Nuclear substate populations near the coulomb barrier in 12C(16O, α)24Mg(21+)

Nuclear Physics A322 (1979) 159-167; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEAR SUBSTATE POPULATIONS NEAR THE COULOMB BARRIER IN 12C(160, ~)24Mg(2+) H. T. KING

Department of Physics, Rutgers University, New Brunswick, NJ 08903 t and

Department of Physics, Stanford University, Stanford, CA 94305 t and M. HASS tt, C. GLASHAUSSER, M. SOSNOWSKI and A. B. ROBBINS

Department of Physics, Rutoers University, New Brunswick, NJ 08903 Received 22 January 1979 Abstract: Nuclear substate parameters have been determined for the 2+ state of 24Mg following the ~2C(160, ~) reaction in the vicinity of the Coulomb barrier. Hauser-Feshbach fits describe the data rather poorly and suggest the need for including exit-channel partial-wave interference, particularly around the "quasimoleeular" resonant anomaly at 9.3 MeV c.m. Fits employing a Breit-Wigner resonance amplitude plus an incoherent compound nucleus background are most consistent with the resonance being 3- or 5-.

E

NUCLEAR REACTIONS 12C(160, ct), (160, ~t),), E = 19.8-23.8 MeV; measured tr(E, 0), ct).(0) coin; 2sSi resonance deduced Jn.

1. Introduction Considerable effort has been focussed on the nature of nonstatistical resonant anamolies in certain heavy-ion scattering cross sections, particularly in the 12C + 12C and 12C+ 160 systems. Generally the resonance angular momentum and parity (for spin-0+spin-0 systems like x2C+ x60) have been deduced from the angular distributions of spin-0 reaction products feeding spin-0 final states; if a resonance of angular momentum J dominates the cross section for such a two-particle final channel the angular distribution will have the form P j(0) 2. The problem of determining J-values in this way becomes much more difficult if the resonance effect is not considerably larger than the background. The present work was undertaken to investigate another technique which might be applicable in cases where the resonant anomaly shows up only weakly in the spin-0+spin-0 final channels. It t Supported in part by the National Science Foundation. tt Present address: Weizmann Institute, Rehovot, Israel. 159

160

H . T . K I N G et al.

involves determining, via particle-~ correlations, the substate populations of a residual nuclear state for which the resonant anomaly is strongly seen. These measurements can also shed light on the reaction mechanisms contributing to the scattering. The data to be described here were taken at energies near the Ec.m. = 9.3 MeV resonance in the 1 2 C - - 1 - 1 6 0 system 1,4). The presence of this resonance in the 12C + 160 fusion cross section 4) suggests the interpretation of an entrance-channel phenomenon such as a quasimolecular resonance. The resonance shows up rather weakly in the ~o channel, and the on-resonance % angular distribution shows no clear P2 signature 1). However, the resonance is seen particularly strongly in the ~t-decay to the 24Mg(2~) state, thus making feasible a fairly detailed ~-~ correlation study. The hope in this work was that the ~-V correlation would prove sensitive to the J-value of the resonance. 2. Angular correlation formalism

The spin-0 + spin-0 ~ spin:0 + spin-2 nature of the 12C(160, ~1) reaction makes an interpretation of the or1-v correlation in terms of substate populations particularly simple. Choosing the z-axis perpendicular to the reaction plane, we find for the in-plane ~-), correlation 5) W(t~),) ~-

¼{[P2,

2 "~- P - 2,

- 2] -

21P2,

-

21cos 4(~b~,- ~/)}.

(1)

The normalization of W is such that a perfectly isotropic correlation would have W = 1 at all angles. In eq. (1) PM~, is the density matrix (a function of energy and of $~) for the 2~ state, normalized such that PMM = 1,

(2)

M

and ~/= ¼arg (P2, - 2).

(3)

The population probability PM for substate M is the diagonal element PMM" If angular momentum and parity are conserved in the reaction, the M = 1 substates cannot be populated. The phase r/ is only defined modulo 90 ° and can therefore be chosen to lie between 0 ° and 90 °. 3. Experimental considerations

Alpha particles for the coincidence data were detected in two 100 pan or 300 pm surface-barrier detectors in a 13 cm scattering chamber. The detectors were collimated to 1.7° half-angle, and had to be significantly underbiased at back angles to prevent

NUCLEAR SUBSTATE POPULATIONS

161

high-energy protons due to the 12C(160, p) reaction from obscuring the 0t-group of interest. The 1.37 MeV de-excitation v-rays from the 24Mg(2~) state were detected in four 12.7 cm × 12.7 cm NaI(T1) detectors placed 17 cm from the target and shielded by 2 mm of Pb to reduce the low-energy count rate. Standard fast-slow electronics were used to record coincidences between either of the two a-detectors and any of the four v-detectors. Accidental coincidences were recorded simultaneously. Three v-ray angle settings provided 12-point angular correlations for each ~t-particle angle. The absolute v-ray efficiency and finite geometry corrections were obtained from an angular correlation taken with an a-detector at 0° to the beam. For this case the angular correlation is known to be W(¢v) -- 1 +#P2(¢~)--~P,,(¢~).

(4)

The measured correlation at 0° is Wme,~(¢r) = e{ 1 +-~Q2P2(¢r)- ~Q,*P4(¢r)}"

(5)

A Legendre fit then yields the efficiency e and the attenuation coefficients Q2 and Q4. The 160 beam was stopped on a 15 mg/cm 2 Au foil placed behind the 12C target, which was 75-100/~g/cm 2 thick and corresponded to a c.m. energy loss o f 3 ~ keV for the beam. The target thickness was thus 3-4 times wider than the compound nucleus (CN) fluctuation width of ~ 100 keV [ref. 6)]. The intention was to average out fine structure fluctuations (which have been extensively studied in ref. 6)) without washing out any broader underlying structure. Prior to the angular correlation measurements, a-particle cross sections were measured (with 100 keV c.m. targets) from Ec.m. ,,~ 8.5-10 MeV, in 10° intervals between 10° and 160° (lab). Excitation functions at three angles are given in fig. 1. The 9.3 MeV anomaly shows up strongly for c.m. angles ~< 15° and ~> 145°. For the middle angular range, the anomaly is lost in the background. Also seen is the weaker 8.8 MeV anomaly reported in ref. 1). The shapes of the angular distributions obtained are in agreement with ref. 1), but in common with ref. 6) we find the cross sections quoted in ref. 1) to be about 7 times too large. Alpha-gamma coincidence data were restricted to ¢~(lab)> 120°, as these are the angles where the resonant anomaly shows up most clearly and also where the cross sections are large enough to obtain reasonable count rates. The angular correlation data were fit to the function

w(¢~) = A - 8 f ( ¢ ~ - ~),

(6)

where A, B and ~/are related to the density matrix as in eqs. (1) and (3), and f(¢) is the function obtained by applying the finite geometry corrections to cos (4¢). Typical results are given in fig. 2. Corrections were made for the change of effective

162

H. T, K I N G et al. I

I

I

I

I

I

Ce 8.5

9.0

i 0.0

9.5

E c.~( MeV ) Fig. 1. Excitation functions at three angles for the lzC(160, c()24Mg(2~) reaction. The dotted line indicates the position of the 9.3 MeV anomaly. Angles shown are c.m. angles.

I

i

I

I

'

f

"P2,2 ÷P_2_2

~* : 155°LAB OI

,

,

,

,

~7

P2,-2 0.5

io.6~ Mg,

0 O. 5

=• i , 9.77 MeV

0 O. 5

,I

°

:• : : 9.70 MeV

,

i

~

i

i

°

°

I

• • °

9.5 MeV

o

i

I

I

I

.•

i

%

i

,

i

,

i

J

i

i

9.52

0 O. 5

: : : 9.32 MeV

°|

•

=,,,,=•,

_l

i

I

I

i

I

-

i

•

0

~

I

~

9.05 MeV

0 0.5

: : : : 8.84 aev

0 0.5

: : : : 8 . 5 3 MeV

,=

:

:

:

:

8.5 MeV c.m. I - 120*

I -60"

t O*

I CoO*

I 120*

'#7 (relotive to 24Mg recoil direction )

Fig. 2. Typical in-plane ~t-7 correlations. A perfectly isotropic correlation would have W -= 1. Curves are fits as described in text.

r

,

,,

° •

0

t

. ° ,

•, •, i

° ° ,

i

0.5

,

I

i

o°°o°,O°

• ° **,"

9.0 MeV

,

MeV

0.5

° , i

I

i

oo °° 0

o

i

130 ° 150" i70"

0 130" 150" 170" ~(=

,

.t½i

,

t

i

130" 150" 170°

(c.m.)

Fig. 3. Summary of ~t-~ correlation data. Except where shown specifically, statistical errors are comparable to the size of the points.

NUCLEAR SUBSTATE POPULATIONS

163

?-ray angle and solid angle due to the recoil velocity (v/c ~, 0.045) of the excited 24Mg nuclei. Fig. 3 summarizes the angular correlation data. Changes with energy in the substate parameters are apparent, though no drastic changes are seen to distinguish the resonance energy, 9.3 MeV. 4. Analysis and results Several fitting procedures were attempted to explain the observed substate parameters. First, to try to characterize the trend of the data with energy, simple HauserFeshbach (HF) calculations were carried out using a smooth-cutoff model for entrance- and exit-channel transmission coefficients. The transmission coefficients were calculated from

Tr = {1 +exp [(L-Lo)/AL]}-',

(7)

where L is the orbital angular momentum and L 0 the"grazing" partial wave computed from simple geometry considerations, The interaction radius for 12C+ a60 was chosen to be 8.5 fm, for 24Mg+at, 7.3 fm. AL reflects the width of the cutoff and was chosen to be 0.7 in both entrance and exit channels. The transmission coefficients so obtained agreed quite well with those from optical-model calculations. The theoretical HF density matrix elements for the 2*Mg(2~) state were obtained from

trpMM, ~ exp (-- 4i~b,,)~ ~

Z~(~b.)g~,(~b,,)*,

(8)

ll' /_~c~c

with X~('" = ~

~

{Ylm(½~'0'Yr's-M(½1t' 0' (m/ M-l' m

_2M/~.exp (imc~)}.

(9,

Here ~ is the differential cross section, I is the entrance and 1' the exit-channel partial wave. No attempt was made to obtain an absolute normalization to the theoretical cross sections. The denominator in eq. (8) was estimated using the prescription of ref. 7):

~. T...~ (2J+ 1) exp [ - J ( J + 1)/2fl2].

(10)

¢

The parameter fl used was 2.5 and the largest value of the entrance-channel ! included in the calculations was 9. Direct comparison of the HF calculations with the data of fig. 3 is not really valid even if the anomalies at 8.8 and 9.3 MeV are statistical in nature, as the assumptions

164

H.T.

KING

et al.

underlying the HF model require energy averaging over many CN fluctuations. Consequently, energy-averaged data were generated from the data of fig. 3 for comparison with the H F predictions. This comparison is given in fig. 4. The agreement is seen to be quite poor; in particular, the H F predictions considerably underestimate the magnitude of P2, - 2- The problems are almost certainly not due to shortcomings in the CN parameters, as these were varied considerably with no real improvement in the agreement. There are, however, several possible explanations for the discrepancies: (a) The non-statistical anomalies clearly present at 9.3 MeV, and to a lesser extent at 8.8 MeV, have been treated simply as statistical fluctuations. Such resonances could give rise to interferences in the exit-channel partial waves that would strongly increase IP2, - 21. (b) No direct reaction contribution has been included (although this is almost certainly very small 6, a)). (c) The energy region being treated is a questionable one in which to attempt H F calculations at all, since the large CN fluctuation width requires a correspondingly large energy-averaging interval, and yet (since the energies involved are near the Coulomb barrier) the CN parameters themselves change dramatically over the same interval. The considerable change in the CN parameters is evident from the differences between the 9.0 MeV and 9.8 MeV predictions in fig. 4. AVERAGE 8 . 4 - 9 . 3 MeV ~ ,

90 °

9.4 - I0. I MeV

lx';

i

I

i

i

I

I

HF 9 . 8 MeV

'

I

i

d 1.0

g 0

140"

160"

180" 140"

160 °

180"

~,, (crn)

Fig. 4. Energy-averageddata and Hauser-Feshbach predictions. Cross section data (not shown) were featureless and fairly well accounted for by the HF predictions. Attempts to fit on-resonance data for the strong 9.3 MeV anomaly also indicate that a HF-like approach (in which all partial-wave interference is ignored) is unsatisfactory. First, there is the fundamental problem that the energy spread due to target thickness (~< 4 CN coherence widths) is not broad enough to justify statistical

NUCLEAR SUBSTATE POPULATIONS (b)

(o)

9O ~

165

,

i

i

i

,

~', 0 / ##.Z. 4

,. 0,2 ,?.

o

I

i

I

i

'

t

I

I

,

1

I

t

',

:

~ - -

+ 0

~

,

I

5 =

t /

--

:

~.5>.-xPP~

0.2 /

120°

I

I

I

t

i

140" 160" 180" 120" 140" 160° 180° ~a (cm)

Fig. 5. (a) D a t a on 9.3 MeV resonance with H F predictions, assuming an enhancement in the J = 3 or J = 5 entrance-channel transmission coefficient. The size of the enhancement was determined from the size o f the resonant effect in the cross section. (b) Data on 9.3 MeV resonance with fits assuming a coherent Breit-Wigner resonance plus an incoherent C N background (see text).

assumptions. Additionally, the entrance-channel transmission coefficients for J < 5 are already nearly 1.0 even in the absence of a resonance; it is therefore difficult to justify the very large enhancements in the transmission coefficients needed to give rise to a significant resonance. The H F predictions of fig. 5(a) (in which one of the entrance-channel transmission coefficients has been strongly enhanced to represent the "doorway" effect of a resonance of particular J) are indeed seen to be poor, bearing out the need for including partial-wave interferences. On-resonance exit-channel interference is expected to arise from a resonant process which feeds different exit-channel partial waves coherently. As an attempt to account for this interference we assume a Breit-Wigner resonant amplitude: Sz,, = exp [i(6, +

Ea + ½iF]-1

(11)

Here Ft (ft) is the ~2C+ 160 entrance-channel partial width (phase); F t, (fit') the or+ Z4Mg exit-channel partial width (phase). For a particular resonant /-value, the entrance channel parameters factor out and the amplitude is simply proportional to F~ exp (/fit')- In the absence of other information, we take the phase fit' to be the Coulomb phase cot,, but allow the signs of the three possible F~ to be either positive or negative to encompass a wide range of possible decay amplitudes. With these approximations the resonant density matrix elements are given by a r. PMMI~ ~exp ros l't"

" -- cot")]rt'rt"XM(~.)XM,(~) , , t'l l"t ['(cot'

,,

(12)

H . T . K I N G et al.

166

where l = J. We estimate the background under the resonance by assuming it to be incoherent CN scattering, described by eq. (8). Then PMMI = [ 6

res res -1.-0-CN ~CN q/I-o.res ..{_o.CN]. DMMt PMMtJ/L

(13)

The free parameters here are the J of the resonance, the three F~ possible for a given J, and the relative normalization of a TM. Resonance J-values from 2-8 were tried, with the F~ varied to yield the best fits. Only d = 3 and 5 gave reasonable fits, with J = 3 perhaps marginally better. Fits for J = 3 and J = 5 are given in fig. 5(b). For neither case is the phase r/well fit, though it is fit better for J = 3. A detailed Zz search was not made for the decay widths Fr, nor did this seem warranted given the uncertainties in the reaction model employed. Likewise, no detailed investigation was made of the effect of interference between the 9.3 MeV resonance and other possible resonances (such as the one at 8.8 Me¥). While such interference with "distant" resonances can noticeably affect the fits at 9.3 MeV, the data given here are clearly insufficient to determine the many new parameters required. In any ease, a considerable improvement in the fits arising from the inclusion of exit-channel partial-wave interference is apparent, particularly for the parameter [P2.-z[. Several conclusions can be drawn from the fitting attempts described here. (a) There is a significant problem in fitting data such as these around the Coulomb barrier, arising from the necessarily limited energy spread with which the experiments must be done in order to see the resonant anomalies. The attempted HF fits indicate the need for more extensive energy averaging of the data before comparisons become meaningful. However, even if such energy averaging is carried out, at the expense of strongly damping the resonance effect, the problem remains that around the Coulomb barrier the compound nuclear parameters themselves can change considerably over the energy-averaging interval employed. (b) On the 9.3 MeV resonant anomaly, a pure "entrance-channel model", in which the ~ + 24Mg channel sees the resonance only via the statistical compound nucleus, is unsatisfactory (fig. 5a). Taking the point of view that the exit-channel interference is due to a Breit-Wigner resonance having non-zero decay widths for the exit as well as entrance channels leads to much improved fits to the data (fig. 5b). However: (c) While the fits suggest a J = 3 or J = 5 assignment to the resonance, the limited angular range and relatively weak structure of the data, plus the complicated reaction mechanisms contributing to the background scattering, make the spin assignment subject to question. It is not clear that any technique can give unambiguous results in cases like this. It may be possible to draw stronger conclusions about similar data on higher energy resonances where the background scattering processes are perhaps more easily described; however, as the higher-lying resonances are generally ones of large angular momentum, rather detailed angular distribution measurements are required.

NUCLEAR SUBSTATE POPULATIONS

167

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