P/1941 Poland

Fusion Chain Reaction

By M. Gryziñski

CHAIN REACTION WITH CHARGED PARTICLES medium, <(c) the mean cross section for the A + В With the discovery of the fission chain reaction with reaction, we can write neutrons, the possibility of obtaining a chain reaction 7>l(dE/dx)~l. (1) with charged particles was abandoned because of the small efficiency of charged particles in nuclear reac- If we assume {a) ~ 10~24 cm2, EQ ~ 10 Mev, we tions. In the most advantageous case, T (D, n) find that the atomic stopping power or stopping factor 4 3 17 1 2 He , the efficiency attained is only ~5 X К)- reactions (dE/dx}/N == EQ would be 10~ ev atom- cm . per 14 Mev deuteron. However, no note was taken Under normal physical conditions it is about a thous- of the fact that the efficiency depends on the physical and times higher,2' 3 and we are far from satisfying conditions and in some cases may be greatly increased. the criterion (1). This is especially true for nuclei of small charge, where The main idea of the problem involves the depen- the factor of Coulomb barrier penetration is not too dence of the atomic stopping factor on the physical high. The development of the chain reaction with conditions. charged particles is therefore possible only for light Under normal physical conditions, in the moderate nuclei, where the release of nuclear energy is due to energy range, the most important losses of energy of the process of fusion. Only highly exoenergetic heavy charged particles are due to scattering from reactions of large cross sections may lead to the fusion electrons. They are about four thousand times chain reaction ; these are the same reactions as those higher than the energy losses in all other processes.3 which are involved in thermonuclear reactions.1 The atomic stopping factor due to scattering from electrons was discussed in detail by the author,4 FORMULATION OF THE PROBLEM and, according to Eq. (6) of Ref.4, for a particle f having a velocity V^ and a charge Z^e, it is The mechanism of a fusion chain reaction, which is due to in statu nascendi reactions, is as follows. In an /dE\sc __ Z? f(Ve);G[(Ve)]dV (2) exoergic reaction A + B, in which weakly bound \ dx /el ~~ e groups of nucléons of nuclei A and В form strongly bound groups of reaction products, we obtain particles where f(Ve) is the momentum distribution of electrons having kinetic energy. Part of their kinetic energy in the medium and G the universal stopping power is transferred in elastic collisions directly to the A and function given by Eq. (8) of Ref. 4. There it was В nuclei of the medium. The recoiling A and В shown that these losses depend mainly on the velocity nuclei, in the process of slowing down to thermal distribution of the electrons, especially in the case energy, have some probability of leading again to the Vg < Ve- In the limiting case V¿

cross section of particle \ from the nuclei A, and N Electron bound in hydrogen! the number of nuclei per cm3. Thefirst ter m in Eq. (6) Plasmo electron» (N = Ю24) represents the Coulomb scattering, and the second the nuclear scattering, which we assumed isotropic in the center-of-mass system. The stopping factor due to inelastic collisions with nuclei is: / (7)

The sum is taken over all channels with the excitation energies AEi. The energy losses connected with the bremsstrah- lung7 of heavy charged particles are very low in comparison with the losses given above; therefore, they can be safely neglected. Finally, the stopping factor of nucleus A and 0.01 1941.1 Proton energy in Mev its ZA electrons is :

Figure 1. Theoretical calculations of the stopping power of dE electrons bound in hydrogen, electrons of plasma at different temperatures, and electrons of a Fermi gas which scattering depends on the state of the medium. The total atomic stopping factor of hydrogen plas- To evaluate the stopping factor of plasma 5 mat for protons and the relative contribution of electrons we have to use the Maxwellian momentum their components in various conditions are plotted in distribution in Eq. (2). We obtain an approximate Fig. 2. dependence on the temperature of the plasma if we Evaluation of Multiplication Factor make the substitution f{Ve) = ô «Fe> — Ve), where (Ve) = (8kT/7im)* is the mean thermal velocity of To determine the exact conditions for the develop- electrons. In the case of interest, V^<^ Ve, we have ment of an avalanche, we shall examine an infinite homogeneous medium formed by a mixture of two /dE\sc (' iimYI* a (4) \^/plasmaelectrons ~ 3m \8kTJ Ч 10-14 Similarly, taking into account the momentum distri- bution of a Fermi gas, we obtain the stopping factor Total stopping power of hydrogen atom of plasma of Fermi gas electrons {cf. Eq. (18), Ref. 4 ; also Ref. 6)

15 J 10Г Scattering from nuclei 1 Scattering from electrons ыFermi electrons s e W.ln (5) -16 m 2 where Fmax = {3n )i{ñ/'m)Neb) Ne the number of electrons per cm3. | The results of exact computations, where for the 10-17 maximum impact parameter we have put Anax = 2VA, E»A) n kinetic energy, obtained from this reaction produce a (dE A/dx) certain number of recoil nuclei. sc If f^(E^°) is the energy distribution of the £ par- NB aBA (E"BEA)q(E'B,E"B (dE" /dx) ' ticles obtained from each reaction A + B, then the B number of £ particles in the energy interval, E^° to (13) 0 0 Eg + dEg°, is fg(Eg°)dEg . Since the major part of If we add the energy distributions of all generations we the reaction A + В in the avalanche occurs in the obtain the energy distribution of the whole cascade moderate energy range (100 — 500 kev) and since the initiated by the particles from the reaction A + B : reaction A + В is strongly exoenergetic, we have assumed that this distribution is independent of the GA{EA) = ХГС£А(И)(£А). (14) energy of the entrance channel. If in the result of reaction A + В we obtain two particles, the function Having obtained the distributions GA{EA) and, in a similar way, GB(EB), we can give the number of fg{Eg°) is the д{Е£ - Ей) function. Owing to the destruction of particles £ on interaction with the A A + B reactions in the slowing-down process of the and В nuclei, the initial number fg(Eg°)dEg° of cascade initiated by the particles from the one reaction particles along the path x drops to the value A + B: q (Eg°,x)fg(Eg°)dEg where N OAB(E' )q k=JGA B A X q (Eg*, x) = exp - JQ (NA agA + NB ^j dx (9) X(EA,E'A)

and о^А(а^в) is the total reaction cross section of the particle £ with the nucleus A (B ). Taking into account X (Ев, Е'в) (15) that the energy of particle £ on the path x drops from (dE'B/dx) E£ to Eg because of energy losses, we can write the last expression in terms of Eg If the reaction A+B has only one exoergic channel the number k is the multiplication factor for the given medium. The condition for the development =exp-j; (10) {dEgjdx of the avalanche, therefore, is k > 1. In the numerical calculations, as long as the slowing where (dEg/dx) denotes the loss of energy of the down process of the products of reaction A + B and particle per unit path length. Introduce recoil nuclei is due to scattering from electrons, we can take into consideration only the first generation of recoil nuclei. Then from Eqs. (4), (5) and (15) we the cross section for the production of recoil nuclei have : A of energy Ex to EA + dEA by the particle £ with energy E^. Then the number of the recoil a. in the case of charged products of the reaction A+B A nuclei, with energy interval EA to EA + dEA 1 forplasma produced by the particles £ from the reaction A + В Y-i * along their paths, is jeiJ |дте2 for degenerate medium b. in the case of neutrons ga(EA)dEA = . , /dE\sc~]-1 (T3I* for plasma (17) dx /el (П) for degenerate medium {dEgldx) a SC £A {Eç> EA) is given by the scattering differential Critical Mass с cross section а^ (Е^, в) and the relation between the All our present considerations concern the condi- 9 angle of scattering and loss of energy in the collision. tions for the development of fusion chain reactions in Summing up all the products of the A + B reaction infinite media. In a finite medium the conditions we obtain: are different and a critical mass exists as in a fission chain reaction. In the first approximation, the critical mass can be We can write a similar expression for the energy distri- estimated very easily if we consider that the mean-free bution of recoil nuclei B. As a result of the elastic path Я with respect to the elastic scattering of the scattering of the first generation of A and В nuclei particles taking part in the reaction has to be com- we obtain the second generation of recoil nuclei of parable with the dimension L of the system. If we the medium. With the help of the above considera- denote by N the number of nuclei of a medium in a FUSION CHAIN REACTION 273

ГЕв/knB d 10 1941.3 gnB{EB) ъ -(ъ

X (21a) 1 +

Jl4.1 ¿NT1 X -. (21b) lio-2 Since the energy losses of deuterons and tritons up to 107 °K, in the case of a plasma, and up to 103 g/cc, in the case of a degenerate medium, are due to scattering from electrons we can write 10" 104 1ОЭ 10° 10е Temperature of DT plasmo in °K Í* 0 NB Figure 3. The multiplication factor in 50% D-T plasma as a dE' function of temperature /

unit volume, by m the mass of a nucleus in grams, NT sc (22) and by the cross section for elasting scattering, 8 sc 3 2 f we obtain mcr ^ W^ZVA = w1(^o') )"~ iV~ . Taking SC 24 2 24 into account that ^ 10- cm and m1 ъ 10~ g, the critical mass in grams is /dEB\sc , 48 2 where a -~z— and a —— are given by Eq. (2). 10 /iV . (18) \ d% /el \ dX /el We see that the critical mass is very strongly depen- The value of the multiplication factor obtained by dent on the density of a medium. For densities the numerical calculations for a 50% D-T mixture N = 1028, 1024, 1020 nuclei/cm3, the critical masses 8 8 under various conditions are plotted in Figs. 3 and 4. are 10~~ , 10°, 10 grams respectively. The cross sections we have put equal to the geometri- cal cross section and авт is taken from Bame and NUMERICAL CALCULATIONS Perry.10 Now, to illustrate the theory given above we shall determine the conditions for the development of a CONCLUSIONS fusion chain reaction in a D-T mixture. The role of the in statu nascendi reactions in the As a result of reaction D + Twe obtain alphas and release of nuclear energy depends on the physical neutrons with energies ~ 3.5 Mev and ~ 14.1 Mev conditions and, in the case of high temperature or respectively. Because of the much greater initial high density, they are decisive. If we denote by £ш energy and much lower energy losses for the neutrons, the energy released in a unit volume in the thermo- most recoil nuclei D and T result from scattering of neutrons; therefore, according to Eq. (12), gi>(1) {Ев) ъ* gni)(Ej)) and £т(1)(£т) ^ £пт(£т). Taking into account the fact that the absorption of fast and 1941.41 intermediate neutrons in the D-T medium is negligibly small we have qn t& 1. Assuming the scattering of neutrons from D and T nuclei isotropic in the center- of-mass system, we obtain

(TnDSC (En) T lio-» J Because, in the first approximation, the slowing down of neutrons is due to elastic scattering from D and T nuclei sc 10" \ Kn n onT NT (20) Density of electrons in 50% DT medium the energy distributions of the first generation of Figure 4. The multiplication factor in 50% D-T medium as a recoil nuclei are, respectively: function of density for T = 0°K 274 SESSION A-5 P/1941 M. GRYZINSKI

APPENDIX As was pointed out previously,4 the maximum impact parameter is, in general, a function of the velocities of interacting particles, their masses and charges, as well as of the external fields. In each problem this parameter must be determined separately. In the case of electrons bound in atoms, or Fermi gas electrons, the determination of the maximum impact parameter does not present any difficulty, but in the case of plasma electrons it is the subject of many discussions. According to Cowling,12 Chandrasekhar 13 and others, it is suitable to put the

T«mp*retur« of DT plasma in °K maximum impact parameter equal to the mean distance between the ions, but according to Landau,14 15 Figure 5. The release of energy in exoergic mixture for the Cohen, Spitzer and Routly and others, it must be usual thermonuclear process, and for a process taking into account equal to the Debye radius. in statu nascendi reactions From Eq. (2) it follows at once that in the limiting case Fg ZZ^e/D)2. Taking into release of energy does not exist above the critical account the mean value of distance between charged temperature or above the critical density; even at a particles in the plasma we finally obtain Dmâx ^ N~l. temperature of absolute zero the exoergic mixture is The assumption that the maximum impact para- explosive. meter is equal to the Debye radius will slightly change A plot of Eth (Reí. 11) and Etot as a function of the numerical results owing to the logarithmic depen- temperature for 50% D-T mixture is given in Fig. 5. dence on Anax-

REFERENCES

1. W. B. Thompson, Proc. Phys. Soc, 70, 1 (1957). 9. E. Segre, Experimental Nuclear Physics, pp. 9, 14, 2. P. K. Weyl, Phys. Rev., 91, 289 (1953). Vol. II, first éd., New York, London (1953). 3. S.K. Allison and S. D. Warshaw, Revs. Modern Phys.. 25, 10. S. J. Bame and J. Perry, Phys. Rev., 707, 1616 (1957). 779 (1953). 11. R. F. Post, Revs. Modern Phys., 28, 338 (1956). 4. M. Gryziñski, Phys. Rev., 707, 1471 (1957). 12. T. G. Cowling, Proc. Roy. Soc, A138, 453 (1945). 5. E. N. Parker, Phys. Rev., 10, 830 (1957). 6. E. Fermi and E. Teller, Phys. Rev., 72, 399 (1957). 13. S. Chandrasekhar, Astrophys. J., 97, 255, 263 (1943). 7. L. Landau and L. Lifshitz, Field Theory, pp. 208, 219, 14. L. D. Landau, Sow. Phys., 70, 154 (1936). sec. éd., Moscow-Leningrad (1948). 15. R. S. Cohen, L. Spitzer and P. M. Routly, Phys. Rev., SO, 8. H. Bethe, Phys. Rev., 55, 434 (1938). 230 (1950).