§L. Reduction of the Collision Problem. the Theory of Collision of Neutrons with Protons Was Developed By

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 contribution 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

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 rb. 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 deuteron 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. cit. (2) Fermi, loc cit. (3) Kikuchi, Aoki and Takeda, Sci. Pap. Inst Phys. Chem. Res. 31, 195, 1937 of the chemical binding of heavy hydrogen atoms in the scatterer,the cross seetion becomes

Hence, the ratio is

548 Hideki YUKAWA and Shoichi SAKATA. [Vol. 19

but our method of estimation is too crude to afford such discussions of finer details. It shoud be noticed that the angular distribution of scattered neutrons is, of course, spherically symmetric in the relative coordinate for small cnergies, so that the probability of it bring systemscattered by an angle between and +d inordinary teh coordinate system becomes

Thus, most of the neutrons are

Fig. 3. Angular Distribution of shown in Fig. 3.. scattered into the forwardscattered direction asNeutrons.

§3. Effect of Chemieal Binding of Deuterons on their Collision with Slow Neutrons. In the above calculations, we assumed the deuterons to be free and initially at rest, but we have to consider the effect of chemical binding of heavy hydrogen atoms with other atones or with one another, if the nentron energy is few volts or smaller. Such an feet in the case of ordinary hydrogen atoms was dealed with in detail by Fermi(1) The procedure can casily be extended to a more general case, in which the scattering nuelei have the mass M' not equal to that of the neutron, at least in the limit of the strong binding

is small compared withNamely, that if the energy of the neutron

and the augular distriibution of the scattered neutrons is spherically/ symmetric in the ordinary coordinate system, where is the cross neutrons, as was calculated in section2. of free deuterons

forthe denteron in confrast to 4 for the proton (M'=M)) and I for

(1) Loc. cit. 1937] On the Theory of Collision of Neutrons with Dcuterons. 549 the heavy nucleus (M'>>M). Thus, the cross section of deuterons in heavy water for thermal beytrons will be nearly twice as large as that for slow neutrons with the energy large compared with that of the chemical binding, as was pointed out already in 2. It should h., noticed, on the, other hand, that the calculation made by the method of Fermi leads, in the limit of the weak binding, to the result which is not consistent with that of the prececding section. Namely, the cross section thus obtained differs from § 2 by a factor

which takes the value 1.365 for the deuteron (M'=2M) and is not equal to 1 except for the proton and the heavy nucleus. Correspon- dingly, the angular distribution of seattered neutrons becomes

in contradistinction to

both in the ordinary coordinate system, which are also at variance with each other except for M' M and M' M. Closer examination shows that the origin of this discrepaney is the illegitimate use of Born's approximation, on the part of Fermi's method, in the limit of the weak binding, the agreement of the result of both methods in the case, of free protons being only accidental. Nevertheless,Fermi's results can be used safely for the case seems to he always true in the limit of the strong binding.

§4. Slowing Down of Neutrons by Multiple Collision With Deuterons.

The energy distribution of neutrons with the initial energy E0 will be constant throughout the interval (E0/9,E0) after the collision with

deuterons initially at rest, if we assume the scattering is spherically symmetric in the coordinate system, in which the centre of mass is at rest. Hence, the mean energy after a single collision is 5/9E0 and re- ducesto (5/9)nafter E0 collisions,n which does not differmuch veryfront centerons and dashen curves to those

using the gencral formulae of Condon and Brcit,(2) has very small valuc

collisious

, for example, are 1.15 10-1 and 0.063 respectively. General state of affairs will be illustrated

y Fig. 4. The ordinateb shows the probability that the neutron has a fraction of its initial energy less than the abscissa after a number of collisions marked on cach curve. Full withcnrves refer to the collisions

1) Condon and Breit, Phys. Rev. 49, 229, 1936. (2) Loc. cit.

(

550 Hikeki YUKAWA and Shoichi SAKATA. [Vol. 19

the corresponding value (1/2)nE0 in the case of collisions with protons.(1)

Thus, the deuteron seems to have an effect on the slowing down of comparable with the proton. neutrons The are, however, many reasons which lead us to the opposite conclusion. Firrtly. the nentron energy can not be smaller than 1/9E0

collision, No that a neutron with the energy of bythe a singleorder should undergo at least 6 collisions with denterons beforeof 10**it a slow neutron of several volts, while this call be effected bybecomes Secondly, the occurrence of such a singlea collision with the proton. ple collision with the former will bo much rarer than that with multi the latter t tiler similar because the cross section of the, former becomes munch smaller than that of the latter as the energy of the neutron decreases. Thirdly, the energy distribution function of valuated byneutrons after n collisions with the former, which can be e

1 " in th e neighborhood of the lower limit(1/9)nEo compared with the cor-

responding value for the latter. The probabilities that the neutron has

i an energy smaller than En/94 after five

with protons. Both the avseissa and the ordinate are in ligaritlunie seale.

Fig 4. l'roi alility that the neutron has Under these circumstances, we can not expect theoretically any ap- an energy smaller than a fraction of preciable contribution of the deuteron the initial energy after mnltiple col- watercompaund to such as heavyli 1937] On the Theory of Collision of Ncutrons with Dcntcrons. 551 the slowing down of neutrons. This conclusion is in good agreement with the recent experimental results of Kikuchi, Aoki and Takeda,(1) which show that the number of neutrons of C-group produced in D2O by using D-D source is at most 1/27 of that produced in the equal volume of H2O, although quantitative comparison of the theory and the experiment is very difficult. Previous results of Dunning and others (2) indicating relative efficiencies about 1 : 5.5 of D2O and H2O for produ- cing slow neutrons by using Rc-Rn source should he attributed to the contribution of B-group as pointed out by Kikuchi and others.(3) In conclusion, the authors want to acknowledge their indebtedness to Prof. S. Kikuchi for valuable discussions.

Institute of Physics, Osaka Imperial University.

(Received April 7, 1937.)

(1) Loc. cit. (2) Loc. cit. (3) Loc. cit.