Final-state interactions in the break-up reaction of deuterons by nucleons

1 1.B:Z.B 1

Nuclear Physics Al85 Not to be reproduced

FINAL-STATE

(1972) 366-384;

by photoprint

@

or microfilm

INTERACTIONS

North-Holland without

written

Publishing

Co., Amsterdam

permission

from the publisher

IN THE BREAK-UP

OF DEUTERONS

REACTION

BY NUCLEONS

F. TAKl%JTCHI Department

of Physics,

University

of Tokyo,

Tokyo, Japan

and Y. SAKAMOTO Laboratoire

de Physique Corpusculaire, Centre de Recherches 67-Strasbourg-Cronenbourg, France

NucGaires,

Received 19 July 1971 Abstract:

The nucleon-nucleon correlation spectra of the cross sections for the break-up reaction of deuterons by nucleons are calculated taking account of final-state interactions between nucleons in pairs. The higher partial waves in the interactions and the D-state wave function of deuterons are included. The partial waves are obtained by solving the Schrijdinger equations with the Hamada-Johnston potential for two-body scattering states. The results are compared with experimental data for 198, 156, 100, 46, 45, 23 and 17 MeV incident protons.

1. Introduction

Recently the break-up reaction of deuterons by nucleons has been extensively investigated ‘) over a wide range of incident energies, as interest in the three-body problem increases. According to the kinematical conditions, the reaction is artificially classified in terms of slightly inelastic and quasifree scattering. The experiments for the reaction were initiated for studying the possibility ‘) of whether or not the deuteron could be used as a free neutron target in the kinematical region of the quasifree scattering and with the aim of extracting “) supplementary information about the nucleon-nucleon scattering parameters from the slightly inelastic scattering data. Although the Faddeev equations “) provide in principle an exact formalism for describing a three-body system, they cannot be, for the time being, solved numerically, unless the energy of the system is so low that only S-waves take part in the behaviour of the system. Some models based on the impulse approximation “) are usually employed for describing the system with high energies. The spectator model (SM) was expected ‘) to be valid in the quasifree scattering region. It is assumed in this model that the incident particle interacts with one of the particles in the deuteron as if the latter were free, and hence that the other particle in the deuteron stands by as a spectator. For high incident energies, this model might give a good prediction of the cross section for the reaction since both the wave length of the incident particle and the range of nuclear forces are smaller than the mean separation distance of the particles in the deuteron. However, the observed cross sec366

FINAL-STATE

INTERACTIONS

367

tions deviate from those predicted with SM even if the energies of incident particles are fairly high. The deviation becomes large as the included angle between two detected particles, einc, becomes larger than 90’. The deviation increases as the incident energies decrease. Everett “) has considered effects due to the double-scattering terms in the multiplescattering formula to improve the agreement between the results calculated with SM and the experimental data. Cromer and Thorndike ‘), and Brown and Thorndike “) have improved the agreement taking account of S-waves in two-body scattering states for the final-state interactions. The latter authors have reproduced well the observed pp correlation spectra of the cross sections at 200 MeV, especially in the region Of large einc where the deviation of the predicted cross sections with SM from the data is very large. L’Huillier ‘) has calculated the cross sections, combining the double-scattering terms and the final-state interactions, including S-waves in two-body scattering states and taking account of the complete anti-symmetrization lo) for three particles. PaiC: rt al. I1912) intro . d uce d a cut-off in the overlap integral of SM to get results which are comparable with the data of the quasifree scattering region. The cut-off in the overlap integral reduces the absolute values and narrows the width of the quasifree scattering peak compared with those calculated by using SM. The cut-off approximation is frequently used to calculate the cross sections for the scattering of particles by nuclei. The use of the cut-off in the integral corresponds to treating interesting nuclear effects as an unknown factor which is a function of the cut-off radius. It is worth-while to see the effects as due to the higher partial waves of two-body scattering states in the final-state interactions for the following reasons: (i) The experimental data 13) at 156 MeV show that the effects due to the final-state interactions depend on the angle between the momenta of the incident and of the spectator particles. (ii) The cross sections for the slightly inelastic scattering of nucleons on deuterons are explained r4) by taking account of the final-state interaction in a pair of nucleons with small relative energies. The effects of the final-state interaction in the slightly inelastic scattering are reflected ‘) on the cross sections for the quasifree scattering and vice versa. When the effects due to the final-state interactions are considered in the region of quasifree scattering, the relative energies of interacting pair particles are high, and hence higher partial waves of two-body scattering states play their roles in the interactions. It is not suitable to assume that only S-waves of two-body scattering states contribute to the final-state interactions. (iii) The D-state function of deuteron has been, till now, neglected in the calculation of the cross sections for the reactions. However, when the function is included in the calculation, it overlaps with the higher partial waves, therefore the higher partial waves should be taken into account. In the present paper, the S-, P- and D-waves of two-body scattering states are included in the final-state interactions for calculating the cross sections, and the Dstate of the deuteron is also considered.

F. TAKkUTCHI

368

AND

Y. SAKAMOTO

A part of the present work 15), the comparison between the calculated results and the data at 156 MeV of the experiment done at Orsay, has already been reported.

2. Formulation 2.1. TRANSITION

OPERATOR

In the problem of inelastic scattering of particles from deuterons, the incident and the target particles are designated as particles 1, 2 and 3, respectively. Particle 2 is of the same kind as particle 1. The Hamiltonian for the system is H = K+V,+V;?+Vs = H,+V,

(1)

where V, and V3 represent the interactions of particle 1 with 3 and 2, respectively; V, represents that between particles 2 and 3; K is the total kinetic energy operator and H,, is the unperturbed Hamiltonian for the initial state given by Ho = K+ VI. Y (*) for the Hamiltonian

The eigenstate

(2)

H with the total energy E is given by

(3) where 1~) satisfies

(4)

Holrl) = Elrl). The transition

matrix element between the initial and final states is Tfi =

(!f+‘lVl~,)

= (rlflT’lrli),

(5)

where T’ = V+V

Fr+Tc

T’ 0

=

The operator

T’ is expanded

v+v

l

v

E-H+k

(6)

’

in the form “)

T’ = t,+t,+t,gt,+t,gt,+

. . .,

(7)

where g = 1--. E-K+ie

)

(8)

FINAL-STATE

and tk is the two-body

t-matrix

in the three-particle

tk = I$+ Vk For the inelastic

process,

INTERACTIONS

369

space;

1

the unperturbed

(9)

v,*

E-K-I/,+ic

Hamiltonian

for the final state is K,

instead of HO. The plane-wave state ~&k,, k, , k3) is an eigenstate of K with a corresponding eigenvalue E, where k 1, k, and k, are the wave vectors of the three particles in the final state. The eigenstate of HO is given in terms of g3(k,, k,, k3) as

(10) The transition

matrix

element

is rewritten

as

(‘1) where

(12) Taking the antisymmetrization between the same kind of particles antisymmetrized transition operator T becomes

T=

I and 2, the

(~+~-&t: )

+(l -P,,)T’

= (1+f~g+t,g)Is+(l+i3g+tlg)tz-(l

+&g+t2g)t,P12+

. . .,

(13)

where 23 = (l-P&3, and PI2 is the exchange wave matrices i6)

operator

between

(14)

the particles

1 and 2. By defining

CO(:)+ 1J = i+t,g, -(-)t 012

Tis rewritten

as, by neglecting

the (15)

= (1-~ld(1+f367)~

terms of order higher than

(16)

g’,

T = T~IA+~;~,

(17)

where GA = &+r~-ri

P12,

(18)

T, = [(~~;‘+-1)+(0~;~+-1)]1~+[~(~):~‘+-(1-~~~))+(~~;~+-1)]t~ -[~(~~,‘+-(l-P,2))+(W:;)+-l)]tlP,2.

(19)

The operator TslA gives the cross sections calculated with the simple impulse approximation (SIA). In the calculation of the cross section with the spectator model (SM),

370

F. TAK&JTCHI

AND

Y. SAKAMOTO

only one term in TslA is considered according to the kinematical conditions. The operator T gives the cross sections calculated by taking account of the final-state interactions between particles in pairs (FSI). The impulse approximation “) consists of replacing the t matrices in T by those for free nucleon-nucleon elastic or inelastic scatterings. 2.2. CROSS

SECTION

The antisymmetrized transition matrix element is given by .F = +$G%i> = $9 SIA + 9,

(20)

3

where 9-

SIA

=

(21)

(~f1131~i)+(~flf21~i)-(~flti p121?i),

f W= ~411’,~‘t~~I?i>~<#~l~3Ipli~+<~~~~l~3lpIi~~~~~l13IYi~ +~~l~~lt~l?i>~~~flt2lYli~+~~l~~ltZtYfi~~~#~lt2lYli~ ~<~l~~lt~P~~lYj~+<~~lt~~~~l~i>~~~’,~’lt~P~~l~~~+~~~lf~~~~lrli~~ (22) The antisymmetrized, scattering-state and antisymmetrized function in eq. (22) are defined to be, respectively,

scattering-state

wave

Id2 = +(I -~IZ)ld& l&‘) = &‘lf#Q) CJ llji’,,‘>

=

4;;

-P,

(29 (24

,;I$$)>

=

4;wf).

(25)

It is noted that ]I/$‘> is the eigenstate for the Hamiltonian IQ- V, with its eigenvalue E: (K+

v,)lt&)j= q+b&)).

(26)

The cross section for the reaction is given by da =

(g /a(k,- k, - k, -

k3)~(~~ - Q

r/Z> !% d% d3k?. 1

(2743

0

(2x)3

(27)

(2743

and hence the two-particle correlation distribution of the cross section becomes d30 ---------= dS2, dQz dE,

Ha2

(28)

_______ KC Px 4h4(2rr}*

where the symbol c means the average on the initial states and the sum over the final states; rn is the nucleon mass and K is the kinematic factor E(

=

m A7243

k, kg____.~ ___ k,

.k,(G,

+ S,)-

2(6, + WC”)

_____--____~___“--8,

k,

cos

0,

+ 672 Et, cos

. (e,

-

8,)

(2%

FINAL-STATE

INTERACTIONS

371

The total energy of the particle i is bi = Ei+mc2 and oi is the angle between the momenta k, and kj as shown in fig. 1. The included angle fjinc is the angle between k, and k, , einc = 01-82. The amplitude A% is written explicitly as dri

= dy* +(dty)

+ J?

_ dy*

* 3)

+ (A;(

_

A:*)

+ (~;(23)

_

A:*)

(30)

_~~“)+(~~(23)_~~)_(~~(12)_~SI*)_(~~(’3)_-~*),

where (31) &;(ij)_-rA

Fig.

1. Kinematic

=

definition

_

(32)

of the reaction.

Angles

are measured

counterclockwise.

When k = 3 and 1, tk are replaced by 7, and t, PI2 respectively, in eqs. (31) and (32) and when ij = 12, (~$2:~ - 1) is reTlaced by +[c$;‘+ - (1 --PI,)] in eq. (32). Using the amplitude (30) the cross section is expressed as d3rr

= K C I~pi12,

d8, dR, dE, 2.3.

OVERLAP

INTEGRALS

The wave function

for the initial

state is

= (27r)+ eiko * ~/dKf,,(K)eiK’

(rZ-rJ)j~n),

(34)

where

hi(K) =

(2n)msjdre-iX’r(x, v&hi);

(35)

here the x are the spin wave functions for the three-nucleon system. The wave function of the deuteron is expressed as

where u and w are respectively the radial wave functions for the S- and D-states and 5’ and At are respectively the tensor and triplet projection operators. The wave function for the final state is

The wave function which includes two-body scattering states is

where g&(H) = (271)~* dre-rH “(x,I,#,?(~)x~). s

The function g(H) has a sharp peak at H = -t(kj-ki)* By the use of eqs. (34) and (379, the matrix element (+J?&Q

becomes

where the impulse approximation is applied. The scattering amplitudes for the twobody systems are calculated by separating the three-nucleon spin wave function into those of two-nucleon q(v),,, and of one-nucleon c(k)+M. The two-body scattering am$itude M:j is

BY

substituting eq. (41) into eq. (40), the amplitude A:rA is reduced to

d~‘A = C My n

5

dr eiks

‘(x, tp&r)~J

FINAL-STATE

Similarly,

&yA

and AyA

are expressed Jzy

373

INTERACTIONS

as

= C MyB(k,; n

ni),

(43) (44)

where B(ki; ni) =

B(k,; ni) =

dre iki “CL pIZ

s

Mf” = -

$!J(3tkjmkk);

M’3” = -

3

I

The matrix

dr eiki ’ ‘(x, qd(r)xi),

s

element

i = 2, 3,

(45)

(46)

(Pd(y)Xi)9

i = 2, 3,

XfItil3(ko+ki); L>,

(47)

W, -kd; xrl’tMko+kd; x2.

(~‘l;‘(fglqi)

x

(48)

is, by using eqs. (34) and (38),

dye3i[k2-4(kl+kJ)-Hl.rfq3neaiCko--H+~(k!+ks)l.r

.

(49)

s

Since the function imated as ($(l;)l131qi)

g(H) has a sharp peak at H = -+(k3 -k,),

the integral

is approx-

+k3))

= (27~)~ ~~d~e~ti~k~~kz~~‘i4;ef”kotk3’~r~dHgt,(H)f.i(~-f(k~

= (2~)~ g
Xql~3lHko+k3); X.>SdHg,(H)f,i(H-f(k,

approximation

J dHgr&H)f,i(H-+(kr

“) is used. The integration

+ k3)) = (2n)-3 s d&



(~~;‘lisl~i)

over H gives

~~~)*(r)~4e~i42 ’ ‘(L (Pcdp)XJ

= (27c)-3Ffi(q2 ) -co,; The matrix element

+kJ))T (50)

qn).

(51)

is reduced to

= z <3(k, -k,);

Xql13l!t(h +k,);

Xn)Ffi(qz > --wz; qn),

(52)

where QJZ = 3(k3 -kA

q2 = k,-k,.

(53)

F. TAKEUTCHI

374

Therefore,

by substituting

AND

Y. SAKAMOTO

eq. (41) into eq. (52), the amplitude A ~‘13’ = C M~F,i(q*) v

+&‘(‘3) is expressed

-Wz; 4n).

as (54)

Similarly 4d23)

=

Jl

=

~‘I’)

,q4’3)

=

_4z ~‘23’

=

C

M43”Ffi((11

) -al

; CJn)>

E

M~FPi(q3

) 03;

4n),

(56)

E

MYFfi((l*

9 w2;

Cln),

(57)

~

M~F,i(q1

) 01;

sn),

(58)

M4,“Ffi(Q3

9 -w3;

(55)

P _qw

=

C

(59)

qn)*

rln where

with qi = k,-ki,

i = 1, 2, 3,

(62)

ai = 3(kj-k,),

i, j, k : cyclic.

(63)

Only one term in TsIA, for example Z3 is taken into account section with SM, and hence only the amplitude ASP

= 1 A4f3”B(k3; ni), ”

for calculating

the cross

(64)

is used in eq. (33). When the D-state of the deuteron is neglected, B(k3; ni) is reduced to B(k,; ii) which is independent of i for the triplet states of the two particles making up the deuteron and becomes zero for the singlet states of the particles. Writing B(k3; ii> = a(k,), the two-particle correlation distribution of the cross section is d30 d& dQ, dE,

= +KIj?(k3)12 5 (Mtj2,

(65)

i= 1

where i runs over the three-nucleon spin functions which contain the triplet states of the two particles. Using the appropriate free two-body scattering cross section in the the two-particle correlation distribution is reduced two-body c.m. system (dc/dQ)ip., to the usual form ‘) d30 da, dQ, dE,

(66)

FINAL-STATE

INTERACTIONS

375

3. Calculation The wave functions of two-body scattering states are obtained by solving the SchrGdinger equations with the Hamada-Johnston potential 17). The S-, P- and Dwaves in the scattering states are included in the final-state interactions between nucleons in pairs. Although the coupling between the 3S, and 3D, states are taken into account exactly, the coupling between the 3P, and 3F, states is neglected. The deuteron wave function is chosen in its analytical form 18) derived from the Schr(idinger equation with the potential. The function chosen has the D-state probability of 7.050 %. The overlap integrals F and B are calculated numerically with these functions. The nucleon-nucleon scattering matrix elements which should be used in F are those off the energy shell and differ from those measured in two-body scattering. The two-body matrix elements used in lfi are obtained from the measurements for inelastic nucleon-nucleon collision such as N+N + N+N+ y. However in the present paper the two-body t-matrices are approximated on the energy shell of the corresponding two-body system, and hence their matrix elements are evaluated with the nucleon-nucleon scattering phase shifts given by Hamada and Johnston 17). The numerical calculations have been performed on UNIVAC 1108 at Orsay and on FACOM 230-60 at the DP Centre of Kyoto University.

4. Results and discussion 4.1. DEPENDENCE

ON INCIDENT

ENERGIES

Many experimental data are now available for different energies of incident particles. The results are compared with the available data at 200 to 17 MeV incident particles. However, it is rather difficult to see how the agreements depend on the incident energies since the experiments have been performed in different experimental conditions, especially at different detection angles 19~and e2. It should be noted that in the present subsection, each two-body t-matrix has been evaluated at the relative energy before the corresponding collision. Results at 198 MeV. Figs. 2 and 3 show the calculated results and data “) at 198 MeV incident proton energy. The effects of the final-state interactions are important, especially for large einc. The agreements are remarkably improved compared with those using SIA. Fig. 3 shows how the calculated results are improved by FSI with S-waves only and with S-, P- and D-waves of the two-body scattering states. Results at 156 MeV. Figs. 4 and 5 show the results at 156 MeV incident proton energy. The results reproduce the experimental data 13), especially for large 0inc. In fig. 5 the peak indicated by E3 ;2: 0 results from the low-momentum components of the S-state wave function of the deuteron, and the peaks denoted by Ez3 z 0 and for the slightly inelastic and neutron E’,, z 0 are caused by the final-state interactions

376

F. TAK&JTCHI

AND Y. SAKAMOTO

80

60

30

50

70

90

110

130 Ej(MeV)

Fig. 2. The pp correlation spectrum of the cross section at 198 MeV incident protons for the angles 6, = -& = 55”. The dashed and solid curves show respectively the results calculated with SIA and with FSI including higher partial waves of two-body scattering%

30

50

70

90

130 110 El (Me’?)

Fig. 3. The correlation spectrum at 198 MeV incident protons for o1 = -02 = 60”. The dashed and solid curves correspond to those in fig. 2. The dotted curve is calculated with FSI including only S-waves of two-body scatterings.

pick-up (or knock-on) processes. The structure of the pp correlation spectrum with these peaks are well reproduced since the reaction is treated without separating it artificially into the quasifree and slightly inelastic scattering. The difference between the results calculated with FSI including S-waves only and including S-, P- and Dwaves of two-body scattering states is not so remarkable as the case of the 198 MeV incident protons.

FINAL-STATE

INTERACTIONS

377

Results at 100 MeV. Figs. 6 and 7 show the results calculated at the 100 MeV incident protons. Unfortunately the data 19) are limited only to the kinematical region corresponding to quasifree scattering. data better than those by SIA.

The

results

by using

FSI agree

with

the

156 Me’/

/’ / I

812

/’

450

02=-60”

I

I

I

50

,

I

I

,

110

90

70

El (MeV) Fig. 4. The correlation

spectrum at 156 MeV incident protons for 0, = 45”, O2 = -60”. correspond respectively to those in fig. 3.

The curves

156MeV 81 : ; iI

\ 1.1

10

I.

30

,.;’

52.5”

e2=-52.5”

/

\

f

,

50

,

I

,

70

,

-

90 El

Fig. 5. The correlation spectrum and solid curves correspond

,

(Mei’)

at 156 MeV incident protons for 8, = -ez = 52.5”. The dashed to those in fig. 2. The dot-dashed curve is calculated with SM.

20

40

10

80

60 El

I

81’

30”

4

1

1

I

7

I\ / t \ \

I I

I

10

I

‘\

:’

92=-30”

I

I

--. \

/’

(_MeV)

Fig. 7. The correlation spectrum at 100 MeV incident protons for fI1 = 40.5” and tr2 = -48.1”. The curves correspond to those in fig. 2.

I

MeV

70 El

Fig. 6. The correlation spectrum at 100 MeV incident protons for ~9~= 35.5” and e2 = -34.5”. The curves correspond to those in fig. 2.

46

50

30

(McV)

20

I

I

30 El ( MeV)

Fig. 8. The correlation spectrum at 46 MeV incident protons for 8r = --Bz = 30”. The curves correspond to those in fig. 2.

El (MeV)

Fig. 9. The correlation spectrum at 46 MeV incident protons for 0r = 50” and &, = - 30’. The curves correspond respectively to those in fig. 3.

FINAL-STATE

379

INTERACTIONS

Reknits at 46 and 45 MeV. Figs. 8, 9 and 10 show the results for 46 and 4.5 MeV incident protons. The results calculated with FSI are smaller than those with SIA by a factor $ to $ and reproduce the observed cross sections 12*20) in the absolute magnitudes around the kinematical conditions for the quasifree scattering. The width of the peak for the quasifree scattering is lye11 reproduced for large Oinc. The calculated width is slightly broader than the observed one for small einc. In the energy I

I

t

I

1

I

i

10

20

30 El

Fig. IO. The correlation

spectrum

(MeV)

at 45 MeV incident protons for 8, = -6, correspond to those in fig. 2.

= 43”. The curves

region around and smaller than 50 MeV incident nucleons, the results calculated with FST including S-waves only do not differ appreciably from those calculated with FSE including higher partial waves in the two-body scattering states. Results at 23 and 17 MeV. Fig. 11 shows the results at 23 MeV incident proton energy. The cross section calculated with FSE is smaller than that with SXA by a factor of 3 and reproduces the observed one 12) in the region of the quasifree scattering. The caterdated cross section around E, 3 zz 0 is larger than that observed. Fig. 12 shows the results for 17 MeV incident protons. The consideration of finalstate interactions reduces strongly the cross section calculated with SIA, although th;

380

F. TAKBUTCHI

AND

Y. SAKAMOTO

O

0

5

10

15 El

The initial

ENERGIES

FOR

15 El

EVALUATING

(Mev’

Fig. 12. The correlation spectrum at 17 MeV incident protons for e1 = -0, = 43”. The curves correspond to those in fig. 2.

results calculated with FSI are still larger than the observed Amado model “) works at these low incident energies. OF

J

n

10

(Mei’)

Fig. 11. The correlation spectrum at 23 MeV incident protons for e1 = -13~ = 43”. The curves correspond to those in fig. 2.

4.2. CHOICE

I

5

cross sections *‘). The

t-MATRICES

and final kinetic energies of the interacting

particles

for tk are different.

For example, when the incident particle 1 is scattered by one of the target particles, the initial kinetic energy of the two colliding particles is greater than the final kinetic energy by an amount AE given by AE, = 2E,+ b,

(67)

where Ek is the energy of the third particle in the final state and b the binding energy of deuteron. Even if tk is approximated on the energy shell of the two-nucleon system, there are several possibilities in choosing the energy for evaluating the matrix elements: (i) the energy before the corresponding two-body collision in its c.m. system, Ei ; (ii) the energy after the collision, E,; (iii) the geometrical average of Ei and Ef. Figs. 13 and 14 show the dependence of the results on the choice of the energies for evaluating the t-matrices for 156 and 45 MeV incident protons. The cross sections

FINAL-STATE

calculated

for 156 MeV incident

381

INTERACTIONS

protons

do not depend

on the choice of the,energies

used for the evaluation in the region of the quasifree scattering. The difference between the cross sections calculated by using the t-matrices evaluated at Ei and Ef becomes large as the momentum transfers differ from those corresponding to the quasifree scattering. The pp correlation spectra are reasonably reproduced by the t-matrices at Ei .

evaluated

r

!

1

I

I

I

/

I

I

156 MeV q=

45”

82=-525”

I

30

50

70

90 El (MeV)

Fig. 13. The correlation spectrum at 156 MeV incident protons for e1 = 45” and 8, = -52.5”. The solid, dotted and dot-dashed curves show respectively the results calculated by using FSI with the two-body r-matrices evaluated at the energy before the corresponding two-body collision, at the energy after the collision and at the geometrical average between the energies before and after the collision.

I

10

I

I

20

1

146

I

30 El (MeV)

Fig. 14. The correlation spectrum at 45 MeV incident protons for 8, = --e2 = 43”. The curves correspond respectively to those in fig. 13.

For the 45 MeV incident protons the calculated cross sections depend on the choice of the energies for evaluating the t-matrices even in the region of the quasifree scattering. In this region the relative energy before the collision corresponding to t, does not differ appreciably from that after the collision. The difference in the cross sections calculated should be caused by the interference terms of t, with t, and t,. The contributions from the terms t, and t, to the results become important for lower energies of the incident particles, and also the energy dependence of the cross sections

382

F. TAKfiUTCHI

AND

Y. SAKAMOTO

for two-body scatterings is strong at low energies. Therefore the difference in the results seems to be reasonable. The results calculated by the t-matrices evaluated at Ei reproduce the shape of the correlation spectrum although the absolute values of the cross sections calculated are slightly larger than those observed. The use of Ei for the evaluation seems to give better agreement with the data than that of Ef. The effects due to off the energy shell of the two-body system on the t-matrix, t, are estimated by Everett “). For around 150 MeV incident protons the effects are very small in the region of the quasifree scattering. It is also shown that the t-matrix is considerably well approximated 23) on the energy shell of the two-body system with the energy before the collision. Therefore the matrices evaluated at the energy before the corresponding collision in the two-body c.m. system are used in the calculations of the preceding subsect. 4.1.

156

MeV

q=

45”

82=-67.5’

30

50

70

90 El (

110 MeV)

Fig. 15. The correlation spectrum at 156 MeV incident protons for 0, = 45” and or = -57.5”. The curves - - - - and -.-. -. - are respectively calculated by using FSI, SIA and SM with the Hamada-Johr;ston deuteron wave function. The curve - - - and . . . . . are respectively calculated by using FSI and SIA with the Hulthen-Sugawara wave function.

4.3.

DEPENDENCE

ON

DEUTERON

WAVE

FUNCTION

CHOSEN

To see the dependence of the calculated cross sections on the deuteron wave functions chosen, the wave function given by HulthCn and Sugawara is used for the calculation. The parameters for this function are adjusted in such a way that the hard core has the same radius as that of the Hamada-Johnston potential and that the function

FINAL-STATE

383

INTERACTIONS

contains the D-state probability of 4 T:,. The results are shown in fig. 15. For high incident particle energies the two-particle correlation distributions of the cross sections are expected to reflect the behaviour of the deuteron wave function at the short-range region. At the 156 MeV incident protons, the difference between the results calculated by using both the wave functions given by Hamada and Johnston, and by HulthCn and Sugawara is small although it is appreciable. Since both the wave functions are almost the same at larger r, the difference between the cross sections calculated with these wave functions is expected to become small as the incident energies of particles become low. For the low incident energies, the difference is mainly caused by the difference between the normalization constants of both the wave functions, which is very small. 5. Conclusions The two-nucleon correlation spectra of the cross sections for the reaction are calculated with FSI which includes higher partial waves in two-body scattering states and the D-state wave function of the deuteron. The results reproduce the data especially for high incident particles energies. For low incident energies, the results agree with the data in their absolute values around the region of the quasifree scattering. The calculated widths of the quasifree scattering peaks are slightly broader than those observed, especially as tIinc becomes small. For evaluating the two-body t-matrices with the scattering phase shifts on-theenergy-shell approximation in the two-body systems, it is favourable to use the phase shifts at the relative energy before each collision. The results do not depend sensitively on the deuteron wave function chosen at the incident particle energies considered in the present paper. Detailed study for the pp correlation spectra of the cross sections at kinematical conditions corresponding to El3 z 0 gives supplementary information on two-body scattering parameters as those “) at Ez3 z 0 did. We would like to express our sincere gratitude to Dr. T. Yuasa whose help and encouragement are invaluable. We gratefully acknowledge the support and encouragement of Professor Y. Nogami. It is our pleasure to thank Dr. K. Kuroda for the helpful discussions. One of us (Y.S.) is greatly indebted to Professor P. Ctier for his kind hospitality. The present work was initiated when we were at the Institut de Physique Nucleaire d’orsay. References 1) For

instance, Proc. Int. Conf. three-body problem, Birmingham, 1969 Amsterdam, 1970) 2) A. F. Kucks, R. Wilson and P. F. Cooper, Jr., Ann. of Phys. 15 (1961) 193

(North-Holland,

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F. TAKEUTCHI

AND Y. SAKAMOTO

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