2) Energy balance and Lawson criterion (p  1 - 2 )

b) Lawson criterion

Let us go back to our energy balance. At stationary state (dW/dt = 0) , we have :

P_alpha + P_external = P_losses = W/ t_E

By replacing P_external by P_fusion /Q and by using the fact that the plasma energy W and fusion power  P_fusion depend on density n (i.e. the number of particles per unit of volume) and temperature T of the plasma, we obtain a relation expressing the constraints on the plasma parameters (density, temperature confinement time of the energy) if we want to obtain a pulse with a given amplification factor Q. This is what is called the Lawson criterion, which gives the value of the product of plasma density multiplied by the energy confinement time t_E at a plasma temperature T to attain an amplification factor Q. How about a little proof for mathematicians? Click here.

In practical terms, for a reactor conditions, we obtain :

n T t_E >  10²¹ (keV m^-3 s) with T between 10 to 20 keV

In other words, to be able to produce energy from fusion reactions, a sufficiently hot (T) and dense (n) plasma must be confined effectively ( t_E , not to be confused once again with pulse duration) * . 

* NB : in the following, we will look at fusion by magnetic confinement, which works at relatively low densities, while trying to obtain long confinement times. There is however possibility, inertial confinement (bombarding a solid deuterium and tritium target with laser beams or very intense particle beams) operating at very high densities (matter compressed by the beams) with very short times.

The difficulty resides in obtaining the three parameters simultaneously. Indeed, for example, when we increase the density (n) by injecting gas into the machine or the temperature T by adding additional power to the plasma, the confinement (t_E) of a tokamak tends to deteriorate.

In a tokamak, achievable plasma densities (i.e. the number of particles per unit of volume) are around 10²⁰ per cubic metre (m^-3); this is in fact very low, much lower than the density of the air surrounding us, and corresponds to conditions near to that of a vacuum. It is not possible to go any further, on account of the appearance of instabilities beyond a certain density threshold, the pressure exerted by the plasma becoming higher than that of the magnetic field. Emphasis is rather on the confinement time t_E , which we try to take further than a second by developing complex physics scenarios (performances attained for the moment are no higher than 0,8 seconds).

 

Progress in fusion research is illustrated by the increase in the triple product n TtE , which may be seen on the figure opposite and which has increased threefold since the first experiments at the end of the 60s up to the current machines (e.g. JET in Europe, TFTR in the United States and JT60U in Japan), closely approaching break-even. There remains only a factor of ten to achieve in order to reach the domain of the reactor.

The next generation machine, ITER,  intended to demonstrate the scientific and technical feasibility of controlled thermonuclear fusion, has been scaled up to attain an amplification factor of 10. The possibility of attaining ignition in certain physics scenarios has not been ruled out. This international project, launched at the end of the 80s, initially with four partners (Europe, Japan, the United States and Russia), has entered the final phase of dimensioning. Now with three partners (Europe, Japan and Russia), it is awaiting the decision for building, with in particular the choice of a site in one of the partner countries. Studies are in progress to assess the potential of Cadarache as a European site candidate.

It is worth noting that the future reactor does not need to be at ignition (infinite amplification factor Q) to work, but simply to reach a Q factor sufficient for the global efficiency h_reactor of the power station to be worthwhile, taking into account the conversion of thermal energy into electricity by conventional methods (turbine etc) and the fact that a part of the energy produced is reused to supply the additional heating systems needed to maintain the plasma.

 

Typical figures for a reactor are amplification factors of several tens, corresponding to a global reactor efficiency hreacteur of 35 % and a fraction of plasma heating by alphas Fa of 90% (i.e. 10 % of the  remaining heating is provided by the additional heating systems).

 

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