By T. Hesselberg Jensen, O. Kofoed-Hansen and С F. Wandel*

Transcription

1 P/2506 Denmark Energy Balance in a Thermonuclear Reacting Plasma containing Deuterium, Tritium and Reaction Products under Isothermal Pulsed or Steady-State Conditions By T. Hesselberg Jensen, O. Kofoed-Hansen and С F. Wandel* The energy balance in a thermonuclear reacting plasma containing deuterium and tritium has been discussed in several places. 1 " 4 By comparing the power escaping as electromagnetic radiation with the power transferred to charged reaction products in the nuclear reactions, a lower limit on the plasma temperature necessary to maintain a self-sustaining reaction is obtained. 1 The limiting temperatures are 350 million K in the case of pure deuterium and 50 million K in the case of deuterium and tritium in equal amounts. Especially when only small fuel burnups are considered, it is important to take into account, in the energy balance, the amount of energy spent in heating the fuel to the reaction temperature. This point has been investigated by Lawson 2 where a pulsed system has been considered in which the fuel is instantaneously heated to the reaction temperature and then allowed to react in a definite time interval after which the plasma is again cooled to essentially zero temperature. If one can make the apparently realistic assumption that the heat content of the plasma, like the radiative power, can be transformed into plasma heat (in a new pulse) only with an efficiency considerably less than unity, then it can be concluded that not only must the temperature exceed a certain limit, but also the reaction must be sustained long enough for a definite fraction of the fuel to be burnt. It is clear that as soon as a considerable fraction of the fuel is burnt the reaction rates will decrease and, because of the higher nuclear charge of the reaction products, the radiative power will increase. Both of these effects will tend to reduce the power economy of the system. The first of these effects has been treated by Lacombe et al., 3 who have considered a pulsed system and investigated the time dependence of the power densities due to primary and secondary nuclear processes. In Ref. 4, we have tried to incorporate all of the above-mentioned effects by treating the problem of a steady-state thermonuclear reaction involving deuterium, tritium and their reaction products. To keep the reacting plasma in a steady state it is in general * Danish Atomic Energy Commission Research Laboratory, Riso. 431 necessary to exchange matter and energy continuously with the surroundings. The rates of these exchanges are completely determined by only three independent parameters. These parameters are the temperature of the plasma, the tritium enrichment in the fuel and the product of the deuterium density and the plasma renewal time. This renewal time corresponds roughly to the pulse time in a pulsed thermonuclear reaction. Instead of the density-time product, a burnup parameter indicating the fraction of the deuterium supplied that is actually burned in the reaction can be used. It is conceivable that a thermonuclear reactor could be constructed so that the charged reaction products would be retained long enough to reach the average particle energy, while it is less likely that the neutrons and the electromagnetic radiation can be prevented from escaping from the reaction region. Thus, an adequate measure of the ability of the thermonuclear reaction to become self-sustaining (in the sense that the amount of energy transferred to the charged reaction products is sufficient to supply the energy required for the electromagnetic radiation and to heat the fresh fuel to the reaction temperature) is given by the reinjection fraction, e, defined as the ratio between the power that must be injected into the plasma to keep the system in a steady state and the total power emitted from the plasma in the form of kinetic energy of neutrons, heat in extracted plasma and electromagnetic radiation. The reaction will obviously be selfsustaining when e is equal to zero, while positive values of e correspond to situations where energy must artificially be supplied to the reaction region in order to keep the plasma in a steady state. When the conditions of a self-sustaining thermonuclear reaction have been reached it may be necessary to extract more energy than is emitted as kinetic energy of neutrons, heat in extracted plasma and electromagnetic radiation in order to maintain the desired reaction temperature. It is possible to extend the definition of e to this case also. In this way, e describes the criticality condition for the thermonuclear reaction: e > 0 meaning subcritical conditions and e < 0 supercritical conditions. A survey of the dependence of e on the three parameters has been made in the case of the steady state

2 432 SESSION A-10 P/2506 T. HESSELBERG JENSEN etal. and it is shown that optimum values of all three parameters can be found. An investigation of the isothermal non-steady state shows great similarity with the behaviour in the steady state, as has been demonstrated by a few examples. The survey of steady-state systems can thus be said to cover the case of pulsed isothermal systems, when they are compared in terms of the deuterium burnup or in terms of the density-time product. Since the actual rate of transfer of kinetic energy between the different kinds of ions and the electrons in the plasma is comparable with the rate of energy production through nuclear reactions and with the rate of energy exchange with the surroundings, it must be expected that corrections must be made to the results derived by assuming the energy exchange between particles to take place instantaneously. Calculations in the steady-state case, with these effects taken into account, show that the most important effect is a lowering of the electron temperature with respect to the ion temperature, in the cases where the reaction is nearly self-sustaining. This effect increases somewhat the range of parameter values for which a self-sustaining reaction is possible. BASIC EQUATIONS The reactions to be considered are listed in Table 1. In this table we have also indicated the notation to be used in the following. For the reaction rate parameters <ot>> we have used Roman numerals according to the numbering of the reactions, and for the average energies of the particles participating in the reactions we have, for example, used E4 1 for the triton in Reaction 1. It is then obvious that for a reaction between particles of kinds i and k giving particles of kinds r and s, = E r +E s. (1) Further, we use щ for the number of particles, per cm 3, of type = D, T, p, 3, 4, e, n for deuterons, tritons, protons, He 3, He 4, electrons and neutrons respectively. The Q values 5» 6 for the reactions are also given in Table 1. Table 1. Reactions of Interest, Q-values, Reaction Rates and Energy Partitions Reaction Q -value, Mev Reaction rates Average energy of particles involved in reactions D + D->T + p X) + D 5- He 3 + n X -f D > He 4 + n He 3 + D > He 4 + p T + T > He 4 + 2n T + He 3 -»He 4 + D T + He 3 -> He 4 + P + n ET 1 n T n s VII _ Reaction 5 of Table 1 needs only to be considered for fuel that is extremely rich in tritium. Reactions б and 7 need only to be considered for very high temperatures and for fuel that is very rich in both tritium and He 3. These reactions can, therefore, be neglected for our present purpose. At low temperatures (i.e. T < 10 kev) even Reaction 4 may be neglected. For the calculation of the quantities mentioned in Table 1 we have assumed a Maxwellian velocity distribution of each kind of particles with the temperature Ti of the th kind of particle. The most important of these quantities, the reaction rate parameters, are functions only of a proper average temperature, T', for the temperatures of the reacting particles given by where mt and nij are the masses of the reacting particles and Ti and Tj are their temperatures. Furthermore, Mij is the reduced mass, and цц defined by (2) will be used later on and termed the temperature reduced mass. In Table 2 the reaction rate parameters are given as functions of T'. In the calculation 4 > 7 of the reaction rate parameters, and the average energy of particles participating in reactions, we have used cross sections compiled by Jarmie and Seagrave 8 and by Bame and Perry. 9 Particle Densities First, we shall give a general set of equations covering the kinetics of the ion densities in the plasma. At time t = 0 one cm 3 of plasma contains щ ions of type i and, if we impose the condition of charge Table 2. Reaction Rate Parameters as Functions of Temperature Temperature T' {kev) I-IO 1? (стз/sec) II. 10 1? (стз/sec) IILIO 1? (стз/sec) IV-10 1? (стз/sec)

3 ENERGY BALANCE IN A PLASMA 433 Table 3. Reaction Kinetic Equations dn v dt = neutrality, n e = TiitiiZi. All summations over г mean summations over ions only. In order to enable us to discuss the steady state situation we may assume fresh fuel to be added at a rate of ^i>a deuterium atoms, and nvrja tritium atoms per cm 3 sec. We may also assume that it is possible to extract a representative mixture of the gas at a rate of щъ for the ions and 8S<w<Z< for the electrons, again per cm 3 sec. Under these conditions, the reaction kinetic equations for the ion densities are as shown in Table 3. Furthermore, for the number of neutrons emitted per cm 3 sec we have and for the electrons we have dn e /dt = - SHiftiZi + w D (l + rj) Д (9) With n e = TiifiiZi this last equation is already contained in Eqs. (3)-(7), but it gives a convenient and often used relation between S and A. Energy Content Next, we shall set up the kinetic equations describing the energy content in each type of charged particles. In order to do this, we assume that the fresh fuel is added at zero temperature and as neutral atoms, and that the ionization energy is negligible. The ions and electrons extracted from the plasma are assumed to have the same average energy as the corresponding type of particles in the plasma. Furthermore, we shall assume that the charged reaction products are retained in the plasma while the neutrons and the bremsstrahlung escape from the reaction region. (8) (3) (4) (5) (6) (7) The charged particles are assumed only to exchange energy by Coulomb encounters. A formula covering this energy exchange is given by the expression 10 A (10) giving the average energy transfer per cm 3 sec to the particles of type i with density щ from particles of type j of density щ. The term In Л is given by 2 In Л = 2(sin -2(cos A) Ci{A) where A = В = (П) (12) (13) and finally the Debye shielding distance, D, is given by D = (kt e l47rn e e2)i (14) In general, it will be necessary to inject or extract energy from the plasma in order to initiate and control the thermonuclear reaction. The plasma will gain energy from nuclear heating and energy will be lost as kinetic energy of neutrons, heat in extracted gas and radiated energy. Let us specify the injected or extracted power as рщ, the sign being positive for injection and negative for extraction. Furthermore, let us assume that this power density is distributed with a fraction, «i, to each type of charged particles. Table 4. Energy Density Equations ctt - Г е )/те + (Г т -Г е )/ре + (Г р = <*3pin-n 3 {ikt z 8 +Z, i n i k[z i {T 3 -T e )f 3e +{T 3 (16) (17) (18) (19) (20)

4 434 SESSION A-10 P/2506 T. HESSELBERG JENSEN etal. With these assumptions, we may write down the equations for the energy densities for the various types of particles as shown in Table 4. The a's fulfil the condition 2 г -а$ + а е = 1 (21) and the radiation power density p v Thompson 11 : p r = о. with kt e in kev. is given by (22) Energy Balance In order to discuss the energy balance of the system we shall add up the Eqs. (15)-(20), taking into account Eqs. (1) and (21), and obtaining pin = (23) where p r is defined by Eq. (22); p n is the neutron kinetic power density, pn = ЪПЪЩЕЦП + ПВПТШЕПШ ; (24) p g is the power emitted as heat in extracted gas, p ë = ÇZàkT&i+tkTe Z<mZ,)8 ; (25) and ^nuci is the nuclear power density,. (26) Finally, UQ stands for the total heat energy present in the gas at any given time t, U G = (27) Power Injection Equation (23) determines the value of the injected power, pm, at any time t if the temperature conditions are defined. The magnitude pi n can assume negative or positive values as a function of time, according to whether the reaction is momentarily self-sustaining or not. In order to analyze the over-all power balance in a period of time t = 0 to t = f we define a power reinjection fraction, e, through the equation (\Pin\ ~pin)dt] (28) The left-hand side of this equation is the total energy that must be injected into the plasma in order to initiate and control the thermonuclear reaction, while the brackets on the right show the total energy emitted by the plasma to the surroundings during the same time interval. When a pulsed system is considered, p in includes the power shot necessary to achieve the high plasma temperature and also the power recovered from the plasma at the end of the pulse. These contributions will appear as positive and negative delta-functions in time at t = 0 and t = f respectively. Solving Eq. (28) for e and using Eq. (23), while assuming UQ to be equal at t = 0 and t', one gets l +pin)df (29) The importance of the power reinjection fraction, e, can most readily be seen for the steady state. In this case, we may simply substitute the integrands for the integrals in Eq. (29) since they are constant in time. It is immediately seen that e can only assume values in the interval 0 < e < 1. e = 0 implies a self-sustaining reaction since in this case p in < 0. Since the e defined by (29) does not reflect to what extent pin difíers from zero when the reaction is selfsustaining, it is convenient to modify (29) by introducing an defined by 1-е' = (29') For positive values of e' the two parameters will be identical in the steady state while for negative values of one will have e = 0. For optimizing the parameters, e' is more convenient than e since it has a smooth variation in changing from positive to negative values, thus permitting a minimum point to be determined instead of a minimum region. In the pulsed isothermal case, e is the most convenient for comparison with the steady state. Although e will always be larger than zero in this case, a momentarily self-sustaining reaction will be achieved in a period of time in which pin < 0. The calculation of e is our ultimate aim and Eqs. (3)-(7) and (15)-(20) are our basic coupled equations. To solve this set of equations in all detail can only be attempted by means of electronic computers. This falls outside our present program and we have only treated problems to which analytical solutions may be found; namely, the steady-state problem and the isothermal kinetics, neglecting all temperature difference T\ T a which means assumption of infinite energy transmission parameters. A survey of this work is presented in the following sections. RESULTS Steady-state Systems In this section we shall assume a steady-state operation in time. This means that Eqs. (3-7) and (15-20) are equated to zero. We shall also make the simplifying assumption that all temperatures are equal to a common temperature T. Mathematically, this means assuming all/ ik -> oo. Thus, Eqs. (15-20) are of no consequences except for their sum leading to the definition of e by Eq. (29'). We then solve Eqs. (3-7) and use the results for the computation of e'. In Ref. 4 we give the explicit expressions for the solutions of Eqs. (3-7) in this case. The solutions depend on three parameters only: the temperature T, the tritium enrichment 77 and the

5 ENERGY BALANCE IN A PLASMA 435 magnitude n-^jh which is a product of density and time since 1/8 is the mean renewal time for the plasma. Instead of the third parameter we may also use the deuterium burnup, p, defined by p = (А-8)/Д. (30) The parameter p is related, in a one to one correspondance, to njy/8 through Eqs. (3-7). Some of the results of the calculations are illustrated in Figs. 1 and 2. In Fig. 1 we shall first discuss the case for77 = 0. For this case, we have shown the e'(j8) curves for the cases T = 25,50, 100 and 150 kev. In each of these cases, an optimum in p is found and if a curve is drawn through these optima an optimum in T results. The total optimum in p is further illustrated by the fat curve which is the envelope of all the e'(js) curves. It is easy to understand the appearance of optimum values for T and p from physical arguments. If the fuel is pumped through the reactor at a fast rate, relative to the reaction rates, the burnup ratio will be low and the heating and cooling of the fuel will dominate the power balance, with the result that ->- 1. On the other hand, if the fuel is permitted to stay for a long time in the reactor the burnup will be high, but there will also be a high buildup of passive reaction products leading to severe radiation loss: the power balance will then be dominated by the radiation losses and the heating necessary to compensate for these losses, again with the result that e > 1. Thus, a minimum in c' must be expected for intermediate burnup ratios. Similarly, for the optimum temperature. For low temperatures, radiation losses dominate over the nuclear heating as demonstrated by Post. 1 On the other hand, for very high temperatures, the reaction rate parameters (ov) level off as a function of temperature while at the same time both the radiation loss from electron-electron collisions, increasing as Г 3^2, and the fuel heating, increasing as T, become more important and finally dominate for the highest temperatures. Again, in the two extremes, e' -> 1 and at intermediate temperatures an optimum must exist. We have already mentioned that an envelope of the f (P) curves for different temperatures can be found. In the remaining part of Fig. 1 we show these envelopes for different values of 77. Again an optimum for this parameter is found, as illustrated by the trend in the entire valley created by these envelopes as functions of77 and p. The slopes of the valley are rather steep for small values of77 and because of this and in order not to have too many intersecting lines in the figure we have chosen to plot the envelopes as a function of 77e rather than 77. Also, in the case of 77, a simple physical interpretation of the optimum can be given. That tritium enrichment improves e is a result of the fact that III > I+11. However, if pure tritium is used, Reaction 3 disappears again and only Reaction 5 contributes to the power balance and again /// > V. Thus, an optimum is found for such values of nn and пт that the condition n^w ^ п-цп^ш and at the same time n^n-^iii ^ пц 2 (1 + 11). However, the 16 n-oo Figure 1. The power reinjection fraction, c', as a function of the deuterium burnup, ft and tritium enrichment, 17, optimized with respect to the temperature, 7. For17 = 0 the individual curves for different temperatures are also given optimum is not very pronounced, since the valley is very flat as a function of 77. In the optimum the D-T reaction dominates, and it is thus obvious that the optimum condition is reached when пъ = пт, which means77 = 1. In Fig. 2 we give a further illustration of the optimization. Here, we have given contour curves for e' in the (»D/8, T)-plane for two values of 77, namely 77 = 0 and 7] = 1. Since we know from Fig. 1 that the optimum77 is close to 1 we see from Fig. 2 that the optimum temperature lies near 25 kev and the optimum in п-о/ь near sec/cm 3 corresponding, for that case, to a burnup of ~20%. From Fig. 2 we also see that in spite of the fact that the reaction rate parameter III for any given temperature is much larger than /+//, only a factor of 10 in the optimum»d/8 is obtained with equal amounts of deuterium and tritium. The region in 77, T and p, in which < ' < 0, is not given with any reasonable precision by the rather unrealistic steady-state calculations with the same temperature for all particles. If we use the extended set of equations, with individual ion and electron kt -^ f 553 ъ A SB Ь5 : ' Figure 2. Contour curves for the power reinjection fraction, c', in the (HD/S, T)-pláne where no is the deuterium ion density, S is the inverse plasma renewal time and Tthe temperature. Two sets of contours are given, one for the tritium enrichment17 = 0 and another for17 = 1 key 02 ai

6 436 SESSION A-10 P/2506 T. HESSELBERG JENSEN et al. temperatures, e' will be diminished; also, if we turn to pulsed operation, the region of e ~ 0 is shifted towards higher burnups. In Refs. 4 and 12 we have given a more detailed description of the steady-state calculations. Effects of Finite Energy Transmission Rates for Charged Particles The aim of this section is to investigate the influence on the reinjection fraction, e', of leaving out the assumption of a common temperature for all particles in the plasma. However, a Maxwellian velocity distribution of each kind of particles is still assumed. We now consider the full set of equations, (3)-(9) and (15)-(21), where expressions (3)-(7) and (15)-(20) are equated to zero. This set of equations is rather complicated and will only be solved approximately. In order to describe a steady state in this case we may again use the parameters rj, пв/s, and a characteristic temperature. As it is seen, however, we must also ascribe definite values to the as, indicating how the injected power ft m is distributed between the different kinds of charged particles. Thus, it is seen that this problem contains more parameters than the previous one. It is possible to avoid a specific choice of the a's by considering only the special case fti n = 0, or e = 0. If we compare two systems with the same average ion temperatures, one in which all particles are assumed to have the same temperature and another where this assumption is not made, then it is found that differs in the two cases mainly because of the following two effects. First, the different ion temperatures modify the reaction rate parameters, thereby shifting the relative density of the different kinds of ions and thus changing the value of e'. Secondly, the electron temperature is lowered, as discussed by Post, 1 which modifies ^> r ad and ft g and, consequently, e'. It is found that the second effect is the stronger. We therefore assume, as a first approximation, that all the ions have the same temperature, which we now choose as our temperature parameter. Eqs. (3)-(8) can then be solved separately as in the previous section, and Eqs. (15) (19) can be neglected, since T e is determined by Eq. (20) alone. In this «p uf I* kt 6'= 0. ) qso E key Figure 3. Contour curves for the power reinjection fraction, c', in the (n D /S, 7)-plane. Curves A include the effects of finite energy transmission rates. Curves В are taken from Fig. 2 for comparison equation it is convenient to put a e = 1; e is then calculated and only the cases where e' = 0 are considered since, for these particular cases, the results are independent of the choice a e = 1. These results are, however, slightly dependent on the density of particles and the temperature through the energy transmission rates between ions and electrons. In the calculations we have set In Л = 14, which, for the actual temperatures, roughly corresponds to particle densities between and cm" 3. For further details see Ref. 10. To justify the assumption of equating all ion temperatures, we have, in some characteristic cases, solved Eqs. (15)-(19) for the ion temperatures in a first order approximation. To solve this set of equations we used the previously found electron temperature. To calculate the energy transmission rate parameters, the mean energy of reacting particles and the reaction rate parameters, it is also necessary to assume a temperature for the ions. This temperature was chosen as the average ion temperature. Using the values of the ion temperatures found in this way one gets a first order correction to e', a correction found to be insignificant. Also, one can, from Eq. 20, find a correction to the previously found electron temperature, but this correction was also found insignificant. For further details, see Ref. 7. The results are given in Fig. 3. Here, contour curves corresponding to e' = 0 are given in the (пт>/8, T)-plane for rj = 0 and 1. Curves A correspond to the assumption of different temperatures for the ions and electrons; and В for comparison, to the case where the same temperature is assumed for all particles. As e' did not reach zero for 77 = 0 in the latter case, the contour curve for e' = 0.1 is shown instead. First, it is seen that modifications are not very significant. For the parameter пт>/8 the interval with e < 0 is somewhat broadened, because both ft g and ftiaa> which dominate the energy balance at large and small values of n^/s, are lowered by the lower electron temperature. This effect is somewhat more pronounced at the higher temperatures, mainly because the relative difference between the ion and electron temperatures is larger at higher temperatures since the energy transmission rate parameters between ions and electrons are proportional to T e ~ 3f2. The Pulsed Isothermal System Since some of the features of the steady-state system seem rather unrealistic from the point of view of present-day ideas and lines of development, we thought it appropriate to investigate a system more closely connected with the physical situation encountered in a pulsed gas discharge. In the model chosen, a mixture of tritium and deuterium in the ratio rj to 1 is instantaneously heated, at time t = 0, to a uniform temperature, Г, and then kept at this temperature until the plasma is instantaneously cooled down to essentially zero temperature at t V. The function ftm in Eqs. (15)-(20)

7 ENERGY BALANCE IN A PLASMA 437 has thus the character of a delta-function at t = 0 and i = V while it is adjusted in between so as to keep the temperature of the mixture of fuel and reaction products constant. Since we do not, in this case, extract or inject matter we have Д = 8 = 0 in Eqs. (3)-(7) and (15)-(20). Accordingly, we wid not have a steady state and àn%\it will, in general, be different from zero. It is further assumed, for the sake of simplicity, that there is a common temperature for all kinds of particles in the plasma. These assumptions considerably simplify Eqs. (3)-(9) and (15)-(20). In Ref. 13 it is shown how Eqs. (3)-(9) can be solved in this case. Only the main results will be presented here. Three linearly independent integrals of the type ^ч^гщ + ХцПп = Nj can be obtained where the A's are constants satisfying the conditions A4 = A + AT+A3 Ap = 2AD AT A n = 2AD A3, (31) where AD, AT and A3 can be chosen arbitrarily, while the Nj's are arbitrary constants to be determined from the initial conditions nj> = Щ, пт = пщ, % = пъ = щ = п п = 0. (32) The integrals can, for example, be chosen to be (a) particle conservation: (b) charge conservation: "ZimZi = n o {l+7 1 ) (c) "tritium conservation": = пщ (33) It is not,possible to express the Щ'Б in terms of real time in any simple form, but if the parameter time, г, defined by T = JO*'(WD/ W O)^ (34) is introduced, three further integrals are obtained, of the type solutions of the form 3 щ = 2 (38) where the K's and L's are constants. In order to compare the pulsed system with the steady state it is necessary to consider the over-all energy balance in the time interval t = 0 to t = t'. This is reflected in the definition of e, in Eqs. (28) and (29) by the fact that time-integrals of the power densities are considered. Thus, it is perfectly possible that p in can be negative during a part of the pulse time, corresponding to nuclear "ignition/' whereas the pulsed system is never self-sustaining in the more exacting over-all sense defined by e = 0 because of the necessity of injecting power initially. All the integrations necessary to determine e can be performed analytically except for the integration of the electromagnetic radiative power which must be evaluated numerically. As in the steady state, e is a function of the temperature, T, and of the tritium enrichment, 77. The third parameter can either be chosen as the burnup, j8, defined by which can directly be compared with the burnup in the steady-state calculations, or it can be chosen as пот, which must then be compared with the parameter пв/й in the steady state. Computations of e as a function of the deuterium burnup, j8, has been made in the two cases: 1. rj = 0 and kt = 100 kev; and 2. rj = and kt = 5 kev. In the last case it was permissible to neglect Reaction 4 because of the relatively low temperature. The results are plotted in Fig. 4, together with the corresponding curves for the steady-state case. The great similarity between the non-steady and the steady case is immediately noticed. The largest differences are to be > T. (35) where y% is any one of the three roots in the equation ЯГ 1 В 2 Ю* 2 б 2 y 3 +(/+//+/// +JF)y 2 +f(j+77).///./f = 0 2 (36) while Ci is an integration constant determined by the initial conditions (32), КГ 1 kts5kev г)* 0383 pulsed q.toqo steady ю- 2 As may be seen from Eq. (34), the parameter time, r, is approximately equal to t' for small deuterium burnups. Solving for щ in Integrals (33) and (35), one gets Figure 4. The power reinjection fraction, e', as a function of the deuterium burnup, j8, for the isothermal pulsed cases where r = 0, kt = 100 kev and1? = 0.983, kt = 5 kev The corresponding steady-state curves for77 = 0, kt = 100 kev and 7) = 1, kt = 5 kev are given for comparison

8 438 SESSION A-10 P/2506 T. HESSELBERG JENSEN etal. found for the large burnups corresponding to long pulse times and slow renewal rates respectively. This is due to the fact that the pulsed system, because of the retarding eñect of the finite reaction times, is relatively richer in the active primary reaction products^ T and He 3, while it is poorer in the passive secondary product He 4 compared with the equilibrium values of these isotopes in the steady-state system. Nevertheless, it is obvious that both the trends and the actual values of e as a function of the parameters T, rj and j8 are quite well reproduced by the steady-state calculations. In particular, it will be true that optimum values for the three parameters will exist also for the pulsed system and that they will not be very much different from the steady-state values. For all practical purposes, at the present time, the analysis of the steady-state problem given above can be taken as valid also for the pulsed system when the comparison is made in terms of the deuterium burnup or the equivalent parameters п^ь and n^r respectively. DISCUSSION To be able to treat the problem above it has been necessary to make some approximations and assumptions that limit the results. In this section we shall discuss some of these limitations. In order to solve the reaction kinetic equations it has been necessary to neglect Reactions 5 to 7. Fortunately, these reactions contribute very little to the power economy in the cases treated here (see Refs. 3, 4 and 12): over most of the parameter ranges considered they contribute less than 10~ 3 in с Since not all of the power losses conceivable have been taken into account in the power balance calculations, the results must be considered as optimistic. The kind of losses which have been neglected are, for instance, energy dissipated in magnetic fields necessary for containment of the plasma, excessive radiation due to contamination of the plasma with heavy atoms, and losses due to particle collisions with the walls in a finite reactor. Also, the optimum conditions can only beràchieved in a restricted region in space and time since temperature gradients and finite heating times are likely to be features of a more realistic fusion reactor. Although we have tried to keep the number of specific assumptions as low as possible, in order to conserve the generality of the results, it has still been necessary to make some more or less arbitrary decisions. This concerns especially the fuel cycling. The power balance would obviously be improved if a nonrepresentative mixture of the plasma could be extracted, containing mainly passive reaction products like protons and He 4 at a temperature lower than the average. Since this possibility did not seem very likely to us we chose to assume a representative mixture at the average temperature to be extracted. The steady-state concept also implies that the fuel is homogeneously and eontinuausly injected in the plasma. The unrealistic character of this assumption is largely removed, we think, by the demonstrated similarity with the pulsed system. It has also been necessary to make assumptions about the energy distribution of the particles involved. For obvious physical reasons, the Maxwellian distribution was chosen; apart from the fact that considerable mathematical simplicity was thereby also achieved. CONCLUSIONS Finally, we state a few conclusions which can be drawn from the material presented. {a) The similarity between results for the steady state and the isothermal pulsed case indicate the generality of the results derived in the former case. Comparisons can be made in terms of the parameters T, r and j8 indicating the temperature, the tritium enrichment and the deuterium burnup respectively. The burnup parameter can be replaced by the parameter щт = Q v njydt which, in the steady state, reduces to njyjb. (b) The effects of finite energy transfer rates between the charged particles slightly increase the parameter range for which self-sustaining thermonuclear reactions can be achieved, the extension being most marked in the high temperature region, above 50 kev. (c) An optimum value of the power-reinjection fraction, e', can be obtained for the following values of the temperature, Г, the tritium enrichment, 77, and the deuterium burnup, p; viz., kt = 25 kev, rj = 1 and j8 = 0.3. A self-sustaining reaction can be achieved at a temperature kt «5 kev in the case of a one-toone deuterium-tritium mixture, with a corresponding value of tf'nvdt «10 16 sec cm~ 3 and j8 = 0.1. (d) Even a slight tritium enrichment considerably lowers the temperature necessary to achieve a selfsustaining reaction. (From about kt = 50 kev for 77 = 0 to about kt = 9 kev for rj = 0.1 in the steadystate case.) In itself, this has not great consequences since, at these temperatures, the reactor will mainly be a tritium burner, but it opens up the possibility of achieving the higher temperatures by nuclear heating and thereby initiating the D-D reaction. The copious neutron production could then be used to produce the tritium necessary for the nuclear ignition. REFERENCES 1. R. F. Post, Controlled Fusion Research An Application of the Physics of High Temperature Plasmas, Rev. Mod. Phys., 28, (1956). 2. J. D. Lawson, Some Criteria for a Power Producing Thermonuclear Reactor, Proc. Phys. Soc, 70 B, 6-10 (1957). 3. E. Lacombe, D. Magnac-Valette and P. Ciier, Evolution Energétique d'un Plasma Thermonuclèaire de Deuterium, Compt. Rend., 246, (1958). 4. T. Hesselberg Jensen, O. Kofoed-Hansen, A. H. Sillesen and C. F. Wandel, Some Criteria for a S elf-sustaining

9 ENERGY BALANCE IN A PLASMA 439 Steady State Thermonuclear Reaction, Rise Report No. 2, 1-20 (1958). 5. F. Ajzenberg and T. bauritsen, Energy Levels of Light Nuclei. V, Rev. Mod. Phys., 27, (1955). 6. D. N. Van Patter and W. Whaling, Nuclear Disintegration Energies, II, Rev. Mod. Phys., 29, (1956). 7. T. Hesselberg Jensen and F. Heikel Vinther, On the Effects on a Steady State Thermonuclear Reaction of Finite Energy Transfer Rates Between the Charged Particles, Riso Report No. 5 (1958), in press. 8. N. Jarmie and J. D. Seagrave, Charged Particle Cross Sections, LA-2014 (1956). 9. S. J. Bame Jr. and J. E. Perry Jr., T(d,n)He* Reaction, Phys. Rev., 107, (1957). 10. C. F. Wandel, Energy Exchange by Coulomb Encounters for Maxwellian Ion Mixtures in the Plasma State, Riso Report No. 4 (1958), in press. 11. W. B. Thompson, Radiation from a Plasma, AERE T/M 73, 1-8 (1957). 12. T. Hesselberg Jensen, On the Optimization of the Tritium Enrichment with Respect to the Energy Balance in Steady State Thermonuclear Reactions, Riso Report, No. 6 (1958), in press. 13. O. Kofoed-Hansen and C. F. Wandel, On the Energy Balance and Reaction Kinetics for Isothermal Thermonuclear Reactions Involving Deuterium and Tritium, Riso Report No. 7 (1958), in press.