Electric quadrupole moment of the first excited state of 12C

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24 February 1983

ELECTRIC QUADRUPOLE MOMENT OF THE FIRST EXCITED STATE OF 12 C W J , VERMEER, M.T. ESAT, J.A. KUEHNER 1 , R.H. SPEAR Department of Nuclear Physics, Australian National University, Canberra, Australia 2600 and

AaM. BAXTER and S. HINDS Physics Department, Australian National University, Canberra, Australia 2600 Received 8 November 1982

The static electric quadrupole moment Q2 + of the first excited state of 12C has been measured using Coulomb excitation of 12C projectiles by a 2os Pb target. The result is Q2 + = +6 +- 3 e fm2, which indicates a substantial oblate deformation, consistent with most theoretical expectations.

The deformation of the 12 C nucleus is a quantity of considerable importance for the understanding of the structure of light nuclei. For example, if alphacluster models have any validity, then the ground state of 12 C would be expected to be oblate, corresponding to negative values of the quadrupole deformation parameter/~2 and the intrinsic electric quadrupole moment Qo. There have been numerous calculations of the 12 C deformation using a variety of models, and the results have ranged from spherical to strongly oblate [ 1]. However, despite much effort, the quantitative experimental evidence is inconclusive. Most of it derives from model-dependent analyses of electron scattering and hadron-scattering data obtained at energies above the Coulomb barrier. Some of these analyses are inherently insensitive to the sign of the deformation. For those which do claim such sensitivity, results for 132 show a large scatter, ranging from +0.3 to - 1 . 3 7 , and there are indications that the values obtained are projectile dependent [ 1 - 4 ] . Quadrupole-moment measurements via Coulomb excitation would clearly be of great value. We present here the first such measurement of Q2 ÷, the static electric quadrupole moment of the first excited state (jTr = 2 +, Ex = 4.438 MeV), using the ] Permanent address: McMaster University, ltamilton, Canada. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

reorientation effect in projectile excitation of 12 C by 2°8pb at bombarding energies below the Coulomb barrier. The excitation probability P, which is the ratio of inelastic to elastic-plus-inelastic scattering cross sections [5], was measured at a scattering angle 01ab = 90 °. The sign and magnitude of Q2 ÷ were determined from the measured excitation probabilities and the known [6] value ofB(E2; 0 ÷ ~ 2 ÷) (38.8 -+ 2.2 e 2 fm4), using the Winther-de Boer multiple Coulomb excitation code. The measurement is very difficult because the relatively high excitation energy of the 2 + state results in inelastic-scattering cross sections five orders of magnitude smaller than the elastic cross section with consequent problems of count rate and background. Furthermore, even very large intrinsic deformations correspond to small values of [Q2 +1 due to the small nuclear radius and charge. Beams of 12 C from the ANU 14UD Pelletron accelerator were used to bombard targets consisting of 50 ~g/cm 2 2°8pbS on thin carbon backings. The isotopic enrichment of 208pb was 98.7%. Scattered ions were momentum-analysed with a split-pole magnetic spectrometer and detected at the focal plane by a position-sensitive, multi-element proportional counter [7]. Position spectra for 12C5,6+ ions were obtained for bombarding energiesE = 53,54, 56 and 58 MeV, 23

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which correspond to distances o f closest approach of the nuclear surfaces, S(Olab = 90°), of 6.5, 6.2, 5.6 and 5.0 fm, respectively (assuming a radius parameter r 0 = 1.25 fm [5]). The choice of01ab = 90 ° was made to maximize the inelastic scattering cross section for a given value of s,within the constraints that, as 0 decreases, (a) aP/aO increases rapidly, thus requiring an accuracy in the determination of 19 which becomes increasingly difficult to achieve, and (b) the sensitivity to the quadrupole moment, aP/aQ, decreases. The angle was measured using the technique described by Kuehner et al. [8]. Typical spectra are shown in figs. 1 and 2a. The latter spectrum shows structure, which may be attributed to a peak corresponding to projectile excitation of the Jn = 2 +, 4.439 MeV level of 12C, the peak being Doppler-broadened due to the emission in flight of T-rays. There is some overlap with peaks corresponding to 208pb states. This attribution is confirmed by inspection of fig. 2b, which shows a corresponding spectrum obtained with 72 MeV 160 projectiles (s = 6.0 fm). Although the resolution achieved with 160 is, because of target thickness effects, somewhat poorer than for 12C,it is clear that the structure attributed to 12 C excitation in fig. 2a is absent from the 160 spectrum. In order to ascertain contributions from states of 208pb with excitation energies similar to that of the 12C(2 +) state, their Coulomb excitation was investigated using beams of 160 w i t h e = 70, 7 1 , 7 2 MeV 10 7 I0 6 I0 5

ELASTIC :. IZ c + 2°spb 55 MeV 90° (LAB)

2°8pb (3-)

10 4

f

1

z °o 10 3

208pb(2, ) .'.

i~.

10 2

c

2 0

..:.,. :--.-:..,.:,-?..:, >.YT-'.: ". ".

"

/

:',."-"

CHANNEL

Fig. 1. Part of a position spectrum for 53 MeV 12 C bombarding a ~os Pb target, including both elastic- and inelasticscattering peaks. For display purposes, the points represent three-channel sums. 24

24 February 1983 F "(a)

I

l

2o8

! Pb (2+)

T- "

/

5° !mC *t°sPb ! 4o 1~5@MeV Z°sPb(4+)~" 3oL 90°(LAB) /'1 ] '2c(2+)'/ ,~.

1 1 [

r90° (LAB) ] 30 b 2°BPb(4+)~,~'t

) !

5.0

4.5

4.0 Ex(MeV)

3.5

Fig. 2. Spectra o b t a i n e d f o r (a) 12 C a n d ( b ) 16 0 p r o j e c t i l e s

on 2°~Pb for excitation energy E x ~ 4.4 MeV. The solid curves show fits to the data obtained as described in the text. Contributions to the fit for each peak are shown as dashed curves, with the arrows indicating peak centroids expected on the basis of the spectrum energy calibration. (s = 6.6, 6.3, 6.0 fro, respectively). Previous studies [ 9 - 1 2 ] of 208pb states with E x = 4 - 5 MeV using (e, e') and ((~, (~') reactions indicate that the 2 +, 4.086 MeV level will have the largest Coulomb excitation probability in the present experiment, with the 4 +, 4.323 MeV level about a factor of 5 smaller and other states negligible. This is consistent with our 16 0 data. These data also established that target contaminants were negligible;elastic scattering from any contaminant would have been clear of inelastic peaks in either the 12C or the 160 spectra. The 12 C 2 + peak area was extracted using two distinctly different methods: (i) the 160 spectra were used to estimate the combined contribution of 2°8pb states to the region of interest in the 12 C spectra, assuming pure Coulomb excitation in each case. To first order, this method is independent of the number of 2°8pb states present and the respective multipolarities of their excitation. Furthermore, the separate contributions of unresolved peaks do not need to be determined since only the total number of counts in the combined inelastic peaks for each spectrum is required. (ii) Least-squares fits to the 12 C

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spectra were performed using, for the 12 C 2 + peak, a Doppler-broadened line-shape, and for the 208pb 2 +, 4 + peaks, the unbroadened elastic peak line shape. The Doppler-broadened line shape was calculated [13], taking into account the velocity of the ion, the 3'-ray energy and angular distribution, and the line shape for an unbroadened peak. The 3'-ray angular distribution was obtained from the Winther-de Boer Coulomb excitation program; the area extracted for the 12 C 2 + peak was relatively insensitive to the angular-distribution parameters. A very good fit was obtained (fig. 2a). Method (ii) also allowed independent extraction of excitation probabilities for the 208pb 2 + , 4 + states. The B(EX) values deduced from the 12C and 160 data were in good agreement with each other. Together these gave B(E2; 0 ÷ ~ 2 +) --- 0.274 +- 0.015 e 2 b 2 for the 4.086 MeV state andB(E4; 0 + ~ 4÷0 = 0.24 + 0.05 e 2 b 4 for the 4.323 MeV state (details of the analysis will be presented elsewhere). Electron scattering results for these are 0.329 -+ 0.016 e 2 b 2 [12] and 0.23 + 0.02 e 2 b 4 [9], respectively. The two different methods of analysis gave excellent agreement for the area of the 12 C 2 + peak. The mean value was adopted, with the error determined mainly by the statistical uncertainty. Table 1 lists the value ofPex p obtained. To extract a reliable value of Q2 ÷, it is essential that the data used be free from the effects of nuclear interactions. It has been shown [ 14] that it is unsatisfactory to use a simple formula to determine the maximum safe bombarding energy for this pupose, nad that it is highly desirable that the safe bombarding energy be determined experimentally in every case. To this end, values of the ratio Pexp/ PCoul are given in table 1 ;PCoul is the excitation probability for Coulomb excitation calculated as described below. The consistency of the three lowestenergy values shows that for these energies nuclear effects are small compared to experimental uncertainties. The ratio is smaller for the highest-energy point, indicating the possible onset of nuclear interference. Therefore, to be conservative, this point was excluded from the analysis; if it were included it would reduce the value obtained for Q2 ÷ by approximately half the uncertainty arising from other sources. Small corrections were applied [5] for energy loss in the target, electron screening, vacuum polarisation and nuclear polarisation. The effects of higher states

24 February 1983

Table 1 Excitation probabilities and values of the ratio Pexp]PCoul for the 2 ÷, 4.439 MeV level of 1:2C;PCouI was calculated using B(E2; 0 ÷ ~ 2 ÷) = 38.8 e 2 fm4 , Q2 + = +6 e fm2 and k = 1 E (_MeV/ 53 54 56 58

s (fm) 6.5 6.2 5.6 5.0

10SPexp

Pexp/PCoul

4.09 _+0.30 5.39 -+0.36 11.1 _+0.7 16.1 -+ 1.2

1.01 + 0.08 0.96 +-0.06 1.06 _+0.07 0.89 +_0.06

in 12C, the hexadecapole moment of the 2 + state, the semi-classical approximation, and uncertainties in the bombarding energy calibration and scattering-angle determination were negligible. No correction has been applied for relativistic effects because no appropriate theory exists. Excitation via the giant-dipole resonance is significant. The magnitude of this effect is usually expressed in terms of the parameter k (defined as in ref. [15]) which is the ratio of the effect to that predicted by the hydrodynamic model. It is known [16] that the value of k varies greatly among the light nuclei. However, shell-model calculations by Barker have had some success in estimating k. Where k has been measured experimentally, his calculated [15,17] values are either in excellent agreement (for 10B, Kexp __. 1.3 +- 0.3, kcalc = 1.22) or the calculated value is too small by at worst a factor of 2 (for 1 7 0 , k e x p = 5.7 + 0.4, kcalc = 2.8). For 12C he calculates [15] k = 0.77. It therefore seems very likely that k = 0 . 5 - 1 . 5 ; hence the usual assumption [16] k = 1 will be made. Fig. 3 displays the value of Q2 *, obtained from a least-squares fit to the measured excitation probabilities, as a function of the assumed values of B(E2; 0 + ~ 2+), for various values of k. For k = 1 and B(E2; 0 + ~ 2 +) = 38.8 --- 2.2 e 2 fm 4 , a value of Q2 ÷ = +6 + 3 e fm 2 is obtained. The contributions to the error are +1.7 e f m 2 f r o m the uncertainties in Pexp, and -+2.3 e fm 2 from the uncertainty in B(E2; 0 -I- __~ 2 + ). A more accurate determination of B(E2; 0 + ~ 2 +) is clearly desirable. No additional allowance has been included for the uncertainty in the value of k, because the choice of the error in k would be rather arbitrary. Instead, the manner in which the deduced value of Q2 ÷ depends on the value assumed for k is shown in fig. 3. The rotational-model prediction for [Q2 ÷i, derived 25

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distribution for 12C has been discussed in ref. [1]. In summary, it has been found that Q2 +(12C) = +6 -+ 3 e fm 2 . This value agrees with those theoretical predictions which imply a large oblate deformation, but appears to exclude the spherical solutions of some H a r t r e e - F o c k calculationL

12

Q2 ÷ 8 (e.fmZ) 4

The authors acknowledge helpful discussions with F.C. Barker.

0 -4.

55 40 B ( E 2 ; O + ~ 2 ÷) (e2.fm 4)

45

Fig. 3. Deduced values for Q2 + plotted as a function of the assumed value ofB(E2; 0 ~ 2 ), for indicated values ofk. Dashed curves indicate the experimental uncertainty for the case ofk = 1. The arrows on the abscissa indicate the value and associated uncertainty ofB(E2; 0 + ~ 2 ÷) adopted in ref. [61. +

+

.

.

from the experimental value o f B ( E 2 ; 0 ÷ ~ 2+), is 5.6 e fm 2 , which agrees with the present result. Excellent agreement is also obtained with several theoretical calculations: shell-model calculations by Barker [15,18] give Q2 ÷ = +5.4 e fm 2, H a r t r e e - F o c k calculations by Abgrall [19] give Q2 ÷ = +6.2 e fm 2, and a calculation by Bassel et al. [20], which assumes that the dominant configuration outside the p-shell comes from mixing with an idealised isoscalar giantquadrupole resonance, gives Q2 + = +6.0 e fm 2. Most model calculations give a value of Q0 rather than Q2 ÷. If it is assumed that the nuclear charge distribution is spheroidal with K = 0, then the present result corresponds to Q0 = - 2 2 -+ 10 e fm 2. An alphacluster calculation by Kamimura [21 ] gives Q0 = - 2 1 . 6 e fm 2. H a r t r e e - F o c k calculations give values of Q0 ranging from 0 to - 2 4 e fm 2, depending on the detailed interactions assumed; the present experiment would appear to invalidate the spherical solutions (Q0 = 0) obtained by Friar and Negele [22] and by Vautherin [23]. The significance of a spherical charge

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References [1] J.P. Svenne and R.S. Mackintosh, Phys. Rev. C18 (1978) 983. [2] S.M. Smith et al., Nucl. Phys. A207 (1973) 273. [3] W.J. Thompson and J.S. Eck, Phys. Lett. 67B (1977) 151. [4] O. Karban et al., Nucl. Phys. A292 (1977) 1. [5] M.P. Fewell et al., Nucl. Phys. A321 (1979) 457. [6] F. Ajzenberg-Seloveand C.L. Busch, Nucl. Phys. A336 (1980) 1. [7] T.R. Ophel and A. Johnston, Nucl. Instrum. Methods 157 (1978) 461. [8] J.A. Kuehner et al., Nucl. Instrum. Methods 200 (1982) 587. [9] M.B. Lewis, Nucl. Data Sheets 5B (1981) 243. [ I0[ J. Friedrich, N. Voegler and H. Euteneuer, Phys. Lett. 64B (1976) 269. [11] H.P. Morsch et al., Phys. Rev. C22 (1980) 489. [12] J. Heisenberg, Adv. Nucl. Phys. 12 (198,1) 61. [13] J.R. Beene and R.M. DeVries, Phys. Rev. Lett. 37 (1976) 1027. [14] R.H. Spear et al., Phys. Lett. 76B (1978) 559. [15] F.C. Barker, Austral. J. Phys. 35 (1982) 291. [16] J.A. Kuehner et al., Phys. Lett. l15B (1982) 437. [17] F.C. Barker, Austral. J. Phys. 35 (1982) 301. [181 F.C. Barker, private communication. [19] Y. Abgrall, B. Morand and E. Caurier, Nucl. Phys. A192 (1972) 372. [20] R.H. Bassel, B.A. Brown, R. Lindsay and N. Rowley, J. Phys. G8 (1982) 1215. [21] M. Kamimura, Nucl. Phys. A351 (1981) 456. [22] J.L. Friar and J.W. Negele, Nucl. Phys. A240 (1975) 301. [23] D. Vautherin, Phys. Rev. C7 (1973) 296.