The Semimicroscopic Algebraic Cluster Model:
II. - Detailed analysis( ) G. L ´EVAI( 1 ), J. CSEH( 1 ), P. VANISACKER( 2 )and W. SCHEID( 3 ) ( 1
) Institute of Nuclear Research of the Hungarian Academy of Sciences - Debrecen, Hungary (
2
) Grand Accelerateur National d’Ions Lourds - Caen, France (
3
) Institut f¨ur Theoretische Physik, Justus-Liebig-Universit¨at - Giessen, Germany
(ricevuto il 30 Luglio 1997; approvato il 15 Ottobre 1997)
Summary. — We present examples for recent applications of the Semimicroscopic
Al-gebraic Cluster Model and also discuss its prospective extensions in various direc-tions. The examples considered here include core+ -particle-type cluster
configura-tions and more complex cluster systems. The former ones are relatively simple in tech-nical terms, so the basics of the model can be demonstrated by them in a transparent way, while the latter ones, though technically more involved, illustrate the ability of the model to handle in a unified way clustering phenomena that are usually treated in terms of essentially different approaches.
PACS 21.60.Fw – Models based on group theory. PACS 21.60.Gx – Cluster models.
PACS 01.30.Cc – Conference proceedings.
1. – Introduction
Previous applications of the Semimicroscopic Algebraic Cluster Model (SACM) [1-3] have shown that this model is able to describe a wide variety of nuclear clustering phe-nomena in a unified way. The examples include “classic” clustering of mainly light nuclei (like core+ -type configurations) [4, 5], nuclear molecular systems observed in heavy-ion
resonances [6, 7], (exotic) cluster radioactivity [8] and fission of heavy nuclei [9]. These examples span wide ranges of mass number, excitation energy, angular momentum and nuclear deformations. The success of the model is mainly due to the group theoretical approach, which allows relatively straightforward formulation of the interplay of the rela-tive motion of the clusters and their internal structure by means of symmetry arguments.
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and
Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
As a result, certain observables (the energy spectrum and the electromagnetic transition probabilities) can be calculated in a relatively simple way.
Here we present selected applications of the SACM. As the first set of examples we review certain core+ systems: this type of clusterization can be formulated in a rather
simple way in the SACM. This is the area where the model is in its most well-developed stage. We have introduced, for example, parameter systematics for the Hamiltonian of these systems, which equipped the model with certain predictive power [10]. We also con-sider more complex cluster systems, where the formalism is less well developed at the moment, nevertheless, a satisfactory description of various cluster configurations can be given in a unified way. This is the case with24
Mg+24
Mg and28
Si+28
Si, which are used to interpret both the ground-state region and highly excited states (seen as nuclear quasi-molecular resonances) of the48
Cr and the56
Ni nuclei. Developing the formalism further by utilizing the concept of multichannel dynamic symmetry [7], different cluster config-urations of the same nucleus can be handled simultaneously. We present here the case of24
Mg as12
C+12
C and20
Ne+ configurations, as well as 32 S as28 Si+ and 16 O+16 O. Combining of theSU(3)dynamic symmetry with the antisymmetry requirement for the
total wavefunction also gives rise to certain predictive power of the model regarding the number and the distribution of states seen in various nuclear reactions.
2. – Core+ -particle systems
The Semimicroscopic Algebraic Cluster Model associates the following group chain and basis states to core+ -particle configurations [1, 2]:
U ST C (4) SU C (3)U R (4) U S C (2)U T C (2)SU C (3)SU R (3) (1) j [f C 1 ;f C 2 ;f C 3 ;f C 4 ]; ( C ; C );[N;0;0;0]; S C ; T C ; (n ;0) U S C (2)U T C (2)SU(3)SU S C (2)SU T C (2)O(3)SU T C (2)SU(2) (;); K ;L; J i:
Note that the spin and isospin quantum numbers of the core nucleusS Cand
T
Cdetermine
these values for the united nucleus as well:S =S C,
T =T C.
In the dynamic symmetry approximation the model Hamiltonian can be constructed from the invariants (Casimir operators) of the groups appearing in eq. (2). If we allow only a single isospin configuration for the cluster system the Hamiltonian does not carry explicitT-dependence, but may include terms that depend onS(when it is non-zero). As
discussed in ref. [10] the Hamiltonian then can be written as
H =+( C , R )C (2) SUC(3) + (2) + R ^ n +( R +)C (2) SU(3) +K 2 + ^ L 2 , ,4 R ^ Q R ^ Q C , 3 2 R ^ L R ^ L C + ^ L ^ S; where we used ^ L = ^ L C + ^ L R, ^ Q = ^ Q C + ^ Q R and C (2) SU(3) = 2 ^ Q ^ Q+ 3 4 ^ L ^ L [1, 2].
This formula allows a straightforward interpretation of the SACM Hamiltonian and its components. Equation (2) includes essentially constants, unless the model space contains
TABLEI. – The quantum numbers associated with the core nuclei and the relative motion of the core and clusters for some nuclei in theA = 16to20mass region. We also displayed theL
C
andJCquantum numbers determined by the(C;C)SUC(3)representation and bySC.LCand JC are not good quantum numbers in theSU(3)dynamical symmetry approximation, but they
identify the actual core states considered implicitly in the model, which include the ground state and some of the lowest-lying excited states.
Core [f C 1 ;f C 2 ;f C 3 ;f C 4 ] TC SC (C;C) LC J C n 12 C [2222] 0 0 (0,4) 0, 2, 4 0 + ,2 + ,4 + 4, 5, ... 14 C [3322] 1 0 (0,2) 0, 2 0 + ,2 + 6, 7, ... 15 N [3332] 1 2 1 2 (0,1) 1 1 2 , , 3 2 , 7, 8, ... 16 O [3333] 0 0 (0,0) 0 0 + 8, 9, ... 13 C [3222] 1 2 1 2 (0,3) 1, 3 1 2 , , 3 2 , 5 2 , , 7 2 , 5, 6, ...
states with different( C
; C
)coreSU(3)quantum numbers, in which case the energy
spectrum is split accordingly. Equation (3) is the most important part containing an har-monic oscillator-like term (
R ^ n
) together with one characteristic of the Elliott model.
This latter one consists of the second-order Casimir operator of theSU(3)group and a
rotational term. Equation (3) also contains the phenomenologicK 2
term. In contrast with eq. (3) the two terms with correlated strength in eq. (3) act on the core and relative motion components of the wave function and represent a coupling between these degrees of freedom, reflecting the cluster nature of the problem. Finally, the remaining term of eq. (3) contains a coupling between the core spinS =S
C(if any) and the orbital angular
momentumL.
The structure of the SACM Hamiltonian written in its new form (2) also facilitates the interpretation of the individual terms and the estimation of the corresponding pa-rameters. This prompts us to choose
R, the strength of the harmonic-oscillator term
compatible with the value appropriate for the nuclei in question. We have used R = 45A ,1=3 ,25A ,2=3
[10]. Although this choice represents a model assumption not directly following from the SACM, it allows systematic determination of the other parameters for a wide range of core+ cluster systems.
In ref. [10] we used the standardized Hamiltonian (2) to reproduce the energy spec-trum of 16
O, 18
O, 19
F and20
Ne in terms of core+ -particle cluster configurations. In
table I we show the quantum numbers associated with the core nuclei and the relative mo-tion of the clusters. We found [10] that the parameters of the Hamiltonian vary smoothly with mass number in the rangeA=16 to 20. This has led us to the idea of determining
the relevant parameters for theA=17 case as well by interpolation of the parameter set
obtained for the neighbouring nuclei withA=16, 18, 19 and 20. This meant that we
con-structed the energy spectrum of the17
O nucleus as a13
C+ cluster confirguration with-out fitting the parameters to the experimental spectrum of this nucleus. As described in
ref. [10] the resulting model spectrum was rather similar to the experimental one both for positive- and negative-parity states, indicating that the Semimicroscopic Algebraic Clus-ter Model is equipped with considerable predicting power.
We have carried out calculations for electromagnetic transition probabilities as well for various core+ -particle systems, such as
14 C+ [4], 34 S+ [5], 15 N+ [11]. The
transi-tion operators usually have two types of terms: one that acts on the relative motransi-tion part of the cluster wavefunction and a second acting on the internal core part, as illustrated
with the electric-quadrupole transition operator T (E2) =q R Q (2) R +q C Q (2) C : (3)
We found that generally the observedB(E2)values are reproduced resonably well, the
situation for electric-dipole transitions is worse, but the agreement between theoretical and experimental transition rates is not worse than in other models, whileB(M1)values
are usually not reproduced very successfully. An especially remarkable finding was that ourB(E2)values in ref. [4] for the
14
C+ system were rather close to those obtained from
a fully microscopic cluster model calculation. The correlation between these two data sets was usually even stronger than that between the experimental values and the results from any of the two calculations. This seems to indicate that the effects of antisymmetrization are fairly well approximated within our semimicroscopic model. We have made the first steps towards a systematic description of electromagnetic transitions in core+ -like
sys-tems [12]. InvestigatingE2transitions within the ground-state bands of about 20 nuclei
in thesd-shell region we found thatq Rand
q
C(in eq. (3)) can be parametrized by a linear
function of the isospin componentT z.
3. – More complex systems: single-cluster configurations
The group structure relevant to cluster configurations consisting of two non-SU(3)
-scalar (i.e. deformed) nuclei is somewhat more complex than that in eq. (2). However, if these areN =Z (or even-even) nuclei, then the role of the groups describing the
spin-isospin structure is only formal and the problem reduces to manipulations with the groups describing the orbital part of the wavefunctions. This requires multiple couplings between theSU
R
(3)and theSU C
i
(3)(i=1;2) representations, giving rise to multiplicities which
are not present in the relatively simple core+ case. Furthermore, if the two clusters
are identical, then the relative motion of the clusters and their internal structure can be coupled in certain combinations only: in practical terms this means that even- and odd-parity relative motion wavefunctions (i.e. even and odd values ofn
) can be combined with
different internalSU C
(3)wavefunctions.
The first such example we considered in the SACM was the24
Mg nucleus as a12
C+12
C configuration [6]. We described about 100 states and 100E2transitions in the
ground-state region together with some 60 quasi-molecular resonance ground-states. This rich spectrum was generated by allowing both12
C clusters to be in the three states of their ground-state bands and considering excitations of the relative motion up to2h! (i.e.n
= 14).
The model was able to reproduce the low-lying band structure of the24
Mg nucleus and the electric-quadrupole transition rates there. We also made estimations for theB(E2)
values within the quasi-molecular resonances [6] and for transitions from these states to the ground-state region [13]. We found that theSU(3) dynamic symmetry assumption
imposes strict selection rules on these transitions, and most of them are rather weak or forbidden. The strongest transitions in the quasi-molecular region are intraband ones within the sameSU(3)multiplets withB(E2) 60W.u. This is weaker than the
pre-diction of other models, but it is still in qualitative agreement with recent experiments [14]. We found a similar transition pattern for the transitions from the quasimolecular resonances to the ground-state band [13]: only selected transitions (i.e. those from the
(;) =(10;4)0and the (9,3)1SU(3)multiplets) were non-negligible withB(E2) 5
TABLEII. – The SACM model space for the28 Si(0;12)+ , 28 Si(12;0)+ and 16 O+ 16 O
configu-rations. TheSU(3)states present in these channels are displayed for the first few shells.
nh ! 28 Si(0;12)+ 28 Si(12;0)+ 16 O+16 O 0 (4,8),(3,7),(2,6),(1,5),(0,4) (4,8) 1 (6,9),(5,8),(4,7),(3,6),(2,5), (9,6),(7,7),(5,8),(3,9) (1,4),(0,3) 2 (8,10),(7,9),(6,8),(5,7),(4,6), (14,4),(12,5),(10,6),(8,7),(6,8) (3,5),(2,4),(1,3),(0,2) (4,9),(2,10) 3 (10,11),(9,10),(8,9),(7,8),(6,7) (19,2),(17,3),(15,4),(13,5),(11,6), (5,6),(4,5),(3,4),(2,3),(1,2),(0,1) (9,7),(7,8),(5,9),(3,10),(1,11) 4 (12,12),(11,11),(10,10),(9,9), (24,0),(22,1),(20,2),(18,3), (24,0) (8,8),(7,7),(6,6),(5,5),(4,4), (16,4),(14,5),(12,6),(10,7),(8,8), (3,3),(2,2),(1,1),(0,0) (6,9),(4,10),(2,11),(0,12)
We applied a similar description to the24
Mg+24
Mg and the 28
Si+28
Si systems re-cently, but since the compound nuclei48
Cr and56
Ni are much less well-known experimen-tally, our results are less conclusive for these systems. The basic band structure could be reproduced in the ground-state region, together with the known quasi-molecular res-onances with angular momenta up to 44h. Here these states were assigned to 8 and 6h!
excitations in the relative motion for48
Cr and56
Ni, respectively.
4. – More complex systems: multiple-cluster configurations
As a further step we considered different cluster configurations of the same nucleus applying the multichannel dynamic symmetry concept. As an illustrative example here we present the case of the32
S nucleus, which we built up as28
Si+ and 16
O+16
O [15]. Contrary to the case of the24
Mg nucleus, the16
O+16
O clusterization becomes relevant only at higher excitations (4h!), which is due to the essentially “structureless” nature
of the16
O clusters. The ground-state region, therefore, is decribed only in terms of the
28
Si+ clusterization. We actually allowed two 28 Si configurations with( C ; C )=(0,12)
and (12,0). The former one contains the ground-state band of28
Si and corresponds to oblate deformation, while the latter one represents an excited band with prolate deforma-tion. We display theSU(3)model space of
32
S in table II. It can be seen that the ground state of32
S, which carries the(;) =(4;8)quantum numbers, can be constructed
us-ing both internal28
Si configurations, but the rest of the states are rather different in the two channels. It is also seen that the lowest allowed band in the16
O+16
O channel with
(;)=(24;0)is also present in the model space built on the prolate 28
Si state, but it is missing from the one built on the oblate structure.
The multichannel dynamic symmetry assumption connects the Hamiltonians of differ-ent cluster systems by requiring that theSU(3)states present in several channels should
appear at the same energy. We found that this multichannel symmetry holds between the
28
Si(12,0)+ and the 16
O+16
O channels, but a reasonable description of the experimental data requires a slight breaking of the multichannel invariance between the28
Si(0,12)+
and the28
Si(12,0)+ configurations [15].
Having fixed the parameters of the Hamiltonian by fitting positive- and negative-parity states in the ground-state region (with0h! and 1h! excitation, respectively) and the
quasimolecular resonances (with4h!) the rest of the spectrum is predicted. We tested
this prediction of the model by comparing the distribution of resonances identified re-cently in elastic -scattering on
28
Si with the spectrum calculated from the model. We found that the number of experimentally identified levels in theJ
=3 , to9 , channels is in rather good agreement with the prediction of the model [15].
We have performed similar analysis of the28
Si nucleus itself by assuming24
Mg+
and12
C+16
O clusterization [7], and extended our study of24
Mg by allowing20
Ne+ and 16
O+2 configurations as well [13]. This latter nucleus is especially interesting, since it
has been the subject of extensive experimental investigations. We plan further study of the24
Mg nucleus based on the concept of multichannel dynamic symmetry, which would allow the description of states seen simultaneously in different reactions.
5. – Summary and outlook
Here we presented selected applications of the Semimicroscopic Algebraic Cluster Model and demonstrated its usefulness in describing cluster systems of various types. We showed that the model is able to account for large amounts of spectroscopic informa-tion in a consistent way and it is also equipped with certain predictive power. We discussed how a standardized SACM Hamiltonian can be introduced in order to establish parameter systematics, which can serve as a guiding line for further studies.
We plan to extend our investigations along several lines: i) We plan to apply various symmetry-breaking terms in the Hamiltonian which might result in more realistic spectra and less restrictive selection rules than those obtained from theSU(3)dynamic
symme-try approximation. These terms can be the Casimir operator of theO R
(4)group, which
mixes states from different shells;O C (3) ^ L 2 C
, which breaks the degeneracy of the states of the core nucleus; and spin-orbit-type interactions like ^
L C
^ L
R. ii) We wish to extend
the formalism to handling the spin-isospin degrees of freedom in a less restricted way. This requires equipping the basis states and the operators withSU
ST C
(4)tensorial
char-acter. By generalizing the model in this way it would be possible to connect states with differentT
Cz, T
C and S
C values, which would mean that the model becomes able to
de-scribe isovector type electromagnetic transitions and -decays, for example. iii) We plan
to search for parameter systematics for electromagnetic transitions as well. We expect that the predictive power of the model could be enhanced in this way too. iv) Another pos-sibility is extending the model to multicluster systems. This also opens the way to discrete symmetries relevant to -particle systems for example [16].
This work was supported by the OTKA Grant No. T14321 and by the MTA-CNRS (Hungarian-French) exchange program.
REFERENCES
[1] CSEHJ., Phys. Lett. B, 281 (1992) 173.
[2] CSEHJ. and L ´EVAIG., Ann. Phys. (N.Y.), 230 (1994) 165.
[3] CSEHJ., L ´EVAIG., ALGORAA., HESSP. O. and KAT¯OK., this issue, p. 921. [4] L ´EVAIG., CSEHJ. and SCHEIDW., Phys. Rev. C, 46 (1992) 548.
[5] F ¨UL¨OPZS., L ´EVAIG., SOMORJAIE., KISSA. Z., C´ SEHJ., TIKKANENP. and KEINONEN J., Nucl. Phys. A, 604 (1996) 286.
[6] CSEHJ., L ´EVAIG. and SCHEIDW., Phys. Rev. C, 48 (1993) 1724. [7] CSEHJ., Phys. Rev. C, 50 (1994) 2240.
[8] CSEHJ., GUPTAR. K. and SCHEIDW., Phys. Lett. B, 299 (1993) 205. [9] ALGORAA. and CSEHJ., J. Phys. G, 22 (1996) L39.
[10] L ´EVAIG. and CSEHJ., Phys. Lett. B, 381 (1996) 1.
[11] L ´EVAIG. and CSEHJ., Phys. Rev. C, 44 (1991) 152; 44 (1991) 166.
[12] L ´EVAI G. and CSEH J., in Proceedings of the 9th International Symposium on Capture Gamma Ray Spectroscopy, Budapest, 1996, in press.
[13] CSEHJ., L ´EVAIG. and KATO¯K., in Proceedings of the 9th International Symposium on Capture Gamma Ray Spectroscopy, Budapest, 1996, in press.
[14] HAASF. et al., this issue, p. 989.
[15] CSEH J., L ´EVAI G., VENTURA A. and ZUFFI L., in Proceedings of the International Conference on Nuclear Data for Science and Technology, Trieste, 1997, in press.
