While the Lawson Criterion represents a reactor criterion, i. e. it refers to the energy viability of the entire plant, we can as well derive a criterion for the
energy viability of the fusion plasma. The latter may be deemed as energetically viable when external power has no longer to be delivered to it. This so-called ignited plasma state can be shown for a homogeneous plasma, as was also assumed for Eq.(8.28), with help of Eqs.(8.10) and (8.11) to be characterized by

ІсЛ Ге’ЦРьг + P;1)te> + 3NT (8.29)

where again Nj = Ne = N, Tj = Te = T was taken and the time integration was performed over a period equal to the global energy confinement time Te*. As in deriving Eq.(8.28), we can similarly substitute for the above power terms by their explicit density-temperature dependent expressions, take Nd = Nt = N/2 and solve

which represents a fusion plasma ignition criterion and does not contain any energy conversion efficiencies. It is displayed in its temperature dependence in Fig. 8.4. A contour plot similar to the Lawson Criterion becomes evident, however featuring quantitative differences. The minimum ignition temperature for N = 1020 m’3 is seen at T~ 30 keV and requires a product of plasma density and global energy confinement time of NTe* = 2.7 X 1020 m’3s and hence Te* = 2.7 seconds.

We note that the denominator in Eq.(8.30) can become negative for high temperatures where the cyclotron radiation terms takes over to dominate the plasma energetics. Such a regime is associated with a negative plasma energy balance and can therefore not be ignited. For that temperature range, Eq. (8.30) has no meaning. The more stringent ignition requirements with increasing T are evident from Fig. 8.4 where the ignition contours tend towards infinity as the plasma temperature approaches the critical value Tcrit, as indicated for the case of N = 1020 m‘3.

Another definition often used for classifying the fusion reactor operation is the scientific (energy) break-even which means that the total fusion energy production amounts to a magnitude equal to the effective plasma energy input,

i. e.

—^r=QP = l• (8-31)

Ліп Ein

This break-even condition can be analogously derived from Eq. (8.9) for Qp=l and is also demonstrated in Fig. 8.4. Recall that the ignition condition is associated with Qp —>

The performance of a high NTe* is not the only goal in fusion reactor research; as the preceding discussion suggests, it is additionally required for a

viable fusion plasma regime that, simultaneously with NTe*, the plasma temperature is established at a sufficiently high level. Hence, presently, the so — called triple product TNTe* is used most often for qualifying recent achievements of fusion experiments. This last ignition criterion is easily obtained by just multiplying Eq.(8.30) with T on both sides. Expectedly, it does not introduce a new characteristic, but rather exhibits a similar temperature dependence as in Fig. 8.4.

Problems

8.1 Reformulate the Lawson Criterion which incorporates the direct conversion of the alpha particle into electricity with 95% efficiency.

8.2 Discuss and compare the plasma energy kT ~ 15 keV associated with the minimum Nte* requirement given by the Lawson Criterion, with the recognition that a kinetic energy of Ek ~ 200 keV is required by hydrogen ions to overcome their Coulomb repulsion in order to fuse.

8.3 Develop Lawson Criteria for a d-d plasma (include both branches of the d-d reaction), and plot Nx vs T. Assume a fusion power to electrical conversion efficiency of 30%.

8.4 Develop Ignition Criteria for a d-t, d-h and d-d plasma, and plot Nx vs T for each. Assume alpha-particle heating of the d-t plasma, alpha-particle and proton heating of the d-h plasma, and triton, proton and h heating of the d-d plasma.

8.5 Derive-analogously to Eq. (8.28)-a more realistic MCF reactor criterion accounting also for cyclotron radiation losses Display graphically its temperature dependence, NexE* (T) for the two cases

(a) d-t: Nd = N, = Ne/2, T) = Te, |//Ne = 10’23 m3, В = 5 T, r|in = 0.5 and tlout=0.35,

(b) d-h: Nd = Nh = N,/2, T, = Te, |//N e= 1022 m3, В = 10 T, riin = 0.5 and

Bout-

Note: For the derivation of the criterion, use fractions of the ion density such that Nj = KjNi with j denoting the considered ion species. You should finally obtain:

ItCjZj

for j = a, b, impurities. Compare the plots from (a) and (b) with those that result if cyclotron radiation losses are neglected.

8.6 Discuss the physical differences between (i) the Lawson reactor criterion, (ii) the ignition criterion and (iii) the break-even condition. What fraction of the entire plasma loss power can be made up for by oc-particle heating in steady state operation, when Qp = 5?