Z. Phys. A 356, 287—291 (1996)

Low energy scattering of a-particles by 28Si

K.-M. Kallman

Department of Physics, As bo Akademi, FIN-20500 Turku, Finland

Received: 23 June 1996Communicated by B. Herskind

Abstract. Excitation functions for 19 angles from 82° to173° have been measured in the laboratory energy range3.6—5.8 MeV using a thick target method. From the meas-ured excitation functions and from data in the literature,parameters for 18 resonances in 32S have been obtained.The found states have been classified according to dynam-ical symmetries of the U(4) model. The thick-targetmethod has been evaluated and found to be applicablealso to experiments with low bombarding energies.

PACS: 25.55

1 Introduction

During the last year extensive investigations of a-particlescattering by 28Si have been carried out, see [1, 2] andreferences therein. Excitation functions for elastic scatter-ing at laboratory energies from about 6 to 34 MeV havebeen measured with the tandem accelerator at the Sved-berg Laboratory in Uppsala (E

-!""6—12 MeV), with the

cyclotron at the As bo Akademi Accelerator Laboratory(E

-!""12—19 MeV) and with the cyclotron at Oslo Uni-

versity (E-!"

"19—34 MeV). In these experiments about200 states in 32S have been found, having excitation ener-gies from about 12 MeV up to 37 MeV. Those excitationfunctions were measured for 21 angles mostly in the back-ward hemisphere and from these the angular distributionsfor the peaks were obtained and the l-values for a majorityof the resonances could be deduced.

As a complement to these measurements and in orderto better understand the analysis of the resonance struc-tures at higher energies, detailed measurements at lowenergies have been performed. The range starting from theeffective Coulomb barrier (at about 3 MeV) up to 6 MeV,where joining the Uppsala data. This energy region hasalso been covered in [4], but by another method, see thenext section. For these energies the resonances in theexcitation function are mostly well separated and narrow.This means that the influence from adjacent peaks is smalland this fact simplifies the analysis of the resonances.

2 Experimental procedure

The measurements were done by using the ‘‘thick-targetmethod’’ [3] that has been developed in our laboratory.The power of the method is that the spectra are continu-ous in energy and the only limitation is channel disper-sion. All the details of the method can be found in [3]. Themethod proved to be applicable also in this energy region.The experiment were done using the 103 cm asynch-ronous cyclotron at As bo Akademi Accelerator Laborat-ory. The excitation functions were obtained in the energyregion from 3.6 to 5.8 MeV for 19 different angles from173° degree down to 82° (in the laboratory system), usingfour surface barrier detectors. In order to minimise thegeometrical broadening of the energy resolution [3],1 mm wide and 7 mm high rectangular collimators wereused in front of the detectors and the beam was collimatedto give a spot with about 1 mm diameter on the target.The energy step between the different measurement pointswas 100 keV. An even larger step length could have beenused but this small step length provided a consistencycheck. The target was a polished plate of natural siliconabout 10 km thick. Due to the thickness of the target itwas not possible, at these low energies, to accuratelyrecord the beam current in the Faraday cup. The nor-malisation of the different angles, and the matching of thedifferent pieces of the excitation function, was achieved byusing a monitor detector in 170° and then summing overa number of channels from the plateau in the beginning ofthe monitor spectrum (Fig. 1). This sum was then usedinstead of the total beam charge to match the differentpieces together to form the excitation function. Thismethod seemed to give a very good result at these lowenergies.

In this experiment the overall resolution is about15 keV. This is mainly due to the resolution of the de-tector. The beam energy resolution after the 115° analys-ing magnet is about 0.1% of the nominal beam energy, i.e.about 4 keV. This energy resolution is not as good ascompared to experiments performed with high resolutionvan de Graaff accelerators. The present experiment is stillvery useful because a lot of information can be obtained in

Fig. 1. Spectrum obtained from the monitor detector at 170°. Thestructure in the spectrum from channel number 110 to 240 corres-ponds to a laboratory energy of 4.8 to 5.4 MeV. The big peak inchannel number 100 is from Carbon deposited on the surface of thetarget. The layer is very thin but the cross section for a-scattering by12C is for this bombarding energy about 30 times larger than theRutherford cross section

Fig. 2. The excitation function obtained with the thick-targetmethod (full line) compared to the excitation function from [4] (dots)

a relatively short time and for the determination of theparameters of the different resonances, the recording ofthe excitation functions for many angles is vital. The steplength between the measurement points is not fixed as ina normal experiment but it is varying depending on wherein the target the scattering has occurred. The average stepbetween points was in this experiment about 7 keV.

The corresponding excitation function has also beenmeasured by Cheng et al. [4], although at 170° laboratoryangle only, and the measurement was done by coveringthe energy region in steps less than or equal to 25 keV.This experiment was done with a very good energy resolu-tion using a tandem accelerator. Their reported resolutionfor the beam was 1.28 keV. They also used a very thintarget (&10 kg SiO#10 kg C) and their overall experi-mental resolution was not depending on the resolution ofthe detector, as is the case when using the ‘‘thick targetmethod’’. The width of the peak at 5.42 MeV, see Fig. 2, inthe excitation function was about 10 keV in their experi-ment whereas in the present experiment the width of this

peak is about 20 keV. The agreement between the twoapproaches is otherwise good as can be seen.

3 Analysis of excitation functions

The excitation functions were analysed and the para-meters for the different resonances were determined byseparating the scattering amplitude in a nonresonant termand a sum of resonant partial waves [5]. The scatteringamplitude f (h) for a spinless particle can be expanded inpartial waves and the differential cross-section can then bewritten as

dpdX

(h, E )"D f (h) D2,

where

f (h )"fc(h)#

i

2k+l

(2l#1)exp(2ial)(1!S

l)P

l(cos h)

and

Sl"exp(2i/

l) G

CaiC

[exp(2ibl)!1]#1H . (1)

Here f#(h) is the Coulomb amplitude, a

lthe relative

Coulomb phase shift, Slis the l : th scattering matrix ele-

ment, C is the width of the resonance and Cai is the partialwidth or single particle width for a given resonance. Thenonresonant part of the nuclear phase shift is /

land the

resonant part is given by

bl"tan~1 C

C/2

Er!ED , (2)

where E is the energy of the a-particles and Eris the energy

for a given resonance with a partial wave having anangular momentum l. The above equation can now besimplified by combining all the nonresonant terms, contri-buting to the cross-section, with the Coulomb term. Fromthe resonant part of the cross-section one can also take thetwo terms not containing [exp(2ib

l)!1] and include

them in the nonresonant term. One then gets the followingformula for the differential cross-section for scattering ofa spinless particle

dpdX

(h, E )"K o (h)exp(is(h))#i

2k+l/lr

(2l#1)

]CaiC

[exp(2ibl)!1] exp[2i/@(h)]P

l(cos h ) K

2(3)

In the first term o (h) is the nonresonant amplitude and itwas assumed to have a linear energy dependence, s (h) isthe phase of the nonresonant term and /@

l(h) are the

nonresonant relative background phases and these canalso be taken to vary linearly with energy. Thus by puttingexp[is (h)]"1, when calculating the excitation functions,one just needs, besides the amplitude o (h), to determinethe relative phases between the resonant terms and thenonresonant background for each angle.

The excitation functions from [4] was fitted accordingto the method outlined above. The fit was done by visual

288

Fig. 3. The excitation function from [4]. The solidline is calculated using the parameters of Table 1

Table 1. 32S levels obtained from 28Si(a, a)28Si

E-!"

[MeV] Jn C [keV] Ca [keV] Ca/C

3.91 0` 5.8 4.0 0.694.06 0` 1.7 1.0 0.594.14 0` 1.2 0.5 0.424.24 1~ 2.3 1.0 0.434.34 0` 8.9 4.0 0.454.42 3~ 4.7 1.4 0.304.48 2` 7.7 0.5 0.064.58 0` 2.9 0.4 0.144.75 2` 67.4 6.0 0.094.78 0` 1.8 1.5 0.834.92 3~ 1.1 0.1 0.095.10 3~ 1.9 0.3 0.165.35 3~ 5.7 1.3 0.235.42 3~ 1.2 0.6 0.55.60 3~ 10.4 2.0 0.195.72 2` 3.2 1.0 0.315.83 3~ 7.0 2.0 0.296.00 3~ 6.4 2.0 0.31

inspection of the calculated functions. The resonanceparameters obtained are listed in Table 1, see also Fig. 3for the quality of the fit. These parameters can then beused when fitting the excitation functions for the otherangles obtained in the present experiment, and from thesethe unambiguous l-values are obtained, as given inTable 1.

Theoretical excitation functions for a number of angleswere calculated using the parameters in Table 1. The onlyadditional parameter that was adjusted when going fromone angle to another was the background parameter o (h)and the whole energy region from 3 to 6 MeV was cal-culated in one piece. When fitting an experiment witha definite energy resolution one must make a convolutionof the calculated excitation function with the assumedenergy resolution in order to be able to compare theexperimental results with the theoretical calculations. Thepeaks in the calculated curves were broadened usinga gaussian with a FWHM of 15 keV. Some of the cal-culated curves are shown in Fig. 4 together with theexperimental excitation functions. As can be seen, all thestructures in the excitation function at 170° by Cheng et al.[4] can be seen also in these experimental curves. There

Fig. 4. Some of the measured excitation functions plotted togetherwith theoretical curves calculated with the parameters from Table 1.The theoretical curves have been convoluted using a gaussian with15 keV FWHM

are even additional peaks in our data that could not beobserved in their experiment because of the large stepbetween the measured points. The structures in the experi-mental excitation functions are presumably smeared outwhen going to smaller angles due to the kinematicalbroadening of the energy resolution.

4 Comparison with the interacting Boson model

The alpha strength of the found states is rather high,implying that they could be of molecular origin, i.e. an

289

Fig. 5. Comparison between the experimentalenergies and a theoretical spectrum obtainedfrom the O(4) limit

a-particle#a 28Si core. A first attempt to explain thepresent states is to try to see if they can be described withthe help of the interacting boson model [6, 7]. The experi-mental energies can be compared with the spectra corres-ponding to the dynamical symmetries of the U(4) model.In this case the O(4) limit seemed to give a better result.The energies for the O(4) limit are given by

E"au(u#2)#b¸ (¸#1)#e, (4)

where u is related to the vibrational quantum numberl"1

2(N!u). N can here be interpreted as the total

number of bosons and was in this case chosen equal to 16according to [7]. The last term in the formula e is anenergy constant determining the location of the molecularbands. The possible values of the quantum numbers aregiven by the following selection rules:

u"N, N!2, N!4,2, 1 or 0,

¸"u, u!1, u!2,2, 0. (5)

The experimental energies were classified according tothe rules above and the parameters a"!0.0042 MeV,b"0.103 MeV and e"10.867 MeV could be obtained.This simple model gives a rather nice correspondencebetween the experimental energies and the calculatedspectrum as can be seen in Fig. 5.

5 Conclusions

There has not previous been much work done with a-particles on 28Si in this energy region besides the measure-ment in [4]. Willard et al. measured excitation functionsfrom 3.0 to 5.3 MeV for four angles [8] and they reportedresonances at 3.9, 4.32, 4.45 and 4.72 MeV. Lawrie et al.[9] started their measurements from 5 MeV and theyrecorded the angular distributions for angles larger than70 in energy steps of 100 keV. Their energy resolution wasnot good enough to be able to accurately analyse thestructures. The alpha strengths of the present resonances

Fig. 6. Excitation energies as a function of J(J#1) for the statesfound in this work (circles) together with some previously foundstates in 32S[1, 2, 10, 11] (see text for more details)

are rather large. The summed reduced alpha widths of thestates, with angular momentum zero and three, are about80% of the Wigner limit (the values were obtained witha channel radius of 5.4 fm). In Fig. 6 are besides from thepresent states (circles) also included some known candi-dates for quasimolecular states [10] (triangles) togetherwith a microscopic a-cluster prediction (dashed line), andsome states obtained from heavy ion a-transfer reactions[11] (squares). The two dotted lines corresponds to rota-tional bands with u"0 and u"16 in the model aboveand the solid line is the band found in [1, 2]. As can beseen from the figure the present states lies under the statesin [1, 2] but they lie in line with the states in [11], and theycould be interpreted as being members of the same rota-tional bands.

When using the thick-target method, the width of thepeaks in the excitation function is mainly given by theenergy resolution of the detector. By some other means ofdetecting the a-particles, as for example a magnetic spec-trograph, a better estimate of the real width of the struc-tures could be obtained.

290

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