Thermonuclear Burn Criteria

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THERMONUCLEAR BURN CRITERIA

Guido Van Oost1 and Roger Jaspers2

1. Department Applied Physics Ghent University St. Pietersnieuwstraat 41, B-9000 Gent, Belgium [email protected]

2. Science and Technology of Nuclear Fusion, Faculty of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

I. INTRODUCTION From this, with E = mv2/2 we immediately obtain

After more than 50 years of fusion research the En/EHe = mHe/mn = 4 (1) time has arrived when fusion processes in experimental plasmas are increasingly getting important. In JET Usually this process is described by the chemical nota- the genuine fuel (deuterium-tritium) of a fusion reaction tor was used for the first time in late 1991, in TFTR the same happened in 1993, and in JET an extended 2D + 3T → 3He(3.5 MeV) + 1n(14.1 MeV) (2) period of experiments of this kind was performed in 1997. Therefore, it is getting more and more rewarding Since the binding energy B of the nucleons (neutrons to deal with the problems related to the ignition and and protons) must be expended for their separation burning of plasmas. from the nucleus, it is released in the reverse process, Nuclear fusion played and still plays an important fusion. And since each nucleus possesses negative bind- role in the Universe. About 1 million years after the big- ing energy, its mass is always smaller than the sum of bang large amounts of 4He were created by the fusion the masses of all neutrons and protons (total number A) of protons on a global scale, and later on heavier ele- of which it consists. In Fig. 1 we see how B/A depends ments were and are created in the huge fusion reactors on A. In the range A ≤ 60 the average binding energy provided by the interior of the stars. On earth, the con- per nucleon can be increased (brought to larger nega- cepts envisaged for a fusion reactor are thermonuclear tive values) by the fusion of smaller nuclei into larger fusion by magnetic plasma confinement in tokamaks ones; in the range A ≥ 60 the same effect is achieved or stellarators, laser- or beam-induced inertial fusion, by the fission of larger nuclei into smaller fragments. and muon catalyzed cold fusion. In this lecture we shall While the first fusion reactions had already been ob- concentrate on magnetic confinement, in particular on served in 1919 by the physicist Ernest Rutherford, nu- the D-T fusion reaction 2D + 3T → 4He + n, which clear fission was only discovered in 1938 by the two + + + has a mass defect ∆m = mD + mT − m4He + mn = chemists Otto Hahn and Fritz Strassmann. Neverthe- 3.1·10−29 kg, i.e. about 4 per thousand of the reactant’s less it was only four years until the physicist Enrico mass, that according to Einstein’s equation E = mc2 Fermi obtained the first controlled chain reactions in corresponds to an energy E = 17.6 MeV released as ki- an experimental fission reactor. On the other hand we netic energy in the reaction products. Starting from the shall have to wait far into this century until the first 5 nucleons in the D&T nuclei, this means 3.5 MeV per fusion reactor will hopefully go into operation. This is nucleon or about 4 times the 0.85 MeV which is released an indication of how much more difficult it is to obtain per nucleon during the fission of U235. The distribu- controlled fusion reactions with an net energy gain. The tion of fusion energy among the reaction products is obstacles in nuclear fusion are well illustrated by the determined by the momentum conservation law. Since following estimate: The energy needed to overcome the the momentum of the reaction products is much larger Coulomb-wall of mutual repulsion for two hydrogen nu- than that of the reaction partners before the reaction, in clei is about 0.4 MeV, and the temperature of a plasma a D-T reaction we essentially have mnvn = −mHevHe. needed

15 In the D-T reaction the main portion of the energy is released to the neutron. Although the fast fusion neutrons created this way lead to secondary radioactivity in some materials surrounding the plasma (first wall, sup- ports, etc.), in magnetic confinement schemes this must, at least at present, be considered an advantage. Since they don’t carry electric charge the neutrons are not held back by the confining magnetic field, and they can also easily penetrate the confinement vessel. Outside of this their energy can be extracted by a moderator.

II. CROSS-SECTIONS, REACTION RATES AND POWER DENSITY OF FUSION REACTIONS

A fusion reaction which releases a lot of energy but occurs very rarely is of little use. Thus the reaction Figure 1: Binding energy per nucleon, B/A, as a func- frequency is a crucial issue. Let us consider a beam of tion of A D-nuclei with density nD, moving at constant relative velocity v through T-nuclei. The number dnD of beam for the particles to achieve this in a classical process particles that is lost due to interaction processes such as with the help of their thermal energies is T ≈ 3 · 109 scattering collisions or fusion reactions when the beam K. Fortunately the tunnelling effect makes consider- advances by a distance ds is proportional to ds, to the ably lower temperatures possible. In a D-T reaction the density nT of target particles and to that of the beam quantum probability for penetrating the Coulomb wall particles, nD: is given by the Gamow factor dnD = σnDnTds =⇒ R :=n ˙ D =n ˙ T = nDnThσ (v) vi , p (4) w ≈ exp −34.4 keV/Ekin (3) where h i denotes the averaging over particles of all For Ekin = 10 keV we obtain a tunnelling probability of possible velocities. R is the reaction rate (or the col- w ≈ 1.9 · 10−5, indicating that markedly lower temper- lision frequency in the case of scattering collisions; for atures than the classically required 3 billion Kelvin can both processes independently a corresponding equation lead to fusion. All this is included in the fusion reaction applies). σ is the D-T fusion cross-section, v the rela- cross-section below (Section II). tive velocity between the reacting particles. The eval- According to the curve of binding energies the di- uation of the average rate coefficient hσ (v) vi requires rect fusion of the 4He-nucleus out of its four nucleons some thermodynamics, involves the Gamow factor, and would be even more energetic than the D-T reaction yields the results shown in Fig. 2 for some typical fusion because a total binding energy of 28 MeV would be re- reactions. It is seen that it assumes by far the largest leased in this process, i.e. 7 MeV per nucleon. However, values in the D-T reaction, and this even at much lower a reaction of this kind would require the simultaneous temperatures than in the other fusion reactions. collision of four nuclei, a process that is so highly im- It is only a small fraction of highly energetic parti- probable at normal densities in magnetic fusion that cles that are reacting and being lost through fusion (see it practically does not occur. Indeed, as we have seen eq. 3). This tail is repopulated by scattering collisions before, the fusion of two reaction partners is already that cause the plasma to approach a Maxwellian distri- a rather improbable process, so only two-particle colli- bution closely. This collisional process for the replace- sions can be envisaged for fusion reactions in a reactor. ment of highly energetic particles lost by fusion is an Altogether more than 80 different fusion reactions essential characteristic of thermonuclear fusion. Thus are currently known. Since singly charged nuclei have while scattering collisions have the unpleasant side ef- the lowest Coulomb repulsion, fusion reactions between fect of causing diffusion and particle losses from the re- hydrogen isotopes require the lowest plasma temper- action vessel on the one hand, on the other hand they atures. The D-T reaction (2) is accompanied by a have the important task of replenishing highly energetic number of side-reactions, the most important of which particles lost by fusion. In a fusion reactor each fusion are D-D and T-T reactions. However, we will neglect collision will be accompanied by a sufficiently high num- these side reactions because of their small fusion cross- ber of scattering collisions. Closer investigation shows sections. that at the temperature of a fusion reactor (≈ 10 keV )

16 Figure 3: Maximum fusion power density Pfus [W/cm3] vs. T . Figure 2: Temperature dependence of the rate coefficient for some typical fusion reactions obtained at a much lower temperature than the maximum value of hσvi. This is due to the factor 1/T 2 in on average for each fusion collision there are about 8000 Pˆfus that, with increasing temperature, causes Pˆfus to scattering collisions (Note: this is the reason that beam- decrease before hσvi has reached its maximum value. target fusion concepts will not produce net energy, and Equation (9) also demonstrates the importance of high a thermonuclear approach is needed). magnetic fields. The quantity which characterizes the efficiency of a fusion reaction is the power density Pfus, the energy released per second in a unit volume: III. BALANCE EQUATIONS

Pfus = REfus = nDnThσviEfus,Efus = 17.6MeV (5) A. Particle balance

Both ions and electrons exert a pressure pI and pe, re- A general particle balance equation has the form spectively, adding to a total pressure p, ∂nk/∂t + div (nkvk) = Qk (10) pI = nIkTI = (nD + nT) kTI, (6) nkvk is the current of particles consisting of a diffusive p = n kT , p = p + p (7) e e e I e and a convective part, and Qk is a local source term. The equation accounts for (a) particle supply, (b) par- Due to stability reasons, there is an upper limit ticle gains and losses through the burning of the fuel, βmax to the ratio between average plasma pressure and and (c) for losses by diffusion and convection. Aver- magnetic pressure aging eq. (10) over the whole plasma volume V (with surface SV) yields β = hpi/hB/2µ0i (8) Z From this and (6)-(7) it follows that there is an up- d¯nk/dt + nkvk · dS/V = Q¯k (11) SV per limit to the fusion particle density that for nD = nTnI/2 and ne = nD +nT (quasi-neutrality) is given by R 2 wheren ¯k = nkdV/V is the average particle density hnÎi ≈ hB βmax/ (8µ0kT )i. The maximum fusion power and Q¯ = R Q dV/V the average source term. R n · density associated with this is k k K dV/V =n ¯k · V is the total number of particles NK (i.e. B4β2 hσvi the particle content). Pˆ = E (9) fus 2 2 2 fus The source term is composed of the fuel losses de- 64µ0k T scribed by (4) and a terms ¯k accounting for the fuel Figure 3 shows Pˆfus for a given βmax as a function of supply: Q¯i = −ninjhσvi +s ¯i. After multiplication with the temperature for several fusion reactions. Compari- V the total loss of particles per second from the plasma, loss R son with Fig. 2 reveals that the highest power output is dNk /dt = nkvk · dS, yields an average particle loss

17 3/2 rate per volume. This leads to the definition of a parti- pα since η ∝ 1/T (this would not be possible cle loss time τk through in tokamaks with extremely strong magnetic fields because in these much stronger currents would be n¯ V k allowed). τk = R (12) nkvk · dS 2. External heating: It is useful to express the ex- Its precise meaning can be seen from the reformulation ternal heating power as a fraction of the fusion Z power through loss τk nkvk · dS = τkdNk /dt =n ¯kV = Nk V pext = pfus/Q = 5pα/Q (17) Under stationary conditions obtained when all parti- Q is called the power enhancement factor (see also cle losses are compensated for by supply, τ is the k IV). It is the ratio of the thermonuclear power pro- time elapsed until just as many particles are lost from duced to the heating power supplied and is a mea- the plasma through diffusion and convection as it mo- sure of the success in approaching reactor condi- mentarily contains. (We assume that effects of parti- tion. cle recycling[12] are included in τk). Assuming approximately equal diffusion loss times, τi = τj = τp, and 3. Alpha particle heating: In our calculations we using the approximations shall assume that the energy released to the alpha particles through fusion processes is fully de- ¯ ninjhσvi ≈ n¯in¯jhσvi, hσvi (T ) ≈ hσvi T (13) livered to the plasma through collisions. The heating power thus obtained is approximately given by from (11) and (12) we obtain the burn equations (5) with Efus replaced by Eα, i.e.

dni/dt = −ni/τp − ninjhσvi + si (14) Pα = nDnThσviEα = nDnThσviEfus/5 (18) where the volume-averaging bars have been omitted for This is only an approximation for the following rea- further convenience. These “zero-dimensional” equa- sons: tions can be improved by taking into account profile effects: For profiles of a given (not self-consistently de- (a) Some alpha particles may already diffuse out, termined) shape each term is modified by a shape factor before they have delivered their surplus en- (see e.g. Ref. 2). ergy to the plasma. Equations (14) must be supplemented by the quasi- (b) The expression for P is a function of the posi- neutrality condition α tion and time of alpha particle creation; how- X ever, the real position and time of energy de- nkZk = ne (15) position are somewhat apart or later respectively. Due to P = E /5 the temperature in which Zk is the charge number of ion species k. α fus dependence of Pα is the same as that of Pfus B. Energy balance shown in Fig. 3. With the simplifying assumption T = T = T = e D T 4. Radiation losses: There are radiation losses T (this implies that all ions created by fusion are ther- through bremsstrahlung, synchrotron radiation, malized) the general energy balance equation has the and through line and recombination radiation. At form the temperatures of a D-T reactor, 10 − 20 keV, ! ∂ 3 X synchrotron radiation can be neglected in com- ne + nλ kT + divJ = pOH + pext + pα + prad parison with bremsstrahlung. For bremsstrahlung, ∂t 2 λ which originates mainly from the acceleration of (16) electrons in the field of ions, we employ the for- where J is the total heat flow current due to heat mula convection and heat conduction, is the ohmic heating 6 r 2 power, pOH the additional external heating power, pα e 2 2 8kTe Z PB = 3 neZ gff (19) the alpha particle heating power; the work ve · ∇pe + 24πe c3m h πm T P 0 e e e λ vλ∇pλ performed by the pressures has been neglected in comparison with the much larger heat source in which gff is a slowly varying function of its argu- terms. ment called Gaunt-factor (accounting for quantum effects). In the D-T reaction there are separate con- 1. Ohmic heating: At fusion temperatures POH = tributions of this kind from D and T with charge ηj2 can usually be neglected in comparison with number Z = 1 and from 4He with Z = 2 (helium

18 is fully ionised under reactor conditions). dnT nT = − − nDnThσviDT + sT (24) In a pure D-T plasma line and recombination ra- dt τp diation do not play an essential role except for the dn n α = − α − n n hσvi (25) much cooler plasma boundary region, because all dt τ D T DT ions are fully ionized and the central plasma is too α hot for recombinations. The situation is different if and from (22) and (18) the energy balance equation the plasma is polluted by nuclei of higher charge d 3 number. We shall only rather crudely take into ac- (ne + nI + nα + nZ ) kT = count such radiation, employing for it again eq. dt 2 (19) with some effective charge number for the im- 3 (26) − (ne + nI + nα + nZ ) kT/τE purities. 2 +nDnThσvi Eα (1 + 5/Q) − PB 5. Transport losses: Integrating the heat flow J DT through diffusion and convection of energy over the where nI is the total number of ions and nZ is the plasma boundary yields the total energy losses by density of impurity ions considered as a single species transport. with effective charge number Z. We shall consider nZ By analogy with (12) we introduce an energy con- as a given parameter. In contrast to our previous inten- finement time τE through tions, we have introduced a separate particle confine- R 3 P ment time τα 6= τp for the alpha particles, the purpose V 2 (ne + λ nλ) kT · dV τE = R (20) being that this will facilitate the transition to a limiting SVJ·dS case to be considered (see IV.A). In addition, we have Frequently, especially by experimentalists, a dif- the quasi-neutrality condition ferent energy confinement time τ ∗ (called global E n + 2n + Zn = n = n /2 (27) confinement time) is used that is defined through I α Z e tot R 3 P For P we have to take into account the radiation caused ∗ V 2 (ne + λ nλ) kT · dV τE = R (21) by hydrogen isotopes (Z = 1), alpha particles (Z = 2) Prad + J · dS SV and impurities (charge number Z), from (19) obtaining It is the time in which the plasma, due to all losses the formula

including radiation, loses the same amount of en- 2 ergy as it presently contains and is easier to mea- PB = ne [cI RI (T ) + cαRα(T ) + cZ RZ (T )] (28) sure than τ . E in which we employed the concentrations 6. Averaged energy balance equation: Integrat- nI nα nZ ing eq.(16) over the whole plasma volume, dividing cI = , cα = , cZ = (29) n n n by V and using (17), (20) plus the same approx- e e e imations as in (13), with omission of the bar for and where averages we obtain √ √ RI = CB T gff (1/T ) ,Rα = 4CB T gff (4/T ) , d 5 e √ (30) e = P 1 + − tot − P = 2 2 dt tot α Q τ rad RZ = Z CB T gff (Z /T ) E (22) 5 e P 1 + − tot with √ α ∗ e6 8k Q τE CB = 3 3 √ (31) P 24π0c meh πme Where etot = (ne + λ nλ) kT is the total energy density and the expressions for Pα and Prad must be evaluated at the average temperature and density. IV. EQUILIBRIA: BREAK-EVEN AND IGNITION

C. Basic equations for the D-T reaction We now want to determine equilibria, i.e. we are looking for stationary solutions d/dt = 0. When an We now make a further approximation in neglecting equilibrium is achieved with Pext = Pfus or Q = 1 all side reactions (D-D, T-T etc.) due to their small resp. this is called break-even. Ignition (notice the anal- fusion cross-sections. With this we obtain from (14) the ogy with the burning of fossil fuels) is achieved when particle balance equations all external heat sources can be turned oﬀ, Pext = 0 or dnD nD Q = ∞ . The conﬁnement conditions are then such that = − − nDnThσviDT + sD (23) dt τp the plasma temperature can be maintained against the

19 energy losses solely by α-particle heating. From (23)- The product neτE is a measure of the quality of the (24) for stationary conditions we get plasma confinement, and the value required according to this formula in order to get an ignited equilibrium or nD sD = + nDnThσviDT + sD , break-even depends only on the temperature. This tem- τp perature dependence is shown in Fig. 4. The minimum n (32) T temperature required for ignition (Q = ∞) is obtained sT = + nDnThσviDT + sD τp by equating the denominator of our result for neτE to zero (becoming infinite). It is given by the smaller tem- the magnitude of the particle sources is fixed by the perature obtained as a solution from requirement of stationarity. The maximum fusion power is obtained for nD = nT = nI /2 (see II), i.e. the particle 1 R (T ) = hσviE (39) sources must satisfy I 4 α 2 nI nI and is typically about 6 keV. sD = sT = + hσvi (33) 2τp 4 We shall now transform the ideal ignition curve into a diagram employing our second energy confinement time With this equations (23) and (24) are satisfied and must ∗ τE defined in (21). Applying this definition, no longer be considered concerning equilibrium. ∗ The remaining equations to be solved are (25), (26) Prad + etot/τE = etot/τE (40) and (15) viz cI + 2cα + ZcZ = 1 (34) to the present situation yields

The latter one is satisfied when we eliminate cI by using 2 ∗ neRI + 3nekT/τE = 3nekT/τE (41) cI = 1 − 2cα − ZcZ . Inserting this we are left with only two equations (for particles and energy respectively), ∗ 3kT τE τE = ∗ (42) 3kT − neτERI cαne 1 2 2 = (1 − 2cα − ZcZ ) nehσvi , (35) τα 4 Since τE must be nonnegative, from this we get the condition 3 ∗ [2 − c − (Z − 1) c ] n kT/τ = neτ ≤ 3kT/RI (43) 2 α Z e E E (36) ∗ 1 2 2 The limit neτE ≤ 3kT/RI is called radiation limit be- (1 − 2cα − ZcZ ) nehσviEα (1 + 5/Q) − PB 4 cause τE = ∞ for it, and all losses are due to radiation. With (42) and multiplication by T the condition (38) A. Ideal ignition condition, minimum burn temper- transforms into ature, and ideal break-even 2 ∗ 12kT In a first quantitative approach we shall neglect the neτ T = (44) E hσviE (1 + 5/Q) presence of impurities as well as that of the helium ash, α i.e. we set cα = 0, cZ = 0. This way we not only get ∗ The so-called fusion product neτET employed in this a widely used result for the ignition condition but also formula is widely used for characterizing the perfor- one which is very easily comprehensible. Of course this mance of a fusion device because it combines the two can only be a rough approximation because the accu- ∗ quantities neτE (also a measure for the quality of con- mulation of helium ash can, in principle, not be avoided. finement) and T , which both have to be large for igni- cα = 0 is compatible with the equilibrium equations if tion, into a single quantity. Its temperature dependence in (35) we set τα and don’t consider this equation any is shown in Fig. 4 together with the radiation limit. longer. (This is the reason why we introduced a sepa- According to (43) only states below the radiation limit rate confinement time τα.) For the bremsstrahlung we are physically meaningful. The two intersection points 2 have PB = neRI (T ) and the only equation left is the between the radiation limit and the ignition curve de- energy equation (36) which becomes scribe radiative equilibria. The temperature at the left point is the minimum temperature for which ignition is 1 3n kT/τ = n2hσviE (1 + 5/Q) − n2R (T ) (37) possible (about 4.4 keV). Please note that this is only e E 4 e α e I true for plasmas which are transparent for the radiation 2 losses considered; the sun burns at lower temperatures. Dividing it by ne and then solving it with respect to neτE we finally obtain the ideal ignition criterion In the temperature range of a fusion reactor a good approximation for hσvi is provided by [13] 3kT n τ = (38) e E 1 hσvi = 1.1 × 10−24T 2 ms−1,T in eV (45) 4 hσviEα (1 + 5/Q) − RI (T )

20 Figure 4: Curves neτET = f1(Q; T ) (dashed lines) and Figure 5: Curves n τ ∗T = f(T ) for non-ideal ignited ∗ e E neτET = f2(Q; T ) (solid lines) for ideal ignition and equilibria, radiation limit, and boundary of radiative ideal break-even. Also shown is the radiation limit (only equilibria assuming an impurity concentration of f = ∗ Z relevant for the description by τE). (Figure adapted 2% beryllium (Z = 4). (Fig. adapted from Ref. [2].) from Ref. [2].)

case (distinct from other species in the plasma) the Inserting this in (44) yields the ideal conditions ∗ 21 −3 particle and energy source profiles are identical. Since neτET = 3 · 10 m keV s for ignition, e.g. reached 20 −3 ∗ particles are somewhat better confined than energy, a with n = 10 m , T = 10 keV and τE = 3 s, and ∗ 21 −3 value ≥ 5 is expected for the ratio. neτET = 0.5 · 10 m keV s for break-even. B. Non-ideal ignition and break-even The statements made above can now be quantified We shall now discuss the influence of the helium by solving equations (35)-(36) together with (28) and ash and impurities on the conditions for ignition and the scaling ansatz (46). After cα is eliminated from the break-even. equations, one can again derive an equation for neτET this time as a function of T and , that can be put into For τα 6= 0 from (35) we also get cα 6= 0, and since according to (15) each α-particle displaces two fuel par- the form [2] ticles and according to (19) radiates twice as much as ρ = ρ (neτET,T ) (47) the two together, too high alpha particle concentrations Figure 5 shows the ignition curves ρ = const nu- will inevitably cause the nuclear fire to suffocate. Thus, merically obtained from this for Q = ∞. For ρ = 0 welcome as they are with respect to heating, the alpha (corresponding to τα = 0) our previous ideal curves are particles may lead to a dangerous fuel dilution and pro- recovered. For ρ > 0 one obtains closed ignition curves, vide a rather unpleasant pollution if they become too and it was shown in Ref. [2] that one also obtains closed numerous. It is therefore important that they disappear ∗ ignition curves for neτET (T ) if the scaling assumption due to diffusion and convection, thereby unfortunately (46) with τE is being kept. The most important outcome being accompanied by fuel particles. of these calculations is that ignited equilibria exist in Diffusion and convection are the only loss mechanisms a pure D-T plasma only for ρleq15 (or ρ ≤ 10 for an for particles, and there is no mechanism that could impurity concentration of 2% beryllium). If ρ becomes be compared with the loss of energy by radiation. larger, the helium concentration becomes too large and Although the mechanisms of particle and of energy dif- ignition is impossible as predicted by our qualitative fusion are quite different, there is a strong coupling be- arguments. However, the helium concentration require- tween them. The scaling ansatz[1, 2] ment is relaxed if elastic scattering by collisions be- τ /τ = ρ = const. (46) tween helium and D/T ions are taken into account[3]. p E The ignition curves shrink in size with ρ even faster appears as a good approximation for the helium ash with increasing Z and cZ . Modelling of impurity seeded particles in the plasma core because in this particular ITER discharges[7] has shown that the interplay be-

21 Figure 6: Ignition curve ρ = 5 (together with curves ρ = 3 and 0 for comparison) (a) in a n, T -plane and (b) in a βN ,T -plane. βN = β/(I/aB) is the so-called normalized beta). In both diagrams the boundary of stability with respect to thermal instabilities is also shown. The stable regime is to the right of all stability curves. (Figures taken from Ref. [11].). tween He and sputtered impurities may under certain different circumstances have been collected and evalu- circumstances result in a rather weak dependence of ated, applying as constraints certain theoretical criteria Q on He confinement. This would relieve the concern [13, 4, 5]. One expects that the energy confinement time about helium ash removal. τE will depend on design parameters according to scaling laws such as ITER 89-P [6],

0.5 0.85 0.2 1.2 0.3 0.5 −0.5 0.1 V. ITER CONFINEMENT SCALING LAWS AND τE = 0.048fH M I B R a κ P ne TRANSFORMATION OF IGNITION CURVES TO (48) THE N, T AND β, T PLANE where fH is the H-mode enhancement factor (fH = 2.0 in Fig. 6), M the isotopic mass (2.5 for a 50:50 D-T mix- The initially designed Ignition ITER has been re- ture), I the plasma current in MA, B the toroidal mag- placed by a High-Q ITER, the construction of which in netic field in Teslas, R and a the major and minor toka- Cadarache has been decided in June 2005. The design mak radius in meters, κ the elongation of the plasma 20 −3 of the High-Q ITER does not preclude the possibility of cross-section, ne the electron density in 10 m , and ignition but the objective is extended burn with Q ≥ 10 P = PQH + Pext + Pα the net heating power in MW. and with a duration sufficient to reach stationary condi- Using the equilibrium equation (36), P can be replaced 3 tions with respect to the characteristic time scales. Fur- by 2 ntotT/τE and (48) rewritten as: thermore, the design also aims at demonstrating steady- 0.5 0.85 0.2 1.2 0.3 0.52 −0.8 −1.0 state operation using non-inductive current drive with τE = 0.048fH M I B R a κ ne T Q > 5. (49) For the planning of a burning plasma experiment With this relation the ignition contours (47) can be like ITER it is important to have some idea about translated from the neτET,T -plane directly into the what confinement properties may be expected. Theo- ne,T -plane (for details see Ref. [10]; note, however, that retically plasma transport is a very difficult and not there the ITER scaling laws where applied to the en- ∗ yet satisfactorily solved problem, so the answers to this ergy confinement time τE including radiation losses). question must be essentially extrapolated from experi- Fig. 6(a) shows the “ignition curve” ρ = 5 (together mental data. In huge international databases the trans- with ρ = 3 and ρ = 0 for comparison) in a ne,T -plane, 2 port properties of many different tokamaks under many and using β = ntotkT B /2µ0 a similar diagram can be

22 obtained in the β, T -plane (see Fig. 6(b)). The advan- released in the form of radiation and fusion energy dur- tage of representing the ignition curve as ne = ne(T ) ing the burn time, all expressed as specific quantities or β = β(T ) is that the impact of plasma stability lim- per volume, its like the β-limit or the Greenwald density limit can immediately be seen.[8] eth + erad + efus (50) √ 2 eth = 3nekT , erad = CBgff ne T τb 1 (51) VI. BURN STABILITY e n2hσviE τ fus 4 e fus b This sum of energies is converted with efficiency ηth, In order to determine the stability of the burn equi- and in a self-sustained power station it supplies the libria with respect to thermal instabilities one has to thermal energy of the plasma and the radiation losses: solve the time dependent equations (23)-(27) for per- turbations of the equilibrium states. This has exten- (eth + erad + efus) ηth = eth + erad (52) sively been done in Ref. [10], and here only the most important results are quoted. One problem arising in After the explicit expressions for the different energy this context is, how to treat the confinement times dur- terms are inserted, one can solve with respect to neτb ing the evolution of instabilities. One possibility would to obtain the Lawson criterion be to keep them constant at their equilibrium values. 12kT neτb √ (53) Since the typical growth times of instabilities turn out hσviE η / (1 − η ) − 4C g n2 T to be several seconds under this assumption, it appears fus th th B ff e reasonable to assume the validity of the scaling law (49) Similarly one can ask the question: When does a reac- also during this time dependent process because the tor yield the efficiency η? In order to answer this ques- plasma has time enough to adapt to these conditions tion we assume a pulsed operation of the reactor with a which were originally derived for equilibrium states. start-up phase of duration τh for heating the plasma to Redoing the stability calculations with these adapted ignition, and a burning time τb with stationary condi- confinement times appreciably changes the stability be- tions at temperature T . In the start-up phase for each haviour, which for this case is shown in Figs. 6 (a) and volume element a heating energy eh must be supplied (b). Stable behaviour is obtained to the right of the sta- externally, from which a fraction bility boundaries shown in the diagram. States to the left are unstable and undergo a transition to some state eα = ηaeh (54) on the right branch of the corresponding ignition curve is absorbed by the plasma for providing its internal en- ρ = const. ergy and compensating all heat losses (transport and radiation). The net efficiency of the power station is defined through: VII. LAWSON CRITERION AND REACTOR EFFI- η = enet/efus (55) CIENCY CRITERION where efus is the total fusion energy gain per volume (at present, probably not all fusion energy delivered to If the plasma of a fusion reactor is ignited, this does the alpha particles can be envisaged for conversion), not imply that there is also a net energy gain, because and enet is the energy per volume that can be supplied there are energy losses during the initial heating phase, to the mains as electricity. Considering all important and also energy is needed for feeding auxiliary devices energy flows in the reactor station, the following reactor to keep the reactor running. The first one to consider efficiency criterion can be derived: problems of this kind was Lawson who, in 1957, formu- ∗ lated the so-called Lawson criterion[9]. He asked the 1 12kT (1 + τh/τE,h) neτb = (56) question: When does a fusion reactor deliver so much ηa(ηeff − η) hσviEfus energy that it can run self-sustained, i.e. when does it where η ≈ η ≈ 1/3. We can combine this efficiency neither need nor deliver energy? However, in this cal- eff th criterion with the corresponding ideal ignition criterion culation Lawson neglected -particle heating, assumed (44). Dividing the first by the second yields (for Q = ∞) that the plasma was heated from an external source, ∗ took for the discharge pulse length, and took only ac- τ (1 + τh/τ ) b = E,h (57) count of hydrogen bremsstrahlung radiation (which is τ ∗ 5η η (1 − η/η ) small in a tokamak plasma). E a eff eff In order to answer Lawsons question, we consider where Eα/Efus = 1/5 was used. This shows that the the sum of the internal plasma energy and the energy factor by which the burning time τb must exceed the

23 ∗ energy confinement time τE is independent of the tem- [10] REBHAN and U. VIETH. Burn stability and safe perature. operating regime of a tokamak reactor with iter ∗ Assuming τh ≈ τE,h and ηa/ηeff ≈ 1/20 we get scaling. Nucl. Fusion, 37:251–270, 1997. ∗ τb ≈ 8τE/(1 − η/ηeff ). ∗ [11] E. REBHAN and U. VIETH. Parameter depen- For η = ηeff = 0.95 this yields τb ≈ 160τE or τb ≈ 560s ∗ dence of the operating regime and performance of for τE = 3.5s as expected in a fusion reactor. In fact much longer burn times will be required for d-t tokamak reactors in a current-versus-size di- other reasons: A reactor must last for about 25 years at agram. In Proc. 24th Eur. Conf. Berchtesgaden, least in order to repay for the large expenses that are volume 21A, page 1029, 1997. needed for its construction. A burn time of 200 s only [12] D. REITER. Recycling and transport of neutrals. 6 would imply about 4 × 10 start-ups and thus changes this volume. between hot and cold during its lifetime. This is more than the reactor will stand according to all technical [13] J. WESSON. Tokamaks. Oxford Engineering Sci- experience. A reasonable number of changes will be no ence Series No. 48. Clarendon Press, third edition more than about 100 000. In that case a burning cycle edition, 2004. would have to last for about 2 h in order to sum up to a life time of 25 years.

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