Energetic Ions in Present-Day Tokamaks
HEATING PLASMA WITH ENERGETIC IONS
After ionisation and starting the plasma, heating must be applied for achieving fusion-grade plasma temperatures to increase fusion reactivity, and alpha particle heating starts playing its role. At the early phase, tokamaks are heated with plasma current I,, induced by the transformer action, with Ohmic power PQhm determined by the plasma current and plasma resistivity R,
At low plasma temperatures, this Ohmic heating is very strong. However, the plasma resistivity depends on electron temperature Te as
so when the plasma becomes hotter, its resistivity becomes smaller, and the Ohmic power becomes insufficiently effective. There is a maximum plasma temperature of ~5 keV, which can be achieved by Ohmic heating on present-day large tokamaks; however, auxiliary heating is needed to obtain -Ю-20 keV temperature of plasma ions, which is optimal for D-T fusion.
One of the most effective and well-tested techniques of auxiliary heating of plasma up to -10-20 keV temperatures is the heating via Coulomb collisions between thermal plasma species (electrons and plasma ions) and a population of low-density, nH«ne, but highly energetic (hot), E„ » Te, Th ions produced in the plasma by auxiliary heating techniques (see Ref. [3.1] and References therein). Although the density of such energetic (hot) ions is small compared to the density of thermal plasma, the energy content of these energetic ions, nH EH, is not necessarily small in comparison to the thermal plasma energy content, nje+njr
The hot ions of different energies transfer their energy to thermal ions and electrons at different proportions via Coulomb collisions. If the energy of hot ions is less than a critical value determined by
the power of the hot ions flows mainly to thermal ions rather than to electrons. If the energy of hot ions is higher than the critical value (3.3), the hot ions mostly heat electrons. Here, Af and A, are the masses of fast ions and thermal ions, respectively, Te is electron temperature, n, and ne are densities of thermal ions and electrons, respectively, and Z, is the atomic number of thermal ions. The amount of energy going from hot ions w'ith initial energy E into plasma thermal ions is given by the Stix formula [3.2]
FIGURE 3.1 Graph of the function G(E/Eml'j given by the Stix formula (3.4).
Figure 3.1 illustrates the function Gt (E/Ecrit).
The value of the critical energy implies that the largest ever existing tokamaks, JET, TFTR, JT-60U, DIII-D, and ASDEX-Upgrade, with neutral beam injection (NBI) as the main auxiliary heating technique have dominant ion heating of plasmas. This conclusion results from (3.3) taking into account the typical NBI energy range of ~ 60-160 keV and typical electron temperatures of -5-10 keV. On the contrary, alpha particles with the energy of 3.52 MeV are well above the critical energy (3.3), and hence provide dominant electron heating. Moreover, NBI on small machines with low electron temperatures most often produce dominant electron heating.
The characteristic slowing-down time of energetic ions due to collisions with plasma electrons was first calculated by Spitzer [3.3]:
where AH, ZH are the mass (in hydrogen mass units) and the charge number of the energetic ions, respectively, /г,.(1тг3) is the electron density, and In Af == 16 is the Coulomb logarithm. The dependence of the slowing-down time on the mass of the energetic ions plays a key role in the experiment. For example, for the same plasma parameters, the slowing-down times of hydrogen, deuterium, and tritium energetic ions differ up to a factor of 3, implying that the tritium ions remain energetic for a much longer time than hydrogen or deuterium ions.
More details on the collisional relaxation of energetic ions are given in Appendix A. A comprehensive review validating the theory of energetic ion distributions in tokamak plasmas could be found in Ref. [3.4].
We now discuss the main mechanisms of plasma heating with energetic ions in tokamaks.
Neutral Beam Injection (NBI)
Among all techniques of auxiliary plasma heating, NBI plays a major role in present-day machines. In particular, all scenarios with high fusion performance rely on NBI-produced energetic ion populations as the principal source of plasma heating. Moreover, NBI is expected to play a key role in plasma heating and current drive in ITER.
In the NBI technique, deuterium ions from an ion source are accelerated via grids to a high energy. Then, they pass through the neutraliser and become neutral high energy atoms. Then, the highly energetic neutral beam is injected into plasma, with the penetration length of the atomic beam that depends on the NBI energy, mass, and the plasma density. Within the plasma, the NBI neutrals are ionised by collisions with thermal electrons and ions, and the resulting energetic ions are trapped by the magnetic field of the machine. Then, the energetic ion beam relaxation begins due to Coulomb collisions with thermal ions and electrons, leading to a steady-state beam distribution of the slowing-down type in energy,
where ©(a) is the step function, V„ is the injection velocity of energetic ions, and И(к) with к = v,, / v is the pitch angle distribution. For an isotropic distribution of, for example, fusion-generated alpha particles, we have Н(к) = 1. The distribution function (3.6) is normalised to the average beta of the energetic ions:
Sometimes, the distribution function of NBI-produced energetic ions deviates from the slowing- down form (3.6) significantly. This happens when mechanisms other than Coulomb collisions affect the beam ions on a time scale shorter than the slowing-down time. For example, if the characteristic time of charge-exchange of the beam ions is shorter than the beam slowing-down time, the beam with a constant source can have a steady-state distribution with a bump-on-tail in the energy.
Apart from deuterium, beams used in present-day machines can inject hydrogen, tritium, He, and He3. Hydrogen gases produce neutral beams in three fractions at E, E/2, and £73 energies, whereas helium beams are produced at energy E only. In addition to plasma heating, NBI also provides fuelling. In particular, tritium NBI at an energy of 160 keV was very effective in penetrating to the plasma core of JET tokamak, thus providing tritium fuelling close to the optimum D:T=50:50 mixture in high fusion power D-T experiments in JET [3.5].
Let us now summarise the advantages and disadvantages of NBI.
Advantages of NBI
Efficient heating of thermal ions in present-day experiments;
High power capability (40 MW on TFTR, >30 MW on JET);
Drives plasma rotation (thus affecting MHD instabilities, i.e., stabilising lock modes); Provides fuelling;
Provides current drive;
Not sensitive to the value of magnetic field.
Needs MeV energy beams for penetrating in ITER —> Negative ion source for NBI is needed; Heating not well localised;