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DWBA analysis of the heavy ion induced transfer reaction
Nuclear Physics A129 (1969) 625----646; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
DWBA ANALYSIS OF THE HEAVY ION INDUCED
TRANSFER
REACTION
TETSUO KAMMURI and HIROSHI YOSHIDAt Osaka University, Toyonaka, Osaka, Japan Received 18 July 1968 Abstract: The heavy-ion induced transfer reactions have been analysed using the finite-range DWBA. The following reactions are treated for incident energies near the Coulomb barrier: X4N(14N,XSN) 15N, 12C(~eO,~sO)x2C, nB(l~O, a6N)12C and 27A10eO, I~N)~sSi. In most cases the transferred particle is assumed to be bound in a harmonic oscillator potential well. Part of the calculations has been performed for a simple shell model with the Woods-Saxon potential. Good agreement with experiments can be obtained for reactions involving one-nucleon transfer. For the ~-particle transfer process, the calculation shows complicated angular distributions, depending on various parameters. 1. Introduction The D W B A m e t h o d has been a p p l i e d successfully to the quantitative u n d e r s t a n d i n g o f direct nuclear reactions involving light particles as projectiles. H o w e v e r , the transfer reactions between c o m p l e x nuclei have been scarcely analysed b y m e a n s o f the D W B A . T h e r e exists several reasons for this situation. First, there is less theoretical justification for the existence o f an o p t i c a l - m o d e l p o t e n t i a l in describing the nucleus-nucleus scattering t h a n the nucleon-nucleus case. Values o f the o p t i c a l - m o d e l p a r a m e t e r s d e t e r m i n e d f r o m the e x p e r i m e n t a l h e a v y - i o n scattering are very few to be used in the D W B A t r e a t m e n t o f the transfer reactions. F u r t h e r , the z e r o - r a n g e a p p r o x i m a t i o n is unjustified except for the transfer f r o m an s-orbit, a n d one has to c a r r y o u t l a b o r i o u s finite-range calculations. Finally, for energies a b o v e the C o u l o m b barrier, the cores come into close c o n t a c t with each o t h e r a n d the r e a c t i o n m a y proceed from virtually excited states o f the nuclei or b y m a n y step processes with exchange of several nucleons. The D W B A p r o c e d u r e s which were a p p l i e d to the h e a v y ion reaction so far, have different f o r m s f r o m the u n m o d i f i e d D W B A in o r d e r t o c i r c u m v e n t these difficulties. In the m e t h o d o f Breit et al. 1), one calculates the p r o b a b i l i t y a m p l i t u d e t h a t a neut r o n tunnels f r o m the p o t e n t i a l well o f one nucleus to a n o t h e r , while the two nuclei are at rest. T h e relative m o t i o n o f the two nuclei is then t a k e n into account a d i a b a t i c a l ly by means o f the d i s t o r t e d - w a v e m e t h o d . This m e t h o d is p a r t i c u l a r l y useful at energies below the C o u l o m b barrier. Starting f r o m the D W B A expression for the transition a m p l i t u d e , D a r a n d others 2) have o b t a i n e d the diffraction plus C o u l o m b scattering m o d e l u n d e r several simplifying a s s u m p t i o n s . This diffraction m o d e l gives a * Present address: Max Planck Institut fiir Kernphysik, Heidelberg, Germany. 625
626
T. K A M M U R I A N D H . Y O S H I D A
reasonably accurate and simple phen0menological description of absorption in the high-energy region, and good fits to a wide variety of the experimental transfer reactions are obtained. Similar model has also been derived by Kammuri and Nakasima [ref. a)] in the case of multi-nucleon transfer reaction. Recently, Buttle and Goldfarb [ref. 4)] have proposed a simple DWBA treatment, in which the wave function of a neutron in a bound state is approximated by its asymptotic form in the region outside the core. At energies below the Coulomb barrier, where there are no ambiguous parameters inherent to the nuclear distortion effects, this method has proved that the heavy ion transfer reactions are valuable tools for extracting the spectroscopic factors. The aim of this paper is to analyse the heavy ion transfer reactions by means of the finite-range DWBA method which does not involve any additional approximations, and to investigate its usefulness and applicability in the region near the Coulomb barrier, where both the Coulomb and absorptive effects are expected to be important. The formalism originally developed by Austern et al. 5) is outlined in sect. 2 for the simplest case. The angular distributions of the reactions we have treated show different shapes from each other, so that they need different theoretical treatments. Therefore we discuss each transfer reaction leading to the ground states of both final nuclei in a separate section. We treat the reactions 14N(14N, laN)lSN, 12C(~60, 160)12C, 11B(160, 15N)12C, and 27A1(160, lSN)2sSi in sects. 3, 4, 5 and 6, respectively. In the calculations, the bound-state wave function of the transferred particle is assumed to be an eigenfunction in a harmonic oscillator or a Woods-Saxon potential well. In the neutron transfer from the system 14N+ 14N, the effect of the identity of the two 14N nuclei is taken into account. For lower energies than the Coulomb barrier, the major contribution to the neutron transfer comes from the region where the neutron is outside the nuclear surfaces. Since the cross section is highly sensitive to the tail of the radial wave function in such a case, we also treated this reaction by using the Woods-Saxon wave functions. In the elastic scattering of 160 from 12C, the angular distributions show strong oscillatory patterns and an increasing cross section at backward angles. We assume that this part of the cross section arises as a result of ~-cluster transfer from 160 to 12C. In the treatment, we utilize the circumstance that the optical model parameters can be determined so as to give good fit to the forward angular distribution. In the 11B(160, 15N)12C reaction, 160 ion is regarded to transfer a proton or an ~-cluster to i xB. Finally, in the proton transfer resulting from the system of 27A1-[-160, we assume 27A1 and 28Si as deformed nuclei. 2. Outline of the theory The transition amplitude for the reaction A(a, b)B in the DWBA has the form
Tfi =- (BbkblTlAaka) = f dra f drbZ(b-)*(kb,rb)(lBMB,SbmblVllAMA,sama)z(~+~(ka,ra),
(2.1)
HEAVY ION INDUCED TRANSFER REACTIONS
627
where the effective interaction is defined by * < I B M a , S b m b I V I I A M A , s . m . ) = J OsObVOAO.d~.*
(2.2)
Here OB, Oh, 0A, and Oa are the internal wave functions for the particles B, b, A, a whose spins and z components are designated by I BMB, Sbmb, IA MA, S, rn,, respectively. The function Z~+) or )~-) describes the relative motion of the pair A, a or B, b, distorted by the optical potential. In eq. (2.2), ~ represents all coordinates independent of r, and rb. The coordinate system for a A(a, b)B reaction is shown in fig. 1, where x is the transferred particle, and a = b + x , B = A + x . x
A Fig. 1. Coordinate vectors describing A(a, b)B reaction in which a = b+x, B = A+x. The effective interaction can be expanded in a multipole series,
lsj X
(SaSbma,
--
mblsm, -- mb)(-- 1)sb--'bi--tGtsi,,(rb, r,).
(2.3)
The interaction V responsible for the transition is taken to be Vbx(rbx) which accounts for the binding of x and b to form a in the entrance channel. In this section we assume that x is a nucleon or a cluster which can be treated like a nucleon, and that all the participating nuclei are of spherical shape. Then G can be factorized as
Glsjm(rb,
ra) ~--- Alsjflsjm(rb, ra),
(2.4)
where A,,j = it[1] [s.]JO i Of W ( j i I i jf If ; Sx 1)3(s, ji)6(j, jf),
flsjm(rb ,
ra ) ~-- L ( l f l i ~ f /ligf
' - ] ~ i l l / ~ ) ( - - 1) m (])/f#f * (rxA)q~/t.i(Fxb) Vbx(rxb).
(2.5) (2.6)
Here [l] = (2/+ 1)~, and the Jacobian of the coordinate transformation is given by J = [ a B / x ( a + A ) ] z, and 0i, Of are real spectroscopic amplitudes corresponding to a bound x-particle with the orbital wave function q~h,,, 4~,,,~ and angular momentum Ji li a n d j d f in the incident ion a and the residual nucleus B, respectively.
628
T. K A M M U R I A N D H. YOSHIDA
We then define the partial amplitude it[1]fllm( kb ,
fllm(kb,'ka)
by
ka) = f draf drb~(-)*(kb, l'b)flm(l'b, l*a)~(a+)(ka,ra).
(2.7)
The differential cross section is given by da-
~a~b
[I.3 1 ~
kb(
(2.8)
where
atsJ(O) = E I E A~jfltml 2 m
ltlf
(2.9)
The method for calculating film is described in ref. 5) and in the code I N S - D W B A - 3 [ref. 6)]. We used the c o m p u t e r code in which the I N S - D W B A - 3 is improved so as to be applicable to the particular cases treated in the following sections 7). The calculations were performed using the H I T A C - 5 0 2 0 c o m p u t e r of T o k y o University. In the numerical calculations, except where otherwise stated, the fo!lowing assumptions are made. (i) In the present study we consider two extreme approximations for the b o u n d state function. In the first, the wave function of the particle x is taken to be the harmonic oscillator type. For a nucleon, the size p a r a m e t e r v takes the standard value 0.96A -~ fm -e corresponding to the nuclear radius 1.2A + fm. For a Charged particle, the effect of the C o u l o m b force on the binding is neglected. A second prescription for the wave function is to consider it an eigenfunction of a Woods-Saxon well with the binding energy equal to the separation energy. (ii) When we use the harmonic oscillator wave function, the interaction is assumed to be a Gaussian form Vbx(r) = -- Vo e-('/°2,
(2.10)
while, for a W o o d s - S a x o n b o u n d state function, it takes the Woods-Saxon form Vbx(r) = - - V o ( l + e ~ ) - ' ,
x = (r--¢)/a.
(2.11)
"Ihe diffuseness p a r a m e t e r a is fixed at 0.65 fro. Choosing the value of the range close to the radius of the nucleus a = b + x or the sum of the radii o r b and x (with the radius p a r a m e t e r ro = 1.25 fro), we determine V 0 in such a way that the potential Vbx gives the correct binding energy of x in the nucleus a. (iii) The optical potentials describing the elastic scattering of a on A, and b on B are assumed to have the W o o d s - S a x o n shape with the same parameters
U(r) = Vc(r) - V(1 + e xR)-I _ i W (1 + e xx)- 1,
(2.12)
where
Xrt = ( r - R)/aR,
x, = (r-- R')/a,,
R = rR(A~+A~2),
R ' = r,(A~+A~),
(2.13)
629
HEAVY ION INDUCED TRANSFER REACTIONS
and (A 1, A2) = (a, A) or (b, B). The Coulomb potential Vc is of a uniform charge distribution with the radius parameter r c. The parameters which give good fits to the experiments in our calculations are summarized in table 1. TABLE 1 Parametric values used in the calculation Reaction
V (MeV)
W (MeV)
rR (fm)
rI (fm)
aR (fro)
aI (fro)
rC (fro)
Reu t (fro)
v (fro -2)
14N(14N, 13N)15N (a) (b)
50 50
5 25
1.25 1.25
1.25 1.25
0.65 0.65
0.65 0.65
1.25 1.25
7 0
0.25 0.25
12CQeO, IEO)12C
59.5
2.38
1.075
1.55
0.45
0.45
1.25
6
0.5
liB(leO, lSN)12C
59
6
1.10
1.4
0.58
0.58
1.25
0 6
0.38(p) 0.5(~)
ZrA1Q60, 15N)2sSi
50
25
1.15
1.15
0.65
0.65
1.25
0
0.96A-1-
C o l u m n s two to eight list the optical-model p a r a m e t e r s in eq. (2.10), and c o l u m n nine gives the value o f the lower radial integration cutoff. T h e last c o l u m n lists the size p a r a m e t e r o f the h a r m o n i c oscillator wave function o f the transferred particle.
(iv) In discussing the absolute magnitude of the cross section, 01 and 0f in eq. (2.5) are set equal to unity.
3. The 14N(14N, laN)lSN reaction 3.1. E F F E C T O F A N T I S Y M M E T R I Z A T I O N
This process requires special consideration on account of the identity of the two in the entrance channel. The required modifications has already been given by Buttle and Goldfarb 4). We write down only the final results. The antisymmetrized transition amplitude is given by laN
T
=
4n(n + 1){ + },
(3.1)
in which n is the number of outer neutrons in the initial nucleus. When we assume that only one set of values ofji l i a n d j i l r is important, we obtain
trs~j = ~ n(n+ 1)[ht~jlz{lfl,m(O)12 + [firm(re--O)[2 + Sl Re
fllm(O)fl~m(~--O)},
(3.2)
•
(3.3)
m
where At~j is given by eq. (2.5) and /1
S, = 2([jl][jf])2UlJi \ Jf 3.2. C O M P A R I S O N
WITH
Ji Jb
IA
IA
IB
EXPERIMENT
We now compare the DWBA predictions with experimental data given by Hiebert, McIntyre and Couch s) on the 14N(14N, 1aN)lSN reaction at incident energies of
630
T. KAMMURI AND H. YOSHIDA
12.3, 14, 16 and 18 MeV. The neutron transfer is assumed to involve capture from a lp~ level in 14N to a lp~ level in the ground state 15N. Optical-model parameters which are obtained from the optical-model fits to the data of the elastic scattering of 14N from ~4N at energies between 15.0 and 21.7 MeV have some ambiguities. The following sets were given by Porter 9) (a) V = 4 0 M e V ,
W=
8+6MeV,
r o = 1.15fm,
a = 0.6fm,
r c = 1.04fm.
(b) V = 2 0 M e V ,
W=
10+6MeV,
r o = 1.25 fm,
a=0.6fm,
rc =
1.04fm.
Kuehner and Almqvist 1o) have fixed V at 50 MeV in the analyses of the elastic scattering of 14N_b 12C, 14N_k_ 9Be, and a 6 0 "{-1 2 C . They have found that this procedure is justified, since it appears that the scattering can determineat most only two parameters of the real potential well. The optical-model parameters which are used in the present calculation are listed in table 1. In the calculation, the force range ~ is fixed at 3 fm. As for the size parameter v, True ~ ) gives v = 0.32 fm -2 from the analysis of ~4N energy levels. This value is smaller than v = 0.96A-+fm -2 = 0.4 f m - 2 . Fig. 2 shows a comparison of the experimental angular distributions with the DWBA predictions obtained by use of the harmonic oscillator (HO) wave functions. For v equal to 0.25 fm -2, the calculated cross sections aca~c are about ½ of the experimental values aexp. When we increase v from 0.25 f m - 2 , the angular distributions remain unchanged, while the magnitudes of the cross sections decrease and the shape of the excitation function becomes worse. For example, at v = 0.32 f m - 2 , acal¢/aexp is about 1~1 o at 12.3 MeV and 1 at 18 MeV. It is necessary to take a lower value ofv than that given by True in order to get agreement with the experimental cross section. This is due to the strong damping of the harmonic oscillator wave function at large distance. Therefore we also tried the calculation using the wave functions of the Woods-Saxon (WS) well. The results are shown in fig. 3. The calculated angular distributions are barely distinguishable from those given in fig. 2. The absolute magnitude of a¢a~ is about 2 of a~xp (table 2). D W B A predictions taken at 12.3 MeV are quite insensitive to wide changes of V and W. Even the V = W = 0 case gives results that are almost the same as in figs. 2 and 3. At this energy, the major contribution to the cross section comes from the region near the classical turning point of about 12 fm, where the effect of nuclear distortion is expected to be negligible. The direct part of the transition amplitude of eq. (3.1) gives backward peaking, characteristic of transfer reactions in a strong Coulomb field. It should be noted that our result does not show such an oscillatory pattern near 90 ° as the one obtained in refs. ~-4). The effect of varying the optical-model parameters is shown in fig. 4. For V = 20 MeV the cross section remains unchanged, while the cross section falls off too rapidly for large angles when we increase V to 100 MeV. The calculation with W = 35 MeV shows no difference from that with W = 20 MeV. Increase of the diffuseness parameter a amplifies the amplitude of the oscillation pattern.
C
0
O.
!(
o"
3O"
6O°
~CM
~ig. 2. Comparison between the experimental ]4N(14N, 1aN)lSN ection and the theoretical curves at E~ab = 12.3, 14, 16 and 18 -Iarmonic oscillator wave function is used for the neutron b o u n d ['he parameters o f the theoretical curves are given in table 1. Each is normalized to experiment arbitrarily.
3
~0~
.
n
.
cross MeV. state. curve
0°
3o °
60"
0o,4
~'ig. 3. Same as in fig. 2 except that the potential for the neutron b o u n d state and the interaction have the Woods-Saxon shapes.
1.(
)C
- o
o
0"
10 -1
I
i
'
)
/
I
I
30"
" ' / -/- "
i
'
i ,,/ ,/
/
•
\
,\
\"
t
.........
eY /-.,
\
,
_ . . - - - "" ~ ',X,~7
: "../
;_" ~-
I
60"
/
,/'\,
I
I
a 0.65 0.65 0.65 0.8
"
",,'
,,
ec M
V W I00 15 50 15 50 25 50 15
\,J'
-"
/ ' "/ . - - - . , k , _ 1
8MeV
Fig. 4. Effect of variations in the optical-model parameters for the 14N04N, IaN)I~N g r o u n d state reaction at Elab = 18 MeV. T h e size parameter o f the h a r m o n i c oscillator function v is 0.49 f m -z a n d $ =3fm.
b "O
%3
3
6
10
1
'<'NCN j'N )"N
~o \
/
[
,
I
\.j'
/
3o*
f
~i,~ ~-.
\ . . . "2" ( a3~5, 6 )
~
.__
I
t
60"
I
I
\
OCt,4
( o=0z, gfrn"~, Rcut=6fm )
._~ / - - - ~
/
I
.~-~
\
T h e parameters are $ = 3.2 fin, V = 50 MeV, W = 5 MeV, r o = 1.25 fm, a = 0.65 fin.
Fig. 5. Effect of variations in the size parameter v and the radial cutoff Rou t for the *4N(I~N, laN)ISN ground state reaction at Ela b = 18 M e V .
,
x.
( o.L5,61 X....
: ",, \\ / ¢o~3L\,.4j ,,, ~, / . . - - - - .
.<..
2\
0°
10"*
-u 1
b
ui
i (0.4,0)
"~N CN"N )"N
=
o
.z
I-O
HEAVY ION INDUCED TRANSFER REACTIONS
633
In fig. 5, we show the effect of the radial cutoff in the calculation with W = 5 MeV. For W equal to 25 MeV, the effect of radial cutoff is negligible because the incident *N wave is strongly damped inside the target nucleus. In the W -- 5 MeV case, we cannot get agreement with experiment without using the radial cutoff. The change of radius of the cutoff around 6 fm does not affect so much. 4. The 12C(160, 160)t2C reaction 4.1. ALPHA
TRANSFER
PROCESS
The angular distributions of the elastic scattering of 160 on 12C at E l a b = 35 and 42 MeV which have been measured by yon Oertzen et al. i2) show strong oscillations and increasing cross sections at backward angles. Such a pronounced structure cannot be fitted by optical-model calculations. They assumed the s-particle transfer from i 6 0 to I2C to explain this structure. With Dar;s diffraction model, good agreement was obtained. Such a transfer process which has zero Q value and contributes to the elastic scattering is named as resonant transfer by Temmer 13). From the analogy with the molecular collision problem, the resonance-like excitation function may arise from the correlation of the incident energy and the frequency that the transferred particle can go back and forth between two colliding nuclei during their contact time. Such a possibility cannot be ruled out in the present reaction, although strong Coulomb force inhibits both nuclei to approach close together at low energies. Indeed, the backward portion of the angular distribution can be fitted with the function (Pi0(cos 0)) 2, characteristic of the process involving compound nucleus formation with spin equal to I0. Kuehner, Almqvist and Bromley i , ) observed resonances in the elastic scattering excitation function for 160 + I2C below 36 MeV at a center-of-mass angle of 90 ° . However, by the analysis of fluctuations observed in the excitation functions at fixed angles, yon Oertzen et al. have concluded that a contribution due to a compound nucleus formation is smaller than 5 % at 139 °. Since we aim at the test of the applicability of the D W B A method, we treat the structure in the backward angle by assuming 2p2n transfer from 160 to 12C in the framework of DWBA. The optical-model parameters can be determined from the forward portion of the angular distribution. The values obtained in ref. 12) are used in our calculation. While this offers a very advantageous situation for applying the D W B A method, we have relied on the s-cluster approximation for the transferred 2p2n system. This approximation implies that the s-cluster is fully space symmetric and the principal quantum number and the orbital angular m o m e n t u m of the motion relative to the core are restricted to those compatible with the simple shell-model wave functions of the constituent nucleons. If the harmonic oscillator wave functions are adopted, 12C and alpha are in a relative 3s state in the 1 6 0 nucleus. The effects of higher shell-model configuration admixture to the 160 ground state wave function will mix states with other quantum numbers. For example, it may be possible to con-
634
T. KAMMURI AND Y. YOSHIDA
sider components in which the 0 + alpha and ~2C are in the 2s state or 2 + excited alpha and 12C are in the 2d state. The calculation shows that both admixed states have nearly the same angular distribution as in the case of 3s motion, but the cross section for the 2s case is very small. We generate the a-cluster wave function as a solution in a harmonic oscillator potential. The size parameter we used are of three types: (i) v1 = 1.143 fm -2. This corresponds to (4(A-4)/A)xO.96A -~ fm -2 given by recoupling four nucleons in shell-model states by using Talmi and other coefficients (A = 16 in the present case). (ii) v2 = 0.71 fm -2. The radius RreI of the corresponding harmonic oscillator well is very close to R~2c+R, = 4.6 fro. (iii) v3 = 0.5 fm -2. The a-particle is assumed to be bound very loosely to 12C with Rre~ = 6.8 fm. The approximation that ~60 can be regarded as a two-body system consisting of 12C and alpha (strongly correlated four-nucleon system) holds only for the region where 12C and alpha do not overlap so much. In the overlapping region, on the one hand, the repulsive force acts due to the effect of the exclusion principle, and on the other hand the ~-cluster tends to dissolve into particles in the shell-model states. Thus the a-cluster wave function damps much more rapidly inside the nucleus and has larger values outside the nuclear surface than in the case of the harmonic oscillator approximation. Under such circumstances it may be natural to use v smaller than vt, and also adopt radial cutoffs in the D W B A calculations of the c~-transfer process. As for the force range ~, we take 3.4 fm and 4.6 fm. The former value corresponds roughly to the radius of ~60, while the latter is equal to R~c+R,. 4.2. C O M P A R I S O N
WITH
EXPERIMENT
Fig. 6 shows comparison between the calculated angular distribution and the experimental result at 35 MeV incident energy 12). The interference between the elastic scattering and the c~-transfer is not taken into account. The calculated a-transfer cross section increases in the backward direction, though the positions of the minima have some deviations from the experimental ones. The cross section at 155 ° obtained by using the harmonic oscillator function with v = 0.5 fm -2 and the interaction range = 3.4 fm is estimated to be about ~ of the experimental value, while that using the function of the Woods-Saxon shape (4 = 4.4 fm) has the right order of magnitude. We have investigated the effects of variations in the cutoff radius, 4, and v. From fig. 7, we see that use of no cutoff smoothes the angular distribution and increases the backward cross section. By using the cutoff we find that the magnitude of the cross section is reduced though very deep minima can be obtained. The use of a radial cutoff at 7.5 fm results in reduction ot the predicted cross section by two orders of magnitude as compared with the case of 6 fm cutoff and the oscillatory pattern becomes weak. It shows that the main contributions to the reaction cross section comes from the
30*
60*
If
90 °
120°
<:.: 150° Oct,4
Fig. 6. Comparison between the cross sections of the experimental ~C(t~O, ~aO)~zC ground state reaction and the theoretical curves at Etch-- 35 MeV (normalized). The dashed and solid curves were obtained using Woods-Saxon (~ = 4 . 4 fm) and harmonic oscillator wave functions, respectively The parameters of the theoretical curves are given in table 1.
<
,/
10
0o
_
,,
•
;
,
,;
/ ',.
/
',.,.,,'
60°
/
{',,,,'1 /"
lit, ,11',
"'.
I
35MeV
Ij
,^,
90"
I
u
I.',"
If
I
'J-
~j
i
',
i
I
J
120" 150°OCM
_/
,!
I
(7.5)
(6}
i/i/
i
~ ~4~ I
(Rcul=Ofm)
Fig. 7. Effect o f variations in the radial cutoff for the ~-transfer part of the lzC(160, 180)12C ground state reaction at E]a ~ = 35 MeV. The optical-model parameters are given in table I, and v = 0.71 fm -2, = 3.4 fm.
10
'13 "b o
""
~
"4
E
"9
cat.
35MeV
~°
~,
't("o/'o)%
z
.4
> .<
636
T. KAMMURI AND H. YOSHIDA
region of the internuclear distances between 6 fm and 7.5 fm, where both nuclei are in close proximity. lncrease of ¢ from 3.4 fm to 4.6 fm increases the peak cross section by almost a factor 5, whereas the angular distribution is little affected. Shifting of the diffraction pattern to larger angles for larger v can be observed. Variation of v from 0.5 to 0.71 and 1.14 f m - 2 , decreases the peak cross section by one and four orders of magnitude, respectively. We have shifted the position of maxima by varying v and have obtained deep minim a by use of radial cutoff. We have found loose binding of the s-cluster in 160 as appropriate, which is reflected in the use of small v or large ¢ in the H O or WS case, respectively. This is contrary to the conclusion of the diffraction model calculation ~2). In the latter, it was necessary to assume a much higher binding energy for the transferred s-cluster in 16 0 of about 25 to 35 MeV. This corresponds to a very rapid decrease of the wave function of the a-cluster outside the nucleus.
5. The UB(160, lSN)12C reaction 5.1. EFFECT OF ANTISYMMETRIZATION The angular distributions of 15N from the b o m b a r d m e n t of ~IB by 27, 30, 32.5 and 35 MeV 160 ion were measured by Bock et al. 15). The 15N cross sections have strong forward and backward peaks. This suggests that the reaction proceeds by two types of transfer process, namely, a proton and an a-cluster transfer from 160 to x~B. G o o d agreements with experiments were obtained by the diffraction model. In the following simple model we represent the angular distribution as a coherent sum of the above two processes. First we treat the transferred 2p2n system like a single a-particle as in sect. 4. Next the nuclei ~6G, ~SN and ~2C are assumed to consist of the 1~B core and the extra p + a, a and p, respectively. The effect of the exclusion principle between the outer particles and the core is ignored. Then the 11B(160, 15N) 12C reaction can be written as 'lB+(11B+p+ct) ~ (t'B+p)+("B+a).
(5.1)
Considering ~IB as a single fermion, we take account of the antisymmetrization of two ~ B (written as X~ and X2). The transition amplitude is given by Tfi =
(f[Vf[P12i)
= (X1 + p , X2 +ctlVflXl, X2 + p + a ) -
( X l + p , X2 +alVflX2, X1 + p + a),
(5.2)
where P12 is the antisymmetrization operator acting o n X 1 and X 2. The interaction Vf is remainder of the interaction between two nuclei in the final state after subtracting the potential U(X 1 + p , X 2 + a ) which generates the distorted wave in the exit
637
HEAVY ION INDUCED TRANSFER REACTIONS
channel. This interaction can be written in two ways, V, = V(Xl + p , X2 + ~ ) - U(X1 + p , X2 +~)
= V(p, X 2 + a ) + { V ( X 1 , X 2 + a ) - U ( X I + p , X2+a)}
(5.3a)
= V(XI+p,a)+{V(XI+p,X:)-U(XI+p,X=+~)}.
(5.3b)
The first term in the transition amplitude, eq. (5.2), corresponds to the proton stripping from 160 = X 2 + p + ~. We adopt the simplest approach in which Vf is approximated by V(p, X 2 + a) = V(p, 1SN) in eq. (5.3a). The contribution from the remaining V(HB, ~ 5 N ) - U ( 1 2 C , 160) is expected to be relatively unimportant in the same way as in the case of the deuteron stripping reaction. In the second term of eq. (5.2) which represents a particle stripping from 160 = X1 + p + ~, we assume that V(~, X 1 + p ) = V(~, ~2C) is of most importance in eq. (5.3b). There is far less basis for the neglect of the contribution V(12C, 1~ B ) - U( 1:C, 160) . In the a-transfer process, 160 and ~SN are assumed to be two-body systems consisting of 12C and a in 3s state, and of 11B and a in 2d state, respectively. 5.2. C O M P A R I S O N
WITH
EXPERIMENT
Comparisons of the D W B A predictions with experimental data at the incident 16 0 energy of 27 and 30 MeV are shown in fig. 8(a) and (b). Use of the same parameters for 27 and 30 MeV (table l) does not reproduce the variations of the angular distributions with energies for the a-transfer part. In the calculation the interference between p and a-cluster transfer process is ignored. 5.2.1. Proton transfer. The optical-model parameters which fit to the elastic scattering of 160 from 11B at the center-of-mass energies ranging from 7.00 MeV to 11.00 MeV are given by O k u m a 16) as V = 59 MeV, W = 6 MeV, rR = rl
=
rc =
1.10 fm,
aR = al = 0.58 fm. For this set of parameters, the angular distribution of p transfer shows smooth forward peaking. When we vary the values of parameters around this set, for example, change a to 0.4 fm or q to 1.4 fm as in table 1, a weak diffractive oscillation can be seen. In fig. 8 where q is taken to be equal to 1.4 fm and v = 0.96A-+fro -2 is used, the cross section calculated with H O wave functions is about 0.3 of the experimental magnitude. In the WS case, these ratios are about 1.4 (27 MeV) and 2 (30 MeV). A potential set with V = 50 MeV, W = 25 MeV, r R = q = r c = 1.25 fm, aR = a~ = 0.65 fin gives practically the same angular distribution as shown in fig. 8. However it reduces the absolute value by almost two orders of magnitude. Since the height of the Coulomb barrier of the system 1 1 B + 1 6 0 is 8.1 MeV for r o = 1.5 fm, the incident energy of 27 MeV (11 MeV in the c.m. system) is sufficiently high for two nuclei to come close together. Therefore one should expect the use of the harmonic oscillator
0°
-
30"
: / ' ~
•
~2
60"
I
, 90"
(a)
, 120°
",,
Z I'~ 153" OCM
o4 transfer
//'t , ~
t{" k
p transfer
27 MeV
~5 15
g(0, N ) g
i
O"
10"z
10 4
"ID
"o
JD E
I
30"
t
60*
I
tl
p transfer
90*
I
I v
30 MeV
"BCo,"N)"C
I
120"
V
V
t'
o~transfer
I
150°0C M
~
',,L/I
(b)
I
Fig. 8. C o m p a r i s o n between the cross sections o f the experimental " B Q 6 0 , a~N)12C g r o u n d state reaction a n d the theoretical curves at Elab = 27 MeV (fig. 8(a)) a n d 30 MeV (fig. 8(b)). T h e d a s h e d a n d solid curves were obtained by using W o o d s - S a x o n a n d h a r m o n i c oscillator wave functions, respectively. For the c~-transfer part, H O calculation uses v = 0.5 f m -~, while W S o n e uses ~ = 3.9 fro. T h e p a r a m e t e r s o f the theoretical curves for p a n d c(-transfer processes are given in table 1. All curves are arbitrarily normalized.
10"
10q
1
H
HEAVY ION INDUCED
639
TRANSFER REACTIONS
wave function to give accurate result also for the absolute cross section within the D W B A framework. 5.2.2. Alpha cluster transfer. Sincethe ~-transfer process contains uncertain parameters, it is not easy to k n o w whether agreement with experimental data is really significant or not. Conversely, the sensitivity of the D W B A predictions to these parameters may be utilized for the investigation of the structure of the four-nucleon cluster. As shown in fig. 8, the angular distributions of a-transfer do not show backward peaking and strong oscillatory behavior, in contrast to the case treated in sect. 4. I''
~. transfer ," zero-range calc. ~ .
E~("0,~N )'2C 27 MeV
10
/ I
,r\ l
~n
l,j~
:=ro 1
If ~\ ,.,
Il
,,j
b "13
Rcut=6fm
10 "1
S
xlO .....,.
;' ',...' / ',, ",j
I 0"
i 30 °
I 60"
V 90"
1 120 °
/
I 150" OCM
Fig. 9. Z e r o - r a n g e D W B A p r e d i c t i o n s a n d effect o f v a r i a t i o n s in radial c u t o f f for the or-transfer p r o c e s s in the l i B ( l e O , lSN)x2C r e a c t i o n at Elab = 27 M e V . T h e p a r a m e t e r s used are V = 50 M e V , 14" = 25 M e V , r0 = 1.25 fm, a = 0.65 f m (a = 0.8 f m for Reu t = 0) a n d v = 1.2 f m -2.
Calculated cross sections (v = 0.5 fm -2 in the HO case) are of the same order of magnitude as the experimentally observed cross sections, apart from the spectroscopic factor for effective a-clustering. The magnitudes of the cross sections are very sensitive to the value of v when we use the harmonic oscillator function. For example, it decreases by about two orders of magnitude for the increase of v from 0.5 to 0.71 fm -2.
We now discuss the effect on the D W B A calculations of variation of the force range and cutoff radius Re, t. Since alpha and 12C are in a relative s-state in 160, it is pos-
o"
104
--
"
/
'\
\"/
JE
¢~
I 30"
,,-.,.
\'
r,,
I 60"
\./
I 90*
,/'"
,' ,,
/
"',-;
/
f',
( 4 )
I 120 °
)
V
/
1
i~ /,t,
/
//
/ [
8cH
"/l
I 150 o
( R c u t= 6 f m
/~,
i
o~ transfer
\,,wf "~\ /."-'. U
.
27 N e V
~ ( O, N ) ~
~ la
Fig. 10. Effect o f variations in radial cutoff for the c~-transfer p r o c e s s in the 11B(~sO, 15N)12C g r o u n d state r e a c t i o n at Elah -- 27 M e V . T h e p a r a m e t e r s used are V = 50 M e V , W = 25 M e V , ro = 1.25 fro, a = 0.65 fm and v = 0.71 fm -~, ~ = 3.4 fm.
"o
m
~T
"E
u~
10
tl
10"
0°
,.
30"
t
...... -"
. "
"'/
60"
I
I
.-xlO =
90 °
Rcut = 8fro
~ . /:-"..-.,/6
I
120 °
..,
I
150"
\ iI
eC M
F",,
"\ ,,j/"~"!/(
,~ transfer
given by O k u m a are used; V - - 59 M e V , W = 6 M e V , r o a -- 0.58 fro, and v = 0.71 f m 2, ~e = 3.4 fro.
1.10 fro,
Fig. 11. Same as in fig. 10 except that the optical-model parameters
b lO
~3
(D
"E
27 b4eV
"B("O,"N)"C
O
£
10 "t
0*
'*\ ///
"1
~
/
30"
,"'
60*
\\
90*
~) = 1 fm
b \ --.~" --"
4.6
120"
\../
c~ transfer
150" eC M
//
F i g . 12. Effect o f v a r i a t i o n s in f o r c e r a n g e ~ f o r t h e co-transfer p r o c e s s in t h e ttB(x~O, t~N)x2C g r o u n d s t a t e r e a c t i o n a t Ela b = 27 M e V . T h e p a r a m e t e r s u s e d a r e I/ = 50 M e V , W = 25 M e V , r0 = 1.25 fro, a = 0 . 6 5 f m a n d v = 1.2 f m -2, Reu t = 0.
b 1D
::, 1
ul E3
10
27 MeV
"B(~):'N)"C
10-'
O"
f5
I 30"
// V
I 60"
I 90"
I ]20 °
Id
~ \
I 150" OCM
, '\
i
6 fm. Effect of variation
-_,---,,
'W
,~ transfer
3.4,0.71 "~"/~'\ /f--,j
12
~:3.4fm,l)=l.2fm-Z
•\!,~,\
27MeV
la
B( O. N) C
F i g . 13. S a m e as in fig. 12 e x c e p t f o r Reu t : i n v is a l s o s h o w n .
b 13
"o
10
II
642
T. KAMMURI AND H. YOSHIDA
sible to use a zero range force for V(~, 1 2 C ) . I n the a-transfer model of the 19F(d, 6Li) 15N reaction, Denes, Daehnick and Drisko 17) have found the similarity between the zero-range and the finite-range calculations. In our case, the calculated angular distributions in the zero-range approximation differs completely from the finite-range treatment. Fig. 9 shows the zero-range results by use of v = 1.1 fm -2 and V = 50 MeV, W = 25 MeV, r o = 1.25 fm, and a = 0.65 fm. Strong dependence on the position of the cutoff radius can be seen. The zero-range approximation overestimates the contribution from the nuclear interior. For instance, the cutoff below 4 fm already changes the angular distribution and the cutoff at 6 fm completely diminishes the cross section. The effect of the variation of the radial cutoff is illustrated in fig. l0 (for W = 25 MeV) and fig. 11 (for Okuma's parameter set 16)) in the finite-range case. In the backward angles, the cutoff below 6 fm has relatively little effect. As is seen, the angular distribution with 6 fm cutoff in fig. l0 agrees well with the experimental data, but the magnitude of the cross section is about ~o of the latter. Finally, the effect of the variation of ~ is shown in figs. 12 and 13 in the case of Rcut = 0 and 6 fm, respectively. Although the variation of ~ produces changes in the backward angular distribution in the case of Rcut = 0, its effect is rather small for Rcut = 6 fm case. F r o m fig. 13, the effect of change in v can also be seen. 6. The 27A1(160, tSN)2sSi reaction 6.1. SPECTROSCOPIC FACTOR FOR DEFORMED NUCLEI We treat the 2TAI(160, 15N)2sSi reaction as the proton transfer process from t 60 to the well.deformed 27A1 nucleus resulting the well-deformed 2SSi nucleus. There have been several arguments about the deformation of both nuclei. According to the Hartree-Fock calculation for 2sSi [refs. 18,19)], the configuration with oblate shape has lower energy than the prolate one. The 27A1 nucleus is usually treated as prolate, inferring from the positive sign of the measured quadrupole moment. Ripka 19) has regarded the 27A1 ground state as the I = ~ state of the K = ½ band corresponding to one hole state of the oblate 2aSi ground state. In this case too, we can expect a positive quadrupole moment for 27A1, and moreover the value of the magnetic moment agrees with the experimental value. However the decoupling parameter must be stronger than that given by the Nilsgon model in order to bring the I = ~ state to the lowest. Accordingly, the ground state configuration composed of 2s½, ld~ and ld~ single-particle states differs from the [220] 1+ state in the Nilsson model. In the strong coupling model the nuclear system has a definite value of deformation and is not an eigengtate of the total angular momentum. In the transition from the 27A1 to the oblate 28Si, more than ten particles j u m p to orbits almost orthogonal to the original orbits. The probability of such a transition is completely negligible. However when we construct an eigenstate of the angular m o m e n t u m from a linear corn-
643
HEAVY ION INDUCED TRANSFERREACTIONS
bination of intrinsic wave functions by means of the projection method 20), the overlap between the system with the oblate shape and that with the prolate shape is not so small. According to Une and Yoshida 2o), the overlap integral between the positive I = 0 state (6 = 0.3) and the negative I = 0 state (6 = - 0 . 3 ) amounts to 0.0987 for the 28Si system. In this subsection, we calculate the spectroscopic factor assuming that the transition occurs for the oblate 27A1 to the oblate 28Si, both having the same value of deformation. Using the Nilsson model, the wave functions are written as tIIIAMAKA(27A1) = ~ _ ~ c~(A) (¢~)[ZKA(C)DMAKA [A IA --K.], + ( - - 1)IA--½I-Izz--K~(C)DMA,
47Z
~ , . M . K . (2S8 . , ) =
~B)(¢c)[Det (x
Ac),
(6.1)
'"
where x and c express the transferred proton and the proton originally contained in 27A1, respectively, and ~v(~c) denotes the vibrational state of the core. The singleparticle states ZK is written in the body-fixed reference frame as XKA = E ClfjfKA ~/lfjfKA(r')" lf jr
(6.2)
The parity of the intrinsic state XK is denoted by fi x = ( - 1 ) z . Using the explicit form for the wave functions, we obtain Gtsis as
Ggsjm = ( - - 1)sx-s' ~ Atsj(lfjf)flsj(rb, ra) ,
(6.3)
lf jr
where in eq. (2.5) for Atsj, Of is taken as
0f
=
~.(A)x [-IA][-l.l- 1/~.(B) \Wv ~ev / c,~y~, _KA(IAjfKA
-- KAIIB0). (B)
In the numerical calculation we put the overlap of the core part <~bv
(6.4) (A)
I~v > as
unity.
6.2. COMPARISON WITH EXPERIMENT
The proton transfer in the 27A1(160, 15N)28Si ground state reaction is assumed to involve capture from alp½ level in 160 to a level composed of 2s~, I d I and ld~ in 2aSi. Taking account of the fact that the ground state spin of 27A1 is ~2, only the ld~ component is effective for the transition. For the transferred angular momentum, l = 2 and 3 are possible. Since the l = 3 transfer is considerably smaller than 1 = 2 case, we discuss only the l = 2 transfer. We used the optical-model parameters listed in table 1. The comparison with the optical-model calculation with the experimental data of the elastic scattering of 32 MeV 160 on A1 target 14) is shown in fig. 14. The experimental results obtained by Newman, Toth and Zucker 2I) are compared in fig. 15 with the DWBA predictions. The calculated cross section are about two orders of magnitude smaller than the experimental values. The origin of the disagree-
I
30"
]
60 °
I
90 °
~
•
E,~0 : 32MeV
120°
I
36(25 )
\30 ( 5 )
150° ec M
I
• \\30 MeV (W:25MeV) ' \ "\.
Fig. 14. C o m p a r i s o n between d a t a for the elastic scattering of 32 MeV ]~O from AI and the predictions o f the optical pot e nt i a l used in ZTAl(leO, ]SN)zsSi calculations.
0°
10-~
10"t
~R
At ('0;'0) At
0"
10"
~ 30"
~,/I
I
60"
I
90"
120"
I~
I
150°
ec.
'LI,~ "/ ~\
!
36MeV
Fig. 15. C o m p a r i s o n between the cross sections o f the ex p er imen tal 2~A1(160, 15N)28Si g r o u n d state re a c t i on an d the theoretical curves at E~a b = 30 a nd 36 MeV. The p a r a m e t e r s o f the theoretical curves are given in table 1. F or 30 MeV case, the da shed curve was o b t a i n e d using V -- 50 MeV, W -- 5 MeV, ro = 1.25 fro, an d a = 0.65 fin. All curves are a r b i t r a r i l y norma lized .
b
E
A
i0"i
645
HEAVY ION INDUCED TRANSFER REACTIONS
ment may be ascribed to the use of the large imaginary potential and the use of the harmonic oscillator wave function at energies near the Coulomb barrier (18.7 MeV for r o = 1.45 fm). Experimental peak cross sections are 1.35 and 1.37 mb for 30.0 and 36.0 MeV energy respectively, while our result increases by a factor 3 in this energy interval. This may be due to our use of small r o (1.15 fm) in order to fit the position of the peak of the angular distribution. The calculation with V = 50 MeV, W = 5 MeV, r 0 = 1.25 fm and ~ = 0.65 fm gives right order for the absolute value, but the position of the peak shifts to smaller angle than the experiment as can be seen from fig. 15. The calculation shows oscillations at forward angles. Such oscillation was also obtained by Dar z). However, in his diffraction model the oscillations are stronger at 30 MeV than 36 MeV, contrary to the present result. For this reaction, calculation using the WS wave function is still incomplete because of the difficulty in the numerical integration in eq. (2.7), and cannot produce reliable results. TABLE 2 C o m p a r i s o n of the absolute m a g n i t u d e s of the e x p e r i m e n t a l cross sections w i t h the D W B A p r e d i c t i o n s using W o o d s - S a x o n wave function Reaction
Lab.energy (MeV)
c.m. angle
80 ° 80 ° 80 ° 80 °
Exp. value (mb)
14N(14N, IaN)I~N
12.3 14 16 18
( 3 . 7 ± 1.4) (3.6±0.4) (2.1 1 0 . 3 ) (3.6±0.2)
x 10 3 X 10 -2 × 10 -1 x 10 1
12C(160, I°O)1~C
35
170 °
14
27 30 27
33 ° 20 ° 141.5 °
1.4 +o.2 -o.1 0.8 ! 0 . 0 6 0.185 i 0 . 0 4 5
Calculation (mb) 1.4 x 10 -a 2.7 X 10 2 1.6 x 10 -1 2 . 9 x 10 -1 12.6
11B(160, 15N)1~C p transfer transfer
1.9 1.7 0.24
7. Concluding remarks We first summarize in table 2 the comparison of the absolute values of experimental cross sections with the calculated ones, putting 0 i and Or in eq. (2.5) equal to unity. A bound state wave function in the Woods-Saxon potential well is used, and the effect of Coulomb force on binding is taken into account. Distorting optical potentials used in table 2 are those listed in table 1. The results of the present investigation indicate that the D W B A theory of the heavy ion transfer reaction is capable of predicting both the shape and magnitudes of these cross sections. For the exceptional case of the 27A1(160, l S N ) 2 8 S i reaction, our analysis is still incomplete. Our ignorance about the properties of the optical-model potential between any complex nuclei offers the greatest obstacle for the application of the D W B A method. This will be partly overcome by the accumulation of more experimental data on elas-
646
T. K A M M U R I A N D H. YOSHIDA
tic scattering. However, it can be expected that the a m b i g u i t y in the values o f the optical-model parameters such as observed i n the elastic scattering of s-particle 22) is also present in the heavy i o n case. W e have a p p r o x i m a t e d the transferred 2p2n system b y a single a-cluster. The ~transfer process in the 11B(160, 15N)12C reaction shows a wide variety of a n g u l a r distributions depending o n the value of the parameters. Such a sensitivity of the cross section to p a r a m e t e r choice will serve as a useful guide for investigating the properties of the s-cluster wave function. The a u t h o r s are grateful to Professor S. Y o s h i d a for stimulating discussions, to Dr. T. U n e for i n f o r m i n g us his calculation of the overlap integrals, a n d to Miss K. M a t s u u r a for preparing the manuscript. They also wish to t h a n k Professor R. Bock for kindly c o m m u n i c a t i n g the results of the experiments at the Max Planck I n s t i t u t ftir Kernphysik, Heidelberg.
References 1) G. Breit, J. A. Polak and D. A. Torchia, Phys. Rev. 161 (1967) 933 and references therein 2) A. Dar, Phys. Rev. 139 (1965) Bl193; A. Dar and B. Kozlowsky, Phys. Rev. Lett. 15 (1965) 1036 3) T. Kammuri and R. Nakasi~a, Proc. Conf. Reactions complex nuclei, Asilomar, 3rd, 1963, 163 4) P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 5) N. Austern, R. M. Drisko, E. C. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3 6) T. Une, T. Yamazaki, S. Yamaji and H. Yoshida, Code INS-DWBA-3 7) H. Yoshida, unpublished 8) J. C. Hiebert, J. A. McIntyre and J. G. Couch, Phys. Rev. 138 (1965) B346 9) C. E. Porter, Phys. Rev. 112 (1958) 1722 10)' J. A. Kuehner and E. Almqvist, Phys. Rev. 134 (1964) B1229 11) W. W. True, Phys. Rev. 130 (1963) 1530 12) W. yon Oertzen, H. H. Gutbrod, M. Mtiller, U. Voos and R. Bock, Phys. Lett. 26B (1968) 291 13) G. M. Temmer, Phys. Lett. 1 (1962) 10 14) J. A. Kuehner, E. Almqvist and D. A. Bromley, Phys. Rev. 131 (1963) 1254 15) R. Bock, M. GroBe-Schulte, W. yon Oertzen and R. Riidel, Phys. Lett. 18 (1965) 45; R. Bock, M. GroBe-Schulte and W. yon Oertzen, Phys. Lett. 22 (1966) 456 16) Y. Okurna, J. Phys. Soc. Japan 24 (1968) 677 17) L. J. Denes, W. W. Daehnick and R. M. Drisko, Phys. Rev. 148 (1966) 1097 18) J. Bar-Tou and I. Kelson, Phys. Rev. 138 (1965) B1035; K. Dietrich, H. J. Mang and J. Pradal, Z. Phys. 190 (1966) 357 19) G. Ripka, Proc. Int. Conf. Nucl. Structure (1966) Gatlinburg 20) T. Une and S. Yoshida, Proc. Int. Conf. Nucl. Structure (1967) Tokyo; T. Une, private communication 21) E. Newman, K. S. Toth and A. Zucker, Phys. Rev. 132 (1963) 1720 22) L. McFadden and G. R. Satchler, Nucl. Phys. 84 (1966) 177; F. P. Brady, J. A. Jungerman and J. C. Young, Nucl. Phys. A98 (1967) 228; W. J. Thompson, G. E. Crawford and R. H. Davis, Nucl. Phys. A98 (1967) 241
DWBA ANALYSIS OF THE HEAVY ION INDUCED
TRANSFER
REACTION
TETSUO KAMMURI and HIROSHI YOSHIDAt Osaka University, Toyonaka, Osaka, Japan Received 18 July 1968 Abstract: The heavy-ion induced transfer reactions have been analysed using the finite-range DWBA. The following reactions are treated for incident energies near the Coulomb barrier: X4N(14N,XSN) 15N, 12C(~eO,~sO)x2C, nB(l~O, a6N)12C and 27A10eO, I~N)~sSi. In most cases the transferred particle is assumed to be bound in a harmonic oscillator potential well. Part of the calculations has been performed for a simple shell model with the Woods-Saxon potential. Good agreement with experiments can be obtained for reactions involving one-nucleon transfer. For the ~-particle transfer process, the calculation shows complicated angular distributions, depending on various parameters. 1. Introduction The D W B A m e t h o d has been a p p l i e d successfully to the quantitative u n d e r s t a n d i n g o f direct nuclear reactions involving light particles as projectiles. H o w e v e r , the transfer reactions between c o m p l e x nuclei have been scarcely analysed b y m e a n s o f the D W B A . T h e r e exists several reasons for this situation. First, there is less theoretical justification for the existence o f an o p t i c a l - m o d e l p o t e n t i a l in describing the nucleus-nucleus scattering t h a n the nucleon-nucleus case. Values o f the o p t i c a l - m o d e l p a r a m e t e r s d e t e r m i n e d f r o m the e x p e r i m e n t a l h e a v y - i o n scattering are very few to be used in the D W B A t r e a t m e n t o f the transfer reactions. F u r t h e r , the z e r o - r a n g e a p p r o x i m a t i o n is unjustified except for the transfer f r o m an s-orbit, a n d one has to c a r r y o u t l a b o r i o u s finite-range calculations. Finally, for energies a b o v e the C o u l o m b barrier, the cores come into close c o n t a c t with each o t h e r a n d the r e a c t i o n m a y proceed from virtually excited states o f the nuclei or b y m a n y step processes with exchange of several nucleons. The D W B A p r o c e d u r e s which were a p p l i e d to the h e a v y ion reaction so far, have different f o r m s f r o m the u n m o d i f i e d D W B A in o r d e r t o c i r c u m v e n t these difficulties. In the m e t h o d o f Breit et al. 1), one calculates the p r o b a b i l i t y a m p l i t u d e t h a t a neut r o n tunnels f r o m the p o t e n t i a l well o f one nucleus to a n o t h e r , while the two nuclei are at rest. T h e relative m o t i o n o f the two nuclei is then t a k e n into account a d i a b a t i c a l ly by means o f the d i s t o r t e d - w a v e m e t h o d . This m e t h o d is p a r t i c u l a r l y useful at energies below the C o u l o m b barrier. Starting f r o m the D W B A expression for the transition a m p l i t u d e , D a r a n d others 2) have o b t a i n e d the diffraction plus C o u l o m b scattering m o d e l u n d e r several simplifying a s s u m p t i o n s . This diffraction m o d e l gives a * Present address: Max Planck Institut fiir Kernphysik, Heidelberg, Germany. 625
626
T. K A M M U R I A N D H . Y O S H I D A
reasonably accurate and simple phen0menological description of absorption in the high-energy region, and good fits to a wide variety of the experimental transfer reactions are obtained. Similar model has also been derived by Kammuri and Nakasima [ref. a)] in the case of multi-nucleon transfer reaction. Recently, Buttle and Goldfarb [ref. 4)] have proposed a simple DWBA treatment, in which the wave function of a neutron in a bound state is approximated by its asymptotic form in the region outside the core. At energies below the Coulomb barrier, where there are no ambiguous parameters inherent to the nuclear distortion effects, this method has proved that the heavy ion transfer reactions are valuable tools for extracting the spectroscopic factors. The aim of this paper is to analyse the heavy ion transfer reactions by means of the finite-range DWBA method which does not involve any additional approximations, and to investigate its usefulness and applicability in the region near the Coulomb barrier, where both the Coulomb and absorptive effects are expected to be important. The formalism originally developed by Austern et al. 5) is outlined in sect. 2 for the simplest case. The angular distributions of the reactions we have treated show different shapes from each other, so that they need different theoretical treatments. Therefore we discuss each transfer reaction leading to the ground states of both final nuclei in a separate section. We treat the reactions 14N(14N, laN)lSN, 12C(~60, 160)12C, 11B(160, 15N)12C, and 27A1(160, lSN)2sSi in sects. 3, 4, 5 and 6, respectively. In the calculations, the bound-state wave function of the transferred particle is assumed to be an eigenfunction in a harmonic oscillator or a Woods-Saxon potential well. In the neutron transfer from the system 14N+ 14N, the effect of the identity of the two 14N nuclei is taken into account. For lower energies than the Coulomb barrier, the major contribution to the neutron transfer comes from the region where the neutron is outside the nuclear surfaces. Since the cross section is highly sensitive to the tail of the radial wave function in such a case, we also treated this reaction by using the Woods-Saxon wave functions. In the elastic scattering of 160 from 12C, the angular distributions show strong oscillatory patterns and an increasing cross section at backward angles. We assume that this part of the cross section arises as a result of ~-cluster transfer from 160 to 12C. In the treatment, we utilize the circumstance that the optical model parameters can be determined so as to give good fit to the forward angular distribution. In the 11B(160, 15N)12C reaction, 160 ion is regarded to transfer a proton or an ~-cluster to i xB. Finally, in the proton transfer resulting from the system of 27A1-[-160, we assume 27A1 and 28Si as deformed nuclei. 2. Outline of the theory The transition amplitude for the reaction A(a, b)B in the DWBA has the form
Tfi =- (BbkblTlAaka) = f dra f drbZ(b-)*(kb,rb)(lBMB,SbmblVllAMA,sama)z(~+~(ka,ra),
(2.1)
HEAVY ION INDUCED TRANSFER REACTIONS
627
where the effective interaction is defined by * < I B M a , S b m b I V I I A M A , s . m . ) = J OsObVOAO.d~.*
(2.2)
Here OB, Oh, 0A, and Oa are the internal wave functions for the particles B, b, A, a whose spins and z components are designated by I BMB, Sbmb, IA MA, S, rn,, respectively. The function Z~+) or )~-) describes the relative motion of the pair A, a or B, b, distorted by the optical potential. In eq. (2.2), ~ represents all coordinates independent of r, and rb. The coordinate system for a A(a, b)B reaction is shown in fig. 1, where x is the transferred particle, and a = b + x , B = A + x . x
A Fig. 1. Coordinate vectors describing A(a, b)B reaction in which a = b+x, B = A+x. The effective interaction can be expanded in a multipole series,
lsj X
(SaSbma,
--
mblsm, -- mb)(-- 1)sb--'bi--tGtsi,,(rb, r,).
(2.3)
The interaction V responsible for the transition is taken to be Vbx(rbx) which accounts for the binding of x and b to form a in the entrance channel. In this section we assume that x is a nucleon or a cluster which can be treated like a nucleon, and that all the participating nuclei are of spherical shape. Then G can be factorized as
Glsjm(rb,
ra) ~--- Alsjflsjm(rb, ra),
(2.4)
where A,,j = it[1] [s.]JO i Of W ( j i I i jf If ; Sx 1)3(s, ji)6(j, jf),
flsjm(rb ,
ra ) ~-- L ( l f l i ~ f /ligf
' - ] ~ i l l / ~ ) ( - - 1) m (])/f#f * (rxA)q~/t.i(Fxb) Vbx(rxb).
(2.5) (2.6)
Here [l] = (2/+ 1)~, and the Jacobian of the coordinate transformation is given by J = [ a B / x ( a + A ) ] z, and 0i, Of are real spectroscopic amplitudes corresponding to a bound x-particle with the orbital wave function q~h,,, 4~,,,~ and angular momentum Ji li a n d j d f in the incident ion a and the residual nucleus B, respectively.
628
T. K A M M U R I A N D H. YOSHIDA
We then define the partial amplitude it[1]fllm( kb ,
fllm(kb,'ka)
by
ka) = f draf drb~(-)*(kb, l'b)flm(l'b, l*a)~(a+)(ka,ra).
(2.7)
The differential cross section is given by da-
~a~b
[I.3 1 ~
kb(
(2.8)
where
atsJ(O) = E I E A~jfltml 2 m
ltlf
(2.9)
The method for calculating film is described in ref. 5) and in the code I N S - D W B A - 3 [ref. 6)]. We used the c o m p u t e r code in which the I N S - D W B A - 3 is improved so as to be applicable to the particular cases treated in the following sections 7). The calculations were performed using the H I T A C - 5 0 2 0 c o m p u t e r of T o k y o University. In the numerical calculations, except where otherwise stated, the fo!lowing assumptions are made. (i) In the present study we consider two extreme approximations for the b o u n d state function. In the first, the wave function of the particle x is taken to be the harmonic oscillator type. For a nucleon, the size p a r a m e t e r v takes the standard value 0.96A -~ fm -e corresponding to the nuclear radius 1.2A + fm. For a Charged particle, the effect of the C o u l o m b force on the binding is neglected. A second prescription for the wave function is to consider it an eigenfunction of a Woods-Saxon well with the binding energy equal to the separation energy. (ii) When we use the harmonic oscillator wave function, the interaction is assumed to be a Gaussian form Vbx(r) = -- Vo e-('/°2,
(2.10)
while, for a W o o d s - S a x o n b o u n d state function, it takes the Woods-Saxon form Vbx(r) = - - V o ( l + e ~ ) - ' ,
x = (r--¢)/a.
(2.11)
"Ihe diffuseness p a r a m e t e r a is fixed at 0.65 fro. Choosing the value of the range close to the radius of the nucleus a = b + x or the sum of the radii o r b and x (with the radius p a r a m e t e r ro = 1.25 fro), we determine V 0 in such a way that the potential Vbx gives the correct binding energy of x in the nucleus a. (iii) The optical potentials describing the elastic scattering of a on A, and b on B are assumed to have the W o o d s - S a x o n shape with the same parameters
U(r) = Vc(r) - V(1 + e xR)-I _ i W (1 + e xx)- 1,
(2.12)
where
Xrt = ( r - R)/aR,
x, = (r-- R')/a,,
R = rR(A~+A~2),
R ' = r,(A~+A~),
(2.13)
629
HEAVY ION INDUCED TRANSFER REACTIONS
and (A 1, A2) = (a, A) or (b, B). The Coulomb potential Vc is of a uniform charge distribution with the radius parameter r c. The parameters which give good fits to the experiments in our calculations are summarized in table 1. TABLE 1 Parametric values used in the calculation Reaction
V (MeV)
W (MeV)
rR (fm)
rI (fm)
aR (fro)
aI (fro)
rC (fro)
Reu t (fro)
v (fro -2)
14N(14N, 13N)15N (a) (b)
50 50
5 25
1.25 1.25
1.25 1.25
0.65 0.65
0.65 0.65
1.25 1.25
7 0
0.25 0.25
12CQeO, IEO)12C
59.5
2.38
1.075
1.55
0.45
0.45
1.25
6
0.5
liB(leO, lSN)12C
59
6
1.10
1.4
0.58
0.58
1.25
0 6
0.38(p) 0.5(~)
ZrA1Q60, 15N)2sSi
50
25
1.15
1.15
0.65
0.65
1.25
0
0.96A-1-
C o l u m n s two to eight list the optical-model p a r a m e t e r s in eq. (2.10), and c o l u m n nine gives the value o f the lower radial integration cutoff. T h e last c o l u m n lists the size p a r a m e t e r o f the h a r m o n i c oscillator wave function o f the transferred particle.
(iv) In discussing the absolute magnitude of the cross section, 01 and 0f in eq. (2.5) are set equal to unity.
3. The 14N(14N, laN)lSN reaction 3.1. E F F E C T O F A N T I S Y M M E T R I Z A T I O N
This process requires special consideration on account of the identity of the two in the entrance channel. The required modifications has already been given by Buttle and Goldfarb 4). We write down only the final results. The antisymmetrized transition amplitude is given by laN
T
=
4n(n + 1){
(3.1)
in which n is the number of outer neutrons in the initial nucleus. When we assume that only one set of values ofji l i a n d j i l r is important, we obtain
trs~j = ~ n(n+ 1)[ht~jlz{lfl,m(O)12 + [firm(re--O)[2 + Sl Re
fllm(O)fl~m(~--O)},
(3.2)
•
(3.3)
m
where At~j is given by eq. (2.5) and /1
S, = 2([jl][jf])2UlJi \ Jf 3.2. C O M P A R I S O N
WITH
Ji Jb
IA
IA
IB
EXPERIMENT
We now compare the DWBA predictions with experimental data given by Hiebert, McIntyre and Couch s) on the 14N(14N, 1aN)lSN reaction at incident energies of
630
T. KAMMURI AND H. YOSHIDA
12.3, 14, 16 and 18 MeV. The neutron transfer is assumed to involve capture from a lp~ level in 14N to a lp~ level in the ground state 15N. Optical-model parameters which are obtained from the optical-model fits to the data of the elastic scattering of 14N from ~4N at energies between 15.0 and 21.7 MeV have some ambiguities. The following sets were given by Porter 9) (a) V = 4 0 M e V ,
W=
8+6MeV,
r o = 1.15fm,
a = 0.6fm,
r c = 1.04fm.
(b) V = 2 0 M e V ,
W=
10+6MeV,
r o = 1.25 fm,
a=0.6fm,
rc =
1.04fm.
Kuehner and Almqvist 1o) have fixed V at 50 MeV in the analyses of the elastic scattering of 14N_b 12C, 14N_k_ 9Be, and a 6 0 "{-1 2 C . They have found that this procedure is justified, since it appears that the scattering can determineat most only two parameters of the real potential well. The optical-model parameters which are used in the present calculation are listed in table 1. In the calculation, the force range ~ is fixed at 3 fm. As for the size parameter v, True ~ ) gives v = 0.32 fm -2 from the analysis of ~4N energy levels. This value is smaller than v = 0.96A-+fm -2 = 0.4 f m - 2 . Fig. 2 shows a comparison of the experimental angular distributions with the DWBA predictions obtained by use of the harmonic oscillator (HO) wave functions. For v equal to 0.25 fm -2, the calculated cross sections aca~c are about ½ of the experimental values aexp. When we increase v from 0.25 f m - 2 , the angular distributions remain unchanged, while the magnitudes of the cross sections decrease and the shape of the excitation function becomes worse. For example, at v = 0.32 f m - 2 , acal¢/aexp is about 1~1 o at 12.3 MeV and 1 at 18 MeV. It is necessary to take a lower value ofv than that given by True in order to get agreement with the experimental cross section. This is due to the strong damping of the harmonic oscillator wave function at large distance. Therefore we also tried the calculation using the wave functions of the Woods-Saxon (WS) well. The results are shown in fig. 3. The calculated angular distributions are barely distinguishable from those given in fig. 2. The absolute magnitude of a¢a~ is about 2 of a~xp (table 2). D W B A predictions taken at 12.3 MeV are quite insensitive to wide changes of V and W. Even the V = W = 0 case gives results that are almost the same as in figs. 2 and 3. At this energy, the major contribution to the cross section comes from the region near the classical turning point of about 12 fm, where the effect of nuclear distortion is expected to be negligible. The direct part of the transition amplitude of eq. (3.1) gives backward peaking, characteristic of transfer reactions in a strong Coulomb field. It should be noted that our result does not show such an oscillatory pattern near 90 ° as the one obtained in refs. ~-4). The effect of varying the optical-model parameters is shown in fig. 4. For V = 20 MeV the cross section remains unchanged, while the cross section falls off too rapidly for large angles when we increase V to 100 MeV. The calculation with W = 35 MeV shows no difference from that with W = 20 MeV. Increase of the diffuseness parameter a amplifies the amplitude of the oscillation pattern.
C
0
O.
!(
o"
3O"
6O°
~CM
~ig. 2. Comparison between the experimental ]4N(14N, 1aN)lSN ection and the theoretical curves at E~ab = 12.3, 14, 16 and 18 -Iarmonic oscillator wave function is used for the neutron b o u n d ['he parameters o f the theoretical curves are given in table 1. Each is normalized to experiment arbitrarily.
3
~0~
.
n
.
cross MeV. state. curve
0°
3o °
60"
0o,4
~'ig. 3. Same as in fig. 2 except that the potential for the neutron b o u n d state and the interaction have the Woods-Saxon shapes.
1.(
)C
- o
o
0"
10 -1
I
i
'
)
/
I
I
30"
" ' / -/- "
i
'
i ,,/ ,/
/
•
\
,\
\"
t
.........
eY /-.,
\
,
_ . . - - - "" ~ ',X,~7
: "../
;_" ~-
I
60"
/
,/'\,
I
I
a 0.65 0.65 0.65 0.8
"
",,'
,,
ec M
V W I00 15 50 15 50 25 50 15
\,J'
-"
/ ' "/ . - - - . , k , _ 1
8MeV
Fig. 4. Effect of variations in the optical-model parameters for the 14N04N, IaN)I~N g r o u n d state reaction at Elab = 18 MeV. T h e size parameter o f the h a r m o n i c oscillator function v is 0.49 f m -z a n d $ =3fm.
b "O
%3
3
6
10
1
'<'NCN j'N )"N
~o \
/
[
,
I
\.j'
/
3o*
f
~i,~ ~-.
\ . . . "2" ( a3~5, 6 )
~
.__
I
t
60"
I
I
\
OCt,4
( o=0z, gfrn"~, Rcut=6fm )
._~ / - - - ~
/
I
.~-~
\
T h e parameters are $ = 3.2 fin, V = 50 MeV, W = 5 MeV, r o = 1.25 fm, a = 0.65 fin.
Fig. 5. Effect of variations in the size parameter v and the radial cutoff Rou t for the *4N(I~N, laN)ISN ground state reaction at Ela b = 18 M e V .
,
x.
( o.L5,61 X....
: ",, \\ / ¢o~3L\,.4j ,,, ~, / . . - - - - .
.<..
2\
0°
10"*
-u 1
b
ui
i (0.4,0)
"~N CN"N )"N
=
o
.z
I-O
HEAVY ION INDUCED TRANSFER REACTIONS
633
In fig. 5, we show the effect of the radial cutoff in the calculation with W = 5 MeV. For W equal to 25 MeV, the effect of radial cutoff is negligible because the incident *N wave is strongly damped inside the target nucleus. In the W -- 5 MeV case, we cannot get agreement with experiment without using the radial cutoff. The change of radius of the cutoff around 6 fm does not affect so much. 4. The 12C(160, 160)t2C reaction 4.1. ALPHA
TRANSFER
PROCESS
The angular distributions of the elastic scattering of 160 on 12C at E l a b = 35 and 42 MeV which have been measured by yon Oertzen et al. i2) show strong oscillations and increasing cross sections at backward angles. Such a pronounced structure cannot be fitted by optical-model calculations. They assumed the s-particle transfer from i 6 0 to I2C to explain this structure. With Dar;s diffraction model, good agreement was obtained. Such a transfer process which has zero Q value and contributes to the elastic scattering is named as resonant transfer by Temmer 13). From the analogy with the molecular collision problem, the resonance-like excitation function may arise from the correlation of the incident energy and the frequency that the transferred particle can go back and forth between two colliding nuclei during their contact time. Such a possibility cannot be ruled out in the present reaction, although strong Coulomb force inhibits both nuclei to approach close together at low energies. Indeed, the backward portion of the angular distribution can be fitted with the function (Pi0(cos 0)) 2, characteristic of the process involving compound nucleus formation with spin equal to I0. Kuehner, Almqvist and Bromley i , ) observed resonances in the elastic scattering excitation function for 160 + I2C below 36 MeV at a center-of-mass angle of 90 ° . However, by the analysis of fluctuations observed in the excitation functions at fixed angles, yon Oertzen et al. have concluded that a contribution due to a compound nucleus formation is smaller than 5 % at 139 °. Since we aim at the test of the applicability of the D W B A method, we treat the structure in the backward angle by assuming 2p2n transfer from 160 to 12C in the framework of DWBA. The optical-model parameters can be determined from the forward portion of the angular distribution. The values obtained in ref. 12) are used in our calculation. While this offers a very advantageous situation for applying the D W B A method, we have relied on the s-cluster approximation for the transferred 2p2n system. This approximation implies that the s-cluster is fully space symmetric and the principal quantum number and the orbital angular m o m e n t u m of the motion relative to the core are restricted to those compatible with the simple shell-model wave functions of the constituent nucleons. If the harmonic oscillator wave functions are adopted, 12C and alpha are in a relative 3s state in the 1 6 0 nucleus. The effects of higher shell-model configuration admixture to the 160 ground state wave function will mix states with other quantum numbers. For example, it may be possible to con-
634
T. KAMMURI AND Y. YOSHIDA
sider components in which the 0 + alpha and ~2C are in the 2s state or 2 + excited alpha and 12C are in the 2d state. The calculation shows that both admixed states have nearly the same angular distribution as in the case of 3s motion, but the cross section for the 2s case is very small. We generate the a-cluster wave function as a solution in a harmonic oscillator potential. The size parameter we used are of three types: (i) v1 = 1.143 fm -2. This corresponds to (4(A-4)/A)xO.96A -~ fm -2 given by recoupling four nucleons in shell-model states by using Talmi and other coefficients (A = 16 in the present case). (ii) v2 = 0.71 fm -2. The radius RreI of the corresponding harmonic oscillator well is very close to R~2c+R, = 4.6 fro. (iii) v3 = 0.5 fm -2. The a-particle is assumed to be bound very loosely to 12C with Rre~ = 6.8 fm. The approximation that ~60 can be regarded as a two-body system consisting of 12C and alpha (strongly correlated four-nucleon system) holds only for the region where 12C and alpha do not overlap so much. In the overlapping region, on the one hand, the repulsive force acts due to the effect of the exclusion principle, and on the other hand the ~-cluster tends to dissolve into particles in the shell-model states. Thus the a-cluster wave function damps much more rapidly inside the nucleus and has larger values outside the nuclear surface than in the case of the harmonic oscillator approximation. Under such circumstances it may be natural to use v smaller than vt, and also adopt radial cutoffs in the D W B A calculations of the c~-transfer process. As for the force range ~, we take 3.4 fm and 4.6 fm. The former value corresponds roughly to the radius of ~60, while the latter is equal to R~c+R,. 4.2. C O M P A R I S O N
WITH
EXPERIMENT
Fig. 6 shows comparison between the calculated angular distribution and the experimental result at 35 MeV incident energy 12). The interference between the elastic scattering and the c~-transfer is not taken into account. The calculated a-transfer cross section increases in the backward direction, though the positions of the minima have some deviations from the experimental ones. The cross section at 155 ° obtained by using the harmonic oscillator function with v = 0.5 fm -2 and the interaction range = 3.4 fm is estimated to be about ~ of the experimental value, while that using the function of the Woods-Saxon shape (4 = 4.4 fm) has the right order of magnitude. We have investigated the effects of variations in the cutoff radius, 4, and v. From fig. 7, we see that use of no cutoff smoothes the angular distribution and increases the backward cross section. By using the cutoff we find that the magnitude of the cross section is reduced though very deep minima can be obtained. The use of a radial cutoff at 7.5 fm results in reduction ot the predicted cross section by two orders of magnitude as compared with the case of 6 fm cutoff and the oscillatory pattern becomes weak. It shows that the main contributions to the reaction cross section comes from the
30*
60*
If
90 °
120°
<:.: 150° Oct,4
Fig. 6. Comparison between the cross sections of the experimental ~C(t~O, ~aO)~zC ground state reaction and the theoretical curves at Etch-- 35 MeV (normalized). The dashed and solid curves were obtained using Woods-Saxon (~ = 4 . 4 fm) and harmonic oscillator wave functions, respectively The parameters of the theoretical curves are given in table 1.
<
,/
10
0o
_
,,
•
;
,
,;
/ ',.
/
',.,.,,'
60°
/
{',,,,'1 /"
lit, ,11',
"'.
I
35MeV
Ij
,^,
90"
I
u
I.',"
If
I
'J-
~j
i
',
i
I
J
120" 150°OCM
_/
,!
I
(7.5)
(6}
i/i/
i
~ ~4~ I
(Rcul=Ofm)
Fig. 7. Effect o f variations in the radial cutoff for the ~-transfer part of the lzC(160, 180)12C ground state reaction at E]a ~ = 35 MeV. The optical-model parameters are given in table I, and v = 0.71 fm -2, = 3.4 fm.
10
'13 "b o
""
~
"4
E
"9
cat.
35MeV
~°
~,
't("o/'o)%
z
.4
> .<
636
T. KAMMURI AND H. YOSHIDA
region of the internuclear distances between 6 fm and 7.5 fm, where both nuclei are in close proximity. lncrease of ¢ from 3.4 fm to 4.6 fm increases the peak cross section by almost a factor 5, whereas the angular distribution is little affected. Shifting of the diffraction pattern to larger angles for larger v can be observed. Variation of v from 0.5 to 0.71 and 1.14 f m - 2 , decreases the peak cross section by one and four orders of magnitude, respectively. We have shifted the position of maxima by varying v and have obtained deep minim a by use of radial cutoff. We have found loose binding of the s-cluster in 160 as appropriate, which is reflected in the use of small v or large ¢ in the H O or WS case, respectively. This is contrary to the conclusion of the diffraction model calculation ~2). In the latter, it was necessary to assume a much higher binding energy for the transferred s-cluster in 16 0 of about 25 to 35 MeV. This corresponds to a very rapid decrease of the wave function of the a-cluster outside the nucleus.
5. The UB(160, lSN)12C reaction 5.1. EFFECT OF ANTISYMMETRIZATION The angular distributions of 15N from the b o m b a r d m e n t of ~IB by 27, 30, 32.5 and 35 MeV 160 ion were measured by Bock et al. 15). The 15N cross sections have strong forward and backward peaks. This suggests that the reaction proceeds by two types of transfer process, namely, a proton and an a-cluster transfer from 160 to x~B. G o o d agreements with experiments were obtained by the diffraction model. In the following simple model we represent the angular distribution as a coherent sum of the above two processes. First we treat the transferred 2p2n system like a single a-particle as in sect. 4. Next the nuclei ~6G, ~SN and ~2C are assumed to consist of the 1~B core and the extra p + a, a and p, respectively. The effect of the exclusion principle between the outer particles and the core is ignored. Then the 11B(160, 15N) 12C reaction can be written as 'lB+(11B+p+ct) ~ (t'B+p)+("B+a).
(5.1)
Considering ~IB as a single fermion, we take account of the antisymmetrization of two ~ B (written as X~ and X2). The transition amplitude is given by Tfi =
(f[Vf[P12i)
= (X1 + p , X2 +ctlVflXl, X2 + p + a ) -
( X l + p , X2 +alVflX2, X1 + p + a),
(5.2)
where P12 is the antisymmetrization operator acting o n X 1 and X 2. The interaction Vf is remainder of the interaction between two nuclei in the final state after subtracting the potential U(X 1 + p , X 2 + a ) which generates the distorted wave in the exit
637
HEAVY ION INDUCED TRANSFER REACTIONS
channel. This interaction can be written in two ways, V, = V(Xl + p , X2 + ~ ) - U(X1 + p , X2 +~)
= V(p, X 2 + a ) + { V ( X 1 , X 2 + a ) - U ( X I + p , X2+a)}
(5.3a)
= V(XI+p,a)+{V(XI+p,X:)-U(XI+p,X=+~)}.
(5.3b)
The first term in the transition amplitude, eq. (5.2), corresponds to the proton stripping from 160 = X 2 + p + ~. We adopt the simplest approach in which Vf is approximated by V(p, X 2 + a) = V(p, 1SN) in eq. (5.3a). The contribution from the remaining V(HB, ~ 5 N ) - U ( 1 2 C , 160) is expected to be relatively unimportant in the same way as in the case of the deuteron stripping reaction. In the second term of eq. (5.2) which represents a particle stripping from 160 = X1 + p + ~, we assume that V(~, X 1 + p ) = V(~, ~2C) is of most importance in eq. (5.3b). There is far less basis for the neglect of the contribution V(12C, 1~ B ) - U( 1:C, 160) . In the a-transfer process, 160 and ~SN are assumed to be two-body systems consisting of 12C and a in 3s state, and of 11B and a in 2d state, respectively. 5.2. C O M P A R I S O N
WITH
EXPERIMENT
Comparisons of the D W B A predictions with experimental data at the incident 16 0 energy of 27 and 30 MeV are shown in fig. 8(a) and (b). Use of the same parameters for 27 and 30 MeV (table l) does not reproduce the variations of the angular distributions with energies for the a-transfer part. In the calculation the interference between p and a-cluster transfer process is ignored. 5.2.1. Proton transfer. The optical-model parameters which fit to the elastic scattering of 160 from 11B at the center-of-mass energies ranging from 7.00 MeV to 11.00 MeV are given by O k u m a 16) as V = 59 MeV, W = 6 MeV, rR = rl
=
rc =
1.10 fm,
aR = al = 0.58 fm. For this set of parameters, the angular distribution of p transfer shows smooth forward peaking. When we vary the values of parameters around this set, for example, change a to 0.4 fm or q to 1.4 fm as in table 1, a weak diffractive oscillation can be seen. In fig. 8 where q is taken to be equal to 1.4 fm and v = 0.96A-+fro -2 is used, the cross section calculated with H O wave functions is about 0.3 of the experimental magnitude. In the WS case, these ratios are about 1.4 (27 MeV) and 2 (30 MeV). A potential set with V = 50 MeV, W = 25 MeV, r R = q = r c = 1.25 fm, aR = a~ = 0.65 fin gives practically the same angular distribution as shown in fig. 8. However it reduces the absolute value by almost two orders of magnitude. Since the height of the Coulomb barrier of the system 1 1 B + 1 6 0 is 8.1 MeV for r o = 1.5 fm, the incident energy of 27 MeV (11 MeV in the c.m. system) is sufficiently high for two nuclei to come close together. Therefore one should expect the use of the harmonic oscillator
0°
-
30"
: / ' ~
•
~2
60"
I
, 90"
(a)
, 120°
",,
Z I'~ 153" OCM
o4 transfer
//'t , ~
t{" k
p transfer
27 MeV
~5 15
g(0, N ) g
i
O"
10"z
10 4
"ID
"o
JD E
I
30"
t
60*
I
tl
p transfer
90*
I
I v
30 MeV
"BCo,"N)"C
I
120"
V
V
t'
o~transfer
I
150°0C M
~
',,L/I
(b)
I
Fig. 8. C o m p a r i s o n between the cross sections o f the experimental " B Q 6 0 , a~N)12C g r o u n d state reaction a n d the theoretical curves at Elab = 27 MeV (fig. 8(a)) a n d 30 MeV (fig. 8(b)). T h e d a s h e d a n d solid curves were obtained by using W o o d s - S a x o n a n d h a r m o n i c oscillator wave functions, respectively. For the c~-transfer part, H O calculation uses v = 0.5 f m -~, while W S o n e uses ~ = 3.9 fro. T h e p a r a m e t e r s o f the theoretical curves for p a n d c(-transfer processes are given in table 1. All curves are arbitrarily normalized.
10"
10q
1
H
HEAVY ION INDUCED
639
TRANSFER REACTIONS
wave function to give accurate result also for the absolute cross section within the D W B A framework. 5.2.2. Alpha cluster transfer. Sincethe ~-transfer process contains uncertain parameters, it is not easy to k n o w whether agreement with experimental data is really significant or not. Conversely, the sensitivity of the D W B A predictions to these parameters may be utilized for the investigation of the structure of the four-nucleon cluster. As shown in fig. 8, the angular distributions of a-transfer do not show backward peaking and strong oscillatory behavior, in contrast to the case treated in sect. 4. I''
~. transfer ," zero-range calc. ~ .
E~("0,~N )'2C 27 MeV
10
/ I
,r\ l
~n
l,j~
:=ro 1
If ~\ ,.,
Il
,,j
b "13
Rcut=6fm
10 "1
S
xlO .....,.
;' ',...' / ',, ",j
I 0"
i 30 °
I 60"
V 90"
1 120 °
/
I 150" OCM
Fig. 9. Z e r o - r a n g e D W B A p r e d i c t i o n s a n d effect o f v a r i a t i o n s in radial c u t o f f for the or-transfer p r o c e s s in the l i B ( l e O , lSN)x2C r e a c t i o n at Elab = 27 M e V . T h e p a r a m e t e r s used are V = 50 M e V , 14" = 25 M e V , r0 = 1.25 fm, a = 0.65 f m (a = 0.8 f m for Reu t = 0) a n d v = 1.2 f m -2.
Calculated cross sections (v = 0.5 fm -2 in the HO case) are of the same order of magnitude as the experimentally observed cross sections, apart from the spectroscopic factor for effective a-clustering. The magnitudes of the cross sections are very sensitive to the value of v when we use the harmonic oscillator function. For example, it decreases by about two orders of magnitude for the increase of v from 0.5 to 0.71 fm -2.
We now discuss the effect on the D W B A calculations of variation of the force range and cutoff radius Re, t. Since alpha and 12C are in a relative s-state in 160, it is pos-
o"
104
--
"
/
'\
\"/
JE
¢~
I 30"
,,-.,.
\'
r,,
I 60"
\./
I 90*
,/'"
,' ,,
/
"',-;
/
f',
( 4 )
I 120 °
)
V
/
1
i~ /,t,
/
//
/ [
8cH
"/l
I 150 o
( R c u t= 6 f m
/~,
i
o~ transfer
\,,wf "~\ /."-'. U
.
27 N e V
~ ( O, N ) ~
~ la
Fig. 10. Effect o f variations in radial cutoff for the c~-transfer p r o c e s s in the 11B(~sO, 15N)12C g r o u n d state r e a c t i o n at Elah -- 27 M e V . T h e p a r a m e t e r s used are V = 50 M e V , W = 25 M e V , ro = 1.25 fro, a = 0.65 fm and v = 0.71 fm -~, ~ = 3.4 fm.
"o
m
~T
"E
u~
10
tl
10"
0°
,.
30"
t
...... -"
. "
"'/
60"
I
I
.-xlO =
90 °
Rcut = 8fro
~ . /:-"..-.,/6
I
120 °
..,
I
150"
\ iI
eC M
F",,
"\ ,,j/"~"!/(
,~ transfer
given by O k u m a are used; V - - 59 M e V , W = 6 M e V , r o a -- 0.58 fro, and v = 0.71 f m 2, ~e = 3.4 fro.
1.10 fro,
Fig. 11. Same as in fig. 10 except that the optical-model parameters
b lO
~3
(D
"E
27 b4eV
"B("O,"N)"C
O
£
10 "t
0*
'*\ ///
"1
~
/
30"
,"'
60*
\\
90*
~) = 1 fm
b \ --.~" --"
4.6
120"
\../
c~ transfer
150" eC M
//
F i g . 12. Effect o f v a r i a t i o n s in f o r c e r a n g e ~ f o r t h e co-transfer p r o c e s s in t h e ttB(x~O, t~N)x2C g r o u n d s t a t e r e a c t i o n a t Ela b = 27 M e V . T h e p a r a m e t e r s u s e d a r e I/ = 50 M e V , W = 25 M e V , r0 = 1.25 fro, a = 0 . 6 5 f m a n d v = 1.2 f m -2, Reu t = 0.
b 1D
::, 1
ul E3
10
27 MeV
"B(~):'N)"C
10-'
O"
f5
I 30"
// V
I 60"
I 90"
I ]20 °
Id
~ \
I 150" OCM
, '\
i
6 fm. Effect of variation
-_,---,,
'W
,~ transfer
3.4,0.71 "~"/~'\ /f--,j
12
~:3.4fm,l)=l.2fm-Z
•\!,~,\
27MeV
la
B( O. N) C
F i g . 13. S a m e as in fig. 12 e x c e p t f o r Reu t : i n v is a l s o s h o w n .
b 13
"o
10
II
642
T. KAMMURI AND H. YOSHIDA
sible to use a zero range force for V(~, 1 2 C ) . I n the a-transfer model of the 19F(d, 6Li) 15N reaction, Denes, Daehnick and Drisko 17) have found the similarity between the zero-range and the finite-range calculations. In our case, the calculated angular distributions in the zero-range approximation differs completely from the finite-range treatment. Fig. 9 shows the zero-range results by use of v = 1.1 fm -2 and V = 50 MeV, W = 25 MeV, r o = 1.25 fm, and a = 0.65 fm. Strong dependence on the position of the cutoff radius can be seen. The zero-range approximation overestimates the contribution from the nuclear interior. For instance, the cutoff below 4 fm already changes the angular distribution and the cutoff at 6 fm completely diminishes the cross section. The effect of the variation of the radial cutoff is illustrated in fig. l0 (for W = 25 MeV) and fig. 11 (for Okuma's parameter set 16)) in the finite-range case. In the backward angles, the cutoff below 6 fm has relatively little effect. As is seen, the angular distribution with 6 fm cutoff in fig. l0 agrees well with the experimental data, but the magnitude of the cross section is about ~o of the latter. Finally, the effect of the variation of ~ is shown in figs. 12 and 13 in the case of Rcut = 0 and 6 fm, respectively. Although the variation of ~ produces changes in the backward angular distribution in the case of Rcut = 0, its effect is rather small for Rcut = 6 fm case. F r o m fig. 13, the effect of change in v can also be seen. 6. The 27A1(160, tSN)2sSi reaction 6.1. SPECTROSCOPIC FACTOR FOR DEFORMED NUCLEI We treat the 2TAI(160, 15N)2sSi reaction as the proton transfer process from t 60 to the well.deformed 27A1 nucleus resulting the well-deformed 2SSi nucleus. There have been several arguments about the deformation of both nuclei. According to the Hartree-Fock calculation for 2sSi [refs. 18,19)], the configuration with oblate shape has lower energy than the prolate one. The 27A1 nucleus is usually treated as prolate, inferring from the positive sign of the measured quadrupole moment. Ripka 19) has regarded the 27A1 ground state as the I = ~ state of the K = ½ band corresponding to one hole state of the oblate 2aSi ground state. In this case too, we can expect a positive quadrupole moment for 27A1, and moreover the value of the magnetic moment agrees with the experimental value. However the decoupling parameter must be stronger than that given by the Nilsgon model in order to bring the I = ~ state to the lowest. Accordingly, the ground state configuration composed of 2s½, ld~ and ld~ single-particle states differs from the [220] 1+ state in the Nilsson model. In the strong coupling model the nuclear system has a definite value of deformation and is not an eigengtate of the total angular momentum. In the transition from the 27A1 to the oblate 28Si, more than ten particles j u m p to orbits almost orthogonal to the original orbits. The probability of such a transition is completely negligible. However when we construct an eigenstate of the angular m o m e n t u m from a linear corn-
643
HEAVY ION INDUCED TRANSFERREACTIONS
bination of intrinsic wave functions by means of the projection method 20), the overlap between the system with the oblate shape and that with the prolate shape is not so small. According to Une and Yoshida 2o), the overlap integral between the positive I = 0 state (6 = 0.3) and the negative I = 0 state (6 = - 0 . 3 ) amounts to 0.0987 for the 28Si system. In this subsection, we calculate the spectroscopic factor assuming that the transition occurs for the oblate 27A1 to the oblate 28Si, both having the same value of deformation. Using the Nilsson model, the wave functions are written as tIIIAMAKA(27A1) = ~ _ ~ c~(A) (¢~)[ZKA(C)DMAKA [A IA --K.], + ( - - 1)IA--½I-Izz--K~(C)DMA,
47Z
~ , . M . K . (2S8 . , ) =
~B)(¢c)[Det (x
Ac),
(6.1)
'"
where x and c express the transferred proton and the proton originally contained in 27A1, respectively, and ~v(~c) denotes the vibrational state of the core. The singleparticle states ZK is written in the body-fixed reference frame as XKA = E ClfjfKA ~/lfjfKA(r')" lf jr
(6.2)
The parity of the intrinsic state XK is denoted by fi x = ( - 1 ) z . Using the explicit form for the wave functions, we obtain Gtsis as
Ggsjm = ( - - 1)sx-s' ~ Atsj(lfjf)flsj(rb, ra) ,
(6.3)
lf jr
where in eq. (2.5) for Atsj, Of is taken as
0f
=
~.(A)x [-IA][-l.l- 1/~.(B) \Wv ~ev / c,~y~, _KA(IAjfKA
-- KAIIB0). (B)
In the numerical calculation we put the overlap of the core part <~bv
(6.4) (A)
I~v > as
unity.
6.2. COMPARISON WITH EXPERIMENT
The proton transfer in the 27A1(160, 15N)28Si ground state reaction is assumed to involve capture from alp½ level in 160 to a level composed of 2s~, I d I and ld~ in 2aSi. Taking account of the fact that the ground state spin of 27A1 is ~2, only the ld~ component is effective for the transition. For the transferred angular momentum, l = 2 and 3 are possible. Since the l = 3 transfer is considerably smaller than 1 = 2 case, we discuss only the l = 2 transfer. We used the optical-model parameters listed in table 1. The comparison with the optical-model calculation with the experimental data of the elastic scattering of 32 MeV 160 on A1 target 14) is shown in fig. 14. The experimental results obtained by Newman, Toth and Zucker 2I) are compared in fig. 15 with the DWBA predictions. The calculated cross section are about two orders of magnitude smaller than the experimental values. The origin of the disagree-
I
30"
]
60 °
I
90 °
~
•
E,~0 : 32MeV
120°
I
36(25 )
\30 ( 5 )
150° ec M
I
• \\30 MeV (W:25MeV) ' \ "\.
Fig. 14. C o m p a r i s o n between d a t a for the elastic scattering of 32 MeV ]~O from AI and the predictions o f the optical pot e nt i a l used in ZTAl(leO, ]SN)zsSi calculations.
0°
10-~
10"t
~R
At ('0;'0) At
0"
10"
~ 30"
~,/I
I
60"
I
90"
120"
I~
I
150°
ec.
'LI,~ "/ ~\
!
36MeV
Fig. 15. C o m p a r i s o n between the cross sections o f the ex p er imen tal 2~A1(160, 15N)28Si g r o u n d state re a c t i on an d the theoretical curves at E~a b = 30 a nd 36 MeV. The p a r a m e t e r s o f the theoretical curves are given in table 1. F or 30 MeV case, the da shed curve was o b t a i n e d using V -- 50 MeV, W -- 5 MeV, ro = 1.25 fro, an d a = 0.65 fin. All curves are a r b i t r a r i l y norma lized .
b
E
A
i0"i
645
HEAVY ION INDUCED TRANSFER REACTIONS
ment may be ascribed to the use of the large imaginary potential and the use of the harmonic oscillator wave function at energies near the Coulomb barrier (18.7 MeV for r o = 1.45 fm). Experimental peak cross sections are 1.35 and 1.37 mb for 30.0 and 36.0 MeV energy respectively, while our result increases by a factor 3 in this energy interval. This may be due to our use of small r o (1.15 fm) in order to fit the position of the peak of the angular distribution. The calculation with V = 50 MeV, W = 5 MeV, r 0 = 1.25 fm and ~ = 0.65 fm gives right order for the absolute value, but the position of the peak shifts to smaller angle than the experiment as can be seen from fig. 15. The calculation shows oscillations at forward angles. Such oscillation was also obtained by Dar z). However, in his diffraction model the oscillations are stronger at 30 MeV than 36 MeV, contrary to the present result. For this reaction, calculation using the WS wave function is still incomplete because of the difficulty in the numerical integration in eq. (2.7), and cannot produce reliable results. TABLE 2 C o m p a r i s o n of the absolute m a g n i t u d e s of the e x p e r i m e n t a l cross sections w i t h the D W B A p r e d i c t i o n s using W o o d s - S a x o n wave function Reaction
Lab.energy (MeV)
c.m. angle
80 ° 80 ° 80 ° 80 °
Exp. value (mb)
14N(14N, IaN)I~N
12.3 14 16 18
( 3 . 7 ± 1.4) (3.6±0.4) (2.1 1 0 . 3 ) (3.6±0.2)
x 10 3 X 10 -2 × 10 -1 x 10 1
12C(160, I°O)1~C
35
170 °
14
27 30 27
33 ° 20 ° 141.5 °
1.4 +o.2 -o.1 0.8 ! 0 . 0 6 0.185 i 0 . 0 4 5
Calculation (mb) 1.4 x 10 -a 2.7 X 10 2 1.6 x 10 -1 2 . 9 x 10 -1 12.6
11B(160, 15N)1~C p transfer transfer
1.9 1.7 0.24
7. Concluding remarks We first summarize in table 2 the comparison of the absolute values of experimental cross sections with the calculated ones, putting 0 i and Or in eq. (2.5) equal to unity. A bound state wave function in the Woods-Saxon potential well is used, and the effect of Coulomb force on binding is taken into account. Distorting optical potentials used in table 2 are those listed in table 1. The results of the present investigation indicate that the D W B A theory of the heavy ion transfer reaction is capable of predicting both the shape and magnitudes of these cross sections. For the exceptional case of the 27A1(160, l S N ) 2 8 S i reaction, our analysis is still incomplete. Our ignorance about the properties of the optical-model potential between any complex nuclei offers the greatest obstacle for the application of the D W B A method. This will be partly overcome by the accumulation of more experimental data on elas-
646
T. K A M M U R I A N D H. YOSHIDA
tic scattering. However, it can be expected that the a m b i g u i t y in the values o f the optical-model parameters such as observed i n the elastic scattering of s-particle 22) is also present in the heavy i o n case. W e have a p p r o x i m a t e d the transferred 2p2n system b y a single a-cluster. The ~transfer process in the 11B(160, 15N)12C reaction shows a wide variety of a n g u l a r distributions depending o n the value of the parameters. Such a sensitivity of the cross section to p a r a m e t e r choice will serve as a useful guide for investigating the properties of the s-cluster wave function. The a u t h o r s are grateful to Professor S. Y o s h i d a for stimulating discussions, to Dr. T. U n e for i n f o r m i n g us his calculation of the overlap integrals, a n d to Miss K. M a t s u u r a for preparing the manuscript. They also wish to t h a n k Professor R. Bock for kindly c o m m u n i c a t i n g the results of the experiments at the Max Planck I n s t i t u t ftir Kernphysik, Heidelberg.
References 1) G. Breit, J. A. Polak and D. A. Torchia, Phys. Rev. 161 (1967) 933 and references therein 2) A. Dar, Phys. Rev. 139 (1965) Bl193; A. Dar and B. Kozlowsky, Phys. Rev. Lett. 15 (1965) 1036 3) T. Kammuri and R. Nakasi~a, Proc. Conf. Reactions complex nuclei, Asilomar, 3rd, 1963, 163 4) P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 5) N. Austern, R. M. Drisko, E. C. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3 6) T. Une, T. Yamazaki, S. Yamaji and H. Yoshida, Code INS-DWBA-3 7) H. Yoshida, unpublished 8) J. C. Hiebert, J. A. McIntyre and J. G. Couch, Phys. Rev. 138 (1965) B346 9) C. E. Porter, Phys. Rev. 112 (1958) 1722 10)' J. A. Kuehner and E. Almqvist, Phys. Rev. 134 (1964) B1229 11) W. W. True, Phys. Rev. 130 (1963) 1530 12) W. yon Oertzen, H. H. Gutbrod, M. Mtiller, U. Voos and R. Bock, Phys. Lett. 26B (1968) 291 13) G. M. Temmer, Phys. Lett. 1 (1962) 10 14) J. A. Kuehner, E. Almqvist and D. A. Bromley, Phys. Rev. 131 (1963) 1254 15) R. Bock, M. GroBe-Schulte, W. yon Oertzen and R. Riidel, Phys. Lett. 18 (1965) 45; R. Bock, M. GroBe-Schulte and W. yon Oertzen, Phys. Lett. 22 (1966) 456 16) Y. Okurna, J. Phys. Soc. Japan 24 (1968) 677 17) L. J. Denes, W. W. Daehnick and R. M. Drisko, Phys. Rev. 148 (1966) 1097 18) J. Bar-Tou and I. Kelson, Phys. Rev. 138 (1965) B1035; K. Dietrich, H. J. Mang and J. Pradal, Z. Phys. 190 (1966) 357 19) G. Ripka, Proc. Int. Conf. Nucl. Structure (1966) Gatlinburg 20) T. Une and S. Yoshida, Proc. Int. Conf. Nucl. Structure (1967) Tokyo; T. Une, private communication 21) E. Newman, K. S. Toth and A. Zucker, Phys. Rev. 132 (1963) 1720 22) L. McFadden and G. R. Satchler, Nucl. Phys. 84 (1966) 177; F. P. Brady, J. A. Jungerman and J. C. Young, Nucl. Phys. A98 (1967) 228; W. J. Thompson, G. E. Crawford and R. H. Davis, Nucl. Phys. A98 (1967) 241