formulasearchengine - User contributions [en]

Jackson network

2014-02-19T16:22:28Z

128.101.152.60: supplying a conspicuously missing comma

== 000|Thousand|500|1000|1 Timberland Outlet ==

Absolute zero is often considered to be the coldest temperature feasible. But now researchers show they can achieve even lower temperature for a strange realm of "negative temperatures." Oddly, another way to take a look at these negative temperatures is to consider them hotter as compared to infinity, researchers added. This uncommon advance could lead to new motors that could technically be more when compared with 100 percent efficient, and shed light on mysteries such as dark energy, the mysterious substance that's apparently pulling our whole world apart. An object's temperature is a pace of how much its atoms proceed the colder an object is actually, the slower the atoms usually are. At the physically impossible to arrive at temperature of zero kelvin, and also minus 459.67 degrees Fahrenheit (minus 273.15 degrees Celsius), atoms would avoid moving. Positive temperatures make-up one part of the circle, although negative temperatures make up the far wall. When temperatures go often below zero or above infinity around the positive region of this scale, they end up in negative area. [ What's That? Your Essential Physics Questions Answered ] With positive temperatures, atoms more often than not occupy low energy declares than high energy says, a pattern known as Boltzmann distribution inside physics. When an object will be heated, its atoms can accomplish higher energy levels. At absolute zero, atoms would not occupy just about any energy states. At an infinite temperature, atoms would occupy many energy states. Negative temperatures then are the opposite of positive temperatures atoms more likely occupy substantial energy states than low energy states. "The inverted Boltzmann submission is the hallmark of damaging absolute temperature, and this is what we should have achieved," said researcher Ulrich Schneider, a physicist at the University or college of Munich in Germany. "Yet the gas is not colder in comparison with zero kelvin, but hotter. It can be even hotter than in any positive temperature a temperature scale simply does not end at infinity, but advances to [http://www.bridgeaustralia.org/webalizer/images/congress.asp?t=39-Timberland-Outlet Timberland Outlet] negative values instead." As one might be expecting, objects with negative conditions behave in very unusual ways. For instance, energy normally flows from objects which has a higher positive temperature to ones with a lower beneficial temperature that is, hotter objects heat up cooler objects, and also colder objects cool down hotter ones, until they accomplish a common temperature. However, power will always flow from items with negative temperature so that you can ones with positive temperatures. In this sense, objects having negative temperatures are always milder than ones with optimistic temperatures. Another odd reaction to negative temperatures has to do with entropy, this is a measure of how disorderly something is. When objects by using positive temperature release energy, they increase the entropy of things all-around them, making them behave additional chaotically. However, when objects using negative temperatures release vitality, they can actually absorb entropy. Damaging temperatures would be thought impossible, since there is typically no superior bound for how much electricity atoms can have, as far as theory presently suggests. (There is a limit about the speed they can travel as outlined by Einstein's theory of relativity, nothing can accelerate to speeds quicker than light.) Wacky physics play with it To generate negative temperatures, research workers created a system where atoms unfavorable reactions a limit to how much energy they can possess. They initial cooled about 100,500 atoms to a positive temperature of a few nanokelvin, or billionth of a kelvin. They refrigerated the atoms within a vacuum slot provided, which isolated them from any environmental influence that could likely heat the atoms up by mistake. The researchers also used a website of laser beams and permanent magnetic fields to very exactly control how these atoms well-socialized, helping to push them in a new temperature realm. [ Twisted Physics: 7 Mind Blowing Conclusions ] "The temperatures we reached are negative nanokelvin," Schneider told LiveScience. Temperature depends on how much atoms shift how much kinetic energy they have. The web of laser beams created a correctly ordered array of millions of shiny spots of light, and in the following "optical lattice," atoms could still go, but their kinetic energy was restricted. Temperature also depends on just how much potential energy atoms have, and just how much energy lies in the particular interactions between the atoms. The researchers limited how much potential energy a atoms had with a system of magnetic fields, and they could also very finely control the particular interactions between atoms, making them both attractive or repulsive. Climate is linked with pressure [http://www.drlindycrocker.com/AWStats/header.asp?n=135-New-Balance-High-Roller-574-Australia New Balance High Roller 574 Australia] the steamy something is, the more this expands outward, and the frigid something is, the more the idea contracts inward. To make sure this gas had a negative heat, the researchers had to give it an unfavorable pressure as well, tinkering with the interactions between atoms until many people attracted each other more than they will repelled each other. "We have created the first adverse absolute temperature state regarding moving particles," stated researcher Simon Braun at the University connected with Munich in Germany. New types of engines Negative temperatures can be used to create heat applications engines that convert warm energy to mechanical do the job, such as combustion engines that are more than 100 percent efficient, a little something seemingly impossible. Such search engines would essentially not only take in energy from hotter substances, but also colder ones. As a result, the work the engine completed could be larger than the energy stripped away from the hotter substance by yourself. Negative temperatures might also assistance shed light on one of the greatest mysteries within science. Scientists had predicted the gravitational pull of topic [http://www.bridgeaustralia.org/webalizer/images/congress.asp?t=112-Timberland-Stockists-Perth Timberland Stockists Perth] to slow down the universe's expansion following the Big Bang, eventually providing it to a dead stop or maybe reversing it for a "Big Emergency." However, the universe's extension is apparently speeding up, accelerated advancement that cosmologists [http://www.indopacificmarine.com.au/mail/copyright.asp?page=133-Air-Max-97 Air Max 97] suggest may be due to be able to dark energy, an confirmed unknown substance that could make-up more than 70 percent of the cosmos. Within much the same way, the negative demand of the cold gas the researchers created should make it collapse. Nevertheless, its negative temperature will keep it from doing so. As such, negative temperatures might have fascinating parallels with dark strength that may help scientists understand this enigma. Damaging temperatures could also shed light on unusual states of matter, building systems that normally might not be stable without them. "A better idea of temperature could lead to new things we have not even thought of yet,In Schneider said. "When you study the principles very thoroughly, you never know where by it may end."<ul>

<li>[http://enseignement-lsf.com/spip.php?article64#forum24618623 http://enseignement-lsf.com/spip.php?article64#forum24618623]</li>

<li>[http://lab.hijike.com/forum.php?mod=viewthread&tid=3211648&extra= http://lab.hijike.com/forum.php?mod=viewthread&tid=3211648&extra=]</li>

<li>[http://swclan.host22.com/index.php?site=clanwars_details&cwID=30 http://swclan.host22.com/index.php?site=clanwars_details&cwID=30]</li>

<li>[http://222.243.160.155/forum.php?mod=viewthread&tid=13050654 http://222.243.160.155/forum.php?mod=viewthread&tid=13050654]</li>

</ul>

Kronecker limit formula

2014-01-28T20:09:21Z

128.101.152.60:

In [[mathematics]], in the theory of [[integrable systems]], a '''Lax pair''' is a pair of time-dependent matrices or [[operator (mathematics)|operator]]s that satisfy a corresponding [[differential equation]], called the ''Lax equation''. Lax pairs were introduced by [[Peter Lax]] to discuss [[soliton]]s in [[continuous media]]. The [[inverse scattering transform]] makes use of the Lax equations to solve such systems.

==Definition==
A Lax pair is a pair of matrices or operators <math>L(t), P(t)</math> dependent on time and acting on a fixed [[Hilbert space]], and satisfying '''Lax's equation''':

:<math>\frac{dL}{dt}=[P,L]</math>

where <math>[P,L]=PL-LP</math> is the [[commutator]].
Often, as in the example below, <math>P</math> depends on <math>L</math> in a prescribed way, so this is a nonlinear equation for <math>L</math> as a function of <math>t</math>.

==Isospectral property==
It can then be shown that the [[eigenvalue]]s and more generally the [[Operator spectrum|spectrum]] of ''L'' are independent of ''t''. The matrices/operators ''L'' are said to be ''[[isospectral]]'' as <math>t</math> varies.

The core observation is that the matrices <math>L(t)</math> are all similar by virtue of

:<math>L(t)=U(t,s) L(s) U(t,s)^{-1}</math>

where <math>U(t,s)</math> is the solution of the [[Cauchy problem]]

:<math> \frac{d}{dt} U(t,s) = P(t) U(t,s), \qquad U(s,s) = I,</math>

where ''I'' denotes the identity matrix. Note that if ''L(t)'' is [[self-adjoint]] and ''P(t)'' is [[skew-adjoint]], then ''U(t,s)'' will be [[unitary operator|unitary]].

In other words, to solve the eigenvalue problem ''Lψ = λψ'' at time ''t'', it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:
:<math>\lambda(t)=\lambda(0)</math> (no change in spectrum)
:<math>\frac{\partial \psi}{\partial t}=P \psi.</math>

=== Link with the inverse scattering method ===
The above property is the basis for the inverse scattering method. In this method, ''L'' and ''P'' act on a [[functional space]] (thus ''ψ = ψ(t,x)''), and depend on an unknown function ''u(t,x)'' which is to be determined. It is generally assumed that ''u(0,x)'' is known, and that ''P'' does not depend on ''u'' in the scattering region where <math>\Vert x \Vert\to \infty</math>.
The method then takes the following form:
# Compute the spectrum of <math>L(0)</math>, giving <math>\lambda</math> and <math>\psi(0,x)</math>,
# In the scattering region where <math>P</math> is known, propagate <math>\psi</math> in time by using <math>\frac{\partial \psi}{\partial t}(t,x)=P \psi(t,x)</math> with initial condition <math>\psi(0,x)</math>,
# Knowing <math>\psi</math> in the scattering region, compute <math>L(t)</math> and/or <math>u(t,x)</math>.

==Example==
The [[Korteweg–de Vries equation]] is
:<math>u_t=6uu_x-u_{xxx}.\,</math>
It can be reformulated as the Lax equation
:<math>L_t=[P,L]\,</math>
with
:<math>L=-\partial_{x}^2+u\,</math> (a [[Sturm–Liouville operator]])
:<math>P= -4\partial_{x}^3+3(u\partial_{x}+\partial_{x} u)\,</math>
where all derivatives act on all objects to the right. This accounts for the infinite number of first integrals of the KdV equation.

==Equations with a Lax pair==
Further examples of systems of equations that can be formulated as a Lax pair include:

* [[Benjamin–Ono equation]]
* One dimensional cubic [[non-linear Schrödinger equation]]
* [[Davey-Stewartson system]]
* [[Kadomtsev–Petviashvili equation]]
* [[Korteweg–de Vries equation]]
* [[KdV hierarchy]]
* [[Modified Korteweg-de Vries equation]]
* [[Sine-Gordon equation]]
* [[Toda lattice]]

==References==
* {{citation|first=P.|last= Lax|title=Integrals of nonlinear equations of evolution and solitary waves|journal=Comm. Pure Applied Math.|volume=21|year=1968|pages= 467–490|doi=10.1002/cpa.3160210503|issue=5 }} [http://archive.org/details/integralsofnonli00laxp archive]
* P. Lax and R.S. Phillips, ''Scattering Theory for Automorphic Functions'', (1976) Princeton University Press.

[[Category:Differential equations]]
[[Category:Automorphic forms]]
[[Category:Spectral theory]]
[[Category:Exactly solvable models]]

Incomplete gamma function

2014-01-28T20:07:45Z

128.101.152.60: fixing some incorrect capitals and some punctuation errors

[[File:Carbon lattice diamond.png|thumb|The [[Diamond cubic|diamond crystal structure]] belongs to the face-centered [[Cubic crystal system|cubic lattice]], with a repeated 2-atom pattern.]]
In [[crystallography]], the terms '''crystal system''', '''crystal family''', and '''lattice system''' each refer to one of several classes of [[space group]]s, [[Bravais lattice|lattice]]s, [[point group]]s, or [[crystal]]s. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the [[trigonal crystal system]] is often confused with the [[rhombohedral lattice system]], and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".

[[Space group]]s and crystals are divided into 7 crystal systems according to their [[point group]]s, and into 7 lattice systems according to their [[Bravais lattice]]s. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems.
The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.

==Overview==
[[Image:Hanksite.JPG|thumb|Hexagonal [[hanksite]] crystal, with three-fold c-axis symmetry]]
A '''lattice system''' is a class of lattices with the same point group. In three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.

A '''crystal system''' is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system,
and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.

A ''' crystal family''' also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.

In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.

The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:
{|class="wikitable" cellpadding=0 style="margin: 1em auto; text-align: center;"
|-
![[Crystal family]]
!Crystal system
!Required symmetries of point group
![[Crystallographic point group|Point groups]]
![[Space group]]s
![[Bravais lattice]]s
![[Lattice system]]
|-
|colspan=2|[[Triclinic]]
|None
|2
|2
|1
|[[Triclinic]]
|-
|colspan=2|[[Monoclinic]]
|1 twofold [[Rotational symmetry|axis of rotation]] or 1 [[Reflection symmetry|mirror plane]]
|3
|13
|2
|[[Monoclinic]]
|-
|colspan=2|[[Orthorhombic]]
| 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.
|3
|59
|4
|[[Orthorhombic]]
|-
|colspan=2|[[Tetragonal]]
| 1 fourfold axis of rotation
|7
|68
|2
|[[Tetragonal]]
|-
|rowspan=3|[[Hexagonal crystal family|Hexagonal]]
|rowspan=2|[[Trigonal]]
|rowspan=2|1 threefold axis of rotation
|rowspan=2|5
|7
|1
|[[Rhombohedral lattice system|Rhombohedral]]
|-
|18
|rowspan=2|1
|rowspan=2|[[Hexagonal lattice system|Hexagonal]]
|-
|[[hexagonal crystal system|Hexagonal]]
|1 sixfold axis of rotation
|7
|27
|-
|colspan=2|[[cubic crystal system|Cubic]]
|4 threefold axes of rotation
|5
|36
|3
|[[cubic crystal system|Cubic]]
|- bgcolor=#e0e0e0
|'''Total:''' 6
|7
|
|32
|230
|14
|7
|}

==Crystal classes==
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 [[crystallographic point group]]s) as shown in the following table:

{| class=wikitable
|-
! crystal family
! crystal system
! [[point group]] / crystal class
! [[Schönflies notation|Schönflies]]
! [[Hermann–Mauguin notation|Hermann-Mauguin]]
! [[Orbifold notation|Orbifold]]
! [[Coxeter notation|Coxeter]]
! Point symmetry
! [[Symmetry number|Order]]
! [[Group_theory#Abstract_groups|Abstract group]]
|-
| rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
| triclinic-pedial
| C1
| 1
| 11
| [ ]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 1
| trivial <math>\mathbb{Z}_1</math>
|-
| triclinic-pinacoidal
| Ci
| {{overline|1}}
| 1x
| [2,1+]
| [[centrosymmetric]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
| monoclinic-sphenoidal
| C2
| 2
| 22
| [2,2]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| monoclinic-domatic
| Cs
| m
| *11
| [ ]
| [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| monoclinic-[[prism (geometry)|prismatic]]
| C2h
| 2/m
| 2*
| [2,2+]
| [[centrosymmetric]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
| orthorhombic-sphenoidal
| D2
| 222
| 222
| [2,2]+
| [[Chirality (chemistry)|enantiomorphic]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| orthorhombic-[[Pyramid (geometry)|pyramidal]]
| C2v
| mm2
| *22
| [2]
| [[Polar point group|polar]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| orthorhombic-[[bipyramid]]al
| D2h
| mmm
| *222
| [2,2]
| [[centrosymmetric]]
| 8
| <math>\mathbb{V}\times\mathbb{Z}_2</math>
|-
| rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
| tetragonal-pyramidal
| C4
| 4
| 44
| [4]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-disphenoidal
| S4
| {{overline|4}}
| 2x
| [2+,2]
| [[non-centrosymmetric]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-dipyramidal
| C4h
| 4/m
| 4*
| [2,4+]
| [[centrosymmetric]]
| 8
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
|-
| tetragonal-trapezoidal
| D4
| 422
| 422
| [2,4]+
| [[Chirality (chemistry)|enantiomorphic]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-pyramidal
| C4v
| 4mm
| *44
| [4]
| [[Polar point group|polar]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| tetragonal-scalenoidal
| D2d
| {{overline|4}}2m or {{overline|4}}m2
| 2*2
| [2+,4]
| [[non-centrosymmetric]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-dipyramidal
| D4h
| 4/mmm
| *422
| [2,4]
| [[centrosymmetric]]
| 16
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
|-
| rowspan=12|[[hexagonal crystal family|hexagonal]] || rowspan=5 | [[trigonal crystal system|trigonal]]
| trigonal-pyramidal
| C3
| 3
| 33
| [3]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 3
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
|-
| rhombohedral
| S6 (C3i)
| {{overline|3}}
| 3x
| [2+,3+]
| [[centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-trapezoidal
| D3
| 32 or 321 or 312
| 322
| [3,2]+
| [[Chirality (chemistry)|enantiomorphic]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-pyramidal
| C3v
| 3m or 3m1 or 31m
| *33
| [3]
| [[Polar point group|polar]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-scalahedral
| D3d
| {{overline|3}}m or {{overline|3}}m1 or {{overline|3}}1m
| 2*3
| [2+,6]
| [[centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| rowspan=7 | [[Hexagonal crystal system|hexagonal]]
| hexagonal-pyramidal
| C6
| 6
| 66
| [6]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-dipyramidal
| C3h
| {{overline|6}}
| 3*
| [2,3+]
| [[non-centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| hexagonal-dipyramidal
| C6h
| 6/m
| 6*
| [2,6+]
| [[centrosymmetric]]
| 12
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
|-
| hexagonal-trapezoidal
| D6
| 622
| 622
| [2,6]+
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-pyramidal
| C6v
| 6mm
| *66
| [6]
| [[Polar point group|polar]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-dipyramidal
| D3h
| {{overline|6}}m2 or {{overline|6}}2m
| *322
| [2,3]
| [[non-centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-dipyramidal
| D6h
| 6/mmm
| *622
| [2,6]
| [[centrosymmetric]]
| 24
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math>
|-
| rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
| tetrahedral
| T || 23
| 332
| [3,3]+
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[alternating group|alternating]] <math>\mathbb{A}_4</math>
|-
| hextetrahedral
| Td
| {{overline|4}}3m
| *332
| [3,3]
| [[non-centrosymmetric]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| diploidal
| Th
| m{{overline|3}}
| 3*2
| [3+,4]
| [[centrosymmetric]]
| 24
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math>
|-
| gyroidal
| O
| 432
| 432
| [4,3]+
| [[Chirality (chemistry)|enantiomorphic]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hexoctahedral
| Oh
| m{{overline|3}}m
| *432
| [4,3]
| [[centrosymmetric]]
| 48
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math>
|}

Point symmetry can be thought of in the following fashion: consider the coordinates which make up the structure, and project them all through a single point, so that (x,y,z) becomes (-x,-y,-z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is '''''centrosymmetric'''''. Otherwise it is '''''non-centrosymmetric'''''. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure. This is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral (enantiomorphic) and its symmetry group is '''''enantiomorphic'''''.<ref>{{cite journal|author=Howard D. Flack|year=2003|title=Chiral and Achiral Crystal Structures|journal=Helvetica Chimica Acta |volume=86|pages= 905–921|doi=10.1002/hlca.200390109}}</ref>

A direction is called polar if its two directional senses are geometrically or physically different. A polar symmetry direction of a crystal is called a polar axis.<ref>E. Koch , W. Fischer , U. Müller , in ‘International Tables for Crystallography, Vol. A, Space-Group Symmetry’, 5th edn., Ed. T. Hahn, Kluwer Academic Publishers, Dordrecht, 2002, Chapt. 10, p. 804.</ref> Groups containing a polar axis are called '''''[[polar point group|polar]]'''''. A polar crystal possess a "unique" axis (found in no other directions) such that some geometrical or physical property is different at the two ends of this axis. It may develop a [[Polarization density|dielectric polarization]], e.g. in [[Pyroelectricity|pyroelectric crystals]]. A polar axis can occur only in non-centrosymmetric structures. There should also not be a mirror plane or 2-fold axis perpendicular to the polar axis, because they will make both directions of the axis equivalent.

The [[crystal structure]]s of chiral biological molecules (such as [[protein]] structures) can only occur in the 11 [[Chirality (chemistry)|enantiomorphic]] point groups (biological molecules are usually [[Chirality (chemistry)|chiral]]).


==Lattice systems ==
The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.

{| class=wikitable
|align=center|'''The 7 lattice systems'''
|colspan=4 align=center| '''The 14 Bravais Lattices'''
|-
|colspan=1 align=center| [[triclinic crystal system|triclinic]] ([[parallelepiped]])
|| [[Image:Triclinic.svg|80px|Triclinic]]
|-
|rowspan=2 align=center| [[monoclinic crystal system|monoclinic]] (right [[prism (geometry)|prism]] with [[parallelogram]] base; here seen from above)
|align=center| simple
|align=center| base-centered
|-
|| [[Image:Monoclinic.svg|80px|Monoclinic, simple]]
|| [[Image:Monoclinic-base-centered.svg|80px|Monoclinic, centered]]
|-
|rowspan=2 align=center| [[orthorhombic crystal system|orthorhombic]] ([[cuboid]])
|align=center| simple
|align=center| base-centered
|align=center| body-centered
|align=center| face-centered
|-
|| [[Image:Orthorhombic.svg|80px|Orthohombic, simple]]
|| [[Image:Orthorhombic-base-centered.svg|80px|Orthohombic, base-centered]]
|| [[Image:Orthorhombic-body-centered.svg|80px|Orthohombic, body-centered]]
|| [[Image:Orthorhombic-face-centered.svg|80px|Orthohombic, face-centered]]
|-
|rowspan=2 align=center| [[tetragonal crystal system|tetragonal]] (square [[cuboid]])
|align=center|simple
|align=center| body-centered
|-
|| [[Image:Tetragonal.svg|80px|Tetragonal, simple]]
|| [[Image:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
|-
|align=center| [[rhombohedral lattice system|rhombohedral]] ([[trigonal trapezohedron]])
| [[Image:Rhombohedral.svg|80px|Rhombohedral]]
|-
|align=center| [[Hexagonal crystal system|hexagonal]] (centered regular [[hexagon]])
| [[Image:Hexagonal lattice.svg|80px|Hexagonal]]
|-
|rowspan=2 align=center| [[Cubic crystal system|cubic]] (isometric; [[cube]])
|align=center| simple
|align=center| body-centered
|align=center| face-centered
|-
|| [[Image:Cubic.svg|80px|Cubic, simple]]
| [[Image:Cubic-body-centered.svg|80px|Cubic, body-centered]]
| [[Image:Cubic-face-centered.svg|80px|Cubic, face-centered]]
|}
{{-}}
In [[geometry]] and [[crystallography]], a '''Bravais lattice''' is a category of [[symmetry group]]s for [[translational symmetry]] in three directions, or correspondingly, a category of translation [[Lattice (group)|lattice]]s.

Such symmetry groups consist of translations by vectors of the form

:<math>\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math>

where ''n''1, ''n''2, and ''n''3 are [[integer]]s and ''a''1, ''a''2, and ''a''3 are three non-coplanar vectors, called ''primitive vectors''.

These lattices are classified by [[space group]] of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.

All crystalline materials must, by definition fit in one of these arrangements (not including [[quasicrystal]]s).

For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the [[primitive cell]]. Depending on the symmetry of a crystal or other pattern, the [[fundamental domain]] is again smaller, up to a factor 48.

The Bravais lattices were studied by [[Moritz Ludwig Frankenheim]] (1801–1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by [[Auguste Bravais|A. Bravais]] in 1848.

==Crystal systems in four-dimensional space==

The four-dimensional unit cell is defined by four edge lengths (<math>a, b, c, d</math>) and six interaxial angles (<math>\alpha, \beta, \gamma, \delta, \epsilon, \zeta</math>). The following conditions for the lattice parameters define 23 crystal families:

1 Hexaclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne \delta \ne \epsilon \ne \zeta \ne 90 ^\circ</math>

2 Triclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne 90 ^\circ, \delta = \epsilon = \zeta = 90 ^\circ</math>

3 Diclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta \ne 90 ^\circ</math>

4 Monoclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

5 Orthogonal: <math>a\ne b \ne c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

6 Tetragonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

7 Hexagonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>

8 Ditetragonal Diclinic: <math>a = d \ne b = c, \alpha = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne 90 ^\circ, \delta = 180 ^\circ - \gamma </math>

9 Ditrigonal (Dihexagonal) Diclinic: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne \delta \ne 90 ^\circ, cos \delta = cos \beta - cos \gamma</math>

10 Tetragonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

11 Hexagonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>

12 Ditetragonal Monoclinic: <math>a = d \ne b = c, \alpha = \gamma = \delta = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ</math>

13 Ditrigonal (Dihexagonal) Monoclinic: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma = \delta \ne 90 ^\circ, cos \gamma = -\color{Black}\tfrac{1}{2} cos \beta</math>

14 Ditetragonal Orthogonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

15 Hexagonal Tetragonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>

16 Dihexagonal Orthogonal: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, </math>

17 Cubic Orthogonal: <math>a = b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

18 Octagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne 90 ^\circ, \beta = \epsilon = 90 ^\circ, \delta = 180 ^\circ - \alpha</math>

19 Decagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne \beta = \delta = \epsilon, cos \beta = -0.5 - cos \alpha</math>

20 Dodecagonal: <math>a = b = c = d, \alpha = \zeta = 90 ^\circ, \beta = \epsilon = 120 ^\circ, \gamma = \delta \ne 90 ^\circ</math>

21 Di-isohexagonal Orthogonal: <math>a = b = c = d, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ</math>

22 Icosagonal (Icosahedral): <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta, cos \alpha = -\color{Black}\tfrac{1}{4}</math>

23 Hypercubic: <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

The names here are given according to Whittaker.<ref name="Whittaker">E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.</ref> They are almost the same as in Brown ''et al'',<ref name="Brown">H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.</ref> with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown ''et al'' are given in parenthesis.

The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.<ref name="Whittaker"/><ref name="Brown"/> Enantiomorphic systems are marked with asterisk. The number of enantiomorphic pairs are given in parentheses. Here the term "enantiomorphic" has different meaning than in table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means, that group itself (considered as geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P31 and P32, P4122 and P4322. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
{|class="wikitable" cellpadding=4 cellspacing=0
|-align=center
!No. of Crystal family
!Crystal family
!Crystal system
!No. of Crystal system
!Point groups
!width=120|Space groups
!Bravais lattices
!Lattice system
|-
| I ||colspan=2| Hexaclinic|| 1
|2
|2
|1
|Hexaclinic P
|-
| II || colspan=2| Triclinic|| 2
|3
|13
|2
|Triclinic P, S
|-
| III ||colspan=2| Diclinic|| 3
|2
|12
|3
|Diclinic P, S, D
|-
|IV || colspan=2| Monoclinic|| 4
|4
|207
|6
|Monoclinic P, S, S, I, D, F
|-
|rowspan=3| V ||rowspan=3| Orthogonal
|rowspan=2|Non-axial Orthogonal|| rowspan=2| 5
|rowspan=2|2
|2
|1
|Orthogonal KU
|-
|112
|rowspan=2|8
|rowspan=2|Orthogonal P, S, I, Z, D, F, G, U
|-
|Axial Orthogonal|| 6
|3
|887
|-
| VI || colspan=2| Tetragonal Monoclinic || 7
|7
|88
|2
|Tetragonal Monoclinic P, I
|-
|rowspan=3| VII ||rowspan=3| Hexagonal Monoclinic
|rowspan=2|Trigonal Monoclinic ||rowspan=2| 8
|rowspan=2|5
|9
|1
|Hexagonal Monoclinic R
|-
|15
|rowspan=2|1
|rowspan=2|Hexagonal Monoclinic P
|-
|Hexagonal Monoclinic || 9
|7
|25
|-
| VIII || colspan=2| Ditetragonal Diclinic* ||10
|1 (+1)
|1 (+1)
|1 (+1)
|Ditetragonal Diclinic P*
|-
|IX || colspan=2| Ditrigonal Diclinic* ||11
|2 (+2)
|2 (+2)
|1 (+1)
|Ditrigonal Diclinic P*
|-
|rowspan=3| X ||rowspan=3| Tetragonal Orthogonal
|rowspan=2|Inverse Tetragonal Orthogonal ||rowspan=2| 12
|rowspan=2|5
|7
|1
|Tetragonal Orthogonal KG
|-
|351
|rowspan=2|5
|rowspan=2|Tetragonal Orthogonal P, S, I, Z, G
|-
|Proper Tetragonal Orthogonal || 13
|10
|1312
|-
|rowspan=3|XI ||rowspan=3| Hexagonal Orthogonal
|rowspan=2|Trigonal Orthogonal ||rowspan=2| 14
|rowspan=2|10
|81
|2
|Hexagonal Orthogonal R, RS
|-
|150
|rowspan=2|2
|rowspan=2|Hexagonal Orthogonal P, S
|-
|Hexagonal Orthogonal || 15
|12
|240
|-
| XII || colspan=2| Ditetragonal Monoclinic* || 16
|1 (+1)
|6 (+6)
|3 (+3)
|Ditetragonal Monoclinic P*, S*, D*
|-
| XIII || colspan=2| Ditrigonal Monoclinic* || 17
|2 (+2)
|5 (+5)
|2 (+2)
|Ditrigonal Monoclinic P*, RR*
|-
|rowspan=3| XIV ||rowspan=3| Ditetragonal Orthogonal
|rowspan=2|Crypto-Ditetragonal Orthogonal ||rowspan=2| 18
|rowspan=2|5
|10
|1
|Ditetragonal Orthogonal D
|-
|165 (+2)
|rowspan=2|2
|rowspan=2|Ditetragonal Orthogonal P, Z
|-
|Ditetragonal Orthogonal ||19
|6
|127
|-
|XV ||colspan=2| Hexagonal Tetragonal || 20
|22
|108
|1
|Hexagonal Tetragonal P
|-
|rowspan=5| XVI || rowspan=5| Dihexagonal Orthogonal
|rowspan=2| Crypto-Ditrigonal Orthogonal* || rowspan=2|21
|rowspan=2|4 (+4)
|5 (+5)
|1 (+1)
|Dihexagonal Orthogonal G*
|-
|5 (+5)
|rowspan=3|1
|rowspan=3|Dihexagonal Orthogonal P
|-
|Dihexagonal Orthogonal || 23
|11
|20
|-
|rowspan=2| Ditrigonal Orthogonal || rowspan=2| 22
|rowspan=2| 11
|41
|-
|16
|1
|Dihexagonal Orthogonal RR
|-
|rowspan=3| XVII ||rowspan=3| Cubic Orthogonal
|rowspan=2|Simple Cubic Orthogonal ||rowspan=2| 24
|rowspan=2|5
|9
|1
|Cubic Orthogonal KU
|-
|96
|rowspan=2|5
|rowspan=2|Cubic Orthogonal P, I, Z, F, U
|-
|Complex Cubic Orthogonal || 25
|11
|366
|-
| XVIII ||colspan=2| Octagonal* || 26
|2 (+2)
|3 (+3)
|1 (+1)
| Octagonal P*
|-
| XIX ||colspan=2| Decagonal || 27
|4
|5
|1
| Decagonal P
|-
| XX ||colspan=2| Dodecagonal* ||28
|2 (+2)
|2 (+2)
|1 (+1)
| Dodecagonal P*
|-
|rowspan=3| XXI ||rowspan=3| Di-isohexagonal Orthogonal
|rowspan=2| Simple Di-isohexagonal Orthogonal || rowspan=2| 29
|rowspan=2|9 (+2)
|19 (+5)
|1
|Di-isohexagonal Orthogonal RR
|-
|19 (+3)
|rowspan=2|1
|rowspan=2|Di-isohexagonal Orthogonal P
|-
|Complex Di-isohexagonal Orthogonal ||30
|13 (+8)
|15 (+9)
|-
|XXII ||colspan=2| Icosagonal|| 31
|7
|20
|2
| Icosagonal P, SN
|-
|rowspan=3| XXIII ||rowspan=3| Hypercubic
|rowspan=2| Octagonal Hypercubic||rowspan=2|32
|rowspan=2|21 (+8)
|73 (+15)
|1
|Hypercubic P
|-
|107 (+28)
|rowspan=2|1
|rowspan=2|Hypercubic Z
|-
|Dodecagonal Hypercubic|| 33
|16 (+12)
|25 (+20)
|- bgcolor=#e0e0e0
|'''Total:'''
|23 (+6)
|33 (+7)
|
|227 (+44)
|4783 (+111)
|64 (+10)
|33 (+7)
|}

==See also==
*[[Crystal cluster]]
*[[Crystal structure]]
*[[Space group#Classification systems for space groups|List of the 230 crystallographic 3D space groups]]
*[[Polar point group]]

==Notes==

<div class="references-small" >
<references/>
</div>

==References==
*{{Cite book | editor1-last=Hahn | editor1-first=Theo | title=International Tables for Crystallography, Volume A: Space Group Symmetry | url=http://it.iucr.org/A/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=5th | isbn=978-0-7923-6590-7 | doi=10.1107/97809553602060000100 | year=2002 | volume=A}}

==External links==
*[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Overview of the 32 groups]
*[http://mineral.galleries.com/minerals/symmetry/symmetry.htm Mineral galleries – Symmetry]
*[http://www.ifg.uni-kiel.de/kubische_Formen all cubic crystal classes, forms and stereographic projections (interactive java applet)]
*[http://reference.iucr.org/dictionary/Crystal_system Crystal system] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
*[http://reference.iucr.org/dictionary/Crystal_family Crystal family] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
*[http://reference.iucr.org/dictionary/Lattice_system Lattice system] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
*[http://materials.duke.edu/awrapper.html Conversion Primitive to Standard Conventional for VASP input files]
*[http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Learning Crystallography]

{{Crystal systems}}
{{Mineral identification}}

[[Category:Symmetry]]
[[Category:Euclidean geometry]]
[[Category:Crystallography]]
[[Category:Morphology]]
[[Category:Mineralogy]]

[[ru:Сингония]]

Complex plane

2014-01-28T19:51:33Z

128.101.152.60: corrections per WP:MOS and WP:MOSMATH

{{Redirect|Carbon burning|combustion of carbon containing compounds|combustion}}

The '''carbon-burning process''' or '''carbon fusion''' is a set of [[nuclear fusion]] reactions that take place in massive [[star]]s (at least 8 [[Solar mass|<math>\begin{smallmatrix}M_\odot\end{smallmatrix}</math>]] at birth) that have used up the lighter elements in their cores. It requires high temperatures (> 5×108 [[kelvin|K]] or 50 [[keV]]) and [[density|densities]] (> 3×109 kg/m3).<ref name=Ryan>{{cite book| author=Ryan, Sean G.; Norton, Andrew J. | title=Stellar Evolution and Nucleosynthesis | year=2010 | page=135| isbn=978-0-521-13320-3|publisher=Cambridge University Press|url=http://books.google.com/?id=PE4yGiU-JyEC&q=carbon+burning#v=onepage&q=carbong%20burning&f=false}}</ref>

These figures for temperature and density are only a guide. More massive stars burn their nuclear fuel more quickly, since they have to offset greater gravitational forces to stay in (approximate) [[hydrostatic equilibrium]]. That generally means higher temperatures, although lower densities, than for less massive stars.<ref name="Clayton">{{cite book|last=Clayton|first=Donald| url=http://books.google.com/books?id=8HSGFThnbvkC|title=Principles of Stellar Evolution and Nucleosynthesis|year=1983|publisher=University of Chicago Press|isbn=978-0-226-10953-4}}</ref> To get the right figures for a particular mass, and a particular stage of evolution, it is necessary to use a numerical [[stellar structure|stellar model]] computed with computer algorithms.<ref name="Siess">{{cite journal
| author=Siess L.
| title = Evolution of massive AGB stars. I. Carbon burning phase
| journal = Astronomy and Astrophysics
| year= 2007
| volume = 476
| issue=2
| pages = 893–909
| doi = 10.1051/0004-6361:20053043
| bibcode = 2006A&A...448..717S}}</ref> Such models are continually being refined based on [[particle physics experiments]] (which measure nuclear reaction rates) and astronomical observations (which include direct observation of mass loss, detection of nuclear products from spectrum observations after convection zones develop from the surface to fusion-burning regions – known as 'dredge-up' events – and so bring nuclear products to the surface, and many other observations relevant to models).<ref>{{cite journal
| title = Rubidium-Rich Asymptotic Giant Branch Stars
| journal = Science
| eprint = arXiv:astro-ph/0611319
| year = 2006
| month = dec,
| volume = 314
| issue = 5806
| pages = 1751–1754
| doi = 10.1126/science.1133706
| bibcode = 2006Sci...314.1751G
| author = Hernandez, G. et al
| pmid = 17095658
|arxiv = astro-ph/0611319 }}</ref>

==Fusion reactions==
The principal reactions are:<ref name="Camiel1992">{{cite book|last=Camiel|first=W. H.|coauthors=de Loore, C. Doom|chapter=Structure and evolution of single and binary stars|editor=Camiel W. H. de Loore|title=Volume 179 of Astrophysics and space science library|year=1992|publisher=Springer|isbn=978-0-7923-1768-5|url=http://books.google.com/?id=LJgNIi0vkeYC&dq=%22carbon+burning%22&q=carbon+burning#v=snippet&q=carbon%20burning&f=false|pages=95–97}}</ref>
:{| border="0"
|- style="height:3em;"
||[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Neon-20|{{nuclide|neon|20}}]] ||+ ||[[Helium|{{nuclide|helium|4}}]] ||+ ||4.617 [[electron volt|MeV]]
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Sodium-23|{{nuclide|sodium|23}}]] ||+ ||[[Hydrogen|{{nuclide|hydrogen|1}}]] ||+ ||2.241 [[electron volt|MeV]]
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Magnesium-23|{{nuclide|magnesium|23}}]] ||+ ||[[Neutron|n]] ||− ||2.599 [[electron volt|MeV]]
|- style="height:3em;"
|colspan=99|Alternatively:
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Magnesium-24|{{nuclide|magnesium|24}}]] ||+ ||{{SubatomicParticle|link=yes|Gamma}} ||+ ||13.933 [[electron volt|MeV]]
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Oxygen-16|{{nuclide|oxygen|16}}]] ||+ ||2 [[Helium|{{nuclide|helium|4}}]] ||colspan=2|−   0.113 [[electron volt|MeV]]
|}

==Reaction products==
This sequence of reactions can be understood by thinking of the two interacting carbon nuclei as coming together to form an [[excited state]] of the Mg-24 nucleus, which then decays in one of the five ways listed above.<ref name=Rose1998>{{cite book|url=http://books.google.com/books?id=yaX0etDmbXMC|last=Rose,|first=William K.|title=Advanced Stellar Astrophysics|publisher=Cambridge University Press|year=1998|isbn=978-0-521-58833-1|pages=227–229}}</ref> The first two reactions are strongly exothermic, as indicated by the large positive energies released, and are the most frequent results of the interaction. The third reaction is strongly endothermic, as indicated by the large negative energy indicating that energy is absorbed rather than emitted. This makes it much less likely, yet still possible in the high-energy environment of carbon burning.<ref name=Camiel1992/> But the production of a few neutrons by this reaction is important, since these neutrons can combine with heavy nuclei, present in tiny amounts in most stars, to form even heavier isotopes in the [[s-process]].<ref name=Rose1998-229>Rose (1998), pp. 229–234</ref>

The fourth reaction might be expected to be the most common from its large energy release, but in fact it is extremely improbable because it proceeds via the electromagnetic interaction,<ref name=Camiel1992/> as it produces a gamma ray photon, rather than utilising the strong force between nucleons as do the first two reactions. Nucleons look a lot bigger to each other than they do to photons of this energy. However, the Mg-24 produced in this reaction is the only magnesium left in the core when the carbon-burning process ends, as Mg-23 is radioactive.

The last reaction is also very unlikely since it involves three reaction products,<ref name=Camiel1992/> as well as being endothermic—think of the reaction proceeding in reverse, it would require the three products all to converge at the same time, which is less likely than two-body interactions.

The protons produced by the second reaction can take part in the [[proton-proton chain reaction]], or the [[CNO cycle]], but they can also be captured by Na-23 to form Ne-20 plus a He-4 nucleus.<ref name=Camiel1992/> In fact, a significant fraction of the Na-23 produced by the second reaction gets used up this way.<ref name=Rose1998/> The oxygen (O-16) already produced by [[helium fusion]] in the previous stage of stellar evolution manages to survive the carbon-burning process pretty well, despite some of it being used up by capturing He-4 nuclei, in stars between 9 and 11 [[solar mass]]es.<ref name=Ryan/><ref name=Camiel1992-97>Camiel (1992), pp.97–98</ref> So the end result of carbon burning is a mixture mainly of oxygen, neon, sodium and magnesium.<ref name="Siess"/><ref name=Camiel1992/>

The fact that the mass-energy sum of the two carbon nuclei is similar to that of an excited state of the magnesium nucleus is known as 'resonance'. Without this resonance, carbon burning would only occur at temperatures one hundred times higher.
The experimental and theoretical investigation of such resonances is still a subject of research.<ref>{{cite journal
| author = Strandberg, E. et al
| title = Mg24(α,γ)Si28 resonance parameters at low α-particle energies
| journal = Physical Review C
| date = May 2008
| volume = 77
| issue = 5
| pages = 055801-+
| doi = 10.1103/PhysRevC.77.055801
| bibcode = 2008PhRvC..77e5801S
}}</ref> A similar resonance increases the probability of the [[triple-alpha process]], which is responsible for the original production of carbon.

==Neutrino losses==
[[Neutrino]] losses start to become a major factor in the fusion processes in stars at the temperatures and densities of carbon burning. Though the main reactions don't involve neutrinos, the side reactions such as the [[proton-proton chain reaction]] do. But the main source of neutrinos at these high temperatures involves a process in quantum theory known as [[pair production]]. A high energy [[gamma ray]] which has a greater energy than the [[rest mass]] of two [[electron]]s ([[mass-energy equivalence]]) can interact with electromagnetic fields of the atomic nuclei in the star, and become a particle and [[anti-particle]] pair of an electron and positron.

Normally, the positron quickly annihilates with another electron, producing two photons, and this process can be safely ignored at lower temperatures. But around 1 in 1019 pair productions<ref name="Clayton"/> end with a weak interaction of the electron and positron, which replaces them with a [[neutrino]] and anti-neutrino pair. Since they move at virtually the speed of light and interact very weakly with matter, these neutrino particles usually escape the star without interacting, carrying away their mass-energy. This energy loss is comparable to the energy output from the carbon fusion.

Neutrino losses, by this and similar processes, play an increasingly important part in the evolution of the most massive stars. They force the star to burn its fuel at a higher temperature to offset them.<ref name="Clayton"/> Fusion processes are very sensitive to temperature so the star can produce more energy to retain [[hydrostatic equilibrium]], at the cost of burning through successive nuclear fuels ever more rapidly. Fusion produces less energy per unit mass as the fuel nuclei get heavier, and the core of the star contracts and heats up when switching from one fuel to the next, so both these processes also significantly reduce the lifetime of each successive fusion-burning fuel.

Up to helium burning, the neutrino losses are negligible, but from carbon burning the reduction in lifetime due to them roughly matches that due to fuel change and core contraction. In successive fuel changes in the most massive stars, the reduction in lifetime is dominated by the neutrino losses. For example, a star of 25 solar masses burns hydrogen in the core for 107 years, helium for 106 years and carbon for only 103 years.<ref name="WoosleyJanka">{{cite journal
| last=Woosley | first=S. | coauthors=Janka, H.-T.
| date=2006-01-12
| bibcode=2005NatPh...1..147W
| title=The Physics of Core-Collapse Supernovae
| journal=Nature Physics
| year=2005 | month=December
| volume=1 | issue=3 | pages=147–154
| doi=10.1038/nphys172|arxiv = astro-ph/0601261 }}</ref>

==Stellar evolution==
{{main|Stellar evolution}}
During [[helium fusion]], stars build up an inert core rich in carbon and oxygen. The inert core eventually reaches sufficient mass to collapse due to gravitation, whilst the helium burning moves gradually outward. This decrease in the inert core volume raises the temperature to the carbon ignition temperature. This will raise the temperature around the core and allow helium to burn in a shell around the core.<ref name="Ostlie">Ostlie, Dale A. and Carrol, Bradley W., [http://www.amazon.com/dp/0805304029 ''An introduction to Modern Stellar Astrophysics''], Addison-Wesley (2007)</ref> Outside this is another shell burning hydrogen. The resulting carbon burning provides energy from the core to restore the star's [[mechanical equilibrium]]. However, the balance is only short-lived; in a star of 25 solar masses, the process will use up most of the carbon in the core in only 600 years. The duration of this process varies significantly depending on the mass of the star.<ref name="Lecture19">Anderson, Scott R.,
[http://www.opencourse.info/astronomy/introduction/19.stars_death_high-mass/ ''Open Course: Astronomy: Lecture 19: Death of High-Mass Stars''], GEM (2001)</ref>

Stars with below 8–9 [[Solar mass]]es never reach high enough core temperature to burn carbon, instead ending their lives as carbon-oxygen [[white dwarf]]s after shell [[helium flash]]es gently expel the outer envelope in a [[planetary nebula]].<ref name="Siess"/><ref name=Ryan-147/>

In the late stages of carbon burning, stars with masses between 8 and 11 solar masses develop a massive stellar wind, which quickly ejects the outer envelope in a [[planetary nebula]] leaving behind an O-Ne-Na-Mg [[white dwarf]] core of about 1.1 solar masses.<ref name="Siess"/> The core never reaches high enough temperature for further fusion burning of heavier elements than carbon.<ref name=Ryan-147/>

Stars with more than 11 solar masses proceed with the [[neon-burning process]] after contraction of the inert (O, Ne, Na, Mg) core raises the temperature sufficiently.<ref name=Ryan-147>Ryan (2010), pp.147–148</ref>

==See also==
*[[Proton–proton chain reaction]]
*[[CNO process]]
*[[Triple alpha process]]
*[[Alpha process]]
*[[Carbon detonation]]
*[[Neon burning]]

==References==
{{reflist|2}}

{{Nuclear processes}}

[[Category:Nucleosynthesis]]