Finite range distorted wave born approximation calculations including recoil for heavy ion particle transfer reactions

2.N

[

Nuclear Physics A212 (1973) 465--477; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

FINITE RANGE DISTORTED WAVE BORN APPROXIMATION CALCULATIONS INCLUDING RECOIL FOR HEAVY ION PARTICLE TRANSFER REACTIONS C. ALEX M c M A H A N and W. TOBOCMAN

Physics Department, Case Western Reserve University, Cleveland, Ohio 44106 t Received 28 May 1973 Abstract: We have developed a computer code to calculate the cross section for particle transfer between heavy ions using finite range DWBA including recoil. This code uses art expansion method which reduces the evaluation o f a six dimensional integral to an infinite sum of products o f one dimensional radial integrals. This method is found to be accurate, stable and it converges rapidly enough to be a useful calculating technique.

1. Introduction

The distorted wave Born approximation transition amplitude for particle transfer I + B = ( A + N ) + B ~ A + ( N + B ) = A + F can be evaluated easily if the interaction potential can be considered to be of zero range. However, when the particle is transferred between heavy ions the zero range approximation cannot be justified. Without this approximation the problem involves a difficult multi-dimensional integral. This multi-dimensional integral problem can be simplified if one makes the "no-recoil" approximation, that is, MI rar=~rln

MN MA + M F MI MA Mr rrqB ~ M~rIB"

This approximation works well for nucleon transfer at low energies but at higher energies recoil effects become important 1). When an o~-particle is transferred recoil becomes a major consideration. Another and perhaps the most important consequence is that certain values of angular momentum transfer are not allowed when using the no-recoil approximation. The exclusion of these values of angular momentum transfer can have important effects. If neither the zero range nor the no-recoil approximation is justified then the full six dimensional integral must be evaluated. A procedure for this evaluation has been given by Austern et aL 2), and an evaluation of the multiple integral by direct numerical integration has been carried out by Bayman 3). Robson and Koshel 4) reduce the problem by expanding the continuum wave functions in plane wave states with * Supported by the National Science Foundation. 465

466

C.A. McMAHAN AND W. TOBOCMAN

the remainder of the problem being an easily evaluated plane wave Born approximation. We use the Sawaguri and Tobocman 5) expansion in terms of harmonic oscillator functions to reduce the multi-dimensional integrals to an infinite sum of products of easily evaluated one dimensional radial integrals. Our calculations indicate this sum converges rapidly enough to be a practical calculational technique. In sect. 2 we discuss the expansion method used to reduce the multi-dimensional integral. Sect. 3 contains a discussion of the computer code and presents results of several test calculations. Sect. 4 is a summary.

2. Expansion method The DWBA transition amplitude is given by 6) Tfl --- (~'k~)~NBI VANI~AN ~IB Ut(+ )\/" The bound states are given by ~NB(r ) = E O2i(jj . M ~BI . • AN(r)

=

Z

{JJB . Jv. MF)(IJN . . mMN IIJN JIt)c~i~(r)zj~NB M~Zj~,M~

I . . . . AN Oyt(JJA/~MA{ JJA SI MI)(IJN rnMNl lSN Jl~)~flm(r)ZSN )~'A'

where (jJAI~MA{JJAJIMI) is a Clebsch-Gordan coefficient, the ;~ are internal wave functions, the 0 are spectroscopic factors and the ~ are single particle wave functions of the bound particle. We assume a single (lj) configuration. The potential VAN is the interaction potential which binds the initial bound state and the ~ are the optical model wave functions in the initial and final channels. The differential cross section is __()da

dO "0"

-

m mtnAF kAy 1 (27rh2)2 k m (2J,+I)(2]B+l)

~ MIMB

(-)

(+) 2

l<~Av ~NBII/AN}~AN '/% >1 ,

MAMr

where m m is the reduced mass and km is the relative motion wave number in the IB channel. More explicitly, (-)

(+)

(~PAF ~Nn] VANI~ANqJm > = 0~ OJl(jJ n fiMBIjJ n 'JF MF)

× (IJNmMNIIJNJ~)(JJA/tMAIJJAJl M~)(IJN mMNI1JN j#)A~F, where (--)*g(¥AF)~2i~ NB*(rNB)VAN(tAN)~jlm(t AN AN)~tYlB (+) (riB)" A,F = f d3 r ANdSrm TJAV

It is with the evaluation of the six dimensional integral A~v that we are concerned. Sawaguri and Tobocman (hereafter referred to as ST) give an expansion of the

DWBA INCLUDING RECOIL

467

function q~(rl-r2) as

x~'tr+ p, P,)F~(2, ,8+ r2) # , ( r , - r e ) = Y, w,1¢p .,, , , ,~.w,,z*¢,+, . ~ , .+,-.,,,+,+,

m2l~n , I+. flj. ~i. m/~nz 12 f12

where

D' l~Im}"~ilt-12-18 L~(21z~(2/-7~

(l!

{

+ 1)(2/+ 1)ld~(121r17zml12111ml)(121001121lt0)'

17 111 l, 172 12 x

/?, =, =~,6(n2-s-s,-½(l+l,-12))4/?1 f12

/?2s,+,,32s+ ,

sst

(_~)+,-+,

(_~).-,

(2_,~,),,,++,+,,++ (2_,~).+s+,++

× [n++r(n~ + h + +)n, +rCn, + l, + +)n+r(n + / + +)]+

F~'(~,/~,) = eU'--=~""%'0,/~'),

F2(1, fir) is the radial harmonic oscillator function, and

t

/?2 = 12[fl2( 2 _ +e)+ fl{(2 - ~l)]J " The method given by ST for evaluating d~r consists of first expressing Atv in terms of the variables rAv and r m as 3 3f (-)* NB* A,F = 2 A ItA d 3 rA~ d3rm ~AF (rAF)~,bjim (~A(,/~AriB - - PAF)) AN" r AF--r mn \ \ I//(m+ )dr x V~$j~.,(J+.(#++ ,+, m~j,

where

}~A =

M[

MA M F

/2A = _ _ MA

M F MI -- MA MB

2B =

Ma Mt M F MI -- M A MB

,

/ta = - -MF . MB

The bound state wave functions ~bji~ NB and VaN•jtm AN then are expanded using ST. This would seem to be the logical procedure since a bound state wave functio n is easily

468

C.A. McMAI,-IAN AND W. TOBOCMAN

expanded in harmonic oscillator functions. This method leads to an infinite series involving the sum of products of four radial integrals. Two of these integrals are overlaps of bound state wave functions with harmonic oscillator functions and the remaining two integrals are overlaps of the continuum wave functions with two harmonic oscillator functions. While this procedure was workable, some cases were found in which the infinite series did not converge fast enough to be a useful and reliable calculational scheme. We now present an alternative approach. Again making a change of variables, this time to VANand rNB, we have

A,F = f darANd3rNB ~-F), (MB --/'NB Mr

~/ NB* X VANtP'jlmk~AN]~tlB ,'kAN[- '(t/(+) ( rNB + -MA - /'AN) " MI

Because the transfer occurs in a limited region of space and hence a knowledge of the wave functions is needed only in a limited region we can expand the continuum wave functions using the ST expansion. This may be regarded as expanding the continuum wave functions over a limited region of space in a series of harmonic oscillator functions. Over this limited region the oscillator expansion can provide a good representation of the continuum wave function. This oscillator representation may then be regarded as the exact continuum function and expanded in the ST expansion. This method of expanding the continuum wave functions is similar to the method of Robson and Koshel 4). Application of the ST expansion to A w yields AIF = E ~fnl'm'IB j~|,~,AF (_pANjv nlltmlfi212~2- cpNB_]~ nlllmln212m2

2 mmm2 2, m')[n2 12 f12

:tn2 ]2

f12

where ~t~n|, m m.

fdarF~'(~,

fir) yv~',(p) ~v}+)(v),

f = n,hr~,,~t2m2 =

r)VANdpjtm(r), Ji,

t',-n,

~,,2-7-.

Yh (P)F.2(2' f12 r)q~j,~ (r).

Now insertion of partial wave expansions for ~u and tk allows us to perform the

DWBA INCLUDING RECOIL

469

angular integrations with the following result: AIF

64 E it'-t2-h +7~Y~(~A)(-- 1)h +'~+" ( 2 L + 1)(2/2 + 1)(212 + 1) - x/4-~ x/(Zr + 1)

~

× (L2OM.[L2LM)(I~ 12O0 II1 12 lO)(l~ 1200111 1210)(/2 L00[12 Lit 0)(/2 L00112 L/t0) x

12 r~[l

12 rJ[L

l

nl

!1 fll

n2

Iz

f12

or1 nl /tn2

11 fll 12

~t

f12

x fl°r 2 dr FL.(e. [lr)f[B(k, r)fo°r 2 dr F}(~. flr)f~V(ka r) x ×

f r drF.~ (Cq,Mt ) 2

72 MA fl~ r F~-2(2, f12 r)VANf~N(r)

h

- -

r d Fn~ ~ , Mvv fll r FI.~(2, f12 r)f~B(r),

where {~_

Ic

b

/

e} = ~/(2e+

d f

J

l)(2f+1)W(abcd; ef),

where W(abcd; ef) is a Racah coefficient and the f are the radial wave functions resulting from the partial wave decomposition of the bound state and continuum wave functions. Thus, rather than a difficult six dimensional integral, we now have an infinite sum of products of one dimensional radial integrals that are easily evaluated. The above expression is similar to the original expansion given by ST but the continuum and bound state wave function have exchanged position in the integrals. The present approach has been more successful in various calculations than the original expansion given by ST. We present the following argument as the reason for this greater success. In the original method the bound state wave functions are expanded in an ST expansion of harmonic oscillator functions of the continuum coordinates rat and r m. The region of exponential decay of the bound state wave functions, that is, large rAN and rNB, must be adequately represented in terms of the continuum coordinates. The bound state coordinates can be expressed as differences of the continuum coordinates. Thus large values of the bound state coordinates imply that one of the continuum coordinates is large or both are of at least intermediate value. Large values of r mean large values of n in the harmonic oscillator functions. The ST expansions require delicate cancellations between terms having large values of n in order to represent the decay of the bound state wave functions. This requirement for cancellation is a difficult numerical problem. In the expansions products of continuum wave functions are integrated over ray and r m. The slow de-

470

c.A. McMAFIAN AND W. TOBOCMAN

cay, r -1, of the continuum wave functions implies that the values of the integrals are sensitive to the large values of r, that is, sensitive to the region where the cancellations are required. N o w consider our present approach. The continuum wave functions are expanded in an ST expansion in terms of the bound state coordinates rAN and rsB. Because we have expanded in a limited region our representations are not valid at large rib and rAr. As before, large values of rib or rAF imply either rAN or rNB is large or both are of at least intermediate value. Thus the expansions for large values of rAN or rNB are not reliable. In this case these expansions are multiplying bound state wave functions and are integrated over rAN and rNB. The exponential decay of these wave functions reduces the contributions from large values of r where the expansions are unreliable and thus avoids the problems associated with the first ST approach. In our expansion for the transition amplitude A w a summation is required over the following indices: L, M., L, 2, n, ~, nx, n2, nx, n2, Ix, /2, I1, /2. The range of L, the number of partial waves included, is determined by the incident energy and the properties of the particles involved. The indices ~t, L and 2 are limited in range by triangle relations which connect them to E,j,j, I and I. The range of values required for n and ~ depends on the volume of configuration space which must be included in the integral which in turn depends on the range of VANtkAN and the range of tkNB. The indices n 2 and ~2 have upper limits of summation:

n2 < n+nx +½(l+lx-12), ~2 < ~+~x +½(i+lx-12). What remains then are infinite limit summations over ~x, nx, 11, Ix, 12 and/2. These indices belong to the harmonic oscillator functions that appear in the radial integrals together with the bound state radial wave functions. It is clear that these radial integrals will become very small when any of the above indices becomes very large. Thus we can expect that the sum will converge. However, the number of terms in this sum is very large. The number of terms grows rapidly as one increase; the number of partial waves (values of l) to accommodate heavier nuclei and higher incident energies. Thus we face the difficulty of accuracy loss due to the accumulation of round-off errors. It is therefore necessary to use double precision for calculating individual terms and also necessary to choose values of fix and fix which give the most rapid convergence. Our test calculations have been carried out for light nuclei and low incident energies to minimize the expenditure of computing time. Comparison with plane wave model calculations, distorted wave recoil calculations of others, and our own distorted wave no-recoil calculations demonstrates that o u r method works. It remains to be seen whether our method is practical for higher incident energy and heavier projectiles and targets.

DWBA INCLUDING RECOIL

471

3. Calculations We now turn to a discussion of the computer code utilizing this expansion method. The code is called H I - D R C (Heavy Ions-Direct Reaction Calculation). Typically, 20 terms in n and in ~ are adequate to give a good representation of the continuum wave functions. The wave functions were reconstructed from the harmonic oscillator expansion and these reconstructed wave functions compared favorably to the original functions. The convergence on n 1 and ~1 is more rapid because these harmonic oscillator functions appear in overlap integrals with a bound state wave function or a potential times a bound state wave function. For large n~ or fi~ the harmonic oscillator function oscillate rapidly at small r and have small overlap with a bound state wave function. Cut-off limits on these n~ and ~x sums are usually set at 15. Once limits are set on n, ~, nl and ~ the limits on l/2 and ~2 are determined. Rapid convergence on t h e / - s u m s is assured because the large angular m o m e n t u m barriers in the harmonic oscillator functions associated with high lvalues produce small overlap with the bound state wave functions. This is especially true of the integral involving the product of a bound state wave function and the interaction potential. Judicious choices of the parameters fl, fl, ill, ill, ~, ~, ~1 and ~1 make the procedure converge more rapidly and the procedure is stable against variation of parameters provided the cut-off values of the n are not set too low. I.O

0.1 P L A N E WAVE X=O -EXACT H I - DRC

.ID

-

-

-

OIOI

0.001

0.0001L o

IO

30

20

• 40

I 50

8c.m.

Fig. 1. Calculated and exact angular distributions for the plane wave Born approximation.

472

C . A . McMAHAN AND W. TOBOCMAN

Several test reactions were studied in order to check the computer code and to examine the accuracy, stability and convergence of the expansion method. These test reactions consisted of three exactly soluble plane wave stripping reactions, three published finite range D W B A results using other computer codes and a single nucleon transfer reaction that was compared with a no-recoil calculation. The plane wave stripping cases were provided by an ~-transfer reaction M B = 13(M I = 17, M A = 13)M v = 17 at an incident energy of 12.25 MeV and with Q = - 2 . 4 MeV. Harmonic oscillator bound state wave functions were used so an analytic expression for the D W B A stripping amplitude was available for comparison with the computer code results. The first case was an angular m o m e n t u m transfer zero case and the comparison of the calculated and exact angular distributions are shown in fig. 1. The continuum wave functions were expanded in 20 harmonic oscillator terms while an upper limit of 15 was used on n t and if1. The results shown contain 10 partial waves and 25 terms in the intermediate/-sums. 1.0

= ~

,

i

I

o.l-x:2 ~

"~\

-

EXACT

--

HI

- DRC

\\,

b

0.01

I

0

IO

1

I

20

30

I |

40

50

8c.m

Fig. 2. Calculated and exact angular distributions for the plane wave Born approximation.

Examination of the expression for AIF shows there are several l which are summed, namely, 11,12, Ix, and 12. These l are not identifiable with any of the physical quantum numbers of the reaction although their range of values is restricted through triangle inequalities with the physical quantum numbers. We call the contribution to A~v from one set of l x , l 2, 11,12 satisfying the triangle inequalities one/-term. Thus we used 25 such terms to compute each partial wave. The computer code is re-

DWBA I N C L U D I N G RECOIL

473

startable for the sum on the/-terms so it is not necessary to recompute the previous /-terms when increasing t h e / - s u m cut-off to obtain convergence. In usual D W B A calculations we have an interaction potential of a Woods-Saxon form multiplying the initial bound state wave function. The largest contributions come f r o m / - t e r m s involving low values of l 1 and 12. Convergence in these cases is very rapid. In the plane wave cases we have instead of a Woods-Saxon potential a harmonic oscillator potential which increases as r z. This diminishes the dominance of low values of 11 and lz and accounts for the large number (25) of/-terms needed. The next two plane wave cases involve angular m o m e n t u m transfer two. In the first case the initial bound state has l = 0 and the final bound state I = 2 whereas in the second case l = 2 and I = 0. The results of these calculations are compared to exact angular distributions in figs. 2 and 3. The calculated angular distribution for the second case is normalized to the exact value. This requires a spectroscopic factor

\\

"e b

k F

PLANE WAVE ~=2 Z=O

"u

I

X=2

~

.01

- -

EXACT

---

HZ-DRC

l

0

IO

i

20

3O

40

50

60

Ocm

Fig. 3. Calculated and exact angular distributions for the plane wave Born approximation.

of 0.85 rather than 1.0. The first case contained 40/-terms and the second contained 50/-terms. This poor convergence is due to the cause mentioned in regard to the angular m o m e n t u m transfer zero case. In cases such as this it is the slow convergence that makes the final result poor in quality due to an accumulation of round-off error and this slow convergence provides a definite indication of when the results cannot be trusted. However, the angular distribution is very close to the correct distribution. This seems to be a general result. The angular distribution takes the correct shape with very few terms and this might be exploited for special application like choosing optical potential parameters. We now examine the reaction 12C(6Li, d ) 1 6 0 6 . o 6 a t ELi = 18 MeV which was

474

C.A. McMAHAN AND W. TOBOCMAN I

~.

I

E

~

I

I

I

I

I

I

2 12 C ( 6 L i , d ) 1 6 0 6 06 1.0~18 MeV X =' 0 5~-~(. - - H I - DRC [ ~'. - - - KUBO a HIRATA 2 I- /"... •....... NO-RECOIL '~./~

_-

......

.01

I 0

I 40

I 80

I

I 1 120

I 160

Oc.m. Fig. 4, Comparison of the calculated angular distribution using H I - D R C with the results of Kubo and I-[irata.

5 -'

' 1 ' 6 0 ( 7 L' i ,

2=t~ 103-~\ '.. 5~*' ~. ~°°"

2-

'

t) 2L O ' Ne ' E = 16 M e V X= I H I - DRC KUBO a HIRATA ......... N O - R E C O I L -

io,~

o

[

o

I

0

I

I

40

I

I

"..,

I

80

I

120

o

1

~1

160

Ocm Fig. 5. Comparison of the calculated angular distribution using H I - D R C with the results of Kubo and I{irata.

analysed by Kubo and Hirata 7). Our results are compared to those of Kubo and I-Iirata in fig. 4. Also presented is the no-recoil result. The limits on the n are the same as the plane wave cases except that the upper limit on ~t was increased to 20. We find convergence after 12 /-terms. We now compare our results to an angular momentum transfer one case also taken from Kubo and Hirata. This is the reaction 1 6 0 ( 7 L i , t)2°Ne at ELi ---- 16 MeV. These calculations are compared in fig. 5. Limits on the program are the same as before except 15/-terms are used.

D W B A I N C L U D I N G RECOIL I001 k [~

OJ 0

I

~

I

475

L

32S ( d , 6 Li)28Si E d : 21 MeV X: 0

I

I

I

I

I

~3

(50

90

120

150

180

ecrn. Fig. 6. Comparison of the calculated angular d i s t r i b u t i o n u s i n g I - U - D R C

,/ -~

o/ 40

60

lOB (14 N I, 3N)II B

E:,o8 M eV ~ H I - DRC

//

80

with the results of Smith.

I00

120

140

x:z

L 160

180

8cm Fig. 7. Comparison of the calculated angular distribution using H i - D R C calculation.

with

the

no-recoil

Fig. 6 gives our comparison with the results of Smith s) for the reaction a2S(d, 6Li)2sSi at Ed = 21 MeV. Our results are normalized to Smith's results at the first peak since no spectroscopic factor was published. We find agreement except

476

C. A. M c M A H A N AND W. TOBOCMAN

for a small discrepancy at forward angles and a large disagreement at backward angles. The destructive interference of partial waves at back angles would magnify any errors in either calculation. Our final test case was the single nucleon transfer reaction 10B(X4N, X3N)I ~B at 10.8 MeV. This case was examined using the no-recoil approximation by Schmittroth et al. 9). This case involves angular momentum transfer two. Recoil effects are not expected to be important for this reaction so our results are compared to the norecoil results. This comparison is shown in fig. 7. As was expected recoil effects are not important. Inclusion o f recoil produced only more back angle peaking. The number of/-terms included in the calculation is 15. TABLE 1 Computer time required for calculations Reaction

2

Number o f partial waves

Number of /-terms

Time (min)

Plane wave 1=/=0

0

10

25

20.8

Plane wave 1=0,/=2

2

8

40

30.8

Plane wave 1=2,/=0

2

8

50

38.4

12C(6Li, d)1606.o6

0

9

12

8.6

160(7Li, t)2°Ne

1

12

15

39.4

32S(d, 6Li)2aSi

0

12

20

21.6

l°B(t 4N, 13N)tIB

2

8

15

20.8

~. is the angular momentum transfer. See text for meaning o f number of/-terms.

Finally we should include a discussion of the computer time required when performing calculations using the code HI-DRC. As is the case with calculations of this type the time requirements are large but within the range of usefulness. All of the calculations presented in this paper were performed on a Univac 1108 and the times required for the various calculations are presented in table 1. The required time increases with angular momentum transfer because of the need to compute more intermediate/-terms since more such terms are allowed by the triangle inequalities. Also we see large time requirements for the plane wave cases because of the previously discussed slow convergence. For some of the distorted wave cases a lower cut-off on the/-sums could have been used but the calculations were continued to higher values to verify convergence.

DWBA INCLUDING RECOIL

477

4. Summary We have developed a computer code, H I - D R C , to calculate cross sections for particle transfer between heavy ions using finite range D W B A including recoil. This code uses an expansion method which reduces the evaluation of a six dimensional integral to an infinite sum of products of one dimensional radial integrals. This method was found to be accurate, stable and converged rapidly enough to be useful as a calculating technique. At the present time the computer time required is large but we believe improvements in the computer code and experience in choosing optimum parameters will reduce the required time. Work is proceeding in this direction. The tests of the calculation were carried out for cases where the incident energies were low and the nuclei involved were light. Applications to heavier nuclei and higher incident energies are being undertaken. For such cases accumulation of round-off error may destroy the accuracy of the numerical calculation. The authors are grateful to R. Siemssen of Argonne National Laboratory and S. Kahana of Brookhaven National Laboratory for arranging use of computer facilities for some of the testing of this computer code.

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

R. M. DeVries and K. I. Kubo, Phys. Rev. Lett. 30 (1973) 325 N. Austern, R. M. Drisko, E. C. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3 B. F. Bayman, Nucl. Phys. A168 (1971) I D. Robson and R. D. Koshel, Phys. Rev. C6 (1972) 1125 T. Sawaguri and W. Tobocman, J. Math. Phys. 8 (1967) 2223 W. Tobocman, Theory of direct nuclear reactions (Oxford University Press, London, 1961) K. I. Kubo and M. Hirata, Nucl. Phys. A187 (1972) 186 W. R. Smith, Phys. Lett. 34B (1971) 252 F. Schmittroth, W. Tobocman and A. A. Golestaneh, Phys. Rev. C1 (1970) 377