Initial-state distortion and final-state interactions in the 2H(3He, 3He p)n and 2H(3He, 3H p)p reactions

Nuclear Physics A255 (1975) 9 5 - 1 0 8 ; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

INITIAL-STATE DISTORTION AND FINAL-STATE INTERACTIONS IN THE 2H(3He, 3Hep)n AND ZH(3He, 3H p)p REACTIONS R. E. WARNER and R. L. RUYLE

Oberlin College, Oberlin, Ohio 44074, USA t and W. G. DAVIES, G. C. BALL, A. J. FERGUSON and J. S. FORSTER

Atomic Energy of Canada limited, Physics Division, Chalk River Nuclear laboratories, Chalk River, Ontario, Canada KOJ I JO Received 8 July 1975 (Revised 24 August 1975) Abstract: The predictions of two fully antisymmetrized reaction theories (DWBA and PWBA-FSI) are compared with absolute coincidence cross sections for the 2H(aHe, aHe.p)n and 2H(3He, 3H p)p

reactions exhibiting final-state interactions (FSI) and quasi-elastic scattering (QES) both with and without charge exchange. The DWBA theory takes into account both the initial 3He-d and the final N-N interactions, while the PWBA-FSI theory includes only the latter. New QES data at Ene =35.9 MeV, as well as previously reported 26.8 and 35.9 MeV data, are fitted, The DWBA theory gives good fits, both in shape and magnitude, to spectra showing N-N final-state interactions but gives somewhat poorer fits to QES spectra whose predicted magnitudes are two to ten times too large. The PWBA-FSI theory always predicts cross sections that are too large; however the predicted shapes are~about as good as those from the DWBA. The initial-state interaction is shown to affect both the width and position of QES peaks from these reactions.

E

I NUCLEAR REACTIONS 2H(3He, 3He p) and 2H(3He, 3H p), E=35.9 MeV; measured absolute I coincidence a(El,01,02). Enriched target. Antisymmetrized DWBA and PWBA-FSI analysis of present and previous 26.8 and 35.9 MeV data for same reactions.

1. Introduction In a previous analysis 1) of the 2H(3He,3H p)p and 2H(3He,3He p)n reactions the effects of nucleon exchange and final-state interactions were studied both in the region of phase space where final-state interactions (FSI) should dominate and in the quasi-elastic scattering (QES) region where spectator effects should dominate. This analysis was an extension of the work Of Warner and Vogt 2) who attempted to understand the ratio of the cross sections at the spectator peaks 3) in the above reactions in terms of a fully antisymmetrized plane wave theory. In the earlier plane wave calculation 2) realistic internal wave functions for the bound nuclei and a finite-range two-body transition potential were used. This t Research supported by the National Science Foundation under grant GP-43518. 95

96

R.E. WARNERet

al.

treatment showed the importance of exchange effects; in particular, knock-out processes were shown to be at least as important in QES as gimple breakup, which in the aH-pp final state is accompanied by charge exchange. Generally the shapes of the QES peaks were accounted for but their centers were somewhat shifted and their magnitudes were greatly overpredicted. Estimates of the effects of FSI and distortion of the incoming and outgoing plane wave indicated that the inclusion of FSI was more likely to effect an improvement in the fits to the spectator poles. Thus a second calculation was undertakenl).that included the nucleon-nucleon FSI as well as full antisymmetry. This extension was facilitated by the use of the Van Oers-Slaus deuteron wave function4) rather than the Hulth6n function. These calculations reproduced the shapes of spectra showing both p-p and p-n FSI but again the shapes of the QES peaks were not correct in detail and the normalizations were unreliable. In this report we include both the 3He-d initial-state interaction and the N - N FSI in a fully antisymmetrized theory. New data showing the transition from the FSI region to the QES region at 35.9 MeV, as well as those given earlier 1'3) at EHe=26.8 and 35.9 MeV, are fitted; the experimental details are given in ref. 1) and will not be discussed here. The FSI effects are usually well described; however, the disappearance of the predicted n-p FSI peak at large helion emission angles shows that the Born approximation theory overemphasizes the pick-up process relative to knock-out and break-up. The normalizations of the FSI spectra are now rather accurately predicted but the QES peaks are still overpredicted. We show that the widths and centres of the QES peaks are affected in an interesting way by the interplay of the initial- and final-state interactions. The importance of initial-state interactions in n-d breakup has been demonstrated by Cahill and Sloan 5) in a model based on the Faddeev equations and explicitly retaining the initial-state interaction. This calculation gives better fits in both the FSI and QES regions of breakup spectra than do simpler theories for reactions among three nucleons. Similar calculations for the five-nucleon reactions discussed in this paper would therefore be of great interest; at present, we know of'no applications of this method beyond the three-nucleon system.

2. Theory and calculations We include here only a synopsis of the antisymmetrized theory of the 3He-d reactions since a reasonably complete description has already been given in ref. 2). Symbolically, we represent the two reactions being studied by t+d~t'+NN,

(1)

where t' represents the final trion (t or 3He) and NN the final dinucleon (pp or pn). In the "post" form of the Born approximation the antisymmetrized matrix element

2H(3He, 3He p), ZH(3He, 3H p)

97

for these reactions is Ti~f= ~(ll)f'lI- ~ (-tl

1)an(Z/-)lW1enlz!+)),

(2)

where the total Hamiltonian H has been divided into two parts, Ha and Wz. The term ~2t in eq. (2) describes transitions caused by Ha. If Ha is chosen to be the sum of the two-body interactions in the final trion plus those between the final free nucleons then this term vanishes since none of these interactions can produce the reactions implied by eq. (1). In the second term, Z~+) is an initial-state eigenfunction of H obeying outgoing-wave boundary conditions, X~-) is a final-state eigenfunction of H~ obeying incoming-wave boundary conditions, and Wa is the sum of the six remaining two-body interactions. The antisymmetrization operator P, in eq. (2) exchange nucleons only between groups (t, d, t' and NN) since we start with wave functions antisymmetric for exchanges within groups; a, is the number of nucleon exchanges needed to obtain P, XI÷) from g~+). Antisymmetrization leads to three types of terms whose coherent sum determines the reaction cross section. These terms, which in a non-antisymmetrized theory would be considered separate processes, are portrayed by Feynman diagrams in fig. 1 ; they are called "break-up", "knock-out", or "pick-up". Their symbols B, K and P, have superscripts showing the parity of the final free nucleons and subscripts identifying the two nucleons whose interaction caused the transition. The first subscript, t or d, indicates whether the interacting nucleon going into the final trion comes from the initial helion or deuteron; likewise, the second subscript indicates the origin of the final free nucleon involved in the interaction. The initial-state eigenfunction Z~+) is the product of internal helion and deuteron functions and a distorted wave describing their relative motion. The final-state INITIAL & FIN~ STA~

INTERACTION MATRIXELEMENTS

BREAKUP

Btd

PICKUP

P.

~t

K~lt Fig. 1. Feynman graphs for the three reaction mechanisms and seven types of interactions which contribute to the reactions 2H(3He, 3He p)n and 2H(3He, 3H p)p. Nucleons of the initial and final trions, initial deuteron, and final nucleon-nucleon pair are labelled t, t', d, and n respectively.

98

R.E. WARNER et al.

eigenfunction %~-) is the product of the trion internal wave function, the N-N scattering function, and a plane wave for the t'-NN relative motion. The trion functions 6) that appear in both initial and final states are Gaussian in all interparticle coordinates with characteristic length ~-a =2.78 fm, a length consistent with electron scattering dataT). The deuteron function is" a sum of three Gaussian functions 4) which behaves like a Hulth6n function for r > 1 fm but approaches the origin in the manner required by hard-core potentials8). The 3He-d distorted waves were computed from a Woods-Saxon central nuclear potential of radius R = r o A ~ = 2 . 8 8 fm and diffuseness a=0.8 fm, and a Coulomb potential for a sphere with radius 2.88 fin and constant volume charge density. The strength Vor2 was taken to be 170 MeV-fm z, about twice the accepted value for nucleon-nucleon scattering. Thus, the real well depth Vo was 42 MeV. The imaginary depth W0 was 6 MeV. The nucleon-nucleon scattering functions, which incorporate the FSI effects, were computed for nuclear potentials 1) which reproduce the N-N scattering lengths and effective ranges9). For the n-p interaction we used Gaussian potentials with Vx=41.0 MeV, V3=72.5 MeV, r l = 1.59 fin, and r3= 1.47 fm, where the subscripts denote singlet or triplet states. The nuclear part of the p-p interaction was taken to be a real central Woods-Saxon well with V0= 33.5 MeV, R = 1.477 fm, and a = 0.5 fm; the Coulomb interaction was assumed to be between point charges. The two-body interactions included in W1 have the'form V(rij) = - ½V~(1 - B P , P~) e - ~2,,~,

(3)

with a depth V~= 72 MeV and characteristic length 6 - 1 = 1.47 fm. The operators P, and P~ exchange spins and charges, respectively. Unless otherwise specified, the exchange parameter B is 1.0; this is a Serber force. Coherent sums of the amplitudes B, K and P determine the cross sections for the two reactions through the following equationsd36 2~pf~ dflt, dflpdE = ~Vo if IJ/g]2' x-',~,h2 L,.~] It + p + p = ' 7 " 1 ~ if

2M I+ + M 2 + Ito.~)+~lMz : ~ - Itx,~), 2

Z[~g[Z~+p+n= 2~s12M~- +M~- + M 2 I (o.½)+ ~ [ if 1 + + +z'rlM1

m +

(4) (5)

2M~- +M~- + 12M~- - 3M2 ](~,,)

: +3M3+ [ll,~,

(6)

M~ = (6 + 6B)Pt~ + 12P~t + 12BK~,

(7)

M ~ = -6B(B~d___2P~)-6(K~t + 2 K ~ ) - 6 ( 1 +B)(K~d+2K~t),

(8)

M ~ = 6B~ + 6 B K ~ .

(9)

2H(3He, 3He p), 2H(3He,3H p)

99

The + superscripts denote the parity of the two final free nucleons and the (S,J) subscripts give the spin quantum numbers for the dinucleon (S) and the five-nucleon system (d). The incident 3He velocity Vo and the density of final states pf are computed to give cross sections as a function of trion lab energy (for the QES data) or proton lab energy (for the FSI data). During the course of this work some errors were found in eqs. (22) through (24) of ref. 2); these are superseded by the corresponding eqs. (6) through (8) above. The integrals defining the amplitudes B, K and P are twelve-dimensional (four inter-particle displacement vectors specify the system after removal of c.m. motion). Since the trion and deuteron wave functions and the two-body potential are Gaussian functions of these coordinates, the only non-Gaussian factors are the radial parts of the two distorted waves. Thus the amplitudes reduce to coupled double integrals, summed over three angular momentum quantum numbers for B, and five for P and K. An efficient computer program was written that produced the distorted-wave calculations given in this paper in about 30 h on a PDP-10 computer. The calculations were checked in two ways. The initial-state distorted wave was computed with the 3He-d potential set at zero and the computed cross sections agreed satisfactorily with those from the program used for FSI calculations in ref. 1). Similar agreement was obtained with results of the PWBA program 2) when both 3He-d and N - N potentials were set to zero. In both cases, the earlier programs were corrected to be consistent with eqs. (6) through (8) above. 3. Results and discussion Our results are shown in figs. 2 through 7. The data in figs. 3 a n d 4 showing QES at 35.9 MeV are new, while all other data have been reported previously 1' a). The QES data (figs. 2 through 4) are fitted by three different theoretical calculations, all of which include complete antisymmetrization [eqs. (4) through (9)]. The distorted wave predictions (solid curves labelled DW) incorporate both the initial heliondeuteron interaction and the final state N-N interaction, while the FSI predictions (dashed curves) include only the latter interaction. PWBA predictions (dash-dot curves), assuming no interaction between unbound particles, are shown for comparison with the FSI predictions. At the spectator peaks, the nucleon-nucleon relative energy eNN is several MeV and so the N-N FSI is not expected to have much effect on the QES yields; in fact the PWBA and FSI predictions are very similar in shape and magnitude. The FSI and PWBA calculations are similar in intent to those reported in refs. 1,2) respectively; however, since the errors mentioned in sect. 2 are now corrected, these calculations supersede our earlier work. Our results at geometries where low eNN causes noticeable FSI effects (figs. 5 through 7) are fitted by D W and FSI calculations. One reaction mechanism not included in our theory is the formation of an excited state of 4He followed by sequential decay into the final trion and undetected nucleon. Arrows mark the locations of possible enhancements due to the first (0 ÷) and second (0-) excited states1°).

100

R . E . WARNER et al,

i

3o

~.H( 3HeeV p)p Eo ~He. " 26.8

d~~, ~: ~ (IS~ ,25~)4He(OA'~")//I//i "~,!)~

,,¢7

,:o

aH(3He ' 3H p) P Eo" 26.8 MeV (,5; ,,I4; )

/] tlp~M~llVl~'-/-I<:

//

\t')

5

.:

'i'-;.J" , IO 15 TRITON

¢#

A > :i

NI

iI

2H(3Heo3He p)n Eo = 26.8 MeV ~

,x,'lo

20 LAB

(,,~,~/,,~,..

,I.l

.

O0 5 ENERGY , MeV

nwxnX .....

,I,,tFSI'~'

I0

,~I It

2O trl=P -~MO-~V

,, .5

z

I

Ji'~'

I \Pw, 04

C

1~\

t"~

( 15H..44~)

i1~"

"7

15

2H(ZHe,3He p)n Eo = 26.8 MeV

60

40

I

I0

.........'

....5,TJ

,~

('pp- MeV

:o He(O-)

I0

¢.

\', F$1 ztI

I

% 5

IO

15 3He

20

LAB

0

5

ENERGY.

IO

15

20

MeV

Fig. 2. Absolute cross sections for the 2H(aHe, 3He p)n and 2H(3He,3H p)p reactions at 26.8 MeV incident energy, at two geometries where the undetected nucleon has low lab energy. Predictions of the theories PWBA, FSI and DW are shown by dash-dot, dashed, and solid lines, respectively. The relative energy of the final nucleon-nucleon pair is indicated at high trion lab energies. Arrows mark the energies where peaks from sequential decay of the first (0 ÷, 20.1 MeV) or second (0-, 21.4 MeV) excited state of 4He into the trion and undetected nucleon may be expected.

Both theories fit the FSI data rather well in the regions of low £NN, where FSI effects (peaks from the n-p nuclear attraction, dips from the p-p Coulomb repulsion) are observed. Discrepancies in shape outside these regions can often, but not always, be attributable to enhancements from 4He* sequential decay. The DW normalization is usually correct within a factor of two, while the FSI theory overpredicts the yields by factors up to fifty. Both theories' worst failure is at (45~,, 31~e) for the 2H(3He, 3He p)n reaction (see fig. 7), a case discussed separately at the end of this section. The fits to the QES data are less good; the peaks are not accurately centered, their widths are overpredicted (though less so by the DW theory than by FSI and PWBA), and the DW normalizations are from two to ten times too large. It was found by removing the nucleon-nucleon FSI and the 3He-d Coulomb interaction, that the improvement in normalization in the DWBA theory was almost entirely due to the initial-state 3He-d nuclear interaction. We then attempted to improve the normalization by varying the 3He-d nuclear well depth. Reducing it from

~H(aHe, aHep),aH(3He, 3H p)

50

50[ p)n

~'H(3He,3He

PWx.05 ~'H(3He. 3He p)n ,.__f"~ Eo = 55.9 MeV i~ItIti~I~'I/~" (14~e,50~)

I .J" I

Eo = 55.9 ieV (18~,,54~) I

101

>

~np - MeV 2 I .3

~."

bJ kO m

I i 25

PWx.04

J "~

25~

:

.:

,,,

-.4

LAB

3He

i5

~"

ENERGY

...."

'i"..,,,,,,.."

20 ,

o

-

".. En

0

I

25

}

~np- MeV

/'

.2

~§ ~

2o

)w~.6

,,'7 •

~:

4He

15

~ ,'.

=

f-..

/'FsI~,os\

4 Z~

- 2 0

25

MeV

Fig. 3. The same as fig. 2, for the 2H(aHe, 3He p)n reaction at 35.9 MeV. 00|

42 to 32 MeV gave somewhat better normalizations at the FSI geometries and somewhat worse ones at the QES geometries while increasing it to 55 MeV had the opposite effect. Thus it was concluded that no overall improvement was to be gained from small variations in this parameter; large variations are not justifiable since the strength Vor2 would then vary substantially from the sum of the values attributable to the two individual nucleons in the deuteron. To show why the predicted cross sections are reduced so much by the initial-state 3He-d interaction, we plot in fig. 8 the real parts of the positive parity matrix elements for both the FSI and DWBA theories at one of the QES geometries. (The imaginary parts show effects similar to those we shall describe, while the negative parity amplitudes are considerably smaller.) These seven matrix elements are, as was noted earlier, integrals in twelve-dimensional space. The 3He-d and ~o |;5.9;oV (,8:.54~) -~ ,o ,.
i

~g

,

t TIM~

".o,o-,i

I

=H( 5H° 5H .)p

\~'N.i'!~l"MIIIV:

,o.

I}ilt~l,I ~:"~.:'<'°=I

't t t;(t I\ :I

3,;

I/~'F.~

I#7' )~,,\5,,,,

=,v

=;'

$7 o;

t7t!77 t/\

,,b 0

o

TRITON

LAB

io

ENERGY,

15

to

MeV

Fig. 4. The same as fig. 2, for the :H(3He, 3H p)p reaction at 35.9 MeV.

25

3(

102

R . E . W A R N E R et al.

.5 .I .001 I~ I I t

40

A

N

30

RELATIVE p-p ENERGY,Epp , MeV .5 .5 J .001 .I .5

.I I

i#

,~L

i

I

-{37~', 35~)

I

i

I

t

FSI / .~t~'

2H( "tHe, 3H p)p E o = 26.8 MeV

x'5/4~wx

(53~ ,25'~]

I

2

i

I

20

4He(O- } I

20 taJ

t

E

h¢.,i

~r

Z t~

OW x I .

"/ io

~f'

40

x,05

.J

lo

.~ .,& .;,

,o

/

,'.o

(25; .35; )/"

,,~...

ofo

.odz J ..~ ?.o ,

. . . p. .,o,, ,58 .

"H.(o')

t

o.

zo

t I0

%. 6

,°

8

PROTON LAB ENERGY, MeV

Fig. 5. Absolute cross sections for the 2H(3He,3H p)p reaction at 26.8 MeV, at several geometries where the c.m. energy of the final nucleon-nucleon pair reaches low values. Predictions of the FSI and distorted wave theories are shown as dashed or solid lines, respectively.

nucleon-nucleon displacements are chosen as six of these coordinates and the scattered waves for these two systems are cut off by Gaussian form factors originating in the trion and deuteron internal wave functions and the residual two-body interaction. The strikingly different behavior of the spectator matrix elements (Ktt , Kdt and Btd) from the non-spectator elements (Kdd, Ktd, Pdt and Pit) results from the different form factors which operate in the two cases. The non-spectator amplitudes are cut off by the trion wave function and/or the nuclear force, whose ranges are comparable to the 3He-d interaction radius. A non-interacting 3He-d pair has a wave-length of about 8 fm at 27 MeV but when their interaction is included (in RELATIVE

,'; ~

15

.,,o?

t

;

,,

=H(3He, ~H p)p E o = 35.9 MeV

ENERGY ,

•pp , MeV

I .... : .... i

} DW /

(45; ,3,; )

p-p

~=

i '5L 152;,26.5~)

15

~4.4/

°

ii {

5

,,

%

,;

,'6

PROTON

LAB

ENERGY ,

i

i

(58.5~, 21~ )

,~

MeV

Fig. 6. The same as fig. 5, for 35.9 MeV incident energy.

i

i

4Hel(O-)

2H(aHe, 3He p), 2H(3He, aH p) RELATIVE

•, ,,

.I ~07 j

p-n

103

ENERGY , Cpn , MeV

;

•',=I

~'H( 3He, aHe p)n Eo =35.9 MeV

;

:

; Eo- 2 6 . 8 MeV

(sa.5;. 2 ~',, )

(53;, 25~,)

15

A

J

>=

I ~~,.~ m FSI x 0.05 0Wx0,75

,"

I

¢

.,, .5,

LU

,,

o!

.,, o,5, .,, .5, ,,

,2 Eo-35.9 MeV

¢

~ 2 i

,'

,~o

.5 ,I .015 .I i i i

.5 i

I i

2 [

Eo-35.9 MeV (52;, 26.5;,.1 k

b

/

"o

"

I~T

4 H e ( 0-1

,,. N "" §~ 4

FSI

0.2

t

t% ! I/

G,

PROTON

16

LAB

-,~

~

,;

ENERGY, MeV

Fig. 7. The same as fig. 5, for the 2H(3He, aHe p)n reaction at both incident energies. The curve labelled "FSI. ,~-1 =0.8 fro", shows the effect of decreasing the range of the residual two-body interaction and is discussed at the end of sect. 3.

the DWBA theory) the s-wave component makes nearly a complete oscillation inside the potential well. This oscillation divides the region where the integrand is large roughly in half. Therefore including this interaction greatly reduces the magnitude of these matrix elements and often changes their signs. The spectator matrix elements describe processes in which one of the final free nucleons originated in the deuteron and did not undergo a two-body interaction. In such cases the aHe-d scattered wave function is cut offonly by the deuteron wave function which is sizable even at 10 or 15 fm. Even though the oscillation of Zh+!d reduo~s these amplitudes about three times by removing a large part of the integrand near the origin, it does not destroy their spectator behavior (peaking near zero lab energy of the undetected nucleon) since the low momentum components come preferentially from large separations of the nucleons in the deuteron. The D W curves in figs. 2 through 4 show that the 3He-d interaction also sharpens the spectator peaks and to a lesser degree displaces their centers. This is associated (see fig. 8) with a sharpening of the peaks for the three spectator matrix elements;

104

R.E. WARNER

et al.

in fact, these amplitudes change sign at large final trion lab energies where the nucleon-nucleon relative energy aNN is small. This results from the coupling of the radial integrals for the 3He-d and N-N scattering wave functions by the form factors of the internal wave functions and residual interaction. For very small eNN the N-N wave function does not change sign throughout the whole region of integration and the sign of the amplitudes is controlled by Z~+2d. As eNN increases the coupled oscillations of the two scattering wave functions reduce the spectator matrix elements and, ultimately, change their signs. This is shown in fig. 9 for the positive-parity direct break-up matrix element at one of the QES geometries. When only the N-N interaction is included (FSI curve) the amplitude comes mainly from small 3He-d separations and eNN does not change sign within the region where the integrand is large, even for moderately large eNN; now the energy dependence of Bt~ is determined mainly by the spectator nucleon's momentum Fourier transform. When the 3He-d interaction is added (DW curve) the effects of the coupled oscillations change the sign of Bt~ when the N-N wavelength is about 15 fm. This effect is removed (dot-dash curve) by replacing just the s-wave part of Z~e!dby its plane-wave counterpart sin kr; with the oscillations of this function near the origin thus removed, Bt~ has about the same energy dependence as in the FSI theory. That this effect results from the behavior of Z~-~at large distances rather than near the origin where the N - N scattered and plane wave show significant differences is shown by the similarity in the energy dependence of the FSI and PWBA matrix elements and the insensitivity of the DWBA to the removal of the N-N interaction. Thus we confirm that the sign

2xlO 4

DWBA MATRIX ELEMENTS ZH(3He,3H pip

/

6xlO 3

o

-" \

(,5;,25;)

o

I.SxlO 4

FSI MATRIX ELEMENTS

Re(M) I.OxlO 4

I

/

,"/.'-. .1,;" =,,

4xlO 't

\ X

~e(M)

-'q.'--. "'~ ........ '

2xlO 3

,g.'/'"" ,;, """....,X '

\

.SxlO 4

--"----

-,,-==-:-_-77"_2__-'27"_£ ........ .

.

.

.

~.--'" .2xlO 3

I

I

I0

15

20

t

I

IO

15

,

20

TRITON LAB ENERGY - MeV Fig. 8. The seven basic matrix elements (see fig. 1) are s h o w n for the 2H(3He, 3 H p ) p reaction at (15~, 25~,) and E = 26.8 M e V for the FSI and D W B A theories.

2H(3He, 3He p), 2H(aHe, 3H p)

:5000

105

[DW-3He-d s-WAVE] x~-

"".~f-" ~,\, •/

..\,

2000

"-~ FsI~-~,;

\ "\

I000 Re(B~d) 0

2H(3He, 3H p)p

(15;, 25; ) 26.8 MeV -I000

-2000 6

)w

\ I I \ I0 15 TRITON LAB ENERGY- MeV

20

Fig. 9. The effect of replacing the 3He-d distorted radial s-wave by its plane-wave counterpart sin kr is shown for the real part of the direct break-up matrix element and is compared with the same matrix element for the full DWBA and FSI theories at (15t, 25~) and 26.8 MeV.

change of the spectator matrix elements is caused by the combined behavior of the initial-state distorted wave near the origin and of Z~-N) at large distances. The initial-state 3He-d interaction sharpens the spectator peaks because it removes much of the contribution to the B, K and P integrals from small radii, where the internucleon wave functions are rich in high-momentum components. The overprediction of QES peak widths has, in fact, been a problem common to most breakup calculations. Impulse approximation fits to p-d breakup data, for example, have been improved 11) by cutting offthe deuteron wave function at small radii. This procedure was first justified empirically on the grounds that QES occurs mainly at large internucleon separations where the deuteron is rich in low momentum components. More recently Cahil112) has shown, using the Faddeev equations, that an effective deuteron wave function rather than the usual bound-state function should be used in nucleon-deuteron breakup matrix elements. This effective function "dilates" the deuteron; that is, it reduces the density at small separations. Thus, for the threenucleon system at least, the phenomenological cutoff procedure has been justified. The disappearance of the FSI peak (see fig. 7c) from both the DWBA and FSI predictions at (45~,, 3 l~e) is a result of the coherence of the seven types of amplitudes. Although the singlet cross section peaks in both theories, the nearly complete cancel-

106

R.E. WARNER et al.

lation among the singlet amplitudes causes it to peak at a much smaller value than the nearly energy-independent triplet cross section. In contrast, at the nearby geometry (52~,, 26.5~e) the singlet cross section exceeds the triplet and the FSI peak is well developed. A related problem arises at the (45~,, 31~) geometry for the 2H(3He, 3H p)p reaction which is the only case where the DWBA prediction is too low by more t h a n a factor of two, again as a result of a particularly low singlet cross section. We first verified that the defect could not be remedied by varying the exchange parameter B (see eq. (3)) even over a rather large range, from 0.5 to 1.5. With this parameter fixed at 1.0 (the Serber value) the singlet squared-and-averaged matrix element simplifies to ~l~'12e+p+n:S=0 ~7~12M~- +M~-[ 2 if

=~-[--Bt+d+4Pt~t.+2P~t+2I(~--2K-~d--4K~t-K~t[ 2,

(10)

where we have neglected the small negative-parity amplitude. The direct breakup amplitude Btd, which is associated with forward scattering and depends on a peripheral process, decreases rapidly with increasing 0r~e while both pickup amplitudes increase as a function of the same angle. All four knockout amplitudes change, perhaps accidentally, so as to enhance the effect; i.e., Ktd increases and the other three decrease with increasing One. All these effects conspire to produce a striking reduction of the singlet cross section. Fig. 7 shows that if we decrease the range of the two-body interaction from 1.47 to 0.8 fm the peak reappears. There is no simple physical explanation for this effect since it turns out that Btd has merely changed by a somewhat larger factor than the other six amplitudes. Similar effects are obtained by decreasing the trion radius or strengthening the deuteron wave function near the origin. No claim is made that these large variations are justifiable; rather, because of the coherence of the many amplitudes even small changes in their relative values can cause dramatic changes in the cross sections. 4. Conclusions

The shapes of those spectra showing mainly FSI effects are usually well fitted by both the FSI and DWBA theories, while the latter also gives nearly the correct normalization. The improvement in the normalization results almost entirely from including the nuclear part of the 3He-d initial-state interaction. One remaining problem is that the antisymmetrized Born approximation calculations still overpredict the magnitudes of the QES peaks, even after the 3He-d initialstate interaction is included. Rogers and Saylor 13) have given an appealing physical reason why first-order theories, such as the Born approximation, over-predict QES intensities in nucleon-deuteron breakup; they count events where, after the initial quasi-elastic scattering, one nucleon scatters again from the would-be spectator nucleon. These authors obtain marked improvement over unmodified first-order

2H(3He, aHe p), 2H(aHe, aH p)

107

theories by introducing an attenuation function which links the re-scattering probability to the initial separation of the two nucleons in the deuteron and to their scattering cross sections. More recently, this procedure has been used successfully by Haracz and Lim 14-17)to calculate cross sections for breakup reactions among other light nuclei. At present we see no computationally feasible way to incorporate it into the antisymmetrized DWBA theory. The shapes of the QES peaks observed in these and similar reactions 18' 19) follow rather closely the momentum distribution of the spectator particle but the two distributions differ in detail; it is worthwhile to identify the effects which cause these deviations. The present DWBA analysis shows that the initial-state interaction affects both the centres and widths of the peaks from the 3He-d reactions. Cahill and SloanS), starting from the Faddeev equations, reached the same conclusion concerning N-d breakup; one specific effect 12) shown to reduce the peak widths was the effective dilation of the deuteron wave function. In general the peak centres are shifted by the interaction potential between the nuclei undergoing QES, in the Born approximation3). Impulse approximation calculations take this effect into account by relating the breakup cross section to the elastic scattering cross section of these two nuclei; the peak centres shift by an impressive amount in the right direction when the elastic cross sections are taken at their relative energy after the collision19). Since the impulse approximation works so much better when the post-collision energy rather than the pre-collision energy is used, it may be worthwhile to expand the present calculations by including the interaction between the final trion and free nucleons as well as the initial 3He-d interaction. This extension would presumably affect both the normalization and QES peak shapes for reasons similar to those discussed in sect. 3 for the production of such effects by the initial-state interaction. One ambiguity arising in such an undertaking is that the Hamiltonian of which the final state function Zt ~ is an eigenfunction (see eq. (2)), would now include the volume-averaged effects of the nucleon-nucleon interactions. Consequently part of the residual interaction currently contained in W~ would be absorbed in the optical potential. The normalization of the cross sections will depend sensitively on the strength and range of the renormalized residual nucleon-nucleon interaction; unfortunately there is no unambiguous way to determine the parameters of this interaction. As in the case of many shell-model calculations, one would probably assume that after the subtraction of the average interaction, one has left potentials with the same strength but considerably shorter range than the complete nucleon-nucleon interactions. Finally one must ask the question "how much should we expect from the Born approximation", especially since the total reaction cross section to all channels is quite large. Amado 2°) has discussed the conditions for convergence of the multiple scattering series, particularly with regard to the p-d breakup work of Durand et al. 2 ~). He concluded that even though first-order diagrams, such as those we considered here, may give a good qualitative description of the physics involved in rearrange-

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ment collisions, the series may hot converge rapidly enough (if it converges at all) for them to give a good approximation to the cross section. It is a special pleasure to acknowledge Prof. Erich Vogt's interest and helpful advice, and his encouragement to undertake this work. One of us (R.E.W.) gladly acknowledges a grant from the Oberlin College Research and Development Committee which enabled us to start the work, and the early participation of James Paget who helped define the computational problems involved. References 1) R. E. Warner, S. A. Gottlieb, G. C. Ball, W. G. Davies, A. J. Ferguson and J. S. Forster, Nucl. Phys. A221 (1974) 593 " 2) R. E. Warner and E. W. Vogt, Nucl. Phys. A204 (1973) 433 3) R.E. Warner, G. C. Ball, W. G. Davies, A. J. Ferguson and J. S. Forster, Phys. Rev. Lett. 27 (1971) 961 4) W.T.H. van Oers and I. Slaus, Phys. Rev. 160 (1967) 853 5) R.T. Cahill and I. H. Sloan, Nucl. Phys. A194 (1972) 589 6) E. M. Henley, F. C. Richards and D. Y. L. Yu, Nucl. Phys. A103 (1967) 361 7) H. Collard, R. Hofstadter, A. Johansson, R. Parks, M. Ryneveld, A. Walker, M. R. Yearian, R. B. Day and R. T. Wagner, Phys. Rev. Lett. 11 (1963) 132 8) H.T. Larson, A. D. Bacher, K. Nagatani and T. A. Tombrello, Nucl. Phys. A149 (1970) 161 9) M.A. Preston, Physics of the nucleus (Addison-Wesley, Reading, Mass., 1962) 10) W. E. Meyerhof and T. A. Tombrello, Nucl. Phys. A109 (1968) 1 11) G. Paic, J. C. Young and D. J. Margaziotis, Phys. Lett. 32B (1970) 437 12) R.T. Cahill, Phys. Lett. 49B (1974) 150 13) J. G. Rogers and D. P.Saylor, Phys. Rev. C6(1972) 734 14) T.K. Lim, Phys. Rev. Lett. 31 (1973) 1258 15) R. D. Haracz and T. K. Lim, Phys. Rev. Lett. 31 (1973) 1263 16) T.K. Lim, Phys. Lett. 47B (1973) 397 17) R.D. Haracz and T. K. Lim, Phys. Rev. C l l (1975) 634 18) I. Slaus, R. G. Alias, L: A. Beach, R. O. Bondelid, E. L. Petersen, J. M. Lambert and D. L. Shannon, Phys. Rev. C8 (1973) 444 19) P, Gaillard, M. Chevalier, J. Y. Grossiord, A. Guichard, M. Gussakow, J.-R. Pizzi and J.-P. Maillard, Phys. Rev. Len. 25 (1970) 593 20) R.D. Amado and M. H. Rubin, Phys. Rev. C7 (1973) 2151 21) J.L. Durand, J. Arvieux, A. Fiore, C. Perrin and M. Durand, Phys. Rev. C6 (1972) 393