Physics:Quantum Fusion reactions and Lawson criterion

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The Lawson criterion is a fundamental condition in nuclear fusion physics that determines when a plasma can produce net energy.[1][2]

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.[3]

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

nTτE

where n is particle density, T temperature, and τE the energy confinement time.

Lawson criterion for various fusion approaches, showing the required triple product for ignition.

Quantum basis of fusion reactions

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

f=n1n2⟨σv⟩

The quantity ⟨σv⟩ depends strongly on temperature and is determined by the interplay between the Maxwell–Boltzmann distribution and the quantum tunnelling probability, producing the so-called Gamow peak.[4][5]

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:

Pfusion=n1n2⟨σv⟩Efusion

Radiative losses (bremsstrahlung) scale as:

PB=1.4⋅10−34N2T1/2

where N is particle density.

Confinement time and nτ formulation

The energy confinement time is defined as:

τE=WPloss

with plasma energy density:

W=3nT

Using these relations gives the Lawson condition:

nτE≥12TEch⟨σv⟩

For the deuterium–tritium reaction:

nτE≳1.5×1020m−3s

at temperatures near:

T≈25–30keV

Triple product

A more useful figure of merit is the triple product:

nTτE

For D–T fusion:

nTτE≳3×1021keVsm−3

Magnetic vs inertial confinement

The Lawson criterion applies to both:

Magnetic confinement

Inertial confinement

  • high density, short confinement

In inertial systems:

ρR≳1g/cm2

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

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Template:Cite tech report
  2. ↑ Lawson, J. D. (1957). "Some Criteria for a Power Producing Thermonuclear Reactor". Proceedings of the Physical Society 70: 6–10. doi:10.1088/0370-1301/70/1/303. 
  3. ↑ Abu-Shawareb, H. (2022). "Lawson Criterion for Ignition Exceeded in an Inertial Fusion Experiment". Physical Review Letters 129. doi:10.1103/PhysRevLett.129.075001. 
  4. ↑ Bethe, H. A.; Critchfield, C. L. (1938). "The Formation of Deuterons by Proton Combination". Physical Review 54: 248–254. doi:10.1103/PhysRev.54.248. 
  5. ↑ Clayton, D. D. (1983). Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press. 
Author: Harold Foppele

Source attribution: Lawson criterion

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