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Dissipation factor

2014-01-03T05:14:31Z

46.226.190.2:

In [[geometric group theory]], '''Gromov's theorem on groups of polynomial growth''', named for [[Mikhail Gromov (mathematician)|Mikhail Gromov]], characterizes finitely generated [[Group (mathematics)|groups]] of ''polynomial'' growth, as those groups which have [[nilpotent group|nilpotent]] subgroups of finite [[index of a subgroup|index]].

The [[Growth rate (group theory)|growth rate]] of a group is a [[well-defined]] notion from [[asymptotic analysis]]. To say that a finitely generated group has '''polynomial growth''' means the number of elements of [[length]] (relative to a symmetric generating set) at most ''n'' is bounded above by a [[polynomial]] function ''p''(''n''). The ''order of growth'' is then the least degree of any such polynomial function ''p''.

A ''nilpotent'' group ''G'' is a group with a [[lower central series]] terminating in the identity subgroup.

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index.

There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of [[Joseph A. Wolf]] showed that if ''G'' is a finitely generated nilpotent group, then the group has polynomial growth. [[Yves Guivarc'h]] and independently [[Hyman Bass]] (with different proofs) computed the exact order of polynomial growth. Let ''G'' be a finitely generated nilpotent group with lower central series

:<math> G = G_1 \supseteq G_2 \supseteq \ldots. </math>

In particular, the quotient group ''G''''k''/''G''''k''+1 is a finitely generated abelian group.

'''The Bass–Guivarc'h formula''' states that the order of polynomial growth of ''G'' is

:<math> d(G) = \sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k+1}) </math>

where:
:''rank'' denotes the [[rank of an abelian group]], i.e. the largest number of independent and torsion-free elements of the abelian group.

In particular, Gromov's theorem and the Bass–Guivarch formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding for example, fractional powers).

In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called the [[Gromov–Hausdorff convergence]], is currently widely used in geometry.

A relatively simple proof of the theorem was found by [[Bruce Kleiner]]. Later, [[Terence Tao]] and [[Yehuda Shalom]] modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds.<ref>http://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/</ref><ref>{{cite arxiv |eprint=0910.4148 |author1=Yehuda Shalom |author2=Terence Tao |title=A finitary version of Gromov's polynomial growth theorem |class=math.GR |year=2009}}</ref>

== References ==
<references/>
* H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, ''Proceedings London Mathematical Society'', vol 25(4), 1972
* M. Gromov, Groups of Polynomial growth and Expanding Maps, [http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1981__53_ ''Publications mathematiques I.H.É.S.'', 53, 1981]
* Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A–B 272 (1971). [http://www.numdam.org/item?id=BSMF_1973__101__333_0]
* {{Cite arxiv | last1=Kleiner | first1=Bruce | year=2007 | title=A new proof of Gromov's theorem on groups of polynomial growth | arxiv=0710.4593}}
* J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, ''Journal of Differential Geometry'', vol 2, 1968

[[Category:Theorems in group theory]]
[[Category:Nilpotent groups]]
[[Category:Infinite group theory]]
[[Category:Metric geometry]]
[[Category:Geometric group theory]]

Continuously variable slope delta modulation

2013-12-26T00:53:07Z

46.226.190.2:

In [[recreational mathematics]], a '''Harshad number''' (or '''Niven number''') in a given [[number base]], is an [[integer]] that is divisible by the [[digit sum|sum of its digits]] when written in that base.
Harshad numbers in base ''n'' are also known as '''''n''-Harshad''' (or '''''n''-Niven''') numbers.
Harshad numbers were defined by [[D. R. Kaprekar]], a [[mathematician]] from [[India]]. The word "Harshad" comes from the [[Sanskrit]] ''{{IAST|harṣa}}'' (joy) + ''{{IAST|da}}'' (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by [[Ivan M. Niven]] at a conference on [[number theory]] in 1977. All integers between [[0 (number)|zero]] and ''n'' are ''n''-Harshad numbers.
To date there appear to be no applications for Harshad numbers, not even within pure mathematics.

Stated mathematically, let ''X'' be a positive integer with ''m'' digits when written in base ''n'', and let the digits be ''ai'' (''i'' = 0, 1, ..., ''m'' − 1). (It follows that ''ai'' must be either zero or a positive integer up to ''n'' − 1.) ''X'' can be expressed as

:<math>X=\sum_{i=0}^{m-1} a_i n^i.</math>

If there exists an integer ''A'' such that the following holds, then ''X'' is a Harshad number in base ''n'':

:<math>X=A\sum_{i=0}^{m-1} a_i.</math>

Harshad numbers in [[base 10]] form the sequence:

: [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[5 (number)|5]], [[6 (number)|6]], [[7 (number)|7]], [[8 (number)|8]], [[9 (number)|9]], [[10 (number)|10]], [[12 (number)|12]], [[18 (number)|18]], [[20 (number)|20]], [[21 (number)|21]], [[24 (number)|24]], [[27 (number)|27]], [[30 (number)|30]], [[36 (number)|36]], [[40 (number)|40]], [[42 (number)|42]], [[45 (number)|45]], [[48 (number)|48]], [[50 (number)|50]], [[54 (number)|54]], [[60 (number)|60]], [[63 (number)|63]], [[70 (number)|70]], [[72 (number)|72]], [[80 (number)|80]], [[81 (number)|81]], [[84 (number)|84]], [[90 (number)|90]], [[100 (number)|100]], [[102 (number)|102]], [[108 (number)|108]], [[110 (number)|110]], [[111 (number)|111]], [[112 (number)|112]], [[114 (number)|114]], [[117 (number)|117]], [[120 (number)|120]], [[126 (number)|126]], [[132 (number)|132]], [[133 (number)|133]], [[135 (number)|135]], [[140 (number)|140]], [[144 (number)|144]], [[150 (number)|150]], [[152 (number)|152]], [[153 (number)|153]], [[156 (number)|156]], [[162 (number)|162]], [[171 (number)|171]], [[180 (number)|180]], [[190 (number)|190]], [[192 (number)|192]], [[195 (number)|195]], [[198 (number)|198]], [[200 (number)|200]], [[201 (number)|201]], ... {{OEIS|id=A005349}}

A number which is a Harshad number in any number base is called an '''all-Harshad number''', or an '''all-Niven number'''. There are only four all-Harshad numbers: [[1 (number)|1]], [[2 (number)|2]], [[4 (number)|4]], and [[6 (number)|6]].

== What numbers can be Harshad numbers? ==

Given the [[divisibility test]] for [[9 (number)|9]], one might be tempted to generalize that all numbers divisible by 9 are also Harshad numbers. But for the purpose of determining the Harshadness of ''n'', the digits of ''n'' can only be added up once and ''n'' must be divisible by that sum; otherwise, it is not a Harshad number. For example, [[99 (number)|99]] is not a Harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.

The base number (and furthermore, its powers) will always be a Harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1.

For a [[prime number]] to also be a Harshad number it must be less than or equal to the base number. Otherwise, the digits of the prime will add up to a number that is more than 1 but less than the prime and, obviously, it will not be divisible. For example: 11 is not Harshad in base 10 because the sum of its digits "11" is 1+1=2 and 11 is not divisible by 2, while in [[hexadecimal]] the number 11 may be represented as "B", the sum of whose digits is also B and clearly B is divisible by B, ergo it is Harshad in base 16.

Although the sequence of [[factorial]]s starts with Harshad numbers in base 10, not all factorials are Harshad numbers. 432! is the first that is not.

== Consecutive Harshad numbers ==

=== Maximal runs of consecutive Harshad numbers ===
Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all Harshad numbers in base 10.<ref>{{citation | zbl=0776.11003 | last1=Cooper | first1=Curtis | last2=Kennedy | first2=Robert E. | title=On consecutive Niven numbers | journal=[[Fibonacci Quarterly]] | volume=31 | number=2 | pages=146–151 | year=1993 | issn=0015-0517 |url=http://www.fq.math.ca/Scanned/31-2/cooper.pdf}}</ref><ref name=HBII382>{{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001|page=382}}</ref> They also constructed infinitely many 20-tuples of consecutive integers that are all 10-Harshad numbers, the smallest of which exceeds 1044363342786.

{{harvs|authorlink=Helen G. Grundman|first=H. G.|last=Grundman|year=1994|txt}} extended the Cooper and Kennedy result to show that there are 2''b'' but not 2''b''+1 consecutive ''b''-Harshad numbers.<ref name=HBII382/><ref>{{citation | last = Grundman | first = H. G. | authorlink=Helen G. Grundman | title = Sequences of consecutive ''n''-Niven numbers | journal = [[Fibonacci Quarterly]] | volume = 32 | issue = 2 | year=1994 | pages = 174–175 | zbl=0796.11002 | issn=0015-0517 |url=http://www.fq.math.ca/Scanned/32-2/grundman.pdf}}</ref>
This result was strengthened to show that there are infinitely many runs of 2''b'' consecutive ''b''-Harshad numbers for ''b'' = 2 or 3 by {{harvs|authorlink=T. Tony Cai|first=T.|last=Cai|year=1996|txt}}<ref name=HBII382/> and for arbitrary ''b'' by [[Brad Wilson (mathematician)|Brad Wilson]] in 1997.<ref>{{citation | last1=Wilson | first1=Brad | title=Construction of 2''n'' consecutive ''n''-Niven numbers | journal=[[Fibonacci Quarterly]] | volume=35 | pages=122–128 | year=1997 | issn=0015-0517 |url=http://www.fq.math.ca/Scanned/35-2/wilson.pdf}}</ref>

In [[binary numeral system|binary]], there are thus infinitely many runs of four consecutive Harshad numbers and in [[ternary numeral system|ternary]] infinitely many runs of six.

In general, such maximal sequences run from ''N · bk - b'' to ''N · bk'' + (''b''-1) , where ''b'' is the base, ''k'' is a relatively large power, and ''N'' is a constant.
Given one such suitably chosen sequence we can convert it to a larger one as follows:
* Inserting zeroes into ''N'' will not change the sequence of digital sums (just as 21, 201 and 2001 are all 10-Harshad numbers).
* If we insert ''n'' zeroes after the first digit, α (worth α''bi''), we increase the value of ''N'' by α''bi(bn - 1)'' .
* If we can ensure that ''bn - 1'' is divisible by all digit sums in the sequence, then the divisibility by those sums is maintained.
* If our initial sequence is chosen so that the digit sums are [[coprime]] to ''b'', we can solve ''bn = 1'' modulo all those sums.
* If that is not so, but the part of each digit sum not coprime to ''b'' divides α''bi'', then divisibility is still maintained.
* ''(Unproven)'' The initial sequence is so chosen.
Thus  our initial sequence yields an infinite set of solutions.

=== First runs of exactly ''n'' consecutive 10-Harshad numbers ===
The smallest naturals starting runs of exactly ''n'' consecutive 10-Harshad numbers (i.e., smallest ''x'' such that ''x'', ''x''+1, ..., ''x''+''n''-1 are Harshad numbers but ''x''-1 and ''x''+''n'' are not) are as follows {{OEIS|id=A060159}}:
{| class="wikitable"
|-
! ''n'': !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
|-
| ''x'': || 12 || 20 || 110 || 510 || 131052 || 12751220 || 10000095 || 2162049150 || 124324220 || 1
|-
| '''''n''''': || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16''' || '''17''' || '''18''' || '''19''' || '''20'''
|-
| ''x'': || 920067411130599 || 43494229746440272890 || 121003242000074550107423034⋅1020 - 10 || 420142032871116091607294⋅1040 - 04 || ? || 50757686696033684694106416498959861492⋅10280 - 9 || 14107593985876801556467795907102490773681⋅10280 - 10 || ? || ? || ?
|}
By the previous section, no such ''x'' exists for ''n'' > 20.

== Estimating the density of Harshad numbers ==

If we let ''N''(''x'') denote the number of Harshad numbers ≤ x, then for any given ε > 0,

:<math>x^{1-\varepsilon} \ll N(x) \ll \frac{x\log\log x}{\log x}</math>

as shown by [[Jean-Marie De Koninck]] and Nicolas Doyon;<ref>{{citation|first1=Jean-Marie|last1=De Koninck|first2=Nicolas|last2=Doyon|title=On the number of Niven numbers up to ''x''|journal=[[Fibonacci Quarterly]]|volume=41|issue=5|date=November 2003|pages=431–440}}.</ref> furthermore, De Koninck, Doyon and Kátai<ref>{{citation|first1=Jean-Marie|last1=De Koninck|first2=Nicolas|last2=Doyon|first3=I.|last3=Katái|title=On the counting function for the Niven numbers|journal=[[Acta Arithmetica]]|volume=106|year=2003|pages=265–275}}.</ref> proved that

:<math>N(x)=(c+o(1))\frac{x}{\log x}</math>

where ''c'' = (14/27) log 10 ≈ 1.1939.

== Nivenmorphic numbers ==

A '''Nivenmorphic number''' or '''Harshadmorphic number''' for a given number base is an integer ''t'' such that there exists some Harshad number ''N'' whose [[digit sum]] is ''t'', and ''t'', written in that base, terminates ''N'' written in the same base.

For example, 18 is a Nivenmorphic number for base 10:

16218 is a Harshad number
16218 has 18 as digit sum
18 terminates 16218

Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except [[11 (number)|11]].<ref>{{citation|first=Sandro|last=Boscaro|title=Nivenmorphic integers|journal=[[Journal of Recreational Mathematics]]|volume=28|issue=3|year=1996–1997|pages=201–205}}.</ref>

== Multiple Harshad numbers ==
{{harvtxt|Bloem|2005}} defines a ''multiple Harshad number'' as a Harshad number that, when divided by the sum of its digits, produces another Harshad number.<ref>{{citation|first=E.|last=Bloem|year=2005|title=Harshad numbers|journal=[[Journal of Recreational Mathematics]]|volume=34|issue=2|page=128}}.</ref> He states that 6804 is "MHN-3" on the grounds that

:<math>
\begin{array}{l}
6804/18=378\\
378/18=21\\
21/3=7
\end{array}
</math>

and went on to show that 2016502858579884466176 is MHN-12. The number 10080000000000 = 1008·1010, which is smaller, is also MHN-12. In general, 1008·10''n'' is MHN-(''n''+2).

== References ==
{{reflist}}

== External links ==
* [http://www.harshad-numbers.com/en/ Harshad Numbers]
* [http://www.numbers-of-harshad.com/ Numbers of Harshad]

{{Classes of natural numbers}}
[[Category:Base-dependent integer sequences]]

Conserved quantity

2013-10-26T20:22:48Z

46.226.188.140:

{{refimprove|date=December 2010}}
'''Muon spin spectroscopy''' is an experimental technique based on the implantation of [[spin polarization|spin-polarized]] [[muon]]s in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion. The motion of the muon [[spin (physics)|spin]] is due to the magnetic field experienced by the particle and may provide information on its local environment in a very similar way to other [[magnetic resonance]]{{disambiguation needed|date=June 2012}}<ref>Resonance techniques are often characterized by the use of resonant circuits, which is not the case for muon spin spectroscopy. However the true resonant nature of all these techniques, muon spectroscopy included, lies in the very narrow, resonant requirement upon any time dependent perturbation in order for it to effectively influence the probe's dynamics: for every excitation interacting with the muon (lattice vibrations, charge and electronic spin waves) only those spectral components very closely matching the muon precession frequency in the specific experimental condition can cause a significant muon spin motion.</ref> techniques, such as [[electron spin resonance]] (ESR or EPR) and, more closely, [[nuclear magnetic resonance]] (NMR).

== Acronym ==
In analogy with the acronyms for these previously established spectroscopies, the muon spin spectroscopy is also known as µSR, which stands for '''muon spin rotation''', or relaxation, or resonance, depending respectively on whether the muon spin motion is predominantly a rotation (more precisely a [[precession]] around a still [[magnetic field]]), or a relaxation towards an equilibrium direction, or, again, a more complex dynamics dictated by the addition of short [[radio frequency]] pulses. The intention of the mnemonic acronym was to draw attention to the analogy with [[NMR]] and [[electron paramagnetic resonance|ESR]]. More generally speaking, the abbreviation covers any study of the interactions of the muon's magnetic moment with its surrounding when implanted into any kind of matter.

== How it works ==

===Introduction===
µSR is an atomic, molecular and condensed matter experimental technique that exploits nuclear detection methods. Despite the fact that particles are used as a probe it is not a diffraction technique. Its two main features are the local nature of the muon probe, due to the short effective range of its interactions with matter, and the characteristic time-window (10−13 - 10−5 s) of the dynamical processes in atomic, molecular and condensed media that can be investigated by this technique. The closest parallel to µSR is "pulsed NMR", in which one observes time-dependent transverse nuclear polarization or the so-called "[[free induction decay]]" of the nuclear
polarization. However, a key difference is the fact that in µSR one uses a specifically implanted spin (the muon's) and does not rely on internal nuclear spins.

In addition, and due to the specificity of the muon, the µSR technique does not require any [[radio-frequency]] technique to align the probing spin. On the other hand, a clear distinction between the µSR technique and those involving neutrons or [[x-rays]] is that scattering is not involved. [[Neutron diffraction]] techniques, for example, use the change in energy and/or momentum of a scattered [[neutron]] to deduce the sample properties. In contrast, the implanted muons are not diffracted but remain in a sample until they decay. Only a careful analysis of the decay product (i.e. a [[positron]]) provides information about the interaction between the implanted muon and its environment in the sample.

As many of the other nuclear methods, µSR relies on discoveries and developments made in the field of particle physics. Following the discovery of the muon by [[Seth Neddermeyer]] and [[Carl D. Anderson]] in 1936, pioneer experiments on its properties were performed with [[cosmic rays]]. Indeed, with one muon hitting each square centimeter of the earth's surface every minute, the muons constitute the foremost constituent of cosmic rays arriving at ground level. However, µSR experiments require muon fluxes of the order of <math>10^4-10^7</math> muons per second and square centimeter. Such fluxes can only be obtained in high-energy [[particle accelerators]] which have been developed during the last 50 years.

===Muon production===
The collision of an accelerated proton beam (typical energy 600 MeV) with the nuclei of a production target produces positive pions (<math>\pi^+</math>) via the possible reactions:
:<math> \begin{array}{lll}
p + p & \rightarrow & p + n + \pi^+\\
p + n & \rightarrow & n + n + \pi^+\\
\end{array}
</math>
From the subsequent weak decay of the [[pions]] (mean lifetime <math>\tau_{\pi^+}</math> = 26.03 ns) positive muons (<math>\mu^+</math>) are formed via the [[Two_body_decay#Two-body_decay|two body decay]]:
:<math>
\pi^+ \rightarrow \mu^+ + \nu_{\mu} .
</math>

[[Parity_(physics)#Parity violation|Parity violation]] in the weak interactions implies that only left-handed neutrinos exist, with their [[spin (physics)|spin]] antiparallel to their linear momentum (likewise only right-handed anti-neutrino are found in nature). Since the pion is spinless both the neutrino and the <math>\mu^+</math> are ejected with spin antiparallel to their momentum in the pion rest frame. This is the key to provide spin-polarised muon beams. According to the value of the pion momentum different types of <math>\mu^+</math>-beams are available for µSR measurements.

====High-energy beam====
The first type of muon beam is formed by the pions escaping the production target at high energies. They are collected over a certain solid angle by [[quadrupole magnets]] and directed on to a decay section consisting of a long superconducting solenoid with a field of several Tesla. If the pion momentum is not too high, a large fraction of the pions will have decayed before they reach the end of the solenoid.

In the laboratory frame the polarization of a high-energy muon beam is limited to about 80% and its energy is of the order of ~40-50MeV. Although such a high energy beam requires the use of suitable moderators and samples with sufficient thickness, it guarantees a homogeneous implantation of the muons in the sample volume. Such beams are also used to study specimens inside of recipients, e.g. samples inside pressure cells.

Such muon beams are available at [[Paul Scherrer Institute|PSI]], [[TRIUMF]], [[J-PARC]] and [http://riken.nd.rl.ac.uk/ral.html RIKEN-RAL].

====Surface beam====
The second type of muon beam is often called the ''surface'' or ''Arizona'' beam (recalling the pioneer works of Pifer ''et al.''<ref>A.E. Pifer, T. Bowen and K.R. Kendall, Nuclear Instruments and Methods '''135''', 39 (1976), http://dx.doi.org/10.1016/0029-554X(76)90823-5</ref> from the [[University of Arizona]]). Here muons are used that arise from pions decaying at rest still inside, but near the surface, of the production target. Such muons, which are 100% polarized, ideally monochromatic and have a very low momentum of 29.8 MeV/c, which corresponds to a kinetic energy of 4.1 MeV, have a range width in matter of the order of 180 mg/cm2. Hence the paramount advantage of this type of beam is the possibility to use relatively thin samples.

Such muon beams are available at [[Paul Scherrer Institute|PSI]] (Swiss Muon Source SµS), [[TRIUMF]], [[J-PARC]], [[ISIS]] and [http://riken.nd.rl.ac.uk/ral.html RIKEN-RAL].

====Low-energy muon beam====
Finally, muon beams of even lower energy (''ultra slow muons'' with energy down to the eV-keV range) can be obtained by further reducing the energy of an Arizona beam using moderators, as a thin layer of a van der Waals gas frozen on a substrate. The tunable energy range of such muon beams corresponds to implantation depths in solids of less than a nanometer up to several hundred nanometers. Therefore the study of magnetic properties as a function of the distance from the surface of the sample is possible.

Up to now, [[Paul Scherrer Institute|PSI]] is the only facility where such low-energy muon beam is available on a regular basis. Technical developments have been also conducted at [[RIKEN-RAL]], but with a strongly reduced low-energy muons rate. [[J-PARC]] is projecting the development of a high-intensity low-energy muon beam.

===Different types of muon sources: ''continuous'' vs. ''pulsed''===
In addition to the above mentioned classification based on energy, muon beams are also divided according to the time structure of the particle accelerator, i.e. continuous or pulsed.

For ''continuous'' muon sources no dominating time structure is present. By selecting an appropriate muon incoming rate, muons are implanted into the sample one by one. The main advantage is that the time resolution is solely determined by the detector construction and the read-out electronics. There are two main limitations for this type of sources: (i) unrejected charged particles accidentally hitting the detectors produce non-negligible random background counts; this compromises measurements after a few muon lifetimes, when the random background exceeds the true decay events; (ii) the requirement to detect muons one at a time sets a maximum event rate. The background problem can be reduced by the use of electrostatic deflectors to ensure that no muons enter the sample before the decay of the previous muon.
[[Paul Scherrer Institute|PSI]] and [[TRIUMF]] host the two continuous muon sources available for µSR experiments.

At ''pulsed'' muon sources [[protons]] hitting the production target are bunched into short, intense and widely separated pulses, that provide a similar time structure in the secondary muon beam. An advantage of pulsed muon sources is that the event rate is only limited by detectors construction. Furthermore detectors are active only after the incoming muon pulse, strongly reducing the accidental background counts. The virtual absence of background allows the extension of the time window for measurements up to about ten times the muon mean lifetime. The reverse of the medal is that the width of the muon pulse limits the time resolution.
[[ISIS]] and [[J-PARC]] are the two ''pulsed'' muon sources available for µSR experiments.

===The technique===

====Muon implantation====
The muons are implanted into the sample of interest where they lose energy very quickly. Fortunately, this deceleration process occurs in such a way that it does not jeopardize a μSR measurement. On one side it is very fast (much faster than 100 ps), which is much shorter than a typical μSR time window (up to 20 μs), and on the other side, all the processes involved during the deceleration are Coulombic ([[ionization]] of atoms, [[electron scattering]], [[electron capture]]) in origin and do not interact with the muon spin, so that the muon is thermalized without any significant loss of polarization.

The positive muons usually adopt interstitial sites of the [[crystallographic lattice]]. In most metallic samples the muon's positive charge is collectively [[screening effect|screened]] by a cloud of conduction electrons. Thus, in metals, the muon is in a so-called diamagnetic state and behave like a free muon. In insulators or semiconductors a collective screening cannot take place and the muon will usually pick-up one electron and form a so-called [[muonium]] (Mu=μ++e-), which has similar size ([[Bohr radius]]), [[Reduced mass|reduced-mass]] and [[ionization energy]] to the [[hydrogen]] atom.

====Detecting the muon polarization====
The decay of the positive muon into a positron and two neutrinos occurs via the weak interaction process after a [[mean lifetime]] of
τμ = 2.197034(21) μs:
:<math>
\mu^+ \rightarrow e^+ + \nu_e + \bar{\nu}_{\mu}~.
</math>
Parity violation in the weak interaction leads in this more complicated case ([[Particle_decay#3-body_decay|three body decay]]) to an anisotropic distribution of the positron emission with respect to the spin direction of the μ+ at the decay time. The positron emission probability is given by
:<math>
W(\theta)d\theta \propto (1 + a\cos\theta)d\theta~,
</math>
where <math>\theta</math> is the angle between the positron trajectory and the μ+-spin, and <math>a</math> is an intrinsic asymmetry parameter determined by the weak decay mechanism. This anisotropic emission constitutes in fact the basics for the μSR technique.

The average asymmetry <math>A </math> is measured over a statistical ensemble of implanted muons and it depends on further experimental parameters, such as the beam spin polarization <math>P_{\mu}</math>, close to one, as [[#High_energy_beams|already mentioned]]. Theoretically <math>A </math> =1/3 is obtained if all emitted positrons are detected with the same efficiency, irrespective of their energy. Practically, values of <math>A</math> ≈ 0.25 are routinely obtained.

The muon spin motion may be measured over a time scale dictated by the [[Muon#Muon_decay|muon decay]], ''i.e.'' a few times τμ, roughly 10 µs. The asymmetry in the muon decay correlates the positron emission and the muon spin directions. The simplest example is when the spin direction of all muons remains constant in time after implantation (no motion). In this case the asymmetry shows up as an unbalance between the positron counts in two equivalent detectors placed in front and behind the sample, along the beam axis. Each of them records an exponentially decaying rate as a function of the time ''t'' elapsed from implantation, according to

:<math>N_\alpha(t)=N_0 \exp(-t/\tau_\mu) (1+\alpha A)</math>

with <math>\alpha=\pm 1</math> for the detector looking towards and away from the spin arrow, respectively. Considering that the huge muon spin polarization is completely outside thermal equilibrium, a dynamical relaxation towards the equilibrium unpolarized state typically shows up in the count rate, as an additional decay factor in front of the experimental asymmetry parameter, ''A''. A magnetic field parallel to the initial muon spin direction probes the dynamical relaxation rate as a function of the additional muon [[Zeeman energy]], without introducing additional coherent spin dynamics. This experimental arrangement is called Longitudinal Field (LF) μSR.

Another simple example is when implanted all muon spins precess coherently around the same magnetic field of modulus <math>B</math>, perpendicular to the beam axis, causing the count unbalance to oscillate at the corresponding [[Larmor]] frequency <math>\omega</math> between the same two detectors, according to

:<math>N_\alpha(t)=N_0 \exp(-t/\tau_\mu) (1+\alpha A\cos\omega t)</math>

Since the Larmor frequency is <math>\omega=\gamma_\mu B</math>, with a gyromagnetic ratio <math>\gamma_\mu=851.616</math> Mrad(sT)−1, the frequency spectrum obtained by means of this experimental arrangement (usually referred to as Transverse Field, TF μSR) provides a direct measure of the internal magnetic field intensity distribution.

== Applications ==
Muon spin rotation and relaxation are mostly performed with positive muons. They are well suited to the study of [[magnetic field]]s at the atomic scale inside matter, such as those produced by various kinds of [[magnetism]] and/or [[superconductivity]] encountered in compounds occurring in nature or artificially produced by modern [[material science]].

The London penetration depth is one of the most important parameters characterizing a [[superconductivity|superconductor]] because its inverse square provides a measure of the density ''ns'' of [[Cooper pairs]]. The dependence of ''ns'' on temperature and magnetic field directly indicates the symmetry of the superconducting gap. Muon spin spectroscopy provides a way to measure the penetration depth, and so has been used to study high-temperature cuprate superconductors since their discovery in 1986.

Other important fields of application of µSR exploit the fact that positive muons capture electrons to form [[muonium]] atoms which behave chemically as light [[isotope]]s of the [[hydrogen]] atom. This allows investigation of the largest known [[kinetic isotope effect]] in some of the simplest types of chemical reactions, as well as the early stages of formation of [[radical (chemistry)|radical]]s in organic chemicals. Muonium is also studied as an analogue of hydrogen in [[semiconductor]]s, where hydrogen is one of the most ubiquitous impurities.

== Facilities ==
µSR requires a [[particle accelerator]] for the production of a muon beam. This is presently achieved at few large scale facilities in the world: the CMMS continuous source at [[TRIUMF]] in Vancouver, Canada; the SµS continuous source at the [[Paul Scherrer Institut]] (PSI) in Villigen, Switzerland; the [[ISIS neutron source|ISIS]] and RIKEN-RAL pulsed sources at the [[Rutherford Appleton Laboratory]] in Chilton, United Kingdom; and the [[J-PARC]] facility in Tokai, Japan, where a new pulsed source is being built to replace that at [[KEK]] in Tsukuba, Japan.
Muon beams are also available at the Laboratory of Nuclear Problems, [[Joint Institute for Nuclear Research]] (JINR) in Dubna, Russia.
The International Society for µSR Spectroscopy (ISMS) exists to promote the worldwide advancement of µSR. Membership in the society is open free of charge to all individuals in academia, government laboratories and industry who have an interest in the society's goals.

==See also==
*[[Muon]]
*[[Muonium]]
*[[Nuclear magnetic resonance]]

==References==

<references />

==External links==
* [http://musr.org/intro/musr/intro.htm introduction to µSR]
* [http://muon.neutron-eu.net/muon/files/muSRBrochure.pdf µSR Brochure] (a 3.2 MB PDF file)
* [http://nmi3.eu Integrated Infrastructure Initiative for Neutron Scattering and Muon Spectroscopy (NMI3)]
* [http://nmi3.eu/about-nmi3/joint-research-activities/muons.html The NMI3 Muon Joint Research Activity]
* [http://www.youtube.com/watch?v=wHCSifl_SGQ Video - What are muons and how are they produced?]

[[Category:Spectroscopy]]
[[Category:Scientific techniques]]

Rusty bolt effect

2013-09-04T00:32:49Z

46.226.186.33: /* Mixing product generation */

In [[mathematics]], specifically in [[algebraic geometry]] and [[algebraic topology]], the '''Lefschetz hyperplane theorem''' is a precise statement of certain relations between the shape of an [[algebraic variety]] and the shape of its subvarieties. More precisely, the theorem says that for a variety ''X'' embedded in [[projective space]] and a [[hyperplane section]] ''Y'', the [[homology (mathematics)|homology]], [[cohomology]], and [[homotopy group]]s of ''X'' determine those of ''Y''. A result of this kind was first stated by [[Solomon Lefschetz]] for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.

== The Lefschetz hyperplane theorem for complex projective varieties ==
Let ''X'' be an ''n''-dimensional complex projective algebraic variety in '''CP'''''N'', and let ''Y'' be a hyperplane section of ''X'' such that {{nowrap begin}}''U'' = ''X'' ∖ ''Y''{{nowrap end}} is smooth. The Lefschetz theorem refers to any of the following statements:<ref>{{Harvnb|Milnor|1969|loc = Theorem 7.3 and Corollary 7.4}}</ref><ref>{{Harvnb|Voisin|2003|loc = Theorem 1.23}}</ref>
# The natural map in {{nowrap begin}}''H''''k''(''Y'', '''Z''') → ''H''''k''(''X'', '''Z'''){{nowrap end}} in singular homology is an isomorphism for {{nowrap begin}}''k'' < ''n'' − 1{{nowrap end}} and is surjective for {{nowrap begin}}''k'' = ''n'' − 1{{nowrap end}}.
# The natural map in {{nowrap begin}}''H''''k''(''X'', '''Z''') → ''H''''k''(''Y'', '''Z'''){{nowrap end}} in singular cohomology is an isomorphism for {{nowrap begin}}''k'' < ''n'' − 1{{nowrap end}} and is injective for {{nowrap begin}}''k'' = ''n'' − 1{{nowrap end}}.
# The natural map {{nowrap begin}}π''k''(''Y'', '''Z''') → π''k''(''X'', '''Z'''){{nowrap end}} is an isomorphism for {{nowrap begin}}''k'' < ''n'' − 1{{nowrap end}} and is surjective for {{nowrap begin}}''k'' = ''n'' − 1{{nowrap end}}.
Using a [[long exact sequence]], one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants. In order, these are:
# The relative singular homology groups {{nowrap begin}}''H''''k''(''X'', ''Y'', '''Z'''){{nowrap end}} are zero for <math>k \leq n-1</math>.
# The relative singular cohomology groups {{nowrap begin}}''H''''k''(''X'', ''Y'', '''Z'''){{nowrap end}} are zero for <math>k \leq n-1</math>.
# The relative homotopy groups {{nowrap begin}}π''k''(''X'', ''Y''){{nowrap end}} are zero for <math>k \leq n-1</math>.

=== Lefschetz's proof ===
Lefschetz<ref>{{harvnb|Lefschetz|1924}}</ref> used his idea of a [[Lefschetz pencil]] to prove the theorem. Rather than considering the hyperplane section ''Y'' alone, he put it into a family of hyperplane sections ''Y''''t'', where {{nowrap begin}}''Y'' = ''Y''0{{nowrap end}}. Because a generic hyperplane section is smooth, all but a finite number of ''Y''''t'' are smooth varieties. After removing these points from the ''t''-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topological trivial. That is, it is a product of a generic ''Y''''t'' with an open subset of the ''t''-plane. ''X'', therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points. Away from the singular points, the identification can be described inductively. At the singular points, the [[Morse lemma]] implies that there is a choice of coordinate system for ''X'' of a particularly simple form. This coordinate system can be used to prove the theorem directly.<ref>{{harvnb|Griffiths|Spencer|Whitehead|1992}}</ref>

=== Andreotti and Frankel's proof ===
Andreotti and Frankel<ref>{{Harvnb|Andreotti|Frankel|1959}}</ref> recognized that Lefschetz's theorem could be recast using [[Morse theory]].<ref>{{Harvnb|Milnor|1969|p=39}}</ref> Here the parameter ''t'' plays the role of a Morse function. The basic tool in this approach is the [[Andreotti–Frankel theorem]], which states that a complex [[affine variety]] of complex dimension ''n'' (and thus real dimension 2''n'') has the homotopy type of a [[CW-complex]] of (real) dimension ''n''. This implies that the [[relative homology]] groups of ''Y'' in ''X'' are trivial in degree less than ''n''. The long exact sequence of relative homology then gives the theorem.

=== Thom's and Bott's proofs ===
Neither Lefschetz's proof nor Andreotti and Frankel's proof directly imply the Lefschetz hyperplane theorem for homotopy groups. An approach that does was found by Thom no later than 1957 and was simplified and published by Bott in 1959.<ref>{{harvnb|Bott|1959}}</ref> Thom and Bott interpret ''Y'' as the vanishing locus in ''X'' of a section of a line bundle. An application of Morse theory to this section implies that ''X'' can be constructed from ''Y'' by adjoining cells of dimension ''n'' or more. From this, it follows that the relative homology and homotopy groups of ''Y'' in ''X'' are concentrated in degrees ''n'' and higher, which yields the theorem.

=== Kodaira and Spencer's proof for Hodge groups ===
Kodaira and Spencer found that under certain restrictions, it is possible to prove a Lefschetz-type theorem for the Hodge groups ''H''''p'',''q''. Specifically, assume that ''Y'' is smooth and that the line bundle <math>\mathcal{O}_X(Y)</math> is ample. Then the restriction map {{nowrap begin}}''H''''p'',''q''(''X'') → ''H''''p'',''q''(''Y''){{nowrap end}} is an isomorphism if {{nowrap|''p'' + ''q'' < n − 1}} and is injective if {{nowrap begin}}''p'' + ''q'' = ''n'' − 1{{nowrap end}}.<ref>{{harvnb|Lazarsfeld|2004|loc = Example 3.1.24}}</ref><ref>{{harvnb|Voisin|2003|loc = Theorem 1.29}}</ref> By Hodge theory, these cohomology groups are equal to the sheaf cohomology groups <math>H^q(X, \textstyle\bigwedge^p\Omega_X)</math> and <math>H^q(Y, \textstyle\bigwedge^p\Omega_Y)</math>. Therefore the theorem follows from applying the [[Akizuki–Nakano vanishing theorem]] to <math>H^q(X, \textstyle\bigwedge^p\Omega_X|_Y)</math> and using a long exact sequence.

Combining this proof with the [[universal coefficient theorem]] nearly yields the usual Lefschetz theorem for cohomology with coefficients in any field of characteristic zero. It is, however, slightly weaker because of the additional assumptions on ''Y''.

=== Artin and Grothendieck's proof for constructible sheaves ===
[[Michael Artin]] and [[Alexander Grothendieck]] found a generalization of the Lefschetz hyperplane theorem to the case where the coefficients of the cohomology lie not in a field but instead in a [[constructible sheaf]]. They prove that for a constructible sheaf ''F'' on an affine variety ''U'', the cohomology groups {{nowrap|''H''''k''(''U'', ''F'')}} vanish whenever {{nowrap|''k'' > ''n''}}.<ref>{{harvnb|Lazarsfeld|2003|loc=Theorem 3.1.13}}</ref>

== The Lefschetz theorem in other cohomology theories ==
The motivation behind Artin and Grothendieck's proof for constructible sheaves was to give a proof that could be adapted to the setting of étale and <math>\ell</math>-adic cohomology. Up to some restrictions on the constructible sheaf, the Lefschetz theorem remains true for constructible sheaves in positive characteristic.

The theorem can also be generalized to [[intersection homology]]. In this setting, the theorem holds for highly singular spaces.

A Lefschetz-type theorem also holds for [[Picard group]]s.<ref>{{harvnb|Lazarsfeld|2003|loc = Example 3.1.25}}</ref>

==Hard Lefschetz theorem==
{{See also| Lefschetz manifold}}

Let ''X'' be a ''n''-dimensional non-singular complex projective variety in '''CP'''''N''.
Then in the [[cohomology ring]] of ''X'', the ''k''-fold product with the [[cohomology class]] of a hyperplane gives an isomorphism between

:''H''''n'' − ''k''

and

:''H''''n'' + ''k''.

This is the '''hard Lefschetz theorem''', christened in French by Grothendieck more colloquially as the ''Théorème de Lefschetz vache''.<ref>{{harvnb|Beauville|}}</ref><ref>{{harvnb|Sabbah|2001}}</ref> It immediately implies the injectivity part of the Lefschetz hyperplane theorem.

The hard Lefschetz theorem in fact holds for '''any compact [[Kähler manifold]]''', with the isomorphism in de Rham cohomology given by multiplication by a power of the class of the Kähler form. It can fail for non-Kähler manifolds: for example, [[Hopf surface]]s have vanishing second cohomology groups, so there is no analogue of the second cohomology class of a hyperplane section.

The hard Lefschetz theorem was proven for [[etale cohomology|''l''-adic cohomology]] of smooth projective varieties over finite fields by {{harvtxt|Deligne|1980}} as a consequence of his work on the [[Weil conjectures]].

== References ==
<references/>

== Bibliography ==
* {{Citation | last1=Andreotti | first1=Aldo | last2=Frankel | first2=Theodore | title=The Lefschetz theorem on hyperplane sections | id={{MathSciNet | id = 0177422}} | year=1959 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=69 | pages=713–717}}
* {{Citation | last1=Beauville | title=The Hodge Conjecture | id = {{citeseerx|10.1.1.74.2423}} }}
* {{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | title=On a theorem of Lefschetz | url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1028998225 | accessdate=2010-01-30 | id={{MathSciNet | id=0215323}} | year=1959 | journal=Michigan Mathematical Journal | volume=6 | issue=3 | pages=211–216}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=La conjecture de Weil. II | url=http://www.numdam.org/item?id=PMIHES_1980__52__137_0 | id={{MathSciNet | id = 601520}} | year=1980 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=52 | pages=137–252}}
* {{Citation | last1=Griffiths | first1=Philip | last2=Spencer | first2=Donald | last3=Whitehead | first3=George | editorlast=National Academy of Sciences | editorfirst=Office of the Home Secretary | title= Biographical Memoirs | volume=61 | chapter=Solomon Lefschetz | publisher=The National Academies Press | year=1992 | isbn= 978-0-309-04746-3}}
* {{Citation | last1=Lazarsfeld | first1=Robert | title=Positivity in algebraic geometry. I | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] | isbn=978-3-540-22533-1 | id={{MathSciNet | id = 2095471}} | year=2004 | volume=48}}
* {{Citation | last1=Lefschetz | first1=Solomon | title=L'Analysis situs et la géométrie algébrique | publisher=Gauthier-Villars | language=French | series=Collection de Monographies publiée sous la Direction de M. Émile Borel | location=Paris | year=1924}} Reprinted in {{Citation | last1=Lefschetz | first1=Solomon | title=Selected papers | publisher=Chelsea Publishing Co. | location=New York | isbn=978-0-8284-0234-7 | id={{MathSciNet | id = 0299447}} | year=1971}}
* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Morse theory | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies, No. 51 | id={{MathSciNet | id = 0163331}} | year=1963}}
* {{Citation | last1=Sabbah | title=Theorie de Hodge et theoreme de Lefschetz "difficile" | year=2001 | url=http://www.math.polytechnique.fr/cmat/sabbah/hodge-str.pdf}}
* {{Citation | last1=Voisin | first1=Claire | title=Hodge theory and complex algebraic geometry. II | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-80283-3 | id={{MathSciNet | id = 1997577}} | year=2003 | volume=77}}

[[Category:Topological methods of algebraic geometry]]
[[Category:Morse theory]]
[[Category:Theorems in algebraic geometry]]
[[Category:Theorems in algebraic topology]]

S/KEY

2013-07-24T01:23:13Z

46.226.190.244:

The '''metabolic theory of ecology''' (MTE) is an extension of [[Kleiber's law]] and posits that the [[Basal metabolic rate|metabolic rate]] of organisms is the fundamental biological rate that governs most observed patterns in ecology.<ref name="Brown04">{{cite journal |author=Brown, J. H., Gillooly, J. F., Allen, A. P., Savage, V. M., & G. B. West |title=Toward a metabolic theory of ecology |journal=Ecology |volume=85 |issue=7 |pages=1771–89 |year=2004 |doi=10.1890/03-9000
}}</ref>

MTE is based on an interpretation of the relationships between body size, body temperature, and [[Metabolism|metabolic rate]] across all organisms. Small-bodied organisms tend to have higher mass-specific metabolic rates than larger-bodied organisms. Furthermore, organisms that operate at warm temperatures through [[Warm-blooded|endothermy]] or by living in warm environments tend towards higher metabolic rates than organisms that operate at colder temperatures. This pattern is consistent from the unicellular level up to the level of the largest animals on the planet.

In MTE, this relationship is considered to be the single constraint that defines biological processes at all levels of organization (from individual up to ecosystem level), and is a [[Macroecology|macroecological]] theory that aims to be universal in scope and application.<ref name="Brown04"/>

== Theoretical background ==

Metabolic rate scales with the mass of an organism of a given species according to [[Kleiber's law]] where ''B'' is whole organism metabolic rate (in watts or other unit of power), ''M'' is organism mass (in kg), and ''B''o is a mass-independent normalization constant (given in a unit of power divided by a unit of mass. In this case, watts per kilogram):

:<math>B = B_oM ^ {3/4}\,</math>

At increased temperatures, chemical reactions proceed faster. This relationship is described by the [[Boltzmann factor]], where ''E'' is [[activation energy]] in [[electronvolt]]s or [[joule]]s, ''t'' is absolute temperature in kelvins, and ''k'' is the [[Boltzmann constant]] in eV/K or J/K:

:<math>e^{-\frac{E}{k\,t}}</math>

While ''B''o in the previous equation is mass-independent, it is not explicitly independent of temperature. To explain the relationship between body mass and temperature, these two equations are combined to produce the primary equation of the MTE, where ''b''o is a normalization constant that is independent of body size or temperature:

:<math>B = b_oM^{3/4}e^{-\frac{E}{k\,t}}</math>

According to this relationship, metabolic rate is a function of an organism’s body mass and body temperature. By this equation, large organisms have proportionally higher metabolic rates (in Watts) than small organisms, and organisms at high body temperatures have higher metabolic rates than those that exist at low body temperatures.However specific metabolic rate (SMR, in Watts/kg) is given by

<math>SMR = (B/M) = b_oM^{-1/4}e^{-\frac{E}{k\,t}}</math>

Hence SMR for large organisms are lower than small organisms.

== Controversy over exponent ==

There is disagreement amongst researchers about the most accurate value for use in the power function, and whether the factor is indeed universal.<ref name="Agutter04">{{cite journal |author=Agutter, P.S., Wheatley, D.N. |title=Metabolic scaling: consensus or controversy? |journal=Theoretical biology and medical modelling |volume=1 |issue= |pages=13 |year=2004 |doi=10.1186/1742-4682-1-13 |pmid=15546492 |pmc=539293}}</ref> The main disagreement is whether metabolic rate scales to the power of 3/4 or 2/3. The majority view is currently that 3/4 is the correct exponent, but a large minority believe that 2/3 is the more accurate value.<ref name="Agutter04"/> Although a rigorous exploration of the controversy over choice of scaling factor is beyond the scope of this article, it is informative to understand the biological justification for the use of either value.

The argument that 2/3 should be the correct scaling factor is based on the assumption that energy dissipation across the surface area of three dimensional organisms is the key factor driving the relationship between metabolic rate and body size.<ref name="Agutter04"/> Smaller organisms tend to have higher surface area to volume ratios, causing them to lose heat energy at a faster rate than large organisms. As a consequence, small organisms must have higher specific metabolic rates to combat this loss of energy over their large surface area to volume ratio.

In contrast, the argument for a 3/4 scaling factor is based on a hydraulic model of energy distribution in organisms, where the primary source of energy dissipation is across the membranes of internal distribution networks. This model is based on the idea that metabolism is essentially the rate at which an organism’s distribution networks (such as circulatory systems in animals or xylem and phloem in plants) deliver nutrients and energy to body tissues.<ref name="West99">{{cite journal |author=West, G.B., Brown, J.H., & Enquist, B.J. |title=The fourth dimension of life: Fractal geometry and allometric scaling of organisms |journal=Science |volume=284 |issue=5420 |pages=1677–9 |year=1999 |pmid=10356399 |doi=10.1126/science.284.5420.167 }}</ref> It therefore takes longer for large organisms to distribute nutrients throughout the body and thus they have a slower metabolic rate. The 3/4 factor is then derived from the observation that selection favors a [[fractal]] or near-fractal distribution network for space-filling circulatory systems.<ref name="West99"/> All fractal networks terminate in identical units (such as capillary beds), and the number of such units in organisms is proportional to a 3/4 power relationship with body size.<ref name="West99"/>

Kolokotrones et al. 2010<ref name="Kolokotrones10">{{cite journal |author=Kolokotrones, T., Van Savage, Deeds, E. J.|title=Curvature in metabolic scaling |journal=Nature |volume=464 |issue=7289 |pages=753–756 |year=2010 |doi=10.1038/nature08920
}}</ref> showed that relationship between mass and metabolic rate has a convex curvature on logarithmic scale. The curvature explains the variations in the power law exponent.

Despite the controversy over the value of the exponent, the implications of this theory might remain true regardless of its precise numerical value.

== Implications of the theory ==
The metabolic theory of ecology’s main implication is that metabolic rate, and the influence of body size and temperature on metabolic rate, provide the fundamental constraints by which ecological processes are governed. If this holds true from the level of the individual up to ecosystem level processes, then life history attributes, population dynamics, and ecosystem processes could be explained by the relationship between metabolic rate, body size, and body temperature.

===Organism level===
Small animals tend to grow fast, breed early, and die young.<ref name="Savage04">{{cite journal |author=Savage V.M., Gillooly J.F., Brown J.H., West G.B. & Charnov E.L. |title=Effects of body size and temperature on population growth |journal=American Naturalist |volume=163 |issue=3 |pages=429–441 |year=2004 |pmid=15026978 |doi=10.1086/381872 }}</ref> According to MTE, these patterns in [[Life history theory|life history]] traits are constrained by metabolism. An organism's metabolic rate determines its rate of food consumption, which in turn determines its rate of growth. This increased growth rate produces trade-offs that accelerate [[senescence]]. For example, metabolic processes produce [[Radical (chemistry)|free radicals]] as a by-product of energy production.<ref>{{cite book |editor=Enrique Cadenas, Lester Packer |title=Understanding the process of ages : the roles of mitochondria, free radicals, and antioxidants |publisher=Marcel Dekker |location=New York |year=1999 |isbn=0-8247-1723-6 }}</ref> These in turn cause damage at the cellular level, which promotes senescence and ultimately death. Selection favors organisms which best propagate given these constraints. As a result, smaller, shorter lived organisms tend to reproduce earlier in their life histories.

===Population and community level===
MTE has profound implications for the interpretation of population growth and community diversity.<ref name="Savage04"/> Classically, species are thought of as being either ''r'' selected (where population size is limited by the exponential rate of population growth) or ''K'' selected (where population size is limited by carrying capacity). MTE explains this diversity of reproductive strategies as a consequence of the metabolic constraints of organisms. Small organisms and organisms that exist at high body temperatures tend to be ''r'' selected, which fits with the prediction that ''r'' selection is a consequence of metabolic rate.<ref name="Brown04"/> Conversely, larger and cooler bodied animals tend to be ''K'' selected. The relationship between body size and rate of population growth has been demonstrated empirically,<ref>{{cite journal |author=Denney N.H., Jennings S. & Reynolds J.D. |title=Life history correlates of maximum population growth rates in marine fishes |journal=Proceedings of the Royal Society of London B |volume=269 |issue= 1506|pages=2229–37 |year=2002 |doi=10.1098/rspb.2002.2138}}</ref> and in fact has been shown to scale to ''M''-1/4 across taxonomic groups.<ref name="Savage04"/> The optimal population growth rate for a species is therefore thought to be determined by the allometric constraints outlined by the MTE, rather than strictly as a life history trait that is selected for based on environmental conditions.

Observed patterns of diversity can be similarly explained by MTE. It has long been observed that there are more small species than large species.<ref>{{cite journal |author=Hutchinson, G., MacArthur, R. |title=A theoretical ecological model of size distributions among species of animals |journal=Am. Nat. |volume=93 |issue= 869|pages=117–125 |year=1959 |doi=10.1086/282063}}</ref> In addition, there are more species in the tropics than at higher latitudes.<ref name="Brown04"/> Classically, the latitudinal gradient in species diversity has been explained by factors such as higher productivity or reduced seasonality.<ref>{{cite journal |author=Rohde, K. |title=Latitudinal gradients in species-diversity: the search for the primary cause |jstor=3545569 |journal=Oikos |volume=65 |issue=3 |pages=514–527 |year=1992 |doi=10.2307/3545569}}</ref> In contrast, MTE explains this pattern as being driven by the kinetic constraints imposed by temperature on metabolism.<ref>{{cite journal |author=Allen A.P., Brown J.H. & Gillooly J.F. |title=Global biodiversity, biochemical kinetics, and the energetic-equivalence rule |journal=Science |volume=297 |issue=5586 |pages=1545–8 |year=2002 |pmid=12202828 |doi=10.1126/science.1072380}}</ref> The rate of molecular evolution scales with metabolic rate,<ref>{{cite journal |author=Gillooly, J.F., Allen, A.P., West, G.B., & Brown, J.H. |title=The rate of DNA evolution: Effects of body size and temperature on the molecular clock |journal=Proc Natl Acad Sci U S A. |volume=102 |issue=1 |pages=140–5 |year=2005 |pmid=15618408 |pmc=544068 |doi=10.1073/pnas.0407735101}}</ref> such that organisms with higher metabolic rates show a higher rate of change at the molecular level.<ref name="Brown04"/> If a higher rate of molecular evolution causes increased speciation rates, then adaptation and ultimately speciation may occur more quickly in warm environments and in small bodied species, ultimately explaining observed [[Body size-species richness|patterns of diversity across body size]] and latitude.

MTE’s ability to explain patterns of diversity remains controversial. For example, researchers analyzed patterns of diversity of New World coral snakes to see whether the geographical distribution of species fit within the predictions of MTE (i.e. more species in warmer areas).<ref>{{cite journal |author=Terribile, L.C., & Diniz-Filho, J.A.F. |title=Spatial patterns of species richness in New World coral snakes and the metabolic theory of ecology |journal=Acta oecologica |volume=35 |issue= 2|pages=163–173 |year=2009 |doi=10.1016/j.actao.2008.09.006}}</ref> They found that the observed pattern of diversity could not be explained by temperature alone, and that other spatial factors such as primary productivity, topographic heterogeneity, and habitat factors better predicted the observed pattern.

===Ecosystem processes===
At the ecosystem level, MTE explains the relationship between temperature and production of biomass. The average production to biomass ratio of organisms is higher in small organisms than large ones.<ref>{{cite journal |author=Banse K. & Mosher S. |title=Adult body mass and annual production/biomass relationships of field populations |jstor=2937256 |journal=Ecol. Monog. |volume=50 |issue=3 |pages=355–379 |year=1980 |doi=10.2307/2937256}}</ref> This relationship is further regulated by temperature, and the rate of production increases with temperature.<ref>{{cite journal |author=Ernest S.K.M., Enquist B.J., Brown J.H., Charnov E.L., Gillooly J.F., Savage V.M., White E.P., Smith F.A., Hadly E.A., Haskell J.P., Lyons S.K., Maurer B.A., Niklas K.J. & Tiffney B. |title=Thermodynamic and metabolic effects on the scaling of production and population energy use |journal=Ecology Letters |volume=6 |issue= 11|pages=990–5 |year=2003 |doi=10.1046/j.1461-0248.2003.00526.x}}</ref> As production consistently scales with body mass, MTE predicts that the primary factor that causes differing rates of production between ecosystems is temperature and not the mass of organisms within the ecosystem.<ref name="Brown04"/> This suggests that regions with similar climatic factors would sustain the same primary production, even if standing biomass is different.<ref name="Brown04"/>

== See also ==
* [[Allometry]]
* [[Constructal theory]]
* [[Dynamic energy budget]]
* [[Ecology]]
* [[Evolutionary physiology]]
* [[Occupancy-abundance relationship]]

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

{{modelling ecosystems}}

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2013-07-17T06:40:52Z

46.226.190.244: /* Mathematical approach */

'''Gamma-ray burst emission mechanisms''' are theories that explain how the energy from a [[gamma-ray burst]] progenitor (regardless of the actual nature of the progenitor) is turned into radiation. These mechanisms are a major topic of research as of 2007. Neither the [[light curve]]s nor the early-time spectra of GRBs show resemblance to the radiation emitted by any familiar physical process.

==Compactness problem==
It has been known for many years that ejection of matter at relativistic velocities (velocities very close to the [[speed of light]]) is a necessary requirement for producing the emission in a gamma-ray burst. GRBs vary on such short timescales (as short as milliseconds) that the size of the emitting region must be very small, or else the time delay due to the finite speed of light would "smear" the emission out in time, wiping out any short-timescale behavior. At the energies involved in a typical GRB, so much energy crammed into such a small space would make the system opaque to photon-photon [[pair production]], making the burst far less luminous and also giving it a very different spectrum from what is observed. However, if the emitting system is moving towards Earth at relativistic velocities, the burst is compressed in time (as seen by an Earth observer, due to the [[relativistic Doppler effect]]) and the emitting region inferred from the finite speed of light becomes much smaller than the true size of the GRB (see [[relativistic beaming]]).

==GRBs and internal shocks==
A related constraint is imposed by the ''relative'' timescales seen in some bursts between the short-timescale variability and the total length of the GRB. Often this variability timescale is far shorter than the total burst length. For example, in bursts as long as 100 seconds, the majority of the energy can be released in short episodes less than 1 second long. If the GRB were due to matter moving towards Earth (as the relativistic motion argument enforces), it is hard to understand why it would release its energy in such brief interludes. The generally accepted explanation for this is that these bursts involve the ''collision'' of multiple shells traveling at slightly different velocities; so-called "internal shocks".<ref>{{
cite journal
|bibcode=1994ApJ...430L..93R
|author=Rees, M.J. and Meszaros, P.
|title=Unsteady outflow models for cosmological gamma-ray bursts|journal=The Astrophysical Journal (Letters)|year = 1994 |volume = 430 |pages = L93–L96
|doi=10.1086/187446 |arxiv = astro-ph/9404038 }}</ref> The collision of two thin shells flash-heats the matter, converting enormous amounts of kinetic energy into the
random motion of particles, greatly amplifying the energy release due to all emission mechanisms. Which physical mechanisms are at play in producing the observed photons is still an area of debate, but the most likely candidates appear to be [[synchrotron|synchrotron radiation]] and [[Compton scattering|inverse Compton scattering]].

As of 2007 there is no theory that has successfully described the spectrum of ''all'' gamma-ray bursts (though some theories work for a subset). However, the so-called Band function (named after [[David Louis Band|David Band]]) has been fairly successful at fitting, empirically, the spectra of most gamma-ray bursts:

<math>N(E)= \begin{cases} {E^\alpha \exp \left( { - \frac{E}{{E_0 }}} \right)}, & \mbox{if }E \le (\alpha - \beta) E_0\mbox{ } \\ {\left[{\left( {\alpha - \beta } \right)E_0 } \right]^{\left( {\alpha - \beta } \right)} E^\beta \exp \left( {\beta - \alpha } \right)}, & \mbox{if }E > (\alpha - \beta) E_0\mbox{ } \end{cases}</math>

A few gamma-ray bursts have shown evidence for an additional, delayed emission component at very high energies (GeV and higher). One theory for this emission invokes [[Compton scattering|inverse Compton scattering]]. If a GRB progenitor, such as a [[Wolf-Rayet star]], were to explode within a [[stellar cluster]], the resulting shock wave could generate gamma-rays by scattering photons from neighboring stars. About 30% of known galactic Wolf-Rayet stars, are located in dense clusters of [[O star]]s with intense ultraviolet radiation fields, and the [[Hypernova#Collapsing star|collapsar model]] suggests that WR stars are likely GRB progenitors. Therefore, a substantial fraction of GRBs are expected to occur in such clusters. As the [[relativistic particle|relativistic matter]] ejected from an explosion slows and interacts with ultraviolet-wavelength photons, some photons gain energy, generating gamma-rays.<ref>{{cite journal | title = Powerful GeV emission from a gamma-ray-burst shock wave scattering stellar photons | author = Giannios, Dimitrios | journal = Astronomy & Astrophysics | volume = 488 | issue = 2 | pages = L55 | year = 2008 | arxiv = 0805.0258 | doi = 10.1051/0004-6361:200810114 | bibcode=2008A&A...488L..55G}}</cite>
</ref>

==Afterglows and external shocks==
The GRB itself is very rapid, lasting from less than a second up to a few minutes at most. Once it disappears, it leaves behind a counterpart at longer wavelengths (X-ray, UV, optical, infrared, and radio) known as the afterglow<ref>{{
cite journal
|bibcode=1997ApJ...476..232M
|author=Meszaros, P. and Rees, M.J.|year=1997|title=Optical and Long-Wavelength Afterglow from Gamma-Ray Bursts
|journal=The Astrophysical Journal
|volume=476
|issue=1
|pages=232–237
|doi=10.1086/303625
|arxiv = astro-ph/9606043 }}</ref> that generally remains detectable for days or longer.

In contrast to the GRB emission, the afterglow emission is not believed to be dominated by internal shocks. In general, all the ejected matter has by this time coalesced into a single shell traveling outward into the [[interstellar medium]] (or possibly the [[stellar wind]]) around the star. At the front of this shell of matter is a shock wave referred to as the "external shock"<ref>{{
cite journal
|bibcode=1992MNRAS.258P..41R
|author=Rees, M.J. and Meszaros, P.
|year=1992
|title=Relativistic fireballs - Energy conversion and time-scales
|journal=MNRAS |volume=258
|pages=41P–43P
}}</ref> as the still relativistically moving matter ploughs into the tenuous interstellar gas or the gas surrounding the star.

As the interstellar matter moves across the shock, it is immediately heated to extreme temperatures. (How this happens is still poorly understood as of 2007, since the particle density across the shock wave is too low to create a shock wave comparable to those familiar in dense terrestrial environments – the topic of "collisionless shocks" is still largely hypothesis but seems to accurately describe a number of astrophysical situations. Magnetic fields are probably critically involved.) These particles, now relativistically moving, encounter a strong local magnetic field and are accelerated perpendicular to the
magnetic field, causing them to radiate their energy via synchrotron radiation.

Synchrotron radiation is well-understood and the afterglow spectrum has been modeled fairly successfully using this template.<ref>{{
cite journal
|bibcode=1998ApJ...497L..17S
|author=Sari, R.; Piran, T.; Narayan, R.
|year=1998
|title=Spectra and Light Curves of Gamma-Ray Burst Afterglows
|journal=Astrophysical Journal Letters
|volume=497
|issue=5
|pages=L17
|doi=10.1086/311269
|arxiv = astro-ph/9712005 }}</ref> It is generally dominated by [[electrons]] (which move and therefore radiate much faster than [[protons]] and other particles) so radiation from other particles is generally ignored.

In general, the GRB assumes the form of a power-law with three break points (and therefore four different power-law segments.) The lowest break point, <math>\nu_a</math>, corresponds to the frequency below which the GRB is opaque to radiation and so the spectrum attains the form Raleigh-Jeans tail of [[blackbody radiation]]. The two other break points, <math>\nu_m</math> and <math>\nu_c</math>, are related to the minimum energy acquired by an electron after it crosses the shock wave and the time it takes an electron to radiate most of its energy, respectively. Depending on which of these two frequencies is higher, two different regimes are possible:<ref>{{
cite journal
|url=http://arxiv.org/astro-ph/0405503
|author=Piran, T
|year=1994
|title=Physics of Gamma-Ray Bursts
|journal=Arxiv astrophysics}}</ref>
* '''Fast cooling''' (<math>\nu_m > \nu_c</math>) - Shortly after the GRB, the shock wave imparts immense energy to the electrons and the minimum electron Lorentz factor is very high. In this case, the spectrum looks like:
<math>F_\nu \propto \begin{cases} {\nu^{2}}, & \nu<\nu_a \\
{\nu^{1/3}}, & \nu_a<\nu<\nu_c \\
{\nu^{-1/2}}, & \nu_c<\nu<\nu_m \\
{\nu^{-p/2}}, & \nu_m<\nu
\end{cases}</math>
* '''Slow cooling''' (<math>\nu_m < \nu_c</math>) – Later after the GRB, the shock wave has slowed down and the minimum electron Lorentz factor is much lower.:
<math>F_\nu \propto \begin{cases} {\nu^{2}}, & \nu<\nu_a \\
{\nu^{1/3}}, & \nu_a<\nu<\nu_m \\
{\nu^{-(p-1)/2}}, & \nu_m<\nu<\nu_c \\
{\nu^{-p/2}}, & \nu_c<\nu
\end{cases}</math>

The afterglow changes with time. It must fade, obviously, but the spectrum changes as well. For the simplest case of [[adiabatic]] expansion into a uniform-density medium, the critical parameters evolve as:

<math>\nu_c \propto t^{1/2}</math>

<math>\nu_m \propto t^{-3/2}</math>

<math>F_{\nu,max} = const</math>

Here <math>F_{\nu,max}</math> is the flux at the current peak frequency of the GRB spectrum. (During fast-cooling this is at <math>\nu_c</math>; during slow-cooling it is at <math>\nu_m</math>.) Note that because <math>\nu_m</math> drops faster than <math>\nu_c</math>, the system eventually switches from fast-cooling to slow-cooling.

Different scalings are derived for radiative evolution and for a non-constant-density environment (such as a [[stellar wind]]), but share the general power-law behavior observed in this case.

Several other known effects can modify the evolution of the afterglow:

==Reverse shocks and the optical flash ==
There can be "reverse shocks", which propagate ''back'' into the shocked matter once it begins to encounter the interstellar medium.<ref>{{
cite journal
|bibcode = 1993ApJ...418L..59M
|author = Meszaros, P. and Rees, M.J.
|year= 1993
|title = Gamma-Ray Bursts: Multiwaveband Spectral Predictions for Blast Wave Models
|journal =The Astrophysical Journal (Letters)|volume = 418
|pages = L59–L62
|doi = 10.1086/187116
|arxiv = astro-ph/9309011 }}</ref><ref>{{
cite journal
|bibcode=1999ApJ...520..641S
|author=Sari, R.; Piran, T.
|year=1999
|title=Predictions for the Very Early Afterglow and the Optical Flash
|journal=Astrophysical Journal
|volume=520
|issue=2
|pages=641–649
|doi=10.1086/307508
|arxiv = astro-ph/9901338 }}</ref> The twice-shocked material can produce a bright optical/UV flash, which has been seen in a few GRBs,<ref>{{
cite journal
|bibcode=1999Natur.398..400A
|author=Akerlof, C. ''et al.''
|year=1999
|title=Observation of contemporaneous optical radiation from a gamma-ray burst
|journal=Nature
|volume=398
|issue=3
|pages=400–402
|doi=10.1038/18837
|arxiv = astro-ph/9903271 }}</ref> though it appears not to be a common phenomenon.

==Refreshed shocks and late-time flares==
There can be "refreshed" shocks if the central engine continues to release fast-moving matter in small amounts even out to late times, these new shocks will catch up with the external shock to produce something like a late-time internal shock. This explanation has been invoked to explain the frequent flares seen in X-rays and at other wavelengths in many bursts, though some theorists are uncomfortable with the apparent demand that the progenitor (which one would think would be destroyed by the GRB) continues to remain active for very long.

==Jet effects==
Gamma-ray burst emission is believed to be released in jets, not spherical shells.<ref>{{
cite journal
|bibcode=1999ApJ...519L..17S
|author=Sari, R.; Piran, T.; Halpern, J. P.
|year=1999
|title=Jets in Gamma-Ray Bursts
|journal=Astrophysical Journal
|volume=519
|issue=1
|pages=L17–L20
|doi=10.1086/312109
|arxiv = astro-ph/9903339 }}</ref> Initially the two scenarios are equivalent: the center of the jet is not "aware" of the jet edge, and due to [[relativistic beaming]] we only see a small fraction of the jet. However, as the jet slows down, two things eventually occur (each at about the same time): First, information from the edge of the jet that there is no pressure to the side propagates to its center, and the jet matter can spread laterally. Second, relativistic beaming effects subside, and once Earth observers see the entire jet the widening of the relativistic beam is no longer compensated by the fact that we see a larger emitting region. Once these effects appear the jet fades very rapidly, an effect that is visible as a power-law "break" in the afterglow light curve. This is the so-called "jet break" that has been seen in some events and is often cited as evidence for the consensus view of GRBs as jets. Many GRB afterglows do not display jet breaks, especially in the X-ray, but they are more common in the optical light curves. Though as jet breaks generally occur at very late times (~1 day or more) when the afterglow is quite faint, and often undetectable, this is not necessarily surprising.

==Dust extinction and hydrogen absorption==
There may be [[interstellar dust|dust]] along the line of sight from the GRB to Earth, both in the host galaxy and in the [[Milky Way]]. If so, the light will be attenuated and reddened and an afterglow spectrum may look very different from that modeled.

At very high frequencies (far-ultraviolet and X-ray) interstellar hydrogen gas becomes a significant absorber. In particular, a photon with a wavelength of less than 91 nanometers is energetic enough to completely ionize neutral hydrogen and is absorbed with almost 100% probability even through relatively thin gas clouds. (At much shorter wavelengths the probability of absorption begins to drop again, which is why X-ray afterglows are still detectable.) As a result, observed spectra of very high-redshift GRBs often drop to zero at wavelengths less than that of where this hydrogen ionization threshold (known as the [[Lyman break]]) would be in the GRB host's reference frame. Other, less dramatic hydrogen absorption features are also commonly seen in high-z GRBs, such as the [[Lyman alpha forest]].

==References==
{{reflist|3}}

[[Category:Gamma-ray bursts]]
[[Category:Astronomical events]]
[[Category:Stellar phenomena]]