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

a) Pasma power balance

As in the example of a cauldron, the plasma energy balance is determined by the energy sources feeding the plasma and the energy losses cooling it down. For the plasma to remain stationary (i.e. not changing over time), the energy balance must be in equilibrium, i.e. the sources must compensate the losses.  

Energy sources : P_fusion and P_external

Fusion power :  Pfusion : The total power produced by the D-T fusion reaction Pfusion is divided between the products of the reaction, the alpha particles, i.e. the helium nuclei (He), and the neutrons. This gives: Pfusion = Palpha + Pneut

The neutrons take away about 80% of the energy, while the heavier alpha particles, keep about 20%. But this energy does not end up in the same place : Palpha : the main source of the plasma energy comes from the alpha particles. Indeed, these charged particles are confined by the tokamak magnetic field, and give their energy to the plasma by collision. Pneut : in contrast, the neutrons (n) from the fusion reaction are not sensitive to the magnetic field, since they have no charge, and thus escape quickly, without having time to give their energy to the plasma. They are stopped by the materials in the components surrounding the tokamak vacuum chamber.

External power P_external :
If the energy from the fusion reactions is not sufficient to compensate losses, it is necessary to supply energy from the outside to maintain the plasma using an additional heating system. This is the external power P_external.

Energy losses: P_losses

• The plasma confinement by the magnetic field is not perfect: particles and heat diffuse from the plasma center towards the outside. The losses connected to this particles and heat transport are considerable.

• As a hot body, the plasma also cools by radiation according to different processes. The electrons emit continuous radiation in their collisions with ions ("Bremsstrahlung"). They also emit synchrotron radiation due to their gyration movement around field lines, which may grow considerably when the plasma is heated up to a very high temperature. Finally, the impurities emitted by the wall surrounding the vacuum chamber produce line radiation due to the different atomic physic processes which have taken place in the plasma. This contribution may become very important if the plasma is strongly polluted and may even lead to an abrupt loss of plasma confinement: this is known as a disruption.

The result of all these terms gives the total power lost by the plasma P_losses.

The balance

The temporal variation of the plasma energy W may thus be written as : dW/dt = P_alpha + P_external - P_losses

Reminder: only the alpha particles give their energy to the plasma, the rest of the fusion power is dissipated into the components surrounding the plasma. If the source term is higher than the loss term (dW/dt >0), the plasma gains energy;in the opposite case (dW/dt <0), it loses it. If the sources precisely compensate the losses (dW/dt =0), the plasma is stationary. Several useful quantities may now be defined.

The energy confinement time  tE

This is the characteristic time of decrease in plasma energy; in other words, it is the time taken by the plasma to empty itself of its energy content if the sources supplying it are abruptly cut off. Thus: W/ tE=  Plosses NB : this time has nothing to do with the pulse duration, which is determined by the capacities of the machine magnetic system or plasma instabilities. For example, on Tore Supra, the energy confinement time is around 200 milliseconds (or 0,2 seconds) while the pulses last tens of seconds and even minutes

 

The amplification factor Q

This is the ratio between the power from fusion reactions and the external power supplied to the plasma by the heating systems:  Q =  Pfusion /  Pexternal This figure thus qualifies the plasma’s energy balance. If it is higher than 1, more energy has been produced with fusion reactions than was necessary to supply to maintain the plasma. NB : the Q factor must not be confused with the overall efficiency of the installation.

 

The Break-even

This is the situation corresponding to Q = 1, i.e. the moment when the quantity of energy produced by the fusion reactions is equal to that supplied to maintain the plasma ( Pfusion= Pexternal) . This is an interesting stage from the scientific point of view, as heating of the plasma is then to a great extent done by the alpha particles and no longer nearly solely by the additional heating, which is close to the situation of the reactor.

 

Ignition

This is the situation where the power supplied by the fusion reactions is enough on its own to compensate losses ( Palpha= Plosses) and where the external power can thus be switched off. This corresponds to an infinite Q amplification factor ( Pexternal = 0). The plasma is thus self-maintained like a candle, which, once it has been ignited by a match (external power), carries on fuelling itself.

Most current experimental machines destined for research and not yet for electricity production, operate at Q<1, i.e. the plasma consumes more energy than it supplies. They only use deuterium as a fuel, enabling the necessary physical studies without the use of radioactive tritium; the results obtained in D-D fusion to D-T fusion are then extrapolated. Only 2 machines have for the moment experimented with the use of tritium: the American machine TFTR, now closed, and the European machine JET, which holds the world record of fusion power in D-T, with 16 MegaWatts produced, corresponding to an amplification factor of 0.64.

 

.