Statistical fluctuations and compound elastic scattering for 12C(12C, 12C)12C and 16O(16O, 16O)16O

I 2.D: 2.N

I

Nuclear Physics A202 (1973) 3 0 - - 3 6 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

STATISTICAL FLUCTUATIONS AND COMPOUND ELASTIC SCATTERING

FOR 12C(12C, 1ZC)12C AND 160(160, 160)160 J. P. B O N D O R F

The Niels Bohr Institute, Copenhagen, Denmark Received 7 N o v e m b e r 1972 Abstract: T h e difference between the o b s e r v e d fluctuations in the elastic excitation functions for c a r b o n - c a r b o n a n d o x y g e n - o x y g e n scattering, just above the C o u l o m b barrier, is interpreted in t e r m s o f the c o m p o u n d nucleus model.

1. Introduction The elastic excitation function for the heavy-ion reaction 12C-{-12C shows strong fluctuations just above the Coulomb barrier in the energy range 7-14 MeV (c.m.), while the elastic excitation function for 1 60 + l 6 0 is very smooth in the corresponding range 10-16 MeV (c.m.). This difference led to the suggestion of quasimolecular states in the carbon-carbon system 1). The quasimolecule interpretation was later questioned 2 , 3 ) because of an apparent compound reaction nature of the cross section in the reaction 12C(12C, ~)24Mg. This reaction exhibits Ericson fluctuations, and the magnitude of the cross section is in agreement with a compound reaction mechanism in the above mentioned interval. In this paper we present an Ericsonfluctuation analysis of the elastic cross sections of the two reactions and compare it with the results of a compound reaction calculation t. An analysis similar to the present one has been performed for 160"-1-160 [ref. 5)] in the higher energy region around 18 MeV (c.m.). We emphasize that the analysis is not concerned with the correlated cross-section bumps in low-energy 12C+ 12C scattering around 6 MeV (c.m.) 6), nor with the regular 3 MeV gross structures observed at higher energies 7).

2. Fluctuatiom The statistical fluctuation of an experimental cross section a can be expressed in terms of the correlation coefficient 2, s)

g =
2

(1)

1" This analysis was first presented at the Hercegnovi meeting 1964 b u t never published. It h a s been t a k e n up n o w because o f its relevance to the c o n t i n u e d discussion o n quasimolecular states. F o r a recent a n d m o s t complete review see ref. 4). 30

:t2C(~2C, 12C), 160(160, z60)

31

which should be used for energy regions in which the smoothed cross section cannot be taken constant in this case and, as always in fluctuation analyses, one there104

a

l

,

i

~

~ ' ~

i

i

,

HEAVY

l

i

i

ION E L A S T I C

i

i

i

J

SCATTERING

O9 t,9 O

10 2

rr"

"' 10

-

zLU (o

.---, 12C O N CARBON -o,.avlE} for12C+12C 4,

l

6,

, ' I'0 ' i14 Ba I~2 ' CENTER OF MASS ENERGY (MeV)

i

1~6

Fig. 1. Differential c r o s s s e c t i o n (90 ° (c.m.)) as f u n c t i o n o f energy for 160(160, a60)160. f

i

r

i

r

t

(

i

i

f

12C(12C, 12C)12C a n d i

RELATIVE DIFFERENTIAL CROSS SECTION

2.0

"

A

I

I

8

I

ct)

12C (12C,~2C)~2C, 90* ('CM;

b)

160 (1eO,t60)160, 90* (CM)

°_z__

..... -2 I

I

IO 12 CENTER OF MASS ENERGY

I

I

14 ( MeV )

I

/

16

Fig. 2. Relative differential cross s e c t i o n s as f u n c t i o n o f energy for the s a m e r e a c t i o n s as in fig. 1. For the oxygen-oxygen s y s t e m the d o t t e d lines r e p r e s e n t upper limits for the relative differential cross section.

32

J.P. BONDORF

fore makes a guess of the possible smooth energy variation of (0>. Having made this guess, t~av(E), one can instead replace cr in eq. (1) by the quantity s = tr/~av (E). This gives an experimentally determined function R which in the following is called the experimentally determined correlation coefficient. We have put ~a~(E) equal to the hand-drawn curve shown in fig. 1. Thus we have chosen a O'av(E) which has a very smooth behaviour. This is in accordance with the idea that the cross section contains only simple one-body shape elastic and purely statistical components. The guess is supported by the results from 2, 3) which show only statistical reaction cross sections.

Io EXPERIMENT o THEORY

z 9

:60 + 160

UJ o

-4 10 I

I 12

I

I 16

CENTER OF MASS ENERGY

I 2O (MeV)

Fig. 3. Experimental (l) and calculated (2) values of the correlation coefficientsR for 12C(l2C, 12C)~2C and 160(160, 160)160 differential cross section at 90° (c.m.).

One could instead have used an optical-model cross section for o-a~(E), or a cross section with intermediate structure such as the one calculated in ref. 9), but the knowledge of H I reaction mechanisms is not yet so detailed that such procedures can be considered more significant than our simpler procedure. The function s for 12C + 1~C is shown in fig. 2. F r o m s one now obtains the experimental values of R as function of energy as shown in fig. 3. In order to have a reasonable statistics the energy intervals are divided into only two regions. We think that this is sufficient for showing

12C(12C, t2C), 160(160, ~60)

33

both the order of magnitude and the trend of R. The oxygen-oxygen fluctuation is small. Not knowing its actual value we are, however, able to give experimental upper limits for R in the case t 6 0 + 160, as shown in fig. 3. 3. Calculation of the compound elastic cross section

The average cross section ( a ) can be divided into a shape elastic part aSE and a fluctuation (or average compound elastic) part aCE SO that we have ( a ) = aSE +aCE. In terms of these quantities the correlation coeificient can be expressed in this case of only one basic cross section 2):

(aSE~ 2

R=

(2)

The energy-averaged cross section ( a ) in (2) is now put equal to the empirically determined aav(E) of sect. 2. We have calculated ace from identifying it with the Hauser-Feshbach expression aCE = rt~2 ~ 4 ( 2 / + 1)7"]2 (elastic) ]:so (0~b)2. ,ore. Ere

(3)

c

The factor 4 and J even come because of the identical particles in the entrance channel. The identification (3) is only correct if the real and imaginary parts of the elastic fluctuation amplitude are uncorrelated. We assume this but we emphasize that the assumption is stronger than the corresponding one for reaction amplitudes between different channels. This is because the partial amplitudes for entrance and exit channels are identical for elastic cross sections which may cause a self-correlation effect which enhances the fluctuation cross section and thus violates (3) [see ref. 1o), p. 160]. In the calculation of aCE we have used the same parameters for 12C-~-12C as in ref. 2). For the 160 + 160 channels we have used the optical-model potentials which __U_.l L~

OPTICAL

~U..

MODEL

FOR

160 4- 160

(a 1.0 z 0.8 o ~n 0.6 to cr

t =0 ~ ' ~ - ~ f f .

y2/

0.4 0.2 6

/ /

14

-

8 10 12 14 16 18 CENTER OF MASS ENERGY {MeV)

Fig. 4. Calculated transmission coefficients f o r ~60-F160 scattering ~1). The potential is given in table 1.

34

J.P. BONDORF

are shown in table 1. V a r i o u s o p t i c a l - m o d e l p o t e n t i a l s for h e a v y - i o n scattering are available ~2). Since we, however, only use the optical m o d e l for calculating transmission coefficients, we believe t h a t o u r results are n o t very sensitive to the choice o f a n y p a r t i c u l a r set o f p o t e n t i a l p a r a m e t e r s . The transmission coefficients for ~ 6 0 + 160 are given in fig. 4. T h e level densities t b r the residual nuclei are given in table 2. I n fig. 3 we have p l o t t e d the theoretical value o f R f r o m eq. (2) for b o t h 1 2 C + ~2C a n d 160"~160. It is w o r t h n o t i n g that for trCE ~( ~SE one has R ~ 2acE/tray(E).

(4)

TABLE 1

Optical-model potential for 160+~60 channels Channel radius (fm) 160+160 0~+2aSi p+alp n-~alS

Shape diffuseness (fm)

Strength real (MeV)

imag. (MeV)

Spinorbit pot.

6.33

0.4

--50

--4

5.33 3.93 3.93

0.576 0.65 0.65

--50 --56.08 +0.55E --51 -~0.5 E

--6.6 (vol.) --4.38--0.12E (surf.) --4.05E (surf.)

(vol.) yes yes

TABLE 2

Level-density parameters for 160 4-~60 channels Nucleus

C

cr2

2sSi alp, al S

0.0310 0.0306

8.22 9.26

The formula 13) 1

co(E*, J) ~ C -(U+t)5/~ exp (2~/~-U) p(J),

with U = E * - - P ( Z ) - - P ( N ) has been used. We have used t ~ 2 MeV and a = 3 MeV -1. The pairing corrections P ( Z ) and t ' ( N ) are from ref. ~). The spin distribution is given by p (J) = norm (2./+ 1) exp \ ~ - - a 2

!

where ~ p ( J ) = I.

4. Conclusions

A s seen f r o m fig. 3 there is g o o d a g r e e m e n t between the calculated a n d e x p e r i m e n t a l c o r r e l a t i o n coefficients R. T h e value o f R for ~6 0 + ~6 0 is b o t h e x p e r i m e n t a l l y a n d theoretically a b o u t three orders o f m a g n i t u d e smaller t h a n t h a t o f ~2C+ ~2C. This fact s u p p o r t s strongly the idea t h a t the fluctuations o b s e r v e d in 1 2 C + 12C are o f statistical nature. T h e calculated a b s o l u t e values o f the c o m p o u n d cross sections are

12C(12C, 12C), 160(160, 160)

35

tested in an independent way for 1 2 c + t 2 c since we know from refs. 2,3) that the statistical model gives correct absolute cross sections for the oc-particle exit channels. The basic reason for the big difference between 12C+12C and 1 6 0 + 1 6 0 in the energy regions investigated is thus the Q-value effect, which gives many more open channels for the compound system formed by oxygen-oxygen than by carboncarbon. It is interesting to compare with 12C + 160 which has Q-values in between and is therefore supposed to show correlation coefficients between 12C+12C and 1 6 0 + 160. This is also confirmed by experiment, as it can be seen from ref. 15). We emphasize, however, that one cannot use the Q-value argument correctly without taking the spins into account. This is because the dominant spins in tree are determined by both the number of exit channels for a given compound spin and by the transmission coefficients for the heavy ions in the entrance channel. The compound nuclear level density has been calculated without an YRAST cut-off. Inclusion of this cut-off may reduce the level density 16) and thus cause an increase in the expected fluctuation cross section. The effect is, however, only observable if the level distance Dj becomes so high that the Ericson condition F~ >> D~ is violated, and this is probably not so since there is no cross correlation between the cross sections in different exit channels. In the calculations we have avoided the question of level densites in the highly excited compound system. Therefore we did not calculate the coherence width which is experimentally determined to be ~ 100 keV in both reactions. In total we stress that this paper checks that the data are consistent with the statistical model but does not rule out other interpretations. It is desirable that more investigations similar to the present one are done both for the reactions studied in this paper and for other "light" heavy-ion reactions. The author wants to thank R. G. Stokstad for inspiring and valuable discussions and R. B. Leachman for fruitful collaboration in the initial stages of this work. References 1) J. A. Kuehner, J. D. Prentice and E. Almqvist, Phys. Lett. 4 (1963) 332 2) J. P. Bondorf, in Proc. advanced course on nuclear physics with thermal neutrons, Kjeller Research Establishment, Norway, report K R 64 (1963); J. Borggreen, B. Elbek, R. B. Leach.man, Mat. Fys. Medd. Dan. Vid. Selsk. 34, no. 9 (1965); J. P. Bondorf and R. B. Leachman, Mat. Fys. Medd. Dan. Vid. Selsk. 34, no. 10 (1965) 3) E. W. Vogt, D. M. Pherson, J. A. Kuehner and E. Almqvist, Phys. Rev. 136 (1964) B86, B99 4) R. G. Stokstad, Europhysics study conference on intermediate processes in nuclear reactions, Plitvice, Yugoslavia, 1972 5) R. W. Shaw, Jr., J. C. Norman, R. Vandenbosch and C. J. Bishop, Phys. Rev. 184 (1969) 1040 6) E. Almqvist, D. A. Bromley and J. A. Kuehner, Phys. Rev. Lett. 4 (1960) 515 7) R. H. Siemssen, J. V. Maher, A. Weidinger and D. A. Bromley, Phys. Rev. Lett. 19 (1967) 639; Phys. Rev. 188 (1969) 1665 8) T. Ericson, Ann. of Phys. 23 (1963) 390 9) W. Scheid, W. Greiner and R. Lemmer, Phys. Rev. Lett. 25 (1970) 176 10) J. P. Bondorf, in Proc. ninth summer meeting of nuclear physicists, Hercegnovi, ed. N. Cindro, Inst. Ruder Boskovid, Zagreb, 1964

36

J.P. BONDORF

11) R. B. Leachman, private communication 12) A. Bisson and R. H. Davis, Phys. Rev. Lett. 22 (1969) 542; R. A. Chatwin, J. S. Eck, D. Robson and A. Richter, Phys. Rev. C1 (1970) 795; R. A. Broglia, S. Landownc and A. Winther, Phys. Lett. 40B (1972) 293 13) K. J. le Couteur and D. W. Lang, Nucl. Phys. 13 0959) 42 14) A. G. W. Cameron, Can. J. Phys. 36 (1958) 1040 15) M. L. Halbert, F. E. Durham, C. D. Moak and A. Zucker, Nucl. Phys. 47 (1963) 353 16) R. G. Stokstad, private communication about calculations in preparation by A. Gobbi, D. Shapira, R. Stokstad and R. Wieland (1972)