A study of break-up processes in neutron-deuteron scattering

I•.L

I

Nuclear Physics A210 (1973) 380--396; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A S T U D Y OF BREAK-UP P R O C E S S E S IN N E U T R O N - D E U T E R O N SCATTERING W. M. K L O E T a n d J. A. T J O N

Institute for Theoretical Physics, University of Utrecht, Utrecht, The Netherlands Received 3 M a y 1973 Abstract: N e u t r o n - d e u t e r o n break-up reactions for kinematically complete a n d incomplete situations are e x a m i n e d for various types o f local t w o - n u c l e o n S-wave potentials a n d for the separable Y a m a g u c h i potential. I n particular, a systematic search has been m a d e in the entire p h a s e space o f three free nucleons for large differences between the results o f the various interactions. Certain parts outside t h e final state peaks a n d the quasifree scattering regions are f o u n d which are significantly sensitive to the t w o - n u c l e o n interactions studied.

1. Introduction Calculations of neutron-deuteron scattering have been performed in the Faddeev tormalism mainly with separable two-nucleon potentials. One of the important reasons for this is that the three-body equations reduce to a relatively simple form if the two-body interactions are separable. In spite of such simplified interactions the calculations describe remarkably well the elastic scattering and integrated break-up data 1- 6). The main purpose of this paper is to explore the possibility that there are more significant differences in the break-up processes in kinematically complete situations. Virtually no experimental work has been done in these situations, mainly because of the complexity of kinematically allowed regions. In view of this we made a direct comparison between the results of various sets of potentials. In this way one might hope to find some guidelines for future experiments. We are in particular interested in finding regions in phase space where the cross section is very sensitive or insensitive to the type of two-nucleon interaction used. In such regions an ultimate comparison of theory and experiment might reveal either which of the two-nucleon interactions are to be preferred or that two-nucleon forces alone do not account completely for the three-nucleon processes. As a result we might get some indication about the nature of three-body forces. In the next section the expressions for the break-up amplitudes are given, while in sect. 3 the two-body interactions for which the calculations were performed, are introduced. These interactions are of a local S-wave spin-dependent type. Also for comparison the separable Yamaguchi potential was studied. The method used in these calculations is the same as in ref. 7) (hereafter to be referred to as A) and consists of 38O

B R E A K - U P PROCESSES

381

applying Pad6 techniques to the multiple scattering series. For elastic n-d scattering it was shown to be a very practical method. Also in the break-up case it can be applied successfully. In sect. 4 it is briefly indicated how the elastic scattering depends on the type of potential while sects. 5 and 6 give the results for the integrated proton spectra, final state interaction peaks and quasifree scattering regions. The energies which were chosen for this study are 14.4 MeV and 50 MeV neutron lab energy. Finally in the last two sections the results of the search of the entire phase space of the three free nucleon final state are described and discussed. 2. The break-up amplitude

Following the notation of A the S-matrix element for the break-up process n + d - - * n + n + p can be written in terms of the partly anti-symmetrized states [ p q f l ) A and Ictiqif~) A as 1 M = x/~ ~m A(pqfl YOre(S+ ie)lctiqifli)Am' (2.1) where the operator Uo,. is defined by

eo,,(z) = Z Vk-- Z VRG(z)V~. k

(2.2)

k l¢~m

Using the energy conservation of the initial and final states eq. (2.1) can be rewritten in terms of the operators Tk~as 1 M(pl , ql , fill) = ~ 1A(pl ql fill ktmZZkl(s+ ie)l~iqlfi) A" (2.3)

k C:m From eq. (2.3) it follows immediately that M(pl, q~, f~) can be expressed in terms of the function U(p, q, f) from A. We have 1

M(p

, ql, f0

=

V(p,,

1 fdpd t

U(p,q, , fl)+ ?,/2

lS). (2.4)

The integrations can now be carried out explicitly. This gives 1 U ( p l , q l , fill)

1 #

(flllfl)(U(Pz, q2, f l ) + ( - I f p'+'p' +~P+'pU(p3, q3, fl)),

(2.5)

where (fxlfl) are recoupling coefficients for spin and isospin and the momentapi and ql are related by p2 = - ½ p l + ½ , / 3 q l ,

p3 =

q2 = --½~/3pl--½qx,

q3 = ½~/3pl--½ql"

(2.6)

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W . M . K L O E T A N D J. A. T J O N

As shown in A the expression U(p, q, fl) satisfies an integral equation. Since we restrict ourselves to pure central S-wave potentials, there are three amplitudes uS~"(p, q) which we denote by U*°(p, q), U~l(p, q) and U~l(p, q). After performing an angular-momentum decomposition, the multiple scattering series is generated for each amplitude and summed by Pad6 technique. This method and the numerical evaluation of the equations were discussed in detail in A. The rate of convergence of the Pad6 approximants for the break-up amplitude is found to be the same as for the elastic amplitude. If we number the three outgoing nucleons in such a way that particle one is the proton, one has ipl = 1. The break-up process can then be described by the three amplitudes

M(pl, qx, S = ½, Sp, = 0, ip, = 1),

M ( p l , qx, S = ½, Sp, = 1, ip, = 1),

M(pl, q l , S = ~:, Sp, = 1, ipl = 1), which we shall denote respectively by

M~°(P,, q,), M½'(Pl, ql) and M~'(pl , q,). F r o m eq. (2.5) we see that these three M-amplitudes are related to the U-amplitudes by

M~°(P,, q,) = - ~

4~/3

(4U½°(Pl, q , ) + U~°(P2, q2) + U~°(P3, q 3 ) - 3 U~'(p2, q 2 ) - 3 U~'(p3, q3)),

M' I(pl, ql) = ¼(U½°(P2, q2)- U °(P3, q3) U I(p2, q2)-U I(p3, q3)), ql) = -½(U X(P2, q2)-

U~I(p3, q3)),

(2.7)

where the recoupling coefficients have been written out explicitly. Hence knowing the functions U from solving the Faddeev equations, the M-amplitudes can be determined with the aid of eq. (2.7). From these the break-up cross section can be found directly through 3~ 4

da = - - l(p, , q,)b(p z + q2 _ s) dpt dql , mqz

(2.8)

where q~ is the relative momentum of the incoming neutron expressed in MeV ~ and m is the nucleon mass. The function I(pl,ql) is the absolute square of the break-up amplitude, averaged over incoming spins and summed over outgoing spins

l(px, q,) = ~tlM½°(pl, q,)12+½1M4e'(p,, qx)12+21M~l(pl, qx)l 2.

(2.9)

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383

3. The two-nucleon potentials As input for the three-nucleon calculations we have to specify the two-nucleon interactions. The potentials used are central S-wave potentials, which represent fairly well the low-energy behaviour of nucleon-nucleon scattering. For convenience, the low-energy parameters of the various potentials studied are given in table 1. TABLE 1 Low-energy parameters o f the two-nucleon potentials

III IV Yamaguchi triplet 2) I I' Reid singlet 9) Yamaguchi singlet 2)

a (fro)

r o (fm)

Ea (MeV)

5.45 5.45 5.41 --23.7 --16.0 --17.1 --23.78

1.8 1.8 1.76 2.8 2.9 2.8 2.67

--2.225 --2.225 --2.225

As a reference potential we take the set (I-11I) from ref. 8). This is a local spin dependent Yukawa-type potential with a soft repulsive core in both the spin singlet and the spin triplet channel. To study the influence of short-range correlations we have also used set (I-IV) from ref. 8) in which the triplet potential has no repulsive core. Since none of the above potentials is realistic from the point of view of twonucleon data, we have also studied the effect of replacing the singlet part of the (1-Ill) potential by the Reid soft-core singlet potential. We will call this set (R-Ill). Recently much effort has been made to determine the value of the neutron-neutron scattering length from final state interaction in neutron-deuteron break-up. Therefore we have also considered a two-nucleon potential in which the n-n scattering length is different from the n-p scattering length in the spin singlet. The Faddeev equations for this case have an additional channel. We have varied the strength of the attractive part of local potential I such that an. = - 16 fm. We shall call this potential 1'. For the singlet n-p channel we retained the reference potential I. Finally we have taken the separable Yamaguchi potential with the same parameters as used in refs. 1 -~) to see where separable potential results differ from those of the local ones. 4. Elastic scattering In this section we will be concerned with the results of the elastic n-d scattering at 14.4 MeV for the six sets of potentials used in the break-up calculations. The differential cross sections are shown in fig. 1 for potentials (I-III), (I-IV), (R-II1) and the separable Yamaguchi potential. Where the result for (I-IV) and (R-III) is not indicated, it coincides with that of set (I-III). The cross sections for potentials (I'-I-III) and (I'-III) both are between the curves for (I-III) and (R-III). As already has been

384

W. M. K L O E T A N D J. A. T J O N

.

200

.

. . . ELfiAB'= 14.4 MeV

I

/ 15'0~

E

/'

lOO

D'

3

"O

li

"t\

0

3'0 6; 9'0 1;~0 1;0 C.f'4. ANGLE IN DEGREES

380

Fig. 1. Elastic n-d cross section at E . jab = 14.4 MeV. Solid curve is the result for potential (I-III). D a s h e d curve is the result for potential (I-IV) a n d dotted line is for (R-Ill). T h e dash-dotted curve is the cross section for the separable Y a m a g u c h i potential.

TABLE 2 D o u b l e t and q u a r t e t phase parameters for elastic n-d scattering at E~ tab = 14.4 MeV for six sets o f potentials a)

26o 26t 2(~2

26a 2r/o 2~71 2~72 2r~3 46 o 46 t *62 46a *~o *J71 4~2 %73

I-II1

I-IV

R-Ill

Yamaguchi

I'-I-III

F-Ill

106.2 14.9 7.05 --1.29 0.427 0.683 0.946 0.989 72.0 33.2 --8.94 3.15 0.973 0.906 0.975 0.997

114.9 14.9 7.25 --1.34 0.368 0.669 0.943 0.989 76.3 32.2 --9.07 3.31 0.967 0.895 0.975 0.996

111.2 13.8 6.93 --1.29 0.490 0.677 0.949 0.990

126.0 13.4 6.46 --1.23 0.464 0.678 0.949 0.990 73.0 29.9 --8.49 2.96 0.976 0.912 0.977 0.997

104.6 13.7 7.00 --1.29 0.488 0.703 0.950 0.990

103.8 13.1 6.97 --1.29 0.495 0.713 0.952 0.990

a) See footnote on pg. 385.

BREAK-UP PROCESSES

385

TABLE 3 Total cross sections at 14.4 MeV for six sets of patentials, cqn,L (b) is derived from the break-up amplitude and tr~net. (r/) follows from the inelasticity parameters in the elastic scattering amplitude

Total cqn~l. (b) o'i.~L (~)

1-tli

V-I-Ill

l'-llI

Reid-Ill

I-IV

Yamaguchi

771.8 165.1 165.3

765.4 160.2 160.4

762.3 158.3 157.9

766.5 165.0 164.5

766.9 176.2 176.4

716.8 163.8 163.5

shown in A, the cross sections for local a n d separable potentials differ c o n s i d e r a b l y in f o r w a r d direction. The differences between the results for the various local potentials are rather small. In table 2 the phase p a r a m e t e r s are given for the lower p a r t i a l waves t. Table 3 shows the calculated reaction a n d total cross sections. There are clearly two ways to o b t a i n the inelastic cross section. In the first place f r o m the elastic scattering a m p l i t u d e a n d secondly by direct integration o f the b r e a k - u p cross section. In view o f the validity o f three-particle unitarity these two ways o f c o m p u t a t i o n s h o u l d yield the same result. F r o m table 3 we indeed see t h a t the results are consistent within 0.5 mb. This is to be c o n t r a s t e d with the results o f the s e p a r a b l e - m o d e l calculations o f ref. ~l). The total cross sections for all local potentials differ very little a m o n g each other, whereas the separable p o t e n t i a l gives a considerably lower cross section. M o r e sensitivity to the type o f p o t e n t i a l used is seen in the reaction cross section. Also a dependence is f o u n d on the value o f the singlet scattering length.

5. Proton spectra Historically the first experimental investigation o f the b r e a k - u p reaction c o n c e r n e d the p r o t o n spectrum. This gives the cross section o f all the events where only the p r o t o n emerging at a fixed angle, is detected. The p r o t o n lab energy is m e a s u r e d , so that the r e m a i n i n g variables should be integrated over. In figs. 2-4 the p r o t o n s p e c t r a at -~lab 4.8 o, 10 o, 20 ° a n d 30 ° are shown at 14.4 M e V for sets (I-III), (I-IV) a n d the -p separable Y a m a g u c h i p o t e n t i a l together with the experimental d a t a 12,13). The s p e c t r u m shows two p r o n o u n c e d peaks. The p e a k at nearly m a x i m u m p r o t o n energy is due to n-n final state interaction a n d therefore is strongly d e p e n d e n t on the n-n scattering length. W e will discuss this feature in m o r e detail in the next section. T h e p e a k in the m i d d l e p a r t o f the s p e c t r u m at small p r o t o n angles is due to n-p final state interaction. F o r potentials which have the same n-p scattering length like the (I-III), (I-IV) and the separable potential the cross sections show differences o f a b o u t ten percent in this region. F o r the separable potential we find a n-p final state p e a k which is slightly higher t h a n in ref. 2). ~---

t For the separable Yamaguchi potential our results differ somewhat for L 0, 1 from the results of ref. lO). This has almost no effect on the differential cross section. Since our estimated errors (see A) in the T-matrix elements are of the order of 1 ~ , there is no inconsistency. Using a one-dimensional version of the Faddeev code, agreement was found with the results of ref. lo).

386

W. M. K L O E T A N D J. A, T J O N

30

OLAB = 4.8 °

25

• data odata

ref 12 ref 13

20

15 2~ E

**%*** °* °* o~

~'II

o oo i o

10

°.o J

0 0

2

4

PROTON

6

8

12

10

LABORATORY ENERGY Ep (MeV)

Fig. 2. Proton-energy spectrum at p r o t o n lab angle o f 4.8 °. Experimental data are from refs. ~2' 13). The data of ref. is) are for 0p '"b -- 4 °. The energy is 14.4 MeV. The solid curve is the result for potential set (I-Ill), the dashed curve is for set (1-IV) and the dash-dotted curve is for the separable Yamaguchi potential.

30 {)LAB = lO* • data o data

25

ref 12 ref 13

20 .Q

E

15

O 13

o

"0

o?'~j

o~O ~o o

10

I

° oo o

o

.. 0

"P~

0

2

n

n

4

6

I

n

,9

10

12

PROTON LABORATORY ENERGY Ep (MeV)

Fig. 3. Proton-energy spectrum at p r o t o n lab angle of 10 °. Experimental data are from refs. 12, za). The theoretical curves are as in fig. 2.

BREAK-UP

PROCESSES

387

0LAB=B0 ° 10 o

Q °o° ° o

0

0LA B =2C .~

15

A eo o

o

5

~

0

2

o

/~,+,-

L

PROTON

oo

eee °e

4

• el

6

. ~ . . . . or.o_~....o._, • eeoeo ee • °° • c

8

10

12

LABORATORY ENERGY Ep (Me.V)

Fig. 4. Proton-energy spectra at p r o t o n l a b a n g l e s o f 2 0 ° a n d 3 0 ° . E x p e r i m e n t a l d a t a a r e f r o m refs. 12.13). T h e t h e o r e t i c a l c u r v e s a r e as in fig. 2.

Another region where the results differ is between 9 and 10 MeV proton lab energy. The separable result here is 10-15 % lower than the cross sections for the local potentials, which practically coincide in this region. The experimental data of refs. 12,13) do not quite agree with each other. They differ considerably as is shown in figs. 2-4. Smearing out of the theoretical cross section due to the experimental energy resolution only will affect the n-n final state peak, but will not modify the rest of the spectrum. 6. Final state interaction and quasifree scattering A more detailed comparison between different theories and experiment can be made in kinematically complete situations. In this case all kinematic variables are known, and no integration is necessary. Experimentally most attention has been focussed on final state interaction (FSI) and quasifree scattering (QFS) of n-n and n-p pairs. A comparison between results of potential sets (I-III) and (I-IV) with some of the experimental data has been published elsewhere 14). Here we mainly study the effect of varying the value of the neutron-neutron scattering length on the FSI peak. In fig. 5 the n-n FSI peak at v.,mab= --n21~lab= 30 ° is shown. If one changes the value of a , . from - 2 3 . 7 fm to - 1 6 fm (potential set (I'-I-III)) the peak decreases with a factor of 2.00. In a simple minded Watson-Migdal type of approximation the heights

388

W . M. K L O E T

ELAB = 1 4 . 4 n

25

~

%

AND

MeV

J. A. T J O N

nn FS 1(30,°30 °}

20

E

tll

g 3

10

3

~

5 ,/ J

0 2

3 Enl

4 in

5

MeV

Fig. 5. N e u t r o n - n e u t r o n final s t a t e i n t e r a c t i o n at 0 ~ ~ 0~2b = 30 °. Solid c u r v e is for p o t e n t i a l ( I - I l l ) , d a s h e d c u r v e for p o t e n t i a l ( R - I l l ) a n d d a s h - d o t t e d c u r v e for p o t e n t i a l ( I ' - I - I I I ) .

2 which means that for a., - 1 6 fm the of the peak would be proportional to ann peak is expected to be 2.19 times smaller than for an. = - 2 3 . 7 fm. The potential set (I'-III) where a.. = a.p = - 1 6 fm gives a peak which is a factor of 2.10 lower than for (I-III). Taking also a.p = - 1 6 fm instead of - 2 3 . 7 fm apparently decreases the n-n FSI peak somewhat more. The set (R-III) which has a.. = a,p = - 1 7 . 1 fm produces a peak which is a factor 1.57 lower than for (I-III). For a fixed value of the n-n scattering length the height of the n-n FSI peak still depends on the type of the interaction used as has been shown in ref. x4). The difference between the result for potential (I-III) and (I-IV) is 12 %. In this kinematical situation the separable result coincides with the result for potential (I-IV). We have recalculated the amplitude for the Yamaguchi potential and find excellent agreement with ref. 3). If one studies merely the shape of the n-n FSI peak one finds for the potentials considered here that the shape does not depend on the type of potential but only on the value of a . , . This is a remarkable fact because, as has been discussed already in

BREAK-UP PROCESSES

ELAB=14.4 MeV

389

nn F£1 (30030° )

1.5

£ and P waves

E

32

.10

08

06

.04

02

0

RELATIVE ENERGY

02

04

Enn

IN

06

08

10

~i2

MeV

Fig. 6. Shape of the separate S- and P-wave contributions to the n-n final state interaction cross section at 0~~ = 09zb = 30°. Curves S and P represent the shape of respectively the L = 0 and L = 1 contribution. Curves S' and P' represent the shape of the L 0 and L = 1 contributions of the direct diagrams only. =

refs. 1, 15), the shape o f the p e a k is a result o f strong interference between direct a n d indirect diagrams. The direct d i a g r a m s alone p r o d u c e quite a different shape. A l s o interference between different p a r t i a l waves is i m p o r t a n t , because the shape o f the s e p a r a t e p a r t i a l wave a m p l i t u d e s is different for each value o f L. In fig. 6 the shapes o f the p a r t i a l waves L = 0 a n d L = 1 are shown for the direct d i a g r a m s alone, d e n o t e d by S' a n d P ' respectively a n d for the sum o f all d i a g r a m s (curves S and P). All amplitudes are n o r m a l i z e d to the value one at zero n-n relative energy. The curve representing the shape o f the total a m p l i t u d e s u m m e d over all p a r t i a l waves w o u l d lie between the curves S and P. I n fig. 7 we show the n-p F S I region at -0- nlab = 30 ° a n d --n2 0 I"b = - 8 0 ° for the p o l tentials (I-III), ( I ' - I - I I I ) a n d ( R - I l l ) . F o r the separable Y a m a g u c h i p o t e n t i a l we find a result t h a t is identical to t h a t o f ref. 3). It coincides with the result o f p o t e n t i a l (I-IV) which is 12 % higher t h a n the cross section o f (I-III). V a r i a t i o n o f a , , f r o m - 2 3 . 7 fm to - 1 6 fm for the local Y u k a w a type potential has a 4 % effect o n the p e a k height. The p e a k height for ( R - I l l ) is a f a c t o r 1.51 lower t h a n the result o f (I-III). I n fig. 8 the n-n Q F S region is shown for 0 nt"b = 30 ° a n d 0- - nlab = - 3 0 °. T h e result l 2 for p o t e n t i a l (I-IV) is again 12 % larger t h a n t h a t o f set (I-III). The difference between the results o f ( R - I I I ) a n d ( I - I I I ) is only 3 %. V a r i a t i o n o f the value o f a . . f r o m - 2 3 . 7 fm to - 1 6 f r o has a 7 % effect. W e also studied the n-p Q F S r e g i o n f o r : - - 3 0 o. The cross sections calculated f r o m potentials ( l - I I I ) a n d nlab = 30 ° a n d ~lab up

390

W. M. K L O E T

AND

J. A . T J O N

14 13 LAB En = 14.4 MeV

o o np F S 1 ( 3 0 , - 8 0 )

12

>

Oa

13 10

E

8 7 w

6

3

5

3

4

~

3

ncJ

0

2

3 Enl

4 in

5

MeV

Fig. 7. Neutron-proton final state interaction for angles 0 ~ b = 30 ° a n d 0 ~ b = - - 8 0 °. Solid curve represents the result for potential (I-III), dashed curve gives the result for potential (R-Ill) and the dash-dotted curve gives the result for potential ( I ' - I - I I I ) .

(I-IV) have been shown in ref. 14). The results for (R-III) and (I'-I-III) are identical with that of potential (I-III). So far we have compared cross sections at a fixed value for the neutron lab energy of 14.4 MeV. The differences between the calculated QFS cross sections are about 10 %. At Elnab = 50 MeV the differences are of the order o f 20 %. This is not surprising since at higher energies also the total inelastic cross section becomes more sensitive to the type of the two-nucleon interaction (see A). As a final remark in this section, very recently it was noticed in analysing p-p QFS data from the p-d break-up reaction that there exists a considerable disagreement between experiment and theoretical predictions of the separable potential model 16). The disagreement appears to depend in a systematic way on the incident energy and 0c.m., where 0c.m. is the scattering angle in the c.m. system of the two nucleons which scatter quasifree. A possible cause for this disagreement is the use of the separable model. We have therefore calculated the QFS cross section at 14.4 MeV and 50 MeV

BREAK-UP PROCESSES

i2

E

LAB n

= 1 4 4 i',4eV

391

n n QFS (30o-30 °)

11 10

£,J L

E

7

: ..--

6

3

-.,:

4

3

2

3

4

5

6

7

8

£

10

Enl in t,4eV

Fig. 8. Neutron-neutron quasifree scattering at 0 ~ b = 3 0 ° a n d 0 ~ b = - - 3 0 °. Solid curve, dashed curve, dash-dotted and dotted curve represent the results of potentials ( I - I I I ) , ( R - I I I ) , ( I ' - I - I I I ) and (I-IV) respectively.

for various values of 0 .... with the local potentials (I-III), (I-IV), (R-III) and the separable Yamaguchi potential. The differences between the various results depend only very weakly on the value of 0c.m., which means that none of the local potentials used here will improve significantly the fit of the p-p QFS experimental data. 7. Search of entire three-nucleon phase space Integrated cross sections like proton spectra and non-integrated cross sections in FSI and QFS regions have shown to be somewhat dependent on the type of twonucleon interaction. One may raise the question whether there are regions in phase space where the dependence on the type of interaction is more significant. We have therefore searched the entire four-dimensional phase space to look for regions in which the cross section depends strongly on the potential used. Of particular interest are areas outside the FSI peaks. As a standard cross section we have chosen that of potential (I-III). Then the relative change of the cross section was calculated if instead of (I-III) respectively the potentials (I-IV), (R-III) and the separable Yamaguchi potential were taken.

392

W. M. KLOET AND J. A. TJON 0

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E13 Fig. 9. Dalitz plot Ex2--EI 3 representing the relative difference between the break-up cross sections for potentials (I-III) and (I-IV) at 02¢'m" = 03¢.". = 78°. The initial neutron energy is 14.4 MeV and hence the c.m. energy s = 7.375 MeV. The numbers in the plot are percentages. In his case the (l-IV) result is always larger than the (I-III) cross section.

The cross section was determined as a f u n c t i o n of two relative energies E12 a n d E13 a n d the n e u t r o n angles 0~"m" and 0~"m'. The angles were varied in steps of ten degrees a n d for each pair o f angles a Dalitz-plot was generated, representing the relative differences in cross section between the searched potential a n d the s t a n d a r d potential for all values of the relative energies which are kinematically allowed. I n this way we find two regions where the cross sections are very sensitive to the two-nucleon potential. There is a region in phase space a r o u n d the p o i n t 0~"m" = 0~"m" = 78 ° a n d E12 = E13 = 2.55 MeV where the potential (I-IV) gives a crosssection which is 52 % larger than for potential (I-III) a n d where the separable potential gives a result which is 64 % larger t h a n that of (I-III). The percentages mentioned are m a x i m u m values. The range of the parameters over which these percentages decrease with 10 % is 5 ° for the angles a n d 0.5 MeV for the energies. C h a n g i n g the angles the p o i n t of m a x i m u m discrepancy may move in the Dalitz plot by a b o u t 1 MeV. Fig. 9 shows the Dalitz plot at 0~"m" = 0~"m" = 78 ° giving the relative discrepancies between the cross sections of potentials (I-III) a n d (I-IV) in percentages. I n this region the result of ( R - I I I ) is 9 % larger t h a n the result of (I-III).

BREAK-UP

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393

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In a second region located around the point 0 ~ "m" = 0~ "m" = 135 °, E l 2 = 2.8 MeV and E13 = 6.5 MeV the result of the separable potential is 22 % smaller than the result for (I-III). In this region the result of potential (I-IV) is 4 % lower than that of (I-III). The cross sections for (R-II[) and (I-III) coincide. The angles 0~m and 0~"m" may be varied by 15 ° still retaining a relative difference of 20 % between the separable result and that of the reference potential. In fig. 10 the cross section near the first mentioned region is shown. The lab angles are 0 l"b = v3nlab = 47 ° and Aq~23 = 159 °. The point with the largest relative difference is at neutron lab energy T 3 = 5.8 MeV. It turns out to be in a region where the cross section has a minimum. F r o m fig. 10 one sees that in this kinematical situation there is an essential difference in shape of the cross section for different potentials. If the angles v2 n~b and O~ab are varied by 3 ° and A(R2 3 by 10 degrees the discrepancy between for instance the results for (I-III) and (I-IV) still amounts to 40 %. In the most sensitive point the only contribution to the cross section comes from the amplitude M ½°. I f one should consider only the doublet contributions the area of large relative differences between the cross sections would be considerably larger. The quartet contribution however wipes out a great deal of the discrepancy. The region where the separable result is 22 % lower than the result for (I-III) also shows up in the integrated proton spectrum. Its coordinates correspond to 0tp"b = 17 °

394

W. M. KLOET

AND

J. A . T J O N

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and -~-ap b = 9.75 MeV. In the proton spectra at 10 ° and 20 ° in figs. 3 and 4 at this value o f -F- pjab the separable result is 10-15 % lower than the result for the local potentials. In this case integration over the neutron variables has not wiped out the discrepancy because the difference between the cross sections extends over a large area. The cross section for the potential (R-Ill) in a very large part o f phase space shows little difference with the result o f potential (I-HI). Outside the FSI peaks we found only a considerable relative difference in a region around the point 0~"m' = 0~"m" = 54 ° and E12 = E1 a = 4.9 MeV. Here the result for (R-Ill) is 21 7o larger than the result o f the reference potential. At the other hand it is close to the result of (I-IV) which is 19 % larger than the result o f (I-III). The separable result is 33 7o larger than for (I-III) in this kinematical situation. The same search o f entire phase space was performed at 50 MeV. As expected the relative differences at this high energy are considerably larger than at 14.4 MeV. The most sensitive region at 14.4 M e V which was located at 0~"m' -- 0~"m" = 78 ° and E l 2 = E~a = 2.55 M e V has developed at 50 M e V a discrepancy o f a factor 3 between the results for potentials (I-IV) and (I-III). It is n o w l o c a t e d at 0~"m"= 0~'m" = 60 ° and E12 = E~ a = 17.9 MeV. The result for the separable potential is a factor 4 larger than that o f (I-III) and the result for (R-III) is only 6 % larger at this point. In fig. 11 the cross section in this region is shown. The lab angles are 0~~b [)lab 33°45 ' and A~P2 3 = 129°35 '. The most sensitive point is again in the minimum region at T 3 = 21.3 MeV. Also here the only contribution to the cross section is due to the

BREAK-UP PROCESSES

395

amplitude M *°. In this sensitive region at 14.4 as well as at 50 MeV the proton angle 0~'m in the three-particle c.m. system is about 140 °, which in this kinematical situation is also the angle between the reaction plane and the incident particles. I f the integrated cross section do-/dg2~"m" is plotted as a function of 0pc . l n . it has a minimum at about 140 °, but this minimum is not very sensitive to the four types of two-nucleon interactions we consider here. 8. Discussion For kinematically complete experiments the n-d break-up cross sections of the four types of two-nucleon potentials we studied here, show considerable differences in some regions of phase space outside the two-nucleon FSI peaks. The largest discrepancy occurs in a region where the cross section has a minimum. The fact that in this region the results for the separable potential and local potential (I-IV) have similar deviations from the reference potential (I-III) and that the results for (I-III) and (R-III) differ considerably less suggests that here we have to do with a core effect. For E ~ b = 14.4 MeV there are certain regions in phase space, where the four potentials considered here, produce nearly the same results. In particular this is the case in the region around a very symmetrical point in the three-body process, namely where E12 = E l 3 = E z 3 = k s and 0~"m" = 0~"m" = 0 ~ "m" = 90 °. In a large region around this point the potentials (I-III), (R-III) and (I-IV) differ at most 2 ~ and the separable potential gives a result which is 9 ~ lower than that of potential (I-III). Intuitively one expects that in this kinematical situation the three-body character of the process will be the most apparent, because in the final state there are no two-nucleon pairs which are privileged and there is as little correlation as possible with the initial state. I f three-nucleon forces do exist, in this symmetrical situation it may be that their effects are less hidden by the two-nucleon forces than in other kinematical situations. The fact that in this region the four types of potentials give nearly identical results, may suggest that here the cross section is not very sensitive to the twonucleon potential and that this could be a possible kinematic situation for an experiment to detect effects of three-nucleon forces. A more definite conclusion about this of course only can be drawn after performing calculations including also higher partial waves in the two-nucleon interaction. At 50 MeV in this region of symmetry the cross section is more sensitive to the type of potential. Taking the result of (I-III) as a reference, the result for (R-III) is 15 ~o larger, for (I-IV) it is 18 ~o larger and the separable potential gives a 29 ~o larger cross section. In view of the overall much larger discrepancies at this energy, these deviations are however relatively small. We have indicated certain regions in phase space which promise to be interesting from the point of view of the cross section being sensitive to the details of the nuclear forces. There is need for experimental data in these regions in order to see how well or how badly the S-wave two-nucleon interactions can fit these data. Also a systematic

396

W . M . KLOET AND J. A. TJON

comparison of experimental data obtained in the whole phase space and the theoretical predictions of S-wave interaction models could be instructive. Concerning the experimental situation for the sensitive regions in phase space discussed in sect. 7 and the three-nucleon symmetrical situation mentioned in this section, one might also think of p-d experiments instead of n-d scattering, since p-d data in general are more accurate. It is expected that Coulomb forces in p-d scattering play a negligible role for relative energies of the p-p pair larger than about 1 MeV. In all situations mentioned here Epp is larger than 1 MeV. This investigation is part of a research program of the "Stichting voor Fundamenteel Onderzoek der Materie" and was made possible by financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek". We are greatly indebted for the computational facilities offered by the Computer Center of Utrecht University. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) l 1) 12) 13) 14) 15) 16)

R. Aaron and R. D. Amado, Phys. Rev. 150 (1966) 857 R. T. Cahill and 1. H. Sloan, Nucl. Phys. A165 (1971) 161 R. Bouchez et al., Nucl. Phys. A185 (1972) 166 I. Slaus e t al., Phys. Rev. Lett. 26 (1971) 789 W. Ebenhoh, Nut1. Phys. A191 (1972) 97 J. M. Wallace, Phys. Rev. C7 (1973) 10 W. M. Kloet and J. A. Tjon, to be published in Ann. of Phys. R. A. Malfliet and J. A. Tjon, Nucl. Phys. A127 (1969) 161 ; Ann. of Phys. 61 (1970) 425 R. V. Reid, Ann. of Phys. 50 (1968) 411 I. H. Sloan, Nucl. Phys. A168 (1971) 211 R. T. Cahill and I. H. Sloan, in Three-body problems, ed. J. S. C. McKee and P. M. Rolph (North-Holland, Amsterdam, 1970) p. 265 K. Ilakovac e t al., Phys. Rev. 124 (1961) 1923; M. Cerineo e t al., Phys. Rev. B133 (1964) 948 N. Koori, J. Phys. Soc. Jap. 32 (1972) 306 W. M. Kloet and J. A. Tjon, in Few-particle problems in the nuclear interaction, Los Angeles, 1972 (North-Holland, Amsterdam, 1973) p. 380 R. D. Amado, Phys. Rev. 158 0967) 1414; I. J. R. Aitchison and C. Kacser, Phys. Rev. 173 (1968) 1700 G. Anzelon et al., Nucl. Phys. A202 (1973) 593