Oceanography

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Oceanography 

Thermohaline circulation

Oceanography (from Ancient Greek ὠκεανός (ōkeanós) 'ocean', and γραφή (graphḗ) 'writing'), also known as oceanology and ocean science, is the scientific study of the oceans. It is an important Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamics; plate tectonics and the geology of the sea floor; and fluxes of various chemical substances and physical properties within the ocean and across its boundaries. These diverse topics reflect multiple disciplines that oceanographers utilize to glean further knowledge of the world ocean, including astronomy, biology, chemistry, climatology, geography, geology, hydrology, meteorology and physics. Paleoceanography studies the history of the oceans in the geologic past. An oceanographer is a person who studies many matters concerned with oceans, including marine geology, physics, chemistry and biology.

History

Map of the Gulf Stream by Benjamin Franklin, 1769–1770. Courtesy of the NOAA Photo Library.

Early history

Humans first acquired knowledge of the waves and currents of the seas and oceans in pre-historic times. Observations on tides were recorded by Aristotle and Strabo in 384-322 BC. Early exploration of the oceans was primarily for cartography and mainly limited to its surfaces and of the animals that fishermen brought up in nets, though depth soundings by lead line were taken.

The Portuguese campaign of Atlantic navigation is the earliest example of a systematic scientific large project, sustained over many decades, studying the currents and winds of the Atlantic.

The work of Pedro Nunes (1502-1578) is remembered in the navigation context for the determination of the loxodromic curve: the shortest course between two points on the surface of a sphere represented onto a two-dimensional map. When he published his "Treatise of the Sphere" (1537), mostly a commentated translation of earlier work by others, he included a treatise on geometrical and astronomic methods of navigation. There he states clearly that Portuguese navigations were not an adventurous endeavour:

"nam se fezeram indo a acertar: mas partiam os nossos mareantes muy ensinados e prouidos de estromentos e regras de astrologia e geometria que sam as cousas que os cosmographos ham dadar apercebidas (...) e leuaua cartas muy particularmente rumadas e na ja as de que os antigos vsauam" (were not done by chance: but our seafarers departed well taught and provided with instruments and rules of astrology (astronomy) and geometry which were matters the cosmographers would provide (...) and they took charts with exact routes and no longer those used by the ancient).

His credibility rests on being personally involved in the instruction of pilots and senior seafarers from 1527 onwards by Royal appointment, along with his recognized competence as mathematician and astronomer. The main problem in navigating back from the south of the Canary Islands (or south of Boujdour) by sail alone, is due to the change in the regime of winds and currents: the North Atlantic gyre and the Equatorial counter current  will push south along the northwest bulge of Africa, while the uncertain winds where the Northeast trades meet the Southeast trades (the doldrums)  leave a sailing ship to the mercy of the currents. Together, prevalent current and wind make northwards progress very difficult or impossible. It was to overcome this problem and clear the passage to India around Africa as a viable maritime trade route, that a systematic plan of exploration was devised by the Portuguese. The return route from regions south of the Canaries became the 'volta do largo' or 'volta do mar'. The 'rediscovery' of the Azores islands in 1427 is merely a reflection of the heightened strategic importance of the islands, now sitting on the return route from the western coast of Africa (sequentially called 'volta de Guiné' and 'volta da Mina'); and the references to the Sargasso Sea (also called at the time 'Mar da Baga'), to the west of the Azores, in 1436, reveals the western extent of the return route. This is necessary, under sail, to make use of the southeasterly and northeasterly winds away from the western coast of Africa, up to the northern latitudes where the westerly winds will bring the seafarers towards the western coasts of Europe.

The secrecy involving the Portuguese navigations, with the death penalty for the leaking of maps and routes, concentrated all sensitive records in the Royal Archives, completely destroyed by the Lisbon earthquake of 1775. However, the systematic nature of the Portuguese campaign, mapping the currents and winds of the Atlantic, is demonstrated by the understanding of the seasonal variations, with expeditions setting sail at different times of the year taking different routes to take account of seasonal predominate winds. This happens from as early as late 15th century and early 16th: Bartolomeu Dias followed the African coast on his way south in August 1487, while Vasco da Gama would take an open sea route from the latitude of Sierra Leone, spending 3 months in the open sea of the South Atlantic to profit from the southwards deflection of the southwesterly on the Brazilian side (and the Brazilian current going southward) - Gama departed in July 1497); and Pedro Alvares Cabral, departing March 1500) took an even larger arch to the west, from the latitude of Cape Verde, thus avoiding the summer monsoon (which would have blocked the route taken by Gama at the time he set sail). Furthermore, there were systematic expeditions pushing into the western Northern Atlantic (Teive, 1454; Vogado, 1462; Teles, 1474; Ulmo, 1486). The documents relating to the supplying of ships, and the ordering of sun declination tables for the southern Atlantic for as early as 1493–1496, all suggest a well-planned and systematic activity happening during the decade long period between Bartolomeu Dias finding the southern tip of Africa, and Gama's departure; additionally, there are indications of further travels by Bartolomeu Dias in the area. The most significant consequence of this systematic knowledge was the negotiation of the Treaty of Tordesillas in 1494, moving the line of demarcation 270 leagues to the west (from 100 to 370 leagues west of the Azores), bringing what is now Brazil into the Portuguese area of domination. The knowledge gathered from open sea exploration allowed for the well-documented extended periods of sail without sight of land, not by accident but as pre-determined planned route; for example, 30 days for Bartolomeu Dias culminating on Mossel Bay, the 3 months Gama spent in the South Atlantic to use the Brazil current (southward), or the 29 days Cabral took from Cape Verde up to landing in Monte Pascoal, Brazil.

The Danish expedition to Arabia 1761-67 can be said to be the world's first oceanographic expedition, as the ship Grønland had on board a group of scientists, including naturalist Peter Forsskål, who was assigned an explicit task by the king, Frederik V, to study and describe the marine life in the open sea, including finding the cause of mareel, or milky seas. For this purpose, the expedition was equipped with nets and scrapers, specifically designed to collect samples from the open waters and the bottom at great depth.

Although Juan Ponce de León in 1513 first identified the Gulf Stream, and the current was well known to mariners, Benjamin Franklin made the first scientific study of it and gave it its name. Franklin measured water temperatures during several Atlantic crossings and correctly explained the Gulf Stream's cause. Franklin and Timothy Folger printed the first map of the Gulf Stream in 1769–1770.

1799 map of the currents in the Atlantic and Indian Oceans, by James Rennell

Information on the currents of the Pacific Ocean was gathered by explorers of the late 18th century, including James Cook and Louis Antoine de Bougainville. James Rennell wrote the first scientific textbooks on oceanography, detailing the current flows of the Atlantic and Indian oceans. During a voyage around the Cape of Good Hope in 1777, he mapped "the banks and currents at the Lagullas". He was also the first to understand the nature of the intermittent current near the Isles of Scilly, (now known as Rennell's Current).

Sir James Clark Ross took the first modern sounding in deep sea in 1840, and Charles Darwin published a paper on reefs and the formation of atolls as a result of the second voyage of HMS Beagle in 1831–1836. Robert FitzRoy published a four-volume report of Beagle's three voyages. In 1841–1842 Edward Forbes undertook dredging in the Aegean Sea that founded marine ecology.

The first superintendent of the United States Naval Observatory (1842–1861), Matthew Fontaine Maury devoted his time to the study of marine meteorology, navigation, and charting prevailing winds and currents. His 1855 textbook Physical Geography of the Sea was one of the first comprehensive oceanography studies. Many nations sent oceanographic observations to Maury at the Naval Observatory, where he and his colleagues evaluated the information and distributed the results worldwide.

Modern oceanography

Knowledge of the oceans remained confined to the topmost few fathoms of the water and a small amount of the bottom, mainly in shallow areas. Almost nothing was known of the ocean depths. The British Royal Navy's efforts to chart all of the world's coastlines in the mid-19th century reinforced the vague idea that most of the ocean was very deep, although little more was known. As exploration ignited both popular and scientific interest in the polar regions and Africa, so too did the mysteries of the unexplored oceans.

HMS Challenger undertook the first global marine research expedition in 1872.

The seminal event in the founding of the modern science of oceanography was the 1872–1876 Challenger expedition. As the first true oceanographic cruise, this expedition laid the groundwork for an entire academic and research discipline. In response to a recommendation from the Royal Society, the British Government announced in 1871 an expedition to explore world's oceans and conduct appropriate scientific investigation. Charles Wyville Thompson and Sir John Murray launched the Challenger expedition. Challenger, leased from the Royal Navy, was modified for scientific work and equipped with separate laboratories for natural history and chemistry. Under the scientific supervision of Thomson, Challenger travelled nearly 70,000 nautical miles (130,000 km) surveying and exploring. On her journey circumnavigating the globe, 492 deep sea soundings, 133 bottom dredges, 151 open water trawls and 263 serial water temperature observations were taken. Around 4,700 new species of marine life were discovered. The result was the Report Of The Scientific Results of the Exploring Voyage of H.M.S. Challenger during the years 1873–76. Murray, who supervised the publication, described the report as "the greatest advance in the knowledge of our planet since the celebrated discoveries of the fifteenth and sixteenth centuries". He went on to found the academic discipline of oceanography at the University of Edinburgh, which remained the centre for oceanographic research well into the 20th century. Murray was the first to study marine trenches and in particular the Mid-Atlantic Ridge, and map the sedimentary deposits in the oceans. He tried to map out the world's ocean currents based on salinity and temperature observations, and was the first to correctly understand the nature of coral reef development.

In the late 19th century, other Western nations also sent out scientific expeditions (as did private individuals and institutions). The first purpose built oceanographic ship, Albatros, was built in 1882. In 1893, Fridtjof Nansen allowed his ship, Fram, to be frozen in the Arctic ice. This enabled him to obtain oceanographic, meteorological and astronomical data at a stationary spot over an extended period.

Ocean currents (1911)

 

Writer and geographer John Francon Williams FRGS commemorative plaque, Clackmannan Cemetery 2019

In 1881 the geographer John Francon Williams published a seminal book, Geography of the Oceans. Between 1907 and 1911 Otto Krümmel published the Handbuch der Ozeanographie, which became influential in awakening public interest in oceanography. The four-month 1910 North Atlantic expedition headed by John Murray and Johan Hjort was the most ambitious research oceanographic and marine zoological project ever mounted until then, and led to the classic 1912 book The Depths of the Ocean.

The first acoustic measurement of sea depth was made in 1914. Between 1925 and 1927 the "Meteor" expedition gathered 70,000 ocean depth measurements using an echo sounder, surveying the Mid-Atlantic Ridge.

In 1934, Easter Ellen Cupp, the first woman to have earned a PhD (at Scripps) in the United States, completed a major work on diatoms that remained the standard taxonomy in the field until well after her death in 1999. In 1940, Cupp was let go from her position at Scripps. Sverdrup specifically commended Cupp as a conscientious and industrious worker and commented that his decision was no reflection on her ability as a scientist. Sverdrup used the instructor billet vacated by Cupp to employ Marston Sargent,a biologist studying marine algae, which was not a new research program at Scripps. Financial pressures did not prevent Sverdrup from retaining the services of two other young post-doctoral students, Walter Munk and Roger Revelle. Cupp's partner, Dorothy Rosenbury, found her a position teaching high school, where she remained for the rest of her career. (Russell, 2000)

Sverdrup, Johnson and Fleming published The Oceans in 1942, which was a major landmark. The Sea (in three volumes, covering physical oceanography, seawater and geology) edited by M.N. Hill was published in 1962, while Rhodes Fairbridge's Encyclopedia of Oceanography was published in 1966.

The Great Global Rift, running along the Mid Atlantic Ridge, was discovered by Maurice Ewing and Bruce Heezen in 1953 and mapped by Heezen and Marie Tharp using bathymetric data; in 1954 a mountain range under the Arctic Ocean was found by the Arctic Institute of the USSR. The theory of seafloor spreading was developed in 1960 by Harry Hammond Hess. The Ocean Drilling Program started in 1966. Deep-sea vents were discovered in 1977 by Jack Corliss and Robert Ballard in the submersible DSV Alvin.

In the 1950s, Auguste Piccard invented the bathyscaphe and used the bathyscaphe Trieste to investigate the ocean's depths. The United States nuclear submarine Nautilus made the first journey under the ice to the North Pole in 1958. In 1962 the FLIP (Floating Instrument Platform), a 355-foot (108 m) spar buoy, was first deployed.

In 1968, Tanya Atwater led the first all-woman oceanographic expedition. Until that time, gender policies restricted women oceanographers from participating in voyages to a significant extent.

From the 1970s, there has been much emphasis on the application of large scale computers to oceanography to allow numerical predictions of ocean conditions and as a part of overall environmental change prediction. Early techniques included analog computers (such as the Ishiguro Storm Surge Computer) generally now replaced by numerical methods (eg SLOSH.) An oceanographic buoy array was established in the Pacific to allow prediction of El Niño events.

1990 saw the start of the World Ocean Circulation Experiment (WOCE) which continued until 2002. Geosat seafloor mapping data became available in 1995.

Study of the oceans is critical to understanding shifts in Earth's energy balance along with related global and regional changes in climate, the biosphere and biogeochemistry. The atmosphere and ocean are linked because of evaporation and precipitation as well as thermal flux (and solar insolation). Recent studies have advanced knowledge on ocean acidification, ocean heat content, ocean currents, sea level rise, the oceanic carbon cycle, the water cycle, Arctic sea ice decline, coral bleaching, marine heatwaves, extreme weather, coastal erosion and many other phenomena in regards to ongoing climate change and climate feedbacks.

In general, understanding the world ocean through further scientific study enables better stewardship and sustainable utilization of Earth's resources.

Branches

Oceanographic frontal systems on the Southern Hemisphere

 

The Applied Marine Physics Building at the University of Miami's Rosenstiel School of Marine and Atmospheric Science on Virginia Key, September 2007

The study of oceanography is divided into these five branches:

Biological oceanography

Main article: Marine life

See also: Marine biology

Biological oceanography investigates the ecology and biology of marine organisms in the context of the physical, chemical and geological characteristics of their ocean environment.

Chemical oceanography

Main article: Chemical oceanography

Chemical oceanography is the study of the chemistry of the ocean. Whereas chemical oceanography is primarily occupied with the study and understanding of seawater properties and its changes, ocean chemistry focuses primarily on the geochemical cycles. The following is a central topic investigated by chemical oceanography.

Ocean acidification

Main article: Ocean acidification

Ocean acidification describes the decrease in ocean pH that is caused by anthropogenic carbon dioxide (CO₂) emissions into the atmosphere. Seawater is slightly alkaline and had a preindustrial pH of about 8.2. More recently, anthropogenic activities have steadily increased the carbon dioxide content of the atmosphere; about 30–40% of the added CO₂ is absorbed by the oceans, forming carbonic acid and lowering the pH (now below 8.1) through ocean acidification. The pH is expected to reach 7.7 by the year 2100.

An important element for the skeletons of marine animals is calcium, but calcium carbonate becomes more soluble with pressure, so carbonate shells and skeletons dissolve below the carbonate compensation depth. Calcium carbonate becomes more soluble at lower pH, so ocean acidification is likely to affect marine organisms with calcareous shells, such as oysters, clams, sea urchins and corals, and the carbonate compensation depth will rise closer to the sea surface. Affected planktonic organisms will include pteropods, coccolithophorids and foraminifera, all important in the food chain. In tropical regions, corals are likely to be severely affected as they become less able to build their calcium carbonate skeletons, in turn adversely impacting other reef dwellers.

The current rate of ocean chemistry change seems to be unprecedented in Earth's geological history, making it unclear how well marine ecosystems will adapt to the shifting conditions of the near future. Of particular concern is the manner in which the combination of acidification with the expected additional stressors of higher temperatures and lower oxygen levels will impact the seas.

Geological oceanography

Main article: Marine geology

Geological oceanography is the study of the geology of the ocean floor including plate tectonics and paleoceanography.

Physical oceanography

Main article: Physical oceanography

Physical oceanography studies the ocean's physical attributes including temperature-salinity structure, mixing, surface waves, internal waves, surface tides, internal tides, and currents. The following are central topics investigated by physical oceanography.

Seismic Oceanography

Main article: Seismic Oceanography

Ocean currents

Further information: Ocean current

Since the early ocean expeditions in oceanography, a major interest was the study of ocean currents and temperature measurements. The tides, the Coriolis effect, changes in direction and strength of wind, salinity, and temperature are the main factors determining ocean currents. The thermohaline circulation (THC) (thermo- referring to temperature and -haline referring to salt content) connects the ocean basins and is primarily dependent on the density of sea water. It is becoming more common to refer to this system as the 'meridional overturning circulation' because it more accurately accounts for other driving factors beyond temperature and salinity.

• Examples of sustained currents are the Gulf Stream and the Kuroshio Current which are wind-driven western boundary currents.

Ocean heat content

Further information: Oceanic heat content

Oceanic heat content (OHC) refers to the extra heat stored in the ocean from changes in Earth's energy balance. The increase in the ocean heat play an important role in sea level rise, because of thermal expansion. Ocean warming accounts for 90% of the energy accumulation associated with global warming since 1971.

Paleoceanography

Main article: Paleoceanography

Paleoceanography is the study of the history of the oceans in the geologic past with regard to circulation, chemistry, biology, geology and patterns of sedimentation and biological productivity. Paleoceanographic studies using environment models and different proxies enable the scientific community to assess the role of the oceanic processes in the global climate by the reconstruction of past climate at various intervals. Paleoceanographic research is also intimately tied to palaeoclimatology.

Oceanographic institutions

Oceanographic Museum

 

See also: List of oceanographic institutions and programs

The first international organization of oceanography was created in 1902 as the International Council for the Exploration of the Sea. In 1903 the Scripps Institution of Oceanography was founded, followed by Woods Hole Oceanographic Institution in 1930, Virginia Institute of Marine Science in 1938, and later the Lamont–Doherty Earth Observatory at Columbia University, and the School of Oceanography at University of Washington. In Britain, the National Oceanography Centre (an institute of the Natural Environment Research Council) is the successor to the UK's Institute of Oceanographic Sciences. In Australia, CSIRO Marine and Atmospheric Research (CMAR), is a leading centre. In 1921 the International Hydrographic Bureau (IHB) was formed in Monaco.

Related disciplines

• Biogeochemistry – Study of chemical cycles of the earth that are either driven by or influence biological activity
• Biogeography – Study of the distribution of species and ecosystems in geographic space and through geological time
• Climatology – Scientific study of climate, defined as weather conditions averaged over a period of time
• Coastal geography – Study of the region between the ocean and the land
• Environmental science – The integrated, quantitative, and interdisciplinary approach to the study of environmental systems.
• Geophysics – Physics of the Earth and its vicinity
• Glaciology – Scientific study of ice and natural phenomena involving ice
• Hydrography – Applied science of measurement and description of physical features of bodies of water
• Hydrology – Science of the movement, distribution, and quality of water on Earth and other planets
• Limnology – Science of inland aquatic ecosystems
• Meteorology – Interdisciplinary scientific study of the atmosphere focusing on weather forecasting
• MetOce

Crystal structure

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Crystal_structure 

 

Crystal structure of table salt (sodium in purple, chloride in green)

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.

The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice.

The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters or cell parameters. The symmetry properties of the crystal are described by the concept of space groups. All possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups.

The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage, electronic band structure, and optical transparency.

Unit cell

Main article: Unit cell

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

• 

  Simple cubic (P)

• 

  Body-centered cubic (I)

• 

  Face-centered cubic (F)

Miller indices

Planes with different Miller indices in cubic crystals

Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices ℓ, m, and n as directional parameters.

By definition, the syntax (ℓmn) denotes a plane that intercepts the three points a₁/ℓ, a₂/m, and a₃/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in (123). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane.

Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes), the distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula

${\displaystyle d={\frac {2\pi }{|\mathbf {g} _{\ell mn}|}}}$

Planes and directions

The crystallographic directions are geometric lines linking nodes (atoms, ions or molecules) of a crystal. Likewise, the crystallographic planes are geometric planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:

• Optical properties: Refractive index is directly related to density (or periodic density fluctuations).
• Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.
• Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.

Dense crystallographic planes

• Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes.
• Cleavage: This typically occurs preferentially parallel to higher density planes.
• Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.

Some directions and planes are defined by symmetry of the crystal system. In monoclinic, rhombohedral, tetragonal, and trigonal/hexagonal systems there is one unique axis (sometimes called the principal axis) which has higher rotational symmetry than the other two axes. The basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis.

Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) lattice planes is (from above):

${\displaystyle d_{\ell mn}={\frac {a}{\sqrt {\ell ^{2}+m^{2}+n^{2}}}}}$

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

• Coordinates in angle brackets such as ⟨100⟩ denote a family of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.
• Coordinates in curly brackets or braces such as {100} denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

Interplanar spacing

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

• Cubic:

  ${\displaystyle {\frac {1}{d^{2}}}={\frac {h^{2}+k^{2}+\ell ^{2}}{a^{2}}}}$

• Tetragonal:

  ${\displaystyle {\frac {1}{d^{2}}}={\frac {h^{2}+k^{2}}{a^{2}}}+{\frac {\ell ^{2}}{c^{2}}}}$

• Hexagonal:

  ${\displaystyle {\frac {1}{d^{2}}}={\frac {4}{3}}\left({\frac {h^{2}+hk+k^{2}}{a^{2}}}\right)+{\frac {\ell ^{2}}{c^{2}}}}$

• Rhombohedral:

  ${\displaystyle {\frac {1}{d^{2}}}={\frac {(h^{2}+k^{2}+\ell ^{2})\sin ^{2}\alpha +2(hk+k\ell +h\ell )(\cos ^{2}\alpha -\cos \alpha )}{a^{2}(1-3\cos ^{2}\alpha +2\cos ^{3}\alpha )}}}$

• Orthorhombic:

  ${\displaystyle {\frac {1}{d^{2}}}={\frac {h^{2}}{a^{2}}}+{\frac {k^{2}}{b^{2}}}+{\frac {\ell ^{2}}{c^{2}}}}$

• Monoclinic:

  ${\displaystyle {\frac {1}{d^{2}}}=\left({\frac {h^{2}}{a^{2}}}+{\frac {k^{2}\sin ^{2}\beta }{b^{2}}}+{\frac {\ell ^{2}}{c^{2}}}-{\frac {2h\ell \cos \beta }{ac}}\right)\csc ^{2}\beta }$

• Triclinic:

  ${\displaystyle {\frac {1}{d^{2}}}={\frac {{\frac {h^{2}}{a^{2}}}\sin ^{2}\alpha +{\frac {k^{2}}{b^{2}}}\sin ^{2}\beta +{\frac {\ell ^{2}}{c^{2}}}\sin ^{2}\gamma +{\frac {2k\ell }{bc}}(\cos \beta \cos \gamma -\cos \alpha )+{\frac {2h\ell }{ac}}(\cos \gamma \cos \alpha -\cos \beta )+{\frac {2hk}{ab}}(\cos \alpha \cos \beta -\cos \gamma )}{1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}}$

Classification by symmetry

Main article: Crystal system

The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals have translational symmetry in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.

Lattice systems

Lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometric arrangement. All crystals fall into one of seven lattice systems. They are similar to, but not quite the same as the seven crystal systems.

Crystal family	Lattice system	Point group (Schönflies notation)	14 Bravais lattices
			Primitive (P)	Base-centered (S)	Body-centered (I)	Face-centered (F)
Triclinic (a)		Ci	aP
Monoclinic (m)		C2h	mP	mS
Orthorhombic (o)		D2h	oP	oS	oI	oF
Tetragonal (t)		D4h	tP		tI
Hexagonal (h)	Rhombohedral	D3d	hR
	Hexagonal	D6h	hP
Cubic (c)		Oh	cP		cI	cF

The simplest and most symmetric, the cubic or isometric system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal, tetragonal, rhombohedral (often confused with the trigonal crystal system), orthorhombic, monoclinic and triclinic.

Bravais lattices

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

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

Crystal systems

See also: Crystallographic point group § Isomorphisms

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

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

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

Point groups

The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include

• Reflection, which reflects the structure across a reflection plane
• Rotation, which rotates the structure a specified portion of a circle about a rotation axis
• Inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point
• Improper rotation, which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

Space groups

In addition to the operations of the point group, the space group of the crystal structure contains translational symmetry operations. These include:

• Pure translations, which move a point along a vector
• Screw axes, which rotate a point around an axis while translating parallel to the axis.
• Glide planes, which reflect a point through a plane while translating it parallel to the plane.

There are 230 distinct space groups.

Atomic coordination

By considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.

Close packing

Main article: Close-packing of equal spheres

 

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

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series:

...ABABABAB...

This arrangement of atoms in a crystal structure is known as hexagonal close packing (hcp).

If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:

...ABCABCABC...

This type of structural arrangement is known as cubic close packing (ccp).

The unit cell of a ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.

APF and CN

Main articles: Atomic packing factor and Coordination number

One important characteristic of a crystalline structure is its atomic packing factor (APF). This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows:

${\displaystyle \mathrm {APF} ={\frac {N_{\mathrm {particle} }V_{\mathrm {particle} }}{V_{\text{unit cell}}}}}$

Another important characteristic of a crystalline structure is its coordination number (CN). This is the number of nearest neighbours of a central atom in the structure.

The APFs and CNs of the most common crystal structures are shown below:

Crystal structure	Atomic packing factor	Coordination number (Geometry)
Diamond cubic	0.34	4 (Tetrahedron)
Simple cubic	0.52	6 (Octahedron)
Body-centered cubic (BCC)	0.68	8 (Cube)
Face-centered cubic (FCC)	0.74	12 (Cuboctahedron)
Hexagonal close-packed (HCP)	0.74	12 (Triangular orthobicupola)

The 74% packing efficiency of the FCC and HCP is the maximum density possible in unit cells constructed of spheres of only one size.

Interstitial sites

Main article: Interstitial site

 

Octahedral (red) and tetrahedral (blue) interstitial sites in a face-centered cubic lattice.

Interstitial sites refer to the empty spaces in between the atoms in the crystal lattice. These spaces can be filled by oppositely charged ions to form multi-element structures. They can also be filled by impurity atoms or self-interstitials to form interstitial defects.

Defects and impurities

Main article: Crystallographic defect

Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials.

Impurities

When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue. Impurities may also manifest as electron spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s.

Dislocations

Main article: Dislocation

Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure.

Grain boundaries

Main article: Grain boundary

Grain boundaries are interfaces where crystals of different orientations meet. A grain boundary is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites, dislocations, and impurities that have migrated to the lower energy grain boundary.

Treating a grain boundary geometrically as an interface of a single crystal cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).

Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall–Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep.

Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion and plasticity. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an amorphous solid.

Prediction of structure

Main article: Crystal structure prediction

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

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

In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": 1⁄2, 1⁄3, 2⁄3, 1⁄4, 3⁄4, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.

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

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

Polymorphism

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

Polymorphism is the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials including polymers, minerals, and metals. According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature. Polymorphism is related to allotropy, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit, amorphous fraction or crystallographic defects. Polymorphs have different stabilities and may spontaneously and irreversibly transform from a metastable form (or thermodynamically unstable form) to the stable form at a particular temperature. They also exhibit different melting points, solubilities, and X-ray diffraction patterns.

One good example of this is the quartz form of silicon dioxide, or SiO₂. In the vast majority of silicates, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral {SiO₄} units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of the 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the {SiO₄} tetrahedra are shared with others, yielding the net chemical formula for silica: SiO₂.

Another example is elemental tin (Sn), which is malleable near ambient temperatures but is brittle when cooled. This change in mechanical properties due to existence of its two major allotropes, α- and β-tin. The two allotropes that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as gray tin and white tin respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa. White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 °C, tin exists in the gray form, which has a diamond cubic crystal structure, similar to diamond, silicon or germanium. Gray tin has no metallic properties at all, is a dull gray powdery material, and has few uses, other than a few specialized semiconductor applications. Although the α–β transformation temperature of tin is nominally 13.2 °C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 °C, and upon addition of Sb or Bi the transformation may not occur at all.

Physical properties

Twenty of the 32 crystal classes are piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All piezoelectric classes lack inversion symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance that has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).