The combinatorial nuclear level densities are used in the orbiting-cluster model calculations. Previous predictions of the model of resonance observability obtained for systems of mass A ≤ 56 with nuclear statistical-model (Bethe formula) level densities are confirmed. The present calculation correctly predicts resonances in barium isotopes, in accordance with recent experimental results, while the statistical model does not.Primijenili smo kombinatornu gustoću nuklearnih stanja u modelu orbitirajućih grozdova. Potvrdili smo ranija predviđanja vjerojatnosti opažanja rezonancija za sustave masa A ≤ 56 koja su bila dobivena rabeći gustoće stanja izračunate nuklearnim statističkim modelom (Betheova formula). Kombinatorni račun ispravno predviđa rezonancije u izotopima barija, u skladu s ishodima nedavnih mjerenja, dok statistički model daje loše rezultate

Basrak, Zoran

Pezer, Robert

Szilner, Suzana

English

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ISSN1330–0016CODEN FIZBE7LETTER TO THE EDITORORBITING-CLUSTER MODEL WITH COMBINATORIAL LEVEL DENSITYZORAN BASRAKa, ROBERT PEZERb and SUZANA SZILNERa,1aRud–er Bošković Institute, Bijenička 54, HR-10002 Zagreb, CroatiabPhysics Department, Faculty of Science, University of Zagreb, Bijenička 32,HR-10000 Zagreb, CroatiaPaper devoted to honour the memory of Professor Nikola CindroReceived 17 June 2003; revised manuscript received 21 July 2003Accepted 28 July 2003 Online 3 March 2004The combinatorial nuclear level densities are used in the orbiting-cluster model cal-culations. Previous predictions of the model of resonance observability obtained forsystems of mass A ≤ 56 with nuclear statistical-model (Bethe formula) level densi-ties are confirmed. The present calculation correctly predicts resonances in bariumisotopes, in accordance with recent experimental results, while the statistical modeldoes not.PACS numbers: 25.70.Ef, 21.10.Ma, 25.70.-z UDC 539.142.3, 539.144Keywords: heavy ions, nuclear molecules, resonances, combinatorial level densityIn the scientific opus of Nikola Cindro, the most cited reference is the theoreticalwork in Ref. [1] describing the conditions a colliding nuclear system has to fulfilto exhibit resonant behaviour. This phenomenological approach to the so-calledmolecular resonances named the orbiting-cluster model (OCM) was introduced inCindro’s earlier works [2]. The aim of the present paper is to check how the use ofmore realistic level densities, a crucial parameter of the model, influences the OCMpredictions.The existence of intermediate-width resonances is a well established feature ofheavy-ion reactions [3]. Owing to the exceptionally small absorption (a property ofseveral nuclei), two nuclei in a low-energy grazing collision stick together and orbit1Present address: Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Legnaro, I-35020 Legnaro (PD), ItalyFIZIKA B 12 (2003) 2, 145–152 145basrak et al.: orbiting-cluster model with combinatorial level densityabout their centre of mass. This simple transient di-nuclear (molecular) configura-tion is a doorway-like state [4] of an intermediate width. A necessary condition thata near-to-grazing entrance-channel partial wave of an angular momentum L shouldexcite a resonance of spin J = L is2 that the above di-nuclear rotator should livelong enough. Colliding nuclei should neither re-separate nor fuse into a compoundnucleus before exhibiting an appreciable portion of a full rotation. A number ofnuclear systems display resonances that for a given system in the E versus J(J +1)plane fall close to a straight median line, as one expects for an energy spectrumof a rigid quantum rotator (see graphical presentations of resonant-system energyspectra, e.g., in Ref. [5])E∗(J) = E0 +h̄22JJ(J + 1), (1)where E∗ is the excitation energy of the resonance J (or, correspondingly, thesystem centre-of-mass energy) and h̄ is the Planck constant divided by 2π. E0 is theband-head corresponding to the touching configuration with no rotation (J = 0).Hence, E0 may be seen as a sum of the binding energy EB of the projectile andthe target into the composite system and the respective Coulomb barrier EC, i.e.E0≈EB+EC. The Coulomb repulsion energy of the two nuclei with atomic numbersZ1 and Z2 and mass numbers A1 and A2, respectively, is calculated according toRef. [6]EC =1.44 Z1Z21.12 (A1/31 + A1/32 ) − 0.94 (A−1/31 + A−1/32 ) + 3.0MeV. (2)The slope of the band is proportional to the moment of inertia J . The experimen-tally deduced effective moment of inertia J eff is in agreement with that calculatedfor the two osculating spherical nuclei A1 and A2 [1]J = 1.036[25(A5/31 + A5/32 ) +A1A2A1 + A2(A1/31 + A1/32 )2]r20 × 10−44 , (3)where J is obtained in units MeV s2 and r0 =1.3 fm.In addition to the representation of orbiting clusters in a molecular-type con-figuration, OCM also relies on the concept of the so-called molecular-resonancewindow [7]. A molecular window is the domain of low (and relatively constant)level density in the (E∗, J) space of the corresponding compound nucleus withinwhich the dissolution of the (simple) molecular configuration into more complex(compound nuclear) states of the system is expected to be strongly reduced. Theprobability of damping of a doorway state into compound-nucleus states is in theformal theory of intermediate structures given by the spreading width [4]Γ↓ = 2π |<CN|V |el> |2 ρ(E∗, J), (4)2Strictly valid only for collisions of spinless nuclei, but they are by far the most frequent casesamong the observed resonating nuclear systems.146 FIZIKA B 12 (2003) 2, 145–152basrak et al.: orbiting-cluster model with combinatorial level densitywhere V is the transition interaction which couples the reaction entrance (elastic)channel |el> and the compound-nucleus states |CN> of the nucleus correspondingto the complete system. ρ(E∗, J) is the density of the final (compound-nucleus)levels. According to this picture, resonances will experimentally show up only whennuclear structure and/or other effects contribute to keep the value of Γ↓ within thelimits of observability (of the order of a few hundred keV), while beyond this limit,the doorway state dumps down its strength into the surrounding continuum andthe intermediate structure is washed out. Obviously, a small compound-nucleuslevel density means a small spreading width which ensures the weak absorptionwithin the molecular window.To estimate Γ↓ as a function of A, E∗ and J in the OCM, it has been as-sumed that a matrix element decreases exponentially with A, without a detaileddependence on nuclear structure|<CN|V |el> |2 = const.× exp(−kA). (5)The matrix element is equal for all nuclear systems of the same mass number Aand varies slowly and smoothly within the molecular window in such a way that itsdependence on E∗ and J may be neglected in the first approximation. Therefore,the reliability with which the ρ(E∗, J) is calculated determines the reliability of themodel predictions. The parameter k in Eq. (5) has been deduced from the fit tosystems with well-established resonances. The constant of proportionality in Eq. (5)has been cancelled by considering the relative damping widths, RΓ = Γ↓/Γ↓C + C. Inthis way, the above rough estimate of the matrix element via Eq. (5) uses the Γ↓C + Cof the 12C+ 12C system as a measure of damping in other systems. Accordingly, thevalue of RΓ classifies the observability of resonances in nuclear systems. A projectile-target combination which, within its molecular window, displays the RΓ value closeto or smaller than unity is likely to exhibit resonating features in the excitationfunctions. The smaller the RΓ, the higher the chance for resonant behaviour.Predictions of resonance observation were given for nuclear systems with A ≤ 58[1] and with A ≤ 72 [8]. These predictions agree remarkably with recent experimen-tal results [5]. Encouraged by the success of the predictions, the OCM calculationwith a small modification of the estimated decrease of the matrix element with A|<CN|V |el> |2 = const.× exp(−kAκ) (6)(κ≤1) was later systematically extended to medium- and high-mass systems withA ≤ 110 [9]. Likelihood for resonant behaviour was evaluated for more than 400target-projectile combinations and several new candidate-systems for resonant be-haviour were identified [9].After the above predictions were published, resonant behaviour was reported foreven heavier systems, 116Ba and 120Ba excited in collisions of 58Ni with 58Ni and62Ni nuclei, respectively [10]. The values of angular momenta estimated from theexperimental angular distributions, measured at energies for which intermediatestructures were identified in the above collisions, were compared with the OCMFIZIKA B 12 (2003) 2, 145–152 147basrak et al.: orbiting-cluster model with combinatorial level densitypredictions. However, the value of the relative damping RΓ, the main OCM criterionof resonance observability, was not reported for these Ba systems.Until now, in all applications of OCM, the standard Bethe level density formula[11] for a given angular momentum J has been usedρ(E∗, J) = 0.0295 (2J + 1) exp[−(J + 1/2)22σ2]exp [2(aE∗)1/2]a1/4 E∗5/4 σ5/2, (7)where a is the level-density parameter and σ the spin cut-off parameter. However,note that the angular momentum distribution underlying this formula is purelya classical result unreliable for the high-spin region. Nuclear structure effects inρ(E∗, J) are accounted for by taking a and the pairing energy correction ∆ accord-ing to the parametrization in Ref. [12]. Values of ratios of the average level densities<ρ> of the reaction systems used in fitting the expression (6) relative to the averagelevel density <ρC + C> of the12C + 12C system plotted against the (shifted) systemmass on κ are shown as open circles in Fig. 1. The systems and the corresponding0102030400 5 10 1512C+12C12C+16O12C+18O14C+16O12C+20Ne16O+16O12C+24Mg12C+28Si16O+24Mg12C+32S,16O+28Si24Mg+24Mg24Mg+28Si28Si+28Si50Cr+70Ge58Ni+62Ni60Ni+60Ni54Fe+64Zn58Ni+60Ni58Ni+58Ni(       A - 24   )   0.62log (<ρ> / <ρC+C >)Fig. 1. Logarithm of the relative level densities calculated by the statistical-modelBethe formula (7) as a function of the (shifted) system mass with the best-fit expo-nent κ=0.62. The heavy full line is a result of the linear best-fit following Eq. (6) ofthe systems with well-established resonant behaviour (open circles). Extrapolationof the best-fit line to heavier systems is represented by the dotted line and theseveral projectile-target combinations leading to the barium isotopes 116,118,120Baare marked with (open and filled) triangles. Filled triangles stand for the experi-mentally studied systems [10].148 FIZIKA B 12 (2003) 2, 145–152basrak et al.: orbiting-cluster model with combinatorial level densitystatistical-model parameters are taken from Ref. [9]. A more careful fitting of Eq.(6) has resulted in a slightly different fit result with k=1.4 and κ=0.62.In the same figure, several projectile-target combinations which lead to thecomposite systems 116,118,120Ba are displayed as open and filled triangles. Thededuced relative damping width RΓ of all studied systems is given in Table 1.Figure 1 and Table 1, show clearly that the criterion of resonance observability(RΓ≈1) is not fulfilled for the systems leading to the Ba isotopes. In contrast to theOCM prediction, the experiment showed the existence of narrow structures in theperipheral collision of nickel isotopes [10]. This discrepancy might be attributed tothe less reliable estimate of level density via the Bethe formula for the Ba isotopes,or to the crudity of the assumption on the matrix element behaviour (6) whoseextension to these rather heavy systems may be unrealistic.TABLE 1. Relative spreading width calculated with statistical-model and combi-natorial level densities, for systems with observed resonant behaviour and systemsleading to barium isotopes.Composite EntranceRstatΓ RcombΓsystem channel24Mg 12C+12C 1.0 1.028Si 12C+16O 0.2 0.230Si 12C+18O 2.4 1.114C+16O 3.7 1.932S 12C+20Ne 0.4 2.416O+16O 0.2 1.036Ar 12C+24Mg 0.4 2.340Ca 12C+28Si 1.8 0.416O+24Mg 1.3 0.344Ti 12C+32S 3.5 0.916O+28Si 3.5 1.048Cr 24Mg+24Mg 6.0 1752Fe 24Mg+28Si 4.6 1.956Ni 28Si+28Si 0.1 0.1116Ba 58Ni+58Ni 800 0.002118Ba 58Ni+60Ni 7800 0.0254Fe+64Zn 2 × 104 0.05120Ba 60Ni+60Ni 2 × 105 0.0558Ni+62Ni 4 × 105 0.150Cr+70Ge 1 × 107 1.7FIZIKA B 12 (2003) 2, 145–152 149basrak et al.: orbiting-cluster model with combinatorial level densityAt higher excitation energies, one expects failure of the Bethe formula (7) whichpredicts an exponential increase of the level density. In addition, this formula can-not reproduce the level density well over a large energy interval assuming a constantlevel density parameter a. In the derivation of level densities in Ref. [12] an ap-proximation is made by replacing the exact single-particle-state density function gby a smooth one. This function is then expanded around the Fermi energy.In order to verify the former hypothesis on the possible cause of the lack of asatisfactory OCM prediction in our calculation, for the same systems and in thesame spirit, a simple approach based on the Bethe formula (7) was replaced byan exact combinatorial approach to the level-density calculation. As a first guess,the combinatorial calculations were performed using the realistic single-particlespectra [13] and a simplified pairing force as a residual interaction of the strengthG. The basic algorithm used was developed in Ref. [14]. In this approach, non-collective excitations are treated by the combinatorial algorithm exploited by thecode SPINDIS [15], while collective excitations are accounted for by the residual-interaction coupling-constant renormalization as described in Ref. [16]. As for thenuclei studied here, when there is no available data on the level densities, we haveused the estimate G = 0.5 MeV that has been proved to account for the miss-ing dynamics in many cases. In other aspects, the calculation is completely ex-act and prompts improvements using more realistic single-particle spectra. Totalparity and angular-momentum dependent level densities are provided in tabularform up to a given excitation energy. The obtained results are shown in Fig. 2and the last column of Table 1. For combinatorial level densities, the matrix ele-01020300 2012C+12C12C+16O14C+16O12C+18O12C+20Ne16O+16O12C+24Mg12C+28Si, 16O+24Mg12C+32S, 16O+28Si24Mg+24Mg24Mg+28Si28Si+28Si50Cr+70Ge58Ni+62Ni60Ni+60Ni54Fe+64Zn58Ni+60Ni58Ni+58Ni(       A - 24   )   0.77log (<ρ> / <ρC+C >)Fig. 2. Same as Fig. 1, but for combinatorial level densities. Parameter κ=0.77.150 FIZIKA B 12 (2003) 2, 145–152basrak et al.: orbiting-cluster model with combinatorial level densityment dependence on A is closer to the original OCM pure exponential behaviour(κ = 0.77), while the parameter k is equal to 1.0. It turns out that the combina-torial level densities do not have an appreciable influence on the OCM estimateof the likelihood of resonant behaviour for systems with unambiguously observedresonances. The only exception is the 24Mg+24Mg system for which the relativedamping RΓ is as high as 17 (see the last column of Table 1). On the other hand,the relative combinatorial level densities of Ba isotopes are drastically lower, soeven the 50Cr+70Ge system with a relatively high binding energy might exhibitresonant behaviour. The predicted likelihood for the observation of resonances incollisions of 58Ni nuclei is extremely high — two orders of magnitude higher thanfor, e.g., collisions of 16O nuclei. The results for the Ba isotopes deduced using thecombinatorial-level-density calculation should, however, be taken with some pre-caution owing to the fact that the OCM estimate for resonant behaviour obtainedfor low- and medium-mass systems was extrapolated to as much as twice highersystem masses.In conclusion, the phenomenological orbiting-cluster model of molecular reso-nances with the combinatorial level densities confirms its predictions of resonanceobservability obtained with the statistical model (Bethe formula) level densitiesand which are in full agreement with experimental observations for A ≤ 56 [5].In contrast to the results obtained using the simple standard Bethe formula, theextrapolated OCM calculation with the combinatorial approach correctly predictsresonances in barium isotopes in accordance with the recent observations [10].AcknowledgementsAs the inspiration of the present paper, the spiritual presence of Nikola Cindrois such that one has a feeling he had participated in it to the extent that oneshould include his name on the author list. His strong personality permanentlyexercises a decisive influence on the present authors. For his contribution as ateacher and researcher, we must also strongly emphasize our feeling of an abidingsense of gratitude for his role in making us accept his internationally recognizedhigh research standards. Thank you Professor Cindro.References[1] N. Cindro and D. Počanić, J. Phys. G: Nucl. Phys. 6 (1980) 356.[2] N. Cindro, J. Phys. G: Nucl. Phys. 4 (1978) L23; D. Počanić and N. Cindro, J. Phys.G: Nucl. Phys. 5 (1979) L25; D. Počanić, Ph. D. Thesis, University of Zagreb, 1981.[3] N. Cindro, Ann. Phys. (France) 13 (1988) 289; R. R. Betts and A. H. Wuosmaa, Rep.Prog. Phys. 60 (1997) 819; R. R. Betts, Proc. of the 7th Int. Conf. on ClusteringAspects of Nuclear Structure and Dynamics, Rab, Croatia, 1999, eds. M. Korolija, Z.Basrak and R. Čaplar, World Scientific (2000) p. 109.[4] H. Feshbach, A. K. Kerman and R. H. Lemmer, Ann. Phys. 41 (1967) 230.[5] U. Abbondanno and N. Cindro, Int. J. Mod. Phys. E 2 (1993) 1.[6] R. Bass, Nuclear Reactions with Heavy Ions, Springer Verlag (1980) p. 403.FIZIKA B 12 (2003) 2, 145–152 151basrak et al.: orbiting-cluster model with combinatorial level density[7] W. Greiner and W. Scheid, J. Physique C 6 (1971) 91; A. Arima, K. W. McVoy andG. Scharff-Goldhaber, Phys. Lett. B 40 (1972) 7.[8] D. Počanić and N. Cindro, Nucl. Phys. A 433 (1985) 531.[9] M. Božin and N. Cindro, Ann. Phys. 192 (1989) 307.[10] L. Vannucci, U. Abbondanno, M. Bettiolo, M. Bruno, N. Cindro, M. D’Agostino, P.M. Milazzo, R. A. Ricci, T. Ritz, W. Scheid and G. Vannini, Z. Phys. A 355 (1996)41.[11] H. Bethe, Rev. Mod. Phys. 9 (1937) 69.[12] A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446.[13] P. A. Seeger and R. C. Perisho, Los Alamos Scientific Report LA-3751 (1957).[14] D. K. Sunko and D. Svrtan, Phys. Rev. C 31 (1985) 1929.[15] D. K. Sunko, Comput. Phys. Commun. 101 (1997) 171.[16] V. Paar, D. K. Sunko, S. Brant, M. G. Mustafa and R. G. Lanier, Z. Phys. A 47 (1993)343.MODEL KRUŽEĆIH GROZDOVA S KOMBINATORNOM GUSTOĆOMSTANJAPrimijenili smo kombinatornu gustoću nuklearnih stanja u modelu orbitirajućihgrozdova. Potvrdili smo ranija predvid–anja vjerojatnosti opažanja rezonancija zasustave masa A ≤ 56 koja su bila dobivena rabeći gustoće stanja izračunate nuk-learnim statističkim modelom (Betheova formula). Kombinatorni račun ispravnopredvid–a rezonancije u izotopima barija, u skladu s ishodima nedavnih mjerenja,dok statistički model daje loše rezultate.152 FIZIKA B 12 (2003) 2, 145–152

Model kružećih grozdova s kombinatornom gustoćom stanja

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