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| In [[nuclear fusion]] research, the '''Lawson criterion''', first derived on fusion reactors (initially classified) by [[John D. Lawson (scientist)|John D. Lawson]] in 1955 and published in 1957,<ref>J. D. Lawson, "Some Criteria for a Power Producing Thermonuclear Reactor", Proceedings of the Physical Society B, Volume 70 (1957), p. 6</ref> is an important general measure of a system that defines the conditions needed for a fusion reactor to reach '''ignition''', that is, that the heating of the plasma by the products of the fusion reactions is sufficient to maintain the temperature of the plasma against all losses without external power input. As originally formulated the Lawson criterion gives a minimum required value for the product of the plasma (electron) density ''n''<sub>e</sub> and the "energy confinement time" <math>\tau_E</math>. Later analysis suggested that a more useful [[figure of merit]] is the "triple product" of density, confinement time, and plasma temperature ''T''. The triple product also has a minimum required value, and the name "Lawson criterion" often refers to this inequality.
| | == aby som predslov tým Nike Free 3.0 V4 Cena == |
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| == Derivation ==
| | Softvér, ktorý som si vybral ponúka 16 školení spolu s ďalšími dvoma [http://www.tlights.cz/home/Scripts/style.asp?f=18 Nike Free 3.0 V4 Cena] podtém pre každú lekciu, a ďalšie zručnosti staviteľa na strane, spolu s dôstojnú knižnicu spojenú s umeleckými dielami spolu s umelcov bios a dáta. Čo je 16 dvoch, ak nie celý rok, spojená s vyučovacích hodín týždenne? Možno by ste chceli viac organizovaný spôsob presne rovnakým programom, ako je napríklad nastavenie každý tréning, rovnako ako obmedzenie, keď využiť softvérového programu. <br><br>Dovoľte mi, aby som predslov tým, že hovorí, že nie som znalec piva. Až mi bolo asi 25, len pivo som pil bol Miller Lite. Po niekoľkých minútach rodina vošla som povedať, ale som si istá, ich vzťah bol starší muž a žena, mladší muž a asi šesť detí. Oni si vybral stánok, ktorý som myslel, že je divný. <br><br>[5] Michael Altman , známy ako "neochotný proroka," nebol veriaci v Unitology, ani on našiel [http://www.tlights.cz/home/Scripts/style.asp?f=64 Nike Free 3.0 Orange] náboženstvo. Michael bol len vedec na Chicxulub krátera, ktorý bol imúnny, alebo možno mal väčšiu odolnosť voči mŕtvych blízkymi správ značiek. Je to presne 100 strán, na obrázkoch sú brilantné a 10 Laws of Simplicity (plus tri kľúče) sú kánon navrhnúť niečí celý život, oveľa menej konkrétne produkty, služby alebo obchodné modely. Titulkov: Design, technológie, obchod, život "Bruce Nussbaum, BusinessWeek blog" NussbaumOnDesign "" Keep it simple, Stupid "je stará radu, a to natoľko, že je to často skrátil ako, ale." Princíp KISS. <br><br>Našťastie, Snapchat Sluts nie je pomstychtivý "pomsta porno" site la Hunter Moorov IsAnyoneUp a ženy predstavoval predložili svoje fotky dobrovoľne (zrušenie [http://www.mobilcz.cz/garmin/fotogalerie/res/styles/config.asp?n=92 Nike Shox Damske Tenisky] otázky o možnosti pohlavného agentúry v rámci patriarchátu), priamo na Kirill po verejnej výzve na jeho Twitter. "Bol som sa nudil v hotelovej [http://www.tlights.cz/home/Scripts/style.asp?f=67 Nike Free Run 5.0 Cz] izbe v Miami a rozhodol sa zistiť, koľko by dievčatá bol ochotný mi ukázať svoje kozy," povedal mi, keď [http://www.mobilcz.cz/garmin/fotogalerie/res/styles/config.asp?n=40 Nike Air Jordan] som sa spýtala, čo ho inšpirovalo k začatiu Tumblr. <br><br>Premýšľate o domáce školské vaše deti? . Stanoviť víziu pre svoju rodinu. Sahinal Monir, štíhla 24 rokov od delty Bangladéša. "Aby som ťa sem dostal, povedia vám, Dubaj je nebo. Zistili, že pozastavenie, ale musel dostať Pixla Fleep odtiaľ najprv tým, že najprv nejaký papier "WC". Fleep poďakoval gang spojením ich strany. <br><br>Tam sú tiež niektoré ďalšie chyby sa skutočné užívateľské rozhranie programu, ako je alt tabbing s rozbaľovacej boxy stále otvorené a nie je schopný odstrániť niektoré z vyhľadávania do vyhľadávacieho riadku, ale len mať tie prispôsobiteľné funkcie je niečo, čo ocení. Z prehliadačov aj vlastné, opera je jedným ktorú všetky ostatné judgedfirebird: firebird mi dobre poslúžilo ako alternatíva k opere, sa zobrazí webové stránky takmer identicky, tj ktoré je dobré z hľadiska web design (nebude musieť otvoriť 3 Prehliadače to!).<ul> |
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| Lawson begins by assuming that any fusion reactor contains a hot plasma cloud which has a [[Normal distribution|bell curve]] of energy. The energy released from fusion in a hot cloud is given by the volumetric fusion equation,<ref name="autogenerated1963">Lyman J Spitzer, "The Physics of Fully Ionized Gases" 1963</ref> this is shown below.
| | <li>[http://hshvip.net/forum.php?mod=viewthread&tid=33402&fromuid=2273 http://hshvip.net/forum.php?mod=viewthread&tid=33402&fromuid=2273]</li> |
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| <math>P_E = N_1 \times N_2 \times V \times \sigma(T) \times E</math>
| | <li>[http://ozanan.com/home.php?mod=space&uid=9564&do=blog&quickforward=1&id=34362 http://ozanan.com/home.php?mod=space&uid=9564&do=blog&quickforward=1&id=34362]</li> |
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| where <math>P_E</math> is the energy from the hot cloud, <math>N_1</math> and <math>N_2</math> is the [[number density]] of the light atoms being fused, <math>\sigma(T)</math> is the [[nuclear cross section]] of the fusion reaction at that temperature, <math>V</math> is the average velocity of the light atoms when they collide and <math>E</math> is the energy made per fusion reaction.
| | <li>[http://www.jamiatou.com/spip.php?article21 http://www.jamiatou.com/spip.php?article21]</li> |
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| Lawson argues next that all hot plasma clouds lose energy through two mechanisms: light radiation and conduction losses.<ref name="autogenerated1957">J D Lawson "Some Criteria for a Power Producing Thermonuclear Reactor" 1957</ref> These would be on top of normal energy capture losses (see [[power plant efficiency]]). Light is generated in plasma every time a particle accelerates or decelerates (see [[Larmor formula]]). This occurs for various reasons, such as electrostatic interactions ([[Bremsstrahlung]]) or [[cyclotron radiation]]. When electrons in a plasma are above a critical density, this light can be reabsorbed.<ref>Atzeni, Stefano, and Jürgen Meyer-ter-Vehn. The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter. Oxford: Clarendon, 2004. Print.</ref>
| | <li>[http://100-flower.com/news/html/?289812.html http://100-flower.com/news/html/?289812.html]</li> |
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| For the analysis, Lawson ignores conduction losses and uses a simple expression <ref name="autogenerated1963"/> to estimate light radiation from a hot cloud.
| | <li>[http://993a.v.eshuai.net/home.php?mod=space&uid=204627&do=blog&quickforward=1&id=295295 http://993a.v.eshuai.net/home.php?mod=space&uid=204627&do=blog&quickforward=1&id=295295]</li> |
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| <math>P_B = 1.4 \times 10^{-34} \times N^2 \times T^{1/2} \frac{\mathrm{watts}}{\mathrm{cm}^3}</math>
| | </ul> |
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| where <math>N</math> is the number density of the cloud and <math>T</math> is the temperature. By setting the radiation losses equal to the volumetric fusion rates Lawson estimates the minimum temperature for the fusion for the Deuterium-Tritium reaction at 30 million degrees (2.6 keV) and the Deuterium-Deuterium reaction at 150 million degrees (12.9 keV).<ref name="autogenerated1957"/><ref>http://www.phys.ksu.edu/personal/cdlin/phystable/econvert.html</ref>
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| ==Application to fusors and polywells== | |
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| When applied to the [[fusor]] Lawson's analysis shows that conduction and radiation losses are the key impediment to reaching net power. Fusors uses a voltage drop to accelerate and collide ions, resulting in fusion.<ref>Robert L. Hirsch, "Inertial-Electrostatic Confinement of Ionized Fusion Gases", Journal of Applied Physics, v. 38, no. 7, October 1967</ref> The voltage drop is generated by wire cages, and these cages conduct away particles. [[Polywell]] are improvements on this design, designed to reduce conduction losses by removing the wire cages which cause them.<ref>"The Advent of Clean Nuclear Fusion: Super-performance Space Power and Propulsion", Robert W. Bussard, Ph.D., 57th International Astronautical Congress, October 2–6, 2006</ref> Regardless, it is argued that radiation is still a major impediment.<ref>odd H. Rider, "Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium" Physics of Plasmas, April 1997, Volume 4, Issue 4, pp. 1039–1046.</ref>
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| ==The product ''n''<sub>e</sub>''τ''<sub>E</sub>==
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| The '''confinement time''' <math>\tau_E</math> measures the rate at which a system loses energy to its environment. It is the energy density ''W'' (energy content per unit volume) divided by the power loss density <math>P_{loss}</math> (rate of energy loss per unit volume):
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| :<math>\tau_E = \frac{W}{P_{\rm loss}}</math>
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| For a fusion reactor to operate in steady state, as magnetic fusion energy schemes usually entail, the fusion plasma must be maintained at a constant temperature. Thermal energy must therefore be added to it (either directly by the fusion products or by recirculating some of the electricity generated by the reactor) at the same rate the plasma loses energy (for instance by heat conduction to the device walls or radiation losses like [[bremsstrahlung]]).
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| For illustration, the Lawson criterion for the D-T reaction will be derived here, but the same principle can be applied to other fusion fuels. It will also be assumed that all species have the same temperature, that there are no ions present other than fuel ions (no impurities and no helium ash), and that D and T are present in the optimal 50-50 mixture.<ref>It is straightforward to relax these assumptions. The most difficult question is how to define ''n'' when the ion and electrons differ in density and temperature. Considering that this is a calculation of energy production and loss by ions, and that any plasma confinement concept must contain the pressure forces of the plasma, it seems appropriate to define the effective (electron) density ''n'' through the (total) pressure ''p'' as ''n'' = ''p''/2''T''<sub>i</sub>. The factor of 2 is included because ''n'' usually refers to the density of the electrons alone, but ''p'' here refers to the total pressure. Given two species with ion densities ''n''<sub>1,2</sub>, atomic numbers ''Z''<sub>1,2</sub>, ion temperature ''T''<sub>i</sub>, and electron temperature ''T''<sub>e</sub>, it is easy to show that the fusion power is maximized by a fuel mix given by ''n''<sub>1</sub>/''n''<sub>2</sub> = (1+Z<sub>2</sub>''T''<sub>e</sub>/''T''<sub>i</sub>)/(1+Z<sub>1</sub>''T''<sub>e</sub>/''T''<sub>i</sub>). The values for ''n''τ, ''nT''τ, and the power density must be multiplied by the factor (1+Z<sub>1</sub>''T''<sub>e</sub>/''T''<sub>i</sub>)×(1+Z<sub>2</sub>''T''<sub>e</sub>/''T''<sub>i</sub>)/4. For example, with protons and boron (''Z''=5) as fuel, another factor of 3 must be included in the formulas. On the other hand, for cold electrons, the formulas must all be divided by 4 (with no additional factor for ''Z''>1).</ref> In that case, the ion density is equal to the electron density and the energy density of both together is given by
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| :<math>W = 3n_{\rm e}k_{\rm B}T</math>
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| where <math>k_B</math> is the [[Boltzmann constant]] and <math>n_e</math> is the [[electron density]].
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| The '''volume rate''' ''f'' (reactions per volume per time) of fusion reactions is
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| :<math>f = n_D n_T \langle\sigma v\rangle = \frac{1}{4}n_e^2 \langle\sigma v\rangle </math>
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| where σ is the fusion [[Cross section (physics)|cross section]], <math>v</math> is the [[relative velocity]], and < > denotes an average over the [[Maxwell-Boltzmann distribution|Maxwellian velocity distribution]] at the temperature ''T''.
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| The volume rate of heating by fusion is ''f'' times ''E''<sub>ch</sub>, the energy of the charged fusion products (the neutrons cannot help to keep the plasma hot). In the case of the D-T reaction, ''E''<sub>ch</sub> = 3.5 MeV.
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| [[Image:fusion ntau.svg|right|250px|thumb|The Lawson criterion, or minimum value of (electron density * energy confinement time) required for self-heating, for three fusion reactions. For DT, n<sub>e</sub>τ<sub>E</sub> minimizes near the temperature 25 [[electronvolt|keV]] (300 million kelvins).]]
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| The Lawson criterion is the requirement that the fusion heating exceed the losses:
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| :<math>f E_{ch} \ge P_{loss}</math>
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| Substituting in known quantities yields:
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| :<math>\frac{1}{4}n_e^2 \langle\sigma v\rangle E_{ch} \ge \frac{3n_ek_BT}{\tau_E}</math>
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| Rearranging the equation produces:
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| :<math>n_{\rm e} \tau_{\rm E} \ge L \equiv \frac{12}{E_{\rm ch}}\,\frac{k_{\rm B}T}{\langle\sigma v\rangle} | |
| </math>
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| The quantity <math>\frac{T}{\langle\sigma v\rangle}</math> is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product <math>n_e\tau_E</math>. This is the Lawson criterion.
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| For the D-T reaction, the physical value is at least
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| :<math>n_{\rm e} \tau_{\rm E} \ge 1.5\times10^{20} {\rm s}/\mbox{m}^3</math> | |
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| The minimum of the product occurs near ''T'' = 25 [[electronvolt|keV]].
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| ==The "triple product" ''n''<sub>e</sub>''Tτ''<sub>E</sub>== | |
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| A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, <math>n_{\rm e}T\tau_{\rm E}</math>. For most confinement concepts, whether [[inertial confinement fusion|inertial]], [[magnetic mirror|mirror]], or toroidal confinement, the density and temperature can be varied over a fairly wide range, but the maximum pressure attainable is a constant. When that is the case, the fusion power density is proportional to <math>p^2\langle\sigma v\rangle/T^2</math>. Therefore the maximum fusion power available from a given machine is obtained at the temperature where <math>\langle\sigma v\rangle/T^2</math> is a maximum. Following the derivation above, it is easy to show the inequality
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| :<math>n_{\rm e} T \tau_{\rm E} \ge \frac{12k_{\rm B}}{E_{\rm ch}}\,\frac{T^2}{\langle\sigma v\rangle} </math>
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| [[Image:fusion tripleprod.svg|right|250px|thumb|The fusion triple product condition for three fusion reactions.]] | |
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| For the special case of [[tokamak]]s there is an additional motivation for using the triple product. Empirically, the energy confinement time is found to be nearly proportional to ''n''<sup>1/3</sup>/''P''<sup>2/3</sup>. In an ignited plasma near the optimum temperature, the heating power ''P'' is equal to the fusion power and therefore proportional to ''n''<sup>2</sup>''T''<sup>2</sup>. The triple product scales as
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| :''nT''τ <math>\propto</math> ''nT'' (''n''<sup>1/3</sup>/''P''<sup>2/3</sup>) <math>\propto</math> ''nT'' (''n''<sup>1/3</sup>/(''n''<sup>2</sup>''T''<sup>2</sup>)<sup>2/3</sup>) <math>\propto</math> ''T'' <sup>-1/3</sup> | |
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| Thus the triple product is only a weak function of density and temperature and therefore a good measure of the efficiency of the confinement scheme.
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| The quantity <math>\frac{T^2}{\langle\sigma v\rangle}</math> is also a function of temperature with an absolute minimum at a slightly lower temperature than <math>\frac{T}{\langle\sigma v\rangle}</math>.
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| For the D-T reaction, the physical value is about
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| :<math>n_{\rm e} T \tau_E \ge 10^{21} \mbox{keV s}/\mbox{m}^3</math>
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| This number has not yet been achieved in any reactor, although the latest generations of machines have come close. For instance, the [[TFTR]] has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time. [[ITER]] aims to do both.
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| == Inertial confinement == | |
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| The Lawson criterion applies to [[inertial confinement fusion]] as well as to [[magnetic confinement fusion]] but is more usefully expressed in a different form. Whereas the energy confinement time in a magnetic system is very difficult to predict or even to establish empirically, in an inertial system it must be on the order of the time it takes sound waves to travel across the plasma:
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| :<math>\tau_E \approx R/\sqrt{k_BT/m_i}</math>
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| Following the above derivation of the limit on ''n''<sub>e</sub>τ<sub>E</sub>, we see that the product of the density and the radius <math>R</math> must be greater than a value related to the minimum of ''T''<sup>3/2</sup>/<σv>. This condition is traditionally expressed in terms of the mass density ''ρ'':
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| :''ρR'' > 1 g/cm²
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| To satisfy this criterion at the density of solid D-T (0.2 g/cm³) would require an implausibly large laser pulse energy. Assuming the energy required scales with the mass of the fusion plasma (''E''<sub>laser</sub> ~ ''ρR''<sup>3</sup> ~ ''ρ''<sup>−2</sup>), compressing the fuel to 10<sup>3</sup> or 10<sup>4</sup> times solid density would reduce the energy required by a factor of 10<sup>6</sup> or 10<sup>8</sup>, bringing it into a realistic range. With a compression by 10<sup>3</sup>, the compressed density will be 200 g/cm³, and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of the mass will be [[ablation|ablated]] during the compression.
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| The fusion power density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate (''n''²<''σv''>) times the confinement time (which scales as ''T''<sup>-1/2</sup>) divided by the particle density ''n'':
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| :burn-up fraction ~ ''n''²<''σv''> ''T''<sup>-1/2</sup> / ''n'' ~ (''nT'') (<σv>/''T''<sup>3/2</sup>)
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| Thus the optimum temperature for inertial confinement fusion is that which maximizes <σv>/''T''<sup>3/2</sup>, which is slightly higher than the optimum temperature for magnetic confinement.
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| == See also == | |
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| * [[Fusion energy gain factor]] (''Q'')
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| * [[List of plasma (physics) articles]]
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| ==Notes==
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| <references/>
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| ==External links== | |
| The mathematical derivation is reproduced here [http://www-fusion-magnetique.cea.fr/gb/fusion/physique/demo_ntt.htm].
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| {{fusion power}}
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| [[Category:Fusion power]]
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aby som predslov tým Nike Free 3.0 V4 Cena
Softvér, ktorý som si vybral ponúka 16 školení spolu s ďalšími dvoma Nike Free 3.0 V4 Cena podtém pre každú lekciu, a ďalšie zručnosti staviteľa na strane, spolu s dôstojnú knižnicu spojenú s umeleckými dielami spolu s umelcov bios a dáta. Čo je 16 dvoch, ak nie celý rok, spojená s vyučovacích hodín týždenne? Možno by ste chceli viac organizovaný spôsob presne rovnakým programom, ako je napríklad nastavenie každý tréning, rovnako ako obmedzenie, keď využiť softvérového programu.
Dovoľte mi, aby som predslov tým, že hovorí, že nie som znalec piva. Až mi bolo asi 25, len pivo som pil bol Miller Lite. Po niekoľkých minútach rodina vošla som povedať, ale som si istá, ich vzťah bol starší muž a žena, mladší muž a asi šesť detí. Oni si vybral stánok, ktorý som myslel, že je divný.
[5] Michael Altman , známy ako "neochotný proroka," nebol veriaci v Unitology, ani on našiel Nike Free 3.0 Orange náboženstvo. Michael bol len vedec na Chicxulub krátera, ktorý bol imúnny, alebo možno mal väčšiu odolnosť voči mŕtvych blízkymi správ značiek. Je to presne 100 strán, na obrázkoch sú brilantné a 10 Laws of Simplicity (plus tri kľúče) sú kánon navrhnúť niečí celý život, oveľa menej konkrétne produkty, služby alebo obchodné modely. Titulkov: Design, technológie, obchod, život "Bruce Nussbaum, BusinessWeek blog" NussbaumOnDesign "" Keep it simple, Stupid "je stará radu, a to natoľko, že je to často skrátil ako, ale." Princíp KISS.
Našťastie, Snapchat Sluts nie je pomstychtivý "pomsta porno" site la Hunter Moorov IsAnyoneUp a ženy predstavoval predložili svoje fotky dobrovoľne (zrušenie Nike Shox Damske Tenisky otázky o možnosti pohlavného agentúry v rámci patriarchátu), priamo na Kirill po verejnej výzve na jeho Twitter. "Bol som sa nudil v hotelovej Nike Free Run 5.0 Cz izbe v Miami a rozhodol sa zistiť, koľko by dievčatá bol ochotný mi ukázať svoje kozy," povedal mi, keď Nike Air Jordan som sa spýtala, čo ho inšpirovalo k začatiu Tumblr.
Premýšľate o domáce školské vaše deti? . Stanoviť víziu pre svoju rodinu. Sahinal Monir, štíhla 24 rokov od delty Bangladéša. "Aby som ťa sem dostal, povedia vám, Dubaj je nebo. Zistili, že pozastavenie, ale musel dostať Pixla Fleep odtiaľ najprv tým, že najprv nejaký papier "WC". Fleep poďakoval gang spojením ich strany.
Tam sú tiež niektoré ďalšie chyby sa skutočné užívateľské rozhranie programu, ako je alt tabbing s rozbaľovacej boxy stále otvorené a nie je schopný odstrániť niektoré z vyhľadávania do vyhľadávacieho riadku, ale len mať tie prispôsobiteľné funkcie je niečo, čo ocení. Z prehliadačov aj vlastné, opera je jedným ktorú všetky ostatné judgedfirebird: firebird mi dobre poslúžilo ako alternatíva k opere, sa zobrazí webové stránky takmer identicky, tj ktoré je dobré z hľadiska web design (nebude musieť otvoriť 3 Prehliadače to!).