formulasearchengine - User contributions [en]

Exponentiation by squaring

2014-02-16T04:22:15Z

129.234.37.155:

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Curie constant

2014-02-07T16:58:58Z

129.234.252.66:

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Einstein notation

2014-02-06T15:58:00Z

129.234.157.101: /* Mnemonics */

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Canonical ensemble

2014-01-28T16:08:39Z

129.234.186.15: /* Properties */

In [[mathematics]], an '''abelian integral''', named after the Norwegian mathematician [[Niels Henrik Abel|Niels Abel]], is an integral in the [[complex plane]] of the form

:<math>\int_{z_0}^z R\left(x,w\right)dx,</math>

where <math>R\left(x,w\right)</math> is an arbitrary [[rational function]] of the two variables <math>x</math> and <math>w</math>. These variables are related by the equation

:<math>F\left(x,w\right)=0, \, </math>

where <math>F\left(x,w\right)</math> is an irreducible polynomial in <math>w</math>,

:<math>F\left(x,w\right)\equiv\phi_n\left(x\right)w^n+\cdots+\phi_1\left(x\right)w+\phi_0\left(x\right), \, </math>

whose coefficients <math>\phi_j\left(x\right)</math>, <math>j=0,1,\ldots,n</math> are [[rational function]]s of <math>x</math>. The value of an abelian integral depends not only on the integration limits but also on the path along which the integral is taken, and it is thus a [[multivalued function]] of <math>z</math>.

Abelian integrals are natural generalizations of [[elliptic integral]]s, which arise when
:<math>F\left(x,w\right)=w^2-P\left(x\right), \, </math>
where <math>P\left(x\right)</math> is a polynomial of degree 3 or 4. Another special case of an abelian integral is a [[hyperelliptic integral]], where <math>P\left(x\right)</math>, in the formula above, is a polynomial of degree greater than 4.

== History ==

The theory of abelian integrals originated with the paper by Abel <ref>a</ref> published in 1841. This paper was written during his stay in Paris in 1826 and presented to [[Cauchy]] in October of the same year. This theory, later fully developed by others, was one of the crowning achievements of nineteenth century mathematics and has had a major impact on the development of modern mathematics. In more abstract and geometric language, it is contained in the concept of [[abelian variety]], or more precisely in the way an [[algebraic curve]] can be mapped into abelian varieties. The Abelian Integral was later connected to the prominent mathematician [[David Hilbert]]'s 16th Problem and continues to be considered one of the foremost challenges to contemporary [[mathematical analysis]].

== Modern view ==

In [[Riemann surface]] theory, an abelian integral is a function related to the [[indefinite integral]] of a [[differential of the first kind]]. Suppose we are given a Riemann surface <math>S</math> and on it a [[Differential form|differential 1-form]] <math>\omega</math> that is everywhere [[holomorphic]] on <math>S</math>, and fix a point <math>P_0</math> on <math>S</math>, from which to integrate. We can regard
:<math>\int_{P_0}^P \omega</math>
as a [[multi-valued function]] <math>f\left(P\right)</math>, or (better) an honest function of the chosen path <math>C</math> drawn on <math>S</math> from <math>P_0</math> to <math>P</math>. Since <math>S</math> will in general be [[multiply connected]], one should specify <math>C</math>, but the value will in fact only depend on the [[homology class]] of <math>C</math>.

In the case of <math>S</math> a [[compact Riemann surface]] of [[genus (mathematics)|genus]] 1, i.e. an [[elliptic curve]], such functions are the [[elliptic integral]]s. Logically speaking, therefore, an abelian integral should be a function such as <math>f</math>.

Such functions were first introduced to study [[hyperelliptic integral]]s, i.e. for the case where <math>S</math> is a [[hyperelliptic curve]]. This is a natural step in the theory of integration to the case of integrals involving [[algebraic function]]s <math>\sqrt{A}</math>, where <math>A</math> is a [[polynomial]] of degree <math>>4</math>. The first major insights of the theory were given by [[Niels Abel]]; it was later formulated in terms of the [[Jacobian variety]] <math>J\left(S\right)</math>. Choice of <math>P_0</math> gives rise to a standard [[holomorphic]] [[function (mathematics)|mapping]]

:<math>S\to J\left(S\right) \, </math>

of [[complex manifold]]s. It has the defining property that the holomorphic 1-forms on <math>S\to J\left(S\right)</math>, of which there are ''g'' independent ones if ''g'' is the genus of ''S'', [[pullback (differential geometry)|pull back]] to a basis for the differentials of the first kind on ''S''.

== References ==
* {{Citation | last1=Appell | first1=Paul | author1-link=Paul Appell | last2=Goursat | first2=Édouard | author2-link=Édouard Goursat| title=Theorie des Fonctions Algebraiques et de Leurs Integrales | publisher=[[Gauthier-Villars]] | location=Paris | year=1895}}.
* {{Citation | last1=Bliss | first1=Gilbert A. | author1-link=Gilbert Ames Bliss| title=Algebraic Functions | publisher=[[American Mathematical Society]] | location=Providence | year=1933}}.
* {{Citation | last1=Forsyth | first1=Andrew R. | author1-link=Andrew Forsyth| title=Theory of Functions of a Complex Variable | publisher=[[Cambridge University Press]] | location=Providence | year=1893}}.
* {{Citation | last1=Griffiths | first1=Phillip | last2=Harris | first2=Joseph | title=Principles of Algebraic Geometry | publisher=[[John Wiley & Sons]] | location=New York | year=1978}}. Lucidly presented modern perspective.
* {{Citation | last1=Neumann | first1=Carl | author1-link=Carl Neumann| title=Vorlesungen über Riemann's Theorie der Abel'schen Integrale| publisher=[[B. G. Teubner]] | edition=2nd | location=Leipzig | year=1884}}.
<References/>
[[Category:Riemann surfaces]]
[[Category:Algebraic curves]]
[[Category:Abelian varieties]]

Leon M. Lederman

2014-01-15T17:17:53Z

129.234.252.65: Moved "He is an atheist." from article introduction to Personal Life, where it belongs.

{{Use mdy dates|date=May 2012}}
[[File:Nematische Phase Schlierentextur.jpg|thumb|300px|[[Schlieren]] texture of liquid crystal [[nematic]] phase]]

'''Liquid crystals (LCs)''' are [[state of matter|matter in a state]] that has properties between those of conventional [[liquid]] and those of solid [[crystal]].<ref name=b2/> For instance, a liquid crystal may flow like a liquid, but its [[molecules]] may be oriented in a crystal-like way. There are many different types of liquid-crystal phases, which can be distinguished by their different [[Optics|optical]] properties (such as [[birefringence]]). When viewed under a [[microscope]] using a [[Polarization (waves)|polarized]] light source, different liquid crystal phases will appear to have distinct [[Texture (crystalline)|textures]]. The contrasting areas in the textures correspond to domains where the liquid-crystal molecules are oriented in different directions. Within a domain, however, the molecules are well ordered. LC materials may not always be in a liquid-crystal phase (just as water may turn into ice or steam).

Liquid crystals can be divided into [[thermotropic]], [[lyotropic]] and metallotropic phases. Thermotropic and lyotropic liquid crystals consist of [[organic molecules]]. Thermotropic LCs exhibit a [[phase transition]] into the liquid-crystal phase as temperature is changed. Lyotropic LCs exhibit phase transitions as a function of both temperature and [[concentration]] of the liquid-crystal molecules in a [[solvent]] (typically water). Metallotropic LCs are composed of both organic and inorganic molecules; their liquid-crystal transition depends not only on temperature and concentration, but also on the inorganic-organic composition ratio.

Examples of liquid crystals can be found both in the natural world and in technological applications. Most contemporary [[Electronic visual display|electronic displays]] use liquid crystals. Lyotropic liquid-crystalline phases are abundant in living systems. For example, many proteins and cell membranes are liquid crystals. Other well-known examples of liquid crystals are solutions of [[soap]] and various related [[detergent]]s, as well as the [[tobacco mosaic virus]].

==History==
In 1888, Austrian botanical physiologist [[Friedrich Reinitzer]], working at the [[Karl-Ferdinands-Universität]], examined the physico-chemical properties of various [[Derivative (chemistry)|derivatives]] of [[cholesterol]] which now belong to the class of materials known as cholesteric liquid crystals. Previously, other researchers had observed distinct color effects when cooling cholesterol derivatives just above the [[freezing point]], but had not associated it with a new phenomenon. Reinitzer perceived that color changes in a derivative [[cholesteryl benzoate]] were not the most peculiar feature.[[File:Cholesteryl benzoate.png|thumb|250 px|Chemical structure of [[cholesteryl benzoate]] molecule]] He found that cholesteryl benzoate does not [[melting|melt]] in the same manner as other compounds, but has two [[melting point]]s. At {{convert|145.5|°C|°F}} it melts into a cloudy liquid, and at {{convert|178.5|°C|°F}} it melts again and the cloudy liquid becomes clear. The phenomenon is reversible. Seeking help from a physicist, on March 14, 1888, he wrote to [[Otto Lehmann (physicist)|Otto Lehmann]], at that time a [[Privatdozent]] in [[Aachen]]. They exchanged letters and samples. Lehmann examined the intermediate cloudy fluid, and reported seeing [[crystallite]]s. Reinitzer's Viennese colleague von Zepharovich also indicated that the intermediate "fluid" was crystalline. The exchange of letters with Lehmann ended on April 24, with many questions unanswered. Reinitzer presented his results, with credits to Lehmann and von Zepharovich, at a meeting of the Vienna Chemical Society on May 3, 1888.<ref>{{cite journal| title = Beiträge zur Kenntniss des Cholesterins| journal=Monatshefte für Chemie (Wien)| volume = 9| issue =1| pages = 421–441|year =1888| doi =10.1007/BF01516710| last1 = Reinitzer| first1 = Friedrich}}</ref>

By that time, Reinitzer had discovered and described three important features of cholesteric liquid crystals (the name coined by Otto Lehmann in 1904): the existence of two melting points, the reflection of [[circular polarization|circularly polarized light]], and the ability to rotate the polarization direction of light.

After his accidental discovery, Reinitzer did not pursue studying liquid crystals further. The research was continued by Lehmann, who realized that he had encountered a new phenomenon and was in a position to investigate it: In his postdoctoral years he had acquired expertise in crystallography and microscopy. Lehmann started a systematic study, first of cholesteryl benzoate, and then of related compounds which exhibited the double-melting phenomenon. He was able to make observations in polarized light, and his microscope was equipped with a hot stage (sample holder equipped with a heater) enabling high temperature observations. The intermediate cloudy phase clearly sustained flow, but other features, particularly the signature under a microscope, convinced Lehmann that he was dealing with a solid. By the end of August 1889 he had published his results in the [[Zeitschrift für Physikalische Chemie]].<ref>{{cite journal| author=Lehmann, O. | title = Über fliessende Krystalle| journal=Zeitschrift für Physikalische Chemie|volume =4| pages = 462–72|year =1889}}</ref>

[[File:Otto Lehmann.jpg|thumb|180px|Otto Lehmann]]
Lehmann's work was continued and significantly expanded by the German chemist [[Daniel Vorländer]], who from the beginning of 20th century until his retirement in 1935, had synthesized most of the liquid crystals known. However, liquid crystals were not popular among scientists and the material remained a pure scientific curiosity for about 80 years.<ref name=b3/>

After World War II work on the synthesis of liquid crystals was restarted at university research laboratories in Europe. [[George William Gray]], a prominent researcher of liquid crystals, began investigating these materials in England in the late 1940s. His group synthesized many new materials that exhibited the liquid crystalline state and developed a better understanding of how to design molecules that exhibit the state. His book ''Molecular Structure and the Properties of Liquid Crystals''<ref>Gray, G. W. (1962) ''Molecular Structure and the Properties of Liquid Crystals'', Academic Press</ref> became a guidebook on the subject. One of the first U.S. chemists to study liquid crystals was Glenn H. Brown, starting in 1953 at the University of Cincinnati and later at Kent State University. In 1965, he organized the first international conference on liquid crystals, in Kent, Ohio, with about 100 of the world’s top liquid crystal scientists in attendance. This conference marked the beginning of a worldwide effort to perform research in this field, which soon led to the development of practical applications for these unique materials.<ref>{{cite journal|doi=10.1080/13583149408628630|title=Professor Horst Sackmann, 1921 – 1993|year=1994|last1=Stegemeyer|first1=H|journal=Liquid Crystals Today|volume=4|pages=1}}</ref><ref>[http://www1.kfupm.edu.sa/phys101/docs/Physics%20Success%20Stories%20-%20Physics%20On%20Display.htm Liquid Crystals]. kfupm.edu.sa</ref>

Liquid crystal materials became a topic of research into the development of flat panel electronic displays beginning in 1962 at [[RCA]] Laboratories.<ref name=castellano/> When physical chemist Richard Williams applied an electric field to a thin layer of a nematic liquid crystal at 125 °C, he observed the formation of a regular pattern that he called domains (now known as Williams Domains). This led his colleague [[George H. Heilmeier]] to perform research on a liquid crystal-based flat panel display to replace the cathode ray vacuum tube used in televisions. Unfortunately the [[para-Azoxyanisole]] that Williams and Heilmeier used exhibits the nematic liquid crystal state only above 116 °C, which made it impractical to use in a commercial display product. A material that could be operated at room temperature was clearly needed.

In 1966, Joel E. Goldmacher and Joseph A. Castellano, research chemists in Heilmeier group at RCA, discovered that mixtures made exclusively of nematic compounds that differed only in the number of carbon atoms in the terminal side chains could yield room-temperature nematic liquid crystals. A ternary mixture of [[Schiff base]] compounds resulted in a material that had a nematic range of 22–105 °C.<ref>Goldmacher, Joel E. and Castellano, Joseph A. “Electro-optical Compositions and Devices,” {{US Patent|3540796}}, Issue date: November 17, 1970.</ref> Operation at room temperature enabled the first practical display device to be made.<ref>{{cite journal|doi=10.1063/1.1652453|title=Dynamic Scattering in Nematic Liquid Crystals|year=1968|last1=Heilmeier|first1=G. H.|journal=Applied Physics Letters|volume=13|pages=46|bibcode = 1968ApPhL..13...46H }}</ref> The team then proceeded to prepare numerous mixtures of nematic compounds many of which had much lower melting points. This technique of mixing nematic compounds to obtain wide [[operating temperature]] range eventually became the industry standard and is used to this very day to tailor materials to meet specific applications.

[[File:MBBA.svg|thumb|Chemical structure of [[MBBA|N-(4-Methoxybenzylidene)-4-butylaniline (MBBA)]] molecule]]
In 1969, Hans Kelker succeeded in synthesizing a substance that had a nematic phase at room temperature, [[MBBA]], which is one of the most popular subjects of liquid crystal research.<ref>{{cite journal| journal=Angew. Chem. Int. Ed.|volume = 8|year =1969| title =A Liquid-crystalline (Nematic) Phase with a Particularly Low Solidification Point| issue =11| doi = 10.1002/anie.196908841| page = 884| last1=Kelker| first1=H.| last2=Scheurle| first2=B.}}</ref> The next step to commercialization of [[liquid crystal displays]] was the synthesis of further chemically stable substances (cyanobiphenyls) with low melting temperatures by [[George William Gray|George Gray]].<ref>{{cite journal| title = New family of nematic liquid crystals for displays| doi = 10.1049/el:19730096| journal=Electronics Lett.|volume =9| issue = 6|year =1973| page = 130| last1 = Gray| first1 = G.W.| last2 = Harrison| first2 = K.J.| last3 = Nash| first3 = J.A.}}</ref> That work with Ken Harrison and the UK MOD ([[Royal Radar Establishment|RRE Malvern]]), in 1973, led to design of new materials resulting in rapid adoption of small area LCDs within electronic products.

In 1991, when liquid crystal displays were already well established, [[Pierre-Gilles de Gennes]] working at the [[Université Paris-Sud]] received the Nobel Prize in physics "for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers".<ref>{{cite news| url = http://nobelprize.org/educational_games/physics/liquid_crystals/history/|publisher=Nobelprize.org| title = History and Properties of Liquid Crystals| accessdate =June 6, 2009}}</ref>

==Design of liquid crystalline materials==
A large number of chemical compounds are known to exhibit one or several liquid crystalline phases. Despite significant differences in chemical composition, these molecules have some common features in chemical and physical properties. There are two types of thermotropic liquid crystals: discotics and rod-shaped molecules. Discotics are flat disc-like molecules consisting of a core of adjacent aromatic rings. This allows for two dimensional columnar ordering. Rod-shaped molecules have an elongated, anisotropic geometry which allows for preferential alignment along one spatial direction.

• The molecular shape should be relatively thin or flat, especially within rigid molecular frameworks. 
• The molecular length should be at least 1.3 nm, consistent with the presence of long alkyl group on many room-temperature liquid crystals. 
• The structure should not be branched or angular. 
• A low melting point is preferable in order to avoid metastable, monotropic liquid crystalline phases. Low-temperature mesomorphic behavior in general is technologically more useful, and alkyl terminal groups promote this.

An extended, structurally rigid, highly anisotropic shape seems to be the main criterion for liquid crystalline behavior, and as a result many liquid crystalline materials are based on benzene rings.<ref name="Chemical Properties of Liquid Crystals">[http://plc.cwru.edu/tutorial/enhanced/files/lc/chem/chem.htm]</ref>

==Liquid-crystal phases==
The various liquid-crystal phases (called [[mesophase]]s) can be characterized by the type of ordering. One can distinguish positional order (whether molecules are arranged in any sort of ordered lattice) and orientational order (whether molecules are mostly pointing in the same direction), and moreover order can be either short-range (only between molecules close to each other) or long-range (extending to larger, sometimes [[macroscopic scale|macroscopic]], dimensions). Most thermotropic LCs will have an [[isotropic]] phase at high temperature. That is that heating will eventually drive them into a conventional liquid phase characterized by random and isotropic molecular ordering (little to no long-range order), and [[fluid]]-like flow behavior. Under other conditions (for instance, lower temperature), a LC might inhabit one or more phases with significant [[anisotropic]] orientational structure and short-range orientational order while still having an ability to flow.<ref name=b2>
{{cite book|author=[[Sivaramakrishna Chandrasekhar|Chandrasekhar, S.]]|title=Liquid Crystals|edition = 2nd|location = Cambridge|publisher=Cambridge University Press|year=1992|isbn=0-521-41747-3|url=http://books.google.com/?id=TxvUxFlQsEsC&printsec=frontcover}}
</ref><ref name=b1>
{{cite book|author=[[Pierre-Gilles de Gennes|de Gennes, P.G.]] and Prost, J|title=The Physics of Liquid Crystals|location = Oxford|publisher=Clarendon Press|year=1993|isbn=0-19-852024-7}}
</ref>

The ordering of liquid crystalline phases is extensive on the molecular scale. This order extends up to the entire domain size, which may be on the order of micrometers, but usually does not extend to the macroscopic scale as often occurs in classical [[crystal]]line solids. However some techniques, such as the use of boundaries or an applied [[electric field]], can be used to enforce a single ordered domain in a macroscopic liquid crystal sample. The ordering in a liquid crystal might extend along only one [[dimension]], with the material being essentially disordered in the other two directions.<ref name=b4>{{cite book|author=Dierking, I. |title=Textures of Liquid Crystals|location=Weinheim|year=2003|publisher=Wiley-VCH|isbn=3-527-30725-7|url=http://books.google.com/?id=p-0xdzDRB2kC&printsec=frontcover}}</ref><ref name=b5>{{cite book|author=Collings, P.J. and Hird, M|title=Introduction to Liquid Crystals|location = Bristol, PA|publisher=Taylor & Francis|year=1997|isbn=0-7484-0643-3}}
</ref>

===Thermotropic liquid crystals===
{{see also|Thermotropic crystal}}
Thermotropic phases are those that occur in a certain temperature range. If the temperature rise is too high, thermal motion will destroy the delicate cooperative ordering of the LC phase, pushing the material into a conventional isotropic liquid phase. At too low temperature, most LC materials will form a conventional crystal.<ref name=b2/><ref name=b1/> Many thermotropic LCs exhibit a variety of phases as temperature is changed. For instance, a particular type of LC molecule (called [[mesogen]]) may exhibit various smectic and nematic (and finally isotropic) phases as temperature is increased. An example of a compound displaying thermotropic LC behavior is [[para-azoxyanisole]].<ref name="Shao">{{cite journal|title=Phase Transitions of Liquid Crystal PAA in Confined Geometries|journal=Journal of Physical Chemistry B|year=1998|volume=102|issue=18|pages=3387–3394|doi=10.1021/jp9734437|last1=Shao|first1=Y.|last2=Zerda|first2=T. W.}}</ref>

====Nematic phase====
{{see also|Biaxial nematic|Twisted nematic field effect}}
[[File:LiquidCrystal-MesogenOrder-Nematic.jpg|thumb|left|120px|Alignment in a nematic phase.]]
[[File:Smectic nematic.jpg|thumb|Phase transition between a nematic (left) and smectic A (right) phases observed between crossed [[polarizers]]. The black color corresponds to isotropic medium.]]

One of the most common LC phases is the nematic. The word ''nematic'' comes from the [[Greek language|Greek]] {{lang|el|νήμα}} (nema), which means "thread". This term originates from the thread-like [[topological defect]]s observed in nematics, which are formally called '[[disclination]]s'. Nematics also exhibit so-called "hedgehog" topological defects. In a nematic phase, the ''calamitic'' or rod-shaped organic molecules have no positional order, but they self-align to have long-range directional order with their long axes roughly parallel.<ref>{{cite journal |url=http://www.csupomona.edu/~jarego/pubs/RD2_LC.pdf |title=Asymmetric synthesis of a highly soluble 'trimeric' analogue of the chiral nematic liquid crystal twist agent Merck S1011 |last=Rego |first=J.A.|last2=Harvey |first2=Jamie A.A. |last3=MacKinnon |first3=Andrew L. |last4=Gatdula |first4=Elysse |journal=Liquid Crystals |volume=37 |issue=1 |date=January 2010 |pages=37–43 |doi=10.1080/02678290903359291}}</ref> Thus, the molecules are free to flow and their center of mass positions are randomly distributed as in a liquid, but still maintain their long-range directional order. Most nematics are uniaxial: they have one axis that is longer and preferred, with the other two being equivalent (can be approximated as cylinders or rods). However, some liquid crystals are [[biaxial nematic]]s, meaning that in addition to orienting their long axis, they also orient along a secondary axis.<ref>{{cite journal|title =Thermotropic Biaxial Nematic Liquid Crystals| doi =10.1103/PhysRevLett.92.145505|journal=Phys. Rev. Lett.|volume = 92| issue =14| page = 145505|year =2004| pmid=15089552|bibcode = 2004PhRvL..92n5505M|last1 =Madsen|first1 =L. A.|last2 =Dingemans|first2 =T. J.|last3 =Nakata|first3 =M.|last4 =Samulski|first4 =E. T. }}</ref> Nematics have fluidity similar to that of ordinary (isotropic) liquids but they can be easily aligned by an external magnetic or electric field. Aligned nematics have the optical properties of uniaxial crystals and this makes them extremely useful in [[liquid crystal display]]s (LCD).<ref name=castellano>{{cite book| author=Castellano, Joseph A. |title =Liquid Gold: The Story of Liquid Crystal Displays and the Creation of an Industry| publisher=World Scientific Publishing|year =2005| isbn = 978-981-238-956-5}}</ref>

====Smectic phases====
[[File:LiquidCrystal-MesogenOrder-SmecticPhases.jpg|thumb|Schematic of alignment in the smectic phases. The smectic A phase (left) has molecules organized into layers. In the smectic C phase (right), the molecules are tilted inside the layers.]]
The smectic phases, which are found at lower temperatures than the nematic, form well-defined layers that can slide over one another in a manner similar to that of soap. The word "smetic" originates from the Latin word "smecticus", meaning cleaning, or having soap like properties.<ref>
{{Cite web| title=smectic|url=http://www.merriam-webster.com/dictionary/smectic |publisher=Merriam-Webster Dictionary}}</ref>
The smectics are thus positionally ordered along one direction. In the Smectic A phase, the molecules are oriented along the layer normal, while in the Smectic C phase they are tilted away from the layer normal. These phases are liquid-like within the layers. There are many different smectic phases, all characterized by different types and degrees of positional and orientational order.<ref name=b2/><ref name=b1/>

====Chiral phases====
[[File:LiquidCrystal-MesogenOrder-ChiralPhases.jpg|thumb|left|Schematic of ordering in chiral liquid crystal phases. The chiral nematic phase (left), also called the cholesteric phase, and the smectic C* phase (right).]]
The [[Chirality (chemistry)|chiral]] [[nematic]] phase exhibits [[Chirality (chemistry)|chirality]] (handedness). This phase is often called the [[Cholesteric liquid crystal|''cholesteric'']] phase because it was first observed for [[cholesterol]] derivatives. Only [[Chirality (chemistry)|chiral molecules]] (i.e., those that have no internal planes of [[Molecular symmetry|symmetry]]) can give rise to such a phase. This phase exhibits a twisting of the molecules perpendicular to the director, with the molecular axis parallel to the director. The finite twist angle between adjacent molecules is due to their asymmetric packing, which results in longer-range chiral order. In the smectic C* phase (an asterisk denotes a chiral phase), the molecules have positional ordering in a layered structure (as in the other smectic phases), with the molecules tilted by a finite angle with respect to the layer normal. The chirality induces a finite azimuthal twist from one layer to the next, producing a spiral twisting of the molecular axis along the layer normal.<ref name=b1/><ref name=b4/><ref name=b5/>

[[File:Cholesterinisch.png|thumb|Chiral nematic phase; p refers to the chiral pitch (see text)]]
The ''chiral pitch'', p, refers to the distance over which the LC molecules undergo a full 360° twist (but note that the structure of the chiral nematic phase repeats itself every half-pitch, since in this phase directors at 0° and ±180° are equivalent). The pitch, p, typically changes when the temperature is altered or when other molecules are added to the LC host (an achiral LC host material will form a chiral phase if doped with a chiral material), allowing the pitch of a given material to be tuned accordingly. In some liquid crystal systems, the pitch is of the same order as the [[wavelength]] of [[visible light]]. This causes these systems to exhibit unique optical properties, such as Bragg reflection and low-threshold [[laser]] emission,<ref>{{cite journal |title=Low threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals |journal=Opt. Lett |volume=23 |pages=1707–1709 |year=1998 |doi=10.1364/OL.23.001707 |author=Kopp, V. I.; Fan, B.; Vithana, H. K. M.; Genack, A. Z. |pmid=18091891 |issue=21|bibcode = 1998OptL...23.1707K }}</ref> and these properties are exploited in a number of optical applications.<ref name=b3>{{cite book|author=Sluckin, T. J.; Dunmur, D. A. and Stegemeyer, H. |title=Crystals That Flow – classic papers from the history of liquid crystals|location=London|year=2004|publisher=Taylor & Francis|isbn=0-415-25789-1|url=http://books.google.com/?id=iMEMAuxrhFcC&printsec=frontcover}}
</ref><ref name=b4/> For the case of Bragg reflection only the lowest-order reflection is allowed if the light is incident along the helical axis, whereas for oblique incidence higher-order reflections become permitted. Cholesteric liquid crystals also exhibit the unique property that they reflect circularly polarized light when it is incident along the helical axis and [[elliptically polarized]] if it comes in obliquely.<ref>{{cite book| title = Introduction to Liquid Crystals| author=Priestley, E. B.; Wojtowicz, P. J. and Sheng, P.| publisher=Plenum Press|year =1974| isbn = 0-306-30858-4}}</ref>

====Blue phases====
'''Blue phases''' are liquid crystal phases that appear in the temperature range between a [[Chirality (chemistry)|chiral]] [[nematic]] phase and an [[Isotropy|isotropic]] liquid phase. Blue phases have a regular three-dimensional cubic structure of defects with [[Crystal structure|lattice]] periods of several hundred nanometers, and thus they exhibit selective [[Bragg's law|Bragg reflections]] in the wavelength range of visible light corresponding to the [[Cubic phase|cubic lattice]]. It was theoretically predicted in 1981 that these phases can possess icosahedral symmetry similar to [[quasicrystal]]s.<ref>
{{cite journal
| title = Lattice Textures in Cholesteric Liquid Crystals
| author=[[Hagen Kleinert|Kleinert H.]] and Maki K.
| journal=Fortschritte der Physik
| volume = 29
| issue = 5
| pages = 219–259
| year = 1981
| doi = 10.1002/prop.19810290503
| url = http://www.physik.fu-berlin.de/~kleinert/75/75.pdf|bibcode = 1981ForPh..29..219K }}
</ref><ref>{{cite journal|url=http://chemgroups.northwestern.edu/seideman/Publications/The%20liquid-crystalline%20blue%20phases.pdf|title=The liquid-crystalline blue phases|journal=Rep. Prog. Phys. |volume=53|year=1990|pages=659–705|bibcode = 1990RPPh...53..659S |doi = 10.1088/0034-4885/53/6/001|last1=Seideman|first1=T|issue=6 }}</ref>

Although blue phases are of interest for fast light modulators or tunable [[photonic crystal]]s, they exist in a very narrow temperature range, usually less than a few [[kelvin]]. Recently the stabilization of blue phases over a temperature range of more than 60 K including room temperature (260–326 K) has been demonstrated.<ref>{{cite journal|title =Liquid crystal 'blue phases' with a wide temperature range|journal=Nature|volume = 436|year =2005| pages = 997–1000| doi =10.1038/nature03932| pmid =16107843| issue =7053|bibcode = 2005Natur.436..997C|last1 =Coles|first1 =Harry J.|last2 =Pivnenko|first2 =Mikhail N. }}</ref><ref>{{cite journal| title =Optical isotropy and iridescence in a smectic blue phase|journal=Nature|volume = 437| issue =7058|year =2005| page = 525|bibcode = 2005Natur.437..525Y|doi = 10.1038/nature04034| last1 =Yamamoto| first1 =Jun| last2 =Nishiyama| first2 =Isa| last3 =Inoue| first3 =Miyoshi| last4 =Yokoyama| first4 =Hiroshi }}</ref> Blue phases stabilized at room temperature allow electro-optical switching with response times of the order of 10−4 s.<ref>{{cite journal| author=Kikuchi H, Yokota M, Hisakado Y, Yang H, Kajiyama T.|title = Polymer-stabilized liquid crystal blue phases|journal=Nature Materials|volume = 1|year = 2002| doi =10.1038/nmat712| pmid =12618852| issue =1|bibcode = 2002NatMa...1...64K| pages=64–8 }}</ref>

In May 2008, the first [[Blue Phase Mode LCD]] panel had been developed.<ref>{{cite news| url = http://www.physorg.com/news129997960.html|title = Samsung Develops World’s First 'Blue Phase' Technology to Achieve 240 Hz Driving Speed for High-Speed Video| accessdate =April 23, 2009}}</ref>

====Discotic phases====
Disk-shaped LC molecules can orient themselves in a layer-like fashion known as the discotic nematic phase. If the disks pack into stacks, the phase is called a [[columnar phase|discotic columnar]]. The columns themselves may be organized into rectangular or hexagonal arrays. Chiral discotic phases, similar to the chiral nematic phase, are also known.

===Lyotropic liquid crystals===
{{see also|Lyotropic liquid crystal|Columnar phase}}
[[File:Lipid bilayer and micelle.svg|thumb|250 px|Structure of lyotropic liquid crystal. The red heads of surfactant molecules are in contact with water, whereas the tails are immersed in oil (blue): bilayer (left) and [[micelle]] (right).]]
A [[lyotropic liquid crystal]] consists of two or more components that exhibit liquid-crystalline properties in certain concentration ranges. In the [[lyotropic]] phases, [[solvent]] molecules fill the space around the compounds to provide [[viscosity|fluidity]] to the system.<ref>{{cite journal |author= Qizhen Liang, Pengtao Liu, Cheng Liu, Xigao Jian, Dingyi Hong, Yang Li.|year= 2005|title= Synthesis and Properties of Lyotropic Liquid Crystalline Copolyamides Containing Phthalazinone Moieties and Ether Linkages|journal= Polymer|volume=46 |issue= 16|pages= 6258–6265|doi= 10.1016/j.polymer.2005.05.059}}</ref> In contrast to thermotropic liquid crystals, these lyotropics have another degree of freedom of concentration that enables them to induce a variety of different phases.

A compound that has two immiscible [[hydrophilic]] and [[hydrophobic]] parts within the same molecule is called an [[amphiphilic]] molecule. Many amphiphilic molecules show lyotropic liquid-crystalline phase sequences depending on the volume balances between the hydrophilic part and hydrophobic part. These structures are formed through the micro-phase segregation of two incompatible components on a nanometer scale. Soap is an everyday example of a lyotropic liquid crystal.

The content of water or other solvent molecules changes the self-assembled structures. At very low amphiphile concentration, the molecules will be dispersed randomly without any ordering. At slightly higher (but still low) concentration, amphiphilic molecules will spontaneously assemble into [[micelle]]s or [[vesicle (biology)|vesicles]]. This is done so as to 'hide' the hydrophobic tail of the amphiphile inside the micelle core, exposing a hydrophilic (water-soluble) surface to aqueous solution. These spherical objects do not order themselves in solution, however. At higher concentration, the assemblies will become ordered. A typical phase is a hexagonal columnar phase, where the amphiphiles form long cylinders (again with a hydrophilic surface) that arrange themselves into a roughly hexagonal lattice. This is called the middle soap phase. At still higher concentration, a lamellar phase (neat soap phase) may form, wherein extended sheets of amphiphiles are separated by thin layers of water. For some systems, a cubic (also called viscous isotropic) phase may exist between the hexagonal and lamellar phases, wherein spheres are formed that create a dense cubic lattice. These spheres may also be connected to one another, forming a bicontinuous cubic phase.

The objects created by amphiphiles are usually spherical (as in the case of micelles), but may also be disc-like (bicelles), rod-like, or biaxial (all three micelle axes are distinct). These anisotropic self-assembled nano-structures can then order themselves in much the same way as thermotropic liquid crystals do, forming large-scale versions of all the thermotropic phases (such as a nematic phase of rod-shaped micelles).

For some systems, at high concentrations, inverse phases are observed. That is, one may generate an inverse hexagonal columnar phase (columns of water encapsulated by amphiphiles) or an inverse micellar phase (a bulk liquid crystal sample with spherical water cavities).

A generic progression of phases, going from low to high amphiphile concentration, is:
* Discontinuous cubic phase ([[micellar cubic]] phase)
* [[Hexagonal phase]] (hexagonal columnar phase) (middle phase)
* [[Lamellar phase]]
* Bicontinuous [[cubic phase]]
* Reverse hexagonal columnar phase
* Inverse cubic phase (Inverse micellar phase)

Even within the same phases, their self-assembled structures are tunable by the concentration: for example, in lamellar phases, the layer distances increase with the solvent volume. Since lyotropic liquid crystals rely on a subtle balance of intermolecular interactions, it is more difficult to analyze their structures and properties than those of thermotropic liquid crystals.

Similar phases and characteristics can be observed in immiscible diblock [[copolymer]]s.

===Metallotropic liquid crystals===

Liquid crystal phases can also be based on low-melting ''inorganic'' phases like [[Zinc chloride|ZnCl2]] that have a structure formed of linked tetrahedra and easily form glasses. The addition of long chain soap-like molecules leads to a series of new phases that show a variety of liquid crystalline behavior both as a function of the inorganic-organic composition ratio and of temperature. This class of materials has been named metallotropic.<ref name=martin>{{cite journal| title =Metallotropic liquid crystals formed by surfactant templating of molten metal halides| doi=10.1038/nmat1610|journal=Nature Materials |volume =5|year =2006| pmid =16547520| issue =4|bibcode = 2006NatMa...5..271M| last1 =Martin| first1 =James D.| last2 =Keary| first2 =Cristin L.| last3 =Thornton| first3 =Todd A.| last4 =Novotnak| first4 =Mark P.| last5 =Knutson| first5 =Jeremey W.| last6 =Folmer| first6 =Jacob C. W.| pages =271–5 }}</ref>

===Laboratory analysis of mesophases===
{{Unreferenced section|date=May 2012}}
Thermotropic mesophases are detected and characterized by two major methods, the original method was use of thermal optical microscopy, in which a small sample of the material was placed between two crossed polarizers; the sample was then heated and cooled. As the isotropic phase would not significantly affect the polarization of the light, it would appear very dark, whereas the crystal and liquid crystal phases will both polarize the light in a uniform way, leading to brightness and color gradients. This method allows for the characterization of the particular phase, as the different phases are defined by their particular order, which must be observed. The second method, Differential Scanning Calorimetry (DSC), allows for more precise determination of phase transitions and transition enthalpies. In DSC, a small sample is heated in a way that generates a very precise change in temperature with respect to time. During phase transitions, the heat flow required to maintain this heating or cooling rate will change. These changes can be observed and attributed to various phase transitions, such as key liquid crystal transitions.

Lyotropic mesophases are analyzed in a similar fashion, through these experiments are somewhat more complex, as the concentration of mesogen is a key factor. These experiments are run at various concentrations of mesogen in order to analyze that impact.

==Biological liquid crystals==
Lyotropic liquid-crystalline phases are abundant in living systems, the study of which is referred to as [[lipid polymorphism]]. Accordingly, lyotropic liquid crystals attract particular attention in the field of biomimetic chemistry. In particular, [[biological membrane]]s and [[cell membranes]] are a form of liquid crystal. Their constituent molecules (e.g. [[phospholipid]]s) are perpendicular to the membrane surface, yet the membrane is flexible. These lipids vary in shape (see page on [[lipid polymorphism]]). The constituent molecules can inter-mingle easily, but tend not to leave the membrane due to the high energy requirement of this process. Lipid molecules can flip from one side of the membrane to the other, this process being catalyzed by [[flippase]]s and floppases (depending on the direction of movement). These liquid crystal membrane phases can also host important proteins such as receptors freely "floating" inside, or partly outside, the membrane, e.g. CCT.

Many other biological structures exhibit liquid-crystal behavior. For instance, the concentrated [[protein]] solution that is extruded by a spider to generate [[spider silk|silk]] is, in fact, a liquid crystal phase. The precise ordering of molecules in silk is critical to its renowned strength. [[DNA]] and many [[peptide|polypeptides]] can also form LC phases and this too forms an important part of current academic research.

==Pattern formation in liquid crystals==
{{See also|Pattern formation}}
Anisotropy of liquid crystals is a property not observed in other fluids. This anisotropy makes flows of liquid crystals behave more differentially than those of ordinary fluids. For example, injection of a flux of a liquid crystal between two close parallel plates ([[viscous fingering]]), causes orientation of the molecules to couple with the flow, with the resulting emergence of dendritic patterns.<ref>{{cite journal| title = Viscous fingering in liquid crystals|doi=10.1103/PhysRevA.36.3984|journal=Phys. Rev. A|volume = 36| issue =8|year =1987| page = 3984|bibcode = 1987PhRvA..36.3984B| last1 = Buka| first1 = A.| last2 = Palffy-Muhoray| first2 = P.| last3 = Rácz| first3 = Z. }}</ref> This anisotropy is also manifested in the interfacial energy ([[surface tension]]) between different liquid crystal phases. This anisotropy determines the equilibrium shape at the coexistence temperature, and is so strong that usually facets appear. When temperature is changed one of the phases grows, forming different morphologies depending on the temperature change.<ref>{{cite journal| title = Phase-field simulations and experiments of faceted growth in liquid crystal| doi = 10.1016/S0167-2789(96)00162-5|journal=Physica D|volume = 99| issue =2–3|year =1996| page = 359| last1 = González-Cinca| first1 = R.| last2 = Ramírez-Piscina| first2 = L.| last3 = Casademunt| first3 = J.| last4 = Hernández-Machado| first4 = A.| last5 = Kramer| first5 = L.| last6 = Tóth Katona| first6 = T.| last7 = Börzsönyi| first7 = T.| last8 = Buka| first8 = Á.}}</ref> Since growth is controlled by heat diffusion, anisotropy in thermal conductivity favors growth in specific directions, which has also an effect on the final shape.<ref>{{cite journal| title = Heat diffusion anisotropy in dendritic growth: phase field simulations and experiments in liquid crystals| doi =10.1016/S0022-0248(98)00505-3|journal=Journal of Crystal Growth|volume = 193| issue =4|year =1998| page = 712|bibcode = 1998JCrGr.193..712G| last1 = González-Cinca| first1 = R| last2 = Ramı́Rez-Piscina| first2 = L| last3 = Casademunt| first3 = J| last4 = Hernández-Machado| first4 = A| last5 = Tóth-Katona| first5 = T| last6 = Börzsönyi| first6 = T| last7 = Buka| first7 = Á }}</ref>

==Theoretical treatment of liquid crystals==
Microscopic theoretical treatment of fluid phases can become quite complicated, owing to the high material density, meaning that strong interactions, hard-core repulsions, and many-body correlations cannot be ignored. In the case of liquid crystals, anisotropy in all of these interactions further complicates analysis. There are a number of fairly simple theories, however, that can at least predict the general behavior of the phase transitions in liquid crystal systems.

===Director===

As we already saw above, the nematic liquid crystals are composed of rod-like molecules with the long axes of neighboring molecules aligned approximately to one another. To allow this anisotropic structure, a dimensionless unit vector '''''n''''' called the ''director'', is introduced to represent the direction of preferred orientation of molecules in the neighborhood of any point. Because there is no physical polarity along the director axis, '''''n''''' and '''''-n''''' are fully equivalent.<ref name=b1/>

===Order parameter===

[[File:Nematic-Director.png|thumb|The ''local nematic director'', which is also the ''local optical axis'', is given by the spatial and temporal average of the long molecular axes]]

The description of liquid crystals involves an analysis of order. A second rank symmetric traceless tensor order parameter is used to describe the orientational order of a nematic liquid crystal, although a scalar order parameter is usually sufficient to describe uniaxial nematic liquid crystals. To make this quantitative, an orientational order parameter is usually defined based on the average of the second [[Legendre polynomial]]:

:<math>S = \langle P_2(\cos \theta) \rangle = \left \langle \frac{3 \cos^2 \theta-1}{2} \right \rangle </math>

where <math>\theta</math> is the angle between the liquid-crystal molecular axis and the ''local director'' (which is the 'preferred direction' in a volume element of a liquid crystal sample, also representing its ''[[optical axis|local optical axis]]''). The brackets denote both a temporal and spatial average. This definition is convenient, since for a completely random and isotropic sample, S=0, whereas for a perfectly aligned sample S=1. For a typical liquid crystal sample, S is on the order of 0.3 to 0.8, and generally decreases as the temperature is raised. In particular, a sharp drop of the order parameter to 0 is observed when the system undergoes a phase transition from an LC phase into the isotropic phase.<ref>{{cite journal|title = A model for the orientational order in liquid crystals| doi =10.1007/BF02453342|journal=Il Nuovo Cimento D|volume = 4| issue =3|year =1984| page = 229|bibcode = 1984NCimD...4..229G|last1 = Ghosh|first1 = S. K. }}</ref> The order parameter can be measured experimentally in a number of ways; for instance, [[diamagnetism]], [[birefringence]], [[Raman scattering]], [[Nuclear magnetic resonance|NMR]] and [[Electron Paramagnetic Resonance|EPR]] can be used to determine S.<ref name=b5/>

The order of a liquid crystal could also be characterized by using other even Legendre polynomials (all the odd polynomials average to zero since the director can point in either of two antiparallel directions). These higher-order averages are more difficult to measure, but can yield additional information about molecular ordering.<ref name=b2/>

A positional order parameter is also used to describe the ordering of a liquid crystal. It is characterized by the variation of the density of the center of mass of the liquid crystal molecules along a given vector. In the case of positional variation along the z-axis the density <math>\rho (z)</math> is often given by:

:<math>\rho (\mathbf{r})=\rho (z)=\rho_0+\rho_1\cos\left (q_sz-\phi\right )+\cdots \, </math>

The complex positional order parameter is defined as <math>\psi (\mathbf{r})=\rho_1 (\mathbf{r})e^{i\phi(\mathbf{r})}</math> and <math>\rho_0</math> the average density. Typically only the first two terms are kept and higher order terms are ignored since most phases can be described adequately using sinusoidal functions. For a perfect nematic <math>\psi=0</math> and for a smectic phase <math>\psi</math> will take on complex values. The complex nature of this order parameter allows for many parallels between nematic to smectic phase transitions and conductor to superconductor transitions.<ref name=b1/>

===Onsager hard-rod model===
{{unsolved|physics|Can the nematic to smectic (A) phase transition in liquid crystal states be characterized as a [[background independence|universal]] phase transition?}}
A simple model which predicts lyotropic phase transitions is the hard-rod model proposed by [[Lars Onsager]]. This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder which the approaching cylinder's center-of-mass cannot enter (due to the hard-rod repulsion between the two idealized objects). Thus, this angular arrangement sees a ''decrease'' in the net positional [[entropy]] of the approaching cylinder (there are fewer states available to it).<ref>{{cite journal| journal=Annals of the New York Academy of Sciences|volume = 51| issue =4|year = 1949| page = 627| doi =10.1111/j.1749-6632.1949.tb27296.x|title=The effects of shape on the interaction of colloidal particles|bibcode = 1949NYASA..51..627O| last1=Onsager| first1=Lars }}</ref><ref name=vroege>{{cite journal| title = Phase transitions in lyotropic colloidal and polymer liquid crystals| doi= 10.1088/0034-4885/55/8/003|journal=Rep. Progr. Phys.|volume = 55| issue =8| year = 1992| page = 1241|bibcode = 1992RPPh...55.1241V| last1 = Vroege| first1 = G J| last2 = Lekkerkerker| first2 = H N W }}</ref>

The fundamental insight here is that, whilst parallel arrangements of anisotropic objects lead to a decrease in orientational entropy, there is an increase in positional entropy. Thus in some case greater positional order will be entropically favorable. This theory thus predicts that a solution of rod-shaped objects will undergo a phase transition, at sufficient concentration, into a nematic phase. Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems.<ref name=vroege/>

===Maier–Saupe mean field theory===
This statistical theory, proposed by [[Alfred Saupe]] and Wilhelm Maier, includes contributions from an attractive intermolecular potential from an induced dipole moment between adjacent liquid crystal molecules. The anisotropic attraction stabilizes parallel alignment of neighboring molecules, and the theory then considers a [[mean-field theory|mean-field]] average of the interaction. Solved self-consistently, this theory predicts thermotropic nematic-isotropic phase transitions, consistent with experiment.<ref>{{cite journal| author=Maier W. and Saupe A.| journal=Z. Naturforsch. A|volume = 13|language=German|title=Eine einfache molekulare theorie des nematischen kristallinflussigen zustandes|page = 564|year =1958|bibcode = 1958ZNatA..13..564M| last2=Saupe }}</ref><ref>{{cite journal| author=Maier W. and Saupe A.|title=Eine einfache molekular-statistische theorie der nematischen kristallinflussigen phase .1|language=German|journal=Z. Naturforsch. A|volume = 14| page = 882|year =1959|bibcode = 1959ZNatA..14..882M| last2=Saupe }}</ref><ref>{{cite journal| author=Maier W. and Saupe A.| journal=Z. Naturforsch. A|volume = 15|language=German|title=Eine einfache molekular-statistische theorie der nematischen kristallinflussigen phase .2|page = 287|year =1960|bibcode = 1960ZNatA..15..287M| last2=Saupe }}</ref>

===McMillan's model===
McMillan's model, proposed by William McMillan,<ref>{{cite journal|title= Simple Molecular Model for the Smectic A Phase of Liquid Crystals|journal=Phys. Rev. A|volume= 4 |year=1971|issue=3|page=1238|doi=10.1103/PhysRevA.4.1238|bibcode = 1971PhRvA...4.1238M|last1= McMillan|first1= W. }}</ref> is an extension of the Maier–Saupe mean field theory used to describe the phase transition of a liquid crystal from a nematic to a smectic A phase. It predicts that the phase transition can be either continuous or discontinuous depending on the strength of the short-range interaction between the molecules. As a result, it allows for a triple critical point where the nematic, isotropic, and smectic A phase meet. Although it predicts the existence of a triple critical point, it does not successfully predict its value. The model utilizes two order parameters that describe the orientational and positional order of the liquid crystal. The first is simply the average of the second [[Legendre polynomials|Legendre polynomial]] and the second order parameter is given by:
:<math>\sigma=\left\langle\cos\left (\frac{2\pi z_i}{d}\right )\left (\frac{3}{2}\cos^2\theta_i-\frac{1}{2}\right )\right\rangle</math>
The values ''zi, θi'', and ''d'' are the position of the molecule, the angle between the molecular axis and director, and the layer spacing. The postulated potential energy of a single molecule is given by:
:<math>U_i(\theta_i,z_i)=-U_0\left (S+\alpha\sigma\cos\left (\frac{2\pi z_i}{d}\right )\right )\left (\frac{3}{2}\cos^2\theta_i-\frac{1}{2}\right )</math>
Here constant α quantifies the strength of the interaction between adjacent molecules. The potential is then used to derive the thermodynamic properties of the system assuming thermal equilibrium. It results in two self-consistency equations that must be solved numerically, the solutions of which are the three stable phases of the liquid crystal.<ref name=b5/>

===Elastic continuum theory===
In this formalism, a liquid crystal material is treated as a continuum; molecular details are entirely ignored. Rather, this theory considers perturbations to a presumed oriented sample. The distortions of the liquid crystal are commonly described by the [[Frank free energy density]]. One can identify three types of distortions that could occur in an oriented sample: (1) '''twists''' of the material, where neighboring molecules are forced to be angled with respect to one another, rather than aligned; (2) '''splay''' of the material, where bending occurs perpendicular to the director; and (3) '''bend''' of the material, where the distortion is parallel to the director and molecular axis. All three of these types of distortions incur an energy penalty. They are distortions that are induced by the boundary conditions at domain walls or the enclosing container. The response of the material can then be decomposed into terms based on the elastic constants corresponding to the three types of distortions. Elastic continuum theory is a particularly powerful tool for modeling liquid crystal devices <ref>{{cite journal|title = Continuum theory for nematic liquid crystals| journal=Continuum Mechanics and Thermodynamics|volume = 4|year= 1992| issue =3| page = 167| doi =10.1007/BF01130288|bibcode = 1992CMT.....4..167L|last1 = Leslie|first1 = F. M. }}</ref> and lipid bilayers.<ref>{{cite journal|title = Determining Biomembrane Bending Rigidities from Simulations of Modest Size| journal=Physical Review Letters|volume = 109|year= 2012| issue =2| page = 028102| doi =10.1103/PhysRevLett.109.028102|last1 = Watson|first1 = M. C.|last2 = Brandt|first2 = E. G.|last3 = Welch|first3 = P. M.|last4 = Brown|first4 = F. L. H. |bibcode = 2012PhRvL.109b8102W }}</ref>

==External influences on liquid crystals==
Scientists and engineers are able to use liquid crystals in a variety of applications because external perturbation can cause significant changes in the macroscopic properties of the liquid crystal system. Both electric and magnetic fields can be used to induce these changes. The magnitude of the fields, as well as the speed at which the molecules align are important characteristics industry deals with. Special surface treatments can be used in liquid crystal devices to force specific orientations of the director.

===Electric and magnetic field effects===
The ability of the director to align along an external field is caused by the electric nature of the molecules. Permanent electric dipoles result when one end of a molecule has a net positive charge while the other end has a net negative charge. When an external electric field is applied to the liquid crystal, the dipole molecules tend to orient themselves along the direction of the field.

Even if a molecule does not form a permanent dipole, it can still be influenced by an electric field. In some cases, the field produces slight re-arrangement of electrons and protons in molecules such that an induced electric dipole results. While not as strong as permanent dipoles, orientation with the external field still occurs.
The effects of magnetic fields on liquid crystal molecules are analogous to electric fields. Because magnetic fields are generated by moving electric charges, permanent magnetic dipoles are produced by electrons moving about atoms. When a magnetic field is applied, the molecules will tend to align with or against the field.

===Surface preparations===
In the absence of an external field, the director of a liquid crystal is free to point in any direction. It is possible, however, to force the director to point in a specific direction by introducing an outside agent to the system. For example, when a thin polymer coating (usually a polyimide) is spread on a glass substrate and rubbed in a single direction with a cloth, it is observed that liquid crystal molecules in contact with that surface align with the rubbing direction. The currently accepted mechanism for this is believed to be an epitaxial growth of the liquid crystal layers on the partially aligned polymer chains in the near surface layers of the polyimide.

===Fredericks transition===
The competition between orientation produced by surface anchoring and by electric field effects is often exploited in liquid crystal devices. Consider the case in which liquid crystal molecules are aligned parallel to the surface and an electric field is applied perpendicular to the cell. At first, as the electric field increases in magnitude, no change in alignment occurs. However at a threshold magnitude of electric field, deformation occurs. Deformation occurs where the director changes its orientation from one molecule to the next. The occurrence of such a change from an aligned to a deformed state is called a Fredericks transition and can also be produced by the application of a magnetic field of sufficient strength.

The Fredericks transition is fundamental to the operation of many liquid crystal displays because the director orientation (and thus the properties) can be controlled easily by the application of a field.

==Effect of chirality==
As already described, [[chirality (chemistry)|chiral]] liquid-crystal molecules usually give rise to chiral mesophases. This means that the molecule must possess some form of asymmetry, usually a [[stereogenic]] center. An additional requirement is that the system not be [[racemic]]: a mixture of right- and left-handed molecules will cancel the chiral effect. Due to the cooperative nature of liquid crystal ordering, however, a small amount of chiral dopant in an otherwise achiral mesophase is often enough to select out one domain handedness, making the system overall chiral.

Chiral phases usually have a helical twisting of the molecules. If the pitch of this twist is on the order of the wavelength of visible light, then interesting optical interference effects can be observed. The chiral twisting that occurs in chiral LC phases also makes the system respond differently from right- and left-handed circularly polarized light. These materials can thus be used as [[polarizer|polarization filters]].<ref>{{cite journal| title = Video camera system using liquid-crystal polarizing filter toreduce reflected light| doi =10.1109/11.735903|journal=IEEE Transactions on Broadcasting|volume = 44| issue =4|year = 1998| page = 419| last1 = Fujikake| first1 = H.| last2 = Takizawa| first2 = K.| last3 = Aida| first3 = T.| last4 = Negishi| first4 = T.| last5 = Kobayashi| first5 = M.}}</ref>

It is possible for chiral LC molecules to produce essentially achiral mesophases. For instance, in certain ranges of concentration and [[molecular weight]], DNA will form an achiral line hexatic phase. An interesting recent observation is of the formation of chiral mesophases from achiral LC molecules. Specifically, bent-core molecules (sometimes called banana liquid crystals) have been shown to form liquid crystal phases that are chiral.<ref>{{cite journal|title = Switching of banana liquid crystal mesophases under field| doi =10.1140/epje/e2003-00016-y|journal=European Physical Journal E| volume = 10|year = 2003| pmid =15011066| issue =2|bibcode = 2003EPJE...10..129A|last1 = Achard|first1 = M.F.|last2 = Bedel|first2 = J.Ph.|last3 = Marcerou|first3 = J.P.|last4 = Nguyen|first4 = H.T.|last5 = Rouillon|first5 = J.C.|pages = 129–34 }}</ref> In any particular sample, various domains will have opposite handedness, but within any given domain, strong chiral ordering will be present. The appearance mechanism of this macroscopic chirality is not yet entirely clear. It appears that the molecules stack in layers and orient themselves in a tilted fashion inside the layers. These liquid crystals phases may be [[ferroelectric]] or anti-ferroelectric, both of which are of interest for applications.<ref>{{cite journal|title = Ferroelectric nematic liquid-crystal phases of dipolar hard ellipsoids|journal=Phys. Rev. A|volume = 40| issue =9| page= 5444| year = 1989| doi =10.1103/PhysRevA.40.5444|bibcode = 1989PhRvA..40.5444B|last1 = Baus|first1 = Marc|last2 = Colot|first2 = Jean-Louis }}</ref><ref>{{cite journal| title = Pressure-Temperature Phase Diagrams of Ferroelectric Liquid Crystals|journal=J. Phys. Soc. Jpn.|volume = 71|year = 2002| issue =2| page = 509| doi =10.1143/JPSJ.71.509|bibcode = 2002JPSJ...71..509U| last1 = Uehara| first1 = Hiroyuki| last2 = Hatano| first2 = Jun }}</ref>

Chirality can also be incorporated into a phase by adding a chiral [[dopant]], which may not form LCs itself. [[Twisted nematic field effect|Twisted-nematic]] or [[Super-twisted nematic display|super-twisted nematic]] mixtures often contain a small amount of such dopants.

==Applications of liquid crystals==
{{see also|Liquid crystal display}}
[[File:LCD layers.svg|thumb|Structure of liquid crystal display: 1 – vertical polarization filter, 2,4 – glass with electrodes, 3 – liquid crystals, 5 – horizontal polarization filter, 6 – reflector]]

Liquid crystals find wide use in liquid crystal displays, which rely on the [[optics|optical]] properties of certain liquid crystalline substances in the presence or absence of an [[electric field]]. In a typical device, a liquid crystal layer (typically 10 μm thick) sits between two [[polarizer]]s that are crossed (oriented at 90° to one another). The liquid crystal alignment is chosen so that its relaxed phase is a twisted one (see [[Twisted nematic field effect]]).<ref name=castellano/> This twisted phase reorients light that has passed through the first polarizer, allowing its transmission through the second polarizer (and reflected back to the observer if a reflector is provided). The device thus appears transparent. When an electric field is applied to the LC layer, the long molecular axes tend to align parallel to the electric field thus gradually untwisting in the center of the liquid crystal layer. In this state, the LC molecules do not reorient light, so the light polarized at the first polarizer is absorbed at the second polarizer, and the device loses transparency with increasing voltage. In this way, the electric field can be used to make a pixel switch between transparent or opaque on command. Color LCD systems use the same technique, with color filters used to generate red, green, and blue pixels.<ref name=castellano/> Similar principles can be used to make other liquid crystal based optical devices.<ref>{{cite journal|title = Integrating liquid crystal based optical devices in photonic crystal| doi = 10.1007/s11082-007-9139-8|journal=Optical and Quantum Electronics|volume = 39| issue =12–13|year =2007| page = 1009|last1 = Alkeskjold|first1 = Thomas Tanggaard|last2 = Scolari|first2 = Lara|last3 = Noordegraaf|first3 = Danny|last4 = Lægsgaard|first4 = Jesper|last5 = Weirich|first5 = Johannes|last6 = Wei|first6 = Lei|last7 = Tartarini|first7 = Giovanni|last8 = Bassi|first8 = Paolo|last9 = Gauza|first9 = Sebastian}}</ref>

[[Liquid crystal tunable filter]]s are used as [[electrooptical]] devices, e.g., in [[hyperspectral imaging]].

[[thermochromism|Thermotropic]] chiral LCs whose pitch varies strongly with temperature can be used as crude [[liquid crystal thermometer]]s, since the color of the material will change as the pitch is changed. Liquid crystal color transitions are used on many aquarium and pool thermometers as well as on thermometers for infants or baths.<ref>Plimpton, R. Gregory "Pool thermometer" {{US Patent|4738549}} Issued on April 19, 1988</ref> Other liquid crystal materials change color when stretched or stressed. Thus, liquid crystal sheets are often used in industry to look for hot spots, map heat flow, measure stress distribution patterns, and so on. Liquid crystal in fluid form is used to detect electrically generated hot spots for [[failure analysis]] in the [[semiconductor]] industry.<ref>{{cite web| url = http://www.acceleratedanalysis.com/LC_hotspotdetection_procedure.html |title =Hot-spot detection techniques for ICs| accessdate=May 5, 2009|work=acceleratedanalysis.com}}</ref>

[[Liquid crystal laser]]s use a liquid crystal in the [[Active laser medium|lasing medium]] as a distributed feedback mechanism instead of external mirrors. Emission at a [[Photonic crystal|photonic bandgap]] created by the periodic dielectric structure of the liquid crystal gives a low-threshold high-output device with stable monochromatic emission.<ref name="Kopp1998">{{cite journal | title = Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals | journal=Optics Express | year = 1998 volume = 23 | issue = 21 | pages = 1707–1709| doi=10.1364/OL.23.001707 | pmid=18091891 | bibcode=1998OptL...23.1707K | last1 = Kopp | first1 = V. I. | last2 = Fan | first2 = B. | last3 = Vithana | first3 = H. K. M. | last4 = Genack | first4 = A. Z. | volume = 23}}</ref><ref name="Dolgaleva2008">{{cite journal | title = Enhanced laser performance of cholesteric liquid crystals doped with oligofluorene dye | journal=Journal of the Optical Society of America | year = 2008 | first = Ksenia | last = Dolgaleva | coauthors = Simon K.H. Wei, Svetlana G. Lukishova, Shaw H. Chen, Katie Schwertz, and Robert W. Boyd | volume = 25 | issue = 9 | pages = 1496–1504| doi=10.1364/JOSAB.25.001496 | bibcode=2008JOSAB..25.1496D}}</ref>

[[Smart glass#Polymer dispersed liquid crystal devices|Polymer Dispersed Liquid Crystal (PDLC)]] sheets and rolls are available as adhesive backed [[Smart film]] which can be applied to windows and electrically switched between transparent and opaque to provide privacy.

Many common fluids, such as [[soap|soapy water]], are in fact liquid crystals. Soap forms a variety of LC phases depending on its concentration in water.<ref>{{cite journal| title = Structure of the Liquid-Crystal Phases of the Soap–water System: Middle Soap and Neat Soap| journal=Nature|volume = 180|year = 1957| issue =4586| page = 600| doi =10.1038/180600a0|bibcode = 1957Natur.180..600L| last1 = Luzzati| first1 = V.| last2 = Mustacchi| first2 = H.| last3 = Skoulios| first3 = A. }}</ref>

==See also==
{{colbegin|3}}
* [[Biaxial nematic]]
* [[Columnar phase]]
* [[Chromonic]]
* [[LCD classification]]
* [[Liquid crystal display]]
* [[Liquid crystal polymer]]
* [[Liquid crystal tunable filter]]
* [[Lyotropic liquid crystal]]
* [[Pattern formation]]
* [[Plastic crystallinity]]
* [[Smart glass]]
* [[Thermochromics]]
* [[Thermotropic crystal]]
* [[Twisted nematic field effect]]
* [[Nematicon]]
* [[Liquid crystal thermometer]]
* [[Mood ring]]
{{colend}}

==References==
{{reflist|35em}}

==External links==
{{commons|Liquid crystal}}
* {{cite news| url = http://nobelprize.org/educational_games/physics/liquid_crystals/history/|publisher=Nobelprize.org| title = History and Properties of Liquid Crystals| accessdate =June 6, 2009}}
* [http://www.iupac.org/publications/pac/2001/7305/7305x0845.html Definitions of basic terms relating to low-molar-mass and polymer liquid crystals (IUPAC Recommendations 2001)]
* [http://plc.cwru.edu/tutorial/enhanced/files/textbook.htm An intelligible introduction to liquid crystals] from Case Western Reserve University
* [http://bly.colorado.edu/lcphysics.html Liquid Crystal Physics tutorial] from the Liquid Crystals Group, University of Colorado
* [http://www.elis.ugent.be/ELISgroups/lcd/lc/lc.php Liquid Crystals & Photonics Group – Ghent University (Belgium)], good tutorial
* [http://www.elis.ugent.be/ELISgroups/lcd/research/bpm.php Simulation of light propagation in liquid crystals], free program
* [http://liqcryst.chemie.uni-hamburg.de/ Liquid Crystals Interactive Online]
* [http://www.lci.kent.edu Liquid Crystal Institute] Kent State University
* [http://www.tandf.co.uk/journals/titles/02678292.asp Liquid Crystals] a journal by Taylor&Francis
* [http://www.tandf.co.uk/journals/titles/15421406.asp Molecular Crystals and Liquid Crystals] a journal by Taylor & Francis
* [http://www.acceleratedanalysis.com/LC_hotspotdetection_procedure.html Hot-spot detection techniques for ICs]
* [http://www.mc2.chalmers.se/mc2/pl/lc/engelska/frame.html What are liquid crystals?] from Chalmers University of Technology, Sweden
* {{cite journal| author=H. Kleinert and K. Maki|title =Lattice Textures in Cholesteric Liquid Crystals| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re75/75.pdf| journal=Fortschritte Physik|volume = 29|year =1981| page = 219| doi=10.1002/prop.19810290503| issue=5|bibcode = 1981ForPh..29..219K }}
* [http://www.beilstein-journals.org/bjoc/browse/singleSeries.htm?sn=5 Progress in liquid crystal chemistry] Thematic series in the Open Access Beilstein Journal of Organic Chemistry
* [http://www.doitpoms.ac.uk/tlplib/liquid_crystals/index.php DoITPoMS Teaching and Learning Package- "Liquid Crystals"]

{{states of matter}}

{{good article}}

[[Category:Liquid crystals]]
[[Category:Soft matter]]
[[Category:Optical materials]]
[[Category:Phase transitions]]

Cross section (physics)

2013-08-07T09:49:42Z

129.234.190.31: /* References */ corrected typo

A '''cross section''' is the effective area that governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the [[Beer–Lambert law]].

In [[nuclear physics|nuclear]] and [[particle physics]], the concept of a '''cross section''' is used to express the likelihood of interaction between particles.

When particles in a beam are thrown against a foil made of a certain substance, the ''cross section'' <math>\sigma</math> is a hypothetical [[area]] measure around the target particles of the substance (usually its atoms) that represents a surface. If a particle of the beam crosses this surface, there will be some kind of interaction.

The term is derived from the purely [[classical mechanics|classical]] picture of (a large number of) [[Point particle|point-like]] projectiles directed to an area that includes a solid target. Assuming that an interaction will occur (with 100% probability) if the projectile hits the solid, and not at all (0% probability) if it misses, the total interaction probability for the single projectile will be the ratio of the area of the section of the solid (the ''cross section'', represented by <math>\sigma</math>) to the total targeted area.

This basic concept is then extended to the cases where the interaction probability in the targeted area assumes intermediate values - because the target itself is not homogeneous, or because the interaction is mediated by a non-uniform field. A particular case is [[scattering]].

==Scattering==
{{main|Scattering cross-section}}

The '''scattering cross-section''', ''σ''scat, is a hypothetical area which describes the likelihood of light (or other radiation) being [[scattering|scattered]] by a particle. In general, the scattering cross-section is different from the [[cross sectional area|geometrical cross-section]] of a particle, and it depends upon the [[wavelength]] of light and the [[permittivity]], shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross-section and the number of particles present. In terms of area, the ''total cross-section'' (σ) is the sum of the cross-sections due to [[absorption cross section|absorption]], [[scattering]] and [[luminescence]]
:<math>\sigma = \sigma_\text{A} + \sigma_\text{S} + \sigma_\text{L}.\ </math>

The total cross-section is related to the [[absorbance]] of the light intensity through [[Beer-Lambert|Beer-Lambert's law]], which says absorbance is proportional to concentration: <math>A_\lambda = C \,\ell\, \sigma</math>, where ''C'' is the concentration as a number density, ''A''λ is the absorbance at a given [[wavelength]] ''λ'', and <math>\ell</math> is the [[path length]]. The extinction or [[absorbance]] of the radiation is the [[logarithm]] ([[Decadic logarithm|decadic]] or, more usually, [[Natural logarithm|natural]]) of the reciprocal of the [[transmittance]]:<ref>{{cite book|chapter=2. Spectrophotometry|isbn=81-219-2633-5|first=P.K.|last=Bajpai|title=Biological Instrumentation and Biology|url=http://books.google.com/?id=THq-cOPO8RQC&pg=PA14&dq=%22extinction+coefficient%22+transmittance+length+concentration}}</ref>
:<math>A_\lambda = - \log \mathcal{T}.\ </math>

==Nuclear physics==
[[File:Cross-section-illustration-simple.svg|right]]
{{Main|neutron cross section}}
In [[nuclear physics]], it is convenient to express the probability of a particular event by a cross section. Statistically, the centers of the atoms in a thin foil can be considered as points evenly distributed over a plane. The center of an atomic projectile striking this plane has geometrically a definite probability of passing within a certain distance <math>r</math> of one of these points. In fact, if there are <math>n</math> atomic centers in an area <math>A</math> of the plane, this probability is <math>(n \pi r^2)/A</math>, which is simply the ratio of the aggregate area of circles of radius <math>r</math> drawn around the points to the whole area. If we think of the atoms as impenetrable steel discs and the impinging particle as a bullet of negligible diameter, this ratio is the probability that the bullet will strike a steel disc, i.e., that the atomic projectile will be stopped by the foil. If it is the fraction of impinging atoms getting through the foil which is measured, the result can still be expressed in terms of the equivalent stopping cross section of the atoms. This notion can be extended to any interaction between the impinging particle and the atoms in the target. For example, the probability that an [[alpha particle]] striking a [[beryllium]] target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.

==Rate (particle physics)==

{{main|Event (particle physics)}}
{{for|the similar quantity in [[chemical kinetics]]|reaction rate}}

In [[scattering theory]], [[particle physics]] and [[nuclear physics]], the '''rate''' at which a specific [[subatomic particle]] reaction occurs is a [[physical quantity]] measuring the number of reactions per unit time.

===Partial cross section===

For a particle beam (say of [[neutron]]s, [[pion]]s) incident on a target ([[liquid hydrogen]]), for each type of reaction in the scattering process labelled by an index ''r'' = 1, 2, 3..., it is calculated from:<ref name="Martin Shaw p 343 - 347">{{cite book|author=B.R. Martin, G. Shaw|year=2009|title=Particle Physics|edition=3rd|publisher=Manchester Physics Series, John Wiley & Sons|pages=343–347|isbn=978-0-470-03294-7|url=http://books.google.co.uk/books?id=7tsVcsm84rEC&pg=PT383&dq=Rate+and+cross+section+particle+physics&hl=en&sa=X&ei=pj2vUajEGsKc0wXRk4GgBg&ved=0CDEQ6AEwAA}}</ref>

:<math>W_r = JN\sigma_r</math>

where ''N'' is the number of target particles, illuminated by the beam containing ''n'' particles per unit volume in the beam ([[number density]] of particles) traveling with average [[velocity]] ''v'' in the [[rest frame]] of the target, and these two quantities combine into the [[flux]] of the beam ''J'' = ''nv''. The cross section of the reaction is ''σr''. Since the beam flux has [[dimensional analysis|dimension]]s of [length]−2·[time]−1 and ''σr'' has dimensions of [length]2 while ''N'' is a dimensionless number, the rate ''W'' has the dimensions of reciprocal time - which intuitively represents a [[frequency]] of recurring events.

The above formula assumes the following:

*the beam particles all have the same [[kinetic energy]],
*the number density of the beam particles is sufficiently low: allowing the interactions between the particles within the beam to be neglected,
*the number density of target particles is sufficiently low: so that only one scattering event per particle occurs as soon as the beam is incident with the target, and multiple scattering events within the target can be neglected,
*the [[matter wave|de Broglie wavelength]] of the beam is much smaller than the inter-particle separations within the target, so that [[diffraction]] effects through the target can be neglected,
*the collision energy is sufficiently high allowing the [[binding energy|binding energies]] in the target particles to be neglected.

These conditions are usually met in experiments, which allows for a very simple calculation of rate.

Sometimes the rate per unit target particle, or '''rate density''', is more useful. For reaction ''r'':<ref>{{cite book|title=Radiation detection and measurement|edition=4th|publisher=Wiley|author=G.F. Knoll|page=55|year=2010|isbn=978-0-470-13148-0|url=http://books.google.co.uk/books?id=4vTJ7UDel5IC&printsec=frontcover&dq=radiation+detection+and+measurement&hl=en&sa=X&ei=kT6vUe_eKZK00QWdloCIBQ&sqi=2&ved=0CDcQ6AEwAA#v=onepage&q=radiation%20detection%20and%20measurement&f=false}}</ref>

:<math>W_r/N = J\sigma_r</math>

===Total cross section===

The cross section ''σr'' is specifically for ''one'' type of reaction, and is called the partial cross section. The total cross section, and corresponding total rate of the reaction, can be found by summing over the cross sections and rates for each reaction:<ref name="Martin Shaw p 343 - 347"/>

:<math>W = \sum_r W_r = JN \sum_r \sigma_r = JN \sigma</math>

===Differential cross section===

In terms of the [[differential cross section]] ''dσr''(''θ'', ''φ'') as a function of [[spherical polar coordinates|spherical polar angles]] ''θ'' and ''φ'' for reaction ''r'', the differential rate is:<ref name="Martin Shaw p 343 - 347"/>

:<math>dW_r = JN d\sigma_r = JN \frac{d\sigma_r}{d\Omega} d\Omega</math>

where dΩ = ''d''(cos''θ'')''dφ'' is the [[solid angle]] element in the vicinity of the event with vertex at the point of scattering. Integrating over ''θ'' and ''φ'' returns the rate for reaction ''r'':

:<math>W_r = JN \int_0^{2\pi} d\varphi \int_{-1}^{+1} d(\cos\theta) \frac{d\sigma_r}{d\Omega} </math>

==See also==

*[[Cross sectional area]]
*[[Differential cross section]]
*[[Luminosity (scattering theory)]]
*[[Neutron cross section]]
*[[Particle detector]]
*[[Radar]]: The (monostatic) [[radar cross section]] is defined as 4 π times the [[radio]] differential cross section at 180 degrees.
*[[Rutherford scattering]]
*[[Scattering amplitude]]

== References ==
{{More footnotes|date=December 2009}}

{{reflist}}

* J.D.Bjorken, S.D.Drell, ''Relativistic Quantum Mechanics'', 1964
* P.Roman, ''Introduction to Quantum Theory'', 1969
* W.Greiner, J.Reinhardt, ''Quantum Electrodynamics'', 1994
* R.G. Newton. ''Scattering Theory of Waves and Particles''. McGraw Hill, 1966.
*{{cite book| author=R.C. Fernow|title=Introduction to Experimental Particle Physics|year=1989|edition=|publisher=Cambridge University Press|pages=|isbn=0-521-379-407|url=http://books.google.co.uk/books?id=WNhMzhm0SscC&pg=PA80&dq=Rate+and+cross+section+particle+physics&hl=en&sa=X&ei=pj2vUajEGsKc0wXRk4GgBg&ved=0CD4Q6AEwAg#v=onepage&q=Rate%20and%20cross%20section%20particle%20physics&f=false}}

==External links==
*[http://hyperphysics.phy-astr.gsu.edu/Hbase/nuclear/nucrea.html#c3 Nuclear Cross Section]
*[http://hyperphysics.phy-astr.gsu.edu/Hbase/nuclear/crosec.html#c1 Scattering Cross Section]
*[http://www-nds.iaea.org/ IAEA - Nuclear Data Services]
*[http://www.nndc.bnl.gov/ BNL - National Nuclear Data Center]
*[http://pdg.lbl.gov/ Particle Data Group - The Review of Particle Physics]
*[http://www.iupac.org/goldbook/R05169.pdf IUPAC Goldbook - Definition: Reaction Cross Section]
*[http://www.iupac.org/goldbook/C01161.pdf IUPAC Goldbook - Definition: Collision Cross Section]

[[Category:Concepts in physics]]
[[Category:Nuclear physics]]
[[Category:Particle physics]]
[[Category:Experimental particle physics]]
[[Category:Scattering]]
[[Category:Scattering theory]]
[[Category:Dimensional analysis]]
[[Category:Measurement]]

Hodge structure

2013-07-10T11:43:07Z

129.234.158.246: /* Definition of Hodge structures */

In [[category theory]], a '''span''', '''roof''' or '''correspondence''' is a generalization of the notion of [[binary relation|relation]] between two objects of a category. When the category has all [[Pullback (category theory)|pullbacks]] (and satisfies a small number of other conditions), spans can be considered as morphisms in a [[Localization of a category|category of fractions]].

== Formal definition ==
A span is a [[Diagram (category theory)|diagram]] of type <math>\Lambda = (-1 \leftarrow 0 \rightarrow +1),</math> i.e., a diagram of the form <math>Y \leftarrow X \rightarrow Z</math>.

That is, let Λ be the category (-1 ← 0 → +1). Then a span in a [[category (mathematics)|category]] C is a [[functor]] S:Λ → C. This means that a span consists of three objects X, Y and Z of C and [[morphisms]] f:X → Y and g:X → Z: it is two maps with common ''domain''.

The [[colimit]] of a span is a [[Pushout (category theory)|pushout]].

== Examples ==

* If ''R'' is a relation between sets ''X'' and ''Y'' (i.e. a subset of ''X'' × ''Y''), then ''X'' ← ''R'' → ''Y'' is a span, where the maps are the projection maps <math>X \times Y \overset{\pi_X}{\to} X</math> and <math>X \times Y \overset{\pi_Y}{\to} Y</math>.
* Any object yields the trivial span <math>A = A = A;</math> formally, the diagram ''A'' ← ''A'' → ''A,'' where the maps are the identity.
* More generally, let <math>\phi\colon A \to B</math> be a morphism in some category. There is a trivial span ''A'' = ''A'' → ''B;'' formally, the diagram ''A'' ← ''A'' → ''B'', where the left map is the identity on ''A,'' and the right map is the given map φ.
* If ''M'' is a [[model category]], with W the set of [[weak equivalence (homotopy theory)|weak equivalence]]s, then the spans of the form <math>X \leftarrow Y \rightarrow Z,</math> where the left morphism is in ''W,'' can be considered a generalised morphism (i.e., where one "inverts the weak equivalences"). Note that this is not the usual point of view taken when dealing with model categories.

== Cospans ==

A cospan K in a category C is a functor K:Λop → C; equivalently, a ''contravariant'' functor from Λ to C. That is, a diagram of type <math>\Lambda^\text{op} = (-1 \rightarrow 0 \leftarrow +1),</math> i.e., a diagram of the form <math>Y \rightarrow X \leftarrow Z</math>.

Thus it consists of three objects X, Y and Z of C and morphisms f:Y → X and g:Z → X: it is two maps with common ''codomain.''

The [[limit (category theory)|limit]] of a cospan is a [[Pullback (category theory)|pullback]].

An example of a cospan is a [[cobordism]] ''W'' between two manifolds ''M'' and ''N'', where the two maps are the inclusions into ''W''. Note that while cobordisms are cospans, the category of cobordisms is not a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that ''M'' and ''N'' form a partition of the boundary of ''W'' is a global constraint.

The category '''nCob''' of finite-dimensional cobordisms is a [[dagger compact category]]. More generally, the category '''Span'''(''C'') of spans on any category ''C'' with finite limits is also dagger compact.

== See also ==
* [[Binary relation]]
* [[Pullback (category theory)]]
* [[Pushout (category theory)]]
* [[Cobordism]]

==References==
* {{nlab|id=span}}

[[Category:Functors]]

Photocatalytic water splitting

2012-06-22T13:30:08Z

129.234.189.203:

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2006-01-24T22:37:42Z

129.234.4.1: Rv to last version by Easyas12c, see WP:DRV

Listed below are the Singapore property measures that the government have launched since 1996. The Authorities's goal for these cooling measures is to ensure a secure and sustainable property market the place prices transfer in step with financial fundamentals. Dennis Wee Group spokesman Lee Sze Teck said those who use as much as 60 per cent of their month-to-month earnings to service their house loans are sometimes incomes decent wages and have a very good credit rating. Engineer James Leow, 26, had plans to purchase a five-room resale flat near his dad and mom' place in Ang [http://fasdchildwelfare.ca/users/tobyfairweather6512 http://fasdchildwelfare.ca] Mo Kio this yr. This measure will bring some sanity to the entire industrial market," mentioned Mr Douglas Foo, chief government of Sakae Sushi, which bought its own 200,000 sq ft industrial constructing in Paya Lebar iPark within the mid-2000s. More than 10 million sq ft of multi-consumer industrial space can also be expected to go on the market within the subsequent few years. If they don't adjust to this rule, they are going to be hit with a hefty extra tax within the type of the ABSD. Skypark Residences EC for a borrower with one or more outstanding residential property loans3; and No SSD can be payable by the vendor if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Finances assistance measures for the property market when the measures expire. Allowing as much as 2 years of property tax deferral for land authorised for improvement. Energetic secondary marketplace for easy exit of property investments. Interest on mortgage cost is tax deductible when the property is rented out. We have progressive property tax rates for residential properties- the higher the worth of the residential property, the upper the tax price. Why do I've to pay property tax on my dwelling when I am an onwer-occupier and never deriving rental revenue? Palacio Cluster Landed Singapore is alreadyas a metropolis with one of the highest quality of residing on the planet. Building on Singapore's sturdy traits as a secure atmosphere with good amenities for elevating households, new developments and events such as the Built-in Resorts, Formula 1 Singtel Singapore Grand Prix and Volvo Ocean Race may even spice up the leisure scene. This concerted drive will additional promote Singapore as a world metropolis. Can get excessive rents Singapore is considered as a prime world metropolis c) Singapore Residents (SC) who already personal # 2 or extra residential properties must pay ABSD of 3% on the purchase or acquisition of one other residential property. Now we have also properties that are open for Sale or Rental. New Condominium Eco Sanctuary Launch 01Dec 2012 c) SSD at 5% if the property is sold in the third year of buy, i.e. the property is held for more than two years and as much as three years from the date of purchase. Property sellers could anticipate potential consumers diverting to extra reasonably priced property inside their borrowing means. For upgraders, their primary concern is the loan quantity they're able to safe for their subsequent property. Property tax is based on property ownership and due to this fact is levied no matter whether the property is owner-occupied, vacant or rented out. Nonetheless, to encourage home possession, we've got a lower tax rate for owner-occupied residential properties. We now have thus kept property tax as our wealth tax based on property ownership. Orchard / Holland (D09-10) Condo For Sale The Hillford, the primary retirement resort in Singapore delivered to you by World Class Land, a subsidiary of the Aspial jewellery group. Located in the status Bukit Timah area at Jalan Jurong Kechil Highway, the hillford development will home a complete six blocks of residences and one bl Trying first hand info for above upcoming Venture depart your contact CLICK HERE New Rental 2014 in February Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September New Apartment SKY GREEN 97% Sold – Final 4Penthouse In case you are focused on buying property in Singapore, previews are a great place to be at. Continuereading "Trilive at Kovan Preview Launch Quickly" Continuereading The Tembusu Kovan upcoming launch by Wing Tai" Ken Pua, seller of Forest Hills Apartment Twin Fountains EC As property tax is a wealth tax based on property possession, AV is reviewed and adjusted to replicate the change in market values of comparable property. The AV may be revised upwards, downwards or kept on the similar level depending available on the market values. There is no such thing as a double taxation here. Property tax is imposed primarily based on property ownership. It is different from earnings tax on the rental income, which is a tax on the revenue which an individual earns. SSD at sixteen per cent (increased than up to three per cent presently), if the property is sold within the first 12 months of buy, i.e. the property is held for 1 yr or much less from its buy date. The size and costs of some EC models have been scorching subjects of late, after the sale of vast, swanky ECs. Search SG Developersale.com