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2014-02-01T07:04:53Z

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{{Unreferenced|date=December 2009}}
The '''job-shop problem''' ('''JSP''') is a problem in [[discrete optimization|discrete]] or [[combinatorial optimization]], and is a generalization of the famous [[travelling salesman problem]]. It is a prominent illustration of a class of problems in [[computational complexity theory]] which are hard to solve.

==Description of the problem==
A number of jobs have to be done and every job consists of using a number of machines for a certain amount of time. The problem is to find the best planning to do all the jobs on all the different machines in the shortest period of time.

Although a job can have any number of operations, the most common formulation of the job shop problem specifies that each job has exactly 'n' operations, one on each machine.

The job shop process differs from flow shop process in that the flow of work is not unidirectional in job shop, hence it is one of the complex scheduling problems.

== References ==
1) principles of sequencing and scheduling by Kenneth R Baker ''et al.''

==Statement of the problem==
Let <math>M = \{ M_{1}, M_{2}, \dots, M_{m} \}</math> and <math>J = \{ J_{1}, J_{2}, \dots, J_{n} \}</math> be two [[Finite set|finite]] [[set (mathematics)|sets]]. On account of the industrial origins of the problem, the <math>\displaystyle M_{i}</math> are called '''machines''' and the <math>\displaystyle J_{j}</math> are called '''jobs'''.

Let <math>\displaystyle \ \mathcal{X}</math> denote the set of all sequential assignments of jobs to machines, such that every job is done by every machine exactly once; elements <math>x \in \mathcal{X}</math> may be written as <math>n \times m</math> matrices, in which column <math>\displaystyle i</math> lists the jobs that machine <math>\displaystyle M_{i}</math> will do, in order. For example, the matrix

: <math>x = \begin{pmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 1 \end{pmatrix}</math>

means that machine <math>\displaystyle M_{1}</math> will do the three jobs <math>\displaystyle J_{1}, J_{2}, J_{3}</math> in the order <math>\displaystyle J_{1}, J_{2}, J_{3}</math>, while machine <math>\displaystyle M_{2}</math> will do the jobs in the order <math>\displaystyle J_{2}, J_{3}, J_{1}</math>.

Suppose also that there is some '''cost function''' <math>C : \mathcal{X} \to [0, + \infty]</math>. The cost function may be interpreted as a "total processing time", and may have some expression in terms of times <math>C_{ij} : M \times J \to [0, + \infty]</math>, the cost/time for machine <math>\displaystyle M_{i}</math> to do job <math>\displaystyle J_{j}</math>.

The '''job-shop problem''' is to find an assignment of jobs <math>x \in \mathcal{X}</math> such that <math>\displaystyle C(x)</math> is a minimum, that is, there is no <math>y \in \mathcal{X}</math> such that <math>\displaystyle C(x) > C(y)</math>.

==The problem of infinite cost==
One of the first problems that must be dealt with in the JSP is that many proposed solutions have infinite cost: i.e., there exists <math>x_{\infty} \in \mathcal{X}</math> such that <math>C(x_{\infty}) = + \infty</math>. In fact, it is quite simple to concoct examples of such <math>x_{\infty}</math> by ensuring that two machines will [[deadlock]], so that each waits for the output of the other's next step.

==NP-hardness==
If one already knows that the [[travelling salesman problem]] is NP-hard (as it is), then the job-shop problem is clearly also NP-hard, since the TSP is special case of the JSP with <math>m = 1</math> (the salesman is the machine and the cities are the jobs).

== See also ==
* [[Job shop scheduling]]
* [[List of NP-complete problems]]

{{DEFAULTSORT:Job-Shop Problem}}
[[Category:NP-complete problems]]
[[Category:Mathematical optimization]]
[[Category:Operations research]]

[[de:Scheduling#Scheduling-Probleme]]

Supernatural numbers

2014-01-03T04:28:36Z

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In the [[Mathematics|mathematical]] field of [[group theory]], the '''Kurosh subgroup theorem''' describes the algebraic structure of [[subgroup]]s of [[free product]]s of [[group (mathematics)|groups]]. The theorem was obtained by [[Aleksandr Gennadievich Kurosh|Alexander Kurosh]], a Russian mathematician, in 1934.<ref>A. G. Kurosh, ''Die Untergruppen der freien Produkte von beliebigen Gruppen.'' [[Mathematische Annalen]], vol. 109 (1934), pp. 647–660.</ref> Informally, the theorem says that every subgroup of a free product is itself a free product of a [[free group]] and of its intersections with the [[Conjugate (group theory)|conjugates]] of the factors of the original free product.

==History and generalizations==
After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Kuhn (1952),<ref>H. W. Kuhn. ''Subgroup theorems for groups presented by generators and relations.'' [[Annals of Mathematics]] (2), vol. 56, (1952), pp. 22–46</ref> [[Saunders Mac Lane|Mac Lane]] (1958)<ref>S. Mac Lane.
''A proof of the subgroup theorem for free products.'' Mathematika, vol. 5 (1958), pp. 13–19</ref> and others. The theorem was also generalized for describing subgroups of [[free product with amalgamation|amalgamated free product]]s and [[HNN extension]]s.<ref>A. Karrass, and D. Solitar. ''The subgroups of a free product of two groups with an amalgamated subgroup.''
[[Transactions of the American Mathematical Society]], vol. 150 (1970), pp. 227–255.</ref><ref>A. Karrass, and D. Solitar. ''Subgroups of HNN groups and groups with one defining relation''. [[Canadian Journal of Mathematics]], vol. 23 (1971), pp. 627–643.</ref> Other generalizations include considering subgroups of free [[pro-finite group|pro-finite]] products<ref>{{cite journal| last=Zalesskii | first=Pavel Aleksandrovich | year=1990 | title=[Open subgroups of free profinite products over a profinite space of indices] | language=Russian | journal=[[Doklady Akademii Nauk SSSR]] | volume=34 | issue=1 | pages=17–20}}</ref> and a version of the Kurosh subgroup theorem for [[topological group]]s.<ref>P. Nickolas. ''A Kurosh subgroup theorem for topological groups.'' [[Proceedings of the London Mathematical Society]] (3), vol. 42 (1981), no. 3, pp. 461–477</ref>

In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural results of [[Bass–Serre theory]] about groups [[group action|acting]] on [[Tree (graph theory)|trees]].<ref name="cohen">Daniel Cohen. ''Combinatorial group theory: a topological approach.'' [[London Mathematical Society]] Student Texts, 14. [[Cambridge University Press]], Cambridge, 1989. ISBN 0-521-34133-7; 0-521-34936-2</ref>

==Statement of the theorem==

Let ''G'' = ''A''∗''B'' be the [[free product]] of groups ''A'' and ''B'' and let ''H'' ≤ ''G'' be a [[subgroup]] of ''G''. Then there exist a family (''A''''i'')''i'' ∈ ''I'' of subgroups ''A''''i'' ≤ ''A'', a family (''B''''j'')''j'' ∈ ''J'' of subgroups ''B''''j'' ≤ ''B'', families ''g''''i'', ''i'' ∈ ''I'' and ''f''''j'', ''j'' ∈ ''J'' of elements of ''G'', and a subset ''X'' ⊆ ''G'' such that

:<math>H=F(X)*(*_{i\in I} g_i A_ig_i^{-1})* (*_{j\in J} f_jB_jf_j^{-1}).</math>

This means that ''X'' ''freely [[Generating set of a group|generates]]'' a subgroup of ''G'' isomorphic to the [[free group]] ''F''(''X'') with free basis ''X'' and that, moreover, ''g''''i''''A''''i''''g''''i''−1, ''f''''j''''B''''j''''f''''j''−1 and ''X'' [[Generating set of a group|generate]] ''H'' in ''G'' as a free product of the above form.

There is a generalization of this to the case of free products with arbitrarily many factors.<ref>William S. Massey, [http://books.google.com/books?id=IX0dhDDHezgC&pg=PA218&dq=%22Kurosh+subgroup+theorem%22&as_brr=3&ei=dQ10S8zsKKasNaSNgJsE&cd=1#v=onepage&q=%22Kurosh%20subgroup%20theorem%22&f=false Algebraic topology: an introduction.] Graduate Texts in Mathematics, [[Springer-Verlag]], New York, 1977, ISBN 0-387-90271-6; pp. 218–225</ref> Its formulation is:

If ''H'' is a subgroup of ∗i∈I''G''i = ''G'', then
:<math>H=F(X)*(*_{j\in J} g_jH_jg_j^{-1}),</math>
where ''X'' ⊆ ''G'' and ''J'' is some index set and ''g''j ∈ ''G'' and each ''H''j is a subgroup of some ''G''i.

==Proof using Bass–Serre theory==

The Kurosh subgroup theorem easily follows from the basic structural results in [[Bass–Serre theory]], as explained, for example in the book of Cohen (1987):<ref name="cohen"/>

Let ''G'' = ''A''∗''B'' and consider ''G'' as the fundamental group of a [[graph of groups]] '''Y''' consisting of a single non-loop edge with the vertex groups ''A'' and ''B'' and with the trivial edge group. Let ''X'' be the Bass–Serre universal covering tree for the graph of groups '''Y'''. Since ''H'' ≤ ''G'' also acts on ''X'', consider the quotient graph of groups '''Z''' for the action of ''H'' on ''X''. The vertex groups of '''Z''' are subgroups of ''G''-stabilizers of vertices of ''X'', that is, they are conjugate in ''G'' to subgroups of ''A'' and ''B''. The edge groups of '''Z''' are trivial since the ''G''-stabilizers of edges of ''X'' were trivial. By the fundamental theorem of Bass–Serre theory, ''H'' is canonically [[Group isomorphism|isomorphic]] to the fundamental group of the [[graph of groups]] '''Z'''. Since the edge groups of '''Z''' are trivial, it follows that ''H'' is equal to the free product of the vertex groups of '''Z''' and the free group ''F''(''X'') which is the [[fundamental group]] (in the standard topological sense) of the underlying graph ''Z'' of '''Z'''. This implies the conclusion of the Kurosh subgroup theorem.

==Extension==
The result extends to the case that ''G'' is the [[amalgamated product]] along a common subgroup ''A'', under the condition that ''H'' meets every conjugate of ''A'' only in the identity element.<ref>{{cite book | title=Trees | first=Jean-Pierre | last=Serre | publisher=Springer | year=2003 | isbn=3-540-44237-5 | pages=56–57 }}</ref>

==See also==
*[[HNN extension]]
*[[Geometric group theory]]

==Notes==
{{reflist}}

[[Category:Geometric group theory]]
[[Category:Theorems in group theory]]

Littelmann path model

2013-11-24T02:08:02Z

209.6.233.136: /* Properties */ Changed the inequality so that the exponent of "e" is positive.

Sir '''(Alfred) George Greenhill''', [[Fellow of the Royal Society|F.R.S.]] (29 November 1847, [[London]] — 10 February 1927, [[London]]), was a [[Great Britain|British]] [[mathematician]].

George Greenhill was educated at [[Christ's Hospital|Christ's Hospital School]] and from there he went up to [[St John's College, Cambridge]] in 1866.<ref>{{acad|id=GRNL866GA|name=Greenhill, George Alfred}}</ref> In 1876, Greenhill was appointed professor of mathematics at the [[Royal Military Academy, Woolwich|Royal Military Academy (RMA) at Woolwich]], London, UK.<ref>School of Mathematics and Statistics, University of St Andrews, Scotland. ''Alfred George Greenhill'' (October 2003). http://www-history.mcs.st-andrews.ac.uk/~history/Biographies/Greenhill.html</ref> He held this chair until his retirement in 1908. His 1892 textbook on applications of [[elliptic functions]] is of acknowledged excellence.

In 1879, Greenhill developed a [[rule of thumb]] for calculating the optimal [[Rifling#Twist rate|twist]] rate for lead-core bullets. This shortcut uses the bullet's length, needing no allowances for weight or nose shape.<ref>Mosdell, Matthew. ''The Greenhill Formula''. http://www.mamut.net/MarkBrooks/newsdet35.htm (Accessed 2009 AUG 19)</ref> Greenhill applied this theory to account for the steadiness of flight conferred upon an elongated projectile by [[rifling]]. The eponymous ''Greenhill Formula'', still used today, is:

<math>Twist = \frac{C D^2}{L} \times \sqrt{\frac{SG}{10.9}}</math>

where:
*C = 150 (use 180 for muzzle velocities higher than 2,800 f/s)
*D = bullet's diameter in inches
*L = bullet's length in inches
*SG = bullet's [[specific gravity]] (10.9 for lead-core bullets, which cancels out the second half of the equation)

The original value of C was 150, which yields a twist rate in inches per turn, when given the diameter D and the length L of the bullet in inches. This works to velocities of about 840 m/s (2800 ft/s); above those velocities, a C of 180 should be used. For instance, with a velocity of 600 m/s (2000 ft/s), a diameter of {{convert|0.5|in|mm}} and a length of {{convert|1.5|in|mm}}, the Greenhill formula would give a value of 25, which means 1 turn in {{convert|25|in|mm}}.

== Textbooks ==
* A. G. Greenhill ''Differential and integral calculus, with applications'' ( London, MacMillan, 1886) [http://www.archive.org/details/differentialinte00greeuoft archive.org]
* A. G. Greenhill, ''The applications of elliptic functions'' (MacMillan & Co, New York, 1892). [http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACQ7072 University of Michigan Historical Mathematical Collection.]
* A. G. Greenhill, ''A treatise on hydrostatics'' (MacMillan, London, 1894) [http://www.archive.org/details/treatiseonhydros00greeuoft archive.org]
* A. G. Greenhill ''The dynamics of mechanical flight'' (Constable, London, 1912) [http://www.archive.org/details/dynamicsofmechan00greerich archive.org]

==External links==
{{Wikisource author}}
* Alfred George Greenhill. [http://www-history.mcs.st-and.ac.uk/Biographies/Greenhill.html The MacTutor History of Mathematics archive]
* Alfred George Greenhill. [http://www.icmihistory.unito.it/portrait/greenhill.php The First Century of the ICMI (1909 - 2008) ]
* School of Mathematics and Statistics, University of St Andrews, Scotland. ''Alfred George Greenhill'' (October 2003). http://www-history.mcs.st-andrews.ac.uk/~history/Biographies/Greenhill.html

==References==
{{reflist}}

{{Authority control|VIAF=49986812}}

{{Persondata 
| NAME = Greenhill, Alfred George
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = British mathematician
| DATE OF BIRTH = 1847
| PLACE OF BIRTH =
| DATE OF DEATH = 1927
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Greenhill, Alfred George}}
[[Category:1847 births]]
[[Category:1927 deaths]]
[[Category:People from London]]
[[Category:People educated at Christ's Hospital]]
[[Category:Alumni of St John's College, Cambridge]]
[[Category:British mathematicians]]
[[Category:Royal Medal winners]]
[[Category:Second Wranglers]]
[[Category:Fellows of the Royal Aeronautical Society]]
[[Category:Fellows of the Royal Society]]
[[Category:Knights Bachelor]]
[[Category:Members of the French Academy of Sciences]]
[[Category:De Morgan Medallists]]
[[Category:Ballistics experts]]
[[Category:ICM 1904 Plenary and Invited Speakers]]

Likelihood ratios in diagnostic testing

2013-10-23T20:00:26Z

209.6.52.109: /* Application to medicine */ Truncated paragraph to only include mathematically correct assertions.

{{Infobox Unit
| name = Tesla
| image = [[File:100RSD front.jpg|300px]]
| caption = The tesla definition {{math|1=T = {{sfrac|Wb|m2}}}} is prominently depicted on the 100 [[Serbian dinar]]s banknote, along with the picture of [[Nikola Tesla]].
| standard = [[SI derived unit]]
| quantity = [[Magnetic flux density]]
| symbol = T
| dimension = M·T−2·I−1
| namedafter = [[Nikola Tesla]]
| extralabel = In [[SI base unit]]s:
| extradata = 1 T = 1 [[kilogram|kg]]·[[second|s]]−2·[[ampere|A]]−1
}}
The '''tesla''' (symbol '''T''') is the [[SI derived unit]] of [[magnetic field strength]] or [[magnetic field|magnetic flux density]], commonly denoted as '''B'''. One tesla is equal to one [[weber (unit)|weber]] per square [[metre]], and it was defined in 1960<ref>{{cite web|url=http://www.sizes.com/units/SI.htm |title=Details of SI units |publisher=sizes.com |date=2011-07-01 |accessdate=2011-10-04}}</ref> in honour of [[Nikola Tesla]]. The strongest fields encountered from permanent magnets are from [[Halbach sphere]]s which can be over 4.5 T.<ref>http://www.magnet.fsu.edu/mediacenter/factsheets/records.html</ref>
The unit was announced during the [[General Conference on Weights and Measures|Conférence Générale des Poids et Mesures]] in 1960.

== Definition ==
A particle carrying a charge of 1 coulomb and passing through a magnetic field of 1 tesla at a speed of 1 meter per second perpendicular to said field experiences a force with magnitude 1 newton, according to the [[Lorentz force law]].
As an [[SI derived unit]], the tesla can also be expressed as

:<math> \mathrm{T} = \mathrm{V{\cdot}s{\cdot}m^{-2}} = \mathrm{N{\cdot}A^{-1}{\cdot}m^{-1}} = \mathrm{Wb{\cdot}m^{-2}} </math>
::<math> = \mathrm{kg{\cdot}C^{-1}{\cdot}s^{-1}} = \mathrm{kg{\cdot}A^{-1}{\cdot}s^{-2}} = \mathrm{N{\cdot}s{\cdot}C^{-1}{\cdot}m^{-1}} </math>

(The 5th equivalent is in [[SI base unit]]s).<ref>''The International System of Units (SI), 8th edition'', [[BIPM]], eds. (2006), ISBN 92-822-2213-6, [http://www.bipm.org/en/si/si_brochure/chapter2/2-2/table3.html Table 3. Coherent derived units in the SI with special names and symbols]</ref>

Units used:
: A = [[ampere]]
: C = [[coulomb]]
: kg = [[kilogram]]
: m = [[meter]]
: N = [[newton (unit)|newton]]
: s = [[second]]
: T = tesla
: V = [[volt]]
: Wb = [[weber (unit)|weber]]

== Electric vs. magnetic field ==

In the production of the [[Lorentz force]], the difference between these types of field is that a force from a [[magnetic field]] on a charged particle is generally due to the charged particle's movement<ref>{{cite book|last=Gregory|first=Frederick|year=2003|title=History of Science 1700 to Present|publisher=The Teaching Company}}</ref> while the force imparted by an electric field on a charged particle is not due to the charged particle's movement. This may be appreciated by looking at the units for each. The unit of [[electric field]] in the [[MKS system of units]] is newtons per coulomb, N/C, while the magnetic field (in teslas) can be written as N/(C·m/s). The dividing factor between the two types of field is meters/second (m/s), which is velocity. This relationship immediately highlights the fact that whether a static [[electromagnetic field]] is seen as purely magnetic, or purely electric, or some combination of these, is dependent upon one's [[frame of reference|reference frame]] (that is: one's velocity relative to the field).<ref>{{cite book|last=Parker|first=Eugene|year=2007|title=Conversations on electric and magnetic fields in the cosmos|publisher=Princeton University press|page=65|url=http://books.google.com/books?id=7gJ_i3CTcpQC&pg=PA65&dq=reference+frame+electric+magnetic+field#v=onepage&q=reference%20frame%20electric%20magnetic%20field&f=false}}</ref><ref>{{cite book|last=Kurt|first=Oughstun|year=2006|title=Electromagnetic and optical pulse propogation|publisher=Springer|page=81|url=http://books.google.com/books?id=behRnNRiueAC&pg=PA81&dq=reference+frame+electric+magnetic+field#v=onepage&q=reference%20frame%20electric%20magnetic%20field&f=false}}</ref>

In [[ferromagnets]], the movement creating the magnetic field is the electron spin<ref>{{cite book|last=Herman|first=Stephen|year=2003|title=Delmar's standard textbook of electricity|publisher=Delmar Publishers|page=97|url=http://books.google.com/books?id=kddgHk0P3NcC&pg=PA97&dq=magnetism+electron+spin#v=onepage&q=magnetism%20electron%20spin&f=false}}</ref> (and to a lesser extent electron orbital angular momentum). In a current-carrying wire (electromagnets) the movement is due to electrons moving through the wire (whether the wire is straight or circular).

== Conversions ==
1 tesla is equivalent to:<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
: 10,000 (or 104) G ([[Gauss (unit)|gauss]]), used in the [[CGS]] system. Thus, 10 G = 1 mT (millitesla), and 1 G = 10−4 T.
: 1,000,000,000 (or 109) γ (gammas), used in [[geophysics]]. Thus, 1 γ = 1 nT (nanotesla)
: 42.6 MHz of the 1H nucleus frequency, in [[NMR]]. Thus a 1 GHz NMR magnetic field is 23.5 teslas.

For those concerned with low-frequency electromagnetic radiation in the home, the following conversions are needed most:

: 1000 nT (nanoteslas) = 1 µT (microtesla) = 10 mG (milligauss)
: 1,000,000 µT = 1 T

Because the tesla is so large in regards to everyday usage, common engineering practice is to report the strength of magnets in gauss. Scientists are split on this issue, with some insisting on proper SI units at all times and some allowing for more practical labeling (though using mT or µT is not less practical than G or mG, and conforms with SI). In publications where fields are reported in teslas, very often they are incorrectly reported with the unit capitalized as "Teslas." However, as with gauss and other units, even those derived from names (watt, joule) the unit is lower-case when written out, but capitalized as a symbol: tesla, T.

For the relation to the units of the [[Effective magnetic field|magnetizing field]] (amperes per meter or [[oersted]]s) see the article on [[Permeability (electromagnetism)|permeability]].

== Examples ==
{{Main|Orders of magnitude (magnetic field)}}

* 31.869 µT (3.1 × 10−5 T) – strength of Earth's magnetic field at 0° latitude, 0° longitude
* 5 mT – the strength of a typical [[refrigerator magnet]]
* 0.3 T – the strength of solar sunspots
* 1.25 T – magnetic field intensity at the surface of a [[neodymium magnet]]
* 1 T to 2.4 T – coil gap of a typical loudspeaker magnet
* 1.5 T to 3 T – strength of medical magnetic resonance imaging systems in practice, experimentally up to 17 T<ref>{{cite web|url=http://www.bruker-biospin.com/uhf-mri.html |title=Ultra-High Field |publisher=Bruker BioSpin |date= |accessdate=2011-10-04}}</ref>
* 4 T – strength of the [[Superconductivity|superconducting]] magnet built around the [[Compact Muon Solenoid|CMS]] detector at [[CERN]]<ref>{{cite web|title=Superconducting Magnet in CMS|url=http://cms.web.cern.ch/news/superconducting-magnet|accessdate=9 February 2013}}</ref>
* 8 T – the strength of [[LHC]] magnets.
* 13 T – strength of the superconducting [[ITER]] magnet system <ref>{{cite web|url=http://www.iter.org/mach/magnets |title=ITER – the way to new energy |accessdate=2012-04-19}}</ref>
* 16 T – magnetic field strength required to levitate a frog,<ref>{{cite web|title=Of Flying Frogs and Levitrons" by M.V. Berry and A.K. Geim, European Journal of Physics, v. 18, 1997, p. 307–13.|url=http://www.webcitation.org/5txsxCmCf|accessdate=12 May 2013}}</ref> per the 2000 [[Ig Nobel Prize]] in Physics.<ref>{{cite web|title=The 2000 Ig Nobel Prize Winners|url=http://www.improbable.com/ig/winners/#ig2000|accessdate=12 May 2013}}</ref>

== Notes and references ==
{{Reflist|2}}

== External links ==
{{Wiktionary|tesla}}
*[http://www.magnii.com/magnetic-units-converter.html Gauss<->Tesla Conversion Tool]

{{SI units}}

{{DEFAULTSORT:Tesla (Unit)}}
[[Category:SI derived units]]
[[Category:Nikola Tesla]]
[[Category:Units of magnetic field]]

Maximum theorem

2013-08-09T23:09:39Z

209.6.193.139: /* Statement of theorem */

{{About|the physical phenomenon|its use as a method in spectroscopy|Nuclear magnetic resonance spectroscopy}}
{{Redirect|NMR}}
[[File:700 lab fix.JPG|thumb|right|Bruker 700 MHz. Nuclear Magnetic Resonance (NMR) spectrometer]]

'''Nuclear magnetic resonance''' ('''NMR''') is a [[physical phenomenon]] in which [[atomic nucleus|nuclei]] in a [[magnetic field]] absorb and re-emit [[electromagnetic radiation]]. This energy is at a specific [[resonance]] frequency which depends on the strength of the magnetic field and the magnetic properties of the [[isotope]] of the atoms; in practical applications, the frequency is similar to [[VHF]] and [[UHF]] [[Television channel frequencies|television broadcasts]] (60–1000 MHz).
NMR allows the observation of specific [[quantum mechanical]] [[magnetic]] properties of the [[atomic nucleus]]. Many scientific techniques exploit NMR phenomena to study [[molecular physics]], [[crystallography|crystal]]s, and non-crystalline materials through [[NMR spectroscopy]]. NMR is also routinely used in advanced [[medical imaging]] techniques, such as in [[magnetic resonance imaging]] (MRI).

All isotopes that contain an odd number of [[proton]]s and/or of [[neutrons]] (see [[Isotope#Even and odd nucleon numbers|Isotope]]) have an intrinsic [[Magnetic moment#Magnetic moment of a nucleus|magnetic moment]] and [[angular momentum]], in other words a nonzero [[Spin (physics)|spin]], while all [[nuclide]]s with even numbers of both have a total spin of zero. The most commonly studied nuclei are {{SimpleNuclide|link|Hydrogen|1}} and {{SimpleNuclide|link|Carbon|13}}, although nuclei from isotopes of many other elements (e.g. {{SimpleNuclide|link|Hydrogen|2}}, {{SimpleNuclide|link|Lithium|6}}, {{SimpleNuclide|link|Boron|10}}, {{SimpleNuclide|link|Boron|11}}, {{SimpleNuclide|link|Nitrogen|14}}, {{SimpleNuclide|link|Nitrogen|15}}, {{SimpleNuclide|link|Oxygen|17}}, {{SimpleNuclide|link|Fluorine|19}}, {{SimpleNuclide|link|Sodium|23}}, {{SimpleNuclide|link|Silicon|29}}, {{SimpleNuclide|link|Phosphorus|31}}, {{SimpleNuclide|link|Chlorine|35}}, {{SimpleNuclide|link|Cadmium|113}}, {{SimpleNuclide|link|Xenon|129}}, {{SimpleNuclide|link|Platinum|195}}) have been studied by high-field NMR spectroscopy as well.

A key feature of NMR is that the [[resonance]] frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonance frequencies of the sample's nuclei depend on where in the field they are located. Since the resolution of the imaging technique depends on the magnitude of magnetic field [[gradient]], many efforts are made to develop increased field strength, often using [[superconductors]]. The effectiveness of NMR can also be improved using [[Hyperpolarization (physics)|hyperpolarization]], and/or using two-dimensional, three-dimensional and higher-dimensional multi-frequency techniques.

The principle of NMR usually involves two sequential steps:
*The alignment (polarization) of the magnetic nuclear spins in an applied, constant [[magnetic field]] '''H'''0.
*The perturbation of this alignment of the nuclear spins by employing an electro-magnetic, usually radio frequency (RF) pulse. The required perturbing frequency is dependent upon the static magnetic field ('''H'''0) and the nuclei of observation.

The two fields are usually chosen to be [[perpendicular]] to each other as this maximizes the NMR signal strength. The resulting response by the total magnetization ('''M''') of the nuclear spins is the phenomenon that is exploited in [[Nuclear magnetic resonance spectroscopy|NMR spectroscopy]] and [[magnetic resonance imaging]]. Both use intense applied magnetic fields ('''H'''0) in order to achieve dispersion and very high stability to deliver [[spectral resolution]], the details of which are described by [[chemical shift]]s, the [[Zeeman effect]], and [[Knight shift]]s (in metals).

NMR phenomena are also utilized in [[low field NMR|low-field NMR]], NMR spectroscopy and MRI in the Earth's magnetic field (referred to as [[#Earth's field NMR|Earth's field NMR]]), and in several types of [[magnetometer]]s.

==History==

Nuclear magnetic resonance was first described and measured in molecular beams by [[Isidor Rabi]] in 1938,<ref>
{{cite journal
|author = Rabi, I.I.; Zacharias, J.R.; Millman, S. and Kusch, P.
|title = A New Method of Measuring Nuclear Magnetic Moment
|journal = [[Physical Review]]
|volume = 53
|year = 1938
|doi = 10.1103/PhysRev.53.318
|pmid=9981980
|issue = 4
|bibcode = 1938PhRv...53..318R
|pages = 318–327 }}</ref> by extending the [[Stern-Gerlach experiment]], and in 1944, Rabi was awarded the [[Nobel Prize in physics]] for this work.<ref>[http://nobelprize.org/nobel_prizes/physics/laureates/1944/rabi-bio.html Biography of I. Rabi at Nobelprize.org]</ref> In 1946, [[Felix Bloch]] and [[Edward Mills Purcell]] expanded the technique for use on liquids and solids, for which they shared the [[Nobel Prize in Physics]] in 1952.<ref name=Filler2009b>{{cite journal|doi=10.1038/npre.2009.3267.5|title=The History, Development and Impact of Computed Imaging in Neurological Diagnosis and Neurosurgery: CT, MRI, and DTI|year=2009|author=Filler, Aaron|journal=Nature Precedings}}</ref><ref>[http://nobelprize.org/nobel_prizes/physics/laureates/1952/ 1952 Nobel Prize for Physics at Nobelprize.org]</ref>

Purcell had worked on the development of [[radar]] during [[World War II]] at the [[Massachusetts Institute of Technology]]'s [[Radiation Laboratory]]. His work during that project on the production and detection of [[radio frequency|radio frequency power]] and on the absorption of such RF power by matter laid the foundation for Rabi's discovery of NMR.

Rabi, Bloch, and Purcell observed that magnetic nuclei, like {{SimpleNuclide|link|Hydrogen|1}} and {{SimpleNuclide|link|Phosphorus|31}}, could absorb RF energy when placed in a magnetic field and when the RF was of a frequency specific to the identity of the nuclei. When this absorption occurs, the nucleus is described as being ''in resonance''. Different atomic nuclei within a molecule resonate at different (radio) frequencies for the same magnetic field strength. The observation of such magnetic resonance frequencies of the nuclei present in a molecule allows any trained user to discover essential, chemical and structural information about the molecule.

The development of NMR as a technique in [[analytical chemistry]] and [[biochemistry]] parallels the development of electromagnetic technology and advanced [[electronics]] and their introduction into civilian use.

==Theory of nuclear magnetic resonance==

===Nuclear spin and magnets===

All nucleons, that is [[neutrons]] and [[protons]], composing any atomic [[nucleus (atomic structure)|nucleus]], have the intrinsic quantum property of [[spin (physics)|spin]]. The overall spin of the nucleus is determined by the [[spin quantum number]] ''S''. If the number of both the protons and neutrons in a given [[nuclide]] are even then {{nowrap|''S'' {{=}} 0}}, i.e. there is no overall spin. Then, just as electrons pair up in [[atomic orbital]]s, so do even numbers of protons or even numbers of neutrons (which are also spin-{{frac|1|2}} particles and hence [[fermion]]s) pair up giving zero overall spin.

However, a proton and neutron will have lower energy when their spins are parallel, '''not anti-parallel''', since this parallel spin alignment does not infringe upon the [[Pauli Exclusion Principle]], but instead it has to do with the [[quark]] structure of these two nucleons. Therefore, the spin ground state for the deuteron (the deuterium nucleus, or the 2H isotope of hydrogen)—that has only a proton and a neutron—corresponds to a spin value of '''1''', ''not of zero''. The single, isolated deuteron therefore exhibits an NMR absorption spectrum characteristic of a quadrupolar nucleus of spin '''1''', which in the "rigid" state at very low temperatures is a characteristic ('[[George Pake|Pake]]') ''doublet'', (not a singlet as for a single, isolated 1H, or any other isolated fermion or dipolar nucleus of spin 1/2). On the other hand, because of the Pauli Exclusion Principle, the [[tritium]] isotope of hydrogen must have a pair of anti-parallel spin neutrons (of total spin zero for the neutron-spin pair), plus a proton of spin 1/2. Therefore, the character of the tritium nucleus is again magnetic dipolar, ''not quadrupolar''—like its non-radioactive deuteron cousin—and the tritium nucleus total spin value is again 1/2, just like for the simpler, abundant hydrogen isotope, 1H nucleus (the ''[[proton]]''). The NMR absorption (radio) frequency for tritium is however slightly higher than that of 1H because the tritium nucleus has a slightly higher [[gyromagnetic ratio]] than 1H. In many other cases of ''non-radioactive'' nuclei, the overall spin is also non-zero. For example, the {{SimpleNuclide|link|Aluminium|27}} nucleus has an overall spin value {{nowrap|''S'' {{=}} {{frac|5|2}}}}.

A non-zero spin is thus always associated with a non-zero magnetic moment (''μ'') via the relation {{nowrap|''μ'' {{=}} ''γS''}}, where ''γ'' is the [[gyromagnetic ratio]]. It is this magnetic moment that allows the observation of NMR absorption spectra caused by transitions between nuclear spin levels. Most nuclides (with some rare exceptions) that have both even numbers of protons and even numbers of neutrons, also have zero nuclear magnetic moments, and they also have zero magnetic dipole and quadrupole moments. Hence, such nuclides do not exhibit any NMR absorption spectra. Thus, {{SimpleNuclide|link|Oxygen|18}} is an example of a nuclide that has no NMR absorption, whereas {{SimpleNuclide|link|Carbon|13}}, {{SimpleNuclide|link|Phosphorus|31}}, {{SimpleNuclide|link|Chlorine|35}} and {{SimpleNuclide|link|Chlorine|37}} are nuclides that do exhibit NMR absorption spectra. The last two nuclei are quadrupolar nuclei whereas the preceding two nuclei ({{SimpleNuclide|link|Carbon|13}} and {{SimpleNuclide|link|Phosphorus|31}}) are dipolar ones.

[[Electron spin resonance]] (ESR) is a related technique in which transitions between [[electron]]ic spin levels are detected rather than nuclear ones. The basic principles are similar but the instrumentation, data analysis, and detailed theory are significantly different. Moreover, there is a much smaller number of molecules and materials with unpaired electron spins that exhibit ESR (or [[electron paramagnetic resonance]] (EPR)) absorption than those that have NMR absorption spectra. ESR has much higher sensitivity than NMR does.

===Values of spin angular momentum===

The [[angular momentum]] associated with nuclear spin is quantized. This means both that the magnitude of angular momentum is quantized (i.e. ''S'' can only take on a restricted range of values), and also that the orientation of the associated angular momentum is quantized. The associated quantum number is known as the [[magnetic quantum number]], ''m'', and can take values from +''S'' to −''S'', in integer steps. Hence for any given nucleus, there are a total of {{nowrap|2''S'' + 1}} angular momentum states.

The z-component of the angular momentum vector ('''S''') is therefore {{nowrap|Sz {{=}} ''mħ''}}, where ''ħ'' is the reduced [[Planck constant]]. The z-component of the magnetic moment is simply:
:<math> \mu_\mathrm{z} = \gamma S_\mathrm{z} = \gamma m\hbar.</math>

====Spin behavior in a magnetic field====
[[Image:NMR splitting.gif|thumb|right|300 px||Splitting of nuclei spin states in an external magnetic field]]
[[Image:NMR EPR.gif|thumb|right|300 px||An intuitive model. Nuclei behave like they had own [[magnetic moment]]s ([[spin magnetic moment]]s). By itself, there is no energetic difference for any particular orientation (only one energy state, on the left), but in external magnetic field there is a high-energy state and a low-energy state depending on the relative orientations of the magnet to the external field, and the orientation of the magnetic moment can [[precession|precess]] relative to it. The external field can be supplied by a large magnet and also by other nuclei in the vicinity.]]

Consider nuclei which have a spin of one-half, like {{SimpleNuclide|link|Hydrogen|1}}, {{SimpleNuclide|link|Carbon|13}} or {{SimpleNuclide|link|Fluorine|19}}. The nucleus has two possible spin states: ''m'' = {{frac|1|2}} or ''m'' = −{{frac|1|2}} (also referred to as spin-up and spin-down, or sometimes α and β spin states, respectively). These states are degenerate, that is they have the same energy. Hence the number of atoms in these two states will be approximately equal at [[Thermodynamic equilibrium|thermal equilibrium]].

If a nucleus is placed in a magnetic field, however, the interaction between the nuclear magnetic moment and the external magnetic field mean the two states no longer have the same energy. The [[energy]] of a magnetic moment '''μ''' when in a magnetic field '''B'''0 is given by:
:<math> E = -\boldsymbol{\mu} \cdot \mathbf{B}_0 = -\mu_\mathrm{x} B_{0x}-\mu_\mathrm{y} B_{0y}-\mu_\mathrm{z} B_{0z} .</math>

Usually the ''z'' axis is chosen to be along '''B'''0, and the above expression reduces to:
:<math> E = -\mu_\mathrm{z} B_0 \ ,</math>
or alternatively:
:<math> E = -\gamma m\hbar B_0 \ .</math>

As a result the different nuclear spin states have different energies in a non-zero magnetic field. In less formal language, we can talk about the two spin states of a spin {{frac|1|2}} as being ''aligned'' either with or against the magnetic field. If γ is positive (true for most isotopes) then {{nowrap|''m'' {{=}} {{frac|1|2}}}} is the lower energy state.

The energy difference between the two states is:
:<math>\Delta{E} = \gamma \hbar B_0 \ ,</math>

and this difference results in a small population bias toward the lower energy state.

====Magnetic resonance by nuclei====
Resonant absorption by nuclear spins will occur only when [[electromagnetic radiation]] of the correct frequency (e.g., equaling the [[Larmor precession]] rate) is being applied to match the energy difference between the nuclear spin levels in a constant magnetic field of the appropriate strength. The energy of an absorbed [[photon]] is then {{nowrap|''E'' {{=}} ''h''ν0}}, where ν0 is the resonance radiofrequency that has to match (that is, it has to be equal to the Larmor precession frequency νL of the nuclear magnetization in the constant magnetic field '''B'''0). Hence, a magnetic resonance absorption will only occur when {{nowrap|Δ''E'' {{=}} ''h''ν0}}, which is when {{nowrap| ν0 {{=}} γ''B''0/(2π)}}. Such magnetic resonance frequencies typically correspond to the [[radio frequency]] (or RF) range of the [[electromagnetic spectrum]] for magnetic fields up to roughly 20 T. It is this magnetic resonant absorption which is detected in NMR. {{Citation needed|date=November 2009}}

====Nuclear shielding====

It might appear from the above that all nuclei of the same nuclide (and hence the same ''γ'') would resonate at the same frequency. This is not the case. The most important perturbation of the NMR frequency for applications of NMR is the "shielding" effect of the surrounding shells of electrons.<ref>[http://nmrcentral.com/2011/08/principle-of-shielding-and-deshielding/ Principle of Shielding and Deshielding | NMRCentral.com]</ref> Electrons, similar to the nucleus, are also charged and rotate with a spin to produce a magnetic field opposite to the magnetic field produced by the nucleus. In general, this electronic shielding reduces the magnetic field ''at the nucleus'' (which is what determines the NMR frequency).

As a result the energy gap is reduced, and the frequency required to achieve resonance is also reduced. This shift in the NMR frequency due to the electronic molecular orbital coupling to the external magnetic field is called [[chemical shift]], and it explains why NMR is able to probe the chemical structure of molecules, which depends on the electron density distribution in the corresponding molecular orbitals. If a nucleus in a specific chemical group is shielded to a higher degree by a higher electron density of its surrounding molecular orbital, then its NMR frequency will be shifted "upfield" (that is, a lower chemical shift), whereas if it is less shielded by such surrounding electron density, then its NMR frequency will be shifted "downfield" (that is, a higher chemical shift).

Unless the local [[symmetry]] of such molecular orbitals is very high (leading to "isotropic" shift), the shielding effect will depend on the orientation of the molecule with respect to the external field ('''B'''0). In [[solid-state NMR]] spectroscopy, [[magic angle spinning]] is required to average out this orientation dependence in order to obtain values close to the average chemical shifts. This is unnecessary in conventional NMR investigations of molecules, since rapid "molecular tumbling" averages out the [[chemical shift anisotropy]] (CSA). In this case, the term "average" chemical shift (ACS) is used.

===Relaxation===
{{details|Relaxation (NMR)}}

The process called population relaxation refers to nuclei that return to the thermodynamic state in the magnet. This process is also called [[Spin-lattice relaxation time|''T''1]], "[[Spin-lattice relaxation time|spin-lattice]]" or "longitudinal magnetic" relaxation, where ''T''1 refers to the mean time for an individual nucleus to return to its thermal equilibrium state of the spins. Once the nuclear spin population is relaxed, it can be probed again, since it is in the initial, equilibrium (mixed) state.

The [[Larmor precession|precessing]] nuclei can also fall out of alignment with each other (returning the net magnetization vector to a non-precessing field) and stop producing a signal. This is called [[Spin-spin relaxation time|''T''2]] or ''transverse relaxation''. Because of the difference in the actual relaxation mechanisms involved (for example, inter-molecular vs. intra-molecular magnetic dipole-dipole interactions ), ''T''1 is usually (except in rare cases) longer than ''T''2 (that is, slower spin-lattice relaxation, for example because of smaller dipole-dipole interaction effects). In practice, the value of <math>T^*_2</math> which is the actually observed decay time of the observed NMR signal, or [[free induction decay]], (to 1/e of the initial amplitude immediately after the resonant RF pulse)-- also depends on the static magnetic field inhomogeneity, which is quite significant. (There is also a
smaller but significant contribution to the observed FID shortening from the RF inhomogeneity of the resonant pulse). In the corresponding FT-NMR spectrum—meaning the [[Fourier transform]] of the [[free induction decay]]—the <math>T^*_2</math> time is inversely related to the width of the NMR signal in frequency units. Thus, a nucleus with a long ''T''2 relaxation time gives rise to a very sharp NMR peak in the FT-NMR spectrum for a very homogeneous ([[Shim (magnetism)|"well-shimmed"]]) static magnetic field, whereas nuclei with shorter ''T''2 values give rise to broad FT-NMR peaks even when the magnet is shimmed well. Both ''T''1 and ''T''2 depend on the rate of molecular motions as well as the gyromagnetic ratios of both the resonating and their strongly interacting, next-neighbor nuclei that are not at resonance.

A [[Spin echo|Hahn echo]] decay experiment can be used to measure the dephasing time, as shown in the animation below. The size of the echo is recorded for different spacings of the two pulses. This reveals the decoherence which is not refocused by the <math>\pi</math> pulse. In simple cases, an [[exponential decay]] is measured which is described by the <math>T_2</math> time.

[[Image:GWM HahnEchoDecay.gif]]

==NMR spectroscopy==
{{Main|NMR spectroscopy}}

[[Image:HWB-NMR - 900MHz - 21.2 Tesla.jpg|thumb|right|300px|900 MHz, 21.2 [[Tesla (unit)|T]] NMR Magnet at HWB-NMR, Birmingham, UK]]

NMR spectroscopy is one of the principal techniques used to obtain physical, chemical, electronic and structural information about [[molecule]]s due to either the chemical shift, [[Zeeman effect]], or the [[Knight shift]] effect, or a combination of both, on the resonant frequencies of the nuclei present in the sample. It is a powerful technique that can provide detailed information on the topology, dynamics and three-dimensional structure of molecules in solution and the solid state. Thus, structural and dynamic information is obtainable (with or without "magic angle" spinning (MAS)) from NMR studies of quadrupolar nuclei (that is, those nuclei with spin {{nowrap|S > {{frac|1|2}}}}) even in the presence of magnetic "[[dipole]]-dipole" interaction broadening (or simply, dipolar broadening) which is always much smaller than the quadrupolar interaction strength because it is a magnetic vs. an electric interaction effect.

Additional structural and chemical information may be obtained by performing double-quantum NMR experiments for quadrupolar nuclei such as {{SimpleNuclide|link|Hydrogen|2}}. Also, nuclear magnetic resonance is one of the techniques that has been used to design quantum automata, and also build elementary [[quantum computer]]s.<ref>[http://planetphysics.org/encyclopedia/QuantumComputers.html Quantum automaton and quantum computation] (see also references therein)</ref><ref name=Vandersypen>
{{cite journal
|doi = 10.1038/414883a
|title = Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance
|year = 2001
|author = Vandersypen, Lieven M. K.
|journal = Nature
|volume = 414
|pages = 883–887
|last2 = Steffen
|first2 = Matthias
|last3 = Breyta
|first3 = Gregory
|last4 = Yannoni
|first4 = Costantino S.
|last5 = Sherwood
|first5 = Mark H.
|last6 = Chuang
|first6 = Isaac L.
|pmid=11780055
|issue = 6866
|arxiv = quant-ph/0112176 |bibcode = 2001Natur.414..883V }}</ref>

===Continuous wave (CW) spectroscopy===
In its first few decades, nuclear magnetic resonance spectrometers used a technique known as [[continuous-wave]] spectroscopy (CW spectroscopy). Although NMR spectra could be, and have been, obtained using a fixed magnetic field and sweeping the frequency of the [[electromagnetic radiation]], this more typically involved using a fixed frequency source and varying the current (and hence magnetic field) in an [[electromagnet]] to observe the resonant absorption signals. This is the origin of the counterintuitive, but still common, "high field" and "low field" terminology for low frequency and high frequency regions respectively of the NMR spectrum.

CW spectroscopy is inefficient in comparison with [[Fourier analysis]] techniques (see below) since it probes the NMR response at individual frequencies in succession. Since the NMR signal is intrinsically weak, the observed spectrum suffers from a poor [[signal-to-noise ratio]]. This can be mitigated by signal averaging i.e. adding the spectra from repeated measurements. While the NMR signal is constant between scans and so adds linearly, the [[random noise]] adds more slowly – [[Proportionality (mathematics)|proportion]]al to the [[square-root]] of the number of spectra (see [[random walk]]). Hence the overall signal-to-noise ratio increases as the square-root of the number of spectra measured.

===Fourier transform spectroscopy===

Most applications of NMR involve full NMR spectra, that is, the intensity of the NMR signal as a function of frequency. Early attempts to acquire the NMR spectrum more efficiently than simple CW methods involved illuminating the target simultaneously with more than one frequency. A revolution in NMR occurred when short pulses of radio-frequency radiation began to be used—centered at the middle of the NMR spectrum. In simple terms, a short pulse of a given "carrier" frequency "contains" a range of frequencies centered about the carrier frequency, with the range of excitation ([[signal bandwidth|bandwidth]]) being inversely proportional to the pulse duration, i.e. the [[Fourier transform]] of a short pulse contains contributions from all the frequencies in the neighborhood of the principal frequency. The restricted range of the NMR frequencies made it relatively easy to use short (millisecond to microsecond) radio frequency pulses to excite the entire NMR spectrum.{{Citation needed|date=October 2011}}

Applying such a pulse to a set of nuclear spins simultaneously excites all the single-quantum NMR transitions. In terms of the net magnetization vector, this corresponds to tilting the magnetization vector away from its equilibrium position (aligned along the external magnetic field). The out-of-equilibrium magnetization vector precesses about the external magnetic field vector at the NMR frequency of the spins. This oscillating magnetization vector [[electromagnetic induction|induces]] a current in a nearby pickup coil, creating an electrical signal oscillating at the NMR frequency. This signal is known as the [[free induction decay]] (FID), and it contains the vector sum of the NMR responses from all the excited spins. In order to obtain the frequency-domain NMR [[spectrum]] (NMR absorption intensity vs. NMR frequency) this time-domain signal (intensity vs. time) must be [[Fourier transform]]ed. Fortunately the development of Fourier Transform NMR coincided with the development of [[digital computer]]s and the digital [[Cooley-Tukey FFT algorithm|Fast Fourier Transform]]. Fourier methods can be applied to many types of spectroscopy. (See the full article on [[Fourier transform spectroscopy]].)

[[Richard R. Ernst]] was one of the pioneers of pulse NMR, and he won a [[Nobel Prize in chemistry]] in 1991 for his work on Fourier Transform NMR and his development of multi-dimensional NMR (see below).

===Multi-dimensional NMR Spectroscopy===

The use of pulses of different shapes, frequencies and durations in specifically designed patterns or ''pulse sequences'' allows the spectroscopist to extract many different types of information about the molecule. Multi-dimensional nuclear magnetic resonance spectroscopy is a kind of FT NMR in which there are at least two pulses and, as the experiment is repeated, the pulse sequence is
systematically varied. In ''multidimensional nuclear magnetic resonance'' there will be a sequence of pulses and, at least, one variable time period. In three dimensions, two time sequences will be varied. In four dimensions, three will be varied.

There are many such experiments. In one, these time intervals allow (amongst other things) magnetization transfer between nuclei and, therefore, the detection of the kinds of nuclear-nuclear interactions that allowed for the magnetization transfer. Interactions that can be detected are usually classified into two kinds. There are ''through-bond'' interactions and ''through-space'' interactions, the latter usually being a consequence of the [[nuclear Overhauser effect]]. Experiments of the nuclear Overhauser variety may be employed to establish distances between atoms, as for example by [[2D-FT NMRI and Spectroscopy|2D-FT NMR]] of molecules in solution.

Although the fundamental concept of 2D-FT NMR was proposed by [[Jean Jeener]] from the [[Free University of Brussels]] at an International Conference, this idea was largely developed by [[Richard R. Ernst|Richard Ernst]] who won the 1991 [[Nobel prize in Chemistry]] for his work in FT NMR, including multi-dimensional FT NMR, and especially 2D-FT NMR of small molecules.<ref>[http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.html "Nuclear Magnetic Resonance Fourier Transform Spectroscopy"] Ernst's Nobel lecture. (Includes mention of Jeener's suggestion.)</ref> Multi-dimensional FT NMR experiments were then further developed into powerful methodologies for studying biomolecules in solution, in particular for the determination of the structure of [[biopolymer]]s such as [[protein]]s or even small [[nucleic acid]]s.<ref name="2D-FT NMR and Magnetic Resonance Imaging">
{{cite web
|author = Baianu, I.C.
|title = Two-dimensional Fourier transforms
|work = 2D-FT NMR and MRI
|publisher = PlanetMath
|url = http://planetmath.org/encyclopedia/TwoDimensionalFourierTransforms.html
|accessdate = 2009-02-22
}}</ref>

In 2002 [[Kurt Wüthrich]] shared the [[Nobel Prize in Chemistry]] (with [[John Bennett Fenn]] and [[Koichi Tanaka]]) for his work with [[protein nuclear magnetic resonance spectroscopy|protein FT NMR]] in solution.

===Solid-state NMR spectroscopy===
{{Main|Solid-state nuclear magnetic resonance}}
This technique complements [[X-ray crystallography]] in that it is frequently applicable to molecules in a liquid or [[liquid crystal]] phase, whereas crystallography, as the name implies, is performed on molecules in a [[solid]] phase. Though nuclear magnetic resonance is used to study solids, extensive atomic-level molecular structural detail is especially challenging to obtain in the solid state. There is little signal averaging by thermal motion in the solid state, where most molecules can only undergo restricted vibrations and rotations at room temperature, each in a slightly different electronic environment, therefore exhibiting a different NMR absorption peak. Such a variation in the electronic environment of the resonating nuclei results in a blurring of the observed spectra—which is often only a broad Gaussian band for non-quadrupolar spins in a solid- thus making the interpretation of such "[[dipolar broadening|dipolar]]" and "[[chemical shift anisotropy]]" (CSA) broadened spectra either very difficult or impossible.

Professor Raymond Andrew at Nottingham University in the UK pioneered the development of high-resolution [[solid-state nuclear magnetic resonance]]. He was the first to report the introduction of the [[magic angle spinning|MAS]] (magic angle sample spinning; MASS) technique that allowed him to achieve spectral resolution in solids sufficient to distinguish between chemical groups with either different chemical shifts or distinct [[Knight shift]]s. In MASS, the sample is spun at several kilohertz around an axis that makes the so-called [[magic angle]] ''θ''m (which is ~54.74°, where cos2''θ''m = 1/3) with respect to the direction of the static magnetic field '''B'''0; as a result of such magic angle sample spinning, the chemical shift anisotropy bands are averaged to their corresponding average (isotropic) chemical shift values. The above expression involving cos2''θ''m has its origin in a calculation that predicts the magnetic dipolar interaction effects to cancel out for the specific value of ''θ''m called the magic angle. One notes that correct alignment of the sample rotation axis as close as possible to ''θ''m is essential for cancelling out the dipolar interactions whose strength for angles sufficiently far from ''θ''m is usually greater than ~10 kHz for [[C-H bond]]s in solids, for example, and it is thus greater than their CSA values.

There are different angles for the sample spinning relative to the applied field for the averaging of quadrupole interactions and paramagnetic interactions, correspondingly ~30.6° and ~70.1°

A concept developed by Sven Hartmann and [[Erwin Hahn]] was utilized in transferring magnetization from protons to less sensitive nuclei (popularly known as cross-polarization) by M.G. Gibby, [[Alexander Pines|Alex Pines]] and [[John S. Waugh]]. Then, Jake Schaefer and Ed Stejskal demonstrated also the powerful use of cross-polarization under MASS conditions which is now routinely employed to detect low-abundance and low-sensitivity nuclei.

===Sensitivity===
Because the intensity of nuclear magnetic resonance signals and, hence, the sensitivity of the technique depends on the strength of the magnetic field the technique has also advanced over the decades with the development of more powerful magnets. Advances made in audio-visual technology have also improved the signal-generation and processing capabilities of newer instruments.

As noted above, the sensitivity of nuclear magnetic resonance signals is also dependent on the presence of a magnetically susceptible nuclide and, therefore, either on the natural abundance of such nuclides or on the ability of the experimentalist to artificially enrich the molecules, under study, with such nuclides. The most abundant naturally occurring isotopes of hydrogen and phosphorus (for example) are both magnetically susceptible and readily useful for nuclear magnetic resonance spectroscopy. In contrast, carbon and nitrogen have useful isotopes but which occur only in very low natural abundance.

Other limitations on sensitivity arise from the quantum-mechanical nature of the phenomenon. For quantum states separated by energy equivalent to radio frequencies, thermal energy from the environment causes the populations of the states to be close to equal. Since incoming radiation is equally likely to cause stimulated emission (a transition from the upper to the lower state) as absorption, the NMR effect depends on an excess of nuclei in the lower states. Several factors can reduce sensitivity, including
*Increasing temperature, which evens out the population of states. Conversely, low temperature NMR can sometimes yield better results than room-temperature NMR, providing the sample remains liquid.
*Saturation of the sample with energy applied at the resonant radiofrequency. This manifests in both CW and pulsed NMR; in the first case (CW) this happens by using too much continuous power that keeps the upper spin levels completely populated; in the second case (pulsed), each pulse (that is at least a 90° pulse) leaves the sample saturated, and four to five times the (longitudinal) relaxation time (5 T1) must pass before the next pulse or pulse sequence can be applied. For single pulse experiments, shorter RF pulses that tip the magnetization by less than 90° can be used, which loses some intensity of the signal, but allows for shorter ''recycle delays''. The optimum there is called an ''Ernst angle'', after [[Richard R. Ernst|the Nobel laureate]]. Especially in solid state NMR, or in samples with very few nuclei with spins > 0, (diamond with the natural 1% of Carbon-13 is especially troublesome here) the longitudinal relaxation times can be on the range of hours, while for proton-NMR they are more on the range of one second.
*Non-magnetic effects, such as electric-[[quadrupole]] coupling of spin-1 and spin-{{frac|3|2}} nuclei with their local environment, which broaden and weaken absorption peaks. {{SimpleNuclide|link|Nitrogen|14}}, an abundant spin-1 nucleus, is difficult to study for this reason. High resolution NMR instead probes molecules using the rarer {{SimpleNuclide|link|Nitrogen|15}} isotope, which has spin-{{frac|1|2}}.

===Isotopes===
Many isotopes of chemical elements can be used for NMR analysis.<ref>[http://chem.ch.huji.ac.il/nmr/techniques/1d/multi.html Multinuclear NMR]</ref>

'''Commonly used nuclei:'''
*{{SimpleNuclide|link|Hydrogen|1}}, the most commonly used spin ½ nucleus in NMR investigation, has been studied using many forms of NMR. Hydrogen is highly abundant, especially in biological systems. It is the nucleus most sensitive to NMR signal (apart from {{SimpleNuclide|link|Hydrogen|3}} which is not commonly used due to its instability and radioactivity). Proton NMR produces narrow chemical shift with sharp signals. Fast acquisition of quantitative results (peak integrals in stoichiometric ratio) is possible due to short relaxation time. The {{SimpleNuclide|link|Hydrogen|1}} signal has been the sole diagnostic nucleus used for clinical magnetic resonance imaging.
*{{SimpleNuclide|link|Hydrogen|2}}, a spin 1 nucleus commonly utilized as signal-free medium in the form of [[:category:deuterated solvent|deuterated solvents]] during proton NMR, to avoid signal interference from hydrogen-containing solvents in measurement of {{SimpleNuclide|link|Hydrogen|1}} solutes. Also used in determining the behavior of lipids in lipid membranes and other solids or liquid crystals as it is a relatively non-perturbing label which can selectively replace {{SimpleNuclide|link|Hydrogen|1}}. Alternatively, {{SimpleNuclide|link|Hydrogen|2}} can be detected in media specially labeled with {{SimpleNuclide|link|Hydrogen|2}}. Deuterium resonance is commonly used in high-resolution [[NMR spectroscopy]] to monitor drifts in the magnetic field strength (lock) and to improve the homogeneity of the external magnetic field.
*{{SimpleNuclide|link|Helium|3}}, is very sensitive to NMR. There is a very low percentage in natural helium, and subsequently has to be purified from {{SimpleNuclide|link|Helium|4}}. It is used mainly in studies of [[endohedral fullerenes]], where its chemical inertness is beneficial to ascertaining the structure of the entrapping fullerene.
*{{SimpleNuclide|link|Boron|11}}, more sensitive than {{SimpleNuclide|link|Boron|10}}, yields sharper signals. Quartz tubes must be used as [[borosilicate]] glass interferes with measurement.
*{{SimpleNuclide|link|Carbon|13}} spin-1/2, is widely used, despite its relative paucity in naturally occurring carbon (approximately 1%). It is stable to nuclear decay. Since there is a low percentage in natural carbon, spectrum acquisition on samples which have not been experimentally enriched in {{SimpleNuclide|link|Carbon|13}} takes a long time. Frequently used for labeling of compounds in synthetic and metabolic studies. Has low sensitivity and wide chemical shift, yields sharp signals. Low percentage makes it useful by preventing spin-spin couplings and makes the spectrum appear less crowded. Slow relaxation means that spectra are not integrable unless long acquisition times are used.
*{{SimpleNuclide|link|Nitrogen|14}}, spin-1, medium sensitivity nucleus with wide chemical shift. Its large [[quadrupole]] moment interferes in acquisition of high resolution spectra, limiting usefulness to smaller molecules and functional groups with a high degree of symmetry such as the headgroups of lipids.
*{{SimpleNuclide|link|Nitrogen|15}}, spin-1/2, relatively commonly used. Can be used for labeling compounds. Nucleus very insensitive but yields sharp signals. Low percentage in natural nitrogen together with low sensitivity requires high concentrations or expensive isotope enrichment.
*{{SimpleNuclide|link|Oxygen|17}}, spin-5/2, low sensitivity and very low natural abundance (0.037%), wide chemical shifts range (up to 2000 ppm). Quadrupole moment causing a line broadening. Used in metabolic and biochemical studies in studies of chemical equilibria.
*{{SimpleNuclide|link|Fluorine|19}}, spin-1/2, relatively commonly measured. Sensitive, yields sharp signals, has wide chemical shift.
*{{SimpleNuclide|link|Phosphorus|31}}, spin-1/2, 100% of natural phosphorus. Medium sensitivity, wide chemical shifts range, yields sharp lines. Spectra tend to have a moderate amount of noise. Used in biochemical studies and in coordination chemistry where phosphorus containing ligands are involved.
*{{SimpleNuclide|link|Chlorine|35}} and {{SimpleNuclide|link|Chlorine|37}}, broad signal. {{SimpleNuclide|link|Chlorine|35}} significantly more sensitive, preferred over {{SimpleNuclide|link|Chlorine|37}} despite its slightly broader signal. Organic chlorides yield very broad signals, its use is limited to inorganic and ionic chlorides and very small organic molecules.
*{{SimpleNuclide|link|Calcium|43}}, used in biochemistry to study calcium binding to DNA, proteins, etc. Moderately sensitive, very low natural abundance.
*{{SimpleNuclide|link|Cadmium|113}}, used to study metal-binding in [[metallothionein]]s and [[alkaline phosphatase]] and other [[metalloprotein]]s.<ref>{{cite book
|last1=Armitage
|first1=Ian M.
|last2=Drakenberg
|first2=Torbjörn
|last3=Reilly
|first3=Brian
|editor=Astrid Sigel, Helmut Sigel and Roland K. O. Sigel
|title=Cadmium: From Toxicology to Essentiality
|series=Metal Ions in Life Sciences
|volume=11
|year=2013
|publisher=Springer
|pages=117-144
|chapter=Chapter 6. Use of 113Cd NMR to Probe the Native Metal Binding Sites in Metalloproteins: an Overview
|doi=10.1007/978-94-007-5179-8_6}}
</ref>

*{{SimpleNuclide|link|Platinum|195}}, used in studies of [[catalyst]]s and complexes.

'''Other nuclei''' (usually used in the studies of their complexes and chemical binding, or to detect presence of the element):
{{div col|cols=5}}
*{{SimpleNuclide|link|Lithium|6}}, {{SimpleNuclide|link|Lithium|7}}
*{{SimpleNuclide|link|Beryllium|9}}
*{{SimpleNuclide|link|Fluorine|19}}
*{{SimpleNuclide|link|Neon|21}}
*{{SimpleNuclide|link|Sodium|23}}
*{{SimpleNuclide|link|Magnesium|25}}
*{{SimpleNuclide|link|Aluminium|27}}
*{{SimpleNuclide|link|Silicon|29}}
*{{SimpleNuclide|link|Phosphorus|31}}
*{{SimpleNuclide|link|Sulfur|33}}
*{{SimpleNuclide|link|Potassium|39}}, {{SimpleNuclide|link|Potassium|40}}, {{SimpleNuclide|link|Potassium|41}}
*{{SimpleNuclide|link|Scandium|45}}
*{{SimpleNuclide|link|Titanium|47}}, {{SimpleNuclide|link|Titanium|49}}
*{{SimpleNuclide|link|Vanadium|50}}, {{SimpleNuclide|link|Vanadium|51}}
*{{SimpleNuclide|link|Chromium|53}}
*{{SimpleNuclide|link|Manganese|55}}
*{{SimpleNuclide|link|Iron|57}}
*{{SimpleNuclide|link|Cobalt|59}}
*{{SimpleNuclide|link|Nickel|61}}
*{{SimpleNuclide|link|Copper|63}}, {{SimpleNuclide|link|Copper|65}}
*{{SimpleNuclide|link|Zinc|67}}
*{{SimpleNuclide|link|Gallium|69}}, {{SimpleNuclide|link|Gallium|71}}
*{{SimpleNuclide|link|Germanium|73}}
*{{SimpleNuclide|link|Arsenic|75}}
*{{SimpleNuclide|link|Selenium|77}}
*{{SimpleNuclide|link|Bromine|81}}
*{{SimpleNuclide|link|Rubidium|87}}
*{{SimpleNuclide|link|Strontium|87}}
*{{SimpleNuclide|link|Molybdenum|95}}
*{{SimpleNuclide|link|Silver|109}}
*{{SimpleNuclide|link|Tin|119}}
*{{SimpleNuclide|link|Tellurium|125}}
*{{SimpleNuclide|link|Iodine|127}}
*{{SimpleNuclide|link|Caesium|133}}
*{{SimpleNuclide|link|Barium|135}}, {{SimpleNuclide|link|Barium|137}}
*{{SimpleNuclide|link|Lanthanum|139}}
*{{SimpleNuclide|link|Tungsten|183}}
*{{SimpleNuclide|link|Mercury|199}}
{{div col end}}

==Applications==

===Medicine===
[[File:MRI-Philips.JPG|thumb|Medical MRI]]
{{See also|Magnetic resonance imaging}}
The application of nuclear magnetic resonance best known to the general public is [[magnetic resonance imaging]] for medical diagnosis and [[magnetic resonance microscopy]] in research settings, however, it is also widely used in chemical studies, notably in NMR spectroscopy such as [[proton NMR]], [[carbon-13 NMR]], deuterium NMR and phosphorus-31 NMR. Biochemical information can also be obtained from living tissue (e.g. human [[brain]] [[tumor]]s) with the technique known as [[in vivo magnetic resonance spectroscopy]] or [[chemical shift]] NMR Microscopy.

These studies are possible because nuclei are surrounded by orbiting electrons, which are charged particles that generate small, local magnetic fields that add to or subtract from the external magnetic field, and so will partially shield the nuclei. The amount of shielding depends on the exact local environment. For example, a hydrogen bonded to an [[oxygen]] will be shielded differently than a hydrogen bonded to a carbon atom. In addition, two hydrogen nuclei can interact via a process known as [[spin-spin coupling]], if they are on the same molecule, which will split the lines of the spectra in a recognizable way.

As one of the two major spectroscopic techniques used in [[metabolomics]], NMR is used to generate metabolic fingerprints from biological fluids to obtain information about disease states or toxic insults.

===Chemistry===

By studying the peaks of nuclear magnetic resonance spectra, chemists can determine the structure of many compounds. It can be a very selective technique, distinguishing among many atoms within a molecule or collection of molecules of the same type but which differ only in terms of their local chemical environment. NMR spectroscopy is used to unambiguously identify known and novel compounds, and as such, is usually required by scientific journals for identity confirmation of synthesized new compounds. See the articles on [[carbon-13 NMR]] and [[proton NMR]] for detailed discussions.

By studying ''T''2 information, a chemist can determine the identity of a compound by comparing the observed nuclear precession frequencies to known frequencies. Further structural data can be [[Structural elucidation|elucidated]] by observing ''[[spin-spin coupling]]'', a process by which the precession frequency of a nucleus can be influenced by the magnetization transfer from nearby chemically bound nuclei. Spin-spin coupling is observed in NMR of hydrogen-1 ({{SimpleNuclide|Hydrogen|1}} NMR), since its natural abundance is nearly 100%; isotope enrichment is required for most other elements.

Because the nuclear magnetic resonance ''timescale'' is rather slow, compared to other spectroscopic methods, changing the temperature of a ''T''2*experiment can also give information about fast reactions, such as the [[Cope rearrangement]] or about structural dynamics, such as ring-flipping in [[cyclohexane]]. At low enough temperatures, a distinction can be made between the axial and equatorial hydrogens in cyclohexane.

An example of nuclear magnetic resonance being used in the determination of a structure is that of [[buckminsterfullerene]] (often called "buckyballs", composition C60). This now famous form of carbon has 60 carbon atoms forming a sphere. The carbon atoms are all in identical environments and so should see the same internal H field. Unfortunately, buckminsterfullerene contains no hydrogen and so {{SimpleNuclide|link|Carbon|13}} nuclear magnetic resonance has to be used. {{SimpleNuclide|link|Carbon|13}} spectra require longer acquisition times since carbon-13 is not the common isotope of carbon (unlike hydrogen, where {{SimpleNuclide|link|Hydrogen|1}} is the common isotope). However, in 1990 the spectrum was obtained by R. Taylor and co-workers at the [[University of Sussex]] and was found to contain a single peak, confirming the unusual structure of buckminsterfullerene.<ref>
{{Cite journal
|author = Taylor, R.; Hare, J.P.; Abdul-Sada, A.K. and Kroto, H.W.
|title = Isolation, separation and characterization of the fullerenes C60 and C70: the third form of carbon
|journal = Journal of the Chemical Society, Chemical Communications
|volume = 20|issue =20|pages =1423–1425
|year = 1990
|doi =10.1039/c39900001423
}}</ref>

===Non-destructive testing===

Nuclear magnetic resonance is extremely useful for analyzing samples non-destructively. Radio waves and static magnetic fields easily penetrate many types of matter and anything that is not inherently [[ferromagnetism|ferromagnetic]]. For example, various expensive biological samples, such as [[nucleic acids]], including [[RNA]] and [[DNA]], or [[protein]]s, can be studied using nuclear magnetic resonance for weeks or months before using destructive biochemical experiments. This also makes nuclear magnetic resonance a good choice for analyzing dangerous samples.

===Acquisition of dynamic information===
In addition to providing static information on molecules by determining their 3D structures in solution, one of the remarkable advantages of NMR over [[X-ray crystallography]] is that it can be used to obtain important dynamic information including the [[low-frequency collective motion in proteins and DNA]], for example in the Ca2+-calmodulin system.<ref>{{cite journal | last1 = Chou | first1 = J. J. | last2 = Li | first2 = S. | last3 = Klee | first3 = C. B. | last4 = Bax | first4 = A. | year = 2001 | title = Solution structure of Ca2+-calmodulin reveals flexible hand-like properties of its domains | url = | journal = Nature Structural Biology | volume = 8 | issue = 11| pages = 990–997 | doi = 10.1038/nsb1101-990 | pmid = 11685248 }}</ref> The low-frequency internal motion in biomacromolecules and its biological functions have been discussed by Chou.<ref>{{cite journal|author=Chou, Kuo-Chen |year=1988|title=Low-frequency collective motion in biomacromolecules and its biological functions|journal=Biophys Chem |volume=30|pages=3–48|pmid=3046672|doi=10.1016/0301-4622(88)85002-6|issue=1}}</ref>

===Data acquisition in the petroleum industry===
{{Main|NMR in porous media}}
Another use for nuclear magnetic resonance is [[data acquisition]] in the [[petroleum industry]] for [[petroleum]] and [[natural gas]] exploration and recovery. A [[borehole]] is drilled into rock and sedimentary strata into which nuclear magnetic resonance logging equipment is lowered. Nuclear magnetic resonance analysis of these boreholes is used to measure rock porosity, estimate permeability from pore size distribution and identify pore fluids (water, oil and gas). These instruments are typically [[low field NMR]] spectrometers.

===Flow probes for NMR spectroscopy===

Recently, real-time applications of NMR in liquid media have been developed using specifically designed flow probes (flow cell assemblies) which can replace standard tube probes. This has enabled techniques that can incorporate the use of high performance liquid chromatography (HPLC) or other continuous flow sample introduction devices.<ref>{{cite journal
|title = Flow Probes for NMR Spectroscopy
|author = Haner, R.L. and Keifer, P.A.
|journal = Encyclopedia of Magnetic Resonance
|year = 2009
|doi= 10.1002/9780470034590.emrstm1085
|isbn = 0470034599
}}</ref>

===Process control===
NMR has now entered the arena of real-time [[process control]] and [[process optimization]] in [[oil refineries]] and [[petrochemical]] plants. Two different types of NMR analysis are utilized to provide real time analysis of feeds and products in order to control and optimize unit operations. Time-domain NMR (TD-NMR) spectrometers operating at low field (2–20 MHz for {{SimpleNuclide|link|Hydrogen|1}}) yield [[free induction decay]] data that can be used to determine absolute [[hydrogen]] content values, [[rheological]] information, and component composition. These spectrometers are used in [[mining]], [[polymer]] production, [[cosmetics]] and food manufacturing as well as [[coal]] analysis. High resolution FT-NMR spectrometers operating in the 60 MHz range with shielded permanent magnet systems yield high resolution {{SimpleNuclide|link|Hydrogen|1}} NMR spectra of [[refinery]] and [[petrochemical]] streams. The variation observed in these spectra with changing physical and chemical properties is modeled using [[chemometrics]] to yield predictions on unknown samples. The prediction results are provided to [[control systems]] via analogue or digital outputs from the spectrometer.

===Earth's field NMR===
{{Main|Earth's field NMR}}
In the [[Earth's magnetic field]], NMR frequencies are in the [[audio frequency]] range, or the [[very low frequency]] and [[ultra low frequency]] bands of the [[radio frequency]] spectrum. Earth's field NMR (EFNMR) is typically stimulated by applying a relatively strong dc magnetic field pulse to the sample and, after the end of the pulse, analyzing the resulting low frequency alternating magnetic field that occurs in the Earth's magnetic field due to [[free induction decay]] (FID). These effects are exploited in some types of [[magnetometer]]s, EFNMR spectrometers, and MRI imagers. Their inexpensive portable nature makes these instruments valuable for field use and for teaching the principles of NMR and MRI.

An important feature of EFNMR spectrometry compared with high-field NMR is that some aspects of molecular structure can be observed more clearly at low fields and low frequencies, whereas other aspects observable at high fields are not observable at low fields. This is because:
* Electron-mediated heteronuclear [[J-coupling]]s ([[spin-spin coupling]]s) are field independent, producing clusters of two or more frequencies separated by several Hz, which are more easily observed in a fundamental resonance of about {{nowrap|2 kHz.}} "Indeed it appears that enhanced resolution is possible due to the long spin relaxation times and high field homogeneity which prevail in EFNMR."<ref name="Robinson">{{cite journal|url=http://www.sfu.ca/~simonw/phys431/references/nmr/robinson_nmr_imaging_JMR2006.pdf | author= Robinson J. N. et al.|year=2006|journal=Journal of Magnetic Resonance |volume=182 |pages=343–347 |title=Two-dimensional NMR spectroscopy in Earth's magnetic field|bibcode = 2006JMagR.182..343R |doi = 10.1016/j.jmr.2006.06.027|issue=2|pmid=16860581 }}</ref>
* Chemical shifts of several [[Parts per million|ppm]] are clearly separated in high field NMR spectra, but have separations of only a few millihertz at proton EFNMR frequencies, so are usually lost in noise etc.

===Quantum computing===
{{Main|Nuclear magnetic resonance quantum computer}}
'''NMR quantum computing''' uses the [[spin (physics)|spin]] states of molecules as [[qubit]]s. NMR differs from other implementations of quantum computers in that it uses an [[Quantum ensemble|ensemble]] of systems, in this case molecules.

===Magnetometers===
{{Main|Magnetometer}}
Various magnetometers use NMR effects to measure magnetic fields, including [[Magnetometer#Proton precession magnetometer|proton precession magnetometer]]s (PPM) (also known as [[proton magnetometer]]s), and [[Magnetometer#Overhauser_magnetometer|Overhauser magnetometer]]s. See also [[Earth's field NMR]].

==Makers of NMR equipment==
Major NMR instrument makers include [[Oxford Instruments]], [[Bruker]], [[Spinlock SRL]], [[General Electric]], [[JEOL]], [[Kimble Chase]], [[Philips]], [[Siemens AG]], and [[Agilent Technologies, Inc.]] (who own [[Varian, Inc.]]).

==See also==
{{colbegin|4}}
*[[Carbon-13 NMR]]
*[[Chemical shift]]
*[[Dynamic nuclear polarisation]] (DNP)
*[[Earth's field NMR]] (EFNMR)
*[[Free induction decay]] (FID)
*[[In vivo magnetic resonance spectroscopy]] (MRS)
*[[J-coupling]]
*[[Larmor equation]] (Not to be confused with [[Larmor formula]]).
*[[Larmor precession]]
*[[Low field NMR]]
*[[Magic angle spinning]]
*[[Magnetometer]]
*[[Magnetic resonance imaging]] (MRI)
*[[NMR crystallography]]
*[[NMR spectra database]]
*[[NMR spectroscopy]]
*[[NMR Microscopy]]
*[[Nuclear magnetic resonance in porous media]]
*[[Nuclear quadrupole resonance]] (NQR)
*[[Protein dynamics]]
*[[Protein NMR]]
*[[Proton NMR]]
*[[Rabi cycle]]
*[[Relaxometry]]
*[[Relaxation (NMR)]]
*[[Spin echo]]
*[[Solid-state NMR]]
*[[Zero field NMR]]
{{colend}}

==References==
{{Reflist|35em}}

==Further reading==
*{{cite book
|author = [[Gary E. Martin]], A. S. Zektzer
|title = Two-Dimensional NMR Methods for Establishing Molecular Connectivity
|publisher = [[Wiley-VCH]]
|location = New York
|year = 1988
|page = 59
|isbn = 0-471-18707-0
|url = http://books.google.com/books?id=9ysYrpe_NoEC&printsec=frontcover
}}
*{{cite book
|author = J.W. Akitt, B.E. Mann
|title = NMR and Chemistry
|publisher =Stanley Thornes
|location = Cheltenham, UK
|year = 2000
|pages = 273, 287
|isbn = 0-7487-4344-8
}}
*{{cite web
|author = J.P. Hornak
|title = The Basics of NMR
|url = http://www.cis.rit.edu/htbooks/nmr/
|accessdate = 2009-02-23
}}
*{{cite book
|author = J. Keeler
|title = Understanding NMR Spectroscopy
|publisher =John Wiley & Sons
|year=2005
|isbn=0-470-01786-4
}}
*{{cite book
|author = [[Kurt Wüthrich]]
|title = NMR of Proteins and Nucleic Acids
|publisher = [[Wiley-Interscience]]
|location = New York (NY), USA
|year = 1986
|isbn = 0-471-11917-2
}}
*{{cite journal
|author = J.M Tyszka, S.E Fraser, R.E Jacobs
|year = 2005
|doi = 10.1016/j.copbio.2004.11.004
|title = Magnetic resonance microscopy: recent advances and applications
|journal = Current Opinion in Biotechnology
|volume = 16 |issue = 1 |pages=93–99
|pmid = 15722021
}}
*{{cite web
|author = J.C. Edwards
|title = Principles of NMR
|url = http://www.process-nmr.com/pdfs/NMR%20Overview.pdf
|publisher=Process NMR Associates
|accessdate = 2009-02-23
}}
*{{cite book
|author = R.L Haner, P.A. Keifer
|title = Encyclopedia of Magnetic Resonance
|publisher = John Wiley
|doi = 10.1002/9780470034590.emrstm1085
|year = 2009
}}

==External links==
{{commons|Nuclear magnetic resonance}}

===Tutorial===
*[http://www.cis.rit.edu/htbooks/nmr/inside.htm NMR/MRI tutorial]
*[http://nmr.chinanmr.cn/guide/eNMR/eNMRindex.html NMR Library] NMR Concepts
*[http://www.grandinetti.org/Teaching/Chem7160/Notes NMR Course Notes]

===Animations and Simulations===
*[http://www.bigs.de/BLH/en/index.php?option=com_content&view=category&layout=blog&id=51&Itemid=222 This animation shows a spin, the modification of spin with magnetic field and HF pulse, spin echo sequences, inversion recovery sequence, gradient echo sequence and relaxation of spin]
*[http://www.chem.queensu.ca/FACILITIES/NMR/nmr/webcourse/precess.htm Animation of NMR spin {{frac|1|2}} precession]
*[http://vam.anest.ufl.edu/forensic/nmr.html A free interactive simulation of NMR principles]

===Video===
*[http://www.youtube.com/watch?v=7aRKAXD4dAg introduction to NMR and MRI]
*[http://www.vega.org.uk/video/programme/21 Richard Ernst, NL – Developer of Multdimensional NMR techniques] Freeview video provided by the Vega Science Trust.
*[http://www.vega.org.uk/video/programme/115 'An Interview with Kurt Wuthrich'] Freeview video by the Vega Science Trust (Wüthrich was awarded a Nobel Prize in Chemistry in 2002 "for his development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution").

; Other
*[http://www.sciencedirect.com/science/article/pii/S109078070700198X Off Magic Angle Spinning]

{{DEFAULTSORT:Nuclear Magnetic Resonance}}
[[Category:Nuclear magnetic resonance| ]]
[[Category:Scientific techniques]]

Key-recovery attack

2013-03-20T02:12:00Z

209.6.74.105:

In [[mathematics]], a '''pre-Lie algebra''' is an [[algebraic structure]] on a vector space, that describes some properties of objects such as [[Tree (graph theory)|rooted trees]] and [[vector fields]] on affine space.

The notion of pre-Lie algebra has been introduced by [[Murray Gerstenhaber]] in his work on deformations of algebras.

Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.

== Definition ==
A pre-Lie algebra <math>(V,\triangleleft)</math> is a vector space <math>V</math> with a bilinear map <math>\triangleleft : V \otimes V \to V</math>, satisfying the relation
<math>
(x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x \triangleleft z) \triangleleft y - x \triangleleft (z \triangleleft y).
</math>

This identity can be seen as the invariance of the [[associator]] <math>(x,y,z) = (x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z)</math> under the exchange of the two variables <math>y</math> and <math>z</math>.

Every [[associative algebra]] is hence also a pre-Lie algebra, as the associator vanishes identically.

== Examples ==
* [[Vector fields]] on the affine space

If we denote by <math>f(x)\partial_x</math> the vector field <math>x \mapsto f(x)</math>, and if we define <math>\triangleleft</math> as <math>f(x) \triangleleft g(x) = f'(x) g(x)</math>, we can see that the operator <math>\triangleleft</math> is exactly the application of the <math>g(x)\partial_x</math> field to <math>f(x)\partial_x</math> field.
<math>(g(x)\partial_x)(f(x)\partial_x) = g(x) \partial_x f(x) \partial_x = g(x) f'(x) \partial_x</math>

If we study the difference between <math>(x \triangleleft y) \triangleleft z</math> and <math>x \triangleleft (y \triangleleft z)</math>, we have
<math>(x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x' y)'z - x'y'z = x'y'z x''yz - z'y'z = x''yz</math>
which is symmetric on ''y'' and ''z''.

* Rooted trees

Let <math>\mathbb{T}</math> be the vector space spanned by all rooted trees.

One can introduce a bilinear product <math>\curvearrowleft</math> on <math>\mathbb{T}</math> as follows. Let <math>\tau_1</math> and <math>\tau_2</math> be two rooted trees.

<math>\tau_1 \curvearrowleft \tau_2 = \sum_{s \in \mathrm{Vertices}(\tau_1)} \tau_1 \circ_s \tau_2</math>

where <math>\tau_1 \circ_s \tau_2</math> is the rooted tree obtained by adding to the disjoint union of <math>\tau_1</math> and <math>\tau_2</math> an edge going from the vertex <math>s</math> of <math>\tau_1</math> to the root vertex of <math>\tau_2</math>.

Then <math>(\mathbb{T}, \curvearrowleft)</math> is a free pre-Lie algebra on one generator.

==References==
*{{citation
| last1 = Chapoton | first1 = F.
| last2 = Livernet | first2 = M.
| year = 2001
| mr = 1827084
| journal = [[International Mathematics Research Notices]]
| title = Pre-Lie algebras and the rooted trees operad
| doi = 10.1155/S1073792801000198
| pages = 395–408
| volume = 8
| issue = 8}}.
*{{citation
| last = Szczesny | first = M.
| year = 2010
| mr =
| journal =
| title = Pre-Lie algebras and incidence categories of colored rooted trees
| volume =1007
| bibcode = 2010arXiv1007.4784S
| pages = 4784
| arxiv = 1007.4784
| class = math.CO }}.

[[Category:Lie groups]]
[[Category:Lie algebras| ]]
[[Category:Non-associative algebra]]

Talk:Piquet

2013-01-20T16:49:10Z

209.6.96.235: /* Carte Blanche */ fix typo

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