Posted on: 3rd December 2005

“In a terrestrial reactor of controllable size (…) it does not seem possible to contain neutrons, but it is not inconceivable that the charged particles could be kept in by suitable electric and magnetic fields. (…) The minimum temperature at which such a system could operate may be found by equating that portion of the reaction energy carried by the charged particles to the radiation loss. This temperature is 3×108 degrees for the D-D reaction and 5×107 degrees for the T-D reaction.””We now define an important parameter R, as the ratio of the energy released in the hot gas to the energy supplied. (…) R is a function of T and nt. (…) It is seen that for a useful reactor T must exceed 108 degrees and nt must exceed 1016. These conditions are very severe. Conditions for a T-D-Li6 reactor (…) are easier though still severe. The corresponding values of temperature and nt are T=3×107 degrees, nt=1014. To conclude we emphasise that these conditions, though necessary are far from sufficient.”

J.D. Lawson in “Some Criteria for a Useful Thermonuclear Reactor”, A.E.R.E. report GP/R 1807, December 1955. declassified April 9th 1957,

Fifty years ago, the young Harwell engineer John D. Lawson – who had joined the then secret British fusion research – wrote the above short and basic report. In it, two criteria were introduced that have to be met in order to achieve a power-generating fusion reactor: minimum temperature and minimum product of density and time. During a recent interview John D Lawson told us that the main motivation for this work was that as an engineer he felt the responsibility to “pin down” the unrealistic expectations of his enthusiastic physics fellows….

In the original article, Lawson considered very short discharges with ideal plasma confinement. However, today’s magnetic fusion research investigates sustained discharges with limited plasma confinement. Therefore, while Lawson introduced t for pulse length, nowadays we use “energy confinement time” τ (tau) instead, which is equal to plasma energy divided by plasma power losses (with plasma in energetic equilibrium). Similarly, Lawson describes fusion gain using the parameter R which relates input and output energies, while nowadays the “fusion gain” factor Q gives the ratio of fusion power to the external power needed to sustain the energetic equilibrium. Also notice that densities n were in particles per cubic centimeter (rather than cubic metre). Nevertheless, the physics behind the two criteria remains perfectly valid, with the numerical values of the nt (or ) limit varying according to the definition being used.

Interestingly, in August 1956 (while the Lawson report was still secret) the criterion was mentioned in the introductory and concluding parts of a talk by Russian fusion physicist L.A. Artsimovich at the ( International Astronomical Union Symposium in Stockholm!). In this talk, Τ is defined as a “time of life of fast particles in the system” which is similar to our current definition. Shortly afterwards, fusion research was declassified in the UK (see our Article of April) and a slightly amended article by Lawson was submitted for publication in November 1956, and published in January 1957 in Proceedings of Physical Society B, vol. 70 (6)..

Notice that the limit is a function of plasma temperature. For D-T reactions, the limit has a minimum around 300 million degrees – however, in magnetic confinement facilities it is easier to achieve higher at lower temperatures. The optimal trade-off appears around 100-200 million degrees, where (to a very good approximation) the limit decreases with increasing temperature. Thus, in this rather narrow temperature interval the triple product nτT sets a constant limit. This limit is today commonly known as the “fusion product”, and for fusion ignition (Q -> infinity) it has the value of

nτT > 3.1028 K m-3 s

Picture of the month

graphic demonstrating the fusion triple procuct

This graphic shows the fusion triple product achieved on different magnetic fusion facilities. Notice that the unit on the temperature scale, one kiloelectronvolt (keV) is equivalent to 11.6 millions of degrees, and that at very high temperatures the difference between Kelvins (K) and degrees Celsius (oC) is negligible. The graph shows clearly that new facilities performed better than previous ones. The present large machines, from the point of view of the fusion product, have now achieved their engineering limits so that only the next step facility, ITER, can bring about decisive progress.