direct and inverse rate constants are k and k−1, vmon is related to the number of adsorption sites through the ideal gas law
22.
Surface tension
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Surface tension is the elastic tendency of a fluid surface which makes it acquire the least surface area possible. Surface tension allows insects, usually denser than water, to float, at liquid-air interfaces, surface tension results from the greater attraction of liquid molecules to each other than to the molecules in the air. The net effect is a force at its surface that causes the liquid to behave as if its surface were covered with a stretched elastic membrane. Thus, the surface becomes under tension from the imbalanced forces, because of the relatively high attraction of water molecules for each other through a web of hydrogen bonds, water has a higher surface tension compared to that of most other liquids. Surface tension is an important factor in the phenomenon of capillarity, Surface tension has the dimension of force per unit length, or of energy per unit area. The two are equivalent, but when referring to energy per unit of area, it is common to use the surface energy. In materials science, surface tension is used for either surface stress or surface free energy, the cohesive forces among liquid molecules are responsible for the phenomenon of surface tension. In the bulk of the liquid, each molecule is pulled equally in every direction by neighboring liquid molecules, the molecules at the surface do not have the same molecules on all sides of them and therefore are pulled inwards. This creates some internal pressure and forces liquid surfaces to contract to the minimal area, Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of water tend to be pulled into a shape by the imbalance in cohesive forces of the surface layer. In the absence of forces, including gravity, drops of virtually all liquids would be approximately spherical. The spherical shape minimizes the necessary wall tension of the surface according to Laplaces law. Another way to view surface tension is in terms of energy, a molecule in contact with a neighbor is in a lower state of energy than if it were alone. The interior molecules have as many neighbors as they can possibly have, for the liquid to minimize its energy state, the number of higher energy boundary molecules must be minimized. The minimized quantity of boundary molecules results in a surface area. As a result of surface area minimization, a surface will assume the smoothest shape it can, since any curvature in the surface shape results in greater area, a higher energy will also result. Consequently, the surface will push back against any curvature in much the way as a ball pushed uphill will push back to minimize its gravitational potential energy. Bubbles in pure water are unstable, the addition of surfactants, however, can have a stabilizing effect on the bubbles
23.
Crystallographic defect
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Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, however, the arrangement of atoms or molecules in most crystalline materials is not perfect. The regular patterns are interrupted by crystallographic defects, point defects are defects that occur only at or around a single lattice point. They are not extended in space in any dimension, strict limits for how small a point defect is are generally not defined explicitly. However, these defects typically involve at most a few extra or missing atoms, larger defects in an ordered structure are usually considered dislocation loops. For historical reasons, many point defects, especially in ionic crystals, are called centers, for example a vacancy in many ionic solids is called a luminescence center and these dislocations permit ionic transport through crystals leading to electrochemical reactions. These are frequently specified using Kröger–Vink Notation, vacancy defects are lattice sites which would be occupied in a perfect crystal, but are vacant. If a neighboring atom moves to occupy the vacant site, the moves in the opposite direction to the site which used to be occupied by the moving atom. The stability of the crystal structure guarantees that the neighboring atoms will not simply collapse around the vacancy. In some materials, neighboring atoms actually move away from a vacancy, a vacancy is sometimes called a Schottky defect. Interstitial defects are atoms that occupy a site in the structure at which there is usually not an atom. They are generally high energy configurations, small atoms in some crystals can occupy interstices without high energy, such as hydrogen in palladium. A nearby pair of a vacancy and an interstitial is often called a Frenkel defect or Frenkel pair and this is caused when an ion moves into an interstitial site and creates a vacancy. Due to fundamental limitations of material purification methods, materials are never 100% pure, in the case of an impurity, the atom is often incorporated at a regular atomic site in the crystal structure. This is neither a vacant site nor is the atom on an interstitial site, the atom is not supposed to be anywhere in the crystal, and is thus an impurity. In some cases where the radius of the atom is substantially smaller than that of the atom it is replacing. These types of defects are often referred to as off-center ions. There are two different types of defects, Isovalent substitution and aliovalent substitution
24.
Sintering
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Sintering is the process of compacting and forming a solid mass of material by heat or pressure without melting it to the point of liquefaction. Sintering happens naturally in mineral deposits or as a process used with metals, ceramics, plastics. The atoms in the materials diffuse across the boundaries of the particles, fusing the particles together, the study of sintering in metallurgy powder-related processes is known as powder metallurgy. An example of sintering can be observed when ice cubes in a glass of water adhere to each other, examples of pressure-driven sintering are the compacting of snowfall to a glacier, or the forming of a hard snowball by pressing loose snow together. The word sinter comes from the Middle High German sinter, a cognate of English cinder, the driving force for densification is the change in free energy from the decrease in surface area and lowering of the surface free energy by the replacement of solid-vapor interfaces. It forms new but lower-energy solid-solid interfaces with a decrease in free energy occurring. On a microscopic scale, material transfer is affected by the change in pressure, If the size of the particle is small, these effects become very large in magnitude. For properties such as strength and conductivity, the area in relation to the particle size is the determining factor. The variables that can be controlled for any material are the temperature. Through time, the radius and the vapor pressure are proportional to 2/3 and to 1/3. The source of power for solid-state processes is the change in free or chemical energy between the neck and the surface of the particle. The pore elimination occurs faster for a trial with many pores of uniform size, for the latter portions of the process, boundary and lattice diffusion from the boundary become important. Sintering is part of the process used in the manufacture of pottery. These objects are made from such as glass, alumina, zirconia, silica, magnesia, lime, beryllium oxide. Some ceramic raw materials have an affinity for water and a lower plasticity index than clay. Sintering is performed at high temperature, besides, second and/or third external force could be used. Commonly used second external force is pressure, so, the sintering that performed just using temperature is generally called pressureless sintering. Pressureless sintering is possible with graded metal-ceramic composites, with a nanoparticle sintering aid, a variant used for 3D shapes is called hot isostatic pressing
25.
Crystallite
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A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. The orientation of crystallites can be random with no preferred direction, called random texture, or directed, possibly due to growth, fiber texture is an example of the latter. Crystallites are also referred to as grains, the areas where crystallite grains meet are known as grain boundaries. Polycrystalline or multicrystalline materials, or polycrystals are solids that are composed of many crystallites of varying size, most inorganic solids are polycrystalline, including all common metals, many ceramics, rocks and ice. The extent to which a solid is crystalline has important effects on its physical properties, sulfur, while usually polycrystalline, may also occur in other allotropic forms with completely different properties. Although crystallites are referred to as grains, powder grains are different, while the structure of a crystal is highly ordered and its lattice is continuous and unbroken. Amorphous materials, such as glass and many polymers, are non-crystalline, Polycrystalline structures and paracrystalline phases are in between these two extremes. Crystal size is measured from X-ray diffraction patterns and grain size by other experimental techniques like transmission electron microscopy. Solid objects that are enough to see and handle are rarely composed of a single crystal. Most materials are polycrystalline, they are made of a number of single crystals — crystallites — held together by thin layers of amorphous solid. The crystallite size can vary from a few nanometers to several millimeters, if the individual crystallites are oriented completely at random, a large enough volume of polycrystalline material will be approximately isotropic. This property helps the simplifying assumptions of continuum mechanics to apply to real-world solids, however, most manufactured materials have some alignment to their crystallites, resulting in texture that must be taken into account for accurate predictions of their behavior and characteristics. When the crystallites are mostly ordered with just some random spread of orientations, material fractures can be intergranular fracture or a transgranular fracture. There is an ambiguity with powder grains, a grain can be made of several crystallites. Thus, the size found by laser granulometry can be different from the grain size found by X-ray diffraction, by optical microscopy under polarised light. Coarse grained rocks are formed very slowly, while fine grained rocks are formed quickly, if a rock forms very quickly, such as the solidification of lava ejected from a volcano, there may be no crystals at all. Grain boundaries are interfaces where crystals of different orientations meet, 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
26.
Plastic deformation
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In materials science, deformation refers to any changes in the shape or size of an object due to- an applied force or a change in temperature. The first case can be a result of forces, compressive forces, shear. The movement or displacement of such mobile defects is thermally activated, deformation is often described as strain. As deformation occurs, internal inter-molecular forces arise that oppose the applied force, a larger applied force may lead to a permanent deformation of the object or even to its structural failure. In the figure it can be seen that the compressive loading has caused deformation in the cylinder so that the shape has changed into one with bulging sides. The sides bulge because the material, although enough to not crack or otherwise fail, is not strong enough to support the load without change. Internal forces resist the applied load, the concept of a rigid body can be applied if the deformation is negligible. Depending on the type of material, size and geometry of the object, the image to the right shows the engineering stress vs. strain diagram for a typical ductile material such as steel. Different deformation modes may occur under different conditions, as can be depicted using a deformation mechanism map and this type of deformation is reversible. Once the forces are no longer applied, the returns to its original shape. Elastomers and shape memory metals such as Nitinol exhibit large elastic deformation ranges, however elasticity is nonlinear in these materials. Normal metals, ceramics and most crystals show linear elasticity and an elastic range. This relationship only applies in the range and indicates that the slope of the stress vs. strain curve can be used to find Youngs modulus. Engineers often use this calculation in tensile tests, the elastic range ends when the material reaches its yield strength. At this point plastic deformation begins, note that not all elastic materials undergo linear elastic deformation, some, such as concrete, gray cast iron, and many polymers, respond in a nonlinear fashion. For these materials Hookes law is inapplicable and this type of deformation is irreversible. However, an object in the plastic deformation range will first have undergone elastic deformation, soft thermoplastics have a rather large plastic deformation range as do ductile metals such as copper, silver, and gold. Steel does, too, but not cast iron, hard thermosetting plastics, rubber, crystals, and ceramics have minimal plastic deformation ranges
27.
Dislocation
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In materials science, a dislocation is a crystallographic defect or irregularity within a crystal structure. The presence of dislocations strongly influences many of the properties of materials, some types of dislocations can be visualized as being caused by the termination of a plane of atoms in the middle of a crystal. In such a case, the planes are not straight. The two primary types of dislocations are edge dislocations and screw dislocations, mixed dislocations are intermediate between these. Mathematically, dislocations are a type of defect, sometimes called a soliton. Dislocations behave as stable particles, they can move around, two dislocations of opposite orientation can cancel when brought together, but a single dislocation typically cannot disappear on its own. Two main types of dislocations exist, edge and screw, dislocations found in real materials are typically mixed, meaning that they have characteristics of both. A crystalline material consists of an array of atoms, arranged into lattice planes. One approach is to begin by considering a 3D representation of a crystal lattice. The viewer may then start to simplify the representation by visualising planes of atoms instead of the atoms themselves, an edge dislocation is a defect where an extra half-plane of atoms is introduced mid way through the crystal, distorting nearby planes of atoms. When enough force is applied from one side of the crystal structure, a simple schematic diagram of such atomic planes can be used to illustrate lattice defects such as dislocations. In an edge dislocation, the Burgers vector is perpendicular to the line direction, the stresses caused by an edge dislocation are complex due to its inherent asymmetry. A screw dislocation is much harder to visualize, imagine cutting a crystal along a plane and slipping one half across the other by a lattice vector, the halves fitting back together without leaving a defect. This is similar to the Riemann surface of the complex logarithm, if the cut only goes part way through the crystal, and then slipped, the boundary of the cut is a screw dislocation. It comprises a structure in which a path is traced around the linear defect by the atomic planes in the crystal lattice. Perhaps the closest analogy is a spiral-sliced ham, in pure screw dislocations, the Burgers vector is parallel to the line direction. Despite the difficulty in visualization, the caused by a screw dislocation are less complex than those of an edge dislocation. This equation suggests a long cylinder of stress radiating outward from the cylinder, please note, this simple model results in an infinite value for the core of the dislocation at r=0 and so it is only valid for stresses outside of the core of the dislocation
28.
Burgers vector
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The vectors magnitude and direction is best understood when the dislocation-bearing crystal structure is first visualized without the dislocation, that is, the perfect crystal structure. In this perfect crystal structure, a rectangle whose lengths and widths are multiples of a is drawn encompassing the site of the original dislocations origin. Once this encompassing rectangle is drawn, the dislocation can be introduced and this dislocation will have the effect of deforming, not only the perfect crystal structure, but the rectangle as well. What was once a rectangle before the dislocation was introduced is now an open geometric figure, whose opening defines the direction, the direction of the vector depends on the plane of dislocation, which is usually on one of the closest-packed crystallographic planes. In edge dislocations, the Burgers vector and dislocation line are perpendicular to one another, in screw dislocations, they are parallel. The Burgers vector is significant in determining the strength of a material by affecting solute hardening, precipitation hardening
29.
Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
30.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
31.
Cubic crystal system
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In crystallography, the cubic 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 Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
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Body-centered cubic
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In crystallography, the cubic 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 Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
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Crystal system
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In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals. Informally, two crystals are in the crystal system if they have similar symmetries, though there are many exceptions to this. Space groups and crystals are divided into seven crystal systems according to their point groups, five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, a lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems, triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, in a crystal system, a set of point groups 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 system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the crystal system. In total there are seven crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, a crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a lattice system. In three dimensions, the families and systems are identical, except the hexagonal and trigonal crystal systems. In total there are six families, triclinic, monoclinic, orthorhombic, tetragonal, hexagonal. Spaces with less than three dimensions have the number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system, in 2D space, there are four crystal systems, oblique, rectangular, square and hexagonal. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the table, Note. To avoid confusion of terminology, the term trigonal lattice is not used, if the original structure and inverted structure are identical, then the structure is centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure and this is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the structure, then the structure is chiral
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Translational symmetry
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In geometry, a translation slides a thing by a, Ta = p + a. In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation, discrete translational symmetry is invariant under discrete translation. More precisely it must hold that ∀ δ A f = A, laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noethers theorem, space translational symmetry of a system is equivalent to the momentum conservation law. Translational symmetry of a means that a particular translation does not change the object. Fundamental domains are e. g. H + a for any hyperplane H for which a has an independent direction. This is in 1D a line segment, in 2D an infinite strip, Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector. In spaces with higher than 1, there may be multiple translational symmetry. For each set of k independent translation vectors the symmetry group is isomorphic with Zk, in particular the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions, in this case the set of all translations forms a lattice. The absolute value of the determinant of the matrix formed by a set of vectors is the hypervolume of the n-dimensional parallelepiped the set subtends. This parallelepiped is a region of the symmetry, any pattern on or in it is possible. E. g. in 2D, instead of a and b we can take a. In general in 2D, we can take pa + qb and ra + sb for integers p, q, r and this ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair, each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object, without further symmetry, this parallelogram is a fundamental domain. The vectors a and b can be represented by complex numbers, for two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group, see lattice. With rotational symmetry of order two of the pattern on the tile we have p2, the rectangle is a more convenient unit to consider as fundamental domain than a parallelogram consisting of part of a tile and part of another one
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Lattice system
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In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals. Informally, two crystals are in the crystal system if they have similar symmetries, though there are many exceptions to this. Space groups and crystals are divided into seven crystal systems according to their point groups, five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, a lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems, triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, in a crystal system, a set of point groups 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 system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the crystal system. In total there are seven crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, a crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a lattice system. In three dimensions, the families and systems are identical, except the hexagonal and trigonal crystal systems. In total there are six families, triclinic, monoclinic, orthorhombic, tetragonal, hexagonal. Spaces with less than three dimensions have the number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system, in 2D space, there are four crystal systems, oblique, rectangular, square and hexagonal. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the table, Note. To avoid confusion of terminology, the term trigonal lattice is not used, if the original structure and inverted structure are identical, then the structure is centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure and this is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the structure, then the structure is chiral
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Triclinic crystal system
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In crystallography, the triclinic crystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors, in the triclinic system, the crystal is described by vectors of unequal length, as in the orthorhombic system. In addition, no vector is at right angles orthogonal to another, the triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It is the lattice type that itself has no mirror planes. There are a total 2 space groups, with each only one space group is associated. Pinacoidal is also known as triclinic normal, pedial is also triclinic hemihedral Mineral examples include plagioclase, microcline, rhodonite, turquoise, wollastonite and amblygonite, all in triclinic normal. Crystal structure Hurlbut, Cornelius S. Klein, Cornelis,1985, Manual of Mineralogy, 20th ed. pp.64 –65, ISBN 0-471-80580-7
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Monoclinic crystal system
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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, while the third vector meets the other two at an angle other than 90°. There is only one monoclinic Bravais lattice in two dimensions, the oblique lattice, two monoclinic Bravais lattices exist, the primitive monoclinic and the centered monoclinic lattices. In this axis setting, the primitive and base-centered lattices interchange in centering type, sphenoidal is also monoclinic hemimorphic, Domatic is also monoclinic hemihedral, Prismatic is also monoclinic normal. Crystal structure Hurlbut, Cornelius S. Klein, Cornelis, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry
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Orthorhombic crystal system
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In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal, there are two orthorhombic Bravais lattices in two dimensions, Primitive rectangular and centered rectangular. The primitive rectangular lattice can also be described by a centered rhombic unit cell, there are four orthorhombic Bravais lattices, primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic. In this axis setting, the primitive and base-centered lattices interchange in centering type, crystal structure Overview of all space groups Hurlbut, Cornelius S. Klein, Cornelis. Hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry
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Tetragonal crystal system
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In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its vectors, so that the cube becomes a rectangular prism with a square base. There is only one tetragonal Bravais lattice in two dimensions, the square lattice, there are two tetragonal Bravais lattices, the simple tetragonal and the centered tetragonal. One might suppose stretching face-centered cubic would result in face-centered tetragonal, BCT is considered more fundamental, so that is the standard terminology. Crystal structure point groups Bravais lattices
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Hexagonal crystal family
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In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral, the hexagonal crystal family consists of two lattice systems, hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice, hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes, the unit cell is a rhombohedron. This is a cell with parameters a = b = c, α = β = γ ≠ 90°. In practice, the description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. However, such a description is rarely used, the hexagonal crystal family consists of two crystal systems, trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves, the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis. The crystal structures of alpha-quartz in the example are described by two of those 18 space groups associated with the hexagonal lattice system. The hexagonal crystal system consists of the seven point groups such that all their groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the crystal system. Note that the atom in the center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif associated with each point in the underlying lattice. The trigonal crystal system is the crystal system whose point groups have more than one lattice system associated with their space groups. The 5 point groups in this system are listed below, with their international number and notation, their space groups in name. The point groups in this system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation
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Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
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Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges