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Fusion cross sections for 6Li+12C and 6Li+13C reactions at low energies
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 635 (1998) 305-324
Fusion cross sections for 6Li+12Cand reactions at low energies
6Li+13C
A. Mukherjee, U. Datta Pramanik, S. Chattopadhyay, M. Saha Sarkar, A. Goswami, P. Basu, S. Bhattacharya, M.L. Chatterjee, B. Dasmahapatra Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta-700 064, India
Received 1 October 1997; revised 11 February 1998; accepted 9 March 1998
Abstract The partial cross sections for the 6Li + lZc and 6Li + ~3C reactions below and above the Coulomb barrier energy have been determined from the cross sections of the characteristic y-rays measured with HPGe detectors and using the branching factor o'r/o'ch obtained from the statistical model calculations. The fusion cross sections obtained from the sum of these partial cross sections are found to be equal to the total reaction cross sections upto an energy well above the Coulomb barrier energy and there appears to be no evidence for limitation of fusion cross sections for these systems at such energies contrary to the evaporation residue measurements. The measured cross sections are also found to agree nicely with the IWBC (Incoming Wave Boundary Condition) and one-dimensional BPM (Barrier Penetration Model) calculation. © 1998 Elsevier Science B.V. PACS: 25.70.Gh; 25.70.Jj Keywords: NUCLEAR REACTIONS 12C(6Li,X), E(cm) = 2.01-8.02 MeV, 13C(6Li,X),
E(cm) = 2.07-8.23 MeV; measured Er, I:,; deduced partial, total fusion o'. Statistical model analysis, Optical model, IncomingWave BoundaryCondition model and one-dimensionalBarrier Penetration Model calculations
1. Introduction In a previous article [ 1 ] from the measurement of the characteristic y-rays of the residual nuclei following fusion of 7Li + 12C and 7Li + 13C systems at energies below and above the Coulomb barrier, the cross sections for different exit channels were determined and from the sum of these, the total fusion cross sections for the two systems were obtained. From the comparison with statistical model calculations it was concluded that the two reactions mainly proceed via compound nucleus formation and 0375-9474/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0375-9474(98) 00 164-X
306
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
its subsequent decay. Among various exit channels om and pn channels were found to be the principal contributors to the fusion cross sections. It was further found that the measured cross sections could nicely be fitted by the total reaction cross sections calculated with the optical model potential with its parameters obtained from fitting the elastic scattering data. This in turn indicates that at subbarrier energies fusion is the dominant reaction mechanism and there is no inhibition of fusion cross sections contrary to the heavy-evaporation residue measurements [2]. In order to see whether similar reaction mechanism prevails also in the case of 6Liinduced reactions, we thought it will be worthwhile to investigate the 6Li + 12C and 6Li + J3C reactions in the similar energy range using the same technique of measurement. Another motivation of the present work was to find out whether the fusion cross sections for the 6Li induced reactions are less than those for the corresponding 7Li induced reactions also at subbarrier energies, which has earlier been observed at higher bombarding energies through the heavy evaporation residue measurements [2]. In fact in this latter measurement the ratio of these cross sections is found to be _~0.80-0.85 throughout the measured energy range [Ec.m. -~ 6 - 2 5 MeV] whereas the optical model calculations predict nearly equal cross sections for them. It may further be mentioned that so far there has been no fusion or total reaction cross section measurement for these systems at subbarrier energies.
2. Experimental details The measurements were done with 6Li 2+ and 6Li 3+ beams obtained from the 3 MV pelletron accelerator of the Institute of Physics, Bhubaneswar. The J2C foils ( ~ 7 0 /.~g/cm 2) were natural as well as enriched (99.9%) and the 13C foils ( ~ 6 0 /.zg/cm 2) were enriched (98.9%). The measurements were done with two scattering chambers. The old chamber used earlier [ 1] has been replaced by a new chamber specifically designed for the cross section measurements. The new scattering chamber having a diameter of 15 cm was thinned down at two places (centred at 125 ° and 90 ° with respect to the beam direction) to facilitate the exit of low energy y-rays without much absorption in the chamber wall. A stainless steel tube, 2.5 cm in diameter and 45 cm long lined inside with a tantalum foil, insulated from a grounded outer cylinder (also made of stainless steel) and the target chamber, and protected with an electron suppression ring at - 2 0 0 V, served as the Faraday cup. In addition, instead of using one heavy ion surface barrier detector (for recording elastic scattering events) at 45 ° in the previous chamber, provisions were made to place four such detectors two on each side of the beam axis. Viewed along the beam direction, the detectors could be placed at 29.6 °, 46.1 °, 30.8 ° and 43.8 °, respectively. This arrangement helped in more accurate monitoring of the beam. The thicknesses of the targets were also accurately determined from the elastic scattering spectra obtained with these detectors using 2-4 MeV Li-ions
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
15
A
A
A
~ ,k
307 A ,k
7,N
0.115 6.24
10
4.7? ]
]/
i
7.12 e.t3
i
2.38
0.72 2.1528131r'O+t• L?O~
__ ,,~d %+z~ tSN+an
v
0
1'g2
_2~
?.03 -5
l] ITF+n I I 0.870.87 2.31 ~ -1' 80~ L_.,,:o÷p
-10
-1,241,
C+ap
N+a
.
18F .-13.3t2 Fig. 1. Energy level diagram for the 6Li + laC reaction. The numbers attached to the ground states give the Q-value of the respective channels. The region investigated is cross-hatched. The 3t-transitions which are seen in the spectrum are shown by vertical lines. The highest levels indicated are particle unstable.
and 10-12 MeV 160 ions. The measured thicknesses of the 4 targets were as follows: ]2C - 68.1 4-4.4, 73.1 + 4 . 7 / z g / c m 2, 13C - 60.7 + 3.9, 64.0 + 4.1 / z g / c m 2. Two HPGe detectors each of ~ 9 0 cm 3 volume were placed at 125 ° and 90 °, respectively, with target-to-detector distance ~ 1 2 cm. The y-ray cross sections were mainly determined from the spectra obtained with the detector at 125 °. The detector at 90 ° gave unshifted y-rays and helped in the proper identification of them. The measurements were done at E]ab = 3.11--12.07 MeV in steps of 500 keV to 1 MeV for both the reactions. These correspond to Ec.m = 2 . 0 1 - 8 . 0 2 MeV for 6 L i + 12C and 2.07-8.23 MeV for 6Li + 13C reactions after making necessary correction for energy loss in the targets. Figs. 1 and 2 show the excitation of the relevant compound nuclei corresponding to the above centre of mass energy and their successive decay to different exit channels.
308
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
1.0-
4
I [
~
]
?.0 4.56
5.51 tl.ol
~o
5,0~g ,,,4t44
0.0113
7.12
t?F+2 n
e Li+mC • ,,
~
-2.669
_~.,,"o+ p
- ~ z nB+2a
-5
2+ap 6.9~
teO+ a H laF+ n 5.27
-10
180+ p
-14,697
-15 1
Fig. 2. Energy level diagram for the 6Li + 13C reaction. For other details see caption of Fig. 1.
3. Analysis and results 3.1. The y-ray cross sections
Fig. 3 shows typical y-ray spectra of the two reactions. In this figure the y-rays other than arising from 6Li + 12C and 6Li + 13C reactions are marked by letters. Most of them are same as observed in the 7Li + 12C and 7Li + 13C reactions studied earlier [ 1 ] and have been identified in Table 2 in that reference. The rest are identified in the caption of Fig. 3. The spectra of the two reactions are found to be nearly identical. This is due to the fact that in the 6Li + 12C reaction the characteristic y-rays mainly originate from pJ70, o~14N and pnl60 exit channels, while in the 6Li at- 13C they come from pnl70, t~nl4N and p2nl60. In addition to these, y-rays of 13C corresponding to apl3C channel in 6Li + 12C reaction and those of liB, 15N, 180 and lSF corresponding to 2t~llB, alSN, p ] 8 0 and nl8F in 6Li + 13C reaction also contribute moderately. The y-ray cross sections for the transitions of the residual nuclei were obtained from the relation Nz, t r e - e~NBNT '
where N r is the number of counts in the y-ray peak, e r is the full energy detection efficiency for the y-ray, NB is the number of beam particles and ArT is the number
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
309
(a) (b)
---
"
t~
-
•
Z
i \
, 0.2
J 0.4
,
, 0.6
. . . .
j
0,8
. . . .
1.0
1.2
t
~
Ii
1,4
1.6
I
i
1.8
2.0
Energy (MeV)
,
t
2,2
Energy
,
i_
2.4
,
i
2,6
2.8
3,0
(MeV)
O
;J
v 3.0
4.0
5.0
6.0
7.0
8.0
90
lO.O
Energy (MeV) Fig. 3. ( a ) - ( c ) . Gamma ray spectra obtained for 6Li + 12C reaction (bottom) and 6Li + 13C reaction (top) at Elab = 9.5 MeV. The contaminant lines are marked by letters and most of them have been identiffed in Table 2 of Ref. [ I ]. Additional lines which have been identified in the present work are as follows: (S):27AI(n, n ~) 1.014 MeV ---*0, (T):7°Ge(n, n ~) (1.041 MeV ---*0), (U):74Ge(n, n I) 1.205 MeV---*0, 7°Ge(n, n ~) 1.216 MeV--*0 and (V):56Fe(n, n ~) 2.658----*0.847 MeV.
310
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324 J
10 2
_ wm_--~-m~-IL~ l 14N2 - 3 1 3 ~ 0 laN 3.948~ 2.313 XI0_I]
6Li+12C 10 ~ 10 0 10-~
//
~ /
10-2 10-3
~
160 6.130~ 0 160 6.919 ~
=2
/
0
~/~xlO-4
104
~ 1 7 0
10 -s 10 ~
0.871 ~
0 X10-6
13C 3.089 ~
c/ ~2 cl~11~
10-7
0 £1~ Q Q Q
-8
10 8
I~
10-9
/
/
13C3"684...... 0
10-1~ 2
3
4
5
6
7
8
Ec.m.(MeV) Fig. 4. y-ray cross sections for different exit channels of the 6Li + ~2C reaction. The experimental points are plotted with total error. The curves drawn through the data represent the SCN (Statistical CompoundNucleus) calculations performed with the code CASCADE. of target nuclei per cm 2. The number of beam particles NB was determined from the charge collected in the Faraday Cup and dividing it by the equilibrium charge value Ze obtained from Marion and Young [3]. The uncertainties in NB and NT are estimated to be ,~5% and ~6.5%, respectively, and in e~, about 7% giving a combined error of ,,~11%. This is added to the statistical error in N z, (which also includes the uncertainty due to the approximation of the background under a y-ray peak) to get the total error in o-~,. Cross sections for some of the prominent y-rays are shown in Figs. 4 and 5. The solid curves represent the SCN (Statistical Compound Nucleus) calculations (Section 3.2). 3.2. The channel cross sections
To get the channel cross sections and hence the fusion cross sections for the two systems, one needs the "branching factors", F~, = O'~,/O'ch giving the fraction of the residual nuclei emitting the characteristic y-ray when left in the bound states. A characteristic y-ray of a residual nucleus results from the deexcitation of a specific bound state either directly or via cascading y-transitions. Thus the evaluation of F r depends on the cross sections for the excitation of each of the bound states of a residual nucleus as well as on the y-ray branching ratios of these bound states. Whereas the latter can be obtained from the literature, the evaluation of the former needs a model calculation. This is done
A. Mukherjeeet aL/Nuclear PhysicsA 635 (1998)305-324 103
I
I
102
6Li+13C
10~
D
~
I
I
I
I
14N 2.313 ~ ~ = ~
10o
~ . ' ~ j
10-1
/
10-2
/
10-8
i /~z
14N 3.948 f
0 -2.313
~ - -
-
~ f
//U
311
XI0-]
t700'871
"0
X103
170 3.841
- 0
X10_4
'SN 5.270, 5 . 2 9 9 ~
0
-
,~/
Xl0 -7 :
407
I0-9
~//~i
~
_
l
O
1 0 lo
10-11
1 0 12
I 2
{ ,
I 3
~
I 4
L
,
5
I 6
~
,
7
l
,
8
Ec.m.(MeV) Fig. 5. y-ray cross sections for different exit channels of the 6Li + 13C reaction. For other details see caption of Fig. 4.
by the Statistical Compound Nucleus (SCN) calculation based on the Hauser-Feshbach formalism using the computer code CASCADE [4]. The calculated branching factors, Fy, for the characteristic T-rays of the residual nuclei in these two reactions are shown in Fig. 6. The parameters used in these calculations are given in Table 1. It may be mentioned in this connection that F r ( = O'r/O-ch) being the ratio of the two cross sections is rather insensitive to the reasonable variation of parameters used in the above calculations [5-7] and this justifies the use of the theoretically evaluated branching factors in the determination of the channel cross sections. In the evaluation of the channel cross sections, uncertainty ~ 1 0 % has been assigned to F~, for most of the T-rays. For some of the T-rays with small F~, and large variation with energy, however, uncertainty ,~25-30% has been assigned. The total uncertainty in the channel cross sections is obtained by adding the above uncertainty with the uncertainty in y-ray cross sections in quadrature. We now briefly discuss some of the individual channel cross sections shown in Figs. 7 and 8.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
312 1.0
• 1 0.5 6Li+13 C
i
'sO 1.982
,
I
'
I
'
I
'
i
F
'
i
6Li+12 C
I
J
0.8
i
!
0.6
~
0.4
t4N 1.635+2.313
@ r~
0.4
'70 0.871 /
1606.130
0.:3
0.2
z
l
0.0
==
I~N 1.635
4 o.2
~C 6-COA
0.1:
i
o.1
0.0~ 7
0.(3, 0.00
I
7I 2
3
4
5
6
7
8
0.0 9
' 2
3
4
5
6
7
8
Ec.m.(MeV) Fig. 6. Theoretical branching factors F~,(= o'~,fitch ) for the decay of the compound nuclei calculated with the code CASCADE and the y-ray branching ratios from Ref. [241.
3.2.1. One-particle evaporation channels In 6Li -4- 13C reaction cross sections for all the one-particle evaporation channels (p + 180, n -4- JSF and cr -t- 15N) are found to be less than those predicted by statistical model calculations (Fig. 8). The energy dependence of cross sections for all the channels are similar and the difference between the experimental and theoretical cross sections appear to decrease with the decrease of bombarding energies. In 6Li -4- 12C reaction, while p -4- 170 channel shows similar behaviour, the ol -4t4N channel shows nice agreement with theory throughout the measured energy range (Fig. 7). The cross sections for the n + 17F channel could not be determined because the characteristic y-ray 0.495 MeV of ~7F is hardly visible in the tail of the strong 0.511 MeV annihilation y-ray (Fig. 3a). It may be mentioned that because of this the cross sections for the 2n -4- 17F channel of 6Li -4- 13C reaction also could not be reliably measured although the y-ray was somewhat prominent in this reaction. It is interesting to note that except a + 14N channel of the 6Li -t- ~2C reaction, the cross sections for all the one-particle evaporation channels are much less compared to the total fusion cross sections in both the reactions. Thus they (one-particle evaporation channels) practically do not affect either the magnitude or the energy dependence of the total fusion cross sections of the two systems.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
313
Table I Parameters used in CASCADE calculations. Parameters of the optical model potentials for 6Li + '2J3C reactions are shown below. For other parameters see Table 3 of Ref. | 1]
V(r)
= - U f ( r , RR, aR) -- i W v f ( r , Rv, av ) + 4iasWs d f ( r , R s, as ) + tr. l(h---~-] 2V-V-~-a d f ( r , RSO, asp) + Vc(r) where, \ nlTrC / r
-- [1 +exp(-:)] r-
f ( r, Ri, ai )
= Z"ZTe2
Vc(r)
=
',
-
r2
ZPZle2 r
Ri
r ) Re '
= ri(A 1/3 + Alp/3),
Ri
i = R(real), V(volume), S(surface), SO(spin-orbit) and C(Coulomb) al/3
6Li+ J.3C
Ri = "i'~T
(1)
rR=rv=rc=
(Ref. 1171 )
aR = as = 0.68 fm
1.50 fm
U = 134 MeV Ws = 9.51 + 0.42Elab MeV
Wv = V s o = 0 (II)
rR=rv=rc=
(Refs. 118,19])
aR = av = 0.40 fm
1.26 fm
U --- 50 MeV
Wv = 10 MeV Ws = Vso = 0 6Li+12C
Ri = ri AI/3
(l)
rR = r c = 1.45 fin
rv = 1.33 fm (Ref. 1171 )
aR = 0.67 fm as = 0.97 fin U = 153 - 0.46EIab MeV WS = 2.20 + 0.25Elab MeV
Wv = Vso = 0 (II)
same as (II) of6Li + 13C
3.2.2. E x i t c h a n n e l w i t h 14N O n e o f t h e m a i n c o n t r i b u t o r s to t h e t o t a l r e a c t i o n c r o s s s e c t i o n s in t h e t w o r e a c t i o n s is t h e c h a n n e l
with
14N a s t h e r e s i d u a l
the an ÷ ]4N channel t h e 6 L i + 12C r e a c t i o n
contributes
nucleus. Whereas
in t h e 6 L i + 13C r e a c t i o n
,-~34% to t h e total r e a c t i o n c r o s s s e c t i o n s , t h a t in
t h e a ÷ JaN c h a n n e l
contributes
,-~28-14%
(in the direction
314
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
10 3
--'
i
,
i
i
,
]
'
i
~
'
i
'
i
6Li+12C 10 2
101
10 o
13 10 2
10 3 ~
~
170+p
Xl04
10 4
1
2
3
4
5
6
7
8
9
10
E~.m.(MeV) Fig. 7. Cross sections for different exit channels of the 6Li + 12C reaction. The Hauser-Feshbach calculations performed with the code CASCADE are shown by the solid curves. For clarity the data have been displaced vertically by the indicated factor. The error bars show the total error.
of increasing bombarding energy) over the energy region investigated in this work. Comparison of the y-ray cross sections for the 2.313 MeV y-ray (2.313 MeV ~ 0) of InN (Fig. 4) as well as the channel cross sections derived from this y-ray (Fig. 7) with the SCN calculations for the 6Li ÷ 12C reaction shows that they agree very nicely over the measured energy range. This is not the case with 6Li ÷ 13C reaction. In this reaction the experimental cross sections are found to be more than the SCN calculations and this difference increases with decrease of bombarding energy (Figs. 5 and 8). The relatively larger cross sections at lower bombarding energies compared to 6Li + 12C reaction could be due to the uncertainty in the evaluation of SCN y-ray cross sections and hence in the estimation of F~,-values of the 2.313 MeV y-ray in this (6Li + 13C) reaction. This is because, whereas in the 6Li ÷ 12C reaction the Fr-value of the 2.313 MeV y-ray remains practically constant ( ~ 3 0 % ) , it varies from --,3 to 17% for the same y-ray in the 6Li + J3C reaction (Fig. 6). This is mainly due to the large population of the ground state compared to other states of 14N in the latter reaction. Thus a small change in the Fz,-value can substantially change the cross sections of the latter (IaN ÷ oln) at lower bombarding energies. Because of this uncertainty, following Wu and Barnes [8], we assumed the energy dependence of the 14N ÷ oln channel cross section to be similar to SCN prediction and obtained the cross sections at lower bombarding energies by
315
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324 10 3
i
I
I
6Li+13C
i
I
I
~ -~ - 4 ] ~ - ! / ~ /
102
I
I
~ 14N+c~ n
////~
170+pn XI0-I 10~
//
1oo / ~ ~ ,-~ I0~ / ~
+
//.--JW-3_ /~//~
10 -7
.
xIo3 x,o
Y
"F+n
XIO-'
I
I
I
I
I
I
I
I
2
3
4
5
6
7
8
9
10
Ec.m.(MeV) Fig. 8. Cross sections for exit channels of the 6Li+ 13C reaction. The Hanser-Feshbach calculations performed with the code CASCADE are shown by the solid curves. The dotted curve in the case of an + 14N channel, represents the theoretical curve normalised to the experimental data at higher bombarding energies (see text). For other details see caption of Fig. 1.
normalizing the theoretical curve to the experimental data at higher bombarding energies (Fig. 8). It is interesting to note that besides the 2.313 MeV 3/-ray, the 1.635 MeV (3.948 MeV --~2.313 MeV) )'-ray of 14N is also observable with good statistics in both the reactions (Fig. 3). Since, in principle, the cross sections for a particular channel can be determined from any characteristic T-ray of the residual nucleus, we determined the same channel cross sections (mentioned above), in the two reactions using this ),-ray. In the evaluation of T-ray and channel cross sections we subtracted the contribution (,,~2-8%) of ~60(6Li, pn)2°Ne (1.634 MeV --~ 0) reaction using the oxygen content of the target (-,~1 /zg/cm 2) obtained from the elastic scattering data and cross sections for the above reaction [9]. Fig. 9 shows the comparison of the cross sections obtained from the two different y-rays in the two reactions. The good agreement proves the validity of SCN calculations in deriving the branching factor Fr and thus shows that the reaction principally proceeds through the compound nucleus formation and its subsequent decay.
316
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305~324
14N channel cross sections 10 3
6Li+I3C 10 2
6Li+12C ~:
101
~
~
o •
I0 °
10-1
~~xl0-'
,
I
I
2
3
L
I
4
,
I
5
florn2.313 MeV from1.635 MeV
,
L
6
,
~_±
7
I
8
,
I ~
9
10
Ec.m.(MeV) Fig. 9. Cross sections for the o~ + 14N channel of 6Li + [2C reaction and ten + ~4N channel of 6Li + 13C reactions obtained from the two y-rays 2.313 and 1.635 MeV of the residual nucleus 14N. See text for details.
3.2.3. 13C + c~p channel in the 6Li + 12C reaction The cross sections for the 13C + ap channel in the 6Li + ]2C reaction could be determined from either the 3.089 MeV y-ray (3.089 MeV--+0) or the 3.684 MeV yray (3.684 MeV-+0) of the 13C residual nucleus. However, the cross sections for the 3.089 MeV y-ray are found to be considerably larger than the SCN calculations (Fig. 4). The channel cross sections obtained from this y-ray are also found to be substantially larger than those obtained from the 3.684 MeV y-ray. The situation appears to be similar to the 9Be + 9Be reaction, where also substantially large cross sections were observed for this 3.089 MeV y-ray in the 9Be + 9Be --+ an + J3C reaction [ 10]. Unfortunately, like 9Be ÷ 9Be reaction, we are unable to find out any reason for such a large cross section for this y-ray in the 6Li + ]2C reaction. Hence the cross sections for the 13C + ap channel of 6Li + 12C reaction were obtained from the 3.684 MeV y-ray and are seen to agree nicely with the SCN calculations. 3.2.4. 160 + pn channel in the 6Li + 12C reaction The 1 6 0 nucleus decays to its ground state predominantly by the emission of 6.130 MeV (6.130 MeV--+0), 6.919 MeV (6.919 MeV-+0) and 7.117 MeV (7.117 MeV--+0) y-rays. All the three y-rays are observable in the spectrum of 6Li+ 12C reaction (Fig. 3). However, the 7.117 MeV y-ray is merged with the stronger 7.030 MeV y-ray of 14N (7.030 MeV --+0) and hence the cross sections for the 7.117 MeV y-ray could not be reliably measured. One further notes that whereas the cross sections for the 6.130 MeV y-ray nicely agree with the SCN calculations those for the 6.919 MeV y-ray differ
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
317
109 ,
~Li+lzC---> t60+pn I0 e
10 7
.<
10 o
160:6.130--->0 (C.N. process) ~ 10 5
10 4
,
I
I
I
J
I
i
I
2
3
4
5
6
7
8
Ec.m.(MeV) Fig. 10. Cross sections in the form of S-factor (S = ~rEc.m.e2,~n, where rI = ZpZTe2/hv is the Sommertield parameter) for the 6.919 and 6.130 MeV y-rays of 160 following 6Li + 12C reaction. While 160:6.919 MeV
(a-transfer) represents the cross sections for the 6.919 MeV 3,-ray of 160 (after subtracting the contribution of the compound nucleus reaction) proceeding through direct reaction, t60:6.130 MeV (compound nucleus) represents the cross sections for the 6.130 MeV 3,-ray produced predominantly through compound nucleus formation. The solid lines serve to guide the eye. See text for details. appreciably from such calculations (Fig. 4). This difference could be attributed to the a-transfer from 6Li projectile to 12C nucleus. Such a reaction has previously been seen to populate the 0+(6.050 MeV), 2+(6.919 MeV), 4+(10.35 MeV) states of 160 preferentially and they are interpreted to be the members of a rotational band built on the 0+(6.050 MeV) state having the character of 4 p - 4 h configuration [ 11]. Fig. 10 shows the cross sections for the 6.919 MeV y-ray plotted in the form of S-factor, S = o'Ec.m.e 2zrn ,
where r/ is the Sommerfield parameter ZpZTe 2 ~7-hv '
and v being the relative velocity. For comparison, we also show the cross sections for the 6.130 MeV y-ray plotted in the same form of S-factor. This kind of energy dependence ( o f 6.919 MeV y-ray) has previously been seen in many subbarrier transfer reactions, particularly, for n-transfer reactions with optimum Q-values [ 12-15]. In view of this the cross sections for the 160 + pn channel were determined from the 6.130 MeV y-ray only. 3.3. Total fusion cross sections
For the determination of total fusion cross sections it is necessary to add the individual channel cross sections proceeding through fusion, as described in the previous sections.
318
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
i
103
i
i
i
6Li+12C
@ ~
i
~
i
E
~
102
8
10a
100
I 2
•
afu ~ f r o m s u m o f the c h a n n e l c r o s s sections
©
cf.~ using
i 3
I 4
s u m o f the "/-ray cross sections
~ 5
I 6
I 7
I 8
9
E,..~(MeV) Fig. 11. Fusion cross sections for the 6Li + J2C reaction obtained from the sum of the channel cross sections and those obtained from the sum of all the observable y-ray cross sections dividing it by the factor, ~YOr~./O'tot from the SCN calculations. See text for details. Whereas in the 6Li + 13C reactions the contribution of the dominant channels could be obtained upto the lowest bombarding energies and thus the fusion cross sections for this system could be obtained at very low energies (Ec.m. = 2.07 MeV), in the 6Li + 12C reaction the cross sections for the characteristic y-rays of 13C ( o f a p + 13C channel) and ~60 ( o f p n + 160 channel) could not be determined below Ec.m ~ 5 MeV. This is because of the predominant population of the ground states and gradual inaccessibility of the excited states of these nuclei with decrease of bombarding energies. In view of this, we determined the fusion cross sections for this system at low energies by summing the cross sections for the observed y-rays corresponding to the transitions to the ground states of the residual nuclei and dividing these by the corresponding branching factors (EO'y/O'tot) obtained from the SCN calculations. It may be mentioned that similar procedure was used by Scholz et al. [ 16] in the determination of fusion cross sections for the 7Li + 160 reaction. Fig. 11 shows the cross sections determined in this way together with those obtained from the sum of the channel cross sections. The fusion cross sections for the two reactions are shown in Figs. 12 and 13, respectively. The uncertainty shown in these cross sections has been obtained by adding the error of the individual channel cross sections (statistical and AF~ added in quadrature) in quadrature and adding this to the total systematic error H11% (Section 3.1) also in quadrature.
A. Mukherjee et al. ~Nuclear Physics A 635 (1998) 305-324
]
i
i
I
10 3
i i Optical Model
319
F
6Li+12C jf
IWBC 102
XI0-L BPM 101
XI0 -2
10 0
IWBC: Vo=50 MeV, rb=l.87 fm, a=0.50 fm BPM: Vo=50 MeV, ro=1.25 fm, a=0.50 fm
10-1
10 2
L
1
2
3
4
5
6
7
8
9
Ec.m.(MeV) Fig. 1:2. Fusion cross sections for 6Li + J2C reaction. The error bars indicate the total error. The two curves at the top show the optical model calculations with parameters given in Table 1. The solid curve shows the calculations with the parameter set i obtained from fitting the elastic scattering data [ 17]. The dashed curve represents the calculated values using the "standard parameter" set II [ 18,19]. The next two curves together with the same set of data as above show the calculated cross sections using 1WBC [20] and I-D BPM [21 ] models, respectively. The values of the parameters of these calculations are shown in the figure. Bc and 2B c denote the positions of the Coulomb barrier energy and two times the Coulomb barrier energy, respectively, where Bc = ZpZTeZ/I.70(Ap U3 + ATU3).
In our recent works [ 1,10] it has been shown that the optical model calculations with parameters obtained from fitting the elastic scattering data gave a good fit to the total fusion cross sections for 7Li + 12"13C, 7Li + riB and 9Be + 9Be reactions at subbarrier energies. For 6Li + ]2j3C reactions such potentials are also available from the work of Poling et al. [ 17]. In fact these potentials have already been used by us in the calculation of Fy-values [Table 1 ]. The solid lines at the top of Figs. 12 and 13 show the total reaction cross sections obtained from such calculations. For comparison we have also shown (dotted lines) the calculated total cross sections using "standard parameters" obtained from a systematic study of a number of reactions [ 18,19]. It is seen that like 7Li + 12C and 7Li + 13C reactions, the cross sections for the 6Li + 12C and 6Li + t3C reactions are also nicely explained by the calculated cross sections using the optical model potential with its parameters from the fitting of elastic scattering data.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
320
Optical M ~ _ _ 10 3
6Li+13C
_
~
IWBC 102
X10-1
BPM 101
X10 -2
E 10 0
IWBC: Vo-50 MeV, rb=l.87 fm, a--0.50 fm BPM: Vo=50 MeV, ro=1.25 fm, a=0.50 fm I0-I
10-2 t 1
2
3
4
5
6
7
8
9
Ec.m.(MeV) Fig. 13. Fusion cross sections for 6Li + 13C reaction. For other details see caption of Fig. 12.
The calculation with standard parameters yield somewhat lower cross sections, a feature which has also been noted in the previous study of 7Li-induced reactions [ 1,10]. Two more calculations which have been successfully applied to many low-mass fusion reactions are Incoming Wave Boundary Condition (IWBC) model [20] and the one-dimensional Barrier Penetration (ID-BPM) model [21]. The fusion cross sections calculated with these models are also shown in the next two curves, together with the data. The parameters of the potentials are also given in these figures. In all the calculations Vo (strength of real potential) and ao (diffuseness parameter) have been kept fixed at 50 MeV and 0.50 fm, respectively. Only the radius parameter, rb in IWBC and r0 in 1-D BPM have been varied to get the above fit. The 1D-BPM model appears to overpredict the cross sections at low bombarding energies. It is possible to get a slightly better agreement for these data by decreasing r0 at the expense of poorer fit at higher energies.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
321
3.4. Comparison with cross sections obtained from the detection of the heavy evaporation residues One of the widely used methods for the determination of fusion cross sections is the detection of heavy evaporation residues following fusion. In this method the heavy residues are detected and identified with the combination of measurement of their flight time, A E / A x and energy. This method has been found to be very much successful for heavy systems at higher bombarding energies. Using this technique Dennis et al. [2] have measured the cross sections for several Li-induced reactions at energies near 2Bc (where Bc is the Coulomb barrier energy) and above. Like 7Li + 12'13C reactions, 6Li +J2,13 C reactions also show that the cross sections for different mass groups obtained by the evaporation residue detection method in general, are much less than those obtained by the y-ray method and the statistical model calculations. In fact the only mass group where the evaporation residue method yields a larger cross sections is A = 15 in the 6Li+ ~3C reactions. However, the cross sections for the channels corresponding to A = 15 (predominantly 15N + ce) are of little significance compared to the total fusion cross sections, shown in Fig. 13 for this (6Li + 13C) reaction. Fig. 14 shows the cross sections for the two 6Li induced reactions together with the cross sections for the 7Li + liB, 7Li + 12C, 7Li + 13C and 7Li + 160 reactions measured by the y-ray method [ 1,10,16] and the evaporation residue technique [2,22]. The solid lines show the optical model calculations using parameters of the potential obtained from fitting the elastic scattering data of the same system in the energy interval similar to the one where fusion cross sections are measured. It should be mentioned, however, that for the 7Li + l JB system because of the non-availability of elastic scattering data the optical model potential of 7Li + 12C reaction has been used with proper scaling [ 10]. The nice agreement of the cross sections for all the Li-induced reactions with the total reaction cross sections calculated using the optical model potential clearly indicates that there is no inhibition of fusion cross sections for such systems at subbarrier energies. However, these calculated cross sections are substantially larger than those measured by the evaporation residue detection method. Similar difference is also observed when the data obtained by the evaporation residue detection method are compared with IWBC and I-D BPM models (discussed in Section 3.3). It is interesting to note that for all the systems the fusion cross sections measured by the y-ray method agree nicely with the total reaction cross sections at lower bombarding energies and appear to decrease slowly at higher bombarding energies as per expectation since at higher bombarding energies, channels other than fusion open up gradually and the fusion cross sections become less than the total reaction cross sections. In this respect the decrease of fusion cross sections from the total reaction cross sections at higher bombarding energies as observed from the measurement of heavy residues is quite likely. It appears that the y-ray data could be matched with the evaporation residue data for these systems provided one disregards the latter data at lower bombarding energies (Ec.m. --~ 6 - 1 0 MeV). The data obtained by the evaporation residue detection technique at lower bombarding energies are observed to be substantially lower than those measured by the y-ray method.
322
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
6Li+12C '
200 [
~ ~, ~
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I
]
8
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16
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E .~(MeV)
Fig. 14. Total fusion cross sections for the 7Li + 160, 7Li + I3C. 7Li + 12C, 7Li + liB, 6Li + 13C and 6Li + ~2C systems measured by the y-ray method (shown by the symbols • and o) and by the detection of heavy evaporation residues (shown by the symbol Fq). The solid lines represent the total reaction cross sections calculated using the optical model potential with parameters obtained from fitting the elastic scattering data. The arrows indicate the positions of two times the Coulomb barrier energy, 2Bc, (where Bc = ZI, ZTe2/1.70(AI, I/3 + AT I/3)) upto which the fusion cross sections are usually seen to be ~ total reaction cross sections.
This discrepancy, however, could not be due to the uncertainty in the measurement of target thickness or beam current, since in both the works proper care was taken for the measurement of these quantities. In the y-ray method used by us the uncertainty in the measurement of NT and NB are estimated to be ~6.5% and ~5%, respectively (Section 3.1 and Ref. [ 1 ] ). In the evaporation residue detection method the uncertainty in NT was ~8% and nothing was mentioned about the beam current measurement [22]. Since the estimated uncertainty in the absolute cross section quoted in this work is 10%, one would expect ~<5% uncertainty in the measurement of the number of beam particles in this work also.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
323
As the uncertainty in the measurement of the target thickness and beam current is not sufficient to account for the large discrepancy (,,~ factor of 2) between the values of cross sections obtained by the two methods at lower bombarding energies, it appears that the difference is due to the underestimation of the yield of the residues at lower bombarding energies in the evaporation residue detection technique. This is because of the fact that as the bombarding energy decreases, the kinetic energy of the residues becomes lower and as a consequence the detection and identification of these residues become more difficult in the background of elastically scattered events the cross sections of which climb up [23]. In fact, because of the limitation in the detection of the low-energy residues, the evaporation residue detection technique has been found to be rather unsuccessful in the measurement of fusion cross sections for the astrophysically important systems, like 12C + 12C or 160 + 160 [23]. On the other hand, such a limitation is not there with the y-ray method and thus the y-ray method has been extensively used for the measurement of fusion cross sections at very low energies for a number of systems including those mentioned above.
4. Concluding remarks In this work cross sections for various channels in 6Li + 12C and 6Li + 13C reactions at subbarrier energies have been measured. Comparison of the y-ray and channel cross sections with statistical model calculations show that like 7Li-induced reactions, the two 6Li-induced reactions studied in this work, also proceed mainly via compound nucleus formation and its subsequent decay. In addition to p n channel the main contribution comes from the a p channel in the 6Li÷ 12C reaction. In the 6Li + 13C reaction the main contribution is from the a n channel though pn, 2 a and p 2 n channels also contribute moderately. All the one-particle evaporation channel cross sections in the 6Li + 13C reaction are found to be substantially smaller than that predicted by statistical model calculations. The measured fusion cross sections nicely agree with the total reaction cross sections calculated by the optical model calculations and fusion cross sections calculated by the Incoming Wave Boundary Condition model and one dimensional Barrier Penetration model. The cross section for the a-transfer reaction 12C(6Li, d)160(6.919 MeV) plotted in the form of S-factor shows a very steep nature with decrease of bombarding energy similar to the n-transfer reaction with optimum Q-value. However, the contribution of the a-transfer channel in the 6Li + 12C system is much less compared to the fusion cross sections and its presence does not influence the determination of fusion cross sections for this system. The fusion cross sections for the 6Li ÷ 12'13C reactions are very nearly equal to those for the 7Li + 12'~3C reactions in agreement with the optical model calculations at low energies but in contradiction with the results obtained by heavy evaporation residue
324
A. Muldzerjee et ak /Nuclear Physics A 635 (1998) 305-324
measurements. The fusion cross sections for all the Li-induced reactions that have been m e a s u r e d by the "y-ray m e t h o d are found to be substantially larger than those m e a s u r e d by the evaporation residue detection method. This discrepancy appears to be due to the limitation o f the latter m e t h o d at lower b o m b a r d i n g energies.
Acknowledgements The authors w o u l d like to thank Prof. S.N. Behera, Prof. D.P. Mahapatra and Dr. Shikha Varma o f the Institute o f Physics, B h u b a n e s w a r for p r o v i d i n g the facilities to p e r f o r m the experiment. They are grateful to Mr. P.K. Das, Mr. S.C Chatterjee and all the students and technical staff o f the Ion B e a m Laboratory, in particular, Dr. G o u t a m Kuri and Mr. A n u p B e h e r a for helping at different stages o f the experiment.
References [ 1 I A. Mukherjee, U. Datta Pramanik, M. Saha Sarkar, A. Goswami, E Basu, S. Bhattacharya, S. Sen, M.L. Chatterjee and B. Dasmahapatra, Nucl. Phys. A 596 (1996) 299. ]2] L.C. Dennis, K.M. Abdo, A.D. Frawley and K.W.Kemper, Phys. Rev. C 26 (1982) 981. 13] J.B. Marion and EC. Young, Nuclear Reaction Analysis: Graphs and Tables (North-Holland, Amsterdam, 1968) p. 38. 141 E PiJhlofer, Nucl. Phys. A 280 (1977) 267; Program CASCADE: Rapport GSI, Darmstadt (1976), unpublished. 151 P.R. Chfistensen, Z.E. Switkowski and R.A. Dayras, Nucl. Phys. A 280 (1977) 189. 16] J.L. Charvet, R. Dayras, J.M. Fieni and J.L. Uzureau, Nucl. Phys. A 376 (1982) 292. 17] C.T. Papadopoulos, R. Vlastou, E.N. Gazis, P.A. Assimakopoulos, C.A. Kalfas, S. Kossionides and C.A. Xenoulis, Phys. Rev. C 34 (1986) 196. 181 S.-C. Wu and C.A. Barnes, Nucl. Phys. A 422 (1984) 373. [91 A. Mukherjee et al. to be published. 1101 A. Mukherjee and B. Dasmahapatra, Nucl. Phys. A 614 (1997) 238. [ 111 K. Bethge, D.J. Pullen and R. Middleton, Phys. Rev. C 2 (1970) 395; K. Bethge, Nucl. Reactions Induced by Heavy Ions, Proc. Int. Conf. Heidelberg, July 15-18, 1969 (North Holland, 1970) p. 231. [121 H.C. Cheung, M. D. High and B. (~ujec, Nucl. Phys. A 296 (1978) 333. [131 B. Cujec, S.-C. Wu and C.A. Barnes, Phys. Lett. B 89 (1979) 151. [ 141 H.A. Roth, J.E. Christiansson and J. Dubois, Nucl. Phys. A 343 (1980) 148. 115] B. Dasmahapatra, B. (~ujec and E Lahlou, Nucl. Phys. A 427 (1984) 186. [16] C.J.S. Scholz, L. Ricken and E. Kuhlman, Z. Phys. A 325 (1986) 203. 117] J.E. Poling, E. Norbeck and R.R. Carlson, Phys. Rev. C 13 (1976) 648. [ 18] R.G. Stokstad, Z.E. Switkowski, R.A. Dayras and R.M. Wieland, Phys. Rev. Lett. 37 (1976) 888. 1191 H. Reeves, Astrophys. J. 146 (1966) 447. [201 P.R. Christensen and Z.E. Switkowski, Nucl. Phys. A 280 (1977) 205. 1211 B. (~ujec and C.A. Barnes, Nucl. Phys. A 266 (1976) 461. 1221 J.F. Mateja, J. Garman, D.E. Fields, R.L. Kozub, A.D. Frawley and L.C. Dennis, Phys. Rev. C 30 (1984) 134. 1231 C.A. Barnes, S. Trentalange and S.-C. Wu, Treatise on Heavy Ion Science, Vol. 6, ed. D.A. Bromley (Plenum, New York, 1985) p. 3, and references therein. [24] E Ajzenberg-Selove, Nucl. Phys. A 506 (1990) 1; A 523 (1991) 1; A 460 (1986) I; A 475 (1987) 1.
Nuclear Physics A 635 (1998) 305-324
Fusion cross sections for 6Li+12Cand reactions at low energies
6Li+13C
A. Mukherjee, U. Datta Pramanik, S. Chattopadhyay, M. Saha Sarkar, A. Goswami, P. Basu, S. Bhattacharya, M.L. Chatterjee, B. Dasmahapatra Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta-700 064, India
Received 1 October 1997; revised 11 February 1998; accepted 9 March 1998
Abstract The partial cross sections for the 6Li + lZc and 6Li + ~3C reactions below and above the Coulomb barrier energy have been determined from the cross sections of the characteristic y-rays measured with HPGe detectors and using the branching factor o'r/o'ch obtained from the statistical model calculations. The fusion cross sections obtained from the sum of these partial cross sections are found to be equal to the total reaction cross sections upto an energy well above the Coulomb barrier energy and there appears to be no evidence for limitation of fusion cross sections for these systems at such energies contrary to the evaporation residue measurements. The measured cross sections are also found to agree nicely with the IWBC (Incoming Wave Boundary Condition) and one-dimensional BPM (Barrier Penetration Model) calculation. © 1998 Elsevier Science B.V. PACS: 25.70.Gh; 25.70.Jj Keywords: NUCLEAR REACTIONS 12C(6Li,X), E(cm) = 2.01-8.02 MeV, 13C(6Li,X),
E(cm) = 2.07-8.23 MeV; measured Er, I:,; deduced partial, total fusion o'. Statistical model analysis, Optical model, IncomingWave BoundaryCondition model and one-dimensionalBarrier Penetration Model calculations
1. Introduction In a previous article [ 1 ] from the measurement of the characteristic y-rays of the residual nuclei following fusion of 7Li + 12C and 7Li + 13C systems at energies below and above the Coulomb barrier, the cross sections for different exit channels were determined and from the sum of these, the total fusion cross sections for the two systems were obtained. From the comparison with statistical model calculations it was concluded that the two reactions mainly proceed via compound nucleus formation and 0375-9474/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0375-9474(98) 00 164-X
306
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
its subsequent decay. Among various exit channels om and pn channels were found to be the principal contributors to the fusion cross sections. It was further found that the measured cross sections could nicely be fitted by the total reaction cross sections calculated with the optical model potential with its parameters obtained from fitting the elastic scattering data. This in turn indicates that at subbarrier energies fusion is the dominant reaction mechanism and there is no inhibition of fusion cross sections contrary to the heavy-evaporation residue measurements [2]. In order to see whether similar reaction mechanism prevails also in the case of 6Liinduced reactions, we thought it will be worthwhile to investigate the 6Li + 12C and 6Li + J3C reactions in the similar energy range using the same technique of measurement. Another motivation of the present work was to find out whether the fusion cross sections for the 6Li induced reactions are less than those for the corresponding 7Li induced reactions also at subbarrier energies, which has earlier been observed at higher bombarding energies through the heavy evaporation residue measurements [2]. In fact in this latter measurement the ratio of these cross sections is found to be _~0.80-0.85 throughout the measured energy range [Ec.m. -~ 6 - 2 5 MeV] whereas the optical model calculations predict nearly equal cross sections for them. It may further be mentioned that so far there has been no fusion or total reaction cross section measurement for these systems at subbarrier energies.
2. Experimental details The measurements were done with 6Li 2+ and 6Li 3+ beams obtained from the 3 MV pelletron accelerator of the Institute of Physics, Bhubaneswar. The J2C foils ( ~ 7 0 /.~g/cm 2) were natural as well as enriched (99.9%) and the 13C foils ( ~ 6 0 /.zg/cm 2) were enriched (98.9%). The measurements were done with two scattering chambers. The old chamber used earlier [ 1] has been replaced by a new chamber specifically designed for the cross section measurements. The new scattering chamber having a diameter of 15 cm was thinned down at two places (centred at 125 ° and 90 ° with respect to the beam direction) to facilitate the exit of low energy y-rays without much absorption in the chamber wall. A stainless steel tube, 2.5 cm in diameter and 45 cm long lined inside with a tantalum foil, insulated from a grounded outer cylinder (also made of stainless steel) and the target chamber, and protected with an electron suppression ring at - 2 0 0 V, served as the Faraday cup. In addition, instead of using one heavy ion surface barrier detector (for recording elastic scattering events) at 45 ° in the previous chamber, provisions were made to place four such detectors two on each side of the beam axis. Viewed along the beam direction, the detectors could be placed at 29.6 °, 46.1 °, 30.8 ° and 43.8 °, respectively. This arrangement helped in more accurate monitoring of the beam. The thicknesses of the targets were also accurately determined from the elastic scattering spectra obtained with these detectors using 2-4 MeV Li-ions
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
15
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.
18F .-13.3t2 Fig. 1. Energy level diagram for the 6Li + laC reaction. The numbers attached to the ground states give the Q-value of the respective channels. The region investigated is cross-hatched. The 3t-transitions which are seen in the spectrum are shown by vertical lines. The highest levels indicated are particle unstable.
and 10-12 MeV 160 ions. The measured thicknesses of the 4 targets were as follows: ]2C - 68.1 4-4.4, 73.1 + 4 . 7 / z g / c m 2, 13C - 60.7 + 3.9, 64.0 + 4.1 / z g / c m 2. Two HPGe detectors each of ~ 9 0 cm 3 volume were placed at 125 ° and 90 °, respectively, with target-to-detector distance ~ 1 2 cm. The y-ray cross sections were mainly determined from the spectra obtained with the detector at 125 °. The detector at 90 ° gave unshifted y-rays and helped in the proper identification of them. The measurements were done at E]ab = 3.11--12.07 MeV in steps of 500 keV to 1 MeV for both the reactions. These correspond to Ec.m = 2 . 0 1 - 8 . 0 2 MeV for 6 L i + 12C and 2.07-8.23 MeV for 6Li + 13C reactions after making necessary correction for energy loss in the targets. Figs. 1 and 2 show the excitation of the relevant compound nuclei corresponding to the above centre of mass energy and their successive decay to different exit channels.
308
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
1.0-
4
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~
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Fig. 2. Energy level diagram for the 6Li + 13C reaction. For other details see caption of Fig. 1.
3. Analysis and results 3.1. The y-ray cross sections
Fig. 3 shows typical y-ray spectra of the two reactions. In this figure the y-rays other than arising from 6Li + 12C and 6Li + 13C reactions are marked by letters. Most of them are same as observed in the 7Li + 12C and 7Li + 13C reactions studied earlier [ 1 ] and have been identified in Table 2 in that reference. The rest are identified in the caption of Fig. 3. The spectra of the two reactions are found to be nearly identical. This is due to the fact that in the 6Li + 12C reaction the characteristic y-rays mainly originate from pJ70, o~14N and pnl60 exit channels, while in the 6Li at- 13C they come from pnl70, t~nl4N and p2nl60. In addition to these, y-rays of 13C corresponding to apl3C channel in 6Li + 12C reaction and those of liB, 15N, 180 and lSF corresponding to 2t~llB, alSN, p ] 8 0 and nl8F in 6Li + 13C reaction also contribute moderately. The y-ray cross sections for the transitions of the residual nuclei were obtained from the relation Nz, t r e - e~NBNT '
where N r is the number of counts in the y-ray peak, e r is the full energy detection efficiency for the y-ray, NB is the number of beam particles and ArT is the number
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
309
(a) (b)
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Energy (MeV) Fig. 3. ( a ) - ( c ) . Gamma ray spectra obtained for 6Li + 12C reaction (bottom) and 6Li + 13C reaction (top) at Elab = 9.5 MeV. The contaminant lines are marked by letters and most of them have been identiffed in Table 2 of Ref. [ I ]. Additional lines which have been identified in the present work are as follows: (S):27AI(n, n ~) 1.014 MeV ---*0, (T):7°Ge(n, n ~) (1.041 MeV ---*0), (U):74Ge(n, n I) 1.205 MeV---*0, 7°Ge(n, n ~) 1.216 MeV--*0 and (V):56Fe(n, n ~) 2.658----*0.847 MeV.
310
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324 J
10 2
_ wm_--~-m~-IL~ l 14N2 - 3 1 3 ~ 0 laN 3.948~ 2.313 XI0_I]
6Li+12C 10 ~ 10 0 10-~
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Ec.m.(MeV) Fig. 4. y-ray cross sections for different exit channels of the 6Li + ~2C reaction. The experimental points are plotted with total error. The curves drawn through the data represent the SCN (Statistical CompoundNucleus) calculations performed with the code CASCADE. of target nuclei per cm 2. The number of beam particles NB was determined from the charge collected in the Faraday Cup and dividing it by the equilibrium charge value Ze obtained from Marion and Young [3]. The uncertainties in NB and NT are estimated to be ,~5% and ~6.5%, respectively, and in e~, about 7% giving a combined error of ,,~11%. This is added to the statistical error in N z, (which also includes the uncertainty due to the approximation of the background under a y-ray peak) to get the total error in o-~,. Cross sections for some of the prominent y-rays are shown in Figs. 4 and 5. The solid curves represent the SCN (Statistical Compound Nucleus) calculations (Section 3.2). 3.2. The channel cross sections
To get the channel cross sections and hence the fusion cross sections for the two systems, one needs the "branching factors", F~, = O'~,/O'ch giving the fraction of the residual nuclei emitting the characteristic y-ray when left in the bound states. A characteristic y-ray of a residual nucleus results from the deexcitation of a specific bound state either directly or via cascading y-transitions. Thus the evaluation of F r depends on the cross sections for the excitation of each of the bound states of a residual nucleus as well as on the y-ray branching ratios of these bound states. Whereas the latter can be obtained from the literature, the evaluation of the former needs a model calculation. This is done
A. Mukherjeeet aL/Nuclear PhysicsA 635 (1998)305-324 103
I
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102
6Li+13C
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14N 2.313 ~ ~ = ~
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~ . ' ~ j
10-1
/
10-2
/
10-8
i /~z
14N 3.948 f
0 -2.313
~ - -
-
~ f
//U
311
XI0-]
t700'871
"0
X103
170 3.841
- 0
X10_4
'SN 5.270, 5 . 2 9 9 ~
0
-
,~/
Xl0 -7 :
407
I0-9
~//~i
~
_
l
O
1 0 lo
10-11
1 0 12
I 2
{ ,
I 3
~
I 4
L
,
5
I 6
~
,
7
l
,
8
Ec.m.(MeV) Fig. 5. y-ray cross sections for different exit channels of the 6Li + 13C reaction. For other details see caption of Fig. 4.
by the Statistical Compound Nucleus (SCN) calculation based on the Hauser-Feshbach formalism using the computer code CASCADE [4]. The calculated branching factors, Fy, for the characteristic T-rays of the residual nuclei in these two reactions are shown in Fig. 6. The parameters used in these calculations are given in Table 1. It may be mentioned in this connection that F r ( = O'r/O-ch) being the ratio of the two cross sections is rather insensitive to the reasonable variation of parameters used in the above calculations [5-7] and this justifies the use of the theoretically evaluated branching factors in the determination of the channel cross sections. In the evaluation of the channel cross sections, uncertainty ~ 1 0 % has been assigned to F~, for most of the T-rays. For some of the T-rays with small F~, and large variation with energy, however, uncertainty ,~25-30% has been assigned. The total uncertainty in the channel cross sections is obtained by adding the above uncertainty with the uncertainty in y-ray cross sections in quadrature. We now briefly discuss some of the individual channel cross sections shown in Figs. 7 and 8.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
312 1.0
• 1 0.5 6Li+13 C
i
'sO 1.982
,
I
'
I
'
I
'
i
F
'
i
6Li+12 C
I
J
0.8
i
!
0.6
~
0.4
t4N 1.635+2.313
@ r~
0.4
'70 0.871 /
1606.130
0.:3
0.2
z
l
0.0
==
I~N 1.635
4 o.2
~C 6-COA
0.1:
i
o.1
0.0~ 7
0.(3, 0.00
I
7I 2
3
4
5
6
7
8
0.0 9
' 2
3
4
5
6
7
8
Ec.m.(MeV) Fig. 6. Theoretical branching factors F~,(= o'~,fitch ) for the decay of the compound nuclei calculated with the code CASCADE and the y-ray branching ratios from Ref. [241.
3.2.1. One-particle evaporation channels In 6Li -4- 13C reaction cross sections for all the one-particle evaporation channels (p + 180, n -4- JSF and cr -t- 15N) are found to be less than those predicted by statistical model calculations (Fig. 8). The energy dependence of cross sections for all the channels are similar and the difference between the experimental and theoretical cross sections appear to decrease with the decrease of bombarding energies. In 6Li -4- 12C reaction, while p -4- 170 channel shows similar behaviour, the ol -4t4N channel shows nice agreement with theory throughout the measured energy range (Fig. 7). The cross sections for the n + 17F channel could not be determined because the characteristic y-ray 0.495 MeV of ~7F is hardly visible in the tail of the strong 0.511 MeV annihilation y-ray (Fig. 3a). It may be mentioned that because of this the cross sections for the 2n -4- 17F channel of 6Li -4- 13C reaction also could not be reliably measured although the y-ray was somewhat prominent in this reaction. It is interesting to note that except a + 14N channel of the 6Li -t- ~2C reaction, the cross sections for all the one-particle evaporation channels are much less compared to the total fusion cross sections in both the reactions. Thus they (one-particle evaporation channels) practically do not affect either the magnitude or the energy dependence of the total fusion cross sections of the two systems.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
313
Table I Parameters used in CASCADE calculations. Parameters of the optical model potentials for 6Li + '2J3C reactions are shown below. For other parameters see Table 3 of Ref. | 1]
V(r)
= - U f ( r , RR, aR) -- i W v f ( r , Rv, av ) + 4iasWs d f ( r , R s, as ) + tr. l(h---~-] 2V-V-~-a d f ( r , RSO, asp) + Vc(r) where, \ nlTrC / r
-- [1 +exp(-:)] r-
f ( r, Ri, ai )
= Z"ZTe2
Vc(r)
=
',
-
r2
ZPZle2 r
Ri
r ) Re '
= ri(A 1/3 + Alp/3),
Ri
i = R(real), V(volume), S(surface), SO(spin-orbit) and C(Coulomb) al/3
6Li+ J.3C
Ri = "i'~T
(1)
rR=rv=rc=
(Ref. 1171 )
aR = as = 0.68 fm
1.50 fm
U = 134 MeV Ws = 9.51 + 0.42Elab MeV
Wv = V s o = 0 (II)
rR=rv=rc=
(Refs. 118,19])
aR = av = 0.40 fm
1.26 fm
U --- 50 MeV
Wv = 10 MeV Ws = Vso = 0 6Li+12C
Ri = ri AI/3
(l)
rR = r c = 1.45 fin
rv = 1.33 fm (Ref. 1171 )
aR = 0.67 fm as = 0.97 fin U = 153 - 0.46EIab MeV WS = 2.20 + 0.25Elab MeV
Wv = Vso = 0 (II)
same as (II) of6Li + 13C
3.2.2. E x i t c h a n n e l w i t h 14N O n e o f t h e m a i n c o n t r i b u t o r s to t h e t o t a l r e a c t i o n c r o s s s e c t i o n s in t h e t w o r e a c t i o n s is t h e c h a n n e l
with
14N a s t h e r e s i d u a l
the an ÷ ]4N channel t h e 6 L i + 12C r e a c t i o n
contributes
nucleus. Whereas
in t h e 6 L i + 13C r e a c t i o n
,-~34% to t h e total r e a c t i o n c r o s s s e c t i o n s , t h a t in
t h e a ÷ JaN c h a n n e l
contributes
,-~28-14%
(in the direction
314
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
10 3
--'
i
,
i
i
,
]
'
i
~
'
i
'
i
6Li+12C 10 2
101
10 o
13 10 2
10 3 ~
~
170+p
Xl04
10 4
1
2
3
4
5
6
7
8
9
10
E~.m.(MeV) Fig. 7. Cross sections for different exit channels of the 6Li + 12C reaction. The Hauser-Feshbach calculations performed with the code CASCADE are shown by the solid curves. For clarity the data have been displaced vertically by the indicated factor. The error bars show the total error.
of increasing bombarding energy) over the energy region investigated in this work. Comparison of the y-ray cross sections for the 2.313 MeV y-ray (2.313 MeV ~ 0) of InN (Fig. 4) as well as the channel cross sections derived from this y-ray (Fig. 7) with the SCN calculations for the 6Li ÷ 12C reaction shows that they agree very nicely over the measured energy range. This is not the case with 6Li ÷ 13C reaction. In this reaction the experimental cross sections are found to be more than the SCN calculations and this difference increases with decrease of bombarding energy (Figs. 5 and 8). The relatively larger cross sections at lower bombarding energies compared to 6Li + 12C reaction could be due to the uncertainty in the evaluation of SCN y-ray cross sections and hence in the estimation of F~,-values of the 2.313 MeV y-ray in this (6Li + 13C) reaction. This is because, whereas in the 6Li ÷ 12C reaction the Fr-value of the 2.313 MeV y-ray remains practically constant ( ~ 3 0 % ) , it varies from --,3 to 17% for the same y-ray in the 6Li + J3C reaction (Fig. 6). This is mainly due to the large population of the ground state compared to other states of 14N in the latter reaction. Thus a small change in the Fz,-value can substantially change the cross sections of the latter (IaN ÷ oln) at lower bombarding energies. Because of this uncertainty, following Wu and Barnes [8], we assumed the energy dependence of the 14N ÷ oln channel cross section to be similar to SCN prediction and obtained the cross sections at lower bombarding energies by
315
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324 10 3
i
I
I
6Li+13C
i
I
I
~ -~ - 4 ] ~ - ! / ~ /
102
I
I
~ 14N+c~ n
////~
170+pn XI0-I 10~
//
1oo / ~ ~ ,-~ I0~ / ~
+
//.--JW-3_ /~//~
10 -7
.
xIo3 x,o
Y
"F+n
XIO-'
I
I
I
I
I
I
I
I
2
3
4
5
6
7
8
9
10
Ec.m.(MeV) Fig. 8. Cross sections for exit channels of the 6Li+ 13C reaction. The Hanser-Feshbach calculations performed with the code CASCADE are shown by the solid curves. The dotted curve in the case of an + 14N channel, represents the theoretical curve normalised to the experimental data at higher bombarding energies (see text). For other details see caption of Fig. 1.
normalizing the theoretical curve to the experimental data at higher bombarding energies (Fig. 8). It is interesting to note that besides the 2.313 MeV 3/-ray, the 1.635 MeV (3.948 MeV --~2.313 MeV) )'-ray of 14N is also observable with good statistics in both the reactions (Fig. 3). Since, in principle, the cross sections for a particular channel can be determined from any characteristic T-ray of the residual nucleus, we determined the same channel cross sections (mentioned above), in the two reactions using this ),-ray. In the evaluation of T-ray and channel cross sections we subtracted the contribution (,,~2-8%) of ~60(6Li, pn)2°Ne (1.634 MeV --~ 0) reaction using the oxygen content of the target (-,~1 /zg/cm 2) obtained from the elastic scattering data and cross sections for the above reaction [9]. Fig. 9 shows the comparison of the cross sections obtained from the two different y-rays in the two reactions. The good agreement proves the validity of SCN calculations in deriving the branching factor Fr and thus shows that the reaction principally proceeds through the compound nucleus formation and its subsequent decay.
316
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305~324
14N channel cross sections 10 3
6Li+I3C 10 2
6Li+12C ~:
101
~
~
o •
I0 °
10-1
~~xl0-'
,
I
I
2
3
L
I
4
,
I
5
florn2.313 MeV from1.635 MeV
,
L
6
,
~_±
7
I
8
,
I ~
9
10
Ec.m.(MeV) Fig. 9. Cross sections for the o~ + 14N channel of 6Li + [2C reaction and ten + ~4N channel of 6Li + 13C reactions obtained from the two y-rays 2.313 and 1.635 MeV of the residual nucleus 14N. See text for details.
3.2.3. 13C + c~p channel in the 6Li + 12C reaction The cross sections for the 13C + ap channel in the 6Li + ]2C reaction could be determined from either the 3.089 MeV y-ray (3.089 MeV--+0) or the 3.684 MeV yray (3.684 MeV-+0) of the 13C residual nucleus. However, the cross sections for the 3.089 MeV y-ray are found to be considerably larger than the SCN calculations (Fig. 4). The channel cross sections obtained from this y-ray are also found to be substantially larger than those obtained from the 3.684 MeV y-ray. The situation appears to be similar to the 9Be + 9Be reaction, where also substantially large cross sections were observed for this 3.089 MeV y-ray in the 9Be + 9Be --+ an + J3C reaction [ 10]. Unfortunately, like 9Be ÷ 9Be reaction, we are unable to find out any reason for such a large cross section for this y-ray in the 6Li + ]2C reaction. Hence the cross sections for the 13C + ap channel of 6Li + 12C reaction were obtained from the 3.684 MeV y-ray and are seen to agree nicely with the SCN calculations. 3.2.4. 160 + pn channel in the 6Li + 12C reaction The 1 6 0 nucleus decays to its ground state predominantly by the emission of 6.130 MeV (6.130 MeV--+0), 6.919 MeV (6.919 MeV-+0) and 7.117 MeV (7.117 MeV--+0) y-rays. All the three y-rays are observable in the spectrum of 6Li+ 12C reaction (Fig. 3). However, the 7.117 MeV y-ray is merged with the stronger 7.030 MeV y-ray of 14N (7.030 MeV --+0) and hence the cross sections for the 7.117 MeV y-ray could not be reliably measured. One further notes that whereas the cross sections for the 6.130 MeV y-ray nicely agree with the SCN calculations those for the 6.919 MeV y-ray differ
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
317
109 ,
~Li+lzC---> t60+pn I0 e
10 7
.<
10 o
160:6.130--->0 (C.N. process) ~ 10 5
10 4
,
I
I
I
J
I
i
I
2
3
4
5
6
7
8
Ec.m.(MeV) Fig. 10. Cross sections in the form of S-factor (S = ~rEc.m.e2,~n, where rI = ZpZTe2/hv is the Sommertield parameter) for the 6.919 and 6.130 MeV y-rays of 160 following 6Li + 12C reaction. While 160:6.919 MeV
(a-transfer) represents the cross sections for the 6.919 MeV 3,-ray of 160 (after subtracting the contribution of the compound nucleus reaction) proceeding through direct reaction, t60:6.130 MeV (compound nucleus) represents the cross sections for the 6.130 MeV 3,-ray produced predominantly through compound nucleus formation. The solid lines serve to guide the eye. See text for details. appreciably from such calculations (Fig. 4). This difference could be attributed to the a-transfer from 6Li projectile to 12C nucleus. Such a reaction has previously been seen to populate the 0+(6.050 MeV), 2+(6.919 MeV), 4+(10.35 MeV) states of 160 preferentially and they are interpreted to be the members of a rotational band built on the 0+(6.050 MeV) state having the character of 4 p - 4 h configuration [ 11]. Fig. 10 shows the cross sections for the 6.919 MeV y-ray plotted in the form of S-factor, S = o'Ec.m.e 2zrn ,
where r/ is the Sommerfield parameter ZpZTe 2 ~7-hv '
and v being the relative velocity. For comparison, we also show the cross sections for the 6.130 MeV y-ray plotted in the same form of S-factor. This kind of energy dependence ( o f 6.919 MeV y-ray) has previously been seen in many subbarrier transfer reactions, particularly, for n-transfer reactions with optimum Q-values [ 12-15]. In view of this the cross sections for the 160 + pn channel were determined from the 6.130 MeV y-ray only. 3.3. Total fusion cross sections
For the determination of total fusion cross sections it is necessary to add the individual channel cross sections proceeding through fusion, as described in the previous sections.
318
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
i
103
i
i
i
6Li+12C
@ ~
i
~
i
E
~
102
8
10a
100
I 2
•
afu ~ f r o m s u m o f the c h a n n e l c r o s s sections
©
cf.~ using
i 3
I 4
s u m o f the "/-ray cross sections
~ 5
I 6
I 7
I 8
9
E,..~(MeV) Fig. 11. Fusion cross sections for the 6Li + J2C reaction obtained from the sum of the channel cross sections and those obtained from the sum of all the observable y-ray cross sections dividing it by the factor, ~YOr~./O'tot from the SCN calculations. See text for details. Whereas in the 6Li + 13C reactions the contribution of the dominant channels could be obtained upto the lowest bombarding energies and thus the fusion cross sections for this system could be obtained at very low energies (Ec.m. = 2.07 MeV), in the 6Li + 12C reaction the cross sections for the characteristic y-rays of 13C ( o f a p + 13C channel) and ~60 ( o f p n + 160 channel) could not be determined below Ec.m ~ 5 MeV. This is because of the predominant population of the ground states and gradual inaccessibility of the excited states of these nuclei with decrease of bombarding energies. In view of this, we determined the fusion cross sections for this system at low energies by summing the cross sections for the observed y-rays corresponding to the transitions to the ground states of the residual nuclei and dividing these by the corresponding branching factors (EO'y/O'tot) obtained from the SCN calculations. It may be mentioned that similar procedure was used by Scholz et al. [ 16] in the determination of fusion cross sections for the 7Li + 160 reaction. Fig. 11 shows the cross sections determined in this way together with those obtained from the sum of the channel cross sections. The fusion cross sections for the two reactions are shown in Figs. 12 and 13, respectively. The uncertainty shown in these cross sections has been obtained by adding the error of the individual channel cross sections (statistical and AF~ added in quadrature) in quadrature and adding this to the total systematic error H11% (Section 3.1) also in quadrature.
A. Mukherjee et al. ~Nuclear Physics A 635 (1998) 305-324
]
i
i
I
10 3
i i Optical Model
319
F
6Li+12C jf
IWBC 102
XI0-L BPM 101
XI0 -2
10 0
IWBC: Vo=50 MeV, rb=l.87 fm, a=0.50 fm BPM: Vo=50 MeV, ro=1.25 fm, a=0.50 fm
10-1
10 2
L
1
2
3
4
5
6
7
8
9
Ec.m.(MeV) Fig. 1:2. Fusion cross sections for 6Li + J2C reaction. The error bars indicate the total error. The two curves at the top show the optical model calculations with parameters given in Table 1. The solid curve shows the calculations with the parameter set i obtained from fitting the elastic scattering data [ 17]. The dashed curve represents the calculated values using the "standard parameter" set II [ 18,19]. The next two curves together with the same set of data as above show the calculated cross sections using 1WBC [20] and I-D BPM [21 ] models, respectively. The values of the parameters of these calculations are shown in the figure. Bc and 2B c denote the positions of the Coulomb barrier energy and two times the Coulomb barrier energy, respectively, where Bc = ZpZTeZ/I.70(Ap U3 + ATU3).
In our recent works [ 1,10] it has been shown that the optical model calculations with parameters obtained from fitting the elastic scattering data gave a good fit to the total fusion cross sections for 7Li + 12"13C, 7Li + riB and 9Be + 9Be reactions at subbarrier energies. For 6Li + ]2j3C reactions such potentials are also available from the work of Poling et al. [ 17]. In fact these potentials have already been used by us in the calculation of Fy-values [Table 1 ]. The solid lines at the top of Figs. 12 and 13 show the total reaction cross sections obtained from such calculations. For comparison we have also shown (dotted lines) the calculated total cross sections using "standard parameters" obtained from a systematic study of a number of reactions [ 18,19]. It is seen that like 7Li + 12C and 7Li + 13C reactions, the cross sections for the 6Li + 12C and 6Li + t3C reactions are also nicely explained by the calculated cross sections using the optical model potential with its parameters from the fitting of elastic scattering data.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
320
Optical M ~ _ _ 10 3
6Li+13C
_
~
IWBC 102
X10-1
BPM 101
X10 -2
E 10 0
IWBC: Vo-50 MeV, rb=l.87 fm, a--0.50 fm BPM: Vo=50 MeV, ro=1.25 fm, a=0.50 fm I0-I
10-2 t 1
2
3
4
5
6
7
8
9
Ec.m.(MeV) Fig. 13. Fusion cross sections for 6Li + 13C reaction. For other details see caption of Fig. 12.
The calculation with standard parameters yield somewhat lower cross sections, a feature which has also been noted in the previous study of 7Li-induced reactions [ 1,10]. Two more calculations which have been successfully applied to many low-mass fusion reactions are Incoming Wave Boundary Condition (IWBC) model [20] and the one-dimensional Barrier Penetration (ID-BPM) model [21]. The fusion cross sections calculated with these models are also shown in the next two curves, together with the data. The parameters of the potentials are also given in these figures. In all the calculations Vo (strength of real potential) and ao (diffuseness parameter) have been kept fixed at 50 MeV and 0.50 fm, respectively. Only the radius parameter, rb in IWBC and r0 in 1-D BPM have been varied to get the above fit. The 1D-BPM model appears to overpredict the cross sections at low bombarding energies. It is possible to get a slightly better agreement for these data by decreasing r0 at the expense of poorer fit at higher energies.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
321
3.4. Comparison with cross sections obtained from the detection of the heavy evaporation residues One of the widely used methods for the determination of fusion cross sections is the detection of heavy evaporation residues following fusion. In this method the heavy residues are detected and identified with the combination of measurement of their flight time, A E / A x and energy. This method has been found to be very much successful for heavy systems at higher bombarding energies. Using this technique Dennis et al. [2] have measured the cross sections for several Li-induced reactions at energies near 2Bc (where Bc is the Coulomb barrier energy) and above. Like 7Li + 12'13C reactions, 6Li +J2,13 C reactions also show that the cross sections for different mass groups obtained by the evaporation residue detection method in general, are much less than those obtained by the y-ray method and the statistical model calculations. In fact the only mass group where the evaporation residue method yields a larger cross sections is A = 15 in the 6Li+ ~3C reactions. However, the cross sections for the channels corresponding to A = 15 (predominantly 15N + ce) are of little significance compared to the total fusion cross sections, shown in Fig. 13 for this (6Li + 13C) reaction. Fig. 14 shows the cross sections for the two 6Li induced reactions together with the cross sections for the 7Li + liB, 7Li + 12C, 7Li + 13C and 7Li + 160 reactions measured by the y-ray method [ 1,10,16] and the evaporation residue technique [2,22]. The solid lines show the optical model calculations using parameters of the potential obtained from fitting the elastic scattering data of the same system in the energy interval similar to the one where fusion cross sections are measured. It should be mentioned, however, that for the 7Li + l JB system because of the non-availability of elastic scattering data the optical model potential of 7Li + 12C reaction has been used with proper scaling [ 10]. The nice agreement of the cross sections for all the Li-induced reactions with the total reaction cross sections calculated using the optical model potential clearly indicates that there is no inhibition of fusion cross sections for such systems at subbarrier energies. However, these calculated cross sections are substantially larger than those measured by the evaporation residue detection method. Similar difference is also observed when the data obtained by the evaporation residue detection method are compared with IWBC and I-D BPM models (discussed in Section 3.3). It is interesting to note that for all the systems the fusion cross sections measured by the y-ray method agree nicely with the total reaction cross sections at lower bombarding energies and appear to decrease slowly at higher bombarding energies as per expectation since at higher bombarding energies, channels other than fusion open up gradually and the fusion cross sections become less than the total reaction cross sections. In this respect the decrease of fusion cross sections from the total reaction cross sections at higher bombarding energies as observed from the measurement of heavy residues is quite likely. It appears that the y-ray data could be matched with the evaporation residue data for these systems provided one disregards the latter data at lower bombarding energies (Ec.m. --~ 6 - 1 0 MeV). The data obtained by the evaporation residue detection technique at lower bombarding energies are observed to be substantially lower than those measured by the y-ray method.
322
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
6Li+12C '
200 [
~ ~, ~
l0
~ ~ '~ ]
T
'
I
'
I
'
|
•~
14
200
k 100 t_ 0
0
100 ~
I 4 ~
16 1248I
8 -m
12
16
a
a
o
12
16
0__~
0
400 I~ 300 t
' -12 -I
t
200
'eli 8
4
12
600~--13rr~
16
0 60
~
,°°
1 4
'
a - T ~
200
0
a a a
16
a
20
t
0i
8
8 12 I Dq-
[
4 [3
~'
4 15~
0
,14° t
n
ll a
0
0
I
h
8
~
I
~
12 '
I
4
8
12
I
I
]
8
12
16 '
16
40 20
0
4
8
12
E~.m.(MeV)
16
4
16
E .~(MeV)
Fig. 14. Total fusion cross sections for the 7Li + 160, 7Li + I3C. 7Li + 12C, 7Li + liB, 6Li + 13C and 6Li + ~2C systems measured by the y-ray method (shown by the symbols • and o) and by the detection of heavy evaporation residues (shown by the symbol Fq). The solid lines represent the total reaction cross sections calculated using the optical model potential with parameters obtained from fitting the elastic scattering data. The arrows indicate the positions of two times the Coulomb barrier energy, 2Bc, (where Bc = ZI, ZTe2/1.70(AI, I/3 + AT I/3)) upto which the fusion cross sections are usually seen to be ~ total reaction cross sections.
This discrepancy, however, could not be due to the uncertainty in the measurement of target thickness or beam current, since in both the works proper care was taken for the measurement of these quantities. In the y-ray method used by us the uncertainty in the measurement of NT and NB are estimated to be ~6.5% and ~5%, respectively (Section 3.1 and Ref. [ 1 ] ). In the evaporation residue detection method the uncertainty in NT was ~8% and nothing was mentioned about the beam current measurement [22]. Since the estimated uncertainty in the absolute cross section quoted in this work is 10%, one would expect ~<5% uncertainty in the measurement of the number of beam particles in this work also.
A. Mukherjee et al./Nuclear Physics A 635 (1998) 305-324
323
As the uncertainty in the measurement of the target thickness and beam current is not sufficient to account for the large discrepancy (,,~ factor of 2) between the values of cross sections obtained by the two methods at lower bombarding energies, it appears that the difference is due to the underestimation of the yield of the residues at lower bombarding energies in the evaporation residue detection technique. This is because of the fact that as the bombarding energy decreases, the kinetic energy of the residues becomes lower and as a consequence the detection and identification of these residues become more difficult in the background of elastically scattered events the cross sections of which climb up [23]. In fact, because of the limitation in the detection of the low-energy residues, the evaporation residue detection technique has been found to be rather unsuccessful in the measurement of fusion cross sections for the astrophysically important systems, like 12C + 12C or 160 + 160 [23]. On the other hand, such a limitation is not there with the y-ray method and thus the y-ray method has been extensively used for the measurement of fusion cross sections at very low energies for a number of systems including those mentioned above.
4. Concluding remarks In this work cross sections for various channels in 6Li + 12C and 6Li + 13C reactions at subbarrier energies have been measured. Comparison of the y-ray and channel cross sections with statistical model calculations show that like 7Li-induced reactions, the two 6Li-induced reactions studied in this work, also proceed mainly via compound nucleus formation and its subsequent decay. In addition to p n channel the main contribution comes from the a p channel in the 6Li÷ 12C reaction. In the 6Li + 13C reaction the main contribution is from the a n channel though pn, 2 a and p 2 n channels also contribute moderately. All the one-particle evaporation channel cross sections in the 6Li + 13C reaction are found to be substantially smaller than that predicted by statistical model calculations. The measured fusion cross sections nicely agree with the total reaction cross sections calculated by the optical model calculations and fusion cross sections calculated by the Incoming Wave Boundary Condition model and one dimensional Barrier Penetration model. The cross section for the a-transfer reaction 12C(6Li, d)160(6.919 MeV) plotted in the form of S-factor shows a very steep nature with decrease of bombarding energy similar to the n-transfer reaction with optimum Q-value. However, the contribution of the a-transfer channel in the 6Li + 12C system is much less compared to the fusion cross sections and its presence does not influence the determination of fusion cross sections for this system. The fusion cross sections for the 6Li ÷ 12'13C reactions are very nearly equal to those for the 7Li + 12'~3C reactions in agreement with the optical model calculations at low energies but in contradiction with the results obtained by heavy evaporation residue
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A. Muldzerjee et ak /Nuclear Physics A 635 (1998) 305-324
measurements. The fusion cross sections for all the Li-induced reactions that have been m e a s u r e d by the "y-ray m e t h o d are found to be substantially larger than those m e a s u r e d by the evaporation residue detection method. This discrepancy appears to be due to the limitation o f the latter m e t h o d at lower b o m b a r d i n g energies.
Acknowledgements The authors w o u l d like to thank Prof. S.N. Behera, Prof. D.P. Mahapatra and Dr. Shikha Varma o f the Institute o f Physics, B h u b a n e s w a r for p r o v i d i n g the facilities to p e r f o r m the experiment. They are grateful to Mr. P.K. Das, Mr. S.C Chatterjee and all the students and technical staff o f the Ion B e a m Laboratory, in particular, Dr. G o u t a m Kuri and Mr. A n u p B e h e r a for helping at different stages o f the experiment.
References [ 1 I A. Mukherjee, U. Datta Pramanik, M. Saha Sarkar, A. Goswami, E Basu, S. Bhattacharya, S. Sen, M.L. Chatterjee and B. Dasmahapatra, Nucl. Phys. A 596 (1996) 299. ]2] L.C. Dennis, K.M. Abdo, A.D. Frawley and K.W.Kemper, Phys. Rev. C 26 (1982) 981. 13] J.B. Marion and EC. Young, Nuclear Reaction Analysis: Graphs and Tables (North-Holland, Amsterdam, 1968) p. 38. 141 E PiJhlofer, Nucl. Phys. A 280 (1977) 267; Program CASCADE: Rapport GSI, Darmstadt (1976), unpublished. 151 P.R. Chfistensen, Z.E. Switkowski and R.A. Dayras, Nucl. Phys. A 280 (1977) 189. 16] J.L. Charvet, R. Dayras, J.M. Fieni and J.L. Uzureau, Nucl. Phys. A 376 (1982) 292. 17] C.T. Papadopoulos, R. Vlastou, E.N. Gazis, P.A. Assimakopoulos, C.A. Kalfas, S. Kossionides and C.A. Xenoulis, Phys. Rev. C 34 (1986) 196. 181 S.-C. Wu and C.A. Barnes, Nucl. Phys. A 422 (1984) 373. [91 A. Mukherjee et al. to be published. 1101 A. Mukherjee and B. Dasmahapatra, Nucl. Phys. A 614 (1997) 238. [ 111 K. Bethge, D.J. Pullen and R. Middleton, Phys. Rev. C 2 (1970) 395; K. Bethge, Nucl. Reactions Induced by Heavy Ions, Proc. Int. Conf. Heidelberg, July 15-18, 1969 (North Holland, 1970) p. 231. [121 H.C. Cheung, M. D. High and B. (~ujec, Nucl. Phys. A 296 (1978) 333. [131 B. Cujec, S.-C. Wu and C.A. Barnes, Phys. Lett. B 89 (1979) 151. [ 141 H.A. Roth, J.E. Christiansson and J. Dubois, Nucl. Phys. A 343 (1980) 148. 115] B. Dasmahapatra, B. (~ujec and E Lahlou, Nucl. Phys. A 427 (1984) 186. [16] C.J.S. Scholz, L. Ricken and E. Kuhlman, Z. Phys. A 325 (1986) 203. 117] J.E. Poling, E. Norbeck and R.R. Carlson, Phys. Rev. C 13 (1976) 648. [ 18] R.G. Stokstad, Z.E. Switkowski, R.A. Dayras and R.M. Wieland, Phys. Rev. Lett. 37 (1976) 888. 1191 H. Reeves, Astrophys. J. 146 (1966) 447. [201 P.R. Christensen and Z.E. Switkowski, Nucl. Phys. A 280 (1977) 205. 1211 B. (~ujec and C.A. Barnes, Nucl. Phys. A 266 (1976) 461. 1221 J.F. Mateja, J. Garman, D.E. Fields, R.L. Kozub, A.D. Frawley and L.C. Dennis, Phys. Rev. C 30 (1984) 134. 1231 C.A. Barnes, S. Trentalange and S.-C. Wu, Treatise on Heavy Ion Science, Vol. 6, ed. D.A. Bromley (Plenum, New York, 1985) p. 3, and references therein. [24] E Ajzenberg-Selove, Nucl. Phys. A 506 (1990) 1; A 523 (1991) 1; A 460 (1986) I; A 475 (1987) 1.