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Repair Quantum Collection B backlink template

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{{Short description|Criterion for ignition in nuclear fusion plasmas}}

{{Quantum book backlink|Plasma and fusion physics}}
'''The Lawson criterion''' is a fundamental condition in [[Physics:Quantum Fusion|nuclear fusion]] physics that determines when a plasma can produce net energy.<ref name="Lawson1955">{{cite tech report |last=Lawson |first=J. D. |title=Some criteria for a useful thermonuclear reactor |date=1955 |institution=Atomic Energy Research Establishment}}</ref><ref name="Lawson1957">{{Cite journal |last=Lawson |first=J. D. |title=Some Criteria for a Power Producing Thermonuclear Reactor |journal=Proceedings of the Physical Society |volume=70 |pages=6–10 |year=1957 |doi=10.1088/0370-1301/70/1/303}}</ref>

It compares the rate of energy generated by fusion reactions to the rate of energy losses from the plasma. When fusion heating exceeds losses, the plasma can reach ignition.<ref>{{Cite journal |last1=Abu-Shawareb |first1=H. |title=Lawson Criterion for Ignition Exceeded in an Inertial Fusion Experiment |journal=Physical Review Letters |volume=129 |year=2022 |doi=10.1103/PhysRevLett.129.075001}}</ref>

In modern form, the Lawson criterion is expressed through the '''triple product''':

<math>n T \tau_E</math>

where <math>n</math> is particle density, <math>T</math> temperature, and <math>\tau_E</math> the energy confinement time.

<div style="float:right; border:1px solid #ccc; padding:4px; background:#ffffe6; margin:0 0 1em 1em; width:400px;">
[[File:Fustion triple-product diagram Horvath.jpg|400px]]
<div style="font-size:90%;">Lawson criterion for various fusion approaches, showing the required triple product for ignition.</div>
</div>

== Quantum basis of fusion reactions ==

Fusion reactions depend on quantum tunnelling through the [[Physics:Quantum Fusion#Coulomb barrier|Coulomb barrier]]. The probability that two nuclei fuse is governed by the fusion cross section and its [[Physics:Quantum Fusion#Maxwellian average|Maxwellian average]]:

<math>
f = n_1 n_2 \langle \sigma v \rangle
</math>

The quantity <math>\langle \sigma v \rangle</math> depends strongly on temperature and is determined by the interplay between the Maxwell–Boltzmann distribution and the quantum tunnelling probability, producing the so-called ''[[Physics:Quantum Fusion#Gamow peak|Gamow peak]]''.<ref>{{Cite journal |last=Bethe |first=H. A. |last2=Critchfield |first2=C. L. |title=The Formation of Deuterons by Proton Combination |journal=Physical Review |volume=54 |year=1938 |pages=248–254 |doi=10.1103/PhysRev.54.248}}</ref><ref>{{Cite book |last=Clayton |first=D. D. |title=Principles of Stellar Evolution and Nucleosynthesis |publisher=University of Chicago Press |year=1983}}</ref>

This quantum-mechanical tunnelling probability ultimately determines the fusion reactivity and therefore sets the conditions required by the Lawson criterion.

== Energy balance ==

The Lawson criterion is derived from plasma energy balance:

'''Net power = Fusion − Radiation loss − Conduction loss'''

Fusion power density:

<math>
P_{fusion} = n_1 n_2 \langle \sigma v \rangle E_{fusion}
</math>

Radiative losses (bremsstrahlung) scale as:

<math>
P_B = 1.4 \cdot 10^{-34} N^2 T^{1/2}
</math>

where <math>N</math> is particle density.

== Confinement time and nτ formulation ==

The energy confinement time is defined as:

<math>
\tau_E = \frac{W}{P_{loss}}
</math>

with plasma energy density:

<math>
W = 3 n T
</math>

Using these relations gives the Lawson condition:

<math>
n \tau_E \ge \frac{12T}{E_{ch}\langle\sigma v\rangle}
</math>

For the deuterium–tritium reaction:

<math>
n \tau_E \gtrsim 1.5 \times 10^{20} \,\mathrm{m^{-3}\,s}
</math>

at temperatures near:

<math>
T \approx 25\text{–}30\,\mathrm{keV}
</math>

== Triple product ==

A more useful figure of merit is the triple product:

<math>
n T \tau_E
</math>

For D–T fusion:

<math>
n T \tau_E \gtrsim 3 \times 10^{21} \,\mathrm{keV\,s\,m^{-3}}
</math>

== Magnetic vs inertial confinement ==

The Lawson criterion applies to both:

=== Magnetic confinement ===
* low density, long confinement
* [[Physics:Quantum Tokamak|tokamaks]]

=== Inertial confinement ===
* high density, short confinement

In inertial systems:

<math>
\rho R \gtrsim 1 \,\mathrm{g/cm^2}
</math>

== Non-thermal systems ==

The Lawson criterion assumes thermal equilibrium. Non-thermal systems such as fusors or Polywell devices accelerate particles directly.

In such systems, energy losses remain the primary limitation.

== Physical interpretation ==

The Lawson criterion combines three essential requirements:

* '''Density''' → collision frequency
* '''Temperature''' → tunnelling probability
* '''Confinement time''' → interaction duration

Only when all three are sufficiently large does fusion become self-sustaining.

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

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

{{Author|Harold Foppele}}

{{Sourceattribution|Lawson criterion|1}}

Physics:Quantum Fusion reactions and Lawson criterion - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template