ever, to reduce it to the form as simple as possible by taking the m any anthors and was able to account for the existing experimental results sufficiently well (1)(2). The problem of theneutron-deuteron colli- to be attacked, is a three body problem already sion, which is the next too complicated to be solved rigorously, It will br, worth while, how- 1, 2, 3 respectively has the general form On the Theory of Collison of Neutrons with Deuterons. By Hideki YUKAWA and Shoichi SAKATA (Read March 13, 1937) ABSTRACT e problem of the neutron-reuteron Th collision was reduced to a simple form by taking the structures of 2II and 3II into account and neglecting the forces depending explicitly on the spin. ( 1.) The form of the effective potential hole was determined by comparing the estimated value of the cross section of scattering of slow neutrons by deuterons with that obtained experimentally The cross section of capture was found to be small in agreement with the experiment. The energy de- pendence of the scattering cross section was also discussed. (§2) The theory of Fermi concerning the effect of the chemical binding on the scattering of slow neutrons was extended to more general case and a necessary limitation of the theory was discussed. The eross section for thermal neutrons was found to he nearly twice as large as that for neutrons of several volts. ( 3) Theoretical reasons for the small con- tribution of the deuteron to the slowing down were considered. (§4). §l. Reduction of the Collision Problem. The theory of collision of neutrons with protons was developed by strnetures of 2II and 3II into account and to find an approximate for neutrons of not too large energies. solution at least valid The wave equation of the system containing a proton and two with the coordinates 1, 2, 3 and the z cnmponents of the spinsneutrons (1) See, for example, the summary report of Bethe and Bacher,Rev. Mod . Phys. 8, 82, 1936 and further Feruri, Re. Scient. VII~2, 1,1936. (2) Note added in proof: Recent experiments on the angular distribation of recoil protons by collision with D P neutrons, however, sterns to in licate the existence of g, forces. (Cf. Bull. Amer Phys Soc. 12, 30, 1937). comparatively lon 1937 On the Theory of Collision of Neutrons with Dcuterons. 5 43 where M is the common mass and V12, V13, V23' are the potentials of forces, each involving the coordinates, spills and exchange operators of two particles in general. If we neglect the terms other than those of Majorana type, they become respectively, where and P*s an, the operators of the coordinate exchange . Further, if we adopt the coordinates of the centre of mass those of the proton relative to the first neutron s aid those of the second neutron relative to the centre of oaks of the former as independent variables, (1) reduces to or after separating s and from R and 's, where and E' is the energy of the system excluding the kinetic energy of the centre of mass. Now, when the energy F of the incident neutron is not large enough to disintegrate the deuteron, i.e. E0 is smaller than its binding energy En=2.2 10 eV, provided drat it has no true excited levels, the wave function can be assumed to have the approximate form(1) (1) In order to avoid the comphcation of the formulac, we want to omit hereafter the doublesign ±, whichshould he suffixedto eachof functions(r), F(r) and G(r). 5 44 Hideki YUKAWA and Shoichi SAKATA. [Vol. 19 where x(s) is the normalized wave function of the deuteron, satisfying the equation and the approximate orthogonality is assumed. The wave function f+ should be multiplied by either of two spin functions antisymmetric with respect to the neutrons, corresponding to 2S, 2P, ... states of 3II, while _ should be multiplied by either of six symmetric spin fumctions corresponding to the mixture of states 2S, 2P, ... , and 4S, 4P .. .. where according as =1 or -1. If we inn-rt (4) in the wave equation (3), multiply both sides by respect to s, we obtain an integro-differential (s) and integrate it with n equatio This can into an integral equation with the asymptotic form(1) (1) Mott and Massey, Theary of Atomic Collisions, Oxford , 1933, Chapt. VI. 1937] On the Theory of Collision of Neutrons with Dcuterons. 545 for large r, where denotes the angle between the vector r and z axis , i.e. the direction of motion of the incident neutron and the angle between r and q. F(r, 0) is an axially symmetric solution of the homogeneous differential equation with the asymptotic form where k=2/ M(E'+E0)/3 and U±(r) are auxiliary potentials, which will be determined in the following section. §2. Estimation of the Scattering Cross Section. In order to obtain an approximate value of the scattering cross section without solving the complicated integral equation, it is necded to choose the auxiliary potentials U±(r) in (9) and (10) such that F±(r, ) become already approximate expressions four the required functions ±(r, ). If we consider the deuteron to be a sphere of diameter 4.36 10-13cm and the nuclear forces to have a de hi ate1/ =range / MED= a=2.32 10-13 cm, the interaction between the neutron and the deuteron can be represented roughly by a potential hole with the radius b, which may take a value between 1/2 =2.18 10-13cm and 1/2 +a=4.5 10-13cm. Than we can put according as r<b or r>b. A legitimate value for U+ can be obtained by assuming that (10), has a solution with the. energy corresponding to the normal state of 2H. Such a procedure was already considered. for slow neutrons in two cases avove 546 Hideki YUKAWA [Vol.and Shoichi 19 SAKATA. and Mohr.(1) Thus we find used by Massey for the extreme case b=4.5 10-13cm or 2.18 10-13cm and conselguently the cross section of the deuterm at rest for slow neutrons become, We have calculated further the cross sectin for several values of neutron cnergy, the results being shown in Fig. 1 and 2. In thethe former case reaches to a large maximum and then decreases as the on account of the presence, of the Virtual P-level, whileenergy increases in the latter care it decreases steadily with the cnergy. On the other hand, U_ can not be determined in like manner, as little is known of the excited states of 3 H, and we can say only that -U _ is much smaller than -U or be negative. Thus the crossmay enen section in tins case can be determined only by making use of further experi- information as follow,.mental Namely, the observed cross section of the deu- teron in heavy water is =4 10-24cm2 for slow neutron, of /'--groupaccording to Dunning, Pegram, Fink and Mit- If we consider the effectchell.(2) of chemical binding of deuterons,(2) the cross section of the free deuteron for slow neutrons should be abouthalf of the above ,so that we obtain a Fig. 1 Thisprobable value 2 10-24cm2. value is to correspond to the average of the cross seetions of the deuterons Fig. 2 Hence, we can choose U_ with a (1) Massey and Mohr, Proc. Roy .Soc. A,148, 206, 1935. (2) Dunning, Pegram, Fink and Mitclell, Phys Rev. 48 , 265, 1935. (3) Compare § 3. be alterd therewith, so that the correct value of b will be determined, when the detailed experimental results for varions values of the neutron 1977] On the Theory of Collision of Neutrons with Dcuterons. 547 given vales of b so as to make _ satisfy the relation for slow neutrons, where +, takes the value determined with the same b. We obtain in this manner the results for b=4.5 10-13cm and for b=2.18 10-13cm. As shown in Fig. 1 and 2, _ decreases with the energy in tin latter case, while it has a small hump due to the virtual S-level in the former case. The average cross section -given by (12) depends on the energy in the manner as indicated by the curve in Figs. 1 or 2. If we change b between the above two extreme values, the shape of the eurve will energy will be obtained. The cross section 1.71.10-24cm2 measured by Dunning and others(1) for fast neutrous from the Rn-Bc souce is an average over a wide energy range and is insuflicient for the determi- of b, although it moms t(i he in favour of the value nationof b nearer to 2.18 10-13cm than to 4.5 10-13cm.. In any case, tike cross section of capture of slow neutrons by den- will be much smaller than by protons, as the above teronsresults show that there is no true or virtual 2S_or 4S_ levels of small encrgy, while, in the latter cane, the existence of the virtual of energy about 12x 104cl was expected from theoretical arguments(2). This seems to be in agrcement with the recent experimental results of Kikuchi, Aoki and Takeda,(3) winch indicate that the cross seetion of the -ray ion by collision of solw neutrons with deaterousemiss is very small and has an upper limit 0.3 10-35cm2. Thus far, we tood only Majorana forees into accomt. The inclu- sion of small Wigner forces will not give rise to any substantial mo- dification, while the presence of Heisenberg forces will result in the further splitting and coupling of two set of states above considred, (1) Loc.