Structure of three-particle nuclei from photodisintegration experiments

2.1

[ I

Nuclear Physics A98 (1967) 437--450; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

STRUCTURE OF THREE-PARTICLE NUCLEI F R O M P H O T O D I S I N T E G R A T I O N EXPERIMENTS Nuclear and Coulomb effects of the final state interaction V. N. F E T I S O V

P. N. Lebedev Physical Institute of the USSR, Academy of Sciences, Moscow, USSR Received 23 J a n u a r y 1967 Abstract: N e w experimental data oll the ~ - q u a n t u m absorption by the 3He a n d 3H nuclei are analysed taking into a c c o u n t the nuclear a n d C o u l o m b interactions in the final state. T h e role o f the asymptotic structure o f the three-body wave functions in the photodisintegration theory is discussed.

1. Introduction

Recent experiments on electron scattering, muon capture and 7-quantum absorption by 3Iffe and 3H nuclei aimed at obtaining information on the structure of ground and excited three-body states. Experimental data on the properties of 3He and 3H become especially important if one takes into account the well-known mathematical difficulties preventing an exact solution of the three-body problem. Due to this, approximate methods are used to describe the states of the three-nucleon system. The theoretical interpretation of the experimental data is a basis for choosing better approximations. Unfortunately, the description of quite a number of experimental effects appeared to be weakly dependent upon the model of three-particle nuclei. So, for example, the electromagnetic form-factors of 3He and 3H measured in experiments on elastic electron scattering can be satisfactorily described by using various radial shapes of the wave functions 1, z). When analysing inelastic electron scattering, one detects a somewhat greater sensitivity of the cross section to the choice of the ground state function. However, even in these calculations, using various models of the 3He and 3H nuclei, deviations of theoretical curves from experimental data do not exceed

20-30 % (ref.3 ). As to the experiments on muon capture, the value of the transition rate in the wellknown reaction 3He(/~-, v)3H appeared to be not sensitive to the details of the structure of the three-particle nuclei. The transition rate of this reaction is determined in fact by the root-mean-square radius of the bound three-nucleon state 4). From the analysis of photonuclear reactions it became known s) that a reasonable description of the integrated cross sections o-o = Sa(E~)dE~ and a_ 1 = Sa(E~)E- 1dE~ can be obtained by using a wide range of ground state wave functions. 437

438

v.N. FETISOV

Detailed calculations performed by many authors have shown that considering only these experiments one cannot make a unique choice of the radial shape of the three-body wave functions. A recent analysis of the experiments on inelastic electron scattering and also 7-quantum absorption in the reaction 7q- 3I-[e(3H) --+ p ( n ) + d ,

(1)

pointed out only the most probable wave function shapes. It became clear that an acceptable description of the nuclear reactions can probably be obtained with Gaussian functions and those suggested by Gunn and Irving 2, 5,6,7). However, the last cross-section measurements 5,8) in the reaction 7+3He ~ p+p+n,

(2)

showed a very sharp discrepancy between theory and experiment and cast much doubt on the validity of our approach concerning the structure of three-particle nuclei. The theoretical cross section of the process (2) calculated with the Gaussian, Irving and Gunn-Irving functions appeared to be three times as much as the experimental value 5, 8,9). It was shown in our preliminary report 1o) that this discrepancy is due to an incorrect asymptotic behaviour of the wave functions used before in the photo-effect theory. It is the asymptotic region that essentially contributes to the transition matrix elements. Wave functions of three-particle nuclei having a correct behaviour in the external nucleus region were suggested by Pappademos 11) Dalitz and Thacker 12). The use of these functions in the photo-effect theory has resulted in a satisfactory description lO) of the reactions (1) and (2). Thus, an investigation of the three-particle break-up of 3He yielded some information about the asymptotic structure of the ground state. In this paper we continue the investigation of the photo-effect on three-particle nuclei. In ref. lo) numerical results were only given for the photodisintegration cross sections and the reasons for the comparatively weak influence of the wave function asymptotics on the cross-section reaction (1) were not explained. This question is considered in sect. 2, and as an example the comparison of the dipole transition matrix elements calculated with the Gaussian wave function and that of Dalitz and Thacker is made. In sect. 3 the role of the nuclear and Coulomb interactions in the continuum for the reaction (2) is discussed. The effects of the nuclear interaction of nucleon and deuteron in the channel (1) are discussed in sect. 4. In the conclusion, an influence of the wave function asymptotics on the quadrupole transitions in the reaction 3He(7, p)d is briefly discussed.

THREE-PARTICLE

NUCLEI

439

2. Role of the asymptotic structure of the aHe and aH wave functions in the photo-effect theory In ref. lO) special attention is paid to the integration region over the variables and r = r 1 - r 2 (rl, r2, r 3 are the radii vectors of the nucleons) in the dipole transition matrix elements. This region has an asymmetric shape due to the existence of the dipole operator proportional to the z-component of the vector p in the overlap integrals. Special calculations showed that the main contribution to the matrix elements is given by that region where [p[ is greater than [r]. If, following Pappademos, Dalitz and Thacker 11,12), we suppose the quasi-deuteron cluster structure of 3He and 3H in the asymptotic region of p, one can for large ]p] factorize the ground-state three-body wave function:

p

= r3--1(rl+r2)

W(p, r) ~ Np-~ exp ( - ~p)(p(r),

(3)

p-*oo

where ~o(r) is the quasi-deuteron function. For a qualitative understanding of the reasons for a sharp decrease of the threebody break-up cross section we substitute in the transition matrix element M = fexp (-iK.

p) exp ( - i k . r)pz ~(p, r)d3pd3r

(4)

the accurate wave function ku(p, r) for the asymptotic expression (3). * Numerical calculations show that the absolute value of the two-body break-up cross section weakly depends upon the choice of the ground-state function. Therefore it is of interest to compare only the ratio of the cross sections of the photodisintegration through the channels (1) and (2) obtained by the function (3) and the Gaussian function ~(p, r) = N~ exp ( - 22p 2) exp ( - ~2r2).

(5)

For the sake of distinction, the normalization constant N is chosen so that the theoretical value of the two-body disintegration cross section approximately corresponds to the experimental data. The parameter e in the wave function (3) is determined by the nucleon binding energy in the three-particle nucleus 11,12). In this paper we take a quantity e = 0.5 f m - 1 . The only parameter 2 of the Gaussian function (5) is chosen, as usual, from the root mean-square radius of the nucleus known from electron scattering. ((R2) ~ ~ 1.7 fm.) For simplicity, we use the Born approximation to describe the final states in the reactions (1) and (2). Combining relations (3), (4) and (5) we can easily obtain a formula for the total t As is known, a similar procedure is also possible in the theory of the deuteron photodisintegration. In the region of low 7-quantum energies the photo-effect cross section can be satisfactorily described by an asymptotic deuteron wave function Nr 1exp(--Kr) if one chooses a correct value for the normalization constant N 13).

440

v.N. FETISOV

three-body break-up cross section in the form of the overlap integral of three functions:

= fo

(m(a)c(mp.

(6)

The functions A, B and Chave a simple physical meaning. The factor A is determined by the final-state density: 46rc2 e 2 [ME,~[ME'~ 2 daKd3k A(fl) - 34x/3 hc \ - ~ - ] \ ~ - ] sin2 fl c°s2 fl ~ (2x)6d~"

(7)

In this expression IKI = (4M(E~-Q)/3h2) ~ sin fl is the nucleon momentum relative to the c.m. of two other nucleons moving' with the relative momentum [kl = (M(E~-Q)/hZ) ~ cos fl, where Er is the y-quantum energy and Q is the reaction threshold. The factor o

N

jx(
(s)

determines the probability of dipole excitation of a nucleon relative to other two nucleons (j~(Kp) is the spherical Bessel function). The function Z(P) describes the bound state of a nucleon in the three-particle nucleus at large Ipl. This function is being determined by eqs. (3) or (5). The coefficient

C(fl) =

jo(kr)q~(r)r 2 dr

(9)

describes the probability of excitation of the residual nucleon pair from the bound S-state into the continuum. The well-known Hulth6n deuteron function was taken as the function ~0(r) in eqs. (3) and (9). In fig. 1 the dependence upon the y-quantum energy of the photodisintegration cross section through the channels (1) and (2) is illustrated. Curves 1, 2 are obtained with the Gaussian ground state function. The cross sections calculated with function (5) are pictured by curves 3,4. The sharp decrease of the three-body break-up cross section obtained with the Dalitz-Thacker function can easily be realized if one considers the dependence of A, B, C on the parameter ft. Fig. 2 shows the factor A and the factors B, C and B 1, C 1 calculated with functions (3) and (5), respectively. The y-quantum energy E~ ~ 37 MeV is taken in the region of curve 2 maximum. From fig. 2 it is seen that the maxima of curves B and C are essentially shifted to the region of small values of the factor A. This shift of the curves B and C relative to curves B1 and C1 is due to the pole structure (in the momentum

441

THREE-PARTICLE NUCLEI

space K and k) of the factors B and C. The poles of the functions B and C arise as a result of the radial shape of the functions p - ~ e x p ( - :¢p) and cp(r) ,~ r - 1 [ e x p ( - ~r) exp(-~/r)]. The shift of maxima of curves B and C to the region of small values of A results in a decrease of the overlap integral (6).

/"

6",m~ 2.0 f

/

/ /

1.0

/

,,~%

3/

/

/

/g

4 ¸'/

!

"',

/// ,/';~

2'0

do

~b

E~ MeY

Fig. 1. The photodisintegration cross section of the three-particle nuclei through the channels (1) and (2) obtained with the ground state wave functions (3) and (5).

If one compares our schematic calculation of the three-body break-up cross section (curve 4) with the experimental data 5,8), it becomes evident that the cross section maximum is shifted to higher energies. This is caused by neglecting the interaction between nucleons in the final state lo). The two-body break-up cross sections are determined only by the factors B and BI. From the aspect of curves B and B 1 depicted in fig. 2 we can conclude that two-body desintegration cross sections would not differ considerably in value, while the cross

442

v . N . FETISOV

section maximum calculated with the Dalitz-Thacker function shifts to small ?quantum energies. This situation is really observed in fig. 1. The cross sections of the reactions (1) and (2) obtained in ref. lO) with exact (non-asymptotic) DalitzThacker functions reveal all cross section details shown in figs. 1 and 2.

8c;~l

~,~)~ fn~

c/#)] "10g

3.0

(3

'q

C

!.(

S,O

,[

/

/ 0,2

0.~

(1G

0.8

1.0

1.2

1,4

/3

Fig. 2. The dependence o f factors B, C, B1, C1 and A on the parameter ft.

3. Influence of the nuclear and Coulomb interactions of protons on photonucleon energy spectra

In refs. J 4) for the three-particle nuclei a "one-particle" mechanism of the excitation by ?-quanta was proposed. In the frame of this mechanism the main features of the experimental differential cross sections are realized if we include two channels of ?-quantum absorption in the reaction (2): 7 7+3He ~

(p+p)s+o+n (p+n)a+o+p

(11)

Previous calculations 14) of cross sections for similar reactions carried out for 3H have demonstrated an important role of the distortion of the wave function of the residual nucleon pairs in the states with the total momentum J = 0. However, due to the absence of appropriate experimental data the theory was compared with experimental data on 3He. The theoretical energy spectra appeared to be somewhat different from the experimental ones in the region of small relative proton momenta. In this paper

THREE-PARTICLE NUCLEI

443

we included both the nuclear interaction of nucleon pairs in the states with J = 0, and the Coulomb proton repulsion in channel (10). Specific calculations have shown a weak sensitivity of the differential cross sections to the choice of the ground state wave function. In this section, using the calculation scheme of the distorted wave method 14), we do not pretend to get the absolute value of the three-body disintegration cross section. It was shown earlier 1,) that the nucleon energy spectra for the Eichmann ground-state function 6) are determined by the expression

dt

--

3 27,j3 he c 1

~2-

+q

~-

(

k)P

x

[1+4~ -1 4

16

F2(Kl,kl)dx

t + ( 1 - t ) ~-,

(12)

K

where K 1 = - ½ K + k , k x = z 3K + ~ k1, x = ( K ' k ) / K k . The physical meaning of momenta K and k was explained in the previous section. By the quantity t = sin 2/1 we designate the ratio of the nucleon energy EN to its maximum energy EM in the c.m. of the target-nucleus. To calculate the neutron spectrum by formula (12), it is necessary to put P = 1, q = ~. On the contrary, at calculating a proton spectrum P = ½, q = 1. The function F(K, k) has the form

(

F(K, k) = exp - ~ x

7*a(r)

exp(-C1¼r2)+g('~tl

exp

\ C 2]

Kz C - C ~ - c 2 ¼ r 2 1 } 4 C2C1

r2dr ,

(13)

where C1 = 0.3 fm -2, C2 = 0.07 fm -2, e = 0,15 are the parameters of the groundstate function. When calculating the neutron spectrum, the function 7~k(r), normalized by the condition

7*k(r) --+ sin ( k r - q In (2kr)+a+~5) --1 r-~ o9

(14)

k]*

describes the singlet S-state of two protons in channel (10) (r/ = Me2/2h2k; c5 and a are the nuclear and Coulomb phase shifts) and the similar function 5gkl(r) describes the singlet S-state of neutron and proton in channel (11). On the contrary, in the case of the proton spectrum calculation the function 7Jk(r) corresponds to the n-p pair, and the function 7Jk~(r) corresponds to the p-p pair. The potential for two protons in the continuum is chosen in the form

V(r)

f-Vo,

r < a,

e2/r,

r > a.

(15)

444

V. N. FETISOV

Since in the region of maximum of the 3He(?, n)2p cross section the energy of the relative movement of two nucleons does not exceed 1 0 - 1 5 MeV, the nuclear phase shift 3 is fairly well determined by the effective range theory 15). This fact saves us from the tedious procedure of very accurate determination of the square well parameters Vo and a. The square well parameters Vo = 10.5 MeV, a = 2.83 fm are taken from ref. 16). The Coulomb functions and transition matrix elements were computed. t~ I

i / i i P J.

i.

'

• I

7

i

I

6.0 -q i

i-. . . . .

,

/r ., I

~.o ~,

r~-- i'/'ti)

I ......

T

go

I

I .~

",

,

T

,

~

I

/

,r 12.0 ,

I'

L - ~

\

~

II

'

r-----~\~' ' /"lJ

,

T

/

O.O

t

/"

I

[[

',T I ~r ,, T _ _ _ ";

-

aul

'

,

,

\

[

h

"

i

0.2

z

,

.

I

,~"

/

i]

-.~,-~-

"

T

J.-J

i

0.~

';' Z

II

".

I

T

T

t.

"

i

06

[

/

i

0.,~

/.0


F i g . 3. E n e r g y s p e c t r a o f p h o t o n u c l e o n s .

In fig. 3 there are listed the proton spectrum (dotted line) and that of neutrons (dotand-dash line) for E ~ - Q = 16 MeV. Theoretical histograms corresponding to these curves are also pictured by dotted and dot-and-dash lines. Points in fig. 3 depict experimental data 5). Comparing the curves in fig. 3 with the results of ref. 14) on the aH nucleus we observe a smearing out of the maximum of the proton spectrum in the region EN/EM ,.~

THREE-PARTICLE

445

NUCLEI

0.25 in comparison with the maximum of the neutron spectrum in the 3H(y, p)2n reaction. In addition, one can note shift of the main maximum in the neutron spectrum to smaller t. The histogram shown by the solid line in fig. 3 depicts the neutron spect r u m at E ~ - Q = 8 MeV. Thus, from fig. 3 it is seen that the distorted wave approximation for the final states correctly describes main features of the photonucleon energy spectra.

',3

.o

',

/z i

L

i /

!

1' /

[~0 I ~

I

~

~ ",

]i.0 r

'

\"~ \

l

", x

\ \\

"'~

]/ / /

.. ~.

I

! ,

/,/ ./2,"

. . . . ,.

0.5

!

i ~,

I.O

.~.

,,

1.5" f3

3.0

Fig. 4. In the upper part are shown the energy spectra o f neutrons in the reaction aHe(~', n)2p (curve 2) and those o f protons in the reaction ZH(y, p)2n (curve 1) calculated with the wave function o f the ZHe nucleus from the paper by Dalitz and ThackerX2). In the lower part are shown the continuum tS-state wave functions for two neutrons and two protons.

The photodisintegration cross section of 3I-[e through channel (10) with the nuclear and Coulomb proton interactions in the continuum was calculated with the DalitzThacker function 12). In this case the neutron spectrum da/dfl is shown in fig. 4 at E ~ - Q = 8 MeV. Curve 1 depicts the spectrum with the nuclear and Coulomb distortion of the wave function of the p-p pair. The substitution of the wave function of the p-p pair for that of the n-n pair results in the spectrum shown by curve 2. In case of the 3He(y, n)2p reaction a suppression of the cross section caused by a decrease in the value of the radial wave function of two protons is observed in the region 1.2 =< fl =< 1.57. This is seen from fig. 4 where by solid line 1 and dotted line 3 radial functions 7Jk(r) of tWO protons and two neutrons at the same energy E = h2k2/M= 0.4 MeV are shown. In the region 0 < fl < 1.2, where the energy of the relative movement of protons is larger than in the region 1.2 < /3< 1,57, an increase occurs in the cross section due to shift to large r of the radial IS function of two protons

446

v.N. FETISOV

(curve 2) with respect to the corresponding aS function of two neutrons (curve 4). Curves 2 and 4 are given at the relative movement energy of two nucleons E = 6.2 MeV. If one takes into account that the wave functions 7Jk(r) are overlapped in the matrix element with the functions of the bound state quickly damping with increase of r, an increase of transition matrix elements is observed for those functions ~Uk(r) which become negative at larger r. From fig. 4 it is evident that the integral So~(da/dfi)dfl equal to the total cross section of the reaction 3He(y, n)2p almost does not change when the Coulomb distortion of the wave function for two protons is included. Coulomb forces essentially influence the total cross section only near the very threshold of the reaction. 4. On the interaction in the final state in the two-particle break-up channel. Quadrupole transitions It is known 6) that dipole transitions are main transitions for 3H and 3He at small y-quantum energies. If one confines oneself to the dipole approximation and neglects P- and D-configurations in the ground state wave function, then it is easy to show that the products of the reaction (1) are emitted in the p-state. There are estimates of the nuclear distortion of the deuteron wave function in the nucleon-deuteron scattering problem a7). In this connection we can expect that for the p-state the distortion of the deuteron wave function due to its interaction with the incident nucleon is small. In solving the nucleon-deuteron scattering problem, we neglect this distortion. Burhop and Massey were the first to use this approximation in photo-effect theory ,a). However, due to the lack of experimental information about the properties of three-particle nuclei these authors made rather rough approximations for the wave functions. Quite recently, Eichmann 6) calculated the cross section of the reaction (1) taking into account the nucleon-deuteron scattering. Eichmann's method of the approximate solution of the Schr6dinger equation is valid only for the Gaussian wave functions. Unfortunately, with these wave functions we obtain too large a value for the cross section s) of the reaction (2). In this section we shall consider the influence of nucleon-deuteron interaction on the cross section of the reaction (1) assuming that the ground state of the target nucleus is described by the DalitzThacker wave function. Following Massey, Burhop and Eichmann we shall also search for the solution of the nucleon-deuteron scattering problem in the form

}Pk(P, r) = ~p ~ft(kp)rp(r)Pl(cos 0),

(16)

where 0 is the scattering angle in the c.m. and q~ the deuteron wave function. Substituting eq. (16) into the SchrSdinger equation and averaging over the deuteron state we obtain for the functionf~(kp) the equation

f/' + uz(k, p)f, =

fo Kt(p, p')ft(kp')dp'.

(17)

447

THREE-PARTICLE NUCLEI

The procedure for obtaining the functions ul(k, p) and Kt(p, p') is well k n o w n 21), therefore the functions are not given in the text because o f the complexity o f the expression. The interaction between the nucleons is chosen in the f o r m 19) v(r) = - v o exp

(-rlr2)(w+bB+hH+rnM).

(18)

For the sake of simplicity o f the calculations the deuteron state ¢p(r) was approximated by two Gaussian functions ¢p(r) = N[exp ( - c ~ r Z ) + C e x p ( - f i r 2 ) ] .

(19)

The parameters Vo = 64.48 MeV, # = 0.4 fm -2, ~ = 0.0356 fm -2, fl = 0.249 fm -2 and C = 2.825 are taken f r o m ref. 19). The m e t h o d o f numerical solution of eq. (17) with the b o u n d a r y conditions

ft(O) = O,

ft(kp) ~ sin (kp-½1n+cS),

(20)

p'-~oO

is given by R o b e r t s o n 20). Employing Robertson's m e t h o d we reduced eq. (17) to a system of algebraic equations which was solved with an electronic computer. Calculations were performed for two variants o f nuclear forces: w = m = 0, 4; h = b = 0, 1 (Serber forces) a n d m = h = 0, w = 0.8, b = 0 , 2 (Wigner-Bartlett forces). The C o u l o m b interaction between the p r o t o n and the deutron was neglected. The wave functionsfl(kp) obtained at E ~ - Q = 6.0 MeV are pictured in fig. 5. Curve 1 depicts the plane wave, curve 2 corresponds to Serber forces and curve 3 to WignerBartlett forces. Curve 3 is considerably shifted to small p as c o m p a r e d to curve 2. '

,/ /

:~e.

/

///

f,f Fig. 5. The continuum wave function describing the nucleon-deuteron scattering in the P-state.

1.0

6-, rn,~

T I0

~

T

i

/~

eb

'

ab

'

~b

r~,/'/eV

Fig. 6. T h e c o m p a r i s o n o f the theoretical curves with the experimental cross section o f the twoparticle disintegration channels. T h e h i s t o g r a m - d a t a o f G o r b u n o v et al. s). ~ - ~ . results o f Stewart et al. 2~), .t.

- results o f B e r m a n et al. 21), t r a n s f o r m e d in ref. 2a), - d a t a o f K o s i e k et al. ~4),

J_

- data o f B6sch et al. ~).

,o p

08

T

i

/

06

Fig. 7. T h e energy dependence o f the a s y m m e t r y coefficient P.

THREE-PARTICLE NUCLEI

449

Owing to this shift the overlap integral of the continuum function f l ( k p ) with that of the bound S-state increases. The larger the overlap integrals and consequently the larger the cross sections appear, the greater is the shift of the continuum functions f l ( k p ) to small p. In fig. 6 curves 1 and 2 represent the photodisintegration cross section of 3He obtained with the Dalitz-Thacker ground-state wave function. The cross section for 3H shown in fig. 6 is calculated with Serber forces. The results of different experiments are shown by histograms and points. From fig. 6 we may conclude that the calculations performed with Serber forces (curve 1) correspond to the experimental data better than those with Wigner-Bartlett forces (curve 2). In connection with the discussion of the role of the asymptotic structure of threenucleon wave functions in the photo-effect theory, we considered the quadrupole transitions in the reaction 3He(~/, p)d. Recently, Corbunov et al. 5) have found an interesting effect: the coefficient of the asymmetry P in the proton angular distribution appeared to be rather great even at small y-quantum energies. The experimental value P in three intervals of photon energy A E 7 = 6-12 MeV, 12-16 MeV and 16-22 MeV is shown by points in fig. 7. Line 1 represents the asymmetry coefficient obtained with the Gaussian ground-state function of Eichmann 6). Line 2 gives P calculated with the Dalitz-Thacker wave function. The nucleon-deuteron interaction was neglected in this calculation. From fig. 7 a high sensitivity of the coefficient P to asymptotic structure of the wave function is evident. However, the theoretical value of P is somewhat smaller than the experimental one. 5. Conclusion

In this paper there are analysed new experimental data on cross sections, energy spectra and partly, quadrupole transitions in the ,/-quantum energy region including cross-section maxima of the reactions (1) and (2). In contrast to other approximations s, 9,21) the three-body wave function suggested by Dalitz and Thacker gives a satisfactory description of the photodisintegration characteristics. At high energies the photo-effect theory should probably be essentially modified by appropriate treatment of tensor and spin-orbit nuclear forces. The main object of a simple photo-effect theory is explaining the principle features of the photon absorption by three-particle nuclei and pointing out those values observed in experiments which are critical to the structure of the wave functions. The results of this paper obtained with phenomenological description of threeparticle states show that the main photodisintegration characteristics can be described with an accuracy of 20-30 % within the usual conceptions of the light nuclei theory without involving many-particle nuclear forces. The correct formulation of the problem of many-particle forces will probably require both a considerable increase in experimental accuracy and further improvement in theoretical treatment of three-nucleon systems.

450

v.N.

FETISOV

I express m y deep gratitude to A. M. Baldin for c o n s t a n t su p p o r t in w o r k and m a n y helpful discussions. I a m t h a n k f u l to A. N. G o r b u n o v an d A. T. V o r f o l o m e e v f or the discussion o f e x p e r i m e n t a l p r o b l e m s and also to A. T. M a t a c h u n , L. Ja. T r e n d e l e v a and V. P. F o m i n a f o r the great assistance in carrying out n u m e r i c a l calculations on an electronic c o m p u t e r .

Note added in proof: While p r e p a r i n g this p a p e r for printing the a u t h o r has got a p r e p r i n t by J. M. Knight, J. S. O ' C o n n e l l a n d F. Prats where the cross section o f the r e a c t i o n 3He (7, n)2p in a z e r o - r a n g e a p p r o x i m a t i o n has been calculated. Th e results o f these authors published in Phys. Lett. 22 (1966) 322 are an al o g o u s to ours given in ref. 1°) and in sect. 2 o f this paper. T h e a u t h o r expresses his deep gratitude to Drs. J. M. Knight, J. S. O ' C o n n e l l and F. Prats f o r letting h i m k n o w their results before publication.

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