Quasi-molecular interpretation of
+ nucleus resonances
()
U. ABBONDANNO(1), N. CINDRO(2)and P. M. MILAZZO(1) (
1
) Dipartimento di Fisica dell’Universit`a di Trieste
INFN, Sezione di Trieste, Italy
( 2
) Rudjer Boˇskovi´c Institute - Zagreb, Croatia
(ricevuto il 18 Luglio 1997; approvato il 15 Ottobre 1997)
Summary. — Models describing the resonant behaviour of heavy-ion reactions are
briefly reviewed. Agreement between experimental data and calculated quantities has been surveyed. The application of these models to the scattering of -particles from
ligth nuclei is discussed. PACS 25.70.Eg – Resonances.
PACS 01.30.Cc – Conference proceedings.
1. – Introduction
The advent of the last generation of large electrostatic accelerators, furnishing a large choice of high-energy heavy-ion beams, has shown that resonances in heavy-ion reactions (RHIR), firstly observed by the group of D. A. Bromley in the far 1961 as correlated structures in various exit channels of12C+12C [1], and considered in the 60s as isolated phenomena, present in only a few nuclear systems, are a general feature of heavy-ion in-teractions. Thus, models capable of describing their physical nature became an important tool in understanding heavy-ion interactions. Among these models, the most widely used are the Orbiting-Cluster Model (OCM), the Group Theoretical Model (GTM) and the An-harmonic Vibrator-Rotator Model (AVRM). Examples of the application of these models to the study of data concerning heavy-ion interactions are reported in ref. [2].
2. – Model description of the resonant behaviour
2.1. The Orbiting-Cluster Model. – The basic ideas underlying the OCM [3] are derived directly from the observation of the main features of data on RHIR: i) all resonance data lie in a narrow zone spread around a straight median line in theE
xvs.
J(J+1)plane,
whereE
xis the excitation energy of the composite system; ii) for each resonant system,
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and
Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
956 U. ABBONDANNO, N. CINDROandP. M. MILAZZO the energy of the observed or projectedJ=0 resonance is close toEB+EC, whereEB
is the binding energy of the projectile plus target in the composite system andEC the
Coulomb barrier.
Resonances are thus seen in a simple and natural way as orbiting-cluster phenomena. This picture yields anE
xvs.
J(J+1)dependence of the form
E x = E0 + h 2 2J J(J+1); (1) where J =0:0104 2 5 (A 5=3 1 +A 5=3 2 )+[A1A2=(A1+A2)](A 1=3 1 +A 1=3 2 )2 r20 10 ,42 MeVs 2 (2)
is the moment of inertia calculated for two osculating nuclei with mass numbersA1and A2, respectively. Also,
E0 = EB + EC;
(3)
where, as mentioned, EB is the binding energy of the colliding nuclei in the composite
system and EC =1:21Z1Z2[r0(A 1=3 1 +A 1=3 2 )+0:5] ,1 MeV (4)
is the Coulomb repulsion energy of the two nuclei. The parameterr0is usually taken as
1.3 fm.
The most impressive result of the application of the OCM to nuclear systems ranging from12C+12Cto58Ni+62Niis that the observed slope ofE
xvs.
J(J+1)yields effective
moments of inertiaJ in good agreement with those calculated from the assumption of two
interacting nuclei as rigid spheres rotating around their centre of mass.
2.2. The Group Theoretical Model. – The Group Theoretical Model (GTM) repairs an intrinsic limitation of the Orbiting-Cluster Model, the latter being able to predict only one resonant level for each spin valueJ. For nuclear systems such as12C+12C,12C+16O, etc.
there is, however, clear experimental evidence for the fragmentation of resonances of a given spinJ into several components with the same spin. One could, thus, say that the
model gives only the mean value of the resonance energy, with the fragments spread in an energy range of several MeV. The first attempt to correctly describe this situation was made by Iachello [4]. The basic idea of his model is that if molecular states are formed in a nuclear collision, it is likely that their structure is more similar to that of diatomic molecules than to that of low-lying nuclear states. Consequently, since it is well known that rotation-vibration spectra of diatomic molecules are dominated by dipole degrees of freedom, quasi-molecular states in heavy-ion collisions may be expected to be also domi-nated by dipole, rather than quadrupole degrees of freedom.
A quantitative study of nuclear quasi-molecular spectra should then be performed in analogy to that of molecular spectra. The energy levels of diatomic molecules are char-acterized by two quantum numbers, the vibrational quantum numbern0and the angular
momentum numberJ. Starting from a purely empirical description of the energy levels
given by a Dunham-like expression [5], Iachello proposed an algebraic approach which is, on one side, relatively simple and, on the other side, very detailed.
Fig. 1. – OCM analysis of the + 12 C, + 16 O, + 20 Ne, + 22 Ne and + 28
Si data. The full circles represent the mean value of the energy of the resonances of a given spin J, the full lines the OCM calculation, the dashed lines the values extracted keeeping both E0and
J as free parameters, the
dotted-dashed lines values extracted keeping onlyJ as free parameter.
The final expression for the energy spectrum can be written as
Ecm =,D+A(n0+ 1 2 ),B(n0+ 1 2 ) 2 +CJ(J+1); (5)
whereD,A,BandCare fitting parameters.
TABLEI. – Results of OCM calculations;J andECare in units of10 ,42
MeVs 2
and MeV, respec-tively. System E C K= h 2 2J J E C ( a ) K( a ) J( a ) K( b ) J( b ) + 12 C 2.62 0.166 1.30 7.94 0.110 1.97 0.900 0.24 + 16 O 3.32 0.125 1.73 5.96 0.122 1.78 0.400 0.54 + 20 Ne 3.97 0.099 2.19 3.79 0.082 2.64 0.057 3.80 + 22 Ne 3.90 0.089 2.43 5.31 0.049 4.42 0.219 0.99 + 28 Si 5.20 0.068 3.19 5.97 0.097 2.23 0.121 1.79 (a) Fit withE Cand Kas free parameters. (b)
958 U. ABBONDANNO, N. CINDROandP. M. MILAZZO
Fig. 2. – GTM fits to the + 12 C, + 16 O, + 20 Ne, + 22 Ne and + 28
Sidata. The full circles
represent the experimental data, the lines are the calculated curves.
2.3. The Anharmonic Vibrator-Rotator Model. – The group-theoretical approach of Iachello is a strong argument in favour of the dipole nature of quasi-molecular reso-nances. A few years later, however, Cindro and Greiner [6] showed that an expression for the resonance energy spectrum equivalent to that given by eq. (5) could be devel-oped from a model that assumes a quadrupole nature of the vibrations characterizing the quasi-molecular configurations, provided an anharmonic term is added to the potential. The combined Hamiltonian yields the energy spectrum
Ecm = 3 2 G+E (n0+ 1 2 )+6G(n0+ 1 2 )2+ 1 2 J(J+1): (6)
Expression (6) gives the energy spectrum of the asymmetric rotator-vibrator in a form essentially identical to that of expression (5), although the basic nature of the vibration is assumed to be quadrupole and not dipole. The correspondence between the parameters of expressions (5) and (6) is straightforwardly given by
C= 1 ; A=E ; B =,6G; D=, 3 G: (7)
Fig. 3. – AVRM fits to the + 12 C, + 16 O, + 20 Ne, + 22 Ne and + 28
Si data. The full circles represent the experimental data, the lines are the calculated curves.
3. – Unified picture of and heavy-ion resonances
The success in describing heavy-ion resonances stimulated interest in the application of the above models to nuclear systems at the borderline of the family of the quasi-molecular states, such as the +nucleus resonant systems. We present here a
quasi-molecular interpretation of resonances in these systems. The experimental data concern-ing +12C, +16O, +20Ne, +22Ne and +28Si, were taken from the literature [7].
The first step of the analysis consisted in applying OCM techniques to the data. Refer-ring to formula (1), the data were fitted and the values of the moments of inertia, both the calculated ones and those extracted from the slopes ofE
xvs.
J(J+1), were compared.
Two kinds of fits were performed, the first keepingE0andJ as free parameters, the
sec-ond keeping free onlyJ. The results are reported in fig. 1 and in table I. The agreement
between extracted and calculated values is not so good as in the case of heavy-ion–induced resonances. A simple explanation could be that for -induced resonances the vibrational
character, which is responsible for the splitting of a resonance with a well-defined spin
J into several components, is more pronounced than for heavy-ion reactions; that fact
renders less applicable the OCM approximation.
four-960 U. ABBONDANNO, N. CINDROandP. M. MILAZZO
TABLEII. – AVRM and GTM fits, all quantities in MeV.
System E G A ,B C ,D + 12 C 2.43 ,0:022 1.828 0.075 0.133 ,1:238 + 16 O 2.99 ,0:040 1.781 0.065 0.121 ,1:297 + 20 Ne 2.08 ,0:029 0.768 ,0:012 0.094 ,2:002 + 22 Ne 2.15 ,0:028 0.713 ,0:020 0.082 ,2:244 + 28 Si 3.66 ,0:066 0.530 0.003 0.101 ,4:160
parameter fit onA;B;CandD) were then applied to the -induced resonance data.
Fig-ures 2 and 3 and table II show that the agreement between the experimental and the cal-culated quantities is satisfactory. Perhaps even more important is the fact that excellent agreement with the data was obtained in both cases (with a minor exception of the AVRM fit for the +28Si system): the use of four free parameters instead of two did not
substan-tially improve the quality of the fit. The values of the parametersE and
Gobtained from
the fit of resonances (AVRM fit) were compared with the values of theA;B;CandD
coefficients obtained in the GTM approach. The consistency of the two sets of parameters for the AVRM and the GTM fits is remarkable, and supports the basic correctness of a unified quasi-molecular picture of both and heavy-ion resonances.
We emphasize two additional points in favour of such a picture. Firstly, we stress the fact that the four-parameter search in the GTM fit yields values of the moment of inertia parameters C very close to the calculated values of12employed in the AVRM fit.
Secondly, the values obtained for the harmonic oscillator quantumE
= h!
(AVRM)
and those obtained for the parameterA(GTM) come out very close. The same is true
for the anharmonicity parameter 6G and the corresponding parameter ,B. The only
difference appears in the constant terms32Gand,D: this, however, causes just a shift in
the absolute values of the resonance energies.
REFERENCES
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[3] CINDRON. and POˇCANIC´D., J. Phys. G, 6 (1980) 359; POˇCANIC´D. and CINDRON., Nucl. Phys. A, 433 (1985) 531.
[4] IACHELLOF., Phys. Rev. C, 23 (1981) 2778. [5] DUNHAMJ. L., Phys. Rev., 41 (1932) 721.
[6] CINDRON. and GREINERW., J. Phys. G, 9 (1983) L175.
[7] AJZENBERG-SELOVEF., Nucl. Phys. A, 460 (1986) 1; 475 (1987) 1; ENDTP. M., Nucl. Phys. A, 521 (1990) 1; ABEGGR. and DAVISC. A., Phys. Rev. C, 43 (1991) 2523; DAVISC. A. and ABEGGR., Nucl. Phys. A, 571 (1994) 265; BRENNERM., private communication.
