Effects of different nuclear reactions on internal tritium breeding in deuterium fusion

Transcription

1 Home Search Collections Journals About Contact us My IOPscience Effects of different nuclear reactions on internal tritium breeding in deuterium fusion This content has been downloaded from IOPscience. Please scroll down to see the full text Nucl. Fusion ( iew the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: This content was downloaded on 02/10/2013 at 18:58 Please note that terms and conditions apply.

2 Effects of different nuclear reactions on internal tritium breeding in deuterium fusion S. Eliezer a,b,z.henis a,j.m.martínez-al b, I. orobeichik a a Plasma Physics Department, Soreq NRC, Yavne, Israel b Instituto de Fusión Nuclear, Madrid, Spain Abstract. A catalytic regime for tritium in deuterium tritium (DT x) fusion is investigated, including ion electron collisions, mechanical expansion, bremsstrahlung radiation and inverse Compton scattering losses. It is shown that the consideration of the three main nuclear reactions of the hydrogen isotopes (D(D,n) 3 He, D(D,p)T and T(D,n) 4 He) only leads to erroneous results for internal tritium breeding. Among the secondary reactions in DT fusion, the reaction 3 He(D,p) 4 He is the most important in the tritium catalytic regime, due to its large contribution to the plasma heating. When all the nuclear reactions are taken into account, internal tritium breeding is found to be possible in a short range of parameters defining the target performance (tritium content parameter x, areal density and ignition temperature). 1. Introduction ery high ignition temperatures are one of the main problems in advanced fuel fusion targets [1]. It has been demonstrated that adding a small amount of tritium can reduce the ignition temperature of deuterium and of advanced fuel inertial fusion targets [2 5]. Tritium will probably be the most significant radiological problem in future DT fusion reactors. Therefore, the possibility of a catalytic regime for tritium, where the need for external tritium breeding is avoided, is very important. It has recently been shown [4] that a small amount of tritium added to a deuterium plasma enables triggering of ignition at less than 10 ke, and internal tritium breeding takes place as the fusion pellet burns up. In this article the effects of different nuclear reactions on the internal tritium breeding in various fuels are investigated. A high density scenario has been chosen for this analysis, i.e. a typical pellet of inertial confinement fusion [6 8]. A full space time numerical simulation of the pellet performance has not been attempted, because this work is aimed at studying the intrinsic features of the nuclear processes taking place inside the plasma. Hence, it is presumed that a pellet of a given fuel has been compressed up to a required density and areal density. The mechanical disassembly of the pellet is taken into account by presuming that the plasma is a sphere whose radius expands at the speed of sound inside the plasma. Therefore, the energy gains computed in this article only refer to the energy generated in the originally heated plasma, without taking into account the fact that much more energy would be generated if fusion ignition were propagated to the surrounding cold fuel. Therefore, the gain for a spark ignition scheme would be much higher than the one calculated in this article. A different ignition model could be used to study these cases, based on the spark model of a central hot core surrounded by cold fuel [5, 9]. In that case, the mechanical expansion loss is negligible if the fusion burn propagation is supersonic, but heat conduction losses to the surrounding cold fuel must be included. Earlier estimates [3, 5] indicated that both models give very similar results on ignition requirements and burnup performance. 2. The simulation model 2.1. Rate equations A time dependent calculation of the fusion process of a DT x pellet, including ion electron collisions, mechanical expansion, bremsstrahlung and inverse Compton scattering losses, was performed. x is the ratio of the tritium to deuterium particle numbers in the initial pellet (at time t = 0).The following nuclear reactions are considered: D+D=n+ 3 He ke (1) D + D = p + T ke (2) T+p=n+ 3 He 764 ke (3) Nuclear Fusion, ol. 40, No. 2 c 2000, IAEA, ienna 195

3 S. Eliezer et al. D+T=n+ 4 He ke (4) T+T=n+n+ 4 He ke (5) 3 He + D = p + 4 He ke (6) 3 He + 3 He = p + p + 4 He ke (7) 3 He+T=n+p+ 4 He ke (8) 3 He + T = D + 4 He ke. (9) The Maxwell averaged reaction rates σv at different temperatures were taken from Ref. [10] and are presented in the Appendix. The total number of particles of species k, N k, is governed by the equation dn k dt = 9 j=1 a j k N j(1)n j(2) σv j 1 (10) where is the volume of the heated plasma, σv j is the Maxwell averaged reaction rate of reaction j and a j k is the number of particles of species k created or destroyed in the reaction j. Six species are considered inthecalculation:n,p,d,t, 3 He and 4 He Energy balance In this context it is important to take into account the different evolutions of the ion and electron temperatures. Ions and electrons in a reacting plasma evolve at different temperatures for two main reasons. First, bremsstrahlung radiation is emitted only by electrons. In previous studies [2, 4], the bremsstrahlung loss was assumed to be the main energy sink. However, at high electron temperatures the electron energy loss by the inverse Compton effect might become important. In this article, we include a simple model for the estimation of the inverse Compton effect, which reduces the electron temperature. Second, at high electron temperatures, the fusion energy will be deposited mainly into the ions. Therefore, the ion temperature will be higher than the electron temperature and an energy flow from the ions to the electrons by collisions is expected. The evolution of ion and electron temperatures is assumed to be governed by the energy balance equations [2, 4]. The equation of energy balance forionsisgivenby 3 d 2 dt (N it i )= 9 j=1 k=1 6 f j k ωj k E jn j(1) N j(2) σv 1 P ie N it i 4πR 2 1 (t)c s (11) where T i is the ion temperature, E j is the energy yield of reaction j, ω j k is the fraction of E j is the energy yield of reaction j, ω j k is the fraction of E j carried by product k (for instance, 0.2 for the alpha particle in the DT reaction), f j k is the fraction of the energy of the product k created in reaction j that is deposited into the plasma ions, P ie is the ion electron energy exchange term, R(t) is the pellet radius, c s is the speed of sound and N i is the total number of ions, N i = 6 N k. (12) k=2 k = 1 stands for the neutron species and, therefore, is not included in Eq. (12). f j 1 in Eq. (11) was assumed to be one third (of the neutron energy) if the areal density ρr was greater than 5 g/cm 2 [11]. In endothermic reactions, the corresponding energy was subtracted from the ion energy. It was also assumed that all ions have the same temperature. The equation of energy balance for electrons is given by 3 d 9 2 dt (N et e )= + P ie j=1 k=2 6 (1 f j k )ωj k E 1 jn j(1) N j(2) σv j P B P C N e T e 4πR 2 1 (t)c s (13) where N e is the number of electrons, T e is the electron temperature, P B is the bremsstrahlung term and P C is the inverse Compton scattering term. The ion electron energy exchange term is given by P ie (ke cm 3 /s) = N e 6 k=2 ln Λ ek Zk 2 N k m k T i(ke) T e (ke) T e (ke) 1.5 (14) where m k and Z k are the mass number and the charge of nuclei k, respectively. Equation (14) has been calculated using Ref. [12] and it is expressed in appropriate units for use in the power balance equations (ke/s) of the plasma. The Coulomb logarithm, ln Λ ek,is [ (Ne ) ] 0.5 ln Λ ek =23 ln Z k T e (e) 1.5. (15) 196 Nuclear Fusion, ol. 40, No. 2 (2000)

4 Article: Internal tritium breeding in deuterium fusion The bremsstrahlung loss, P B,isgivenby P B (ke cm 3 /s) = N e N k Zk 2 T e(ke) 0.5 k=2 ( 1+ 2T ) e(ke) (16) where 2T e (ke)/511.1 is a relativistic correction term. In the mechanical expansion term, the speed of sound is computed using γp c s = (17) ρ where ρ is the density, γ =5/3 andp is the total pressure, P = 1 (N it i + N e T e ). (18) The radius of the pellet is governed by R(t) =R(t t)+φc s t (19) where φc s is the effective outer radius speed which simulates the stagnation phase after the implosion of the pellet, such that { φ(t) for φ(t) <φ0 φ = (20) 1 for φ(t) φ 0 where φ(t) =1 N D (t)/n D (0) is the burnup fraction and φ 0 =0.5, in accordance with pellet performance simulations [13 15]. The fraction f j k of the energy of the fusion product k created in reaction j that is deposited into the plasma ions was calculated according to Dawson [16]. As the reaction product energy E j is much larger than the electron temperature, it can be shown that [16, 4] f j k = 1 ω j k E j ( me ω j k Ej m k T i dw [ ) 1/2 N e π ( W T e ) 3/2 m k i N iz 2 i /m i ] 1. (21) Equation (21) can be integrated analytically to give ( f j α 2 k = 1 α 2W tan 3 α 3 α2 3 ln α2 αw + W 2 ) W =(ω j k Ej)1/2 (α + W ) 2 (22) W =T 1/2 i where [ ( ) ] 1/2 1/3 1.6 me N e 1 α = π m k m k i N izi 2/m. (23) i Te 1/2 The energy gain is defined as the ratio between the total energy created by the fusion reactions, 9 Efus tot = 1 dt E j σv j N j(1) N j(2) (24) 0 j=1 and the energy contained in the originally heated plasma, E input = 3 2 [N e(0)t e (0) + N i (0)T i (0)] (25) i.e. GAIN = Etot fus. (26) E input The catalytic regime of tritium is obtained when the final amount of tritium is slightly higher than the initial amount. In this case, external tritium breeding is avoided, and is replaced by internal tritium breeding. The internal tritium breeding (ITB) is defined as ITB = N T (t = ) N T (t =0). (27) 2.3. The inverse Compton effect At high electron temperatures, the electron energy loss by the inverse Compton effect might be important to both the internal tritium breeding ratio and the pellet gain. Gsponer and Hurni [17] suggested that the three temperature Hurwitz model [18] can be used to calculate the radiation temperature evolution due to bremsstahlung and Compton effects. The main assumption of this model is that for any electron temperature all the created bremsstrahlung radiation is assumed to become Planck like. This assumption might be reasonable in the limit of optically thick plasma, but is unjustified for the optically thin plasma. For the optically thick plasma the mean free path of bremsstrahlung photons is smaller than the plasma radius and radiation can be treated as a photon gas in thermal equilibrium. Thus, a black body radiation spectrum is obtained and the radiation temperature can be defined. In this case, the inverse Compton effect can significantly reduce the electron temperature and increase accordingly the radiation temperature. However, in the limit of optically thin plasma, radiation created by bremsstrahlung is unlikely to become Planckian. Nuclear Fusion, ol. 40, No. 2 (2000) 197

5 S. Eliezer et al. Figure 1. The solid line shows the ratio between the number of photons with hν > T e (Eq. (34)) obtained from the bremsstrahlung radiation spectrum (Eq. (31)) and from the black body radiation spectrum (Eq. (32)) as a function of the electron temperature. The chain line shows the bremsstrahlung optical depth (Eq. (28)) as a function of the electron temperature. An areal density ρr = 10 g/cm 2, a density ρ = 5000 g/cm 3 and a ratio of tritium to deuterium particle numbers x =0.02 (n e = n i = cm 3 ) were used for this computation. For the optically thin plasma, the photon spectrum will be determined by the bremsstrahlung emission spectrum. Thermal equilibration of photons and electrons does not take place, since hot photons are leaving the plasma before the equilibrium is established. One can estimate the optical depth of plasma to bremsstrahlung as [12] τ B (ρr)2 (28) RTe 7/2 where ρr is measured in g/cm 2, R is measured in microns and the electron temperature, T e,ismeasured in kiloelectronvolts. Plasmas are optically thick (opaque) for τ B 1 and optically thin (transparent) for τ B 1. Using Eq. (28) one finds that for typical initial plasma parameters used in this article, ρr = 10 g/cm 2, R = 50 µm andt e = 10 ke, the plasma is initially optically thick but rapidly becomes optically thin as the electron temperature rises (Fig. 1). Therefore, for the cases considered here, the three temperature model is not realistic and will greatly overestimate radiation losses due to the inverse Compton effect. An accurate treatment of the inverse Compton effect can be obtained by using a radiation transport model which would span the optically thick to optically thin plasma transition. Such a model, however, would involve a solution of complicated equations of radiation transport [19 21], which is out of the scope of this article. To estimate the radiation losses due to the inverse Compton effect without fully solving the complicated problem of radiation transport, a simple model based on the optically thin limit is adopted. Nevertheless, we will also see that our model is also accurate for the optically thick plasma in the early stages of the microexplosions, because the electron and radiation temperatures are very similar then. In the optically thin plasma limit, the total radiation energy density, E r, can be estimated in terms of the bremsstrahlung loss rate, P B, if one assumes that the average travelling time of the bremsstrahlung photons through a thin plasma of radius R is t R/c. Then, E r = P B R 2 (29) c where c is the speed of light in vacuum. The bremsstrahlung emission spectrum is given by [22] J ν (ke/cm 3 ) = Z 2 1 n e n i T e (ke) 1/2 e hν/kt e (30) where n e = N e / and n i = N i /. Integration over the whole spectrum of Eq. (30) gives the total intensity of bremsstrahlung radiation in Eq. (8) (up to a relativistic correction). Using the same approximation as that in Eq. (29) we define the bremsstrahlung radiation energy density as E Br R (ν) =J ν c. (31) We would like to compare this radiation energy density with the radiation energy density of a black body given by E BB (ν) = 8π (hν) 3 c 3 h 2 (32) e hν/kt r 1 where T r is the radiation temperature. For the black body radiation, the total radiation energy density is E r (ke/cm 3 )= 4σ c T 4 r (33) where σ is the Stefan Boltzmann constant. For the inverse Compton effect, the radiation energy density of hot photons with hν > T e is important: E IC = hν>t e E(ν)dν. (34) 198 Nuclear Fusion, ol. 40, No. 2 (2000)

6 Article: Internal tritium breeding in deuterium fusion In Fig. 1 we compare the number of photons with hν > T e obtained from the bremsstrahlung radiation spectrum (Eq. (31)) and the black body radiation spectrum (Eq. (32)) as a function of the electron temperature. As one can see from Fig. 1, for T e < 60 ke the black body radiation spectrum overestimates the number of hot photons. For T e > 60 ke, however, the number of hot photons in the bremsstrahlung spectrum is larger than that in the black body radiation spectrum. To estimate the radiation losses due to the inverse Compton effect in the optically thin plasma limit we define the instantaneous radiation temperature, T r, of bremsstrahlung photons in terms of E r by equating Eq. (29) and Eq. (33). The system defined in this way has the same radiation energy as that created by the bremsstrahlung emission, but with a Planckian distribution. Clearly, this assumption is not correct for the optically thin plasma. In fact, in the framework of this approximation, for T e. 60 ke, the electron energy losses due to the inverse Compton scattering are overestimated since in this electron temperature range the number of hot photons in the black body spectrum is larger than that in the bremsstrahlung spectrum. Therefore, our model can provide an upper limit for the electron energy losses due to the inverse Compton scattering in an optically thin plasma with relatively low electron temperatures. For T e & 60 ke the black body distribution underestimates the number of hot photons. However, we assume that for such high electron temperatures only a fraction of the total number of photons are in instantaneous equilibrium with a temperature T r, whereas the rest is leaving the plasma without having a chance to be scattered by electrons. This fraction is given by the Planck distribution and it becomes smaller as the electron temperature rises. This assumption is consistent with the fact that the plasma optical thickness also decreases very rapidly with T e (see τ B in Fig. 1). From the Planck distribution for the photons we can easily estimate the inverse Compton scattering contribution. Provided the photons are of low energy, hν/mc 2 1, and the electron temperature moderate, kt e /mc 2 1, the energy lost by the electrons in inverse Compton scattering is given by Ref. [23] as ( ) 8 [Te (ke) T r (ke)] P C (ke/s) = 4E r N e c 3 πr2 e (35) where 8 3 πr2 e is the Thomson cross-section and r e is the classical electron radius. From Eq. (27) it follows that the electron photon energy exchange grows as the difference between the electron and photon temperatures increases. This is compensated, however, by a small radiation energy density that, in our model, is proportional to the bremsstrahlung radiation rate. In addition, the instantaneous radiation temperature is kept low because most of the photons leave the system without interaction in an optically thin plasma. Although this simplified model is based on the optically thin plasma limit, using Eqs (16), (29) and (33) one can verify that for the typical initial parameters used in this article, ρr = 10 g/cm 2, ρ = 5000 g/cm 3 and T e = 10 ke (for which the plasma is optically thick), the radiation temperature is about equal to the electron temperature. For an optically thick plasma, this can be understood by assuming a time independent thermal equilibrium between the bremsstrahlung source and the black body losses, P B = σt r 4 4πR2. (36) This equation yields a value of T r equal to the one derived by Eqs (29) and (33) up to a factor of (4/3) 1/4 =1.07. Therefore, this model gives a satisfactory approximation to the initial conditions of an optically thick plasma. 3. The 3 He D reaction effect The aim of our simulations was to investigate the importance of the different nuclear reactions taking place in the DT burn (Eq. (1)) to the catalytic tritium regime. In addition, the contribution of the inverse Compton effect, which was ignored in previous studies [2, 4], is analysed. We perform our numerical calculations for three different simulation schemes. First, only three main nuclear reactions between deuterium and tritium are considered (reactions (1), (2) and (4) in Eq. (1)). The results of these simulations are compared with those of the second scheme, where an additional secondary reaction between deuterium and 3 He is considered (i.e. reactions (1), (2), (4) and (6)). Finally, all nine reactions (1) (9) are taken into consideration. The rate equations (10), (11) and (13) were integrated numerically using the fourth order Runge Kutta method [24]. The number of particles of each species and the ion, the electron and the radiation temperatures were calculated as a function of time. A time step of 0.01 ps was used. Exact sum rules [4] N D (0) + N T (0) = N D (t)+n T (t)+n p (T ) +2N3 He(t)+2N4 He(t) (37) Nuclear Fusion, ol. 40, No. 2 (2000) 199

7 S. Eliezer et al. and N T (0) = N n (t)+n T (t) N p (T ) N3 He(t) (38) were used to check the numerical accuracy. The results were checked to converge with respect to the time step. The energy gain and the internal tritium breeding as a function of the ratio of the tritium to deuterium particle numbers in the initial pellet are shown in Fig. 2. As one can see, the three primary DD and DT reactions contribute mainly to both the gain and the amount of created tritium for x>0.2. Therefore, when non-catalytic amounts of tritium are considered, there is no need to take into account secondary nuclear reactions. However, for small initial tritium to deuterium ratios, the results are quite different. For x<0.1, when only the main three reactions are considered, the internal tritium breeding is much lower than in the case of all the nine reactions. The inclusion of the additional 3 He D reaction improves the results dramatically. The 3 He D reaction changes both the concentration of deuterium and the amount of energy released from fusion. When the 3 He D reaction is considered, the concentration of deuterium is reduced. Since tritium production depends on ND 2 (reaction (2)) and tritium burn depends on N D (reaction (4)), from this argument alone, the tritium content should decrease. However, the 3 He D reaction affects tritium production by making an important contribution to the charged particle energy deposition. Thus, the temperature is increased and more tritium is produced in the D(D,p)T reaction. For small values of x, the results obtained with four and nine reactions are very similar, except that in the four reactions case the amount of created tritium is slightly overestimated, since other secondary reactions consume tritium. To illustrate the effect of the 3 He D reaction in the catalytic tritium regime, the temperature evolution for small and large initial amounts of tritium for all the three cases considered (three, four and nine reactions) is depicted in Fig. 3. As one can see in Figs 3(d, e, f) for x = 0.2, the temperature evolution differs only slightly when different numbers of reactions are considered. However, for x = (Figs 3(a, b, c)), the maximal ion temperature is more than 100 ke lower when only primary reactions are considered (Fig. 3(c)). Therefore, for a large enough initial amount of tritium the energy deposition of the DT reaction is dominant. When the initial tritium content is reduced, the contribution of 3 He D reaction becomes very important. Figure 2. (a) Energy gain (Eq. (26)) plotted versus ratio of tritium to deuterium particle numbers in the initial pellet when nine reactions (Eqs (1) (9), solid line), four reactions (three primary reactions and the 3 He D reaction, dashed line) and three primary reactions (chain line) are considered. The areal density of the initial pellet was ρ 0R 0 =12.5 g/cm 2 and the initial temperature of the ions and electrons T i(0) = T e(0) = 10 ke. (b) The internal tritium breeding as defined in Eq. (27) as a function of the ratio of the tritium to deuterium particle numbers in the initial pellet. The inset is zoomed from near small x values. This argument holds also when other plasma parameters change. In Figs 4, 5 and 6 we compare the energy gain and the internal tritium breeding for x = (which corresponds to maximal internal tritium breeding in Fig. 2) as a function of the initial areal density, the initial density and the initial temperature, respectively. As one can see in Figs 4, 5 and 6, the primary reactions alone give a very poor estimate for the internal tritium breeding for all plasma values in the tritium catalytic regime. However, since the effect of the 3 He D reactions become significant only after the fusion process begins, the ignition conditions (smallest values of the areal density, density and initial temperature for which fusion is obtained) are very well reproduced by the primary reactions 200 Nuclear Fusion, ol. 40, No. 2 (2000)

8 Article: Internal tritium breeding in deuterium fusion Figure 3. Temperature evolution of ions T i (solid line), electrons T e (dashed line) and photons T r (chain line) as a function of time: (a) x = with nine reactions considered, (b) x = with four reactions, (c) x = with three reactions, (d) x =0.2 with nine reactions, (e) x =0.2 with four reactions, (f) x =0.2 with three reactions. The initial density, areal density, ion and electron temperatures are as in Fig. 2. alone. To illustrate this, in Fig. 7 we plot the ratio between the tritium content at time t and the initial tritium content as a function of time. As one can see, the deviation of the tritium content obtained from the primary reactions takes place only after about 8 ps, when the plasma is heated up to about 50 ke (Fig. 3(a)). At this temperature, the crosssection of the 3 He D reaction is already large enough to give a significant contribution to the energy equation. Therefore, it is clear that the 3 He D reaction must be taken into consideration in the studies of DT x fusion with small amounts of tritium. In order to gain an insight into the dependence of the internal tritium breeding on the initial conditions, energy gain and internal tritium breeding are plotted in Figs 8 13 as a function of the initial temperature, the initial areal density and the ratio of the tritium to deuterium particle numbers in the initial pellet. As one can see in Figs 8 and 9, for x =0.0112, in the region of low initial temperature (10 ke) and areal density of about 12.5 g/cm 2 the amount of the produced tritium is equal to the initial tritium content and, at the same time, the energy gain is high. One can achieve higher values of the internal tritium breeding for higher initial temperatures, but it will be at the expense of the energy gain. Figure 10 demonstrates that a higher internal tritium breeding is obtained for smaller initial tritium to deuterium ratios. In addition, our simulations show that the areal density of 12.5 g/cm 2 is an optimal one for the internal tritium breeding (Figs 8 and 10). Figures 12 and 13 show that a high internal tritium breeding is obtained in the range of 1% of initial tritium content. The internal tritium breeding grows as the initial temperature increases (up to ITB = 2.73 for x =0.006 and T i =50ke.However, for high initial ion temperatures, the energy gain decreases (Fig. 13). Another question addressed in this article is the effect of inverse Compton scattering on the internal tritium breeding. At very high temperatures, various relativistic and radiation processes such as inverse bremsstrahlung, electron electron bremsstrahlung and Compton scattering may become important. As pointed out by Gsponer and Hurni [17], the most important of these phenomena is the electron energy Nuclear Fusion, ol. 40, No. 2 (2000) 201

9 S. Eliezer et al. Figure 4. (a) Energy gain (Eq. (18)) plotted versus areal density of the initial pellet for nine reactions (solid line), four reactions (dashed line) and three primary reactions (chain line). The initial tritium to deuterium ratio was x = , the density of the initial pellet ρ 0 = 5000 g/cm 3 and the initial temperature of ions and electrons T i(0) = T e(0) = 10 ke. (b) The internal tritium breeding as defined in Eq. (19) as a function of the density of the initial pellet. loss by inverse Compton scattering. To illustrate the importance of the inverse Compton scattering, the contributions of different terms to the energy balance are compared in Fig. 14. These terms are: the total energy created by fusion defined as E fusion = 9 j=1 E j N j(1) N j(2) σv j 1. (39) The fusion energy deposited on charged particles defined as E fusion (charged) 9 6 = j=1 k=2 ω j k E jn j(1) N j(2) σv j 1. (40) The plasma expansion work defined as (N i T i + N e T e )4πR 2 1 (t)c s (41) Figure 5. (a) Energy gain (Eq. (26)) plotted versus density of the initial pellet for nine reactions (solid line), four reactions (dashed line) and three primary reactions (chain line). The initial tritium to deuterium ratio was x = , the areal density of the initial pellet ρ 0R 0 =12.5 g/cm 2 and the initial temperature of ions and electrons T i(0) = T e(0) = 10 ke. (b) The internal tritium breeding as defined in Eq. (19) as a function of the density of the initial pellet. and the bremsstrahlung and inverse Compton terms defined in Eqs (16) and (35), respectively. As one can see in Fig. 14, the inverse Compton scattering term (chain line) becomes the dominant energy loss factor after t 9pswhenT e > 70 ke (Fig. 3(a)). Therefore, at these electron temperatures, the electron photon energy exchange is very significant. The difference between the total fusion energy (solid line) and the energy deposited into the charged particles (dashed line) indicates that a large fraction of the fusion energy is taken out by neutrons. The discontinuity in the expansion work term (dotted line) is due to the stagnation phase simulation. After this time, the pellet is assumed to expand at the speed of sound (φ = 1 in Eq. (20)). In Fig. 15 the ion temperature and the relative tritium content are compared with and without the inverse Compton scattering contribution. As one can see, the inverse Compton scattering significantly lowers the temperature and, therefore, the 202 Nuclear Fusion, ol. 40, No. 2 (2000)

10 Article: Internal tritium breeding in deuterium fusion Figure 6. (a) Energy gain (Eq. (26)) plotted versus initial ion temperature for nine reactions (solid line), four reactions (dashed line) and three reactions (chain line). The initial electron temperature and the initial ion temperature are assumed to be equal, T e(0) = T i(0). The initial tritium to deuterium ratio was x = , the density of the initial pellet ρ 0 = 5000 g/cm 3 and the areal density of the initial pellet ρ 0R 0 = 12.5 g/cm 2. (b) The internal tritium breeding as defined in Eq. (27) as a function of the initial ion temperature. internal tritium breeding. Thus, inverse Compton scattering must be included in the DT pellet simulations. Indeed, for the parameters considered in Ref. [4] (x =0.02, ρ = 5000 g/cm 3, ρ 0 R 0 = 10 g/cm 2 and T i (0) = T e (0) = 10 ke), the final ratio of tritium content becomes smaller than 1 (0.65, according to our simulations), when the inverse Compton scattering is included. This is in agreement with the results reported in Ref. [17]. However, according to our simulations, one can find plasma parameters for which the internal tritium breeding effect does exist. Figures 2, 4, 5 and 6 show that one should use a higher compression (ρ 0 R 0 =12.5 g/cm 2 instead of ρ 0 R 0 = 10 g/cm 2 used in Refs [4, 17]) and a lower initial tritium content (x = instead of x =0.02 used in Refs [4, 17]). It must be taken into account that the inverse Compton effect does not create photons. It only heats them up. Photons are created in a plasma Figure 7. Ratio between the number of tritium particles at time t and the initial number of tritium particles as a function of time for nine reactions (solid line), four reactions (dashed line) and three reactions (chain line). The initial tritium to deuterium ratio was x = , the density of the initial pellet ρ 0 = 5000 g/cm 3,theareal density ρ 0R 0 =12.5 g/cm 2 and the initial temperature of ions and electrons T i(0) = T e(0) = 10 ke. Figure 8. Internal tritium breeding as defined in Eq. (27) as a function of the initial ion temperature and of the initial areal density. The initial electron temperature and the initial ion temperature are assumed to be equal, T e(0) = T i(0). The initial tritium to deuterium ratio was x = and the density of the initial pellet ρ 0 = 5000 g/cm 3. The nine reactions given by Eqs (1) (9) are considered. by bremsstrahlung, and they can be upscattered by the hot electrons before leaving. At the same time, some radiation reabsorption can take place inside the plasma mainly in the surrounding cold fuel, which will be preheated in that way, so stimulating Nuclear Fusion, ol. 40, No. 2 (2000) 203

11 S. Eliezer et al. Figure 9. Energy gain as defined in Eq. (26) as a function of the initial ion temperature and of the initial areal density. The initial electron temperature and the initial ion temperature are assumed to be equal, T e(0) = T i(0). The initial tritium to deuterium ratio was x = and the density of the initial pellet ρ 0 = 5000 g/cm 3. The nine reactions given by Eqs (1) (9) are considered. Figure 11. Energy gain as defined in Eq. (26) as a function of the ratio of the tritium to deuterium particle numbers in the initial pellet and the initial areal density. The density of the initial pellet was ρ = 5000 g/cm 3 and the initial temperature of ions and electrons T e(0) = T i(0) = 15 ke. The nine reactions given by Eqs (1) (9) are considered. Figure 10. Internal tritium breeding as defined in Eq. (27) as a function of the ratio of the tritium to deuterium particle numbers in the initial pellet and the initial areal density. The density of the initial pellet was ρ = 5000 g/cm 3 and the initial temperature of ions and electrons T e(0) = T i(0) = 15 ke. The nine reactions given by Eqs (1) (9) are considered. Figure 12. Internal tritium breeding as defined in Eq. (27) as a function of the ratio of the tritium to deuterium particle numbers in the initial pellet and the initial ion temperature. The initial electron temperature and the initial ion temperature are assumed to be equal, T e(0) = T i(0). The density of the initial pellet was ρ 0 = 5000 g/cm 3 and the initial areal density ρ 0R 0 =12.5 g/cm 2. The nine reactions given by Eqs (1) (9) are considered. ignition propagation [25]. An accurate calculation of all radiation transport phenomena is out of the scope of this article, but the main mechanisms affecting the temperature evolution of deuterium fuelled pellets have been included in our computational model. 4. Conclusions We have shown that the secondary nuclear reaction 3 He (D,p) 4 He should be taken into account in studies of DT fusion with small initial tritium to deuterium ratios. The 3 He D reaction influences the 204 Nuclear Fusion, ol. 40, No. 2 (2000)

12 Article: Internal tritium breeding in deuterium fusion Figure 13. Energy gain as defined in Eq. (26) as a function of the ratio of the tritium to deuterium particle numbers in the initial pellet and the initial ion temperature. The initial electron temperature and the initial ion temperature are assumed to be equal, T e(0) = T i(0). The density of the initial pellet was ρ 0 = 5000 g/cm 3 and the initial areal density ρ 0R 0 =12.5 g/cm 2. The nine reactions given by Eqs (1) (9) are considered. Figure 14. Total energy created by fusion as defined in Eq. (39) (solid line), fusion energy deposited on charged particles as defined in Eq. (40) (dashed line), bremsstrahlung radiation loss as defined in Eq. (16) (long dashed line), inverse Compton scattering loss as defined in Eq. (35) (chain line) and plasma expansion work as defined in Eq. (41) (dotted line) as a function of time. All nine nuclear reactions are considered. The plasma parameters are as in Fig. 7. amount of produced tritium since it makes an important contribution to the charged particle energy deposition and to the temperatures. This effect is particularly important when the initial tritium to Figure 15. (a) Ion temperature as a function of time with (solid line) the inverse Compton scattering term (P C in Eqs (13) and (35)) and without it (dashed line). All nine nuclear reactions are considered. The plasma parameters are as in Fig. 7. (b) The ratio between the number of tritium particles at time t and the initial number of tritium particles as a function of time with the inverse Compton scattering term (solid line) and without it (dashed line). deuterium ratio is small. In addition, the energy loss of electrons by inverse Compton scattering is important at high electron temperatures. The ion temperature and the internal tritium breeding are significantly reduced when the inverse Compton scattering is taken into account. However, a careful choice of the plasma parameters allows one to reach internal tritium breeding, even when the Compton scattering losses are dominant. In general, the higher the initial density and the initial temperature, the higher the internal tritium breeding. The latter also increases as the initial tritium to deuterium ratio decreases, but in this case the ignition requirements become very demanding. An advisable range for this ratio is found slightly above 1% of initial tritium content in the capsule fuel. The importance of tritium internal breeding is not only restricted to the case of the actual catalytic regime, i.e. when the internal breeding index is 1 or slightly higher. Even when the index is below 1, Nuclear Fusion, ol. 40, No. 2 (2000) 205

13 S. Eliezer et al. internal breeding can have a very positive effect in reducing the tritium inventory in a fusion reactor, and therefore it would reduce the technical complexity of the blanket and the radiological risk, which will be mainly related to the tritium inventory. It must be taken into account that the reprocessing time τ y of the pellet debris (to recover tritium) will be much shorter than the reprocessing time τ z of the tritium produced in the outer blanket, because the latter would involve a much higher volume of breeder material (a lithium compound) and a much lower tritium concentration. The total tritium inventory I could be expressed as the addition of the inventory in the internal stream I y and that in the outer blanket I z. Let us assume the reactor consumes n mg/s of deuterium (in several pellets, of course). The mass flow rate of tritium in the pellet factory will be nx, ifthefuelisdt x. If i stands for the internal breeding ratio, the mass flow rate of tritium in the pellet debris reprocessing would be nix. In order to feed the pellet factory, the outer blanket has to supply n(1 i)x. So, the total inventory can be expressed as I = nx[iτ y +(1 i)τ z ] (42) Of course, the inventory is proportional to the power, whichisinturnproportionalton. Inourcase,1 x>0, with a typical value of a few per cent. In a futuristic scenario of reactors, τ y will be of the order of hours, while τ z could be longer than one month. Hence, even for i 0.9, very large reductions of tritium inventory will be found, as compared with the standard case (x =1)wherei is very small. A numerical example can help underline the importance of this effect. As a reference point, let us take stoichiometric DT (x = 1) with 17.6 Me/fusion of effective yield. In this case i 0.6 (because not all the tritium is burned up) and the burnup fraction ϕ =0.4 for the initial nuclei. Let us assume τ y =3handτ z =15d.Inthiscase,thetritium inventory is found to be 2.5 g/mw (thermal) of fusion power. Close to the breeding regime, we can have x = 0.02, with i =0.9 and again a burnup fraction (for deuterium) of 0.4. Now, the effective fusion yield is about 6 Me per fusion (taking into account DD secondary reactions). For the same values of τ y and τ z, a tritium inventory of 35 mg/mw is found, i.e. 70 times lower than the one formerly found for stoichiometric DT. The result is still much more spectacular for the case of actual internal breeding (i =1),wherean inventory as low as 1 mg/mw is found, i.e times lower than the value of the stoichiometric case. Because of this, future work on conceptual designs of inertial fusion reactors would have to pay attention to this feature. By using more sophisticated numerical simulation tools, more accurate values of i can be computed, and a trade-off process can be carried out to determine the optimum working point for a given reactor, taking into account not only driver requirements and energy gain but also the beneficial effects of high internal tritium breeding. Appendix The Maxwell averaged reaction rates (in units of cm 3 /s) versus temperature taken from Ref. [10], and used in the simulations presented here, are given in Table 1. Table 1. Maxwellian reactivities (in units of cm/s) T D D T p D T T T D 3 He 3 He 3 He 3 He T 3 He T (ke) (1) or (2) (3) (4) (5) (6) (7) (8) (9) e e-5 3.3e-6 1e e e e-3 7.1e-5 1.4e-7 0 1e e e-4 6.7e e e e-2 2.3e e e e-2 3.8e e-3 1e e e-3 5.3e e e e e e Nuclear Fusion, ol. 40, No. 2 (2000)

14 Article: Internal tritium breeding in deuterium fusion The reaction rates of nuclear reactions of identical particles ((1), (2), (5) and (7)) were divided by 2. At temperatures different from those given in Table 1, the reaction rates were calculated by linear interpolation on the logarithmic scale of both σv and the temperature. References [1] Martínez-al, J.M., elarde, G., Ronen, Y., Nuclear Fusion by Inertial Confinement (elarde, G., et al., Eds), CRC Press, Boca Raton, FL (1993) Ch.1. [2] Eliezer, S., Henis, Z., Martínez-al, J.M., Nucl. Fusion 37 (1997) 985. [3] Tahir, N.A., Hoffmann, D.H.H., Fusion Technol. 33 (1998) 164. [4] Eliezer, S., Henis, Z., Martínez-al, J.M., Piera, M., Phys. Lett. A 243 (1998) 311. [5] Martínez-al, J.M., Eliezer, S., Henis, Z., Piera, M., Nucl. Fusion 38 (1998) [6] Duderstadt, J.J., Moses, G., Inertial Confinement Fusion, Wiley, New York (1981). [7] Guskov, S.Yu., Rosanov,.B., Nuclear Fusion by Inertial Confinement (elarde, G., et al., Eds), CRC Press, Boca Raton, FL (1993) Ch. 12. [8] Brueckner, K.A., Jorna, S., Rev. Mod. Phys. 46 (1974) 325. [9] Tabak, M., Nucl. Fusion 36 (1996) 147. [10] Feldbacher, R., Alternate Physics Program Datalib, Rep. NDC (ANS)-12/G, Tech. Univ. of Graz (1987). [11] Martínez-al, J.M., Fusion Technol. 17 (1990) 476. [12] Huba, J.D., NRL Plasma Formulary, Naval Research Lab., Washington, DC (1994). [13] elarde, G., et al., Laser Part. Beams 4 (1986) 349. [14] Taihr, N.A., Long, L.A., Nucl. Fusion 23 (1983) 887. [15] Martínez-al, J.M., Eliezer, S., Piera, M., Laser Part. Beams 12 (1994) 681. [16] Dawson, J.M., Fusion, ol. 1, Part B, Academic Press, New York (1981) Ch. 16. [17] Gsponer, A., Hurni, J.P., Comment on Deuterium tritium fusion reactors without external tritium breeding, Reprint, Independent Scientific Research Institute, Geneva (1998). [18] Fraley, G.S., Phys. Fluids 17 (1974) 474. [19] Pomraning, G.C., The Equations of Radiation Hydrodynamics, Pergamon Press, Oxford (1973). [20] Kershaw, D.C., Prasad, M.K., Beason, J.D., J. Quant. Spectrosc. Radiat. Transfer 36 (1986) 4. [21] Rose, S.J., J. Quant. Spectrosc. Radiat. Transfer 55 (1996) 707. [22] Zel dovich, Ya.B., Raiser, Yu.P., Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, New York (1996). [23] Weymann, R., Phys. Fluids 8 (1965) [24] Koonin, S.E., Meredith, D.C., Computational Physics, Addison-Wesley, New York (1990). [25] Eliezer, S., Martínez-al, J.M., Laser Part. Beams 16 (1998) 581. (Manuscript received 10 December 1998 Final manuscript accepted 7 October 1999) address of J.M. Martínez-al: mval@etsii.upm.es Subject classification: A0, Ii Nuclear Fusion, ol. 40, No. 2 (2000) 207