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Fix Matter see-also invoke

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Fix Matter see-also invoke

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{{Short description|Criterion for igniting a nuclear fusion chain reaction}}

{{Quantum matter backlink|Plasma and fusion physics}}

The '''Lawson criterion''' is a [[figure of merit]] used in [[Physics:Quantum Fusion|nuclear fusion]] research. It compares the rate of energy generated by fusion reactions within fusion fuel to the rate of energy losses from the plasma environment. When the rate of energy production exceeds the rate of energy loss, the fusion system can produce net energy. If enough of this energy is retained within the plasma to sustain further reactions, the plasma reaches [[Physics:Quantum Fusion#Ignition|ignition]].<ref name="Lawson"/>

The concept was first developed by [[John D. Lawson (scientist)|John D. Lawson]] in a classified 1955 report at the Atomic Energy Research Establishment in Harwell, United Kingdom.<ref>{{cite tech report
|last=Lawson
|first=J. D.
|title=Some criteria for a useful thermonuclear reactor
|date=December 1955
|issue=A.E.R.E. GP/R 1807
|institution=Atomic Energy Research Establishment, Harwell, Berkshire, U. K.
|url=[https://www.euro-fusion.org/fileadmin/user_upload/Archive/wp-content/uploads/2012/10/dec05-aere-gpr1807.pdf](https://www.euro-fusion.org/fileadmin/user_upload/Archive/wp-content/uploads/2012/10/dec05-aere-gpr1807.pdf)
}}{{Dead link|date=November 2023}}</ref> The work was later declassified and formally published in 1957.<ref name="Lawson">{{Cite journal
|last=Lawson
|first=J. D.
|title=Some Criteria for a Power Producing Thermonuclear Reactor
|journal=Proceedings of the Physical Society, Section B
|volume=70
|number=1
|pages=6–10
|date=1957
|doi=10.1088/0370-1301/70/1/303
|bibcode=1957PPSB...70....6L
}}</ref>

As originally formulated, the Lawson criterion defines a minimum value for the product of plasma density and energy confinement time required to obtain net fusion power. Later refinements introduced the more useful '''fusion triple product''', which combines plasma density, temperature, and confinement time into a single performance measure.<ref name="Lawson"/>

On 8 August 2021, researchers at the [[National Ignition Facility]] at [[Lawrence Livermore National Laboratory]] announced an inertial confinement fusion experiment that exceeded the Lawson criterion for ignition conditions.<ref>{{cite web
|url=[https://www.sciencealert.com/scientists-achieved-self-sustaining-nuclear-fusion-but-now-they-cant-replicate-it](https://www.sciencealert.com/scientists-achieved-self-sustaining-nuclear-fusion-but-now-they-cant-replicate-it)
|title=Scientists Achieved Self-Sustaining Nuclear Fusion… But Now They Can't Replicate It
|publisher=ScienceAlert
|date=August 16, 2022
}}</ref><ref>{{Cite journal
|last1=Abu-Shawareb
|first1=H.
|last2=Acree
|first2=R.
|last3=Adams
|first3=P.
|last4=Adams
|first4=J.
|last5=Addis
|first5=B.
|date=2022-08-08
|title=Lawson Criterion for Ignition Exceeded in an Inertial Fusion Experiment
|journal=Physical Review Letters
|volume=129
|issue=7
|article-number=075001
|doi=10.1103/PhysRevLett.129.075001
|pmid=36018710
|bibcode=2022PhRvL.129g5001A
|hdl=10044/1/99300
|hdl-access=free
|doi-access=free
}}</ref>
<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">
[[File:The Lawson Criterion.png|400px]]
<div style="font-size:90%;">Lawson criterion and triple-product performance of major magnetic confinement fusion experiments.</div>
</div>
== Energy balance ==

The Lawson criterion is based on the energy balance of a fusion plasma. A fusion reactor can only produce net power when the fusion heating exceeds all energy losses.

'''Net power = Efficiency × (Fusion − Radiation loss − Conduction loss)'''

#'''Fusion''' is the energy generated by nuclear fusion reactions.
#'''Radiation loss''' is the energy lost through electromagnetic radiation such as [[X-rays]].
#'''Conduction loss''' is the energy carried away by escaping plasma particles.
#'''Efficiency''' measures how effectively the reactor converts fusion energy into useful power.

Lawson estimated the fusion power using a thermalized plasma obeying a [[Maxwell–Boltzmann distribution]].<ref name="autogenerated1963">{{Cite journal
|last1=Spitzer
|first1=Lyman
|last2=Seeger
|first2=Raymond J.
|title=Physics of Fully Ionized Gases
|journal=American Journal of Physics
|volume=31
|issue=11
|pages=890–891
|date=1963
|doi=10.1119/1.1969155
|bibcode=1963AmJPh..31..890S
}}</ref>

The volumetric fusion reaction rate is approximately:

<math>
\text{Fusion rate}
=
n_A n_B \langle \sigma v \rangle E
</math>

where:

* <math>n_A</math> and <math>n_B</math> are the number densities of the fusion fuels,
* <math>\sigma</math> is the fusion [[Cross section (physics)|cross section]],
* <math>v</math> is the relative particle velocity,
* <math>E</math> is the fusion energy released per reaction.

Lawson also estimated radiation losses using the bremsstrahlung relation:

P_B = 1.4 \times 10^{-34} N^2 T^{1/2} \frac{\mathrm{W}}{\mathrm{cm}^3}

where <math>N</math> is the plasma number density and <math>T</math> is the plasma temperature.<ref name="Lawson"/>

By balancing fusion power against radiation losses, Lawson estimated minimum ignition temperatures of approximately:

* 30 million K (2.6 keV) for the [[deuterium]]–[[tritium]] reaction,
* 150 million K (12.9 keV) for the deuterium–deuterium reaction.<ref name="Lawson"/><ref>{{Cite web
|title=Energy Converter
|publisher=Kansas State University
|url=[https://www.phys.ksu.edu/personal/cdlin/phystable/econvert.html](https://www.phys.ksu.edu/personal/cdlin/phystable/econvert.html)
|access-date=2023-02-17
}}</ref>

== Confinement time ==

The '''energy confinement time''' <math>\tau_E</math> measures how long a plasma retains its energy before losses dominate.

\tau_E = \frac{W}{P_{\mathrm{loss}}}

where:

* <math>W</math> is the plasma energy density,
* <math>P_{\mathrm{loss}}</math> is the power loss density.

For a steady-state fusion reactor, fusion heating must at least balance energy losses:

f E_{\rm ch} \ge P_{\rm loss}

For a 50–50 deuterium–tritium plasma, the Lawson criterion becomes:

n\tau_E \ge \frac{12T}{E_{\rm ch}\langle\sigma v\rangle}

For the D–T reaction, the minimum required value is approximately:

n\tau_E \ge 1.5 \times 10^{20}\ \frac{\mathrm{s}}{\mathrm{m}^3}

The minimum occurs near a plasma temperature of approximately 26 keV.<ref name="Lawson"/>

== Triple product ==

A more useful performance parameter is the '''fusion triple product''':

nT\tau_E

This combines plasma density, temperature, and confinement time. For the D–T reaction, the minimum required triple product is approximately:

nT\tau_E \ge 3 \times 10^{21}\ \mathrm{keV\ s\ m^{-3}}

The optimum temperature for this condition is near 14 keV.<ref>{{Cite journal
|url=[https://pdfhost.io/view/NMf0wmfoq_Wesson_J_Tokamaks_3Ed_Oxford_2004_K_T_755Spdf](https://pdfhost.io/view/NMf0wmfoq_Wesson_J_Tokamaks_3Ed_Oxford_2004_K_T_755Spdf)
|first=J.
|last=Wesson
|title=Tokamaks
|journal=Oxford Engineering Science Series
|issue=48
|publisher=Clarendon Press
|location=Oxford
|edition=3
|year=2004
}}</ref>

Major fusion experiments such as [[JT-60]], [[TFTR]], [[JET]], and [[ITER]] are often evaluated using the triple-product diagram shown above.<ref>{{cite web
|url=[http://www-jt60.naka.jaea.go.jp/english/html/exp_rep/rep36.html](http://www-jt60.naka.jaea.go.jp/english/html/exp_rep/rep36.html)
|title=World Highest Fusion Triple Product Marked in High-βp H-mode Plasmas
|archive-url=[https://web.archive.org/web/20130106002319/http://www-jt60.naka.jaea.go.jp/english/html/exp_rep/rep36.html](https://web.archive.org/web/20130106002319/http://www-jt60.naka.jaea.go.jp/english/html/exp_rep/rep36.html)
|archive-date=2013-01-06
}}</ref>

== Inertial confinement ==

The Lawson criterion also applies to [[Physics:Quantum Magnetic confinement fusion|magnetic confinement fusion]] and [[inertial confinement fusion]]. In inertial confinement systems, confinement time is determined by the expansion time of the compressed fuel pellet.

For inertial confinement fusion, the criterion is often written as:

\rho R \ge 1\ \mathrm{g/cm^2}

where:

* <math>\rho</math> is the fuel density,
* <math>R</math> is the fuel radius.

Achieving this condition generally requires extremely high compression ratios produced by powerful laser systems.<ref>{{Cite journal
|last=Hirsch
|first=Robert L.
|title=Inertial-Electrostatic Confinement of Ionized Fusion Gases
|journal=Journal of Applied Physics
|volume=38
|issue=11
|pages=4522–4534
|date=1967
|doi=10.1063/1.1709162
|bibcode=1967JAP....38.4522H
}}</ref>

== Non-thermal systems ==

The Lawson criterion was originally derived for thermalized plasmas, but some fusion concepts use non-thermal particle distributions. Examples include the [[fusor]], [[polywell]], and [[migma]] concepts.

In such systems, energy losses from radiation and particle conduction remain major obstacles to net energy production.<ref>{{Cite journal
|last=Rider
|first=Todd H.
|title=Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium
|journal=Physics of Plasmas
|volume=4
|issue=4
|pages=1039–1046
|date=1997-04-01
|doi=10.1063/1.872556
|bibcode=1997PhPl....4.1039R
|hdl=1721.1/11412
|hdl-access=free
}}</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Lawson criterion|1}}

Physics:Quantum Lawson criterion - Revision history

imported>WikiHarold: Fix Matter see-also invoke

imported>WikiHarold: Fix Matter see-also invoke