Lawson criterion

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The Lawson criterion (after John Lawson ) is a physical condition that a nuclear fusion reaction occurring in a plasma is self- sustaining , i.e. H. self-sustaining itself in a macroscopic amount of fuel. In simple terms, the proportion of the fusion power that remains in the plasma must be at least as great as the power dissipation of the plasma.

The criterion was originally formulated for the fusion of deuterium and tritium (DT), but can in principle also be generalized to other fusion fuels. The type of plasma confinement, such as fusion by means of magnetic confinement or inertial fusion , does not matter.

All attempts to achieve the Lawson criterion in a controlled reaction have failed so far (2016) because the plasma volumes were too small and cooled too quickly to enable a permanent fusion process. This should be achieved with the original draft for ITER , which, however, was not approved in this size. With his successor, DEMO , it should be possible. Even in inertial fusion experiments, the criterion, ignition, has not yet been met.

DT plasma in equilibrium

The criterion results from an equilibrium. In a DT plasma , the proportion of the fusion power carried by the alpha particles produced must be equal to the power dissipation of the plasma.

The neutrons released during the fusion reaction leave the plasma immediately because they are electrically neutral and their kinetic energy of 14.1  MeV is then used to generate energy. The electrically charged alpha particles, on the other hand, give off their kinetic energy of 3.5 MeV in the plasma through collisions, so they heat the plasma with one power . At the same time, the plasma loses energy through bremsstrahlung and transport; this is the power dissipation . Is in balance . If a DT plasma fulfills this condition, it “ignites”, “burns” then without supplying energy and in turn supplies energy as kinetic neutron energy. The criterion must be met in nuclear weapons and also in inertial fusion reactors. In magnetic confinement fusion reactors , however, it does not have to be completely fulfilled; a certain external heating that is constantly required (e.g. with a few percent of the neutron energy obtained) would even have the advantage of offering an additional means of controlling the reaction.

The alpha particle power is in a DT plasma

with the particle densities of the two reactants, the nuclear reaction rate averaged over the speed distribution of the particles (particle speed multiplied by the speed-dependent cross-section ), the part of the energy released per fusion and the plasma volume due to the alpha particle .

The thermal energy contained in the plasma is

with the electron density , the Boltzmann constant and the temperature .

Radiation and particle transport processes cause power loss . The quotient of the thermal energy and the power loss has the dimension of a time and is called the energy containment time:


To achieve automatic burning, the following must apply:


With the assumption that both reaction partners are present in the same quantities, i.e. have the same particle density and are virtually completely ionized

follows the Lawson criterion :


At a given temperature, the minimum value of the product of particle density n and energy inclusion time results for the automatically burning fusion reaction. This product is a function of temperature, which is slightly different for each fusion reaction, but always has an absolute minimum. For the DT reaction, for example, one obtains

the minimum being at a temperature of approximately 25 keV.

The triple product

Instead of , the so-called triple product is usually used as a measure for reaching the ignition condition. The Lawson criterion is then


This has the advantage that the minimum of as a function of the temperature is around 14 keV , the value that is approximately necessary to operate a fusion reactor.

Losses from bremsstrahlung

In particular, highly ionized impurities in the plasma (e.g. ) lead to a loss of energy due to bremsstrahlung . The value of the triple product required for ignition is therefore higher.

The bremsstrahlung losses are given by

with the constant and the effective charge (the sum runs over all ion species in the plasma).

For a fusion reaction that takes place independently, the criterion arises from the condition (whereby here only describes the loss due to transport processes)


Without impurities, i.e. H. , this gives the value at the minimum . Does the plasma contain e.g. B. 0.5% contamination by , d. H. , the value of the triple product increases at the minimum . It is therefore more difficult to achieve the conditions necessary for ignition.


  • JD Lawson: Some Criteria for a Power Producing Thermonuclear Reactor. In: Proceedings of the Physical Society. Section B. 70, 1957, pp. 6-10, doi : 10.1088 / 0370-1301 / 70/1/303 . Extended version of the AERE report GP / R 1807, December 1955, declassified April 9th ​​1957

Individual evidence

  1. ^ TJM Boyd, JJ Sanderson: The Physics of Plasmas . Cambridge University Press, 2003, ISBN 0-521-45290-2 , pp. 3-4
  2. ^ Weston M. Stacey: Fusion. An Introduction to the Physics and Technology of Magnetic Confinement Fusion. Wiley-VCH, 2010, ISBN 978-3-527-40967-9 , limited preview in Google Book Search, pages 8–9