Physics:Quantum Lawson criterion: Difference between revisions
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{{Short description|Criterion for igniting a nuclear fusion chain reaction}} | {{Short description|Criterion for igniting a nuclear fusion chain reaction}} | ||
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The '''Lawson criterion''' is a | <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;"> | ||
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 | The concept was first developed by John D. Lawson in a classified 1955 report at the Atomic Energy Research Establishment in Harwell, United Kingdom.<ref>{{Dead link|date=November 2023}}</ref> The work was later declassified and formally published in 1957.<ref name="Lawson">{{Cite journal | ||
|last=Lawson | |last=Lawson | ||
|first=J. D. | |first=J. D. | ||
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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"/> | 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 | 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) | |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 | |title=Scientists Achieved Self-Sustaining Nuclear Fusion… But Now They Can't Replicate It | ||
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|doi-access=free | |doi-access=free | ||
}}</ref> | }}</ref> | ||
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[[File:The Lawson Criterion.png|thumb|280px|Lawson criterion and triple-product performance of major magnetic confinement fusion experiments.]] | |||
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== Energy balance == | == Energy balance == | ||
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#'''Fusion''' is the energy generated by nuclear fusion reactions. | #'''Fusion''' is the energy generated by nuclear fusion reactions. | ||
#'''Radiation loss''' is the energy lost through electromagnetic radiation such as | #'''Radiation loss''' is the energy lost through electromagnetic radiation such as X-rays. | ||
#'''Conduction loss''' is the energy carried away by escaping plasma particles. | #'''Conduction loss''' is the energy carried away by escaping plasma particles. | ||
#'''Efficiency''' measures how effectively the reactor converts fusion energy into useful power. | #'''Efficiency''' measures how effectively the reactor converts fusion energy into useful power. | ||
Lawson estimated the fusion power using a thermalized plasma obeying a | Lawson estimated the fusion power using a thermalized plasma obeying a Maxwell–Boltzmann distribution.<ref name="autogenerated1963">{{Cite journal | ||
|last1=Spitzer | |last1=Spitzer | ||
|first1=Lyman | |first1=Lyman | ||
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* <math>n_A</math> and <math>n_B</math> are the number densities of the fusion fuels, | * <math>n_A</math> and <math>n_B</math> are the number densities of the fusion fuels, | ||
* <math>\sigma</math> is the fusion | * <math>\sigma</math> is the fusion cross section, | ||
* <math>v</math> is the relative particle velocity, | * <math>v</math> is the relative particle velocity, | ||
* <math>E</math> is the fusion energy released per reaction. | * <math>E</math> is the fusion energy released per reaction. | ||
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By balancing fusion power against radiation losses, Lawson estimated minimum ignition temperatures of approximately: | By balancing fusion power against radiation losses, Lawson estimated minimum ignition temperatures of approximately: | ||
* 30 million K (2.6 keV) for the [[deuterium]] | * 30 million K (2.6 keV) for the [[Physics:Quantum deuteron|deuterium]]–tritium reaction, | ||
* 150 million K (12.9 keV) for the deuterium–deuterium reaction.<ref name="Lawson"/><ref>{{Cite web | * 150 million K (12.9 keV) for the deuterium–deuterium reaction.<ref name="Lawson"/><ref>{{Cite web | ||
|title=Energy Converter | |title=Energy Converter | ||
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}}</ref> | }}</ref> | ||
Major fusion experiments such as | 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) | |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 | |title=World Highest Fusion Triple Product Marked in High-βp H-mode Plasmas | ||
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== Inertial confinement == | == Inertial confinement == | ||
The Lawson criterion also applies to [[Physics:Quantum Magnetic confinement fusion|magnetic confinement fusion]] and | 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: | For inertial confinement fusion, the criterion is often written as: | ||
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== Non-thermal systems == | == Non-thermal systems == | ||
The Lawson criterion was originally derived for thermalized plasmas, but some fusion concepts use non-thermal particle distributions. Examples include the | 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 | In such systems, energy losses from radiation and particle conduction remain major obstacles to net energy production.<ref>{{Cite journal | ||
Latest revision as of 11:34, 22 May 2026
The Lawson criterion is a figure of merit used in 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 ignition.[1]
The concept was first developed by John D. Lawson in a classified 1955 report at the Atomic Energy Research Establishment in Harwell, United Kingdom.[2] The work was later declassified and formally published in 1957.[1]
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.[1]
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.[3][4]
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.[5]
The volumetric fusion reaction rate is approximately:
where:
- and are the number densities of the fusion fuels,
- is the fusion cross section,
- is the relative particle velocity,
- 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 is the plasma number density and is the plasma temperature.[1]
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.[1][6]
Confinement time
The energy confinement time measures how long a plasma retains its energy before losses dominate.
\tau_E = \frac{W}{P_{\mathrm{loss}}}
where:
- is the plasma energy density,
- 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.[1]
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.[7]
Major fusion experiments such as JT-60, TFTR, JET, and ITER are often evaluated using the triple-product diagram shown above.[8]
Inertial confinement
The Lawson criterion also applies to 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:
- is the fuel density,
- is the fuel radius.
Achieving this condition generally requires extremely high compression ratios produced by powerful laser systems.[9]
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.[10]
See also
Table of contents (84 articles)
Index
Full contents
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Lawson, J. D. (1957). "Some Criteria for a Power Producing Thermonuclear Reactor". Proceedings of the Physical Society, Section B 70 (1): 6–10. doi:10.1088/0370-1301/70/1/303. Bibcode: 1957PPSB...70....6L.
- ↑ [|permanent dead link|dead link}}]
- ↑ [[1](https://www.sciencealert.com/scientists-achieved-self-sustaining-nuclear-fusion-but-now-they-cant-replicate-it) "Scientists Achieved Self-Sustaining Nuclear Fusion… But Now They Can't Replicate It"]. ScienceAlert. August 16, 2022. [2](https://www.sciencealert.com/scientists-achieved-self-sustaining-nuclear-fusion-but-now-they-cant-replicate-it).
- ↑ Abu-Shawareb, H.; Acree, R.; Adams, P.; Adams, J.; Addis, B. (2022-08-08). "Lawson Criterion for Ignition Exceeded in an Inertial Fusion Experiment". Physical Review Letters 129 (7). doi:10.1103/PhysRevLett.129.075001. PMID 36018710. Bibcode: 2022PhRvL.129g5001A.
- ↑ Spitzer, Lyman; Seeger, Raymond J. (1963). "Physics of Fully Ionized Gases". American Journal of Physics 31 (11): 890–891. doi:10.1119/1.1969155. Bibcode: 1963AmJPh..31..890S.
- ↑ [[3](https://www.phys.ksu.edu/personal/cdlin/phystable/econvert.html) "Energy Converter"]. Kansas State University. [4](https://www.phys.ksu.edu/personal/cdlin/phystable/econvert.html).
- ↑ Wesson, J. (2004). [[5](https://pdfhost.io/view/NMf0wmfoq_Wesson_J_Tokamaks_3Ed_Oxford_2004_K_T_755Spdf) "Tokamaks"]. Oxford Engineering Science Series (Oxford: Clarendon Press) (48). [6](https://pdfhost.io/view/NMf0wmfoq_Wesson_J_Tokamaks_3Ed_Oxford_2004_K_T_755Spdf).
- ↑ [[7](http://www-jt60.naka.jaea.go.jp/english/html/exp_rep/rep36.html) "World Highest Fusion Triple Product Marked in High-βp H-mode Plasmas"]. [8](http://www-jt60.naka.jaea.go.jp/english/html/exp_rep/rep36.html).
- ↑ Hirsch, Robert L. (1967). "Inertial-Electrostatic Confinement of Ionized Fusion Gases". Journal of Applied Physics 38 (11): 4522–4534. doi:10.1063/1.1709162. Bibcode: 1967JAP....38.4522H.
- ↑ Rider, Todd H. (1997-04-01). "Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium". Physics of Plasmas 4 (4): 1039–1046. doi:10.1063/1.872556. Bibcode: 1997PhPl....4.1039R.
Source attribution: Physics:Quantum Lawson criterion
