Approximate recoil approach to heavy-ion induced one-nucleon transfer reactions

Nuclear Physics A242 (1975) 406-422; ~

North-Holland Publishing Co., Amsterdam

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

APPROXIMATE

RECOIL

APPROACH

TO HEAVY-ION INDUCED ONE-NUCLEON

TRANSFER REACTIONS

W. REISDORF* Laboratoire Ren~-Bernas du Centre de Spectrom~trie Nucl~aire et de Spectrom~trie de Masse, Orsay, France

Received 30 August 1974 (Revised 25 November 1974) Abstract: It is shown that a distorted wave Born approximation theory combining the Hankel function

approach for the form factor with a linear recoil correction gives satisfactory results when compared to exact finite-range with recoil calculations for a large variety of heavy-ion induced one-nucleon transfer reactions. The test cases presented involve incident projectile energies in the vicinity of the Coulomb barrier as well as energies several MeV/nucleon above the barrier, strongly absorbing optical potentials as well as recently used more transparent potentials, and transfer with large recoil corrections. The exact calculations are generally reproduced with an accuracy of the order of 10%. Computing times are a factor 2+0.5 larger than typical times needed for a no-recoil calculation. The limits of the method are discussed and the importance of the tails of bound states is stressed. I. Introduction

In the past years strong interest has arisen in heavy-ion induced transfer reactions. A very large n u m b e r o f data were obtained a n d first steps in the quantitative interpretation were taken. It was soon f o u n d that a p p r o x i m a t i o n s which are often used in light-particle induced transfer reactions, such as the zero-range distorted wave Born a p p r o x i m a t i o n ( D W B A ) neglecting recoil terms (due to the fact that the transferred mass is not equal to zero), c a n n o t be used in heavy-ion induced transfer reactions. A n u m b e r o f theories were developed to o v e r c o m e these difficulties. Sawaguri and T o b o c m a n 1) developed m e t h o d s to p e r f o r m full finite-range calculations, but in the first applications 2), recoil continued to be neglected. It was not until exact finiterange calculations including recoil ( E F R ) were p e r f o r m e d 3, 4) that some essential features o f existing transfer data were explained: it was f o u n d that recoil effects can, in some cases, change the predicted cross sections by an order o f m a g n i t u d e and, m o r e strikingly, can also influence the overall appearance o f the angular distributions o f the transfer products. Thus, for example, in a n u m b e r o f transfer reactions the angular distributions are f o u n d to be structureless 5) in complete contrast to the predictions o f no-recoil D W B A calculations. The inclusion o f recoil in the theory was shown to resolve this discrepancy with the data 4). One d r a w b a c k o f exact Also at Gesellschafl ffir Schwerionenforschung mbH., Darmstadt, Germany (present address). 406

APPROXIMATE RECOIL APPROACH

407

calculations was of practical nature: prohibitive computing times were needed. Another perhaps less serious drawback is the difficulty one has to understand intuitively the workings of the exact theory for simple interpretive purposes. Thus a number of approximate theories were soon developed. A review of most of these approximate theories together with a comparative evaluation of their different merits has been given elsewhere 6), and some of these methods will also be mentioned later in the text. The main purpose of this work is to present an approximate approach to transfer reactions which is determined by the philosophy of obtaining reasonable accuracy for the predictions while keeping maximum simplicity of both the theory and the computational techniques. We have chosen two approximations to simplify the exact theory: first we apply the Hankel function approximation to the final bound state of the transferred nucleon, a method introduced in 1965 by Buttle and Goldfarb 7), and which allows us to obtain an elegant and simple solution to the finite-range problem, avoiding multidimensional integrals; and second, we use a linear recoil approximation, i.e. terms to first order in ran~me will be taken into account exactly. Here m, is the mass of the transferred nucleon, mc the mass of the donor or acceptor core. A formalism using both of these approximations has been given in ref. 8). In sect. 2 we present the theory in a slightly different form, which is suitable, firstly, to show the modifications which have to be introduced into standard zero-range, no-recoil DWBA programs, secondly, to discuss coherent and incoherent contributions of recoil to the no-recoil amplitude, a question where we draw conclusions differing somewhat from those in ref. s); and thirdly, to show the possibility of generalisation to two-nucleon transfer. In sect. 3 we show some applications of the method, presenting, above all, limiting cases: transfer with large recoil effects and relatively transparent optical model potentials. The success of the Hankel function method implies an important physical conclusion: the paramount and almost exclusive importance of "nuclear tails" in transfer reactions. 2. Theory 2.1. LINEAR RECOIL APPROXIMATION

We shall consider the one-nucleon transfer reaction (Cl + n ) + c 2 al

--, c l + ( c 2 + n

).

a2

If we assume that cores cl and c2 are inert, first-order DWBA theory predicts the following for the transition amplitude in the post representation:

T~p = S~S~(J~,M~ljl ml IJ~ M~)(J~: M~2J2 m2lJ~ M J x fdrdr 1X~-)*(r~,KB)dp'~(r+ rl)V~,(rl)d?l(rl))~+)(r~,K~), d

(1)

408

W. R E I S D O R F n

c2

Fig. 1. Coordinates used in the transfer theory, where cs and c2 are the cores, and n is the transferred nucleon.

where q~(ri) is a single-particle state with quantum numbers (hi, li,j~, m0, Vc, is the interaction of the transferred nucleon with the core Cl, X~-) and X~+) are the outgoing and incoming distorted waves respectively. The various coordinates are defined in fig. 1 and are linked by the following relations: mc 2 mn r# ~ - - r - - - - r l , ma z ma 2

r~

= r q- -mn - rl,

(2)

ma,

rl-{-r-

Ir2 =

The spectroscopicamplitudes are defined by: S~ - f ~ba,(~Cl, rl)[c~o,(~,)d?l(rt)]'a'Ma'd~,drt,

fqSa,(~ , re)[c~c~(~Jdpe(r2)]sa~Ua~d~¢zdr2, [

o,

12

lr

=

~

:

m

,r

M

~kdc~Mc~'fijtml

.-i

(3) ,

Ml,ml

where ~Cl and ~c2 are the core degrees of freedom. Using the abbreviation r' = (rn¢2/ma2)r, we make the linear approximations mn

~-)*(r#, g#) ~ X~-)*(r', K#)- - - rl grad,,g~-:(r', K#), maz

(4)

X~+)(r~, K~) ~ X~+)(r, K~)+ mn rlgrad, z~+)(r, K,). real

The second term on the right hand side (r.h.s.) of eq. (4) will be called the "linear recoil term". "Non-linear recoil", which would be represented by higher-order terms in the Taylor expansions of the distorted waves around r' or r, is ignored. The no-recoil approximation includes the first term on the r.h.s, only.

A P P R O X I M A T E RECOIL A P P R O A C H

409

For Z~+) and Z~-) we introduce the partial wave expansions

X~+)(r,K~) = 4n E iL'e'°L"fLf(K~, r)Y~*.(I~JY~'(~), K ct Lo~M,~

4reL#u. E i-L"ei°LafL,fK #, r)YLU#*"f[~#)YLU#'fr"), Z~-)(r,K#) = K#

(5)

and use for the evaluation of the recoil terms the following relation which can be derived by introducing spherical coordinates:

rx grad,{ fL(r)Y~(~) } = ra ~ gLL,(r)(EM'lpILM)Y~(~x)Y~,(~),

(6)

L'p

where L' = L - 1, L + 1 and # = - 1, O, + 1 and (fL - rfL)

lC

gLL-1 = r kL fLr + ~-r ]k,2(2~~] gLL+I

- - r1

(

(L+I)

fL

OfL~( 2 L + 2 ~r/\~]

k-3-]' ~½:4n~½ k,3]"

The integral in eq. (1) can then be written

I~#= fdrz~-)'(r ',K#)F12(r)z~+)(r, K~)

fdr roof mn

ma 2

-t-

ma2

rz~;)'(F, K#)F~;(r)]x~ +)(r, K~)

drz~-)*(r ', K#)E [F12(r)z~. " (+)(r, K~).

(7)

p

The first term corresponds to the no-recoil approximation, the other two terms represent recoil in the outgoing and incoming channels respectively. The definition of eq. (7) is completed by the following expressions for the recoil distorted waves X~ )*, X~ ), the no-recoil form factor F12 and the recoil form factor F¢2: Z(+)/r •~, ,-, K~) -- ~4rr E iL~eioL~E gL,L~(K~, r)Y~*(I~)YL~'.(~XE~M'~1/~[L~M~), L~M,~

L'~

4~ V i-L~d °L. ~,gL~L'~ (K#, r')Y~}(I~p)YL~'~(~)(£pM'plpIL#Mp), Z~)*(r ', K#) = K# L~-~p L~

(8)

¢1

F12(r ) = / d r x ~b~(r1 + r)V~l(rl)~bx(rl), F~2(r) = t d r l ~b~(rl + r)Y~(rl)cPl(rl)rl Y~( ~1)J

(9) (10)

410

W. REISDORF

As will be shown later, the form factors can be written in the following forms, if one uses Hankel function approximations for the radial part of the bound state in the final channel 7):

(l~.jl ml lJ2 m2)Yta*(~)i- 112ft(r),

(11)

(lAj~mllJ2m2) ~ (-i)-"-l(lld'2'll2)Yr~'*(~)~2fw(r ).

(12)

F12(r) = ~ 12

F~2(r) = ~ 12

l'A'

We note here that the equivalent expressions for two-nucleon transfer can be written in the same form as eqs. (11) and (12). This will be shown in more detail in a later publication. The subsequent equations in this subsection are therefore also valid for two-nucleon transfer. Using eqs. (8), (11) and (12) and some Racah algebra, one obtains

T~p = S~S~(J~IM~,jxmxIJ~ Ma~XJ¢2M~2j2m2IJaeM~2M~) ~ (12jlml[j2m2)t t~,

(13)

la

where t tz = tn. tz + t a, lz + t~.tx and (l, 2) is the transferred angular momentum. The first term represents the nO-recoil amplitude with a form familiar to that from zero-range DWBA theory (we write ~ = (2x+ 1)½): t ~ z - K, (4n)Ka ~

~" eiaL~,+iaLP~£ f l iL'~-La-t(LaOIOILc,O)R~,,Lp Za x ~ (Lz-Mal21L~M~)(-)z+M.Y~Z(g,)Y~(I~a)(-) L~+*+L*,

L~,La

(14)

M~M#

where

RtL=L# = f fLtj(Kp, r') 1 2Z(r _) fL.( K ~, r)r2 dr. The recoil amplitude in the fl-channel becomes

eiaL=+iaLpLa'~ K~Kp L~L# ~ L.

,4 _ (4n) ~

ta"

×

-- ~

Z iL'-LP - r - 1( _ ) r £ , f l , ( E a 0 1 , 0 l L ~ 0 ) W ( L a I'L'#

1Ea l';

Lal)RI~.LpL,

× ~ (La--Mal21L~M~X.--)~+M'Y~I(g~)Y~(gaX--)L~+'+L',

(15)

MeM~

where

R'~'.L.L', = f g.L'.(Ka, r') ' 2I,,,(r)L.(K., r)r2dr. From eq. (6) we know that the summation over L~ for a given outgoing orbital

APPROXIMATE RECOIL APPROACH

411

momentum Lp is limited to L a + 1. The recoil amplitude has been written in a form as close as possible to the more familiar term eq. (14). A comparison of eqs. (14)and (15) shows which modifications have to be made to existing zero-range DWBA programs, such as DWUCK9), to include recoil corrections. The equation for ta~ is obtained by interchanging 0e,--, ~ (except that iL',--, i -L") and substituting m.

for

mn

ma I

ma2 "

Finally, the cross section is obtained using

(13) and

/i~## Kaj~Ej~]2 E IT~al2. try# -- (2rCh2)2 K~ MaIM¢I

(16)

Ma2Mc 2

After carrying out the summations in (16) this can be written #~/~a Ka ,-'2 ~-2--2 0"~# -- (2/th2)2 ~-Ja2Je2 J1 --~

E T-21tUl2"

(17)

l)`

Here #~ and K~ are the reduced mass and wave number in channel ct, and the summation is over all transferred angular momenta I compatible with the triangular conditions (jlj2 l) and (1112l). 2.2. THE FORM FACTORS IN THE HANKEL FUNCTION APPROXIMATION The transition from eqs. (9) and (10) to (11) and (12) is performed as follows. The bound state wave functions are (i is the intrinsic spin) ~bl(rl) -- ~ (ll 21 il vlljx ml)ut~(rl)i l' Yt~I(~)Z]'~, ),tvl

and similarly for t~2(r2). We make a Hankel function approximation for the radial part of the final bound state,

utz(r2) ~ N 2h~)(iK2 r2).

(18)

This approximation allows us to make a very useful expansion 7):

u~(r2)Yl~(r2) ~ (4~) ½• (_i)12+l_rr2~ ,r

l'~(l 0 o}\2 22 - 2 '

(-Y'+~Y~a(~)Y~'~'(~3

),2'

x [N*hll)*(ix2r)jr(ix2rl)],

(19)

which is valid for r < rl. For r > rl the expression in the square brackets is replaced by

[ N* j*( i~c2r)hl,1)'( i~2 r ~)].

412

W. R E I S D O R F

The Hankel function approximation is meaningful only for the outer tail of the radial amplitude ul~(r2) which is evaluated by numerical integration of the corresponding bound state Schr6dinger equation. We use central, single-particle potentials of Woods-Saxon shape with a standard spin-orbit term. The constants N2 and r2 are matched to the tail of u~(r2) by a least squares fitting procedure. We shall come back to this point in sect. 3. The integration over the coordinate space rl, for a given value of r, can now be performed without difficulty. After some Racah algebra the no-recoil form factor can be written in the form of eq. (11), with

12ft(r)=12H~(rX4~)½(_)il+h+,l+t{Jl12 J211 ill}(l'Ol201llO)~2"fffl'

(20)

where 12Ht(r) = N2 hll)(ix2 r)~ojl~(ix 2 rl)ul~(rl)V~,(rl)r 2 dr 1

+ N2 jz(ir.2 r)~hz,(ix2 rl)ul~(rl)~(rl)r 2 dr1. In the most frequent case il = ½, and eq. (20) simplifies to 12ft(r) = 12H,(rX4~)~)" 1)'2 1(Jl ½101J2½)T.

(21)

This expression is well known since the work of Buttle and Goldfarb 7). A similar procedure is used for the recoil form factor. In addition one uses

yt~lvu , -1

=

~(3)~

Tll'I-

l(1101011,lOX11211pll,12,1)yl~,,

and obtains for 12fH, in eq. (12)

12fu, = 3½)'1Tl"[Z(-}il+J~W(lj112i 1 ;jz11)'[(- )t × ~ i'~-q+l~(l'Ol2Oll'~O)(llOlOll'~O)W(ll ll21'; l'~l)12G*,,, r~

(22)

with

12Grtl _~N2 h}l)(i/¢ 2 r

li(iI~2 rl)ul,(rl)~l(rl)r adr 1 + N2Jr(iX2r)

hI~)(ix2rl)utl(rl)V~(rl)r~ drl •

We close this section with a few comments. Both expressions (20) and (22) imply, aside from some angular momentum coefficients, the evaluation of spherical Hankel and Bessel functions and a one-dimensional quadrature. Thus the computation of the form factors is both simple and fast. A look at the various angular momentum

APPROXIMATE RECOIL APPROACH

413

coefficients shows that recoil has introduced contributions from "non-normal" /-transfers (ll +12 + l = odd), which add incoherently to the no-recoil terms (see eq. (17)), and can be traced back to the angular part o f grad Zt±~ in eqs. (4). They represent "transverse recoil". Linear transverse recoil is approximately evaluated in the local wavenumber approach o f Nagarajan 8), but is ignored in the simple Buttle-Goldfarb approach lo). There are also recoil contributions coming from the radial part of grad Xt ±~ ("longitudinal recoil") at "normal"/-values (ll + 12+ l = even), the only allowed ones in the no-recoil treatment. They must be added coherently to the no-recoil amplitude and the transfer cross section is therefore often found to be particularly sensitive to comparatively small longitudinal recoil amplitudes. Many cases exist where longitudinal recoil is important and transverse recoil negligible, but the reverse is hardly ever true. The approach in ref. lo) reproduces longitudinal recoil effects rather well [see ref. 6) for more details] as long as the differential cross sections are peaked at or close to 180 ° (i.e., for transfer below or in the immediate vicinity of the Coulomb barrier). In ref. a) it is concluded that first-order recoil theory leads to angular momentum transfer of parity opposite to the no-recoil term, and that non-linear recoil effects can therefore be estimated by comparing the norecoil amplitude with the normal angular momentum transfer amplitude calculated with an exact theory. Our comments above show that we do not agree to these conclusions since longitudinal recoil contributes to normal transfer even in first order. The quantity defined by l in ref. a) has a different significance for the no-recoil terms, where it is identical to our l, i.e. to the transferred angular momentum, and in the recoil terms where it corresponds to our quantity l', which cannot be directly identified with the transferred angular momentum. 3. Calculations and discussion

In most o f the calculations we have concentrated on cases where it was possible to compare our results with those obtained from more elaborate calculations 4, 6,11). In particular we define the approach o f DeVries 4) as "exact". 3.1. MATCHING RULES If one aims at a 10 ~ accuracy of the predicted cross sections the parameters N 2 and x2 in eq. (18) have to be chosen such that the Hankel function approximates the bound state wave function u2(r~) within 5 ~o over the range of r2 values important for the transfer amplitude. A least squares fit o f that quality can only be achieved over a limited range o f r2 values since (a) at r2 = r2 ..... the radial wave function will change the sign o f its curvature and rapidly deviate from any Hankel function as r2 < r2 .... (ra .... is generally quite close to the radius o f the single-particle potential), and (b) for large values oft2 a limit is given by the fact that for bound protons the wave function asymptotically approaches a Whittacker function, rather than a Hankel

414

W. R E I S D O R F

function (for neutrons this limitation does not exist). It is therefore necessary to establish reasonable working rules as to which range of r2 should be matched with a Hankel function. Taking plots of calculated total transfer cross section versus rmax, the upper integration limit in r-space, as guidelines, we recommend the following matching rules: Above the Coulomb barrier the r-integration should be performed out to at least rma x = r l n -Jr-r2n -q- 4.6(~-

1 + x~ 1),

(23)

F¢1 = K1-.I-a - 1 .

Here r~n is the radius o f the bound state potential, tq is the effective inverse wave length of the bound state, approximately given by x~ = 0.219(Berf#)~,

(24)

with Bee f ~,

B i + Ze, Zn/m~,,

where # is the reduced nucleon mass in a.m.u., Z , its charge (zero for neutrons), Zc, is the charge of core ci, B~ is the nucleon binding energy in MeV, and a is the diffuseness (a standard value is 0.65 fm) of the Woods-Saxon interaction Ve,(rO. The above minimum value for rmax insures generally a 1% stabilisation of ate against further increases o f the integration limit. Furthermore O'tr(rmax) plots indicate that the asymptotic value of ate is reached within + 5 % at approximately r = 0.Brma x.

If one wants the overlap calculation to be still reasonably accurate for this core-core distance, the wave function u2(r2) should be well fitted out to approximately r2 max :

0"8rmax - - r l n q- K 2 1.

(25)

Below and in the immediate vicinity of the Coulomb barrier one should replace Rn = rln + rzn by Re, a radius which is correlated to the Coulomb distance of closest approach d. For projectiles in the mass region 10 < A < 40 and single-particle radial parameters ro close to the standard value of 1.25 fm, the transition from Rc to R, is done fairly smoothly at the Coulomb barrier if one chooses Rc = 0.85d.

(26)

In cases where the Q-value of the reaction is such that d is very different for the outgoing and the ingoing channels the larger of the two values should be preferred. Small deviations from the above rules forr2m~x(+ 20 %) are not critical. Fig. 2 shows some plots of ate versus the upper integration limit, plots which helped us establish the above matching rules. For strongly bound protons (in the 11B.~2C system, see

APPROXIMATE RECOIL APPROACH

1.5

/~-~

415

NB(12C'lB)12C

!1.o ,

2,/,, 6

8

|

*

i

10

i

12

rrt~ (~)

-" t

i

14

t

1(}

Fig. 2. Variation of the total transfer cross section as a function of the upper integration limit rmax. Solid curve: l = 2 part of the reaction 11B(12C, 11B)12C at 87 MeV. Dashed line: l = 1 part of the reaction 12C(14N, 13C)13N at 78 MeV. u2 15

zsD

10

r2

5

i -10

l r, .1.26 fm a -,O.8S fm

-15 i I

*

i

Fig. 3. Percentage deviation of Hankel function fits to the ld~ bound proton state wave function in 31p calculated in a Woods-Saxon potential. Fits were obtained in the range from r z m i , to 14 fm, where r2mi. = 4.4 fm for the solid curve, and 3.4 fm for the other curves. The dashed curve is a two Hankel function fit. The origin of the abscissa is the radius of the outermost maximum (2.6 fm) of the wave function as shown in the schematic inset and the vertical line marks the location of the zero curvature point r2 . . . . (3.9 fm).

fig. 2) t h e t r a n s f e r r e a c t i o n is r e m a r k a b l y l o c a l i s e d i n r - s p a c e a n d c o n s e q u e n t l y a l s o in angular momentum

space.

F o r s m a l l v a l u e s o f t 2 w e h a v e f o u n d t h a t fits w i t h i n t h e d e s i r e d 5Yo l i m i t c a n a l m o s t a l w a y s b e o b t a i n e d i f r2min ~ r2 . . . . + 0 . 5 f m . F i t s i n c l u d i n g s m a l l e r d i s t a n c e s o f r2 will a l w a y s b e a t t h e e x p e n s e o f t h e a c c u r a c y a t l a r g e r d i s t a n c e s . T h i s is i l l u s t r a t e d i n fig. 3. A s is a l s o s h o w n i n t h e s a m e f i g u r e , it is p o s s i b l e t o e x t e n d t h e l i m i t o f 5 ~ fits t o

416

W. R E I S D O R F

approximately r2curv-0.5 if a two Hankel function fit is done. This is possible with a minor increase in computational effort and time. The use of a second Hankel function rE), though very attractive at first sight, was found to be rewarded by a modest improvement only. Fits for r 2 < r2eurv- 0.5 are not necessary for the large majority of existing data, as will be shown later. As a conclusion, when one Hankel function is used, we choose r2 rain = r2 . . . . +

0.5,

(27)

or r2

rain

=

d

m~2

m~l + m ~ , '

whichever is larger. 3.2. T R A N S F E R IN THE VICINITY OF THE C O U L O M B B A R R I E R

The approximations introduced are expected to work best at energies below, or close to, the Coulomb barrier for two reasons: (a) the Coulomb repulsion keeps the two cores far apart justifying the Hankel function description of the tails of the bound state wave functions, and (b) recoil effects, though not negligible, are generally less important than at higher incident projectile energies, justifying the linear recoil approximation. Since a detailed numerical comparison of various approximate DWBA calculations, including the one described here, with exact recoil plus finiterange results has been given elsewhere 6) we will mention only results for the experimentally well studied 13,14) proton transfer reaction 88Sr(160, tSN)s9Y to the p~ ground state of 89y induced by oxygen ions incident with lab energies from 42.5 to 59 MeV. Table 1 displays the products of spectroscopic factors S~,Sa deduced at four different energies (corrected for target thickness) from the data of ref. :3). The TABLE 1 Spectroscopic factors extracted from the data of ref. 13) for the ground slate transfer reaction 8SSr(160, lSN)S9yb Projectile energy (MeV) a)

Maximum in angular distribution (deg)

45.75 47.75 51.75 55.75 58.75

exp.

theory

180 134+2 98 + 2 80 _+2 70+2

180 138 100 81 71

S~,Sp

4.70 4.95 4.60 4.65 4.81 average = 4.74 + 0.15

a) Corrected for target thickness.

APPROXIMATE RECOIL APPROACH

417

products S~ Sp are found to be independent of the projectile energy within 3 ~ in contrast to a more than 1 0 0 ~ variation if deduced from a no-recoil code13). Assuming S~ = 2, we find Sp = 2.35 +0.06 which is in good agreement with exact calculations 6). 3.3. TRANSFER WELL ABOVE THE COULOMB BARRIER, BUT WITH STRONG ABSORPTION: NON-LINEAR RECOIL

While at very low energies the Coulomb barrier was keeping the two cores apart, we now use the fact that in many reactions, whenever the two cores move to separations where the Hankel function approximation should break down, they are absorbed into non-transfer, presumably fusion, channels. Thus again transfer takes place at relatively large core distances. In fig. 4 we show results from two reactions well above the Coulomb barrier which have been previously calculated 4). We have kept all the optical model and bound state parameters of the exact calculation. As was pointed out at the time by DeVries, recoil was an important feature both quantitatively and qualitatively. In the reaction 12C(14"N, 13C) only the addition of recoil leads to a structureless angular distribution in accord with experimental data 5). We reproduce all these features of the exact calculation very well. We find, in particular, the same relative importance of various contributions corresponding to different transferred angular momenta including the non-normal transfers (! = 1 in both reactions). We also find in complete agreement with ref. 4) the somewhat unusual result of an about equal contribution from l = 2 and l = 0 in the 11B_12C system. Overall our results were about 20 ~o in excess of the exact calculation. However, if one compares with the no-recoil result (see fig. 4) one can see that agreement with the exact calculation (and the data) has improved by an order of magnitude. It is interesting to note that similarly successful fits to the same transfer data have 10

.

.

.

rlO

.

"e( cYe) ~

I

,

~ 1

o.o~

]

16 ~

30 ~

I

I

~Ol

ff~l

,~0 ecrn 10 20 30 40

Fig. 4. Calculation ofthevariouscontributions to the reactions 11B(12C, llB)12Cand 12C(14N, 13C)13N. The non-normal transfers have l = 1 (dashed curves).

418

W. REISDORF

been obtained recently by Braun-Munzinger et al. 26) using the local wavenumber approach to recoil. Since absorption is quite strong in these reactions, the most probable source of error is expected to originate from the linear recoil approximation. It is rather surprising that recoil effects of the magnitude shown in fig. 4 should be so well reproduced by a simple linear approximation. Undoubtedly, non-linearities in recoil must exist. This problem has also been discussed by Baltz and Kahana 11), and they have given rough estimates of second-order effects. In a more recent development xx) they have extended the Taylor expansion in eqs. (4) to second order, an extension which leads to considerable complication in the formalism. In keeping with our philosophy of greatest possible simplicity we will show here how, in the linear recoil approximation, one can attempt to minimize non-linear effects and test if they are important. For a well-behaved function x(r) and sufficiently small Ar there exists a vector such that one can write exactly z(r + Ar) = z(r)+ Ar grad, [z(r+8)],

with 181 ~ Ihrl. Generally, 8 is a function of r, but one can try to find an approximate value 8'. In the linear approximation one sets e' = 0. For oscillating sine-type functions it is easy to convince oneself that the choice 5'-1/2A7 will give a substantially better approximation. In our case m n

Ar

---- +

--

m~l

or

r I

m n

--

m~2

r 1.

The most probable magnitude of these "recoil shifts" can be estimated with the following assumptions: the largest contributions to the transfer amplitude are expected to occur (a) along the line joining the two core centers (i.e. 0 ~ 0 in fig. 1) and (b) within the range of the interaction Vxc in the post representation. Hence one can set 1

mn

2 real

r 71c

]r[

| or

--

mn r -71c

2 ma2

]r]

For the DWBA calculations this means that for a given core separation 7, the gradient in eq. (4) is not evaluated at r (or r') but at radially shifted values, 1 m.~ 7--

2

- -

real

tic ,

r'+

1 mn ~ ma2 rio,

in the entrance and exit channels respectively. We have essentially followed above the suggestions given by Buttle and Goldfarb 10) with an important difference: we

APPROXIMATE RECOIL APPROACH

419

use the approximations to correct the gradient o f the functions X~+~ and X~-) rather than the functions themselves. The assumptions used affect therefore a correction of higher order in the recoil terms. The usefulness of these approximations has been shown in ref. 10) by comparing post. and prior representation calculations, which were found to be in better agreement when applying these corrections. A quantitative justification can be done also by comparison with exact calculations. For the cases discussed in this subsection we have found a 10-15 ~ improvement when applying the "shifted gradient" technique. For cases with modest recoil corrections the results were not affected as one would expect. We remark that the shifts A r are quite small ( ~ 0.25 fm for the cases shown in fig. 4) and that one should use in the numerical evaluations integration step sizes smaller than 0.1 fm in such cases. The sensitivity of predictions in strong recoil cases is not surprising since the large recoil effects are precisely caused by a high sensitivity o f the transfer amplitude to comparatively small differences in the values of the vectors r~ and ra relative to r, and r'. The linear approximation is no longer justified if one finds that the shifted gradient technique modifies the results considerably. 3.4. TRANSPARENT OPTICAL POTENTIALS

One can imagine an optical potential with weak enough absorption that transfer will take place well inside the range o f core-core distances where the Hankel function approach breaks down. More relevant however is the question whether such transparencies are unambiguously required by existing data. Several types of transparency indicators have shown up in heavy-ion scattering data, the earliest example being the very shallow potentials introduced to reproduce the striking structure in the 160-160 elastic excitation function 16,17) and also used for the adjacent 160_ is O and 1sO-1 s o systems is, 20). More recent studies involve a number of transfer reactions among which we shall choose, as an example, #SCa(I#N, 13C) reactions at 50 MeV projectile energy 11, 21). Errors in the calculated overlap of initial and final bound states will start to b e c o m e important at core distances small enough that the region of r2 space becomes relevant in which the Hankel function fit to the final state is inadequate, i.e. approximately if r < rc with re = r l .... + r2 ..... When using two Hankel functions, the limit o f validity can be pushed about 1 fm further inside, which may be useful in a few limiting cases. F o r smaller core separations the Hankel function approximation strongly overestimates the correct form factor, a fact which can give rise to spurious "inside" contributions to the transfer amplitude if the steep monotonic rise of the Hankel function form factor (HFF) is not overcome by a correspondingly strong absorption of the distorted partial waves. In order to avoid these spurious contributions it is generally sufficient to keep it constant at the value calculated at ro. This flattening out o f the H F F for r < r¢ has the following advantage over the complete cut-off (setting H F F equal zero) : even though the correct form factor is

420

W. REISDORF

not well approximated for r < re, plots such as those shown in fig. 1 for the total transfer cross section as a function o f the upper integration limit can be used as rough indicators as to whether contributions from the region r < rc might be important. If they are, the Hankel function approach is not reliable. When using the unmodified H F F this indicator is unduly overestimated.

3'I E . .oaM,v

]

ZO

a

A illio

50

eem Fig. 5. The reaction 4SCa(14N, t3C)49Sc populating the 3.08 MeV state. Dotted curve, unmodified Hankel function form factor. Solid curve, the form factor is kept constant for radii smaller than 7 frd, the breakdown radius of the Hankel function approximation in this case.

We show in fig. 5 results for the *SCa(I*N, 1 3 C ) reaction populating the 3.08 MeV state in 49Sc. The dotted line represents a calculation allowing the H F F to rise indefinitely for r < re. The calculated function O'tr(rmax)indicates a 7 % contribution from the inside region which is almost certainly unrealistic and due only to a wild overestimate of the correct form factor in that region. When flattening out the H F F at r = re, we predict the full curve which is, in contrast to the dotted curve, in excellent agreement with the full finite-range calculation and the data ~1, 21). The contribution from the fiat part of the H F F is now less than 1%. Using a one or a two Hankel function fit to the final bound state did not make much difference (less than 10 %), convincing us o f the stability of the procedure. Our conclusion is: One can evaluate even fairly transparent cases with the Hankel function approach if one flattens out the H F F at approximately re = rleurv-+-1"2. . . . • The resulting predictions can be trusted if the calculated total transfer cross section integrated from r = 0 out to rc does not substantially exceed 1% of the asymptotic value (r ~ ~ ) . Finally, we comment on the shallow potentials used in the 160-16 0 region. If used as such, we find that our above criterion for the validity of the Hankel function approach is violated and these potentials are therefore not suited for this method. It remains to be seen if they are realistic potentials for transfer reactions. Full finite-

APPROXIMATE RECOIL APPROACH

421

range calculations done so far 22) indicate that such potentials give rise to strongly overblown transfer cross sections in disagreement with data. There are at least two procedures to modify these potentials while still keeping the relative transparency for grazing partial waves required by the elastic scattering data. Firstly one replaces the shallow imaginary part by a folded-type potential, as was done in ref. la). These potentials coincide at large radii with the original shallow potential, but are much more absorptive for small radii. Alternatively one introduces angular momentum dependent absorption 23) which allows the selective absorption of the lower partial waves. With these modifications the use of the H F F becomes again justifiable. 4. Conclusion

The Hankel function plus linear recoil approach to one-nucleon transfer reactions seems to be the simplest approach compatible with reasonable accuracy. We gave clear prescriptions on its use and showed how to test if the approximations are valid. The extension to two-nucleon transfer reactions is possible. Here the simplicity of the theory is an advantage. Computing times needed are only about a factor two larger than in any standard light-particle no-recoil DWBA code. Our interest in this field was first aroused during a very enjoyable stay at the Nuclear Physics Laboratory of the University of Washington in Seattle and in particular through collaboration in various experiments with Profs. R. Vandenbosch and J. S. Blair. We wish to thank in particular J. S. Blair for repeated criticism and for helping us eliminate a number of errors which had encroached into the earlier stages of this study. We are also indebted to L. Remsberg for reading the manuscript. References 1) 2) 3) 4) 5) 6)

7) 8) 9) 10) 11)

12) 13)

T. Sawaguri and W. Tobocman, J. Math. Phys. 8 (1967) 2223 F. Schmittroth, W. Tobocman and A. A. Golestaneh, Phys. Rev. C1 (1970) 377 N. Austern, R. Drisko, E. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3 R. M. DeVries, Phys. Rev. C8 (1973) 951 M. Liu, W. von Oertzen, J. C. Jacmart, F. Pougheon, M. Riou, J, C. Roynette and C. Stephan, Nucl. Phys. A143 (1970) 34 J. S. Blair, R. M. DeVries, K. G. Nair, W. Reisdorf and A. J. Baltz, Int. Conf. on reactions between complex nuclei, Nashville, Tennessee, 1974, ed. R. L. Robinson, F. K, McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam, 1974) p. 58; Phys. Rev. C, submitted (1974) P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 M. A. Nagarajan, Nucl. Phys. A196 (1972) 34; A209 (1972) 485 P. D. Kunz, DWBA code DWUCK, Department of Physics, University of Colorado, Boulder, Colorado P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. A176 (1971) 299 A. J. Baltz and S. Kahana, Phys. Rev. C9 (1974) 2243; A. J. Baltz, Int. Conf. on reactions between complex nuclei Nashville, Tennessee, 1974, ed. R. L. Robinson, F. K. McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam, 1974) p. 60 D. P. Bouldin and L. J. B. Goldfarb, Phys. Lett. 47B (1973) 203 N. Anantaraman, Phys. Rev. C8 (1973) 2245

422

W. REISDORF

14) K. G. Nair, J. S. Blair, W. Reisdorf, W. R. Wharton, W. J. Braithwaite and M. K. Mehta, Phys. Rev. C8 (1973) 1129 15) P. Braun-Munzinger and H. L. Harney, Nucl. Phys. A223 (1974) 381 ; P. Braun-Munzinger, H. L. Harney and S. Wenneis, Int. Conf. on reactions between complex nuclei, Nashville, Tennessee, 1974, ed. R. L. Robinson, F. K. McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam, 1974) p. 59 16) R. H. Siemssen, J. V. Maher, A. Weidinger and D. A. Bromley, Phys. Rev. Lett. 19 (1967) 369 17) J. V. Maher, M. W. Sachs, R. H. Siemssen, A. Weidinger and D. A. Bromley, Phys. Rev. 188 (1969) 1665 18) R. H. Siemssen, H. T. Fortune, A. Richter and J. W. Tippie, Phys. Rev. C5 (1972) 1839 19) R. W. Shaw, Jr., R. Vandenbosch and M. K. Mehta, Phys. Rev. Lett. 25 (1970) 457 20) R. Vandenbosch, W. N. Reisdorf and P. H. Lau, Nucl. Phys. A230 (1974) 59 21) M. J. Schneider, C. Chasman, E. H. Auerbach, A. J. Baltz and S. Kahana, Phys. Rev. Lett. 31 (1973) 320; C. Chasman, S. Kahana and M. J. Schneider, Phys. Rev. Lett. 31 (1973) 1074 22) K. I. Kubo, Int. Conf. on reactions between complex nuclei, Nashville, Tennessee, 1974, ed. R. L. Robinson, F. K. McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam, 1974) p. 13 23) R. A. Chatwin, J. S. Eck, D. Robson and A. Richter, Phys. Rev. C1 (1970) 795