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121-457: A crystal or crystalline solid is a solid material whose constituents (such as atoms , molecules , or ions ) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape , consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation

242-930: A grain boundary . Like a grain boundary, a twin boundary has different crystal orientations on its two sides. But unlike a grain boundary, the orientations are not random, but related in a specific, mirror-image way. Mosaicity is a spread of crystal plane orientations. A mosaic crystal consists of smaller crystalline units that are somewhat misaligned with respect to each other. In general, solids can be held together by various types of chemical bonds , such as metallic bonds , ionic bonds , covalent bonds , van der Waals bonds , and others. None of these are necessarily crystalline or non-crystalline. However, there are some general trends as follows: Metals crystallize rapidly and are almost always polycrystalline, though there are exceptions like amorphous metal and single-crystal metals. The latter are grown synthetically, for example, fighter-jet turbines are typically made by first growing

363-514: A molten condition nor entirely in solution, but the high temperature and pressure conditions of metamorphism have acted on them by erasing their original structures and inducing recrystallization in the solid state. Other rock crystals have formed out of precipitation from fluids, commonly water, to form druses or quartz veins. Evaporites such as halite , gypsum and some limestones have been deposited from aqueous solution, mostly owing to evaporation in arid climates. Water-based ice in

484-618: A molten fluid, or by crystallization out of a solution. Some ionic compounds can be very hard, such as oxides like aluminium oxide found in many gemstones such as ruby and synthetic sapphire . Covalently bonded solids (sometimes called covalent network solids ) are typically formed from one or more non-metals, such as carbon or silicon and oxygen, and are often very hard, rigid, and brittle. These are also very common, notable examples being diamond and quartz respectively. Weak van der Waals forces also help hold together certain crystals, such as crystalline molecular solids , as well as

605-515: A rock sample is a random aggregate of minerals and/or mineraloids , and has no specific chemical composition. The vast majority of the rocks of the Earth's crust consist of quartz (crystalline SiO 2 ), feldspar, mica, chlorite , kaolin , calcite, epidote , olivine , augite , hornblende , magnetite , hematite , limonite and a few other minerals. Some minerals, like quartz , mica or feldspar are common, while others have been found in only

726-416: A "crystal" is based on the microscopic arrangement of atoms inside it, called the crystal structure . A crystal is a solid where the atoms form a periodic arrangement. ( Quasicrystals are an exception, see below ). Not all solids are crystals. For example, when liquid water starts freezing, the phase change begins with small ice crystals that grow until they fuse, forming a polycrystalline structure. In

847-541: A composite made up of a thermoplastic matrix such as acrylonitrile butadiene styrene (ABS) in which calcium carbonate chalk, talc , glass fibers or carbon fibers have been added for strength, bulk, or electro-static dispersion. These additions may be referred to as reinforcing fibers, or dispersants, depending on their purpose. Thus, the matrix material surrounds and supports the reinforcement materials by maintaining their relative positions. The reinforcements impart their special mechanical and physical properties to enhance

968-1043: A few locations worldwide. The largest group of minerals by far is the silicates (most rocks are ≥95% silicates), which are composed largely of silicon and oxygen , with the addition of ions of aluminium, magnesium , iron, calcium and other metals. Ceramic solids are composed of inorganic compounds, usually oxides of chemical elements. They are chemically inert, and often are capable of withstanding chemical erosion that occurs in an acidic or caustic environment. Ceramics generally can withstand high temperatures ranging from 1,000 to 1,600 °C (1,830 to 2,910 °F). Exceptions include non-oxide inorganic materials, such as nitrides , borides and carbides . Traditional ceramic raw materials include clay minerals such as kaolinite , more recent materials include aluminium oxide ( alumina ). The modern ceramic materials, which are classified as advanced ceramics, include silicon carbide and tungsten carbide . Both are valued for their abrasion resistance, and hence find use in such applications as

1089-553: A fraction of a millimetre to several centimetres across, although exceptionally large crystals are occasionally found. As of 1999, the world's largest known naturally occurring crystal is a crystal of beryl from Malakialina, Madagascar , 18 m (59 ft) long and 3.5 m (11 ft) in diameter, and weighing 380,000 kg (840,000 lb). Some crystals have formed by magmatic and metamorphic processes, giving origin to large masses of crystalline rock . The vast majority of igneous rocks are formed from molten magma and

1210-580: A material can absorb before mechanical failure, while fracture toughness (denoted K Ic ) describes the ability of a material with inherent microstructural flaws to resist fracture via crack growth and propagation. If a material has a large value of fracture toughness , the basic principles of fracture mechanics suggest that it will most likely undergo ductile fracture. Brittle fracture is very characteristic of most ceramic and glass-ceramic materials that typically exhibit low (and inconsistent) values of K Ic . For an example of applications of ceramics,

1331-558: A material that indicates its ability to conduct heat . Solids also have a specific heat capacity , which is the capacity of a material to store energy in the form of heat (or thermal lattice vibrations). Electrical properties include both electrical resistivity and conductivity , dielectric strength , electromagnetic permeability , and permittivity . Electrical conductors such as metals and alloys are contrasted with electrical insulators such as glasses and ceramics. Semiconductors behave somewhere in between. Whereas conductivity in metals

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1452-440: A meltdown of the metallic parts. Work is also being done in developing ceramic parts for gas turbine engines . Turbine engines made with ceramics could operate more efficiently, giving aircraft greater range and payload for a set amount of fuel. Such engines are not in production, however, because the manufacturing of ceramic parts in the sufficient precision and durability is difficult and costly. Processing methods often result in

1573-578: A metallic conductor, current is carried by the flow of electrons, but in semiconductors, current can be carried either by electrons or by the positively charged " holes " in the electronic band structure of the material. Common semiconductor materials include silicon, germanium and gallium arsenide . Many traditional solids exhibit different properties when they shrink to nanometer sizes. For example, nanoparticles of usually yellow gold and gray silicon are red in color; gold nanoparticles melt at much lower temperatures (~300 °C for 2.5 nm size) than

1694-422: A non-crystalline intergranular phase. Glass-ceramics are used to make cookware (originally known by the brand name CorningWare ) and stovetops that have high resistance to thermal shock and extremely low permeability to liquids. The negative coefficient of thermal expansion of the crystalline ceramic phase can be balanced with the positive coefficient of the glassy phase. At a certain point (~70% crystalline)

1815-607: A perfect, exactly repeating pattern. However, in reality, most crystalline materials have a variety of crystallographic defects : places where the crystal's pattern is interrupted. The types and structures of these defects may have a profound effect on the properties of the materials. A few examples of crystallographic defects include vacancy defects (an empty space where an atom should fit), interstitial defects (an extra atom squeezed in where it does not fit), and dislocations (see figure at right). Dislocations are especially important in materials science , because they help determine

1936-529: A periodic lattice. Mathematically, the potential of the ion cores can be treated by various models, the simplest being the nearly free electron model . Minerals are naturally occurring solids formed through various geological processes under high pressures. To be classified as a true mineral, a substance must have a crystal structure with uniform physical properties throughout. Minerals range in composition from pure elements and simple salts to very complex silicates with thousands of known forms. In contrast,

2057-404: A piezoelectric response several times larger than the traditional piezoelectric material quartz (crystalline SiO 2 ). The deformation (~0.1%) lends itself to useful technical applications such as high-voltage sources, loudspeakers, lasers, as well as chemical, biological, and acousto-optic sensors and/or transducers. Space group In mathematics , physics and chemistry , a space group

2178-499: A resin during processing, which is then shaped into a final form. Polymers that have been around, and that are in current widespread use, include carbon-based polyethylene , polypropylene , polyvinyl chloride , polystyrene , nylons, polyesters , acrylics , polyurethane , and polycarbonates , and silicon-based silicones . Plastics are generally classified as "commodity", "specialty" and "engineering" plastics. Composite materials contain two or more macroscopic phases, one of which

2299-442: A significant portion of the fields of solid-state chemistry, physics, materials science and engineering. Metallic solids are held together by a high density of shared, delocalized electrons, known as " metallic bonding ". In a metal, atoms readily lose their outermost ("valence") electrons , forming positive ions . The free electrons are spread over the entire solid, which is held together firmly by electrostatic interactions between

2420-434: A single crystal of titanium alloy, increasing its strength and melting point over polycrystalline titanium. A small piece of metal may naturally form into a single crystal, such as Type 2 telluric iron , but larger pieces generally do not unless extremely slow cooling occurs. For example, iron meteorites are often composed of single crystal, or many large crystals that may be several meters in size, due to very slow cooling in

2541-718: A single solid. Polycrystals include most metals , rocks, ceramics , and ice . 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 . Despite the name, lead crystal, crystal glass , and related products are not crystals, but rather types of glass, i.e. amorphous solids. Crystals, or crystalline solids, 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

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2662-488: A subgroup Z . This table gives the number of space group types in small dimensions, including the numbers of various classes of space group. The numbers of enantiomorphic pairs are given in parentheses. In addition to crystallographic space groups there are also magnetic space groups (also called two-color (black and white) crystallographic groups or Shubnikov groups). These symmetries contain an element known as time reversal. They treat time as an additional dimension, and

2783-404: A translation along the direction of the axis. These are noted by a number, n , to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of

2904-692: A way that the strain is directly proportional to the stress ( Hooke's law ). The coefficient of the proportion is called the modulus of elasticity or Young's modulus . This region of deformation is known as the linearly elastic region. Three models can describe how a solid responds to an applied stress: Many materials become weaker at high temperatures. Materials that retain their strength at high temperatures, called refractory materials , are useful for many purposes. For example, glass-ceramics have become extremely useful for countertop cooking, as they exhibit excellent mechanical properties and can sustain repeated and quick temperature changes up to 1000 °C. In

3025-471: A wide distribution of microscopic flaws that frequently play a detrimental role in the sintering process, resulting in the proliferation of cracks, and ultimate mechanical failure. Glass-ceramic materials share many properties with both non-crystalline glasses and crystalline ceramics . They are formed as a glass, and then partially crystallized by heat treatment, producing both amorphous and crystalline phases so that crystalline grains are embedded within

3146-435: A wide range of properties. Polyamorphism is a similar phenomenon where the same atoms can exist in more than one amorphous solid form. Crystallization is the process of forming a crystalline structure from a fluid or from materials dissolved in a fluid. (More rarely, crystals may be deposited directly from gas; see: epitaxy and frost .) Crystallization is a complex and extensively-studied field, because depending on

3267-414: A wide variety of polymers and plastics . Wood is a natural organic material consisting primarily of cellulose fibers embedded in a matrix of lignin . Regarding mechanical properties, the fibers are strong in tension, and the lignin matrix resists compression. Thus wood has been an important construction material since humans began building shelters and using boats. Wood to be used for construction work

3388-480: Is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's eighteenth problem . Zassenhaus (1948) showed that conversely any group that

3509-474: Is a natural organic material consisting primarily of cellulose fibers embedded in a matrix of organic lignin . In materials science, composites of more than one constituent material can be designed to have desired properties. The forces between the atoms in a solid can take a variety of forms. For example, a crystal of sodium chloride (common salt) is made up of ionic sodium and chlorine , which are held together by ionic bonds . In diamond or silicon,

3630-445: Is a noncrystalline form. Polymorphs, despite having the same atoms, may have very different properties. For example, diamond is the hardest substance known, while graphite is so soft that it is used as a lubricant. Chocolate can form six different types of crystals, but only one has the suitable hardness and melting point for candy bars and confections. Polymorphism in steel is responsible for its ability to be heat treated , giving it

3751-686: Is also a fundamental feature of many biological materials and the manner by which the structures are assembled from the molecular level up. Thus, self-assembly is emerging as a new strategy in the chemical synthesis of high performance biomaterials. Physical properties of elements and compounds that provide conclusive evidence of chemical composition include odor, color, volume, density (mass per unit volume), melting point, boiling point, heat capacity, physical form and shape at room temperature (solid, liquid or gas; cubic, trigonal crystals, etc.), hardness, porosity, index of refraction and many others. This section discusses some physical properties of materials in

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3872-439: Is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R. The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for

3993-399: Is also the n {\displaystyle n} glide, which is a glide along the half of a diagonal of a face, and the d {\displaystyle d} glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of

4114-469: Is caused by electrons, both electrons and holes contribute to current in semiconductors. Alternatively, ions support electric current in ionic conductors . Many materials also exhibit superconductivity at low temperatures; they include metallic elements such as tin and aluminium, various metallic alloys, some heavily doped semiconductors, and certain ceramics. The electrical resistivity of most electrical (metallic) conductors generally decreases gradually as

4235-416: Is commonly known as lumber or timber . In construction, wood is not only a structural material, but is also used to form the mould for concrete. Wood-based materials are also extensively used for packaging (e.g. cardboard) and paper, which are both created from the refined pulp. The chemical pulping processes use a combination of high temperature and alkaline (kraft) or acidic (sulfite) chemicals to break

4356-578: Is impossible for an ordinary periodic crystal (see crystallographic restriction theorem ). The International Union of Crystallography has redefined the term "crystal" to include both ordinary periodic crystals and quasicrystals ("any solid having an essentially discrete diffraction diagram"). Quasicrystals, first discovered in 1982, are quite rare in practice. Only about 100 solids are known to form quasicrystals, compared to about 400,000 periodic crystals known in 2004. The 2011 Nobel Prize in Chemistry

4477-483: Is its visible external shape. This is determined by the crystal structure (which restricts the possible facet orientations), the specific crystal chemistry and bonding (which may favor some facet types over others), and the conditions under which the crystal formed. By volume and weight, the largest concentrations of crystals in the Earth are part of its solid bedrock . Crystals found in rocks typically range in size from

4598-586: 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 ( krustallos ), meaning both " ice " and " rock crystal ", from κρύος ( kruos ), "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

4719-473: Is mechanically very strong, the sheets are rather loosely bound to each other. Therefore, the mechanical strength of the material is quite different depending on the direction of stress. Not all crystals have all of these properties. Conversely, these properties are not quite exclusive to crystals. They can appear in glasses or polycrystals that have been made anisotropic by working or stress —for example, stress-induced birefringence . Crystallography

4840-411: Is no long-range order in the position of the atoms. These solids are known as amorphous solids ; examples include polystyrene and glass. Whether a solid is crystalline or amorphous depends on the material involved, and the conditions in which it was formed. Solids that are formed by slow cooling will tend to be crystalline, while solids that are frozen rapidly are more likely to be amorphous. Likewise,

4961-464: Is often ceramic. For example, a continuous matrix, and a dispersed phase of ceramic particles or fibers. Applications of composite materials range from structural elements such as steel-reinforced concrete, to the thermally insulative tiles that play a key and integral role in NASA's Space Shuttle thermal protection system , which is used to protect the surface of the shuttle from the heat of re-entry into

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5082-456: Is primarily concerned with the physical and chemical properties of solids. Solid-state chemistry is especially concerned with the synthesis of novel materials, as well as the science of identification and chemical composition . The atoms, molecules or ions that make up solids may be arranged in an orderly repeating pattern, or irregularly. Materials whose constituents are arranged in a regular pattern are known as crystals . In some cases,

5203-470: Is rather easy to picture in the two-dimensional, wallpaper group case. Some of the point groups have reflections, and the reflection lines can be along the lattice directions, halfway in between them, or both. These correspond to conjugacy classes of lattice point groups in GL n ( Z ), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains

5324-604: Is the symmetry group of a repeating pattern in space, usually in three dimensions . The elements of a space group (its symmetry operations ) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups . In crystallography , space groups are also called

5445-409: Is the ability of crystals to generate a voltage in response to an applied mechanical stress. The piezoelectric effect is reversible in that piezoelectric crystals, when subjected to an externally applied voltage, can change shape by a small amount. Polymer materials like rubber, wool, hair, wood fiber, and silk often behave as electrets . For example, the polymer polyvinylidene fluoride (PVDF) exhibits

5566-465: Is the extension of Z by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Z by a finite group acting faithfully. It is essential in Bieberbach's theorems to assume that

5687-459: Is the science of measuring the crystal structure (in other words, the atomic arrangement) of a crystal. One widely used crystallography technique is X-ray diffraction . Large numbers of known crystal structures are stored in crystallographic databases . Solid The branch of physics that deals with solids is called solid-state physics , and is the main branch of condensed matter physics (which also includes liquids). Materials science

5808-400: Is the study of the behavior of solid matter under external actions such as external forces and temperature changes. A solid does not exhibit macroscopic flow, as fluids do. Any degree of departure from its original shape is called deformation . The proportion of deformation to original size is called strain. If the applied stress is sufficiently low, almost all solid materials behave in such

5929-424: Is the type of impurities present in a corundum crystal. In semiconductors , a special type of impurity, called a dopant , drastically changes the crystal's electrical properties. Semiconductor devices , such as transistors , are made possible largely by putting different semiconductor dopants into different places, in specific patterns. Twinning is a phenomenon somewhere between a crystallographic defect and

6050-483: Is used in capacitors. A capacitor is an electrical device that can store energy in the electric field between a pair of closely spaced conductors (called 'plates'). When voltage is applied to the capacitor, electric charges of equal magnitude, but opposite polarity, build up on each plate. Capacitors are used in electrical circuits as energy-storage devices, as well as in electronic filters to differentiate between high-frequency and low-frequency signals. Piezoelectricity

6171-480: The crystallographic or Fedorov groups , and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography Hahn (2002) . Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after

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6292-447: The mechanical strength of materials . Another common type of crystallographic defect is an impurity , meaning that the "wrong" type of atom is present in a crystal. For example, a perfect crystal of diamond would only contain carbon atoms, but a real crystal might perhaps contain a few boron atoms as well. These boron impurities change the diamond's color to slightly blue. Likewise, the only difference between ruby and sapphire

6413-513: The periodic table moving diagonally downward right from boron . They separate the electrical conductors (or metals, to the left) from the insulators (to the right). Devices made from semiconductor materials are the foundation of modern electronics, including radio, computers, telephones, etc. Semiconductor devices include the transistor , solar cells , diodes and integrated circuits . Solar photovoltaic panels are large semiconductor devices that directly convert light into electrical energy. In

6534-436: The 32 crystallographic point groups with the 14 Bravais lattices , each of the latter belonging to one of 7 lattice systems . What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering ),

6655-405: The Earth's atmosphere. One example is Reinforced Carbon-Carbon (RCC), the light gray material that withstands reentry temperatures up to 1,510 °C (2,750 °F) and protects the nose cap and leading edges of Space Shuttle's wings. RCC is a laminated composite material made from graphite rayon cloth and impregnated with a phenolic resin . After curing at high temperature in an autoclave,

6776-427: The aerospace industry, high performance materials used in the design of aircraft and/or spacecraft exteriors must have a high resistance to thermal shock. Thus, synthetic fibers spun out of organic polymers and polymer/ceramic/metal composite materials and fiber-reinforced polymers are now being designed with this purpose in mind. Because solids have thermal energy , their atoms vibrate about fixed mean positions within

6897-428: The air ( ice fog ) more often grow from a supersaturated gaseous-solution of water vapor and air, when the temperature of the air drops below its dew point , without passing through a liquid state. Another unusual property of water is that it expands rather than contracts when it crystallizes. Many living organisms are able to produce crystals grown from an aqueous solution , for example calcite and aragonite in

7018-493: The atoms share electrons and form covalent bonds . In metals, electrons are shared in metallic bonding . Some solids, particularly most organic compounds, are held together with van der Waals forces resulting from the polarization of the electronic charge cloud on each molecule. The dissimilarities between the types of solid result from the differences between their bonding. Metals typically are strong, dense, and good conductors of both electricity and heat . The bulk of

7139-537: The axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:( Palistrant 2012 )( Souvignier 2006 ) Table of the wallpaper groups using the classification of the 2-dimensional space groups: For each geometric class, the possible arithmetic classes are Note: An e plane is a double glide plane, one having glides in two different directions. They are found in seven orthorhombic, five tetragonal and five cubic space groups, all with centered lattice. The use of

7260-510: The case of most molluscs or hydroxylapatite in the case of bones and teeth in vertebrates . The same group of atoms can often solidify in many different ways. Polymorphism is the ability of a solid to exist in more than one crystal form. For example, water ice is ordinarily found in the hexagonal form Ice I h , but can also exist as the cubic Ice I c , the rhombohedral ice II , and many other forms. The different polymorphs are usually called different phases . In addition,

7381-630: The cell times the order of the point group. This ranges from 1 in the case of space group P1 to 192 for a space group like Fm 3 m, the NaCl structure . The elements of the space group fixing a point of space are the identity element, reflections, rotations and improper rotations , including inversion points . The translations form a normal abelian subgroup of rank 3, called the Bravais lattice (so named after French physicist Auguste Bravais ). There are 14 possible types of Bravais lattice. The quotient of

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7502-409: The cell, the glides occur in two perpendicular directions simultaneously, i.e. the same glide plane can be called b or c , a or b , a or c . 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 space groups have been modified: A screw axis is a rotation about an axis, followed by

7623-663: The chemical bonds of the lignin before burning it out. One important property of carbon in organic chemistry is that it can form certain compounds, the individual molecules of which are capable of attaching themselves to one another, thereby forming a chain or a network. The process is called polymerization and the chains or networks polymers, while the source compound is a monomer. Two main groups of polymers exist: those artificially manufactured are referred to as industrial polymers or synthetic polymers (plastics) and those naturally occurring as biopolymers. Monomers can have various chemical substituents, or functional groups, which can affect

7744-467: The chemical properties of organic compounds, such as solubility and chemical reactivity, as well as the physical properties, such as hardness, density, mechanical or tensile strength, abrasion resistance, heat resistance, transparency, color, etc.. In proteins, these differences give the polymer the ability to adopt a biologically active conformation in preference to others (see self-assembly ). People have been using natural organic polymers for centuries in

7865-404: The conditions, a single fluid can solidify into many different possible forms. It can form a single crystal , perhaps with various possible phases , stoichiometries , impurities, defects , and habits . Or, it can form a polycrystal , with various possibilities for the size, arrangement, orientation, and phase of its grains. The final form of the solid is determined by the conditions under which

7986-497: The converses are not true. An inversion and a mirror implies two-fold screw axes, and so on. There are at least ten methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names. The viewing directions of the 7 crystal systems are shown as follows. There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in

8107-409: The crystal can shrink or stretch it. Another is birefringence , where a double image appears when looking through a crystal. Moreover, various properties of a crystal, including electrical conductivity , electrical permittivity , and Young's modulus , may be different in different directions in a crystal. For example, graphite crystals consist of a stack of sheets, and although each individual sheet

8228-410: The crystal is one grain in a polycrystalline solid. The flat faces (also called facets ) of a euhedral crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal : they are planes of relatively low Miller index . This occurs because some surface orientations are more stable than others (lower surface energy ). As a crystal grows, new atoms attach easily to

8349-532: The crystals may form hexagons, such as ordinary water ice ). Crystals are commonly recognized, macroscopically, by their shape, consisting of flat faces with sharp angles. These shape characteristics are not necessary for a crystal—a crystal is scientifically defined by its microscopic atomic arrangement, not its macroscopic shape—but the characteristic macroscopic shape is often present and easy to see. Euhedral crystals are those that have obvious, well-formed flat faces. Anhedral crystals do not, usually because

8470-606: The degree of crystallization depends primarily on the conditions under which they solidified. Such rocks as granite , which have cooled very slowly and under great pressures, have completely crystallized; but many kinds of lava were poured out at the surface and cooled very rapidly, and in this latter group a small amount of amorphous or glassy matter is common. Other crystalline rocks, the metamorphic rocks such as marbles , mica-schists and quartzites , are recrystallized. This means that they were at first fragmental rocks like limestone , shale and sandstone and have never been in

8591-411: The early 1980s, Toyota researched production of an adiabatic ceramic engine with an operating temperature of over 6,000 °F (3,320 °C). Ceramic engines do not require a cooling system and hence allow a major weight reduction and therefore greater fuel efficiency. In a conventional metallic engine, much of the energy released from the fuel must be dissipated as waste heat in order to prevent

8712-453: The eight faces of the octahedron belong to another crystallographic form reflecting a different symmetry of the isometric system. A crystallographic form is described by placing the Miller indices of one of its faces within brackets. For example, the octahedral form is written as {111}, and the other faces in the form are implied by the symmetry of the crystal. Forms may be closed, meaning that

8833-543: The elements in the periodic table , those to the left of a diagonal line drawn from boron to polonium , are metals. Mixtures of two or more elements in which the major component is a metal are known as alloys . People have been using metals for a variety of purposes since prehistoric times. The strength and reliability of metals has led to their widespread use in construction of buildings and other structures, as well as in most vehicles, many appliances and tools, pipes, road signs and railroad tracks. Iron and aluminium are

8954-578: The extreme hardness of zirconia is utilized in the manufacture of knife blades, as well as other industrial cutting tools. Ceramics such as alumina , boron carbide and silicon carbide have been used in bulletproof vests to repel large-caliber rifle fire. Silicon nitride parts are used in ceramic ball bearings, where their high hardness makes them wear resistant. In general, ceramics are also chemically resistant and can be used in wet environments where steel bearings would be susceptible to oxidation (or rust). As another example of ceramic applications, in

9075-710: The final block of ice, each of the small crystals (called " crystallites " or "grains") is a true crystal with a periodic arrangement of atoms, but the whole polycrystal does not have a periodic arrangement of atoms, because the periodic pattern is broken at the grain boundaries . Most macroscopic inorganic solids are polycrystalline, including almost all metals , ceramics , ice , rocks , etc. 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,

9196-531: The fluid is being solidified, such as the chemistry of the fluid, the ambient pressure , the temperature , and the speed with which all these parameters are changing. Specific industrial techniques to produce large single crystals (called boules ) include the Czochralski process and the Bridgman technique . Other less exotic methods of crystallization may be used, depending on the physical properties of

9317-401: The following table. Each classification system is a refinement of the ones below it. To understand an explanation given here it may be necessary to understand the next one down. Arithmetic crystal classes may be interpreted as different orientations of the point groups in the lattice, with the group elements' matrix components being constrained to have integer coefficients in lattice space. This

9438-414: The form can completely enclose a volume of space, or open, meaning that it cannot. The cubic and octahedral forms are examples of closed forms. All the forms of the isometric system are closed, while all the forms of the monoclinic and triclinic crystal systems are open. A crystal's faces may all belong to the same closed form, or they may be a combination of multiple open or closed forms. A crystal's habit

9559-399: The form of snow , sea ice , and glaciers are common crystalline/polycrystalline structures on Earth and other planets. A single snowflake is a single crystal or a collection of crystals, while an ice cube is a polycrystal . Ice crystals may form from cooling liquid water below its freezing point, such as ice cubes or a frozen lake. Frost , snowflakes, or small ice crystals suspended in

9680-433: The form of waxes and shellac , which is classified as a thermoplastic polymer. A plant polymer named cellulose provided the tensile strength for natural fibers and ropes, and by the early 19th century natural rubber was in widespread use. Polymers are the raw materials (the resins) used to make what are commonly called plastics. Plastics are the final product, created after one or more polymers or additives have been added to

9801-535: The glass-ceramic has a net coefficient of thermal expansion close to zero. This type of glass-ceramic exhibits excellent mechanical properties and can sustain repeated and quick temperature changes up to 1000 °C. Glass ceramics may also occur naturally when lightning strikes the crystalline (e.g. quartz) grains found in most beach sand . In this case, the extreme and immediate heat of the lightning (~2500 °C) creates hollow, branching rootlike structures called fulgurite via fusion . Organic chemistry studies

9922-673: The gold slabs (1064 °C); and metallic nanowires are much stronger than the corresponding bulk metals. The high surface area of nanoparticles makes them extremely attractive for certain applications in the field of energy. For example, platinum metals may provide improvements as automotive fuel catalysts , as well as proton exchange membrane (PEM) fuel cells. Also, ceramic oxides (or cermets) of lanthanum , cerium , manganese and nickel are now being developed as solid oxide fuel cells (SOFC). Lithium, lithium-titanate and tantalum nanoparticles are being applied in lithium-ion batteries. Silicon nanoparticles have been shown to dramatically expand

10043-475: The group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain

10164-743: The group elements can include time reversal as reflection in it. They are of importance in magnetic structures that contain ordered unpaired spins, i.e. ferro- , ferri- or antiferromagnetic structures as studied by neutron diffraction . The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D ( Kim 1999 , p.428). It has also been possible to construct magnetic versions for other overall and lattice dimensions ( Daniel Litvin's papers , ( Litvin 2008 ), ( Litvin 2005 )). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and

10285-420: The groups with a different method, but omitted four groups (Fdd2, I 4 2d, P 4 2 1 d, and P 4 2 1 c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is incorrect. Burckhardt (1967) describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of

10406-434: The identity. The matrices M form a point group that is a basis of the space group; the lattice must be symmetric under that point group, but the crystal structure itself may not be symmetric under that point group as applied to any particular point (that is, without a translation). For example, the diamond cubic structure does not have any point where the cubic point group applies. The lattice dimension can be less than

10527-556: The incoming light prior to capture. Here again, surface area of the nanoparticles (and thin films) plays a critical role in maximizing the amount of absorbed radiation. Many natural (or biological) materials are complex composites with remarkable mechanical properties. These complex structures, which have risen from hundreds of million years of evolution, are inspiring materials scientists in the design of novel materials. Their defining characteristics include structural hierarchy, multifunctionality and self-healing capability. Self-organization

10648-457: The interlayer bonding in graphite . Substances such as fats , lipids and wax form molecular bonds because the large molecules do not pack as tightly as atomic bonds. This leads to crystals that are much softer and more easily pulled apart or broken. Common examples include chocolates, candles, or viruses. Water ice and dry ice are examples of other materials with molecular bonding. Polymer materials generally will form crystalline regions, but

10769-448: The ions and the electron cloud. The large number of free electrons gives metals their high values of electrical and thermal conductivity. The free electrons also prevent transmission of visible light, making metals opaque, shiny and lustrous . More advanced models of metal properties consider the effect of the positive ions cores on the delocalised electrons. As most metals have crystalline structure, those ions are usually arranged into

10890-499: The laminate is pyrolized to convert the resin to carbon, impregnated with furfural alcohol in a vacuum chamber, and cured/pyrolized to convert the furfural alcohol to carbon. In order to provide oxidation resistance for reuse capability, the outer layers of the RCC are converted to silicon carbide. Domestic examples of composites can be seen in the "plastic" casings of television sets, cell-phones and so on. These plastic casings are usually

11011-471: The lengths of the molecules usually prevent complete crystallization—and sometimes polymers are completely amorphous. A quasicrystal consists of arrays of atoms that are ordered but not strictly periodic. They have many attributes in common with ordinary crystals, such as displaying a discrete pattern in x-ray diffraction , and the ability to form shapes with smooth, flat faces. Quasicrystals are most famous for their ability to show five-fold symmetry, which

11132-418: The matrix properties. A synergism produces material properties unavailable from the individual constituent materials, while the wide variety of matrix and strengthening materials provides the designer with the choice of an optimum combination. Semiconductors are materials that have an electrical resistivity (and conductivity) between that of metallic conductors and non-metallic insulators. They can be found in

11253-543: The much more difficult classification of space groups had largely been completed. In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality . More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer Evgraf Fedorov and the German mathematician Arthur Moritz Schoenflies noticed that two of them were really

11374-409: The ordered (or disordered) lattice. The spectrum of lattice vibrations in a crystalline or glassy network provides the foundation for the kinetic theory of solids . This motion occurs at the atomic level, and thus cannot be observed or detected without highly specialized equipment, such as that used in spectroscopy . Thermal properties of solids include thermal conductivity , which is the property of

11495-477: The overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension): The 65 "Sohncke" space groups, not containing any mirrors, inversion points, improper rotations or glide planes, yield chiral crystals, not identical to their mirror image; whereas space groups that do include at least one of those give achiral crystals. Achiral molecules sometimes form chiral crystals, but chiral molecules always form chiral crystals, in one of

11616-409: The parallel lattice vector. So, 2 1 is a twofold rotation followed by a translation of 1/2 of the lattice vector. The general formula for the action of an element of a space group is where M is its matrix, D is its vector, and where the element transforms point x into point y . In general, D = D ( lattice ) + D ( M ), where D ( M ) is a unique function of M that is zero for M being

11737-425: The point group symmetry operations of reflection , rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The number of replicates of the asymmetric unit in a unit cell is thus the number of lattice points in

11858-418: The point group. Conway , Delgado Friedrichs, and Huson et al. ( 2001 ) gave another classification of the space groups, called a fibrifold notation , according to the fibrifold structures on the corresponding orbifold . They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 wallpaper groups , and

11979-413: The process of forming a glass does not release the latent heat of fusion , but forming a crystal does. A crystal structure (an arrangement of atoms in a crystal) is characterized by its unit 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

12100-499: The production of polycrystalline transparent ceramics such as transparent alumina and alumina compounds for such applications as high-power lasers. Advanced ceramics are also used in the medicine, electrical and electronics industries. Ceramic engineering is the science and technology of creating solid-state ceramic materials, parts and devices. This is done either by the action of heat, or, at lower temperatures, using precipitation reactions from chemical solutions. The term includes

12221-439: The purification of raw materials, the study and production of the chemical compounds concerned, their formation into components, and the study of their structure, composition and properties. Mechanically speaking, ceramic materials are brittle, hard, strong in compression and weak in shearing and tension. Brittle materials may exhibit significant tensile strength by supporting a static load. Toughness indicates how much energy

12342-507: The regular ordering can continue unbroken over a large scale, for example diamonds, where each diamond is a single crystal . Solid objects that are large enough to see and handle are rarely composed of a single crystal, but instead are made of a large number of single crystals, known as crystallites , whose size can vary from a few nanometers to several meters. Such materials are called polycrystalline . Almost all common metals, and many ceramics , are polycrystalline. In other materials, there

12463-419: The remaining 35 irreducible groups are the same as the cubic groups and are classified separately. In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n -dimensional Euclidean space with a compact fundamental domain. Bieberbach ( 1911 , 1912 ) proved that the subgroup of translations of any such group contains n linearly independent translations, and

12584-431: The requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries (230 is commonly cited, but this treats chiral equivalents as separate entities), called crystallographic space groups . These are grouped into 7 crystal systems , such as cubic crystal system (where the crystals may form cubes or rectangular boxes, such as halite shown at right) or hexagonal crystal system (where

12705-423: The rougher and less stable parts of the surface, but less easily to the flat, stable surfaces. Therefore, the flat surfaces tend to grow larger and smoother, until the whole crystal surface consists of these plane surfaces. (See diagram on right.) One of the oldest techniques in the science of crystallography consists of measuring the three-dimensional orientations of the faces of a crystal, and using them to infer

12826-444: The same atoms may be able to form noncrystalline phases . For example, water can also form amorphous ice , while SiO 2 can form both fused silica (an amorphous glass) and quartz (a crystal). Likewise, if a substance can form crystals, it can also form polycrystals. For pure chemical elements, polymorphism is known as allotropy . For example, diamond and graphite are two crystalline forms of carbon , while amorphous carbon

12947-495: The same. The space groups in three dimensions were first enumerated in 1891 by Fedorov (whose list had two omissions (I 4 3d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies (whose list had four omissions (I 4 3d, Pc, Cc, ?) and one duplication (P 4 2 1 m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. William Barlow  ( 1894 ) later enumerated

13068-527: The solid state. The mechanical properties of materials describe characteristics such as their strength and resistance to deformation. For example, steel beams are used in construction because of their high strength, meaning that they neither break nor bend significantly under the applied load. Mechanical properties include elasticity , plasticity , tensile strength , compressive strength , shear strength , fracture toughness , ductility (low in brittle materials) and indentation hardness . Solid mechanics

13189-414: The space group by the Bravais lattice is a finite group which is one of the 32 possible point groups . A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a {\displaystyle a} , b {\displaystyle b} , or c {\displaystyle c} , depending on which axis the glide is along. There

13310-415: The space groups that permit this. Among the 65 Sohncke groups are 22 that come in 11 enantiomorphic pairs. Only certain combinations of symmetry elements are possible in a space group. Translations are always present, and the space group P1 has only translations and the identity element. The presence of mirrors implies glide planes as well, and the presence of rotation axes implies screw axes as well, but

13431-447: The specific crystal structure adopted by a crystalline solid depends on the material involved and on how it was formed. While many common objects, such as an ice cube or a coin, are chemically identical throughout, many other common materials comprise a number of different substances packed together. For example, a typical rock is an aggregate of several different minerals and mineraloids , with no specific chemical composition. Wood

13552-479: The storage capacity of lithium-ion batteries during the expansion/contraction cycle. Silicon nanowires cycle without significant degradation and present the potential for use in batteries with greatly expanded storage times. Silicon nanoparticles are also being used in new forms of solar energy cells. Thin film deposition of silicon quantum dots on the polycrystalline silicon substrate of a photovoltaic (solar) cell increases voltage output as much as 60% by fluorescing

13673-442: The structure, properties, composition, reactions, and preparation by synthesis (or other means) of chemical compounds of carbon and hydrogen , which may contain any number of other elements such as nitrogen , oxygen and the halogens: fluorine , chlorine , bromine and iodine . Some organic compounds may also contain the elements phosphorus or sulfur . Examples of organic solids include wood, paraffin wax , naphthalene and

13794-743: The substance, including hydrothermal synthesis , sublimation , or simply solvent-based crystallization . Large single crystals can be created by geological processes. For example, selenite crystals in excess of 10  m are found in the Cave of the Crystals in Naica, Mexico. For more details on geological crystal formation, see above . Crystals can also be formed by biological processes, see above . Conversely, some organisms have special techniques to prevent crystallization from occurring, such as antifreeze proteins . An ideal crystal has every atom in

13915-448: The symbol e became official with Hahn (2002) . The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system

14036-508: The temperature is lowered, but remains finite. In a superconductor, however, the resistance drops abruptly to zero when the material is cooled below its critical temperature. An electric current flowing in a loop of superconducting wire can persist indefinitely with no power source. A dielectric , or electrical insulator, is a substance that is highly resistant to the flow of electric current. A dielectric, such as plastic, tends to concentrate an applied electric field within itself, which property

14157-794: The two most commonly used structural metals. They are also the most abundant metals in the Earth's crust . Iron is most commonly used in the form of an alloy, steel, which contains up to 2.1% carbon , making it much harder than pure iron. Because metals are good conductors of electricity, they are valuable in electrical appliances and for carrying an electric current over long distances with little energy loss or dissipation. Thus, electrical power grids rely on metal cables to distribute electricity. Home electrical systems, for example, are wired with copper for its good conducting properties and easy machinability. The high thermal conductivity of most metals also makes them useful for stovetop cooking utensils. The study of metallic elements and their alloys makes up

14278-429: The underlying crystal symmetry . A crystal's crystallographic forms are sets of possible faces of the crystal that are related by one of the symmetries of the crystal. For example, crystals of galena often take the shape of cubes, and the six faces of the cube belong to a crystallographic form that displays one of the symmetries of the isometric crystal system . Galena also sometimes crystallizes as octahedrons, and

14399-619: The vacuum of space. The slow cooling may allow the precipitation of a separate phase within the crystal lattice, which form at specific angles determined by the lattice, called Widmanstatten patterns . Ionic compounds typically form when a metal reacts with a non-metal, such as sodium with chlorine. These often form substances called salts, such as sodium chloride (table salt) or potassium nitrate ( saltpeter ), with crystals that are often brittle and cleave relatively easily. Ionic materials are usually crystalline or polycrystalline. In practice, large salt crystals can be created by solidification of

14520-464: The wear plates of crushing equipment in mining operations. Most ceramic materials, such as alumina and its compounds, are formed from fine powders, yielding a fine grained polycrystalline microstructure that is filled with light-scattering centers comparable to the wavelength of visible light . Thus, they are generally opaque materials, as opposed to transparent materials . Recent nanoscale (e.g. sol-gel ) technology has, however, made possible

14641-399: Was awarded to Dan Shechtman for the discovery of quasicrystals. Crystals can have certain special electrical, optical, and mechanical properties that glass and polycrystals normally cannot. These properties are related to the anisotropy of the crystal, i.e. the lack of rotational symmetry in its atomic arrangement. One such property is the piezoelectric effect , where a voltage across

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