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Energy dependence of the 8Be + 13C interaction
Nuclear Physics A 660 Ž1999. 267–279
Energy dependence of the 8 Be q 13 C interaction A.T. Rudchik a , O.A. Momotyuk a , A. Budzanowski b, A. Szczurek b, V.K. Chernievsky a , A.V. Mokhnach a , V.A. Ziman a , E.I. Koshchy c , S. Kliczewski b, R. Siudak b, I. Skwirczynska ´ b, J. Turkiewicz d a
b
Institute for Nuclear Research, 252028 KieÕ, Ukraine H. Niewodniczanski ´ Institute of Nuclear Physics, PL-31-342 Cracow, Poland c KharkiÕ State UniÕersity, 310077 KharkiÕ, Ukraine d A. Soltan Institute for Nuclear Research, PL-00-681 Warsaw, Poland
Received 21 July 1999; received in revised form 17 September 1999; accepted 27 September 1999
Abstract Angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction were measured at the beam energy of E labŽ 12 C. s 65 MeV for the transitions to the ground, 3.09 MeV Ž1r2q. and 3.68 MeV Ž3r2y. q 3.85 MeV Ž5r2q. excited states of the 13 C nucleus. These data together with experimental data at the energies of E labŽ 12 C. s 12, 15 MeV and ElabŽ 9 Be. s 20 MeV were analyzed within the coupled reaction channels ŽCRC. model. The elastic and inelastic scattering as well as one- and two-step transfers were included in the coupled channel scheme. It was found that n- and a-transfers dominate in these reactions. The energy dependence of the OM potential for the 8 Be q 13 C channel was deduced. A good description of all sets of experimental data was achieved. q 1999 Elsevier Science B.V. All rights reserved. PACS: 24.50.qg; 24.10.Eq; 25.45.Hi
Keywords: NUCLEAR REACTIONS 9 BeŽ 12 C, 13 C., E lab Ž 12 C. s 65 MeV; Measured s Ž u .; Coupled reaction channels model analysis of the reaction data at E labŽ 12 C. s 12, 15, 65 MeV and E labŽ 9 Be. s 20 MeV; Deduced optical model parameters for the 8 Be q 13 C interaction; Analysis of the energy dependence of the 8 Be q 13 C interaction
1. Introduction The interaction of radioactive heavy-ion beams with atomic nuclei is one of the most interesting problems of nuclear physics studied in many nuclear research centers. Unfortunately, the experiments with radioactive beams are expensive and strongly restricted by the life time of radioactive ions and beam intensities. At present, only elastic scattering of radioactive ions can be effectively investigated experimentally. To study an interaction of very short-lived ions with atomic nuclei one needs an alternative method. One of such methods is to use nuclear reactions with radioactive Žor unstable. 0375-9474r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 4 1 1 - X
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ions in the exit channels Žwe shall call them the reactions with radioactive channels or RC reactions for brevity.. The quasi-elastic RC reactions with well-known mechanisms are preferable for this purpose. The RC reactions can be successfully investigated within the coupled reaction channels ŽCRC. model. In this case the optical model ŽOM. potential parameters of the exit channel are fitted to describe the reaction angular distributions. The RC reactions can be effectively used to study the interaction of unstable nuclei provided that the OM potential for the entrance channel was already obtained in the analysis of elastic, inelastic scattering and other reaction channels in a broad energy range, spectroscopic amplitudes were determined experimentally or calculated in a model and the mechanism of the reaction was very carefully studied. These conditions can be easily fulfilled for the 9 BeŽ 12 C,13 C. 8 Be reaction for which the energy-dependent OM potential for the 9 Be q 12 C elastic scattering was recently obtained in the work w1x and where the n-transfer evidently dominates. This article is devoted to the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of E lab Ž 12 C. s 65 MeV. The angular distributions for the transitions to the ground and low-energy excited states of the 13 C and 8 Be nuclei were measured using the Kiev cyclotron U-240. These data together with the available data for the 9 BeŽ 12 C,13 C. 8 Be reaction at Elab Ž 12 C. s 12, 15 MeV w2x and 12 CŽ 9 Be,8 Be.13 C reaction at Elab Ž 8 Be. s 20 MeV w3x were analyzed within the CRC model using the energy-dependent OM potential found in Ref. w1x for the 9 Be q 12 C interaction. The spectroscopic amplitudes were calculated within translation-invariant shell model ŽTISM. w5x, which was tested in the analysis of various reaction data. The mechanism of the 9 BeŽ 12 C,13 C. 8 Be reaction was carefully studied and the results are presented in this article. In the CRC calculations we used two OM potentials for the 8 Be q 13 C interaction: the same as for the entrance channel and the OM potential obtained in the fitting procedure. It is shown that the first potential failed in the description of the experimental data and the second possibility must be considered. The energy dependence of this potential was studied using the approach described in detail in Ref. w1x. It was compared with the energy dependence of the OM potential for the 9 Be q 12 C elastic scattering. Some differences were found. The present paper is organized as follows. In Section 2 we present the experimental procedure and experimental data for the angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of Elab Ž 12 C. s 65 MeV. Results of analysis of all available experimental data at Elab s 12–65 MeV are presented in Section 3. The energy-dependent OM potential obtained in the analysis for the 8 Be q 13 C interaction is discussed in Section 4. A summary and conclusions close our paper.
2. Experimental procedure The angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction for the transitions to the ground and low-energy excited states of the 13 C and 8 Be nuclei were measured using the Kiev U-240 cyclotron at the beam energy Elab Ž 12 C. s 65 MeV in the angular range ucm f 248–958. The spread of the beam energy on the target was ; 0.6%. The 9 Be target was prepared on the thin nickel foil Ž; 300 m grcm2 .. The 9 Be layer was equal to ; 300 m grcm2 . Two D E y E spectrometers with D E Ž40 m m. and E Ž1 mm. silicon
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
269
Fig. 1. Angular distribution of the 9 BeŽ12 C,13 C. 8 Be reaction at the energy Elab Ž12 C. s 65 MeV for the transitions to the ground states of the 13 C and 8 Be nuclei. The dashed curves ²n:, ² a :, ² 3 Heqn:, ²dqd: and solid curve S show the angular distributions corresponding to the n, a , sequential 3 He q n, dq d transfers and a coherent sum of these processes calculated within the CRC model, respectively. The dotted curve S 0 corresponds to the CRC cross section obtained with the same OM potentials for the entrance and exit channels.
detectors and standard electronics were used in the experiment. Two-dimensional E = D E-spectra for Z s 3–6 were accumulated. The energy spectra of 11 C, 12 C and 13 C were obtained from the two-dimensional spectra with special computer codes prepared for this purpose. The energy resolution of the spectrometers was sufficient to separate the transition to the 2.94 MeV Ž2q. Žstate of 8 Be. q 3.09 MeV Ž1r2q. Žstate of 13 C. from the transitions to the 3.68 MeV Ž3r2y. q MeV 3.85 Ž5r2q. states of the 13 C nucleus. The angular distributions for the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of Elab Ž 12 C. s 65 MeV for the transitions to the ground states of the 8 Be and 13 C nuclei and to the 2.94 MeV Ž2q. Žstate of 8 Be. q 3.09 MeV Ž1r2q. Žstate of 13 C. and 3.68 MeV Ž3r2y. q 3.85 MeV Ž5r2q. excited states of the 13 C nucleus are shown in Figs. 1 and 2.
3. Analysis of the data The angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of Elab Ž 12 C. s 65 MeV Ž Ecm s 27.86 MeV. for the transitions to the ground and excited states of 13 C and 8 Be nuclei together with similar data at E lab Ž 12 C. s 12, 15 MeV Ž Ecm s 5.14, 6.43 MeV. w2x and E lab Ž 9 Be. s 20 MeV Ž Ecm s 11.43 MeV. for the 12 CŽ 9 Be,8 Be.13 C reaction w3x were analyzed within the CRC model using the code FRESCO w4x. The oneand two-step transfer reactions corresponding to the diagrams are presented in Fig. 3, and the elastic and inelastic scattering for the transitions to the 1.68 MeV Ž1r2q. and 2.43 MeV Ž5r2y. excited states of the 9 Be nucleus were included in the coupled channel scheme. The excited states of 9 Be were assumed to be rotational and deformation parameters obtained in Ref. w1x were used in the CRC calculations.
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
270
Fig. 2. Angular distributions of the 9 BeŽ12 C,13 C. 8 Be reaction at the energy Elab Ž13 C. s 65 MeV for the transitions to the 2.94 MeV Ž2q . q 3.09 MeV Ž1r2q . excited states of the 8 Be and 13 C nuclei, respectively Župper panel. and to the 3.68 MeV Ž3r2y . q 3.85 MeV Ž5r2q . excited states of the 13 C nucleus Žlower panel.. The dashed curves S 2.94 , S 3.09 , S 3.68 , S 3.85 show the angular distributions calculated within the CRC model for the transitions to the 2.94 MeV, 3.09 MeV, 3.68 MeV, 3.85 MeV excited states of the 8 Be and 13 C nuclei, respectively, and solid curves S 2.94q3.09 , S 3.68q3.85 show incoherent sums of the correspondent distributions.
The Woods–Saxon optical potential V0
VŽ r. s 1 q exp
ž
r y RV aV
WS
qi
/
1 q exp
ž
r y RW aW
Ž 1.
/
and Coulomb potential of the uniformly charged sphere
°h VC Ž r . E
~ kR
s
C
2h
¢ kr
ž
3y
r2 R C2
/
for r ( R C
Ž 2. for r 0 R C
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271
Fig. 3. Diagrams of one- and two-step transfer processes relevant for the 9 BeŽ12 C,13 C. 8 Be reaction.
1r3 . were used for the entrance and exit reaction channels. Here R i s ri Ž A1r3 P q AT Ž i s V,W,C ., A P and AT are the masses of the projectile and target, respectively, h s 0.157454ZP ZT mrE , m s A P ATrŽ A P q AT . and k is the wave vector length in the corresponding channel. The energy-dependent OM potential for the elastic scattering 9 Be q 12 C obtained recently in Ref. w1x was used for the entrance reaction channel. The parameters of this potential are given in Table 1. The OM potential parameters for the exit 8 Be q 13 C
'
Table 1 3 1r3 . Parameters of Woods–Saxon OM potentials Ž Rs ri Ž A1r , isV,W,C . T q AP System 9
12
Beq C
Ecm wMeVx V wMeVx r V wfmx aV wfmx WCRC wMeVx r W wfmx aW wfmx rC wfmx Ref.
5.14 6.43 11.43 27.86 8 ) Beq13 C 3.85 4.57 8 ) Beq13 C 3.09 5.34 8 ) Beq13 C 3.85 5.86 8 ) Beq13 C 3.09 6.62 8 Beq13 C 8.42 8 Beq13 C 9.71 8 ) Beq13 C 3.85 10.86 8 ) Beq13 C 3.68 11.03 8 13 ) Beq C 3.09 11.62 8 Beq13 C 14.71 8 ) Beq13 C 3.85 27.28 8 ) Beq13 C 3.68 27.45 8 ) Beq13 C 3.09 28.05 8 ) Be 2.94 q13 C 28.20 8 Beq13 C 31.14 10 Bq11 B 18.00
108.0 115.0 167.1 181.4 91.3 100.0 105.5 110.0 129.0 147.0 148.0 149.0 159.0 170.0 170.9 170.9 171.5 171.5 170.2 169.6
0.843 0.810 0.789 0.789 1.146 1.127 1.114 1.030 0.950 0.950 0.894 0.885 0.870 0.830 0.793 0.793 0.793 0.793 0.793 0.788
0.760 0.790 0.800 0.760 0.354 0.371 0.385 0.390 0.440 0.660 0.664 0.671 0.723 0.750 0.760 0.760 0.760 0.760 0.760 0.760
3.50 3.80 8.00 14.50 0.52 0.64 0.70 0.80 1.70 2.40 2.76 2.81 3.50 5.50 6.97 6.97 7.00 7.00 7.00 10.00
1.350 1.340 1.270 1.250 1.517 1.500 1.490 1.451 1.400 1.370 1.336 1.333 1.325 1.280 1.250 1.250 1.250 1.250 1.250 1.250
0.600 0.600 0.680 0.680 0.354 0.371 0.385 0.390 0.440 0.670 0.664 0.671 0.723 0.750 0.760 0.760 0.760 0.760 0.760 0.760
0.843 0.810 0.789 0.789 1.146 1.127 1.114 1.030 0.950 0.950 0.894 0.885 0.870 0.830 0.793 0.793 0.793 0.793 0.793 0.788
w1x w1x w1x w1x
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272
channel were fitted to the reaction data using the OM potential of the entrance channel as the starting potential in the fitting procedure. The spectroscopic amplitudes ŽSA. for the 1p-shell states of nuclei were calculated within the translation-invariant shell model ŽTISM. w5x using the code DESNA w6,7x and Boyarkina’s wave functions w8x. The spectroscopic amplitudes for the 3.09 MeV Ž1r2q. and 3.85 MeV Ž5r2q. excited sd-shell states of 13 C were fitted to the corresponding data at E lab Ž 12 C. s 12, 15 MeV w2x. The fitting procedure is reasonable because the CRC angular distributions of these reactions at small angles are only slightly dependent on the variation of the OM potential parameters for the exit channels of these reactions. This means that a good description of the angular distributions at forward angles is evidence for the correctness of the spectroscopic amplitudes. The values of the spectroscopic amplitudes are collected in Table 2. The angular distributions calculated within the CRC model are shown in Figs. 1, 2, 4–6 by the solid and dashed curves. In Figs. 1, 4–6 the dotted curves marked by S 0 display illustrate the CRC cross sections obtained with the same potentials in the entrance and exit channels. The evident failure in the description of experimental data is clear evidence for the necessity to use different potentials in the exit and entrance channels. Figs. 1, 4–6 show that the n-transfer Ždashed curves ²n:. dominates in the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions at all beam energies considered here. The a transfer Ždashed curves ² a :. contributes to the angular distributions only at large angles and at some energies. Fig. 1 shows that the two-step transfers 3 He q n and d q d Ždashed curves ² 3 He q n: and ²d q d:. in these reactions are negligible. Other two-step reactions give also small contributions to the cross section for these reactions. Therefore the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions are suitable for studying the 8 Be q 13 C interaction at different energies. The values of the OM potential parameters V,WS s WCRC ,r V ,r W ,aV ,aW , rC s r V obtained in the fitting procedure for the interaction of the 8 Be and 13 C nuclei in their Table 2 Spectroscopic amplitudes of clusters x in the AsC q x systems A
C
8
7
Be Be 9 Be
8
9
Be
7
10
Be Be 10 B 11 B
9
10
8
9
Be Be 8 ) Be 2.94
n n n
Be
2n
Be Be 8 Be 9 Be
n 2n d d
12
11
12
10
C C 12 C 12 C
x
C B 9 Be 8 Be
n d 3 He a
nlj 1 P3r 2 1 P3r 2 1 P1r2 1 P3r 2 2 S0 1 D2 1 P3r 2 2 S0 1 D3 2 S1 1 D1 1 D3 1 P3r 2 1 D3 2 P3r 2 3S0
Sx
A
C
y1.234 0.866 y0.573 0.573 0.247 0.430 1.405 y0.833 0.811 y0.607 y0.109 0.610 1.706 1.780 1.224 0.822
12
C C 13 ) C 3.088 13 ) C 3.684 13 ) C 3.854 13 C 13 C
8
13
9
13
9
13
C ) C 3.088 13 ) C 3.684
13
) C 3.854 C 14 C 14
) Be 2.94 12
C 12 C 12 C 12 C 11 C 11 B
Be Be 9 Be 9
Be C 12 C 13
x
nlj
Sx
a n n n n 2n d
2 D2 1 P1r2 2 S1r2 1 P3r2 1 D5r2 1 D2 2 S1 1 D1 1 D2 2 D2 3 P1 3S0 2 D2 3 P1 1 P1r2 2 S0
y0.919 0.601 0.550 0.601 0.550 y0.559 y0.263 y0.162 y0.485 0.504 0.500 y0.539 y0.356 y0.400 y1.094 0.615
a a a a n 2n
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273
Fig. 4. Angular distributions of the 9 BeŽ12 C,13 C. 8 Be reaction at the energy E lab Ž12 C. s 12 MeV w2x for the transitions to the ground states of the 13 C and 8 Be nuclei Župper panel. and to the 3.09 MeV Ž1r2q . and 3.85 MeV Ž5r2q . excited states of the 13 C nucleus Žmiddle and lower panels, respectively.. The dashed curves ²n:, ² a : and solid curve S show the angular distributions corresponding to the n, a transfers and a coherent sum of these processes calculated within the CRC model, respectively. The meaning of the dotted curve S 0 is the same as in Fig. 1 but at Elab Ž12 C. s 12 MeV.
ground and excited states at the energies Ecm s 4.57–31.14 MeV are given in Table 1. The parameters aV and aW were fitted independently from each other. It was found that the CRC angular distributions are more sensitive to the variation of the parameter aV than to the variation of the parameter aW . The best description of the data was observed for almost identical values of aV and aW . Thus, in the following we put aV s aW . The CRC cross sections calculated as a coherent sum of the n- and a y transfer transition amplitudes for the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions are shown in Figs. 1, 2 and Figs. 4–6 by the solid curves S , S 2.94q3.09 and S 3.68q3.85 . As can be seen from the figures a good description of all experimental data is achieved. We note that the OM potentials for the interactions of the 8 Be and 13 C nuclei in their ground and excited states can differ only by their spin-dependent parts. The depth of the spin-dependent part of the OM potential for the nucleus–nucleus interaction is considerably smaller than the corresponding depth of its central part. For this reason we used the values of OM potential parameters obtained for excited states of nuclei Žignoring spin
274
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
Fig. 5. The same as in Fig. 2 at the energy Elab Ž12 C. s 15 MeV w2x.
effects. to study the energy dependence of the 8 Be q 13 C OM potential at the corresponding relative center-of-mass energies Žsee Table 1..
4. Energy dependence of OM potential parameters The values of OM potential parameters listed in Table 1 for the 8 Be q 13 C interaction are plotted as functions of the center-of-mass energy in Figs. 7 and 8. The errors D X i of the parameters X i s V0 , WS s WCRC , r V , r W shown in Figs. 7 and 8, were estimated by using the following simple criterion w1x: min N x 2 Ž X i . y x 2 Ž X i q D X i . N ,N x 2 Ž X i . y x 2 Ž X i y D X i . N 4
x 2 Ž Xi .
f 1.
If the error D X i determined in such a way was smaller than 0.1 = X i , then it was assumed to be D X i s 0.1 = X i . In the region of the Coulomb barrier one may expect a fast evolution of the parameters of nucleus–nucleus potential. The following simple parametric forms of the
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275
Fig. 6. Angular distributions of the 9 BeŽ12 C,8 Be.13 C reaction at the energy E lab Ž 8 Be. s 20 MeV w3x for the transitions to the ground states of the 13 C and 8 Be nuclei Župper panel. and to the 3.09 MeV Ž1r2q . and 3.68 MeV Ž3r2y . q 3.85 MeV Ž5r2q . excited states of the 13 C nucleus Žmiddle and lower panels, respectively.. The dashed curves ²n:, ² a : and solid curve S show the angular distributions corresponding to the n, a transfers and coherent sum of these processes calculated within the CRC model, respectively. The dashed curves S 3.68 , S 3.85 and solid curve S 3.68q3.85 show the angular distributions calculated within the CRC model for the transitions to the 3.68 MeV, 3.85 MeV excited states of the 13 C nucleus and incoherent sum of these distributions, respectively. The meaning of the dotted curve S 0 is the same as in Fig. 1 but at Elab Ž12 C. s 20 MeV.
Woods–Saxon type were chosen to parameterize the observed energy dependence of the OM parameters w1x:
WS Ž E .
° ¶ 1 ~1 y • ¢ 1 q exp E y E ß,
s WSmax
ž
ri Ž E .
s rimin q
W
DW
rimax y rimin 1 q exp
ž
E y Er i
Dr i
,
/
Ž 3.
/
Ž 4.
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
276
a max y a min i i
a i Ž E . s a min q i
1 q exp
ž
E y Eai
D ai
.
Ž 5.
/
Here E s Ecm , i s V,W, ri s r V ,r W and ai s aV ,aW . The parameters X imax s WSmax ,rimax ,amax and X imin s rimin ,a min correspond to the values of the upper and lower i i limits for the variation of OM potential parameter X i s WS , r V , r W , aV , aW . Taking into account the dispersion relation, connecting V and W, one gets for V Ž E . w9x V Ž E . s V0 Ž E . q D V Ž E . ,
Ž 6.
where
DV Ž E . s
P
p
`
H0
WS Ž E X . EX y E
dEX ,
Ž 7.
P denoting the principal value of the integral.
Fig. 7. Energy dependence of Woods–Saxon potential parameters V Ž Ecm . and WS Ž Ecm . for the 8 Be q interaction. The markers are described in the text.
13
C
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
Fig. 8. Energy dependence of Woods–Saxon OM potential parameters for the 8 Be q Be q 12 C Ždashed curves w1x. interactions.
277
13
C Žsolid curves. and
9
To calculate D V Ž E ., WS Ž E . was approximated by two straight lines, WSmax for E ) Eb ,
° ¢
W Ž E . s~ W S
max S
Ž E y Ea .
Ž 8.
for‘E ( Eb .
Eb y E a
This is illustrated in Fig. 7 Žlower panel. by the dashed lines. In the interval Ž Ea , Eb . the straight line is the tangent to the WS Ž E . at the point E s EW s Ž Ea q Eb .r2 For such an approximation w9x WSmax E y Ea E y Ea E y Eb E y Eb DV Ž E . s ln y ln . Ž 9. p DW DW DW DW In practice, V0 Ž E . is often assumed to be energy-independent w9x. In our case we could not describe the energy dependence of V at low energies using this approximation. A better description was obtained using the energy-dependent form w1x
½
V0 Ž E .
° ¢
5
1
~1y
s V0max
1 q exp
ž
E y EV
DV
/
¶ • ß.
Ž 10 .
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
278
Table 3 Energy dependence of the 8 Be q
13
C OM potential parameters
Xi
V0 wMeVx
WS sWCRC wMeVx
r V wfmx
r W wfmx
aV wfmx
aW wfmx
X imin X imax
0.00 199.38 3.76 5.48
0.00 7.00 11.39 2.56
0.793 1.300 6.850 2.650
1.250 1.620 7.480 3.020
0.37 0.76 9.57 0.78
0.37 0.76 9.57 0.78
E X i wMeVx D X i wMeVx
The values of the energy dependence parameters X imin , X imax , EX i and D X i obtained from the fitting procedure are listed in Table 3. The corresponding curves of the energy dependence of V Ž E .,WS Ž E .,r V Ž E .,r W Ž E .,aV Ž E .,aW Ž E . are plotted in Figs. 7 and 8. For comparison, in Fig. 8 the energy dependence of the OM potential parameters for the 8 Be q 13 C and 9 Be q 12 C w1x interactions is shown by the solid and dashed curves, respectively. It can be seen that the differences between the real parts V Ž E . of the OM potentials are rather small at the energies Ecm - 50 MeV. The curve V Ž E . for the 8 Be q 13 C interaction shows only a smaller threshold anomaly in comparison with its counterpart for the 9 Be q 12 C interaction. This is due to the fact that WS Ž E . s WCRC Ž E . for the 8 Be q 13 C OM potential is smaller than for the OM potential of 9 Be q 12 C interaction Žthe second panel of Fig. 8.. Large differences are observed between values of the parameters r V Ž E ., r W Ž E . for the 8 Be q 13 C and 9 Be q 12 C interactions Žthe third panel of Fig. 8. at energies Ecm - 10 MeV. However, the parameters for both systems take similar values at larger energies. Very large differences are also observed for the energy dependence of aV Ž E ., aW Ž E . for the 8 Be q 13 C and 9 Be q 12 C systems at the energies Ecm - 15 MeV Žlower panel of Fig. 8.. For the 9 Be q 12 C interaction the functions aV Ž E . and aW Ž E . are close to constant whereas for the 8 Be q 13 C interaction these functions show a rapid variation at the energy Ecm f 9 MeV Žcorresponding to the Coulomb barrier energy.. These differences can be explained by the difference in the structure and the breakup threshold for the 8 Be and 9 Be nuclei.
5. Summary and conclusions The angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction were measured at the energy E lab Ž 12 C. s 65 MeV for the transitions to the ground states of the 8 Be and 13 C nuclei and to the 2.94 MeV Ž2q. Žstate of 8 Be. q 3.09 MeV Ž1r2q. Žstate of 13 C. and 3.68 MeV Ž3r2y. q 3.85 MeV Ž5r2q. excited states of 13 C nucleus in the angular range ucm f 248 - 908. These data together with the angular distribution of the 9 BeŽ 12 C,13 C. 8 Be reaction at the energies Elab Ž 12 C. s 12, 15 MeV w2x and the 12 CŽ 9 Be,8 Be.13 C reaction at E lab Ž 9 Be. s 20 MeV w3x for the transitions to the ground states of the 8 Be and 13 C nuclei and to the 3.09 MeV, 3.85 MeV excited states of the 13 C nucleus were analyzed within the coupled reaction channels ŽCRC. model w4x using the energy-dependent OM potential found recently in Ref. w1x for the 9 Be q 12 C channel and spectroscopic amplitudes calculated within the translation-invariant shell model w5x. The one- and two-step transfer reactions as well as elastic and inelastic scattering for the transitions to
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the low energy excited states of 9 Be were included in the coupled channel scheme. It was found that the n-transfer dominates in the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions at all energies studied. This simplified the fit of the OM potential to the 8 Be q 13 C channel experimental data. A good description of the experimental data is achieved at all energies. The energy dependence of the OM potential for the 8 Be q 13 C interaction was analyzed using V0 Ž E .,WS Ž E .,r V Ž E .,r W Ž E ., aV Ž E ., aW Ž E . forms of the Woods–Saxon type. The dispersion relation between the real and imaginary parts of the OM potential w9x was taken into account to deduce the energy dependence of V Ž E .. A regular behaviour of the energy dependence of the OM potential parameters for the 8 Be q 13 C interaction was found. A rapid variation of the aV Ž E . and aW Ž E . parameters was found at the energy close to the Coulomb barrier Ž Ecm f 9 MeV.. Energy dependences of the OM potentials for the 8 Be q 13 C and 9 Be q 12 C interactions were compared. Large differences were found for WS Ž E . in the whole energy range and for r V Ž E .,r W Ž E ., aV Ž E ., aW Ž E . at the energies Ecm - 15 MeV. These differences can be explained by the difference in the structure and breakup threshold for the 8 Be and 9 Be nuclei. In summary, we would like to note that a systematic study of the OM potentials of the nucleus–nucleus interaction of unstable and radioactive nuclei is very important to obtain new information about the structure of nuclei and properties of their interaction.
Acknowledgements This work was supported in part by the Polish State Committee for Scientific Research.
References w1x w2x w3x w4x w5x w6x
A.T. Rudchik et al., Nucl. Phys. A, in print. P.H. Barker et al., Nucl. Phys. A 155 Ž1970. 401. L. Jarczyk et al., Nucl. Phys. A 316 Ž1979. 139. I.J. Thompson, Comp. Phys. Rep. 7 Ž1988. 167. Yu.F. Smirnov, Yu.M. Tchuvil’sky, Phys. Rev. C 15 Ž1977. 84. A.T. Rudchik, Yu.M. Tchuvil’sky, Code DESNA, Kiev Institute for Nuclear Research report KIYAI-82-12 Ž1982.. w7x A.T. Rudchik, Yu.M. Tchuvil’sky, Ukrainian Phys. J. 30 Ž1985. 819. w8x A.N. Boyarkina, Structure of 1p-shell nuclei ŽMoscow State University, 1973.. w9x C. Mahaux et al., Nucl. Phys. A 449 Ž1986. 354.
Energy dependence of the 8 Be q 13 C interaction A.T. Rudchik a , O.A. Momotyuk a , A. Budzanowski b, A. Szczurek b, V.K. Chernievsky a , A.V. Mokhnach a , V.A. Ziman a , E.I. Koshchy c , S. Kliczewski b, R. Siudak b, I. Skwirczynska ´ b, J. Turkiewicz d a
b
Institute for Nuclear Research, 252028 KieÕ, Ukraine H. Niewodniczanski ´ Institute of Nuclear Physics, PL-31-342 Cracow, Poland c KharkiÕ State UniÕersity, 310077 KharkiÕ, Ukraine d A. Soltan Institute for Nuclear Research, PL-00-681 Warsaw, Poland
Received 21 July 1999; received in revised form 17 September 1999; accepted 27 September 1999
Abstract Angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction were measured at the beam energy of E labŽ 12 C. s 65 MeV for the transitions to the ground, 3.09 MeV Ž1r2q. and 3.68 MeV Ž3r2y. q 3.85 MeV Ž5r2q. excited states of the 13 C nucleus. These data together with experimental data at the energies of E labŽ 12 C. s 12, 15 MeV and ElabŽ 9 Be. s 20 MeV were analyzed within the coupled reaction channels ŽCRC. model. The elastic and inelastic scattering as well as one- and two-step transfers were included in the coupled channel scheme. It was found that n- and a-transfers dominate in these reactions. The energy dependence of the OM potential for the 8 Be q 13 C channel was deduced. A good description of all sets of experimental data was achieved. q 1999 Elsevier Science B.V. All rights reserved. PACS: 24.50.qg; 24.10.Eq; 25.45.Hi
Keywords: NUCLEAR REACTIONS 9 BeŽ 12 C, 13 C., E lab Ž 12 C. s 65 MeV; Measured s Ž u .; Coupled reaction channels model analysis of the reaction data at E labŽ 12 C. s 12, 15, 65 MeV and E labŽ 9 Be. s 20 MeV; Deduced optical model parameters for the 8 Be q 13 C interaction; Analysis of the energy dependence of the 8 Be q 13 C interaction
1. Introduction The interaction of radioactive heavy-ion beams with atomic nuclei is one of the most interesting problems of nuclear physics studied in many nuclear research centers. Unfortunately, the experiments with radioactive beams are expensive and strongly restricted by the life time of radioactive ions and beam intensities. At present, only elastic scattering of radioactive ions can be effectively investigated experimentally. To study an interaction of very short-lived ions with atomic nuclei one needs an alternative method. One of such methods is to use nuclear reactions with radioactive Žor unstable. 0375-9474r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 4 1 1 - X
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ions in the exit channels Žwe shall call them the reactions with radioactive channels or RC reactions for brevity.. The quasi-elastic RC reactions with well-known mechanisms are preferable for this purpose. The RC reactions can be successfully investigated within the coupled reaction channels ŽCRC. model. In this case the optical model ŽOM. potential parameters of the exit channel are fitted to describe the reaction angular distributions. The RC reactions can be effectively used to study the interaction of unstable nuclei provided that the OM potential for the entrance channel was already obtained in the analysis of elastic, inelastic scattering and other reaction channels in a broad energy range, spectroscopic amplitudes were determined experimentally or calculated in a model and the mechanism of the reaction was very carefully studied. These conditions can be easily fulfilled for the 9 BeŽ 12 C,13 C. 8 Be reaction for which the energy-dependent OM potential for the 9 Be q 12 C elastic scattering was recently obtained in the work w1x and where the n-transfer evidently dominates. This article is devoted to the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of E lab Ž 12 C. s 65 MeV. The angular distributions for the transitions to the ground and low-energy excited states of the 13 C and 8 Be nuclei were measured using the Kiev cyclotron U-240. These data together with the available data for the 9 BeŽ 12 C,13 C. 8 Be reaction at Elab Ž 12 C. s 12, 15 MeV w2x and 12 CŽ 9 Be,8 Be.13 C reaction at Elab Ž 8 Be. s 20 MeV w3x were analyzed within the CRC model using the energy-dependent OM potential found in Ref. w1x for the 9 Be q 12 C interaction. The spectroscopic amplitudes were calculated within translation-invariant shell model ŽTISM. w5x, which was tested in the analysis of various reaction data. The mechanism of the 9 BeŽ 12 C,13 C. 8 Be reaction was carefully studied and the results are presented in this article. In the CRC calculations we used two OM potentials for the 8 Be q 13 C interaction: the same as for the entrance channel and the OM potential obtained in the fitting procedure. It is shown that the first potential failed in the description of the experimental data and the second possibility must be considered. The energy dependence of this potential was studied using the approach described in detail in Ref. w1x. It was compared with the energy dependence of the OM potential for the 9 Be q 12 C elastic scattering. Some differences were found. The present paper is organized as follows. In Section 2 we present the experimental procedure and experimental data for the angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of Elab Ž 12 C. s 65 MeV. Results of analysis of all available experimental data at Elab s 12–65 MeV are presented in Section 3. The energy-dependent OM potential obtained in the analysis for the 8 Be q 13 C interaction is discussed in Section 4. A summary and conclusions close our paper.
2. Experimental procedure The angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction for the transitions to the ground and low-energy excited states of the 13 C and 8 Be nuclei were measured using the Kiev U-240 cyclotron at the beam energy Elab Ž 12 C. s 65 MeV in the angular range ucm f 248–958. The spread of the beam energy on the target was ; 0.6%. The 9 Be target was prepared on the thin nickel foil Ž; 300 m grcm2 .. The 9 Be layer was equal to ; 300 m grcm2 . Two D E y E spectrometers with D E Ž40 m m. and E Ž1 mm. silicon
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
269
Fig. 1. Angular distribution of the 9 BeŽ12 C,13 C. 8 Be reaction at the energy Elab Ž12 C. s 65 MeV for the transitions to the ground states of the 13 C and 8 Be nuclei. The dashed curves ²n:, ² a :, ² 3 Heqn:, ²dqd: and solid curve S show the angular distributions corresponding to the n, a , sequential 3 He q n, dq d transfers and a coherent sum of these processes calculated within the CRC model, respectively. The dotted curve S 0 corresponds to the CRC cross section obtained with the same OM potentials for the entrance and exit channels.
detectors and standard electronics were used in the experiment. Two-dimensional E = D E-spectra for Z s 3–6 were accumulated. The energy spectra of 11 C, 12 C and 13 C were obtained from the two-dimensional spectra with special computer codes prepared for this purpose. The energy resolution of the spectrometers was sufficient to separate the transition to the 2.94 MeV Ž2q. Žstate of 8 Be. q 3.09 MeV Ž1r2q. Žstate of 13 C. from the transitions to the 3.68 MeV Ž3r2y. q MeV 3.85 Ž5r2q. states of the 13 C nucleus. The angular distributions for the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of Elab Ž 12 C. s 65 MeV for the transitions to the ground states of the 8 Be and 13 C nuclei and to the 2.94 MeV Ž2q. Žstate of 8 Be. q 3.09 MeV Ž1r2q. Žstate of 13 C. and 3.68 MeV Ž3r2y. q 3.85 MeV Ž5r2q. excited states of the 13 C nucleus are shown in Figs. 1 and 2.
3. Analysis of the data The angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction at the energy of Elab Ž 12 C. s 65 MeV Ž Ecm s 27.86 MeV. for the transitions to the ground and excited states of 13 C and 8 Be nuclei together with similar data at E lab Ž 12 C. s 12, 15 MeV Ž Ecm s 5.14, 6.43 MeV. w2x and E lab Ž 9 Be. s 20 MeV Ž Ecm s 11.43 MeV. for the 12 CŽ 9 Be,8 Be.13 C reaction w3x were analyzed within the CRC model using the code FRESCO w4x. The oneand two-step transfer reactions corresponding to the diagrams are presented in Fig. 3, and the elastic and inelastic scattering for the transitions to the 1.68 MeV Ž1r2q. and 2.43 MeV Ž5r2y. excited states of the 9 Be nucleus were included in the coupled channel scheme. The excited states of 9 Be were assumed to be rotational and deformation parameters obtained in Ref. w1x were used in the CRC calculations.
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
270
Fig. 2. Angular distributions of the 9 BeŽ12 C,13 C. 8 Be reaction at the energy Elab Ž13 C. s 65 MeV for the transitions to the 2.94 MeV Ž2q . q 3.09 MeV Ž1r2q . excited states of the 8 Be and 13 C nuclei, respectively Župper panel. and to the 3.68 MeV Ž3r2y . q 3.85 MeV Ž5r2q . excited states of the 13 C nucleus Žlower panel.. The dashed curves S 2.94 , S 3.09 , S 3.68 , S 3.85 show the angular distributions calculated within the CRC model for the transitions to the 2.94 MeV, 3.09 MeV, 3.68 MeV, 3.85 MeV excited states of the 8 Be and 13 C nuclei, respectively, and solid curves S 2.94q3.09 , S 3.68q3.85 show incoherent sums of the correspondent distributions.
The Woods–Saxon optical potential V0
VŽ r. s 1 q exp
ž
r y RV aV
WS
qi
/
1 q exp
ž
r y RW aW
Ž 1.
/
and Coulomb potential of the uniformly charged sphere
°h VC Ž r . E
~ kR
s
C
2h
¢ kr
ž
3y
r2 R C2
/
for r ( R C
Ž 2. for r 0 R C
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
271
Fig. 3. Diagrams of one- and two-step transfer processes relevant for the 9 BeŽ12 C,13 C. 8 Be reaction.
1r3 . were used for the entrance and exit reaction channels. Here R i s ri Ž A1r3 P q AT Ž i s V,W,C ., A P and AT are the masses of the projectile and target, respectively, h s 0.157454ZP ZT mrE , m s A P ATrŽ A P q AT . and k is the wave vector length in the corresponding channel. The energy-dependent OM potential for the elastic scattering 9 Be q 12 C obtained recently in Ref. w1x was used for the entrance reaction channel. The parameters of this potential are given in Table 1. The OM potential parameters for the exit 8 Be q 13 C
'
Table 1 3 1r3 . Parameters of Woods–Saxon OM potentials Ž Rs ri Ž A1r , isV,W,C . T q AP System 9
12
Beq C
Ecm wMeVx V wMeVx r V wfmx aV wfmx WCRC wMeVx r W wfmx aW wfmx rC wfmx Ref.
5.14 6.43 11.43 27.86 8 ) Beq13 C 3.85 4.57 8 ) Beq13 C 3.09 5.34 8 ) Beq13 C 3.85 5.86 8 ) Beq13 C 3.09 6.62 8 Beq13 C 8.42 8 Beq13 C 9.71 8 ) Beq13 C 3.85 10.86 8 ) Beq13 C 3.68 11.03 8 13 ) Beq C 3.09 11.62 8 Beq13 C 14.71 8 ) Beq13 C 3.85 27.28 8 ) Beq13 C 3.68 27.45 8 ) Beq13 C 3.09 28.05 8 ) Be 2.94 q13 C 28.20 8 Beq13 C 31.14 10 Bq11 B 18.00
108.0 115.0 167.1 181.4 91.3 100.0 105.5 110.0 129.0 147.0 148.0 149.0 159.0 170.0 170.9 170.9 171.5 171.5 170.2 169.6
0.843 0.810 0.789 0.789 1.146 1.127 1.114 1.030 0.950 0.950 0.894 0.885 0.870 0.830 0.793 0.793 0.793 0.793 0.793 0.788
0.760 0.790 0.800 0.760 0.354 0.371 0.385 0.390 0.440 0.660 0.664 0.671 0.723 0.750 0.760 0.760 0.760 0.760 0.760 0.760
3.50 3.80 8.00 14.50 0.52 0.64 0.70 0.80 1.70 2.40 2.76 2.81 3.50 5.50 6.97 6.97 7.00 7.00 7.00 10.00
1.350 1.340 1.270 1.250 1.517 1.500 1.490 1.451 1.400 1.370 1.336 1.333 1.325 1.280 1.250 1.250 1.250 1.250 1.250 1.250
0.600 0.600 0.680 0.680 0.354 0.371 0.385 0.390 0.440 0.670 0.664 0.671 0.723 0.750 0.760 0.760 0.760 0.760 0.760 0.760
0.843 0.810 0.789 0.789 1.146 1.127 1.114 1.030 0.950 0.950 0.894 0.885 0.870 0.830 0.793 0.793 0.793 0.793 0.793 0.788
w1x w1x w1x w1x
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
272
channel were fitted to the reaction data using the OM potential of the entrance channel as the starting potential in the fitting procedure. The spectroscopic amplitudes ŽSA. for the 1p-shell states of nuclei were calculated within the translation-invariant shell model ŽTISM. w5x using the code DESNA w6,7x and Boyarkina’s wave functions w8x. The spectroscopic amplitudes for the 3.09 MeV Ž1r2q. and 3.85 MeV Ž5r2q. excited sd-shell states of 13 C were fitted to the corresponding data at E lab Ž 12 C. s 12, 15 MeV w2x. The fitting procedure is reasonable because the CRC angular distributions of these reactions at small angles are only slightly dependent on the variation of the OM potential parameters for the exit channels of these reactions. This means that a good description of the angular distributions at forward angles is evidence for the correctness of the spectroscopic amplitudes. The values of the spectroscopic amplitudes are collected in Table 2. The angular distributions calculated within the CRC model are shown in Figs. 1, 2, 4–6 by the solid and dashed curves. In Figs. 1, 4–6 the dotted curves marked by S 0 display illustrate the CRC cross sections obtained with the same potentials in the entrance and exit channels. The evident failure in the description of experimental data is clear evidence for the necessity to use different potentials in the exit and entrance channels. Figs. 1, 4–6 show that the n-transfer Ždashed curves ²n:. dominates in the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions at all beam energies considered here. The a transfer Ždashed curves ² a :. contributes to the angular distributions only at large angles and at some energies. Fig. 1 shows that the two-step transfers 3 He q n and d q d Ždashed curves ² 3 He q n: and ²d q d:. in these reactions are negligible. Other two-step reactions give also small contributions to the cross section for these reactions. Therefore the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions are suitable for studying the 8 Be q 13 C interaction at different energies. The values of the OM potential parameters V,WS s WCRC ,r V ,r W ,aV ,aW , rC s r V obtained in the fitting procedure for the interaction of the 8 Be and 13 C nuclei in their Table 2 Spectroscopic amplitudes of clusters x in the AsC q x systems A
C
8
7
Be Be 9 Be
8
9
Be
7
10
Be Be 10 B 11 B
9
10
8
9
Be Be 8 ) Be 2.94
n n n
Be
2n
Be Be 8 Be 9 Be
n 2n d d
12
11
12
10
C C 12 C 12 C
x
C B 9 Be 8 Be
n d 3 He a
nlj 1 P3r 2 1 P3r 2 1 P1r2 1 P3r 2 2 S0 1 D2 1 P3r 2 2 S0 1 D3 2 S1 1 D1 1 D3 1 P3r 2 1 D3 2 P3r 2 3S0
Sx
A
C
y1.234 0.866 y0.573 0.573 0.247 0.430 1.405 y0.833 0.811 y0.607 y0.109 0.610 1.706 1.780 1.224 0.822
12
C C 13 ) C 3.088 13 ) C 3.684 13 ) C 3.854 13 C 13 C
8
13
9
13
9
13
C ) C 3.088 13 ) C 3.684
13
) C 3.854 C 14 C 14
) Be 2.94 12
C 12 C 12 C 12 C 11 C 11 B
Be Be 9 Be 9
Be C 12 C 13
x
nlj
Sx
a n n n n 2n d
2 D2 1 P1r2 2 S1r2 1 P3r2 1 D5r2 1 D2 2 S1 1 D1 1 D2 2 D2 3 P1 3S0 2 D2 3 P1 1 P1r2 2 S0
y0.919 0.601 0.550 0.601 0.550 y0.559 y0.263 y0.162 y0.485 0.504 0.500 y0.539 y0.356 y0.400 y1.094 0.615
a a a a n 2n
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273
Fig. 4. Angular distributions of the 9 BeŽ12 C,13 C. 8 Be reaction at the energy E lab Ž12 C. s 12 MeV w2x for the transitions to the ground states of the 13 C and 8 Be nuclei Župper panel. and to the 3.09 MeV Ž1r2q . and 3.85 MeV Ž5r2q . excited states of the 13 C nucleus Žmiddle and lower panels, respectively.. The dashed curves ²n:, ² a : and solid curve S show the angular distributions corresponding to the n, a transfers and a coherent sum of these processes calculated within the CRC model, respectively. The meaning of the dotted curve S 0 is the same as in Fig. 1 but at Elab Ž12 C. s 12 MeV.
ground and excited states at the energies Ecm s 4.57–31.14 MeV are given in Table 1. The parameters aV and aW were fitted independently from each other. It was found that the CRC angular distributions are more sensitive to the variation of the parameter aV than to the variation of the parameter aW . The best description of the data was observed for almost identical values of aV and aW . Thus, in the following we put aV s aW . The CRC cross sections calculated as a coherent sum of the n- and a y transfer transition amplitudes for the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions are shown in Figs. 1, 2 and Figs. 4–6 by the solid curves S , S 2.94q3.09 and S 3.68q3.85 . As can be seen from the figures a good description of all experimental data is achieved. We note that the OM potentials for the interactions of the 8 Be and 13 C nuclei in their ground and excited states can differ only by their spin-dependent parts. The depth of the spin-dependent part of the OM potential for the nucleus–nucleus interaction is considerably smaller than the corresponding depth of its central part. For this reason we used the values of OM potential parameters obtained for excited states of nuclei Žignoring spin
274
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
Fig. 5. The same as in Fig. 2 at the energy Elab Ž12 C. s 15 MeV w2x.
effects. to study the energy dependence of the 8 Be q 13 C OM potential at the corresponding relative center-of-mass energies Žsee Table 1..
4. Energy dependence of OM potential parameters The values of OM potential parameters listed in Table 1 for the 8 Be q 13 C interaction are plotted as functions of the center-of-mass energy in Figs. 7 and 8. The errors D X i of the parameters X i s V0 , WS s WCRC , r V , r W shown in Figs. 7 and 8, were estimated by using the following simple criterion w1x: min N x 2 Ž X i . y x 2 Ž X i q D X i . N ,N x 2 Ž X i . y x 2 Ž X i y D X i . N 4
x 2 Ž Xi .
f 1.
If the error D X i determined in such a way was smaller than 0.1 = X i , then it was assumed to be D X i s 0.1 = X i . In the region of the Coulomb barrier one may expect a fast evolution of the parameters of nucleus–nucleus potential. The following simple parametric forms of the
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
275
Fig. 6. Angular distributions of the 9 BeŽ12 C,8 Be.13 C reaction at the energy E lab Ž 8 Be. s 20 MeV w3x for the transitions to the ground states of the 13 C and 8 Be nuclei Župper panel. and to the 3.09 MeV Ž1r2q . and 3.68 MeV Ž3r2y . q 3.85 MeV Ž5r2q . excited states of the 13 C nucleus Žmiddle and lower panels, respectively.. The dashed curves ²n:, ² a : and solid curve S show the angular distributions corresponding to the n, a transfers and coherent sum of these processes calculated within the CRC model, respectively. The dashed curves S 3.68 , S 3.85 and solid curve S 3.68q3.85 show the angular distributions calculated within the CRC model for the transitions to the 3.68 MeV, 3.85 MeV excited states of the 13 C nucleus and incoherent sum of these distributions, respectively. The meaning of the dotted curve S 0 is the same as in Fig. 1 but at Elab Ž12 C. s 20 MeV.
Woods–Saxon type were chosen to parameterize the observed energy dependence of the OM parameters w1x:
WS Ž E .
° ¶ 1 ~1 y • ¢ 1 q exp E y E ß,
s WSmax
ž
ri Ž E .
s rimin q
W
DW
rimax y rimin 1 q exp
ž
E y Er i
Dr i
,
/
Ž 3.
/
Ž 4.
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
276
a max y a min i i
a i Ž E . s a min q i
1 q exp
ž
E y Eai
D ai
.
Ž 5.
/
Here E s Ecm , i s V,W, ri s r V ,r W and ai s aV ,aW . The parameters X imax s WSmax ,rimax ,amax and X imin s rimin ,a min correspond to the values of the upper and lower i i limits for the variation of OM potential parameter X i s WS , r V , r W , aV , aW . Taking into account the dispersion relation, connecting V and W, one gets for V Ž E . w9x V Ž E . s V0 Ž E . q D V Ž E . ,
Ž 6.
where
DV Ž E . s
P
p
`
H0
WS Ž E X . EX y E
dEX ,
Ž 7.
P denoting the principal value of the integral.
Fig. 7. Energy dependence of Woods–Saxon potential parameters V Ž Ecm . and WS Ž Ecm . for the 8 Be q interaction. The markers are described in the text.
13
C
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
Fig. 8. Energy dependence of Woods–Saxon OM potential parameters for the 8 Be q Be q 12 C Ždashed curves w1x. interactions.
277
13
C Žsolid curves. and
9
To calculate D V Ž E ., WS Ž E . was approximated by two straight lines, WSmax for E ) Eb ,
° ¢
W Ž E . s~ W S
max S
Ž E y Ea .
Ž 8.
for‘E ( Eb .
Eb y E a
This is illustrated in Fig. 7 Žlower panel. by the dashed lines. In the interval Ž Ea , Eb . the straight line is the tangent to the WS Ž E . at the point E s EW s Ž Ea q Eb .r2 For such an approximation w9x WSmax E y Ea E y Ea E y Eb E y Eb DV Ž E . s ln y ln . Ž 9. p DW DW DW DW In practice, V0 Ž E . is often assumed to be energy-independent w9x. In our case we could not describe the energy dependence of V at low energies using this approximation. A better description was obtained using the energy-dependent form w1x
½
V0 Ž E .
° ¢
5
1
~1y
s V0max
1 q exp
ž
E y EV
DV
/
¶ • ß.
Ž 10 .
A.T. Rudchik et al.r Nuclear Physics A 660 (1999) 267–279
278
Table 3 Energy dependence of the 8 Be q
13
C OM potential parameters
Xi
V0 wMeVx
WS sWCRC wMeVx
r V wfmx
r W wfmx
aV wfmx
aW wfmx
X imin X imax
0.00 199.38 3.76 5.48
0.00 7.00 11.39 2.56
0.793 1.300 6.850 2.650
1.250 1.620 7.480 3.020
0.37 0.76 9.57 0.78
0.37 0.76 9.57 0.78
E X i wMeVx D X i wMeVx
The values of the energy dependence parameters X imin , X imax , EX i and D X i obtained from the fitting procedure are listed in Table 3. The corresponding curves of the energy dependence of V Ž E .,WS Ž E .,r V Ž E .,r W Ž E .,aV Ž E .,aW Ž E . are plotted in Figs. 7 and 8. For comparison, in Fig. 8 the energy dependence of the OM potential parameters for the 8 Be q 13 C and 9 Be q 12 C w1x interactions is shown by the solid and dashed curves, respectively. It can be seen that the differences between the real parts V Ž E . of the OM potentials are rather small at the energies Ecm - 50 MeV. The curve V Ž E . for the 8 Be q 13 C interaction shows only a smaller threshold anomaly in comparison with its counterpart for the 9 Be q 12 C interaction. This is due to the fact that WS Ž E . s WCRC Ž E . for the 8 Be q 13 C OM potential is smaller than for the OM potential of 9 Be q 12 C interaction Žthe second panel of Fig. 8.. Large differences are observed between values of the parameters r V Ž E ., r W Ž E . for the 8 Be q 13 C and 9 Be q 12 C interactions Žthe third panel of Fig. 8. at energies Ecm - 10 MeV. However, the parameters for both systems take similar values at larger energies. Very large differences are also observed for the energy dependence of aV Ž E ., aW Ž E . for the 8 Be q 13 C and 9 Be q 12 C systems at the energies Ecm - 15 MeV Žlower panel of Fig. 8.. For the 9 Be q 12 C interaction the functions aV Ž E . and aW Ž E . are close to constant whereas for the 8 Be q 13 C interaction these functions show a rapid variation at the energy Ecm f 9 MeV Žcorresponding to the Coulomb barrier energy.. These differences can be explained by the difference in the structure and the breakup threshold for the 8 Be and 9 Be nuclei.
5. Summary and conclusions The angular distributions of the 9 BeŽ 12 C,13 C. 8 Be reaction were measured at the energy E lab Ž 12 C. s 65 MeV for the transitions to the ground states of the 8 Be and 13 C nuclei and to the 2.94 MeV Ž2q. Žstate of 8 Be. q 3.09 MeV Ž1r2q. Žstate of 13 C. and 3.68 MeV Ž3r2y. q 3.85 MeV Ž5r2q. excited states of 13 C nucleus in the angular range ucm f 248 - 908. These data together with the angular distribution of the 9 BeŽ 12 C,13 C. 8 Be reaction at the energies Elab Ž 12 C. s 12, 15 MeV w2x and the 12 CŽ 9 Be,8 Be.13 C reaction at E lab Ž 9 Be. s 20 MeV w3x for the transitions to the ground states of the 8 Be and 13 C nuclei and to the 3.09 MeV, 3.85 MeV excited states of the 13 C nucleus were analyzed within the coupled reaction channels ŽCRC. model w4x using the energy-dependent OM potential found recently in Ref. w1x for the 9 Be q 12 C channel and spectroscopic amplitudes calculated within the translation-invariant shell model w5x. The one- and two-step transfer reactions as well as elastic and inelastic scattering for the transitions to
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the low energy excited states of 9 Be were included in the coupled channel scheme. It was found that the n-transfer dominates in the 9 BeŽ 12 C,13 C. 8 Be and 12 CŽ 9 Be,8 Be.13 C reactions at all energies studied. This simplified the fit of the OM potential to the 8 Be q 13 C channel experimental data. A good description of the experimental data is achieved at all energies. The energy dependence of the OM potential for the 8 Be q 13 C interaction was analyzed using V0 Ž E .,WS Ž E .,r V Ž E .,r W Ž E ., aV Ž E ., aW Ž E . forms of the Woods–Saxon type. The dispersion relation between the real and imaginary parts of the OM potential w9x was taken into account to deduce the energy dependence of V Ž E .. A regular behaviour of the energy dependence of the OM potential parameters for the 8 Be q 13 C interaction was found. A rapid variation of the aV Ž E . and aW Ž E . parameters was found at the energy close to the Coulomb barrier Ž Ecm f 9 MeV.. Energy dependences of the OM potentials for the 8 Be q 13 C and 9 Be q 12 C interactions were compared. Large differences were found for WS Ž E . in the whole energy range and for r V Ž E .,r W Ž E ., aV Ž E ., aW Ž E . at the energies Ecm - 15 MeV. These differences can be explained by the difference in the structure and breakup threshold for the 8 Be and 9 Be nuclei. In summary, we would like to note that a systematic study of the OM potentials of the nucleus–nucleus interaction of unstable and radioactive nuclei is very important to obtain new information about the structure of nuclei and properties of their interaction.
Acknowledgements This work was supported in part by the Polish State Committee for Scientific Research.
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