The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 adopt this structure, including α-tin, the semiconductors silicon and germanium, silicon/germanium alloys in any proportion. Diamond's cubic structure is in the Fd3m space group, which follows the face-centered cubic Bravais lattice; the lattice describes the repeat pattern. The diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by 1/4 of the width of the unit cell in each dimension. Many compound semiconductors such as gallium arsenide, β-silicon carbide, indium antimonide adopt the analogous zincblende structure, where each atom has nearest neighbors of an unlike element. Zincblende's space group is F43m, but many of its structural properties are quite similar to the diamond structure; the atomic packing factor of the diamond cubic structure is π√3/16 ≈ 0.34 smaller than the packing factors for the face-centered and body-centered cubic lattices.
Zincblende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms. The first-, second-, third-, fourth- and fifth-nearest-neighbor distances in units of the cubic lattice constant are √3/4, √2/2, √11/4, 1 and √19/4, respectively. Mathematically, the points of the diamond cubic structure can be given coordinates as a subset of a three-dimensional integer lattice by using a cubic unit cell four units across. With these coordinates, the points of the structure have coordinates satisfying the equations x = y = z, x + y + z = 0 or 1. There are eight points that satisfy these conditions:, All of the other points in the structure may be obtained by adding multiples of four to the x, y, z coordinates of these eight points. Adjacent points in this structure are at distance √3 apart in the integer lattice; this structure may be scaled to a cubical unit cell, some number a of units across by multiplying all coordinates by a/4. Alternatively, each point of the diamond cubic structure may be given by four-dimensional integer coordinates whose sum is either zero or one.
Two points are adjacent in the diamond structure if and only if their four-dimensional coordinates differ by one in a single coordinate. The total difference in coordinate values between any two points gives the number of edges in the shortest path between them in the diamond structure; the four nearest neighbors of each point may be obtained, in this coordinate system, by adding one to each of the four coordinates, or by subtracting one from each of the four coordinates, accordingly as the coordinate sum is zero or one. These four-dimensional coordinates may be transformed into three-dimensional coordinates by the formula →; because the diamond structure forms a distance-preserving subset of the four-dimensional integer lattice, it is a partial cube. Yet another coordinatization of the diamond cubic involves the removal of some of the edges from a three-dimensional grid graph. In this coordinatization, which has a distorted geometry from the standard diamond cubic structure but has the same topological structure, the vertices of the diamond cubic are represented by all possible 3d grid points and the edges of the diamond cubic are represented by a subset of the 3d grid edges.
The diamond cubic is sometimes called the "diamond lattice" but it is not, mathematically, a lattice: there is no translational symmetry that takes the point into the point, for instance. However, it is still a symmetric structure: any incident pair of a vertex and edge can be transformed into any other incident pair by a congruence of Euclidean space. Moreover, the diamond crystal as a network in space has a strong isotropic property. Namely, for any two vertices x and y of the crystal net, for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the ordered y-edge. Another crystal with this property is the Laves graph; the compressive strength and hardness of diamond and various other materials, such as boron nitride, is attributed to the diamond cubic structure. Truss systems that follow the diamond cubic geometry have a high capacity to withstand compression, by minimizing the unbraced length of individual struts.
The diamond cubic geometry has been considered for the purpose of providing structural rigidity though structures composed of skeletal triangles, such as the octet truss, have been found to be more effective for this purpose. Allotropes of carbon Crystallography Laves graph Triakis truncated tetrahedral honeycomb Media related to Diamond cubic at Wikimedia Commons Software to construct self avoiding random walks on the diamond cubic lattice
Low-energy electron diffraction
Low-Energy electron diffraction is a technique for the determination of the surface structure of single-crystalline materials by bombardment with a collimated beam of low energy electrons and observation of diffracted electrons as spots on a fluorescent screen. LEED may be used in one of two ways: Qualitatively, where the diffraction pattern is recorded and analysis of the spot positions gives information on the symmetry of the surface structure. In the presence of an adsorbate the qualitative analysis may reveal information about the size and rotational alignment of the adsorbate unit cell with respect to the substrate unit cell. Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I-V curves. By comparison with theoretical curves, these may provide accurate information on atomic positions on the surface at hand; the theoretical possibility of the occurrence of electron diffraction first emerged in 1924 when Louis de Broglie introduced wave mechanics and proposed the wavelike nature of all particles.
In his Nobel laureated work de Broglie postulated that the wavelength of a particle with linear momentum p is given by h/p, where h is Planck's constant. The de Broglie hypothesis was confirmed experimentally at Bell Labs in 1927 when Clinton Davisson and Lester Germer fired low-energy electrons at a crystalline nickel target and observed that the angular dependence of the intensity of backscattered electrons showed diffraction patterns; these observations were consistent with the diffraction theory for X-rays developed by Bragg and Laue earlier. Before the acceptance of the de Broglie hypothesis diffraction was believed to be an exclusive property of waves. Davisson and Germer published notes of their electron diffraction experiment result in Nature and in Physical Review in 1927. One month after Davisson and Germer's work appeared and Reid published their electron diffraction work with higher kinetic energy in the same journal; those experiments revealed the wave property of electrons and opened up an era of electron diffraction study.
Though discovered in 1927, low energy electron diffraction did not become a popular tool for surface analysis until the early 1960s. The main reasons were that monitoring directions and intensities of diffracted beams was a difficult experimental process due to inadequate vacuum techniques and slow detection methods such as a Faraday cup. Since LEED is a surface sensitive method, it required well-ordered surface structures. Techniques for the preparation of clean metal surfaces first became available much later. Nonetheless, H. E. Farnsworth and coworkers at Brown University pioneered the use of LEED as a method for characterizing the absorption of gases onto clean metal surfaces and the associated regular adsorption phases, starting shortly after the Davisson and Germer discovery into the 1970s. In the early 1960s LEED experienced a renaissance, as ultra high vacuum became available and the post acceleration detection method was introduced by none less than Germer and his coworkers at Bell Labs using a flat phosphor screen.
Using this technique diffracted electrons were accelerated to high energies to produce clear and visible diffraction patterns on a fluorescent screen. The post-acceleration method had been proposed by Ehrenberg in 1934. In 1962 Lander and colleagues introduced the modern hemispherical screen with associated hemispherical grids. In the mid sixties, modern LEED systems became commercially available as part of the ultra-high vacuum instrumentation suite by Varian Associates and triggered an enormous boost of activities in surface science. Notably future Nobel prize winner Gerhard Ertl started his studies of surface chemistry and catalysis on such a Varian system, it soon became clear that the kinematic theory, used to explain X-ray diffraction experiments, was inadequate for the quantitative interpretation of experimental data obtained from LEED. At this stage a detailed determination of surface structures, including adsorption sites, bond angles and bond lengths was not possible. A dynamical electron diffraction theory which took into account the possibility of multiple scattering was established in the late 1960s.
With this theory it became possible to reproduce experimental data with high precision. In order to keep the studied sample clean and free from unwanted adsorbates, LEED experiments are performed in an ultra-high-vacuum environment; the most important elements in a LEED experiment are: A sample holder with the prepared sample An electron gun A display system a hemispherical fluorescent screen on which the diffraction pattern can be observed directly A sputtering gun for cleaning the surface An Auger electron spectroscopy system in order to determine the purity of the surface. A simplified sketch of an LEED setup is shown in figure 2; the sample is prepared outside the vacuum chamber by cutting a slice of around 1 mm in thickness and 1 cm in diameter along the desired crystallographic axis. The correct alignment of the crystal can be achieved with the help of x-ray methods and should be within 1° of the desired angle. After being mounted in the UHV chamber the sample is chemically flattened. Unwanted surface contaminants are removed by ion sputtering or by chemical processes such as oxidation and reduction cycles.
The surface is flattened by annealing at high temperatures. Once a clean and well-defined surface is prepared, monolayers can be adsorbed on the surface by exposing it to a gas consisting of the desired adsorbate atoms or molecules; the annealing process will let bulk impurities diffuse to the surface and
Gerd Binnig is a German physicist, who won the Nobel Prize in Physics in 1986 for the invention of the scanning tunneling microscope. He was played in the ruins of the city during his childhood, his family lived in Frankfurt and in Offenbach am Main, he attended school in both cities. At the age of 10, he decided to become a physicist, but he soon wondered whether he had made the right choice, he concentrated more on music. He started playing the violin at 15 and played in his school orchestra. Binnig studied physics at the J. W. Goethe University in Frankfurt, gaining a bachelor's degree in 1973 and remaining there do a PhD with in Werner Martienssen's group, supervised by Eckhardt Hoenig. In 1969, he married Lore Wagler, a psychologist, they have a daughter born in Switzerland and a son born in California, his hobbies are reading and golf. In 1978, he accepted an offer from IBM to join their Zürich research group, where he worked with Heinrich Rohrer, Christoph Gerber and Edmund Weibel. There they developed the scanning tunneling microscope, an instrument for imaging surfaces at the atomic level.
The Nobel committee described the effect that the invention of the STM had on science, saying that "entirely new fields are opening up for the study of the structure of matter." The physical principles on which the STM was based were known before the IBM team developed the STM, but Binnig and his colleagues were the first to solve the significant experimental challenges involved in putting it into effect. The IBM Zürich team were soon recognized with a number of prizes: the German Physics Prize, the Otto Klung Prize, the Hewlett Packard Prize and the King Faisal Prize. In 1986, Binnig and Rohrer shared half of the Nobel Prize in Physics, the other half of the Prize was awarded to Ernst Ruska. From 1985-1988, he worked in California, he was at IBM in Almaden Valley, was visiting professor at Stanford University. In 1985, Binnig invented the atomic force microscope and Binnig, Christoph Gerber and Calvin Quate went on to develop a working version of this new microscope for insulating surfaces.
In 1987 Binnig was appointed IBM Fellow. In the same year, he started the IBM Physics group Munich, working on creativity and atomic force microscopy In 1994 Professor Gerd Binnig founded Definiens which turned in the year 2000 into a commercial enterprise; the company developed Cognition Network Technology to analyze images just like the human eye and brain are capable of doing.in 2016, Binnig won the Kavli Prize in Nanoscience. He became a fellow of the Norwegian Academy of Letters; the Binnig and Rohrer Nanotechnology Center, an IBM-owned research facility in Rüschlikon, Zürich is named after Gerd Binnig and Heinrich Rohrer. Pioneers in Electricity and Magnetism - Gerd Binnig National High Magnetic Field Laboratory Autobiography of Gerd Binnig Astra Zeneca acquires Definiens
Atomic force microscopy
Atomic force microscopy or scanning force microscopy is a very-high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit. AFM is a type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit; the information is gathered by "feeling" or "touching" the surface with a mechanical probe. Piezoelectric elements that facilitate tiny but accurate and precise movements on command enable precise scanning; the AFM has three major abilities: force measurement and manipulation. In force measurement, AFMs can be used to measure the forces between the probe and the sample as a function of their mutual separation; this can be applied to perform force spectroscopy, to measure the mechanical properties of the sample, such as the sample's Young's modulus, a measure of stiffness.
For imaging, the reaction of the probe to the forces that the sample imposes on it can be used to form an image of the three-dimensional shape of a sample surface at a high resolution. This is achieved by raster scanning the position of the sample with respect to the tip and recording the height of the probe that corresponds to a constant probe-sample interaction; the surface topography is displayed as a pseudocolor plot. In manipulation, the forces between tip and sample can be used to change the properties of the sample in a controlled way. Examples of this include atomic manipulation, scanning probe lithography and local stimulation of cells. Simultaneous with the acquisition of topographical images, other properties of the sample can be measured locally and displayed as an image with high resolution. Examples of such properties are mechanical properties like stiffness or adhesion strength and electrical properties such as conductivity or surface potential. In fact, the majority of SPM techniques are extensions of AFM.
The major difference between atomic force microscopy and competing technologies such as optical microscopy and electron microscopy is that AFM does not use lenses or beam irradiation. Therefore, it does not suffer from a limitation in spatial resolution due to diffraction and aberration, preparing a space for guiding the beam and staining the sample are not necessary. There are several types of scanning microscopy including scanning probe microscopy. Although SNOM and STED use visible, infrared or terahertz light to illuminate the sample, their resolution is not constrained by the diffraction limit. Fig. 3 shows an AFM, which consists of the following features. Numbers in parentheses correspond to numbered features in Fig. 3. Coordinate directions are defined by the coordinate system; the small spring-like cantilever is carried by the support. Optionally, a piezoelectric element oscillates the cantilever; the sharp tip is fixed to the free end of the cantilever. The detector records the motion of the cantilever.
The sample is mounted on the sample stage. An xyz drive permits to displace the sample and the sample stage in x, y, z directions with respect to the tip apex. Although Fig. 3 shows the drive attached to the sample, the drive can be attached to the tip, or independent drives can be attached to both, since it is the relative displacement of the sample and tip that needs to be controlled. Controllers and plotter are not shown in Fig. 3. According to the configuration described above, the interaction between tip and sample, which can be an atomic scale phenomenon, is transduced into changes of the motion of cantilever, a macro scale phenomenon. Several different aspects of the cantilever motion can be used to quantify the interaction between the tip and sample, most the value of the deflection, the amplitude of an imposed oscillation of the cantilever, or the shift in resonance frequency of the cantilever; the detector of AFM measures the deflection of the cantilever and converts it into an electrical signal.
The intensity of this signal will be proportional to the displacement of the cantilever. Various methods of detection can be used, e.g. interferometry, optical levers, the piezoresistive method, the piezoelectric method, STM-based detectors. Note: The following paragraphs assume that'contact mode' is used. For other imaging modes, the process is similar, except that'deflection' should be replaced by the appropriate feedback variable; when using the AFM to image a sample, the tip is brought into contact with the sample, the sample is raster scanned along an x-y grid. Most an electronic feedback loop is employed to keep the probe-sample force constant during scanning; this feedback loop has the cantilever deflection as input, its output controls the distance along the z axis between the probe support and the sample support. As long as the tip remains in contact with the sample, the sample is scanned in the x-y plane, height variations in the sample will change the deflection of the cantilever; the feedback adjusts the height of the probe support so that the deflection is restored to a user-d
Van der Waals force
In molecular physics, the van der Waals force, named after Dutch scientist Johannes Diderik van der Waals, is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; the Van der Waals force vanishes at longer distances between interacting molecules. Van der Waals force plays a fundamental role in fields as diverse as supramolecular chemistry, structural biology, polymer science, surface science, condensed matter physics, it underlies many properties of organic compounds and molecular solids, including their solubility in polar and non-polar media. If no other force is present, the distance between atoms at which the force becomes repulsive rather than attractive as the atoms approach one another is called the van der Waals contact distance; the van der Waals force has the same origin as the Casimir effect, arising from quantum interactions with the zero-point field. The term van der; the term always includes the London dispersion force between instantaneously induced dipoles.
It is sometimes applied to the Debye force between a permanent dipole and a corresponding induced dipole or to the Keesom force between permanent molecular dipoles. Van der Waals forces include attraction and repulsions between atoms and surfaces, as well as other intermolecular forces, they differ from covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby particles. Being the weakest of the weakest chemical forces, with a strength between 0.4 and 4kJ/mol, they may still support an integral structural load when multitudes of such interactions are present. Such a force results from a transient shift in electron density; the electron density may temporarily shift more to one side of the nucleus. This generates a transient charge to which a nearby atom can be either repelled; when the interatomic distance of two atoms is greater than 0.6 nm the force is not strong enough to be observed. In the same vein, when the interatomic distance is below 0.4 nm the force becomes repulsive.
Intermolecular forces have four major contributions: A repulsive component resulting from the Pauli exclusion principle that prevents the collapse of molecules. Attractive or repulsive electrostatic interactions between permanent charges, quadrupoles, in general between permanent multipoles; the electrostatic interaction is sometimes called the Keesom interaction or Keesom force after Willem Hendrik Keesom. Induction, the attractive interaction between a permanent multipole on one molecule with an induced multipole on another; this interaction is sometimes called Debye force after Peter J. W. Debye. Dispersion, the attractive interaction between any pair of molecules, including non-polar atoms, arising from the interactions of instantaneous multipoles. Returning to nomenclature, different texts refer to different things using the term "van der Waals force"; some texts describe the van der Waals force as the totality of forces. All intermolecular/van der Waals forces are anisotropic, which means that they depend on the relative orientation of the molecules.
The induction and dispersion interactions are always attractive, irrespective of orientation, but the electrostatic interaction changes sign upon rotation of the molecules. That is, the electrostatic force can be attractive or repulsive, depending on the mutual orientation of the molecules; when molecules are in thermal motion, as they are in the gas and liquid phase, the electrostatic force is averaged out to a large extent, because the molecules thermally rotate and thus probe both repulsive and attractive parts of the electrostatic force. Sometimes this effect is expressed by the statement that "random thermal motion around room temperature can overcome or disrupt them"; the thermal averaging effect is much less pronounced for the attractive induction and dispersion forces. The Lennard-Jones potential is used as an approximate model for the isotropic part of a total van der Waals force as a function of distance. Van der Waals forces are responsible for certain cases of pressure broadening of spectral lines and the formation of van der Waals molecules.
The London-van der Waals forces are related to the Casimir effect for dielectric media, the former being the microscopic description of the latter bulk property. The first detailed calculations of this were done in 1955 by E. M. Lifshitz. A more general theory of van der Waals forces has been developed; the main characteristics of van der Waals forces are: They are weaker than normal covalent and ionic bonds. Van der Waals forces can not be saturated, they have no directional characteristic. They are all short-range forces and hence only interactions between the nearest particles need to be considered. Van der Waals attraction is greater. Van der Waals forces are independent
Gold is a chemical element with symbol Au and atomic number 79, making it one of the higher atomic number elements that occur naturally. In its purest form, it is a bright reddish yellow, soft and ductile metal. Chemically, gold is a group 11 element, it is solid under standard conditions. Gold occurs in free elemental form, as nuggets or grains, in rocks, in veins, in alluvial deposits, it occurs in a solid solution series with the native element silver and naturally alloyed with copper and palladium. Less it occurs in minerals as gold compounds with tellurium. Gold is resistant to most acids, though it does dissolve in aqua regia, a mixture of nitric acid and hydrochloric acid, which forms a soluble tetrachloroaurate anion. Gold is insoluble in nitric acid, which dissolves silver and base metals, a property that has long been used to refine gold and to confirm the presence of gold in metallic objects, giving rise to the term acid test. Gold dissolves in alkaline solutions of cyanide, which are used in mining and electroplating.
Gold dissolves in mercury, forming amalgam alloys. A rare element, gold is a precious metal, used for coinage and other arts throughout recorded history. In the past, a gold standard was implemented as a monetary policy, but gold coins ceased to be minted as a circulating currency in the 1930s, the world gold standard was abandoned for a fiat currency system after 1971. A total of 186,700 tonnes of gold exists above ground, as of 2015; the world consumption of new gold produced is about 50% in jewelry, 40% in investments, 10% in industry. Gold's high malleability, resistance to corrosion and most other chemical reactions, conductivity of electricity have led to its continued use in corrosion resistant electrical connectors in all types of computerized devices. Gold is used in infrared shielding, colored-glass production, gold leafing, tooth restoration. Certain gold salts are still used as anti-inflammatories in medicine; as of 2017, the world's largest gold producer by far was China with 440 tonnes per year.
Gold is the most malleable of all metals. It can be drawn into a monoatomic wire, stretched about twice before it breaks; such nanowires distort via formation and migration of dislocations and crystal twins without noticeable hardening. A single gram of gold can be beaten into a sheet of 1 square meter, an avoirdupois ounce into 300 square feet. Gold leaf can be beaten thin enough to become semi-transparent; the transmitted light appears greenish blue, because gold reflects yellow and red. Such semi-transparent sheets strongly reflect infrared light, making them useful as infrared shields in visors of heat-resistant suits, in sun-visors for spacesuits. Gold is a good conductor of electricity. Gold has a density of 19.3 g/cm3 identical to that of tungsten at 19.25 g/cm3. By comparison, the density of lead is 11.34 g/cm3, that of the densest element, osmium, is 22.588±0.015 g/cm3. Whereas most metals are gray or silvery white, gold is reddish-yellow; this color is determined by the frequency of plasma oscillations among the metal's valence electrons, in the ultraviolet range for most metals but in the visible range for gold due to relativistic effects affecting the orbitals around gold atoms.
Similar effects impart a golden hue to metallic caesium. Common colored gold alloys include the distinctive eighteen-karat rose gold created by the addition of copper. Alloys containing palladium or nickel are important in commercial jewelry as these produce white gold alloys. Fourteen-karat gold-copper alloy is nearly identical in color to certain bronze alloys, both may be used to produce police and other badges. White gold alloys can be made with nickel. Fourteen- and eighteen-karat gold alloys with silver alone appear greenish-yellow and are referred to as green gold. Blue gold can be made by alloying with iron, purple gold can be made by alloying with aluminium. Less addition of manganese, aluminium and other elements can produce more unusual colors of gold for various applications. Colloidal gold, used by electron-microscopists, is red. Gold has only one stable isotope, 197Au, its only occurring isotope, so gold is both a mononuclidic and monoisotopic element. Thirty-six radioisotopes have been synthesized, ranging in atomic mass from 169 to 205.
The most stable of these is 195Au with a half-life of 186.1 days. The least stable is 171Au. Most of gold's radioisotopes with atomic masses below 197 decay by some combination of proton emission, α decay, β+ decay; the exceptions are 195Au, which decays by electron capture, 196Au, which decays most by electron capture with a minor β− decay path. All of gold's radioisotopes with atomic masses above 197 decay by β− decay. At least 32 nuclear isomers have been characterized, ranging in atomic mass from 170 to 200. Within that range, only 178Au, 180Au, 181Au, 182Au, 188Au do not have isomers. Gold's most stable isomer is 198m2Au with a half-life of 2.27 days. Gold's least stable isomer is 177m2Au with a half-life of only 7 ns. 184m1Au has three decay paths: β+ decay, isomeric
A crystal or crystalline solid is a solid material whose constituents are arranged in a ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations; the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification; the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both "ice" and "rock crystal", from κρύος, "icy cold, frost". Examples of large crystals include snowflakes 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 and ice. A third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever.
Examples of amorphous solids include glass and many plastics. Despite the name, lead crystal, crystal glass, related products are not crystals, but rather types of glass, i.e. amorphous solids. Crystals are used in pseudoscientific practices such as crystal therapy, 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 microscopic arrangement of atoms inside it, called the crystal structure. A crystal is a solid where the atoms form a periodic arrangement.. 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 the final block of ice, each of the small crystals 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 all metals, ice, etc. Solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids called glassy, vitreous, or noncrystalline; these have no periodic order 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 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 the requirement that the unit cells stack with no gaps. There are 219 possible crystal symmetries, called crystallographic space groups; these are grouped into 7 crystal systems, such as hexagonal crystal system. Crystals are recognized 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 present and easy to see. Euhedral crystals are those with well-formed flat faces. Anhedral crystals do not because the crystal is one grain in a polycrystalline solid; the flat faces of a euhedral crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal: they are planes of low Miller index. This occurs; as a crystal grows, new atoms attach to the rougher and less stable parts of the surface, but less 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. One of the oldest techniques in the science of crystallography consists of measuring the three-dimensional orientations of the faces of a crystal, using them to infer the underlying crystal symmetry.
A crystal's habit is its visible external shape. This is determined by the crystal structure, the specific crystal chemistry and bonding, 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 range in size from a fraction of a millimetre to several centimetres across, although exceptionally large crystals are found; as of 1999, the world's largest known occurring crystal is a crystal of beryl from Malakialina, Madagascar, 18 m long and 3.5 m in diameter, weighing 380,000 kg. 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 the degree of crystallization depends on the conditions under which they solidified. Such rocks as granite, which have cooled slowly and under great pressures, have crystallized.