Distorted wave calculations of low-energy deuteron stripping reactions

I

~

NuclearPhysics 86 (1966) 535--544; (~) North-Holland Publishing Co., Amsterdam

!

Not to be reproduced by photoprint of microfilm without written permission from the publisher

D I S T O R T E D WAVE CALCULATIONS OF LOW-ENERGY DEUTERON S T R I P P I N G REACTIONS G. L. S T R O B E L

University of California, Davis, California Received 9 M a y 1966 Abstract: Distorted wave calculations for low-energy deuteron stripping reaction cross sections

are compared to experiment. Possible competition of various direct reaction mechanisms is considered for various parameterizations of the distorting optical potentials.

1. Introduction

Deuteron stripping reactions have been successfully analysed in terms of the distorted wave Born approximation method (DWB) for many targets and deuteron energies 1). For low-Q Ieactions and for deuteron energies below the Coulomb barrier, Coulomb stripping has been shown 2) to reproduce the experimental differential cross sections. At low energies, but above the Coulomb barrier energy, the DWB has failed to reproduce experimental data for various targets. Possible explanations for this failure have been the inapplicability of the optical model for low-A targets, the use of single-particle shell model wave functions to describe the internal motion of the nuclei involved, the use of the Born approximation, ambiguities in the optical model parameterization and alternate reaction mechanisms such as compound nucleus formation, exchange effects or competition from other direct reaction mechanisms. Here we investigate only the last points, ambiguities in the optical model parameterization and the possible competition of other reaction mechanisms with the direct stripping reaction mechansism, for low deuteron energies on small-A targets. Appeal to optical potential uncertainties or to alternate reaction mechanisms is often made when a direct stripping calculation fails to repioduce a reaction differential cross section 3,4). In an antisymmetrized calculation of a deuteion stripping reaction, the Born approximation transition amplitude consists of four telms, labelled 5) direct stripping, exchange knock-out, heavy particle knock-out and heavy particle stripping. Previously the exchange knock-out mechanism has been shown to be negligibly small 6) in comparison to the direct stripping contribution for a distorted wave calculation of a deuteron stripping reaction. The effects of other mechanisms * W o r k s u p p o r t e d in part by the U. S. A t o m i c Energy C o m m i s s i o n . 535

536

G. L. STROBEL

such as heavy particle knock-out or heavy particle stripping should also be small for a distorted wave calculation, as they involve the matrix element of a nucleon-core potential minus an optical model potential that attempts to reproduce the gross effects of the nucleon-core potential in question. When trying to ascertain the reaction mechanism for low-energy deuteron stripping reactions, one is confronted with a second problem. For energies below the Coulomb barrier energy, parameters for the optical potential cannot be extracted from an analysis of the relevant elastic scattering. However, the stripping matrix element is sensitive to the optical potential paramete~ s if the reaction Q-value is high enough. Indeed, there is some question if the parameters that best reproduce elastic scattering are the parameters that best reproduce deuteron stripping reactions 7). The problem of possible competition from other reaction mechanisms remains to be solved in the presence of the problem of the uncertainty of which of many optical potential parameterizations should be used. One has a choice of local or non-local potentials, volume or surface absorption potentials and parameters for these potentials selected by elastic scattering analyses or selected by reaction best fit parameter searches. The 4°Ca(d, p)41Ca reaction has been studied extensively, and at deuteron energies of 7 to 12 MeV very good agreement has been found between experiment and a distorted wave Born calculation using a direct reaction mechanism 8), suggesting both the optical potentials and reaction mechanism are known. Fodor et al. 9) at a deuteron energy of about 2 MeV, obtained an optical potential which was able to fit the (d, p) stripping reaction to the first excited state of 4a'Ca (1.95 MeV excitation). This optical potential however, could not reproduce the (d, p) reaction to the ground state of 4aCa. In the next section an optical potential extrapolated from an analysis of higher energy deuteron elastic scattering is used to calculate the 4°Ca(d, p)4~Ca reaction to both the ground state and the first excited state of 41Ca. Low-energy deuteron stripping reactions on 11B are then calculated again using optical potentials extrapolated from an analysis of higher energy deuteron elastic scattering. The ~lB(d, n)~ac reaction is also calculated using a modified non-local optical potential to study the effect of this parameterization of the optical model. This is done as previously non-local optical potentials have been successfully used ~o) to reproduce neutron scattering over a wide range of experimental situations. 2. Optical Potential Parameterization

The (local) optical potential used is of the form (1)

Vop(r) = Vofws(r) + iWofw~(r) + iW1 g(r), where

fw~(r) = g(r) = 4 exp

1 + exp 1 +exp

, ,

DEUTERON STRIPPING REACTIONS

537

a n d R = ro A+. The shape p a r a m e t e r s ro a n d ~ are allowed to be different for the real a n d i m a g i n a r y terms in eq. (1). N o s p i n - o r b i t p o t e n t i a l is i n c l u d e d for the scattering wave functions. A n o n - l o c a l optical p o t e n t i a l was used to calculate certain scattering wave functions. By m e a n s o f the local energy a p p r o x i m a t i o n 11), it has been shown t h a t the only difference b e t w e e n a wave f u n c t i o n defined via a n o n - l o c a l optical p o t e n t i a l a n d the wave f u n c t i o n a s y m p t o t i c a l l y equal to it defined b y a local optical p o t e n t i a l is a r a d i a l l y d e p e n d e n t factor. Thus, if q5L a n d qSNLare respectively the local a n d non-local wave functions then 12) qSNL = [1 +mfl z V/Zh2]-~ (aL, where fl is the range o f the n o n - l o c a l i t y o f the n o n - l o c a l p o t e n t i a l . A s s u m i n g only the real p a r t o f the optical p o t e n t i a l is non-local, this factor 13) has been i n c l u d e d for certain o f the wave functions utilized herein. B o u n d n u c l e o n wave functions are c a l c u l a t e d b y a d j u s t i n g the d e p t h o f a real W o o d s - S a x o n local p o t e n t i a l while keeping the s h a p e fixed, until the b i n d i n g energy is m a t c h e d . T h e C o u l o m b p o t e n t i a l has been t r e a t e d as due to t h a t o f a u n i f o r m l y c h a r g e d sphere o f r a d i u s R = rc where rc is the same as ro A+ for the real p a r t o f the nucleon optical potential.

3. Results 3.1. THE 4°Ca(d, p)41Ca REACTION F o r d e u t e r o n energies o f 7-12 MeV, o p t i c a l m o d e l p o t e n t i a l s are available t h a t s i m u l t a n e o u s l y r e p r o d u c e d e u t e r o n a n d p r o t o n elastic scattering a n d the (d, p ) s t r i p p i n g r e a c t i o n cross section. H o l d i n g Vo a n d ro constant, the m a j o r energy d e p e n d ence o f the p a r a m e t e r s was a slight increase o f Wo with energy 14). This d e u t e r o n optical p o t e n t i a l is e x t r a p o l a t e d to a d e u t e r o n energy o f 2 M e V using the p o t e n t i a l p a r a m e t e r s which best fit the 7 M e V (d, p ) reaction. The d e u t e r o n p o t e n t i a l p a r a TABLE 1 Optical model parameters used. Reaction

Fig. Projectile fl

1 Set A 4°Ca(d, p)41Ca 1 azC(d, n)l~N 13C(d, n)14N ~lB(d, n)12C

2-4 3

liB(d, n)a2*c

4

d p d p d n d n d n

0 0 0 0 0 0 1 1 1 1

Vo

ro

So

Wo

W1

rl

~1

110 56.2 145.l 56.0 77 49.2 142 71 142 71

1.2 1.25 0.803 1.25 1.4 1.24 1.25 1.25 1.25 1.25

0.9 0.65 0.987 0.65 0.6 0.51 0.65 0.65 0.65 0.65

0 7.5 0 0 10 8.5 17 8.5 17 15

20 0 9.6 10.5 0 0 0 0 0 0

1.55 1.25 1.718 1.25 1.4 1.24 1.25 1.25 1.25 1.25

0.47 0.65 0.578 0.47 0.6 0.51 0.65 0.65 0.65 0.65

fl, ro, %, rl, and ~1 are in fm. V0, Wo and W1 are expressed in MeV.

538

G.L. STROBEL

meters used are listed in table 1. The principle difference between these parameters and the parameters previously obtained by a parameter search is the depth of the real optical potential which is 145 MeV compared to the fixed value of 110 MeV previously used. Using the extrapolated deuteron optical potential, and the proton optical potential which fit 7 MeV proton scattering 8), also listed in table 1, the 4°Ca(d, p)41Ca reaction cross section was calculated for deuteron energies of 1.9 and 2.0 MeV. The Q-value for the ground state reaction is 6.14 MeV, and the neutron is assumed captured into a lf_~_shell-model level by an inert core of 4°Ca. The Q-value I

I

E d = 2.0

1.0~

Ed= 2.0

1,0~- o

/ o - h-

9 s

.-.

///

~\\ \\\\

0 ' 5 ~ ~

i

/

I / I / I

[~ 0

I

t

60

120

I

0

I 60

I 120

Ed =1.9

IOL

/0

9s

1.0 0

i/

± 180

Ed=l.9

o A.~-

I

05 ~

I

180

//

\\

\\\

0 o

/

0.~

o o

oo o

/o o /o o 0

__£_.

I

I

60

120

180

60

120

180

Fig. 1. Calculated and experimental cross sections for the 4°Ca(d, p)42Ca ground state and first excited state reactions. The solid line represents the calculation of this paper. The dashed line is the result of using potential set A of table 1. The experimental cross section data are shown as circles.

for the (d, p) reaction to the first excited state of 41Ca is 4.19 MeV, and for the reaction the neutron is assumed captured into a 2p~ level. In fig. 1, the calculated cross sections are compared to experiment. All cross sections have been normalized to unity at the first peak of the experimental cross section. The curves are drawn in arbitrary units. Using potentials extrapolated from higher energy deuteron elastic scattering to calculate a low deuteron energy (d, p) reaction cross section, for a 4°Ca target, results in a calculation in agreement with experiment for both the ground state and first excited state of 41Ca for deuteron energies of 1.9 and 2.0 MeV in

539

DEUTERON STRIPPING REACTIONS

contrast to previous distorted wave calculations 9). Therefore, for W o = 145 MeV, a distorted wave calculation with a direct stripping reaction mechanism is capable of reproducing the experimental reaction cross sections at these low deuteron energies. 3.2. THE riB(d, n)12C REACI'ION The (d, n) reaction cross section has been obtained for a variety of deuteron energies ~5) between 1.96 MeV and 7 MeV. Smith and Ivash have obtained a deuteron optical potential 3) that fits the ~°B(d, p)ttB reaction cross section for 9 MeV deuterons. This optical potential also reproduces fairly well the experimental 16) elastic scattering of 5.5 MeV deuterons. This potential is extrapolated for a deuteron 1 OF ~c,. [ £/ o~, /

13C(d,n)14N Ed=3.9 6 ~

o

a \~

0.~

I

60

,.o[[o_o° /

o.2

__

//

!

Io 0.5

\L oo °°

t

180

~_

1.0 Ed=2.815

ii

60

\

120

'o

\

/

12C(d,n)13N

/ca.

~ Io \

o o Oo

\ o

\\

t

120

I

180

0

o

0

I

60

I

120

,

180

Fig. 2. Calculated and experimental cross sections for 12 13C(d n)13,X4N reactions. .energy of 1.96 MeV by reducing the imaginary depth parameter of the optical potential. The first neutron optical potential considered here was obtained from an analysis 3) of proton elastic scattering on t z c at 14 MeV and assuming the Coulomb potential is the only difference between the proton and neutron optical potentials. This set of potentials is also capable of reproducing the experimental differential cross sections iv) for the 12C(d, n)t3N and the t3C(d, n)lgN reactions, see fig. 2, indicating these optical potentials should be at least approximately suited for the 1 t B(d, n) 12C reaction. The 11B(d, n) ~2,C (4.43 MeV excited) reaction cross section was then calculated. The Q-value for the reaction is 9.297 MeV; the ploton is assumed

540

G. L. STROBEL

c a p t u r e d into a lp~ shell m o d e l level b y a 11B core. T h e C o u l o m b p o t e n t i a l is a s s u m e d to be t h a t o f a n u n i f o r m l y c h a r g e d sphere o f r a d i u s equal to 1.24A ~. T h e c a l c u l a t e d cross section, i n d i c a t e d by a d a s h e d line, is c o m p a r e d with e x p e r i m e n t for d e u t e r o n energies f r o m 1.96 to 7 M e V in fig. 3. The calculation r e p r o d u c e s only the m a j o r features o f the e x p e r i m e n t a l cross section for the lower d e u t e r o n energies. F o r d e u t e r o n energies o f 7 MeV, even this small success is denied the calculation.

1'0 ~ k \ \

Ed=1"96

\ \ ',,\ \

0

L~

10 f~ , , \ \

60

~

1.0

6o

\\\

~

7

180

I

...... 0

1

60

1.0[- ~"

\\\\ \

o sd

I

120

I

180

0

3

120

180

Ed=7. o

I / _\

!,7 I',

0.5

i

/ ~ - "X

ix k

!20

Ed = 55

60

180

1.0 I'- ~-\\ Ed= 4.11 °/'/ \ ° '\ 7 \\\

Ed= 3.78 °\~)k \

0

120

/

',.

I

60

/o," V -

I

120

180

Fig. 3. Calculated and experimental cross sections for the liB(d, n)12C (4.43 MeV excited) reaction for various deuteron energies. Result of a local optical potential calculation is the dashed line. Result of a non-local optical potential calculation is the solid line. Experimental values and typical uncertainties are shown as circles and error flags. Reaction Q-value is 9.297 MeV, the proton is assumed captured into a lp~ level.

DEUTERON STRIPPING REACTIONS

541

The cross section for the reaction liB(d, n ) 1 2 C ground state was also calculated (see fig. 4). The Q-value of this reaction is 13.73 MeV, the proton is here assumed to be captured into the lp~ shell model level. This set of optical potential parameters extrapolated from higher energy elastic scattering was unable to closely reproduce the experimental cross section for any deuteron energy between 1.96 and 7 MeV. A limited parameter search at selected energies was not encouraging, so parameters resulting in a "best fit" to experiment are not presented. Ed = 1.96

/

_/

%o/° 0.5-

\

\\ I 60

0

/

Ed = 3.78

T

1° ~JT~ ~ \ \

o

I

1.C

\x\, x \

I

0

,-

- / - \ Ed=4"11

o sk-

I

120

/.~o/-,,

180

.[ o

NX

{//'\

t

/../

\\\ I

\', ,'I

\

J 60

0

1.0

P\

\\\

I 120

I 180

-~

i II \\\

~X

~

0

\x~X ,

I

60

/~I I

0.5

1"01

I 180

A

o:i i : ¢ [i

I 120

\

//

[

'\\

.1/ \ \

~ \\ I

', X\

/ //--.

0.5

I

o

I

I

60 120 180 0 60 120 180 Fig. 4. Calculated and experimental cross sections for l~B(d, n)12C ground state reaction for various deuteron energies. Result o f a local optical potential calculation is the dashed line. Result o f a non-local optical potential is the solid line. Experimental values are shown as circles. Reaction Qvalue is 13.731 MeV, the proton is assumed captured into a lp~ level.

542

G.L. STROBEL

Perey and Buck ~o) have shown a non-local optical potential can be found with energy independent parameters that will reproduce neutron elastic scattering on a variety of targets. The non-local potential parameters Perey and Buck used to reproduce neutron scattering on S6Fe for neutron energies of 4 to 26 MeV are used for a non-local potential to describe neutrons scattering on x2C. These parameters were used rather than the more fully exploited set A of ref. 1o) because available stripping codes could not include the spin-orbit term in that optical potential. Only the ieal part of the neutron optical potential was assumed to be non-local ~3). The real part of the deuteron optical potential was also assumed to be non-local. The relation 2

(Vo ro)no,_loo,l :

1.5

2 ( V o r o ) e q u i v a l e n t local potential,

which is consistent with neutron calculations as), is assumed to be also true for nonlocal deuteron potentials and their corresponding equivalent local potentials. Hence, for non-local calculations the depth of the real part of the deuteron optical potential was taken to be 142 MeV, consistent with a radius of 1.25 A ~. The imaginary part of all optical potentials was assumed to be of the local, volume absorption type and possibly energy dependent. Using the non-local potentials in table 1 the liB(d, n)12C cross section was recalculated. The depth of the neutron imaginary (local) potential was fixed at 8.5 MeV for the ~lB(d, n)12C ground state reaction. A slightly larger value, 10 MeV, would improve the fit for Ea = 1.96 MeV. The calculated cross section fit the experimental cross section for all deuteron energies. It is interesting to see that using non-local optical potentials the DWB method is capable of fitting the reaction cross sections utilizing only the direct stripping reaction mechanism. This is for the same deuteron energy range that Owen and Madansky considered 4) when they included heavy particle stripping within the plane wave Butler theory. It has p:eviously been shown that the knock-out exchange term is small 6) compared to the direct stripping term for deuteron reactions when distorted waves are employed. The fit to experiment obtained here indicates the heavy particle stripping exchange mechanism also need not be considered in distorted wave calculations of stripping reactions, at least for the liB(d, n)12C reaction. The peaks in the experimental differential cross section at large scattering angles, the original motivation for including the heavy particle stripping exchange mechanism in plane wave calculations, are seen to be reproduced by a distorted wave calculation using a direct stripping reaction mechanism only. For the 4.43 MeV excited state of ~2C, the experimental differential cross section was also calculated for a range of deuteron energies using the above non-local potential for the real part of the optical potential. The depth of the neutron imaginary local potential W o was set to 15 MeV. The resulting cross sections consistently fit experiment better than using Wo equal to 8.5 which had been utilized for neutrons scattering from 12C in the ground state reaction. At a deuteron energy of 1.96 MeV, the calculated cross section (not shown) would better fit experiment if Wo were

D E U T E R O N S T R I P P I N G REACTIONS

543

increased to 20 MeV. Using Wo = 15 MeV, the a~B(d, n)12"C calculated cross section agrees with experiment fairly well, except for 7 MeV deuterons. For 7 MeV deuterons, the maj or features of the experimental cross section are reproduced by the calculation, but the calculated peak at large scattering angles is at too large an angle to fit the large angle experimental cross section peak. Moderate parameter variations gave no hint of shifting the calculated peak to the correct angle. The calculation was repeated with both local and non-local real optical potentials, using derivative type imaginary potentials (i) for the neutron only and (ii) for both the neutron and the deuteron. The fits to experiment obtained in a brief parameter search were considerably poorer than those obtained using a non-local real potential and volume absorption imaginary potentials. 4. Conclusions

As a result of these calculations, we see it is possible to fit the (d, p) reactions leading to both the ground state and the first excited state of 41Ca for deuteron energies of 1.9 and 2.0 MeV. Direct stripping is the only reaction mechanism required to obtain calculations in agreement with experiment. The real optical potential depth parameter Vo that was utilized here was larger than the depth usually utilized in the optical potential for low-energy deuterons. The (d, n) reaction cross sections leading to the ground state and the 4.43 MeV excited state of 12C were calculated for a range of deuteron energies. A local potential that fit the l~B(d, n)12"C and 12't3C(d, n)~a'~4N reactions gave a qualitative fit only for the XXB(d, n)12C ground state reaction. A non-local optical potential was also used which did fit both the ~B(d, n)12C ground state and 4.43 MeV excited state reaction cross sections. Peaks in the experimental differential cross section at large angles can be reproduced by distorted wave direct stripping calculations without recourse to the heavy particle stripping exchange mechanism. Finally it is also noted that a non-local potential for the real part of the optical potential and a volume absorption term for the imaginary part of the optical potential were used for the calculation that fit experiment. For the considerable parameter range searched, all other optical potentials, local and/or derivative absorption, resulted in poorer fits to the experimental data. I wish to thank John Young for several elastic scattering calculations, W. W. True for resolving some computer difficulties and especially warm thanks to W. R. Smith for the use of his deuteron stripping code.

544

O. L. STROBEL

References 1) W. R. Smith and E. V. Ivash, Phys. Rev. 128 (1962) 1175; N. K. Glendenning, Ann. Rev. Nucl. Sci. 13 (1963) 191 2) F, B. Morinigo, Nuclear Physics 62 (1965) 373; L, C. Biedenharn, K. Boyer and M. Goldstein, Phys. Rev. 104 (1956) 383; K. A. Ter-Martirosain JETP (Soy. Phys.) 2 (1956) 620; W, R. Smith, Nuclear Physics 72 (1965) 593 3) W. R. Smith and E. V. Ivash, Phys. Rev. 131 (1963) 304 4) G. E. Owen and L. Madansky, Phys. Rev. 105 (1957) 1766; O. Ames, G. E. Owen and C. D. Swartz, Phys. Rev. 106 (1956) 775; B. Zeidmau and J. M. Fowler, Phys. Rev. 112 (1958) 2020 5) M. Tanifuji, Nuclear Physics 40 (1963) 357 6) G. L. Strobel and B. L. Scott, Phys. Rev. 140 (1965) B311 7) S. Hinds, R. Middleton and D. J. Pullen, Phys. Lett. 1 (1962) 12; D. W. Miller, H. E. Wegner, W. S. Hall, Phys. Rev. 125 (1962) 2054; P. E. Hodgsen, Proc. Padua Conf., 1962 (Gordon and Breach, New York, 1963); K. R. Greider and G. L. Strobel, Bull. Am. Phys. Soc. 11 (1966) 355 8) L. L. Lee et al., Phys. Rev. 136B (1964) 971 9) I. Foder et al., Nuclear Physics 73 (1965) 155 10) F. Perey and B. Buck, Nuclear Physics 32 (1962) 353; P. J. Wyatt, J. G. Wills and A. E. S. Green, Phys. Rev. 119 (1960) 1031 l l ) F. Perey and D. S. Saxon, Phys. Lett. 10 (1964) 107 12) F. Perey, in Proc. Padua Conf., 1962 (Gordon and Breach, New York, 1963) 13) W. R. Smith, Oak Ridge National Laboratory report ORNL-TM-1151 (1965) 14) L. L. Lee et al., Phys. Rev. 136B (1964) 978 15) E. M. Class et al., Nuclear Physics 72 (1965) 436 16) A. Gallmann et al., Phys. Rev. 138 (1965) 560 17) R. E. Benenson and B. Yaramis, Phys. Rev. 129 1(962) 720; J. R. Sawers Jr. et al., Phys. Rev. 141 (1966) 825 18) F. Perey and B. Buck, Nuclear Physics 32 (1962) 353, table 1, p. 358