Nucleon-deuteron breakup quantities calculated with separable interactions including tensor forces and P-wave interactions

Nuclear Physics A292 (1977) 1-28 ; © North-Holland Publishing Co., Anuterdsm Not to be reproduced by photoprint or microfilm without written parmiuion from the publisher

NUCLEON-DEUTERON BREAKUP QUANTITIES CALCULATED WITH SEPARABLE INTERACTIONS INCLUDING TENSOR FORCES AND P-WAVE INTERACTIONS l . BRUINSMA and R. VAN WAGENINGEN Natuurkundig Laboratorium der Vrije Universiteit, Amsterdam, The Netherlands Received 14 December 1976 Abstract : Nucleon-deuteron breakup calculations at a nucleon bombarding energy of 22.7 MeV have been performed with separable interactions including a tensor force and P-wave interactions . Differential cross sections and a selection of polarization quantities have been computed for special regions of the phase space. The influence of a tensor force and P-wave interactions on the differential cross section is of the order of 20 % . Large discrepancies between theory and experiment occur for the vector analyzing powers, both for the kinematically complete and for the incomplete situation . The calculations show that there are kinematical situations in which the differential cross sections and the tensor analyzing powers are sufficiently large to make measurements feasible .

1. Introduction During the last twenty years the three-nucleon system has been studied intensively. Theoretically it is, on the one hand, interesting to give a description of the threenucleon system, on the other hand one has tried to gain information about the nucleon-nucleon (N-N) interaction, which cannot or only with much difficulty be obtained from two-nucleon data. Attention has been focussed on off-shell behaviour, neutron-neutron interaction, etc . Research in this fieldwas greatly stimulated by the possibility of solving the non-relativistic Schr6dinger three-body problem exactly with the help of the famous Faddeev equations. In this paper we concentrate ourselves on the nucleon-deuteron (N-d) breakup reaction. A variety of calculations has already been performed with separable S-wave interactions [cf. e.g. ref. 1 )] and with the S-wave part of local potentials [cf. the work of Kloet ')] . Up to now hardly any calculations with interactions including a tensor force and/or P-wave interactions have been performed. Doleschall s) has done some work, and we reported some preliminary results a . s). To obtain a general view of the N-d breakup results we have to consider a large variety of possible situations . The breakup case is complicated for several reasons : (i) There are many different kinematical situations, since we have five independent variables. (ii) Ifwehave polarized nucleons and/or vector or tensor polarized deuterons in the initial or final state a whole host of quantities, like analyzing powers, polarizations, spin-spin correlations, should be calculated. I

2

J . BRUINSMA AND R . VAN WAGENINGEN

(iii) A fully-fledged program should also include the full structure of the N-N interaction. This program is much too presumptuous at the moment. On the one hand, experiments in this field, particularly polarization experiments, are hard to perform accurately. On the other hand the computer programs are very complicated, take a lot of computer time and computer resources. In this investigation we have made a start with breakup calculations with an interaction including a tensor force and an interaction in all four possible P-states . We limit ourselves to separable interactions in all these states . Not only cross sections in various kinematical situations have been obtained, but also a few polarization quantities have been calculated . The choice of these quantities is rather arbitrary, although we restrict ourselves to some of the most simple ones, in the hope that one might be able to determine them experimentally . We concentrate our attention on a few . regions in three-particle phase space, namely the final state interaction (FSI) regions, quasi-free scattering (QFS) and certain loci with deep interference minima inthe cross section (the symmetric constant relative energy loci). We also study the kinematically incomplete situation, which is particularly appropriate in the case of polarized incoming and/or outgoing particles, since limited experimental possibilities often make a kinematically complete experiment very difficult. We solve the Faddeev equations, using the form in which they have been cast by Alt, Grassberger and Sandhas 6); the so-called AGS equations. We shall not give the formalism in this paper. A detailed account may be found in the thesis of one of us (J.B.) 7). Only a few remarks are made in sect . 2, about the method of solution . In sect. 3 we introduce the N-N potentials which have been used. Calculations have been performed at 22 .7 MeV nucleon bombarding energy . The amplitudes obtained from the (exact or iterative) solution of the integral equations are used to calculate cross sections and polarization quantities . The way in which these quantities are constructed is described in sect . 4. In sect . 5 the results of the calculations are presented and discussed. 2. Some remarks about the calculations By applying an angular momentum decomposition to the AGS equations they reduce to one-dimensional integral equations if the nucleon-nucleon interaction is separable. In this way we get, for each set of values J' for the total angular momentum J and parity r a set of coupled integral equations. We use Doleschall's method $) ofangular momentum decomposition. The number of coupled equations depends on the complexity of the interaction and on the value ofJ. This is shown in table 1, where Y-Y0, Y-Y7 and Y-Y7-PdO, indicate a pure S-wave interaction, an S-wave interaction

N-d BREAKUP

3

TABLE I

Number of coupled integral equations for charge independent rank-one separable interactions

J

Y-YO

Y-Y7

Y- Y7-P,,,

}

2 2 2

3, 4 4

10 15 16

>}

plus a 3St -3 Dt tensor force, and an interaction in S- and P-states plus a tensor force in the 3St -3 Dt state respectively (cf. sect. 3). We developed a set of computer programs to solve these sets of integral equations. The singularities of the equations were taken care of by the Fuda-Stuivenberg method 9 , '0). For small values of J the equations had to be solved exactly, but for J > j it appeared to be adequate to solve the equations by iterating them once, which means a considerable reduction of computer time. We solve the integral equations up to J'° = ~- and furthermore we restrict ourselves to a maximum 1-value of8. The computer codes are suited for handling separable interactions of arbitrary rank with all kinds of form factors at any nucleon bombarding energy . In this investigation we restrict ourselves to the relatively simple rank-one interactions to be described in sect. 3, and a nucleon bombarding energy of 22.7 MeV, for the following reasons. (i) The complexity of the computer codes makes it highly desirable to test them as well as possible . By choosing the interaction of sect. 3, we can compare at least our elastic scattering amplitudes with those obtained by Doleschall 8), and to a certain extent, with those obtained by Lamot t t). (ii) The energy of 22 .7 MeV was chosen because the elastic scattering calculations of Doleschall e) and of Lamot t t) were carried out at this energy, so we could make the necessary comparisons. In addition, some experimental data on breakup quantities at or near this energy are available, namely differential cross sections at 23 MeV by Petersen et al. 12), polarization quantities at 22 .7 MeV by the Berkeley group t 3) and at 20 .4 and 21 .4 MeV by the Texas A & M group 14,15) . The accuracy of our results was tested in two ways. (i) The number of integration points, which was 21 in our main calculations, was increased in various regions separately by two in certain test rums . (ii) The results of our calculations were compared with results obtained in a different way or by others. We compared the amplitudes for a simple S-wave interaction, obtained with our codes, with those obtained with the help of Stuivenberg's program t °), which is especially made for such interactions, and therefore much simpler. On the other hand the elastic scattering amplitudes were compared with those calculated by Doleschall B) and by Lamot t t ).

4

J . BRUINSMA AND R . VAN WAGENINGEN

From these checks we found, that the inaccuracy of the amplitudes is at most something like 5 %, but usually smaller. From this it can be inferred that also the cross sections and polarization quantities in general have a satisfactory accuracy. A few remarks should be made here . (a) No method is as yet available to take into account the Coulomb interaction in the p-d system in a reliable way. Therefore we are forced to compare experimental p-d results, with n-d calculations . (b) Apart from this we have used a charge independent nucleon-nucleon interaction, although we know from the work of Stuivenberg et al. te) that the effects of charge dependence are important in certain regions. (c) Our potentials are only of rank one, and could therefore only be adjusted to the low-energy scattering parameters of the nucleon-nucleon scattering, and do not reproduce the high-energy phase shifts. Especially the 313 , phase shift is far from realistic. For these reasons one should be very careful in comparing our results with experimental data . In sect. 5 we shall present a method with which we can at least make an estimate of the influence on the cross sections of charge dependence and of the high-energy phase shifts . It is hoped that in the near future we can perform additional calculations with more realistic nucleon-nucleon interactions, also at other energies . 3. The two-nucleon potentials We make calculations with a number of rank-one charge independent interactions which we call : Y-Y7-Pd,,,, Y-Y7, Y-Y0 . The first character indicates the Yamaguchi form factor in the 'S o state, the second one represents the 3 S, -313 1 state and the number gives the percentage D-state. For the P-wave interactions we take the old Doleschall set e ) . The 'S o and P-wave interactions are given by V,,L(p, 9) = A n

z

Y~t

(ß~L+P

+t

z

2

YqL

(PAL+4

+t

TABLE 2 The parameters of the separable rank-one nucleon-nucleon interaction Interaction state I SO 3s , 313, 3S, (no tensor) 3Po 'P, 33P2 P,

AL (MeV 112 ) 7 .555408 7 .993328 12 .54254 9 .052250 7 .274097 8 .45944 8 .673697 14.82735

A"In 28 .2114 McV 3 /4 27 .4548 McV 3/4 -123 .4057 McV 3i4 44 .8636 MCV 3/4 233 .7608 McV 3 /4 368 .8672 McV 3 /4 417 .4671 McV 3/4 1491 .3706 McV 314

-1 -1 -1 -1 -1 1 1 -I

N-d BREAKUP

5

150 w w

60

w100 0

30 w w x w _o

N

I

SO PHASE SHIFTS - Y " EXF Mac Gregor "t al.

x so N

w a x 0 0.

0

t-30

w w a x o.

30

_ t1 i - Y7

0 -20

Fig. l a. The 'So, 3S, and 3D, N-N phase shifts and mixing parameters e, for the Y-Y7 and Y-YO potential. The experimental points are from MacGregor et al. (table 7) 's) .

3

P0

- Pdol

Fig. 1b. The P-wave N-N phase shifts for the Pd,, potential. The experimental points arefrom MacGregor et al. [the 'P, from table 7 and the 3Po, 3P 3Pz from table 4, ref. 111)] .

6

J . BRUINSMA AND R. VAN WAGENINGEN

whereas the 351-3131 interaction is taken in the old Yamaguchi form ") : 2) L +p ) c+1 (ßi-+4 .+1 "(ßi When the tensor interaction is not included, the 351 interaction has the same form as the 1 S. potential. In eqs. (1) and (2) L is the relative orbital angular momentum of the two nucleons, and q = Sjn means the spin, total angular momentum and parity of the two-nucleon system . The parameters ofthe potentials are adjusted to the following low-energy sacttering parameters : a, = -20.34 fm, r, = 2.7 fm, a, = 5 .397 fm and r, = 1 .722 fm. The parameters for the P-wave interactions have been obtained by fitting them to the nucleon-nucleon phase shift up to 100-150 MeV [ref. ls)] . In table 2 we give the parameters of the interactions. The tensor force gives the following deuteron quantities : Ed = 2.225 MeV, Pd = 7 % and Qd = 0.283 fm2. In fig. 1 we compare the phase shifts and mixing parameters with the experimental points taken from MacGregor et al. l e) . It is clear that the interactions do not reproduce well the nucleon-nucleon scattering quantities, especially in the 3S1-3131 channel. 4. Crass sections and polarization quantities All cross sections and polarization quantities can be conveniently calculated with the help of the density matrix p with respect to the spins of the initial particles, the breakup amplitude M, cf. ref. '), and the operators Q corresponding to the quantities measured after the breakup. With respect to the spins the breakup amplitude M is a matrix with eight rows and six columns, where the rows are numbered by the spin magnetic quantum numbers of the three final nucleons ml , m2, m3 = ± J, and the columns are labelled by the spin magnetic quantum numbers of the incoming nucleon and the deuteron m2 = t 1, mJ = 1, 0, -1 . In almost any conceivable experimentally realizable situation the spins of the two initial particles will be uncorrelated . Therefore the density matrix p for the initial state will have the form P = PN ® Pd, (3) where pN and pd are the density matrices of the nucleon (spin jr) and of the deuteron (spin 1) respectively . The most general form of these density matrices has been discussed and given explicitly by Ohlsen 19) . The spin operators Q corresponding to polarization quantities to be measured in the final state are 8 x 8 matrices, where the rows and columns are labeled by the three spin-i magnetic quantum numbers. There are quite a lot of such quantities, but in this investigation we shall restrict ourselves to just two examples . In general

N-d BREAKUP

7

the Q-matrix involves the spins of all three particles in a correlated way. But if we restrict ourselves to the description of the polarization of one of the nucleons, Q becomes very simple . All measurable quantities can be derived in an obvious way from the general expression Tr(MpMt) = Tr(MpMtQ) . First the quantities for kinematically complete situations are given, followed by those for kinematically incomplete cases . For the choice of the coordinate system, we follow the Madison Convention 2°) . This means that (i) The positive z-axis is chosen along the direction of motion of the projectile (k,o). We shall always take the nucleon as the bombarding particle. (ii) The y-axis is chosen along k ,o x ko,. Here ko, gives the direction of the detected particle. This is unambiguous if one measures only one particle (incomplete experiment). This is not the case if one detects the two identical particles, which we call 2 and 3, in a kinematically complete experiment . Here one detector defines the direction of k 2 and the other one the direction of k 3 . We choose the y-axis in the direction k in x k2 . Now we summarize the quantities which we shall calculate. (a) The unpolarized differential cross section [d(n, nn)p] da 2n = (MMt). dQL2 dl2L,dELz hVi . f6Tr

27r/hv,n is due to flux conservation . The quantity f is the phase space factor [see e.g. Ohlsen 2t )], and M the breakup amplitude, which has been explicitly given by Bruinsma') . The subscript L refers to lab quantities . (b) The neutron analyzing power [d(f, nn)p]

(M(o'y (9 A° = Tr Id)Mt) ' Tr (MMt) where ey is the usual Pauli spin operator and Id the 3 x 3 unit matrix . (c) The deuteron vector analyzing power [d(n, nn)p] (9 PY)MI)

Ad = Tr (M(I" Tr (MM t)

(6)

Tr (M(I" ® Pil)MI) Aa -_ `J Tr (MMt) '

(7)

(d) The deuteron tensor analyzing power [a(n, nn)p]

where i andj stand for x, y and z. The spin operators P used in eqs. (6) and (7) are the ones explicitly given by Ohlsen [ref. I'), eq. (2.17)] t. The experimental results are frequently expressed in a spherical form, therefore we t The operators P.. and P should be exchanged and a factor of three should be added to the righthand side of Ohlsen's P., equation.

8

J. BRUINSMA AND R. VAN WAGENINGEN

give the relations between Cartesian deuteron analyzing powers and spherical ones iT1 1

= 1fAy°+

Tzo = VjA=e

Tzz = j~A,,-Ay). -,/SAX, In an incomplete experiment only one of the three outgoing particles will be observed . The differential cross section for the reaction d(n, p)nn is given by zz) T2I =

dQ __ 2n P i 1 d(cos 02) dQL ,dE L , itvio 2 1

zx

o

dWz

Here the same quantities as in eq. (4) appear, and

n=

Pi 6Tr (MMt). I 2Pz+PIxI

x = cos 0 1 cos 02 + sin 0 1 sin 02 cos 912'

(9)

(10)

The summation in eq. (9) runs over the physically allowed solutions of Pz obtained from the kinematical locus (11) PZ+PlPzx+P1-E = 0, for fixed p1, x and E. For details about the way this formula can be calculated, cf. ref. '). Now we are going to consider measurable quantities with polarization in the final and/or initial state. Such quantities all have the form 1

Pz Tr (MAi1M1B), .) I2Pz+PIxI J -I o p= R= ~ I rzx z Pz d(cos 0z)J d(P2 Tr (MMI) 1 o r2 I2Pz+PIxI d(cos 02)

z1c

d(pz

z

( 12)

We calculate the following cases: (i) The neutron analyzing power Any : Aid = aly ® Id, By. = I. (ii) The deuteron vector analyzing power Ay: A ir = I° ® Py, By. = I. (iii) The deuteron tensor analyzing power AÛ : A ir = I° ® Pig, By, = I. (iv) The proton polarization Py. : A ,j = I, By. = QyP. ® I°°. (v) The n to p transfer coefficient KY' : A id = ßi ® Id, By. = ay ®I°° . The method of calculation is the same as for eq . (9). 5. Results and conclusions

For the calculations we make a special choice from all allowed regions in the phase space. We restrict ourselves to the final state interaction (FSI) region, the quasi-free scattering (QFS) and the symmetric constant relative energy loci. In view of the possibility of performing experiments, it is of interest to look at regions in the phase space where both the cross section and the polarization quantities are large. Moreover, it is recommendable to take regions where the quantities

9

N-d BREAKUP

show a structure, since it is likely that the results in such regions might be sensitive to details of the N-N interaction. 5.1 . RESULTS ALONG KINEMATICAL LOCI

5.1.1. The differential cross section. Here we give the n-d breakup differential cross sections along some kinematical loci for calculations with the Y-Y7-Paa the Y-Y7 and the Y-YO potentials . In figs . 2 and 3 we take angle combinations, selected by Petersen et al. t Z), with typical QFS bumps. In fig. 3 the results for the Y-YO potential for the second half of the curve have not been given. They coincide with the Y-Y7 nn QUASI FREE SCATTERING + np FINAL STATE INTERACTION

Figs . 2 and 3. The n-d breakupcross section for the Y-Y7-P,,,, Y-Y7 and Y-YO interactions . The arrows indicate the positions of the extrema in the differential cross section and will often be repeated in the figures for the analyzing powers . In the FSI peak the minimum relative energy is given.

np FINAL STATE INTERACTION 92 .40' 0,=55.910 vp.1600

r- rT

---0 Y_ YO

np

Enpi 96 keV

10

15

0

5

ARC LENGTH (MeV)

10

15

~E

20

i

25

Figs . 4 and 5. Breakup cross sections for two angle combinations with pure FSI peaks (E.P = 0 MeV) . The Y-YO results are only indicated in the extrema.

10

J. BRUINSMA AND R. VAN WAGENINGEN

results. In figs. 4 and 5 FSI peaks with relative energy Enp = 0 are given. Here the Y-YO results are only indicated in the maxima and minima. In all these figures we indicate in the FSI point the minimum relative energy . With the exception of the QFS bumps, the Y-Y7-Pd., results give larger cross sections than the Y-YO calculations, while in all four figures the Y-Y7-Pd., results are never smaller than those of the Y-Y7 calculations. So it looks as if the P-wave interactions always raise the cross section . In addition it is clear that the influence of P-wave interactions and a tensor force is not negligible . nn QUASI

FREE SCATTERING + np FINAL STATE INTERACTION 62 .53 .1 ° 93 .30 ° y23' 1800

40

Y n

h â E eY t â

10

20

0

0 2 4 6 E2

6

10

10 (M9V)

12

14

Fig. 6. The differential cross sections projected on the energy axis E_ . The experimental points are from Petersen et al . at EP = 23 MeV [ref. ' 2 )] .

In fig. 6 we give for an angle combination the experimental points of Petersen et al. 12) . One should be careful in comparing these points with the theoretical curves. In the first place Petersen and coworkers performed p-d experiments whereas we did n-d calculations ; secondly our calculations are charge independent. By performing charge dependent calculations for the Y-YO potential the FSI peak increases by 3 %, whereas the QFS bump decreases by 7.5 %. (The parameters for this charge dependent potential are adjusted to the following low-energy twonucleon singlet parameters : ann = -16 fin, anp = -23.68 fin, ronn = 2.76 fm and r onp = 2.67 fm.) In the third place we use very simple potentials, which do not well reproduce the N-N scattering quantities . Other form factors, such as the exponential one (E) '), give better agreement with the experimental nucleon-nucleon phase shifts . We already pointed out') that for

N-d BREAKUP

11

S-wave interactions the breakup differential cross section is sensitive to the form ofthe interaction. Charge dependent calculations with the E-EO potential, where the parameters are adjusted to the low-energy scattering parameters mentioned before, give a growth with respect to charge independent Y-YO calculations of 14 % in the FSI peak and a decrease of 12 % in the QFS bump. We expect that charge dependent calculations with the exponential form factors for the S-wave interactions and furthermore a tensor force and P-wave interactions, give differences with the Y-Y7-Pd., results which are of the same order of magnitude. This expectation is based on the assumption that the effects of charge dependence on the one hand and of the use of an other S-wave form factor on the other hand are additive. This assumption is certainly not valid for all quantities, especially when the interference of the amplitudes is important, like with the polarization . However, for the differential cross section the additivity might be more or less a reasonable approximation, at least in certain regions. The application of the above mentioned correction of charge dependent calculations with more realistic S-wave interactions increases the agreement with the experimental results. For a good comparison exact calculations with more realistic potentials are necessary. DEUTERON TENSOR ANALYZING, POWERS e, . %- 41.70 1h, . ISO*

020 010 (n 0: W

3 0 a cD _z N

Y-Y7-Pda

/

\

0 -0.10 010

--- Y-Y7

i~

E n .22.7 MsV

/

~~

20

0

RIM

a -0.10 0.10

~~ T21 /~

0 -0.10 0

+4111110

/1C -

T22

npFSI t ~

4

nn OFS ~?

\\"--/

npFSI t

8 12 16 20 24 -" ARC LENGTH (MW) Fig. 7 . The deuteron tensor analyzing powers for the same symmetrical angle combination as in fig. 2 . 5 .1 .2 . Deuteron tensor analyzing power . For the same angle combinations as above the deuteron tensor analyzing powers aregiven (figs. 7-10). The upward arrows in the figures indicate the positions of the peaks in the differential cross section and the downward arrows the positions of the minima. No curve is given for the Y-YO case, since of course for S-wave interactions the analyzing powers are zero . The structure is dominated by the tensor force, but the influence of the P-wave

J . BRUINSMA AND R. VAN WAGENINGEN

12

DEUTERON TENSOR ANALYZING POWERS J,, 9= " Sii o e 3 " 30 0 , 180 0 Y-Y7-P dal

\-___ Y-Y7 E n . 22 .7

MeV

nn tiFS 0

4

8

ARC LENGTH (MsV) Fig . 8 . The deuteron tensor analyzing powers for the same angle combination as in fig. 3 . Q40

U7

w 3 â c9 z_ N

a

DEUTERON ANALYZING POWERS 9= " 900 e3 " 220 923 " 150°

020

-

___

\

Y-Y7Y-Y7

Plot

0.10 T 22 -0.10-

~~

_-

\\ En " 22 .7 MsV \

0

~

\

2

\ \

020

-Q40

w

np FSI

0

np FSI

4

8

np FSI

T21

\

i

12

16 4 8 12 16 ARC LENGTH (MeV) Fig. 9 . The deuteron tensor analyzing powers for the same angle combination as in fig . 4 . The upward arrow on the right-hand side of the figures indicates the pure FSI . -40

interactions is not to be neglected either. No bumps occur in the QFS region, whereas in the neighbourhood of the FSI position there often appears a peak . In fig. 7 the analyzing powers are given for a situation in which the two identical

N-d BREAKUP

13

DEUTERON ANALYZING POWERS e, . W e, . sssl' %, .Ieo' Y -Y 7- Pda ___- Y- Y7

Q30 0.20

E n .22.7 MsV

0.10 0

\

-030

/

\

/ /

T20

a -Q20 w

3

0 N

a z

Q20 0.10

Q

0.10

i

T22

0

l\ \\

-010 np FSI

-020 0

4

rip FSI

e

12

16

20

24

-" ARC LENGTH (MCV) Fig . 10 . The deuteron analyzing powers for the same angle combination as in fig. 5 . The upward arrow on the right-hand side of the figure indicates the pure FS] .

particles are measured in a kinematically symmetrical situation. The T20 and T22 are symmetric, whereas T2, is antisymmetric, which is the result of the change of the direction of the y-axis in the QFS point. In figs. 9 and 10 we look at situations with pure FSI (i.e. E.P = 0). As mentioned before we see a dip (or bump) in the analyzing powers near the FSI point . In the case of the angle combination 02 = 90°, 03 = 27.4° the shape of the curves is rather peculiar. For the T20 there is a narrow dip on the slope of the bump . This situation can be clarified by changing the kinematical situation a little bit, so that we have no pure FSI. In fig. 11 the results for two other angle combinations are presented 02 = 90°, 03 = 24' and 02 = 90°, 03 = 20', which means that the minimal relative energy is 49 and 218 keV respectively. We find that the dip on the slope for the T20 is less pronounced. Obviously, this dip is caused by the interference between amplitudes which change slowly and amplitudes which change rapidly in the FSI region. In fig. 11 we also give the corresponding effects for T2t and T22 . In these figures there is a minimum beside which a narrow dip grows when the relative energy of one pair ofparticles decreases.

14

J . BRUINSMA AND R. VAN WAGENINGEN -0 .18 DEUTERON ANALYZING POWERS

-0.22

N w 3 Zz N Q 2 Q

Y-Y7 90°- 2240 90P- 24.00 900- 20.00

-0 .26

-030 -Oi .D6 0.32

-0 .08

-036

-010

-0.40

-0.12 10

14

18 10 ARC LENGTH (MsV)

14

18

Fig. 11 . Deuteron tensor analyzing powers for three angle combinations in the FSI region . The minimum relative energy in the FSI point is for the solid line 0 keV, for the dashed line 49 keV and for the dashdotted line 218 keV . The whole curve for the angle combination 90°/27 .4° has been given in fig. 9 .

N w z N Q Z Q

1

24 8 12 16 20 -0- ARC LENGTH (MeV) Fig. 12 . Neutron vector analyzing power (A) and deuteron vector analyzing power (iT ) for the same angle combination as in fig. 2 .

N-d BREAKUP

15

We see that the contribution of the amplitudes of the pair of particles with small relative energy gives a narrow dip on a general, usually pronounced structure. The same phenomenon occurs for the angle combination in fig. 10. Here we do not see the same striking behaviour, due to the fact that the FSI point is situated in the extrema of the analyzing powers.

0.10 005 W

3

0 -005

z_ -0 .l0 N

a z a

005 0

I -005 -0.10 ARC LENGTH (MeV)

Fig. 13 . Vector analyzing powers for the same angle combination as in fig. 4. The upward arrow on the right-hand side indicates the pure FSI.

5.1.3. Neutron and deuteron vector analyzing powers. For two angle combinations, the neutron vector analyzing powers (Ay) and the deuteron vector analyzing powers (iTtt ) are given in figs. 12 and 13. The Y-Y7 calculations give results which are in general small and show little structure. Just as in the case of n-d elastic scattering the P-wave interactions are essential. For the Y-Y7-Pd., calculations the bumps are found near the valleys of the differential cross section, which means that the measurements are hard to perform. 5.2 . THE FSI PEAKS

Here we only look at points on the kinematical curve with zero relative energy . This situation may be considered as a quasi two-body process, which can be specified, as in the elastic scattering case, by the scattering angle Od.. Here Od ., usually called the production angle, is the angle of the third particle in the c.m. system. In fig. 14 we give the FSI peak cross sections for the different interactions as a function ofthe production angle. We notice that the influence of a tensor force and P-wave interactions is considerable . The calculations with the Y-Y7-Pd., interaction always give for the FSI points the largest differential cross section.

16

J . BRUINSMA AND R . VAN WAGENINGEN 3.5

Y N

N â

25

60

80

100 120 140 ---* 9,* (DEGREES)

160

Fig . 14 . The n-p FSI cross section as a function of the production angle Od .. 0 .8

0

60

80

120

ISO

1 e (DEGREES) in the peaks of the FSI breakup cross sections for several potentials compared Fig. 15 . The difference with the Y-YO calculations. The solid line gives the difference of charge dependent E-EO calculations.

In fig. 15 we compare our results with charge independent Y-YO calculations . In the same figure we also give the differences with the charge independent E-E0 potential (the parameters for the exponential form factors are adjusted to the same low-energy two-nucleon scattering parameters as for the Y-YO potential) . The influence of the use of a more realistic potential on the FSI peaks may be estimated in a first approximation by adding the difference d = QE-EO-ffY-YO (fig. 15) to the results ofthe Y-Y7-Pdo1 calculations . In that case the FSI cross sections become smaller for production angles larger than 75°.

N-d BREAKUP

17

Performing charge dependent calculations for the Y-YO potential we obtain differences up to 30 % with the charge independent Y-YO calculations. The same differences occur in charge dependent and charge independent calculations with the E-EO potential. Let us also assume additivity of the contributions of charge dependence. Then we may correct our calculations by adding to our results the difference of the charge dependent E-EO and the charge independent Y-YO calculations. In that case the differences given by the solid line of fig. 15 should be added to the results of fig. 14. The curves rise except for Od, = 126° up to 152° . Q20 0 .10

DEUTERONS TENSOR ANALYZING POWERS FOR np FSI y

0 -0 .10 W

â

-0.20 -Q30

0 i y -0.10 Q â -0.20

\Tzo \

_ _Y-Y7 -Pda\\ ----Y-Y7 \

0

I

En . 22 .7 M@V

\

\

\

\\

-0 .10

, i

T2 1

LM I

60 100 140 --* 8d* (DEGREES) Fig. 16. Deuteron tensor analyzing powers for n-p FSI peaks .

Fig. 16 shows the deuteron tensor analyzing powers in the FSI points . The results are obviously dominated by the tensor interaction . There is no similarity with the elastic scattering case a). This is not astonishing, since the differential cross sections for the FSI and the elastic scattering do not agree either . The influence of the P-wave interactions on the vector analyzing powers is significant (fig . 17). The experimental points are from Rad et al. t 3), who measured the proton analyzing powers in the 2H($, p)a* reaction at EP = 22 .7 MeV, and the deuteron vector analyzing powers in the 'H(d, p)d* reaction at Ed = 45 .4 MeV. Here d* denotes final state n-p pairs with relative energy less than 1 MeV. We notice that the iTt 1 and A. calculated with both the Y-Y7 and Y-Y7-Pd., interaction do

18

J. BRUINSMA AND R. VAN WAGENINGEN

0.20

VECTOR ANALYZING POWERS ~' *\ FOR np FSI E n = 22 .7 MeV

OJ5 0.10 005

3-OAS â co z_ N 015

Y -Y 7 -Pdol Y-Y 7 ---Y-!

---_

zz a a10 exp. } Rad et OL

/ /

\ ~}

\

G05 0 -005 -0.10 60

60

100 120 140 ---* 6d* (DEGREES)

160

Fig. 17 . Deuteron vector analyzing powers (iT) and neutron analyzing powers (A,) for n-p FSI peaks and for n-d elastic scattering . The experimental points are from Rad et al . l a) .

TABLE 3 Vector analyzing powers in the FSI for three production angles calculated with the Y-Y7-Pda interaction Bd, (deg)

0s-03 (deg)

B.om, . (keV)

A~FSI)

iT,,(FSI)

122.8

90-32 90-27.4 90-20

148 0 218

0.046 0.031 0.064

0.096 0.089 0.090

94.6

65-35 65-40.7 65-50

94 0 288

0.031 -0 .006 0.052

0.095 0.091 0.116

60 .4

40-48 40-55.91 40-66

86 0 156

0.020 -0 .025 0.034

0.089 0.084 0.089

N-d BREAKUP

19

not describe the experimental points. A part of the discrepancy might be explained from the fact that Ay and iT vary with the relative energy . As already mentionedour calculations are for n-p FSI pairs with a relative energy of 0 keV, while the measurements allow an n-p relative energy from 0 to 1 MeV. In table 3 we give a few examples for the variation of the vector analyzing powers for three production angles ed " = 122.8', 94.6° and 60 .4' as a function of the minimum relative energy . For non-coplanar kinematical situations similar effects appear. It is clear that the distribution of the experimental points in the area from 0 to 1 MeV relative energy should be taken into account in the theoretical calculations . From the experiments of Rad and coworkers it appeared that there is a similarity between the experimental inelastic and elastic scattering results. As they point out themselves this similarity is unexpected. Indeed, the calculations show little similarity as can be seen in fig. 17 where the elastic scattering results for the Y-Y7-Pd., calculations have also been given. The same contradiction with Rad et al. has been found experimentally by Fisher et al. s3) at 14.5 neutron bombarding energy . In connection with these results it is interesting to look at the n-d elastic scattering results of Doleschall za) and of Lamot i') . When they included the P-wave interactions in the calculations, they found that the vector analyzing powers improved when they used tensor forces which reproduced better than the Y-Y7 potential the experimental low-energy 3D, nucleon-nucleon phase shifts as well as the mixing parameter E, . But even for the less realistic Y-Y7-Pd,,, interaction (bad 3D, and E,), the shape of the n-d elastic vector analyzing powers agrees with the shape given by the experimental results. However, the experimental behaviour is not reproduced at all by the breakup calculations, except possibly in the backward peak for the Ay. One should keep in mind that a change might occur by integrating the theoretical results over the relative energies from 0 to 1 MeV. On the other hand it is possible that a large change in the interaction is required to get agreement with the experimental results. Maybe, we have here a breakup situation from which interesting information about the nucleon-nucleon interaction can be obtained. 5 .3 . THE QFS BUMPS

In fig. 18 the QFS peak cross sections, obtained by projection on theenergy axis, are given as a function of one of the polar angles in the lab system . We only look at situations in which the proton is at rest. The experimental points from Petersen et al. ' z) at 23 MeV proton bombarding energy are below the theoretical curves . The influence of the P-wave interactions and the tensor force are too small to get agreement with the experimental results. Calculations with other S-wave interactions show changes in the height of the curve, e.g. the E-EO calculations decrease the QFS cross section by 5 %. This sensitivity is mainly due to on-shell differences, since Petersen et al. Z s) and also Stuivenberg '°) pointed out that off-shell effects are unimportant.

20

J . BRUINSMA AND R . VAN WAGENINGEN 45

40

3.5 E â Y Y

â b 2.5

2 .0 20

30 40 50 80 --- " 92 (DEGREES) Fig . 18 . The n-n QFS peak cross sections, obtained by projection on the energy axis E_, as a function of the polar angle 0. in the lab system. The experimental points are from Petersen et al. at EP = 23 MeV [ref. 12 )] . The CHE curve gives the Y-Y7-Pa,, results corrected for charge dependence and more realistic S-wave interactions. O FS ANALYZING POWERS Y-Y 7-Pda --- Y- Y 7

-0.04

N C W 3 g 0 z N a z a

1

4004

-0.032 I .20

T21

Q02

Z 40

, 80

\

0.008

0

0 .004

-0.02

0

Oa4

\',,] _0M4

_008 I 20

\

-0008

Q04

x

Ay

-0004

408

-0.024

\

0

-0.08

-0.12 0 .018

w

0.008

.

40

I

I

80

-0008

20

40

e2(DEGREES) Fig . 19 . Analyzing powers for the n-n QFS as a function of the polar angle 0 . in the lab system.

80

N-d BREAKUP

21

Charge dependent calculations for the Y-YO potential give results which are about 8 % lower. The calculations with the charge dependent E-EO potential give the same reduction, relative to the charge independent E-EO calculations . If we assume additivity, it is possible to correct the results for charge dependence and the use of a more realistic S-wave interaction. In fig. 18 we add the difference between the charge dependent E-EO and the charge independent Y-YO calculations to the Y-Y7-Pdo, results (CHE curve). The agreement with the experimental points improves . In general the polarization quantities for the QFS are small (fig . 19). The same is true for the free nucleon-nucleon scattering with which process the quasi free scattering is sometimes compared . The comparison is only rough, since in the QFS case the whole multiple scattering series takes part. From the work of Petersen 12) we know that not only the first term in the multiple scattering series is important. Evidently the interference is of the kind that the analyzing powers are small . They show a clear structure, and in addition the shape of the curves does not change when we include P-wave interactions in our calculations . The influence of the P-wave interactions on the magnitude is large for both the tensor and vector analyzing powers . However the absolute value is so small that experiments will be difficult. 5 .4. THE SYMMETRIC CONSTANT RELATIVE ENERGY LOCI

Calculations using several S-wave local and separable N-N potentials have shown a large sensitivity to the potential parameters in regions of interference minima 2 " 26 ) . Kloet and Tjon 2) found that the largest differences occur in regions, where both the polar angles of the two identical particles (02 = 03) and the relative energies E, 2 and E, 3 are equal . In this situation, if merely S-wave interactions are used in the calculations, the only amplitude which contributes to the cross section is that doublet amplitude in which the spins of the two identical nucleons are coupled to zero. We shall study this situation by looking at cases with 02 = 03 and E2 = E3, which means that the relative energies E12 and E,3 are equal too. By choosing the relative energies fixed we obtain, as suggested by Van Oers 26), the differential cross section along these "symmetric constant relative energy loci" as a function of the energy of the non-identical particle . In fig. 20 we present results for a locus in which all relative energies are equal. Here again the Y-Y7-Pd., differential cross sections are higher than the other ones, whereas the Y-Y7 and Y-YO results do not differ much. The minimum for the Y-YO results is raised by 30 % when we use the Y-Y7-Pd., potential. This is an improvement in the direction ofthe experimental results of Van Oers 26). Here too, one should be careful to compare with the experiments, since the more realistic S-wave potential E-EO gives a deeper minimum (0.045 mb/sr2 . MeV). The same has been found by Van

22

J. BRUINSMA AND R. VAN WAGENINGEN 1.0

SYMMETRIC CONSTANT RELATIVE ENERGY LOCUS

0.5 d

Z

N

E 1 2=E23 " E31 " 8.454 MeV 8 2 =9 3 Y - Y7 - Pdol Y-Y7

I

ô Y

I1

EXP von Oers

I

E n . 22.7 MeV

II Il 4

2

6 -->

8 10 El ( MeV)

12

Fig. 20 . Differential cross sections along a symmetric constant relative energy locus. The experimental points are from Van Oers 26) at EP = 23.0 MeV.

0.2

Y d E

0 0

8 .85

e

1182

3

5

7

9

E 23( MeV )

11

Fig. 21 . Differential cross sections for minima of the Van Oers loci as a function of the relative energy of the n-n pair .

Oers with local S-wave potentials. Furthermore we note that our calculations, as those of Van Oers, are charge independent. Charge dependent n-d calculations for the E-EO potential fill up the minimum obtained for the charge independent E-EO calculations by about 30 %. So, the use of a more realistic S-wave potential will deepen the minimum, but this effect will be partly compensated by performing

N-d BREAKUP

23

charge dependent calculations. It should be noted, however, that at other points on the curve there is almost no sensitivity for charge dependence . Also the use of the exponential form factor hardly changes the curve except in the minimum. When the relative energy of the n-n pair is varied, the effect of the inclusion of the tensor force and the P-wave interactions changes. In fig. 21 the minima of the symmetric constant relative energy loci are shown as a function of the relative energy ofthe n-n pair. The very deep minimum in the Y-YO cross section increases by a factor of 3, when we include the tensor force and still another factor of 3 when we also include the P-wave interactions . Also in the n-d elastic scattering the minimum in the differential cross section rises by including a tensor force [see e.g. Stolk and Tjon Z')] . The addition of P-wave potentials to the S-wave and tensor interactions hardly change the minimum. However, for the minimum in fig. 21 it is essential to include P-wave interactions in the calculations . 5.5 . THE KINEMATICALLY INCOMPLETE SITUATION

In a kinematically incomplete experiment the energy of only one of the three particles is measured at a fixed angle of the detector . Of course, then the kinematical locus is not fixed. The measured quantities are obtained by integration over all kinematically allowed situations, which are compatible with the momentum of the measured particle . We compare our n-d calculations with the experimental p-d results from the Texas A & M group. They determine the following quantities for the reaction 2H(p, n)pp at a neutron angle of 18' :

15

10

DIFFERENTIAL CROSS SECTIONS y-Y7-Pdol ___ y _y7 G P . 180 E n . 22 .7 MeV } EXF Graves et al. l E P - 21.9 MeV)

E 5

1

12

I

I

I

I

I

14 16 -i> E p ( MeV )

I

18

I

I

I

20

Fig. 22 . The differential cross section for the reaction =H(n, p)nn at a proton lab angle of 18°. The experimental points are from Graves et al . _°).

J. BRUINSMA AND R . VAN WAGENINGEN

24

15A 13A

DIFFERENTIAL CROSS SECTIONS Y-Y7 Y-YO -"-CY- YO ----- CE - EC

ep .1!°

Y

11.0

â 6

En "22 .71AOV

9.0 /.

OY

bâ

// "

7.0

i 18D

i 185

i 19A --+ Ep(MeV)

19"5

Fig. 23 . The n-n FSI differential cross section for several potentials at 0, = 18' .

by Rad et al. "), EP = 21 .4 MeV . 2e (ü) abra.kap by Graves et al. ), EP = 21 .9 MeV. (iii) Ay and K.y' by Graves et al. 14), EP = 20.4 MeV. ., and Y-Y7 In fig. 22 we present the differential cross section for the Y-Y7-P. potential. The pronounced peak at nearly maximum proton energy is due to n-n final state interaction. This peak is strongly dependent on the n-n scattering length 22). Since the p-p scattering length deviates from the n-n one, and moreover the Coulomb effects will be important, a comparison with p-d experiments in this region is dubious. The calculations agree remarkably well with the experimental results, except the last point. The main reason for this discrepancy is the difference between the experimental bombarding energy of 21 .9 MeV and the 22.7 MeV used in the calculation (the maximum proton energy in our calculation is 19 .38 MeV, whereas for the experimental bombarding energy we obtain 18.62 MeV). By performing calculations at 21 .9 MeV the n-n FSI peak will shift to the left . (i) Joy,

TABLE

4

Differential cross sections for different potentials for the n-n FSI peak and at Eo - 12 MeV Potential

aP-k

aie

Y-Y7- Pdd Y-Y7 Y-YO CY-YO E-EO CE-EO

14.3 14.1 10.7 9.2 11 .4 9,9

3 .42 3 .05 2 .99 2.95 2.97 2.95

N-d BREAKUP

25

When we perform calculations with the Y-YO potential, we obtain the same results as for the Y-Y7 potential, except for the n-n FSI peak . The same is true for the more realistic charge independent E-EO potential. Also charge dependent calculations with the Y-YO and E-EO potentials (CY-YO and CE-EO) only give differences in the n-n FSI peak. In table 4 we present the differential cross sections for 0.40

w w

Q20

z 0

ap .

âN 4 -

18e E, " 22 .7 MeV Y - Y7 - Pd ol -- - -Y-Y7 - -Y-YO _- . ._Y-Y0 (E, " 20.4MeV) " EXP. Graves et al . ( Ep " 20 .4 MeV)

010

w w

0a

Ay

0

CD z N

EXP . Graves et at. (Ep " 20.4 MeV 1

4

-1110 .04 0

1102

j EXP. Rad et al . IEp " 21 .4MsV)

Py t

z

0 t4

0 a

0 0A2

-004 10

12

14 16 -" Ep (MeV)

18

Fig . 24. The polarization quantities P ., A~ and Jq' for the reaction IH(n, p)nn at 9 0 = 18° as a function ofthe proton energy . The experimental points for the Py are from Rad et al . t °),and for the A~ and 1C, from Graves et al. ") .

26

J . BRUINSMA AND R. VAN WAGENINGEN

the different potentials in the n-n FSI peak and in a different arbitrarily chosen other point (EP = 12 MeV) . It is clear that the tensor force is of great importance for the FSI peak . The width ofthe n-n FSI peak is often used todetermine the n-n scattering length 22} In our case the ano is equal for all charge independent calculations. Nevertheless the widths are not equal, as can be seen in fig. 23, where only n-n FSI peaks have been given. The addition of a tensor force makes the peak higher and more narrow. The Y-Y7-P d ., calculation gives nearly the same results as the Y-Y7 . It is clear that the determination of the n-n scattering length with only S-wave interactions gives incorrect results. Furthermore the charge dependent calculations with S-wave potentials decrease the peak cross section and widen the peak . In fig. 24 we give the Y-Y7-Pd ., and Y-Y7 results for the measured polarization quantities PY ., A Y and Ky.'. The calculations deviate significantly from the experimental results. For the Ky' we also give the results for the Y-YO calculation at 22.7 MeV and at 20.4 MeV. Again a great part of the discrepancy comes from the inequality of the bombarding energy . The influence of the tensor interaction is considerable, but in the wrong direction. Maybe, this is a consequence of the tensor force used, which does not reproduce the N-N phase shifts (see fig. 1). For this reason it is worthwhile to perform calculations with a more realistic tensor force. The addition of P-waves gives a small improvement for the low-energy proton energies. For the S-wave interactions we also perform charge dependent calculations . The influence on the KY is very small, just as the use of the more realistic exponential form factor for S-wave potentials. Although the agreement of the exact calculations using simple potentials with the experimental Ky' is not good, there is a qualitative agreement. This is at least M60

.08 0

N w O.AO 3

OAR 0

CD z_ N t

-0.04

Q20

-0.06

Z

-0.12 8

12

16 ---* Ep (MeV]

20

-0.16

8

12

16 -" Ep (MeV)

Fig. 25 . The deuteron tensor analyzing powers for the reaction 2 H(n, p)nn at OP = 18°.

20

N-d BREAKUP

27

much better than the predictions based on the impulse approximation [see e.g. Dass and Queen s9 )]. In fig. 25 the deuteron tensor analyzing powers are given. The structure and magnitude, especially for the TZ0, are such that it seems possible and desirable to determine them experimentally . The T22 is very small and in addition there is no difference between Y-Y7 and Y-Y7-Pd., results. 5.6 . CONCLUSIONS

A large variety of results concerning n-d breakup calculations with the Y-Y7-Pdo,, the Y-Y7 and some S-wave interactions has been presented. The changes in the shape of the differential cross sections are small. In all cases which have been investigated the addition of P-wave interactions to the Y-Y7 potential increases the differential cross section. A simple statement like that, however, cannot be made for the tensor force. Merely for the QFS peak cross sections the results for the Y-Y7 potential are always lower than those for the Y-YO interaction. A comparison with p-d breakup experiments is difficult, since we perform charge independent n-d calculations with very simple potentials . Considerable differences in the results occur, when we use the more realistic E-EO potential instead of the Y-YO interaction. However, it is not sure whether calculations with a so-called realistic potential will resolve the discrepancies with the experimental results. In this connection it should be noted that for exact calculations with the super soft core potential the elastic scattering results do not agree everywhere 30). Also the use of charge dependent potentials is of importance . By adding the differences of charge dependent E-EO calculations and charge independent Y-YO calculations to the Y-Y7 and Y-Y7-Pd,,, results we obtain better agreement with the experiment . This conclusion is only very qualitative since it is well known that changes in the potential give results which are not always additive . The deuteron tensor analyzing powers are clearly dominated by the tensor interaction, but the influence of P-wave interactions is not negligible. The general behaviour along the kinematical locus is determined only for a small part by the FSI and QFS situation. The pure FSI gives narrow dips or peaks on a general structure. We have shown that the tensor analyzing powers are so large that it seems feasible to do experiments. The Y-Y7 calculations give small vector analyzing powers, while the influence of P-wave interactions is large. This does not mean that the tensor force is unimportant. We expect that just as in the n-d elastic scattering, the influence of the tensor force is considerable when P-wave interactions are present. The calculated vector analyzing powers are not in agreement with the experimental data . Since the structure is so different, we have the impression that maybe a large change of the interaction is necessary to get agreement. Possibly we have

28

J. BRUINSMA AND R. VAN WAGENINGEN

here a breakup situation from which interesting information can be obtained about the N-N interaction. The results on which we have reported, and the conclusions drawn from them, make it clear that more calculations should be performed using different sets of S-, P- and tensor interactions . Also the inclusion of charge dependence in the calculations seems necessary.

1) 2) 3) 4)

References

J. Bruinsma, W. Ebenh6h, J. H. Stuivenberg and R. van Wageningen, Nucl . Phys. A228 (1974) 52 W. M. Kloet and J. A. Tjon, Nucl . Phys . A210 (1973) 380 P. Doleschall, private communication J. Bruinsma and R. van Wageningen, Proc. 7th Int. Conf. on few body problems in nuclear and particle physics, ed . A. N. Mitra et al. (North-Holland, Amsterdam, 1976) p. 206 5) J. Bruinsma and R. van Wageningen, Phys . Lett . 63B (1976) 19 6) E. O. Alt, P. Grassberger and W. Sandhas, Nucl . Phys. B2 (1967) 167 7) J. Bruinsma, thesis, Vrije Universiteit, Amsterdam, 1976 8) P. Doleschall, Nucl . Phys . A201 (1973) 264 9) M. G. Fuda, Phys . Rev. Lett . 32 (1974) 620 10) J. H. Stuivenberg, thesis, Vrije Universiteit, Amsterdam, 1976 11) G. H. Lamot, thesis, Universit6 Claude Bernard, Lyon, 1975 12) E. L. Petersen, M. 1 . Haftel, R. G. Allas, L. A. Beach, R. O. Bondelid, P. A. Treado, J. M . Lambert, M. Jain and J. M . Wallace, Phys . Rev . C9 (1974) 508 13) F. N. Rad, H. E. Conzett, R. Roy and F. Seiler, Phys . Rev. Lett. 35 (1975) 1134 14) R. G. Graves, M. Jain, H. D. Knox, E. P. Chamberlin and L. C. Northcliffe, Phys . Rev. Lett . 35 (1975) 917 15) F. N. Rad, L. C. Northcliffe, D. P. Saylor, J. G. Rogers and R. G. Graves, Phys. Rev. C8 (1973) 1248 16) J. H. Stuivenberg, J. Bruinsma and R. van Wageningen, Proc . Int. Conf. on few body problems in nuclear and particle physics, ed . R. J. Slobodrian et al . (Les presses de l'Universit6 Laval, Quebec, 1975) p. 505 17) Y. Yamaguchi and Y. Yamaguchi, Phys . Rev. 95 (1954) 1635 18) M. H. MacGregor, R. A. Arndt and R. M. Wright, Phys. Rev . 182 (1969) 1714 19) G. G. Ohlsen, Rep. Prog. Phys. 35 (1972) 717 20) H. H. Barschall and W. Haeberli, ed ., Polarization phenomena in nuclear reactions (Univ. of Wisconsin Press, Madison) 1971, p. xxv-xxix 21) G. G. Ohlsen, Nucl . Instr. 37 (1965) 240 22) R. Aaron and R. D. Amado, Phys. Rev. 150 (1966) 857 23) R. Fischer, H. Dobiasch, H. O. Klages, R. Maschuw and B. Zeitnitz, preprint, 1976 24) P. Doleschall, Nucl . Phys. A220 (1974) 491 25) E. L. Petersen, M. 1. Haftel and J. M. Lambert, Proc. Int. Conf. on few body problems in nuclear and particle physics, ed . R. J. Slobodrian et al. (Les presses de l'Université Laval, Quebec, 1975) p. 758 26) W. T. H. van Oers, Proc. 7th Int. Conf. on few body problems in nuclear and particle physics, ed . A. N. Mitra et al. (North-Holland, Amsterdam, 1976) p. 746 27) C. Stolk and J. A. Tjon, Phys. Rev. Lett . 35 (1975) 985 28) R. G. Graves, M. Jain, L. C. Northcliffe and F. N. Rad, Proc. Int. Conf. on few body problems in nuclear and particle physics, ed . R. J. Slobodrian et al. (Les presses de l'Universit6 Laval, Quebec, 1975) p. 754 29) G. V. Dass and N. M. Queen, J. of Phys. Al (1968) 259 30) J. J. Benayoun, J. Chauvin, C. Gignoux and A. Laverne, Phys. Rev. Lett. 36 (1976) 1438