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DWBA analysis of the reaction 12C(α, n)15O
Nuclear Physics A202 (1973) 377--384; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
D W B A A N A L Y S I S O F T H E R E A C T I O N 12C(~, n)lSO v. CORCALCIUC*, M. DUMA, R. DUMITRESCU, I. M]NZATU and A. CIOCANELtt Institute of Atomic Physics, Bucharest, Romania ttt Received 20 November 1972 Abstract: Angular distributions of neutrons from the reaction 12C(~, n)lSO(g.s.) have been measured at lab energies from 18.4 to 23.1 MeV and angles ranging from 0 ° to 130°, using a time-of-flight technique. The experimental curves generally show a forward peaking and a strong dependence on the incident energy. The data were compared with the angular distributions predicted by the distorted-wave theory of direct nuclear reactions, and no agreement could be obtained when only a stripping mechanism was taken into account.
E[ 1
NUCLEAR REACTION 12C(~, n)150, E~ = 18-23 MeV; measured or(0). DWBA calculation. Natural target.
1. Introduction D u r i n g the past few years the multinucleon transfer processes have become an important source o f information concerning reaction mechanisms and nuclear structure. Trinucleon transfer, viewed as knock-out, pick-up or stripping in reactions like (~, N ) and (N, c~), is particularly suited for the study of nuclear levels in those nuclei for which it is difficult to get data by other means. Unfortunately, multiple transfer in the region o f light and very light nuclei is by no means as clearly understood as single-nucleon transfer; therefore, comparison between experimental results and a direct-reaction theory such as D W B A presents many difficulties. A m o n g these we should emphasize the laborious mathematics and programming, the insufficiency o f the optical model in describing 7-particle reactions, and the scarcity o f experimental data over a wide mass-energy range. It is evident from the above considerations that a more careful investigation is necessary, both experimentally and theoretically, o f c~-particle reactions on light nuclei. The available experimental data indicate that (c~, N) reactions at sufficiently high energy (say above 10 MeV), proceed by direct interaction. The angular distributions frequently include a strong forward peaking which cannot be interpreted in the spirit o f the compound-nucleus theory. t Present address: N e u t r o n Physics Laboratory, Studsvik, Sweden. tt N o w at Cyclo Lab, Nuclear Research Center, Karlsruhe, G e r m a n y . t+t W o r k s u p p o r t e d by C S E N u n d e r contract no 558/7B, 1972. 377
378
V. CORCALCIUC et al.
The reaction mechanism is usually considered as a stripping process a + A ~ B + b (a --- b + x , B = A + x where a, b, x, A and B are nucleon numbers) in which the core excitations are negligible and the simplest configurations of the final nucleus are those for which ~(B) = ~(A)~b(x), with appropriate angular-momentum coupling. If one assumes this simplified picture of the reaction process then one can hope that the experimental data obtained from the (~, no) reaction on the simple nucleus laC(0 +) should be more amenable to a physical interpretation and, therefore, such an experiment and the corresponding theoretical analysis can test the applicability of the DI theory to a-particle reactions on light nuclei.
2. Experimental results and DWBA analysis The Y-120 cyclotron at the Institute of Atomic Physics in Bucharest was used to provide a 0.5 #A a-particle beam of 12-26 MeV, with an energy resolution of about 1.5 % and a natural modulation of about 2 ns. The neutrons emitted from a 1.6 mg/cm 2 target of pure self-supporting graphite (0.48 MeV energy loss) were analysed by means of our time-of-flight spectrometer (fig. 1), described previously 1). There were no difficulties with time-resolution and peak-separation for the ground state neutrons (fig. 2), since the first excited level in the final nucleus is located at 5.195 MeV, though a continuum background has to be taken into account. The a-particle energy was chosen sufficiently high (above 18 MeV) to assure the predominance of the direct effects and to correspond to some salient intervals (resonances or valleys in fig. 3) of the excitation function 2). The relative differential cross sections a(O) were measured at incident energies of 18.4, 18.9, 20.8, 21.2, 22.2 and 23.1 MeV (average values from the time-of-flight spectra) at 10° intervals from 0 ° to
IW
V ....
m --
j
TPHC ]
tAFj
~i A
I
l
I
C l-----
CB .J
[i
~4A
E
FD-~a~ . . . .lYi. m-e-----~C ] SA-40 ~
|
[. . . . l , I ,, I '
Analyzer I
1
SA-40 AnacIRYzer
i .=a
Fig. 1. Experimental set-up for time-of-flight spectrometry. D-mobile detector, M-fixed monitor (plastic+56 AVP surrounded by lead and borated paraffin), FM-fast mixer, TPHC-time-to-pulseheight converter, CB-command block, A-amplifier, D-discriminator (common threshold), F-interface RF-radio frequency.
a2C(% n) 1~0 DWBA ANALYSIS
379
Pulses 150C
100(
Y2
71
xlO2
3 ns
xlO2
II 0
50
I00
150
200
Channels
Fig. 2. A typical time-of-flight spectrum from the 12C(~, n)~SO reaction, at E a = 18.4 MeV and 0 = 30° (L = 3m). C~mb
3oL
13
] 1
15
17
19
21
23
E~,MeV
Fig. 3. Excitation function of the reaction 12C(~, n) 150. The arrows indicate the incident energies in the present experiment. 130 ° . The errors affecting the experimental data are mainly due to statistics and backg r o u n d subtraction and have been estimated at about 5 ~ , in three independent runs. O u r results are generally in agreement with those o f K o n d o et al. a) measured in 1963 at incident energies o f 20.0, 20.9 and 21.8 MeV, by using a different technique and a rather thick target (1.3 MeV energy loss). The theoretical analysis o f the present experimental angular distributions has been done assuming 3He stripping, in order to c o m p u t e with the help of the K u n z p r o g r a m [ref. 4)] the quantity:
aLSJ(o)_
4n 1 . 0 x l 0 4 ( 2 L + l ) Ea Eo 2Sa + 1
k b ~ Cl~.[t~'a~'b2 fm2/sr, ~-a ,',vbXt ~ L S J
380
et al.
V. C O R C A L C [ U C TABLE 1
Optical-model parameters Ei,¢.
Set
Particle
VR
WI
18.4
1
18.9
2
20.8
3
21.2
4
2.22
5
23.1
6
18.4
1A
18.4
1B
18.4
1C
~ n ~ n ~ n ~ n ~ n ~ n cz n g n :~ n
--212.63 - - 47.63 -- 212.50 -- 4 7 . 5 0 --212.04 -- 47.04 --211.96 -- 46.96 --211.66 -- 46.66 --211.48 - - 46.48 - - 2 0 2 63 - - 47.63 --192.63 --192.63 --182.63 - - 182.63
4V~.o.a
- - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 -- 16.75 - - 5.75 --17.75 - - 5.75 --18.75 - - 5.75
4W~.o.t
0.0 --37.00 0.0 --36.92 0.0 --36.60 0.0 --36.56 0.0 --36.36 0.0 --36.24 0.0 --37.00 0.0 --37.00 0.0 --37.00
R0R :
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Rol
1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25
a :
b
0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55
E n e r g i e s in M e V , l e n g t h s in f m .
Uopt (r) = Vc (r)~
VR
r - - R ~_
X r
a at ~ VR
v3He --R
V~
n :t - - VR' WI =
ikVl
t • S,
rdr~ ) I xl e " : ' ~ + eX'-}-~ --(V~.o.R -:-iWs.o.O r( ~e T
r--Rl --- , RR -~ RoRAJ, Rl = RoIA{, b
W 3He -t, --I W~,
49.3--0.31En,
Vn
s.o.R
=
V 3He --R
=
165 M e V ,
9.52--0.05E~,
"W' 13 H e
~
10 M e V ,
W~ = 5.75 M e V .
where t~ 2 ~--- 2 a +
e~QgLSJ
=
Z
1,
"la--lb--L~2t~lasaJa I~lbsbJb f'jbJJa plbLla 1 ~b ~"~OVama~JOVbmb ~"~mo-MMma "~000
jajblalb
x ~ 7j~Tb
S
Jjalajbtb
mb--M'
~laSaJa
The last expression involves the distorted waves and form factor in the form of the stripping integrals:
jojbt.tb = (A/B) dr, z*blb(kb, r.A/B)fLss(r,)zj.,.(ka, yLJ f Thus, the differential cross section becomes: do" _ 2JB+ 1 dr2
[BLss[ 2
o"Lss
2JA+I 1.0xl0 4 2J+l'
r.).
12C(~,
n) 150 D W B A A N A L Y S I S
381
I
151 ~(e) I
i
,
I0 [ / ~
I
[
I
Ea=23.1±0.5 MeV Sei6
¢
5//
I 1
15
I
, Ea:20,.8_+0.3 WMVe Set 3
,
~et18.9-+0.4 MeV
l
1
~
l
l
~et2~.2-+0.4 MeV
I0
l
l
l
15 ,
I
I
I
1
l
Ea=18.4±0.3 MeV Set 1
÷
l
l
Ec~:21.2±0.4 MeV Set 4
I0
5
0
l
30
1
60
l
90
l
120
1
150
i
30
l
60
i
90
l
120
l
150 0o
Fig. 4. DWBA fits to the experimentalangular distributions from the ~2C(~,n)~50 reaction using the p a r a m e t e r sets 1-6 f r o m table 1. T h e calculated, values are normalized to experimental ones at 45 ° (c.m.).
382
V. CORCALCIUC et i
15
u
i
al.
,
i
I
Ea=18.4 MeV
Set 1A
~(e)
V~:-202.63 MeV
10
W?:-I6.75 MeV
5
~
~ i 1' ¢
I
I
15
I
I
I
Ea=18.4 MeV Set 1B
+
V~-192.63 MeV
10
W?=-I7. 75 MeV
5
,b
I
15
I
,
I
~
I
I
|
÷
Ea:18.4 MeV Set IC 101
~ +
W~:-18.75 MeV
I
I
I
I
0
30
60
90
120
I
150 eo
Fig. 5. DWBA fit to the 18.4 MeV angular distribution from the reaction 12C(~, n) j 50 using the parameter sets 1A-IC from table 1 (the depths of the real and imaginary wells for =(-particles vary in 10 and 1 MeV steps respectively).
12C(~,n)lSO D W B A ANALYSIS
383
where BLsj = (~/~,)ALsj and ALsj is the usual spectroscopic coefficient of the J U L I E code. This expression for the differential cross section reduces the triple stripping to the transfer of a "heavy nucleon" of mass 3 and charge 2 with J= and T as well-defined quantum numbers. On the other hand, one knows that the general D W B A theory of multiple transfer 5) involves a coherent sum over the N, n, n' principal quantum numbers of the c.m. and relative motions. Therefore, the single-nucleon transfer formula cannot be correct for (c~, N) reactions unless it can be argued that only one value of N contributes to the cross section. Since the D W U C K program does not add the amplitudes for different N we have taken a unique value N = 2, assuming a symmetric s-state for the relative motions (n = n' = 1 and l = l' --- 0) in the helion. The nucleons transferred from the projectile to the ground state (0 +) of the target nucleus are in the l p state (n 1 = t'/2 = r/3 = 1, 11 = 12 = 13 = l) and the final nucleus is obtained in its ground state (½-). The form factor is calculated by D W U C K in the zero-range approximation: =
y
N FLsj
N
where the radial form factor F~ss(N = 2, 2 J = l, L = l, 2S = 1) is the radial wave function of the helion considered as a point object bound in a Saxon-Woods well, with an appropriate energy (the separation energy of 3He and ~2C in ~50 is 12.122 MeV). In the framework of the drastic simplifications made above, a very difficult problem arises in addition, namely, the calculation of the distorted waves. It is well known that the geometrical and dynamical parameters of the optical model, for the interaction of the ~-particle with the nucleus, present some ambiguities, and they have not yet been well determined or verified from elastic scattering experiments. Owing to the lack of a realistic criterion in the choice of the z-particle optical-model parameters, we have adopted the suggestion of Stock et al. 6) assuming that the well for an :~-particle (table 1) would be that for 3He [in our case that given by Fortune 7)], to which could be added that of the neutron [in our case that given by Rosen et aL 8), which also applies to the exit channel]. The results of the D W U C K calculations on an IBM 360/40 with the parameters giver in table 1 are shown in figs. 4 and 5. 3. Conclusions The reaction 12C(e, no) 150 has been analysed at bombarding energies between 18 and 23 MeV. In this energy range the stripping zero-range D W B A calculations do not reproduce the observed angular distributions, with the exception of the 18.4 and 18.9 MeV incident energies (minimum of the excitation function) at forward angles ranging from 0 ° to about 80 °. It is evident from the excitation function (fig. 3) that compoundnuclear effects are not negligible in the energy region we have studied. On the other
384
v. CORCALCIUC et al.
hand, the f o r w a r d p e a k i n g in the differential cross sections is very unlikely to be caused by c o m p o u n d resonances. These facts suggest a s t r o n g interference between r e s o n a n t a n d direct processes. A p r o p e r finite-range D W B A t h e o r y o f the triple-stripping a n d a statistical evaluation o f the c o m p o u n d - n u c l e u s c o n t r i b u t i o n (with the same o p t i c a l - m o d e l p a r a m e t e r s for the calculation o f the t r a n s m i s s i o n coefficients a n d d i s t o r t e d waves) might at least be useful to o b t a i n better a g r e e m e n t between experiment and theory.
References 1) A. CiocS.nel, thesis, Bucharest, 1971, unpublished; M. Molea, preprint IFA, CRD-37, Bucharest, 1968 2) J. L. Black et aL, N'ucl. Phys. A l l 5 (1968) 683; R, D. Carpenter et al., Phys. Rev. 125 (1962) 282; A. V. Spasski et al., Atom. Energ. 26 (1969) 303 3) M. Kondo et aL, J. Phys. Soc. Jap. 18 (1963) 22 4) P. D. Kunz, Code DWUCK, Bucharest version, 1972, unpublished 5) A. Y. Abul-Magd et aL, Prog. Theor. Phys. 38 (1967) 366; N. K. Glendenning, Phys. Rev. 156 (1967) 1344; E. M. l~enley and D. U. L. Yu, Phys. Rev. 133 (1964) 1445 6) R. Stock et al., Nucl. Phys. A104 (1967) 136 7) H. T. Fortune et aL, Bull. Am. Phys. Soc. 12 (1967) 1197; P. E. Hodgson, Adv. Phys. 17 (1968) 563 8) L. Rosen et aL, Ann. of Phys. 34 (1965)96; D. Wilmore and P. E. Hodgson, Nucl. Phys. 55 (1964) 673
D W B A A N A L Y S I S O F T H E R E A C T I O N 12C(~, n)lSO v. CORCALCIUC*, M. DUMA, R. DUMITRESCU, I. M]NZATU and A. CIOCANELtt Institute of Atomic Physics, Bucharest, Romania ttt Received 20 November 1972 Abstract: Angular distributions of neutrons from the reaction 12C(~, n)lSO(g.s.) have been measured at lab energies from 18.4 to 23.1 MeV and angles ranging from 0 ° to 130°, using a time-of-flight technique. The experimental curves generally show a forward peaking and a strong dependence on the incident energy. The data were compared with the angular distributions predicted by the distorted-wave theory of direct nuclear reactions, and no agreement could be obtained when only a stripping mechanism was taken into account.
E[ 1
NUCLEAR REACTION 12C(~, n)150, E~ = 18-23 MeV; measured or(0). DWBA calculation. Natural target.
1. Introduction D u r i n g the past few years the multinucleon transfer processes have become an important source o f information concerning reaction mechanisms and nuclear structure. Trinucleon transfer, viewed as knock-out, pick-up or stripping in reactions like (~, N ) and (N, c~), is particularly suited for the study of nuclear levels in those nuclei for which it is difficult to get data by other means. Unfortunately, multiple transfer in the region o f light and very light nuclei is by no means as clearly understood as single-nucleon transfer; therefore, comparison between experimental results and a direct-reaction theory such as D W B A presents many difficulties. A m o n g these we should emphasize the laborious mathematics and programming, the insufficiency o f the optical model in describing 7-particle reactions, and the scarcity o f experimental data over a wide mass-energy range. It is evident from the above considerations that a more careful investigation is necessary, both experimentally and theoretically, o f c~-particle reactions on light nuclei. The available experimental data indicate that (c~, N) reactions at sufficiently high energy (say above 10 MeV), proceed by direct interaction. The angular distributions frequently include a strong forward peaking which cannot be interpreted in the spirit o f the compound-nucleus theory. t Present address: N e u t r o n Physics Laboratory, Studsvik, Sweden. tt N o w at Cyclo Lab, Nuclear Research Center, Karlsruhe, G e r m a n y . t+t W o r k s u p p o r t e d by C S E N u n d e r contract no 558/7B, 1972. 377
378
V. CORCALCIUC et al.
The reaction mechanism is usually considered as a stripping process a + A ~ B + b (a --- b + x , B = A + x where a, b, x, A and B are nucleon numbers) in which the core excitations are negligible and the simplest configurations of the final nucleus are those for which ~(B) = ~(A)~b(x), with appropriate angular-momentum coupling. If one assumes this simplified picture of the reaction process then one can hope that the experimental data obtained from the (~, no) reaction on the simple nucleus laC(0 +) should be more amenable to a physical interpretation and, therefore, such an experiment and the corresponding theoretical analysis can test the applicability of the DI theory to a-particle reactions on light nuclei.
2. Experimental results and DWBA analysis The Y-120 cyclotron at the Institute of Atomic Physics in Bucharest was used to provide a 0.5 #A a-particle beam of 12-26 MeV, with an energy resolution of about 1.5 % and a natural modulation of about 2 ns. The neutrons emitted from a 1.6 mg/cm 2 target of pure self-supporting graphite (0.48 MeV energy loss) were analysed by means of our time-of-flight spectrometer (fig. 1), described previously 1). There were no difficulties with time-resolution and peak-separation for the ground state neutrons (fig. 2), since the first excited level in the final nucleus is located at 5.195 MeV, though a continuum background has to be taken into account. The a-particle energy was chosen sufficiently high (above 18 MeV) to assure the predominance of the direct effects and to correspond to some salient intervals (resonances or valleys in fig. 3) of the excitation function 2). The relative differential cross sections a(O) were measured at incident energies of 18.4, 18.9, 20.8, 21.2, 22.2 and 23.1 MeV (average values from the time-of-flight spectra) at 10° intervals from 0 ° to
IW
V ....
m --
j
TPHC ]
tAFj
~i A
I
l
I
C l-----
CB .J
[i
~4A
E
FD-~a~ . . . .lYi. m-e-----~C ] SA-40 ~
|
[. . . . l , I ,, I '
Analyzer I
1
SA-40 AnacIRYzer
i .=a
Fig. 1. Experimental set-up for time-of-flight spectrometry. D-mobile detector, M-fixed monitor (plastic+56 AVP surrounded by lead and borated paraffin), FM-fast mixer, TPHC-time-to-pulseheight converter, CB-command block, A-amplifier, D-discriminator (common threshold), F-interface RF-radio frequency.
a2C(% n) 1~0 DWBA ANALYSIS
379
Pulses 150C
100(
Y2
71
xlO2
3 ns
xlO2
II 0
50
I00
150
200
Channels
Fig. 2. A typical time-of-flight spectrum from the 12C(~, n)~SO reaction, at E a = 18.4 MeV and 0 = 30° (L = 3m). C~mb
3oL
13
] 1
15
17
19
21
23
E~,MeV
Fig. 3. Excitation function of the reaction 12C(~, n) 150. The arrows indicate the incident energies in the present experiment. 130 ° . The errors affecting the experimental data are mainly due to statistics and backg r o u n d subtraction and have been estimated at about 5 ~ , in three independent runs. O u r results are generally in agreement with those o f K o n d o et al. a) measured in 1963 at incident energies o f 20.0, 20.9 and 21.8 MeV, by using a different technique and a rather thick target (1.3 MeV energy loss). The theoretical analysis o f the present experimental angular distributions has been done assuming 3He stripping, in order to c o m p u t e with the help of the K u n z p r o g r a m [ref. 4)] the quantity:
aLSJ(o)_
4n 1 . 0 x l 0 4 ( 2 L + l ) Ea Eo 2Sa + 1
k b ~ Cl~.[t~'a~'b2 fm2/sr, ~-a ,',vbXt ~ L S J
380
et al.
V. C O R C A L C [ U C TABLE 1
Optical-model parameters Ei,¢.
Set
Particle
VR
WI
18.4
1
18.9
2
20.8
3
21.2
4
2.22
5
23.1
6
18.4
1A
18.4
1B
18.4
1C
~ n ~ n ~ n ~ n ~ n ~ n cz n g n :~ n
--212.63 - - 47.63 -- 212.50 -- 4 7 . 5 0 --212.04 -- 47.04 --211.96 -- 46.96 --211.66 -- 46.66 --211.48 - - 46.48 - - 2 0 2 63 - - 47.63 --192.63 --192.63 --182.63 - - 182.63
4V~.o.a
- - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 - - 15.75 - - 5.75 -- 16.75 - - 5.75 --17.75 - - 5.75 --18.75 - - 5.75
4W~.o.t
0.0 --37.00 0.0 --36.92 0.0 --36.60 0.0 --36.56 0.0 --36.36 0.0 --36.24 0.0 --37.00 0.0 --37.00 0.0 --37.00
R0R :
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Rol
1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25 1.65 1.25
a :
b
0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55 0.70 0.55
E n e r g i e s in M e V , l e n g t h s in f m .
Uopt (r) = Vc (r)~
VR
r - - R ~_
X r
a at ~ VR
v3He --R
V~
n :t - - VR' WI =
ikVl
t • S,
rdr~ ) I xl e " : ' ~ + eX'-}-~ --(V~.o.R -:-iWs.o.O r( ~e T
r--Rl --- , RR -~ RoRAJ, Rl = RoIA{, b
W 3He -t, --I W~,
49.3--0.31En,
Vn
s.o.R
=
V 3He --R
=
165 M e V ,
9.52--0.05E~,
"W' 13 H e
~
10 M e V ,
W~ = 5.75 M e V .
where t~ 2 ~--- 2 a +
e~QgLSJ
=
Z
1,
"la--lb--L~2t~lasaJa I~lbsbJb f'jbJJa plbLla 1 ~b ~"~OVama~JOVbmb ~"~mo-MMma "~000
jajblalb
x ~ 7j~Tb
S
Jjalajbtb
mb--M'
~laSaJa
The last expression involves the distorted waves and form factor in the form of the stripping integrals:
jojbt.tb = (A/B) dr, z*blb(kb, r.A/B)fLss(r,)zj.,.(ka, yLJ f Thus, the differential cross section becomes: do" _ 2JB+ 1 dr2
[BLss[ 2
o"Lss
2JA+I 1.0xl0 4 2J+l'
r.).
12C(~,
n) 150 D W B A A N A L Y S I S
381
I
151 ~(e) I
i
,
I0 [ / ~
I
[
I
Ea=23.1±0.5 MeV Sei6
¢
5//
I 1
15
I
, Ea:20,.8_+0.3 WMVe Set 3
,
~et18.9-+0.4 MeV
l
1
~
l
l
~et2~.2-+0.4 MeV
I0
l
l
l
15 ,
I
I
I
1
l
Ea=18.4±0.3 MeV Set 1
÷
l
l
Ec~:21.2±0.4 MeV Set 4
I0
5
0
l
30
1
60
l
90
l
120
1
150
i
30
l
60
i
90
l
120
l
150 0o
Fig. 4. DWBA fits to the experimentalangular distributions from the ~2C(~,n)~50 reaction using the p a r a m e t e r sets 1-6 f r o m table 1. T h e calculated, values are normalized to experimental ones at 45 ° (c.m.).
382
V. CORCALCIUC et i
15
u
i
al.
,
i
I
Ea=18.4 MeV
Set 1A
~(e)
V~:-202.63 MeV
10
W?:-I6.75 MeV
5
~
~ i 1' ¢
I
I
15
I
I
I
Ea=18.4 MeV Set 1B
+
V~-192.63 MeV
10
W?=-I7. 75 MeV
5
,b
I
15
I
,
I
~
I
I
|
÷
Ea:18.4 MeV Set IC 101
~ +
W~:-18.75 MeV
I
I
I
I
0
30
60
90
120
I
150 eo
Fig. 5. DWBA fit to the 18.4 MeV angular distribution from the reaction 12C(~, n) j 50 using the parameter sets 1A-IC from table 1 (the depths of the real and imaginary wells for =(-particles vary in 10 and 1 MeV steps respectively).
12C(~,n)lSO D W B A ANALYSIS
383
where BLsj = (~/~,)ALsj and ALsj is the usual spectroscopic coefficient of the J U L I E code. This expression for the differential cross section reduces the triple stripping to the transfer of a "heavy nucleon" of mass 3 and charge 2 with J= and T as well-defined quantum numbers. On the other hand, one knows that the general D W B A theory of multiple transfer 5) involves a coherent sum over the N, n, n' principal quantum numbers of the c.m. and relative motions. Therefore, the single-nucleon transfer formula cannot be correct for (c~, N) reactions unless it can be argued that only one value of N contributes to the cross section. Since the D W U C K program does not add the amplitudes for different N we have taken a unique value N = 2, assuming a symmetric s-state for the relative motions (n = n' = 1 and l = l' --- 0) in the helion. The nucleons transferred from the projectile to the ground state (0 +) of the target nucleus are in the l p state (n 1 = t'/2 = r/3 = 1, 11 = 12 = 13 = l) and the final nucleus is obtained in its ground state (½-). The form factor is calculated by D W U C K in the zero-range approximation: =
y
N FLsj
N
where the radial form factor F~ss(N = 2, 2 J = l, L = l, 2S = 1) is the radial wave function of the helion considered as a point object bound in a Saxon-Woods well, with an appropriate energy (the separation energy of 3He and ~2C in ~50 is 12.122 MeV). In the framework of the drastic simplifications made above, a very difficult problem arises in addition, namely, the calculation of the distorted waves. It is well known that the geometrical and dynamical parameters of the optical model, for the interaction of the ~-particle with the nucleus, present some ambiguities, and they have not yet been well determined or verified from elastic scattering experiments. Owing to the lack of a realistic criterion in the choice of the z-particle optical-model parameters, we have adopted the suggestion of Stock et al. 6) assuming that the well for an :~-particle (table 1) would be that for 3He [in our case that given by Fortune 7)], to which could be added that of the neutron [in our case that given by Rosen et aL 8), which also applies to the exit channel]. The results of the D W U C K calculations on an IBM 360/40 with the parameters giver in table 1 are shown in figs. 4 and 5. 3. Conclusions The reaction 12C(e, no) 150 has been analysed at bombarding energies between 18 and 23 MeV. In this energy range the stripping zero-range D W B A calculations do not reproduce the observed angular distributions, with the exception of the 18.4 and 18.9 MeV incident energies (minimum of the excitation function) at forward angles ranging from 0 ° to about 80 °. It is evident from the excitation function (fig. 3) that compoundnuclear effects are not negligible in the energy region we have studied. On the other
384
v. CORCALCIUC et al.
hand, the f o r w a r d p e a k i n g in the differential cross sections is very unlikely to be caused by c o m p o u n d resonances. These facts suggest a s t r o n g interference between r e s o n a n t a n d direct processes. A p r o p e r finite-range D W B A t h e o r y o f the triple-stripping a n d a statistical evaluation o f the c o m p o u n d - n u c l e u s c o n t r i b u t i o n (with the same o p t i c a l - m o d e l p a r a m e t e r s for the calculation o f the t r a n s m i s s i o n coefficients a n d d i s t o r t e d waves) might at least be useful to o b t a i n better a g r e e m e n t between experiment and theory.
References 1) A. CiocS.nel, thesis, Bucharest, 1971, unpublished; M. Molea, preprint IFA, CRD-37, Bucharest, 1968 2) J. L. Black et aL, N'ucl. Phys. A l l 5 (1968) 683; R, D. Carpenter et al., Phys. Rev. 125 (1962) 282; A. V. Spasski et al., Atom. Energ. 26 (1969) 303 3) M. Kondo et aL, J. Phys. Soc. Jap. 18 (1963) 22 4) P. D. Kunz, Code DWUCK, Bucharest version, 1972, unpublished 5) A. Y. Abul-Magd et aL, Prog. Theor. Phys. 38 (1967) 366; N. K. Glendenning, Phys. Rev. 156 (1967) 1344; E. M. l~enley and D. U. L. Yu, Phys. Rev. 133 (1964) 1445 6) R. Stock et al., Nucl. Phys. A104 (1967) 136 7) H. T. Fortune et aL, Bull. Am. Phys. Soc. 12 (1967) 1197; P. E. Hodgson, Adv. Phys. 17 (1968) 563 8) L. Rosen et aL, Ann. of Phys. 34 (1965)96; D. Wilmore and P. E. Hodgson, Nucl. Phys. 55 (1964) 673