L 1 ~ ехр гы. * = Am) Thermodynamics of Deuterium-Tritium Mixtures. By G. Boulegue, P. Chanson, R. Combe, M, Feix and P.

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1 P/2495 France Thermodynamics of Deuterium-Tritium Mixtures By G. Boulegue, P. Chanson, R. Combe, M, Feix and P. Strasman ;î We propose to study deuterium-tritium mixtures which, when heated to an elevated temperature, produce a considerable amount of thermonuclear energy. Such mixtures will constitute the active part of a fusion reactor. This reactor could operate in a stationary manner, a feed of fusionable materials replacing the material used up by the thermonuclear reaction, and the inert waste products being removed in such a manner that the various concentrations as well as the temperature remain constant. It is, however, doubtful that one could construct an apparatus of this type and interest at present seems to be in the direction of a cycle in which the mixture is heated to a given temperature with liberation of thermonuclear energy during the period in which the temperature is sufficiently high. After cooling of the system the cycle begins all over again. We will present rather briefly a problem of a stationary plasma and will then study in greater detail the time behaviour of a D-T mixture of known initial composition taken to a temperature Го. In particular, we will calculate the energy released during such a cycle as well as the time necessary for its completion. We wish to treat as thoroughly as possible the following two points: (i) the importance of "secondary" reactions, i.e., reactions between initial nuclei and products of the primary reaction. (In particular, in those plasmas having no tritium initially, the D-T reaction between deuterons and the tritons formed during the reaction D-D plays a decisive role as will be shown later. The importance of these secondary reactions has been pointed out by Lacombe et al. 1 ) : (ii) the importance of radiation, which results in a cooling of the system, thereby limiting the time during which the release of thermonuclear energy is important. Basic Assumptions The following hypotheses are therefore made: (a) There is a Maxwell distribution of particle velocities for nuclei and electrons, corresponding to a unique temperature which characterizes the environment. (b) The environment is transparent to neutrons: this Original language : French. * Laboratoire Central de l'armement. 49 hypothesis is quite justified in view of the small density of thermonuclear plasmas. The neutrons escape from the system, therefore, carrying with them a part of the reaction energy in the form of kinetic energy. (c) The surroundings are not in radiative equilibrium. At the high temperatures considered ( > 6 K) the plasma will be totally ionized. The processes of radiation emission or absorption are discussed below. (d) The pinch effect is perfect, i.e. no charged particle can escape from the plasma, whose actual dynamic behaviour is thus ignored. Radiation Most of the radiative energy loss, see (c) above, is due to bremsstrahlung (mainly of the electrons in the field of the ions), an emission process corresponding to free-free absorption (in French : absorption par une particule libre). The absorption cross section is given by Spitzer. 2 For a photon of frequency v the inverse of the mean free path K v is given by K * = Am) Г /M L 1 ~ ехр гы where N e, N\ are the densities of electrons and nuclei of charge Ze, and the other letters have the usual definitions. With the usual values of N e ~ Ni ~ 15 to 18 electrons (or nuclei) per cm 3, the mean free path \ K V is in the range 6 to 8 cm for photons whose energy is of the order of kt, i.e. "average" plasma photons (kt = kev). Therefore, in view of the inadequate size of the apparatus used in the laboratory, no equilibrium can be obtained. The Compton effect might also contribute to energy losses. It is usually considered to be an absorption effect, since the photon energy is ordinarily far greater than the thermal energy of the electrons which undergo the Compton collision; but the situation is different in a thermonuclear plasma, since both the electron and photon energy are of the order of kt. If one takes into account the electron motion, one realises that the impact may cause the photon either to lose or gain energy. The Compton effect should, therefore, be capable of inducing an equilibrium, according to Planck's law for photons. Nevertheless, because of the low plasma densities, the mean free paths are far greater than the dimensions of the system

2 4 SESSION A- P/2495 G. BOULEGUE et al. and, unlike conditions in the stars, there is not sufficient space for an equilibrium to be established. We will take into account, therefore, only the bremsstrahlung: the power radiated is proportional to the volume, and varies with temperature as the average speed of the electrons, that is, in proportion to TK Reactions We have said that we would take into consideration the ' 'secondary" reactions, but, of course, if we had to consider all possible reactions we should steadily be led on to study all nuclear reactions, which would make our calculation impossible. We also confine ourselves to reactions with reasonably large cross sections, taking into account only the following five: > 1T 3 + ipi+4.3 Mev гнез + о^ + З^Б Mev (2.44 Mev carried away by the neutron) id 2 + it 3 -> 2 He4 + on Mev (14.6 Mev carried away by the neutron) id He3 -> 2 He4 + ip Mev it 3 + T3 -> He4 + 2 X 2 n! Mev (5 Mev carried away by the neutrons) The rates of the reactions (ov} will be called respectively a, js, y, 8, (in the same order as in the above list). Calculation of the Reaction Rates If a stands for the cross section, and v for the relative speed of the two nuclei it is known that the number of reactions per unit volume and time will be n\n^iavy ) where n\ and n are the particle densities; or, if one is dealing with reactions between identical nuclei of density n, the reaction rate will be \n\ovy. The evaluation of <cny> has been given in an article by Thompson. 3 It is obtained by calling the cross section o(v) y a function of the relative speeds of the two nuclei, and taking into consideration the Maxwell distribution of velocities, whose direction is assumed to be isotropic: В in such a way as to conform to the experimental results of Arnold et al. Unfortunately, it is difficult to present these results exactly in an energy range large enough for the proper calculation of the integral in Eq. (2). We have, for our part rejected Gamow's formula, and have integrated (2) numerically with the help of the experimental values of a. These values extend down to about 13 kev. If we study more closely the variations of the product v 3 a(v) exp( tnv 2 /2kT), which becomes Eo(E) exp(-elkt) by change of variable, we find that it is impossible to calculate lower than kt = kev.f Table 1 shows our results for the two D-D reactions and the D-T reaction, temperatures varying from to kev. These values are slightly lower than those given by Thompson, but we use his results for reactions for which we have no accurate cross section values and for energies lower than kev (,.1,.1, 1 and kev) ; for intermediate values we interpolate in a log-log representation. Calculation of the Radiated Power There still remains to be calculated the power lost by radiation, taking into account the bremsstrahlung spectrum and the Maxwell distribution of the particles. Spitzer has obtained the equation: Post 5 ' has given a numerical formula derived from the results of Heitler. One finds, on applying the formula to a non-relativistic electron: Prad = corresponding to Spitzer's expression multiplied by a factor 2л/3/тт ~ 1.9. The discrepancy is due to the fact that the first equation is derived from a semiclassical calculation making use of a uniform energy spectrum, for the bremsstrahlung of the electron, while Heitler treats the phenomenon in a quantum electrodynamical manner. 3m Table 1. Calculation of Reaction Rates «ш>> in units of 18 cm 3 /sec) (4) Temperature [energy kt), kev a D(Dp)T D(Dn)He* T(Dn)He* where m is the reduced mass, mim 2 /(mi+w 2 ). The quantity (av) is thus given by an integral of the curve G(V). Some experimental results are given by Arnold et ala In Eq. (2) the integral can be completely evaluated, if one assumes that a depends on the relative energy ^mv 2 = E, as in Gamow's formula: a = (A/E) exp (Б/Е*). (3) Therefore, it is only necessary to determine A and B y which is what Thompson has done, by choosing A and f We express the temperature in units of energy kt (with 1 kev as unity). It should be noted that this does not refer to the mean energy of the particles, which is f k T. One kev corresponds to 11.6 x 6 C. Particles of such a plasma (at a temperature of 1 kev) will have a mean energy of 1.5 kev.

THERMODYNAMICS OF D-T MIXTURES 411 We shall use the equation of Post which, for a mixture of nuclei, becomes: Prad = 3.

3 THERMODYNAMICS OF D-T MIXTURES 411 We shall use the equation of Post which, for a mixture of nuclei, becomes: Prad = X -«(*T) WeSiAW- (6) The radiated power is expressed in Mev/sec cm 3 if kt is expressed in kev. Equations of the System We designate the densities of the nuclei of D, T, He 3, H, and He 4 by x, y, z, и and w, respectively. There may be sources (positive or negative) of relative intensities S x, S y...s w (number of nuclei/sec cm 3 ). Using the reaction rates, a,... e, previously defined, the equations may be written: x = S x -(d+p)x 2 -yxy-8xz У = Sy + ax 2 -yxy- y 2 z = S z + ^x 2-8xz (7) Ù = S w = STUDY OF DEUTERIUM-TRITIUM MIXTURES Static Case The first problem investigated is that of a stationary mixture. It is assumed that the sources are regulated in such a way that the concentrations and temperature are constant; this eliminates the left sides of Eqs. (7). Under these conditions one can calculate the concentrations as a function of the sources. It is, of course, necessary for the concentration to be positive, in order to make sense physically. It is therefore necessary to have available a positive source of deuterium and a negative source of protons and of helium-4. This was foreseeable a priori. We will now determine the intensity of the sources, subject to other conditions. First, the energy flows must be balanced; that is, the radiated power must be equal to the released thermonuclear power. In addition, we will assume that the feed and extraction, i.e. the concentrations, can be controlled at will. This will lead, then, to the complete elimination of protons and helium-4, inert products which contribute only to the radiation without taking part in the thermonuclear reaction. The helium-3 case is different since it reacts with the deuterium, liberating a large quantity of energy which, moreover, is completely transmitted to the plasma (although in the case of the D-T reaction a considerable part is carried off by the neutron). Unfortunately, helium-3 makes an important contribution to the radiation while having only a small cross section for Reaction 8, He 3 (D, p)he 4, so that it is only for very high temperatures that it becomes advantageous not to extract it completely. The temperature limit is calculated to be k T = 36 kev. For lower temperatures we will assume, therefore, that the concentrations of H, He 3 and He 4 are zero. We calculate, then, the feed of deuterons and tritons as a function of operating temperature, for a given release of energy. The values Table 2. Temperature Dependence of Feed Rates Temperature (energy kt), kev Deuterons per Mev Tritons per Mev of the feed will be expressed in deuterons or tritons per Mev. The minimum temperature below which there is no possible solution is in the range kev. Table 2 gives the values of the feed for temperatures above this limit. In particular, operation without a supply of tritium takes place for kt = 24.6 kev. For higher temperatures the reactor will be able to produce tritium. A part of this tritium, produced by the D(D, p) reaction could be extracted from the reactor; whereas, up to this temperature, it was necessary not only to leave all the tritium formed, but even to supply some of it. This problem corresponds, unfortunately, to not very realistic experimental conditions. Its only interest is to point out rather simply the economic conditions of operation for a stationary reactor. Dynamic Case The following problem is much closer to the planned laboratory experiments. A deuterium-tritium mixture is taken to an initial temperature To. We make the same assumptions as previously. The temperature varies as the reaction proceeds, the system becoming progressively poisoned by the accumulation of the waste products He 4 and H. When the power dissipated by bremsstrahlung is greater than the thermonuclear power the system cools itself very quickly. During this process a certain amount of energy has been liberated and it is interesting to compare this with the energy necessary to heat the plasma to its initial temperature. Introducing into the mixture a certain proportion of tritium is very advantageous and we will see that a rather small quantity permits a considerable reduction of the initial temperature necessary. The equations of the problem are the five equations (7), with the terms representing the sources eliminated, together with the following expressions for the energy balance:

411 SESSION A- P/2495 G. BOULEGUE et ai. j t (NkT) = +Y*yWy+&xzW B +te*w (8) where W a, Wp, W y, W d, and W are the energies given to the charged particles; i.e., the energies of reaction minus the energy carried off by the neutrons.

4 411 SESSION A- P/2495 G. BOULEGUE et ai. j t (NkT) = +Y*yWy+&xzW B +te*w (8) where W a, Wp, W y, W d, and W are the energies given to the charged particles; i.e., the energies of reaction minus the energy carried off by the neutrons. If JV is the total number of particles and N e is the electron density, which stays constant, then JV = Ne + x+y+z+u + w; (9) the neutrons escaping from the plasma. In the following it will be helpful to introduce two reduced variables, the product N e t and the quotient W/Ne, W being the energy released at time t. Since we are looking at deuterium-tritium mixtures, this quantity WjN e is the average energy released per nucleus. One usually considers, for the initial temperature (at t = ), the critical temperature, T' c, above which Ръ > W t Table 3 shows T' c as a function of, the concentration of tritium at t =, expressed as an atomic percentage. Table 3. Dependence of Critical Temperature on Initial Tritium Concentration Cone Temp. kt' e kev Cone Temp. vr. kev The notion of a critical temperature is interesting, but allows the evolution of the system to be followed only during the initial phase. Later, this evolution can continue in various ways. Let us suppose, for example, that a plasma is composed initially only of deuterium. If the initial temperature To is less than 42.4 kev, the radiated power is greater than the thermonuclear power, and the temperature decreases. Nevertheless some tritium is formed after a certain time, the power Рш increases because of the large cross section of the DT reaction, and the temperature rises after having passed through a minimum. On the other hand, let us consider an environment relatively rich in tritium ( ^ 1 percent) and heated to the temperature T o = 15 kev. The derivative f is X This temperature is not exactly that above which the derivative Ф is positive. In fact, according to (8) it is the expression d(nkt)ldt, i.e., Nkiï+kTN, which is positive. Meanwhile the neutrons escape and the number of particles decreases: Ñ is negative. But since one has ktñ <^ Nkf, the temperature at which the derivative Ф becomes zero is essentially equal to i c Table 4. Energy Release and Reaction Time of D-T Mixtures for Selected Initial Temperatures, kt CT kt =7 kev WjNe kev nucleus kt = 9 kev кт = 15 kev W Ne sec/cm seclcm sec cm WjNe kt u^kev kt = kev W Ne kevfnucleus seclcm* seclcm* sec/cm Table 5. Effect of Initial Tritium Concentration on Critical Temperature, r CTF,kev positive initially, but the complete investigation shows that the tritium ''burns" before the temperature attains a value sufficient to '"ignite" the deuterium. After having passed through a maximum this time, the temperature decreases and, in all, the released thermonuclear energy is small, in this case 171 kev/ nucleus. The yield of the process is mediocre, since it was necessary to supply 45 kev in order to heat the plasma: if the energy is extracted in heat form, these 216 ( ) kev will just allow the production of the 45 kev of electrical energy necessary for the heating of the system. We have calculated the thermal evolution of a deterium-tritium mixture for different values of kto and - Table 4 shows the results, namely the thermonuclear energy released, W/N e, and the reduced time, N e t, necessary to complete the process. The calculations were discontinued when the temperature fell below 1 kev. We can now define a new critical temperature, Tew (or rather, a family of new critical temperatures). TQW is> for a given initial concentration of tritium, the initial temperature necessary in order that the released energy have a given value W. It is, in general, a function of and W. An examination of Table 4 shows that, at low tem-

$THERMODYNAMICS OF D-T MIXTURES 413 5 ДО kt(kev) i v \ \ \4 LT 4 -^ \ >^ 2495 1 33kT(kev) 32 31 29 28 27 _^ " ^ " " 15 26 " ^»>д; * Ci 25,2 /,6,8 1. 1,2 \k 1.6 1,8 2 WT N a tx- 15 w(t)/ Ne Figure 1.$

5 THERMODYNAMICS OF D-T MIXTURES ДО kt(kev) i v \ \ \4 LT 4 -^ \ >^ kT(kev) _^ " ^ " " " ^»>д; * Ci 25,2 /,6,8 1. 1,2 \k 1.6 1,8 2 WT N a tx- 15 w(t)/ Ne Figure 1. Effect of initial tritium concentration on critical temperatures kt' c and kt G w peratures (kto < 9 kev), W/N e varies very rapidly in the neighbourhood of a certain critical concentration. Tew is therefore a function of but practically independent of W/N e. Further, in comparing Tables 3 and 4, one notes that Tew T' c. The classical notion of a critical temperature T' G is therefore still applicable under these conditions. Table 5 shows ktcw as a function of for W/N e = 1 Mev. Figure 1 shows the variation of both kt'c and ktcw as functions of. For = О, kt'c = 42.4 kev and kt C w = 27.7 kev. The curves Table 6. Temperature and Energy Release in Pure Deuterium XlO" 1 * sec/cm* a In kev. b In Mev/nucleus. kt* *» b кт kt = 2Skev WIN, кт = 27 kev kt Figure 2. Temperature and energy release for pure deuterium intersect for =.24 and become identical above = 6. Their initial portions ( <.24), where Tew < T f c, correspond to a mixture whose tritium concentration is less than the equilibrium concentration oc/2y ^.24. (At this equilibrium concentration the amount of tritium formed by the D(D, n) reaction is equal to that consumed by the D-T reaction, the T-T reaction being practicauy negligible.) The formation of tritium allows the temperature to rise again, after passing through a minimum, and gives a reasonable yield (a considerable fraction of the deuterium being burned), if Го > TQW even though To < Г с. In the cases corresponding to the following portion of the curves (.24 < < 6), if T' c < T o < T C w, the temperature rises, at first, but the tritium disappears before attainment of a temperature sufficiently high to assure even a partial combustion of the deuterium. Finally, for the high concentrations ( > 6), the two curves overlapping, one can really speak of a critical temperature T c = T' c = Tew', the mixture heats itself as soon as the thermonuclear power is greater than the power radiated and there is enough tritium for the combustion to be more or less complete. Finally, we present some results (Table 6 and Fig. 2) concerning the evolution of a pure deuterium plasma, i.e. the values of the temperature and released energy, as functions of the reduced time, for three values of kt. 1. E. Lacombe, D. Magnac-Valette and P. Cuer, Compt. Rend., 246, p. 744 (1958). 2. L. Spitzer, Physics of Fully Ionized Gases, pp Interscience Press, New York. REFERENCES 3. W. B. Thompson, Proc. Phys. Soc. (London), B, p. 1 (1957). 4. W. R. Arnold, J. A. Philips, G. A. Sawyer, E. J. Stovall and J. L. Tuck, Phys. Rev., 93, p. 483 (1954). 5. E. Post, Rev. Mod. Phys., 28, p. 338 (1956).

Slowing-down of Charged Particles in a Plasma

P/2532 France Slowing-down of Charged Particles in a Plasma By G. Boulègue, P. Chanson, R. Combe, M. Félix and P. Strasman We shall investigate the case in which the slowingdown of an incident particle