A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei
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A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei
I: One heavy and one light cluster
Peter O. Hess and Leonardo J. Ch´avez-Nu˜nez Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, C.U., A.P. 70-543, 04510M´exico D.F., Mexico Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universit¨at, Ruth-Moufang-Str. 1, 60438 Frankfurt amMain, GermanyReceived: date / Revised version: date
Abstract.
An extension of the
Semimicroscopic Algebraic Cluster Model (SACM) is proposed, based onthe pseudo- SU (3) model ( g SU (3)). The Hamiltonian and the spectroscopic factor operator of the model arepresented and a procedure of constructing the model space. Because a huge number of SU (3) irreduciblerepresentations (irrep) appear, one has to be careful in designing a practical, consistent path to reduce theHilbert space. The concept of forbiddenness , taking into account excitations of the clusters, is introducedand applied. The applications are to two systems with a low forbiddenness, namely to U → Pb + Ne and Ra → Pb + C, and to U → Xe + Sr, which appears in the fission of U ,which requires a large forbiddenness. Energies, electromagnetic transitions and spectroscopic factors arecalculated. PACS.
The
Semimicroscopic Algebraic Cluster Model (SACM)was proposed in [1,2] for light nuclei (see, for example,[5]), with the intend as an alternative to also describe nu-clear molecules, i.e., an alegraic version of their geometri-cal description [3,4]. A first attempt to extend it to heavynuclei is published in [6], using the g SU (3) (pseudo- SU (3))model [7,8]. While the heavy cluster is treated within the g SU (3) the light cluster is described by the SU (3) standardmodel. In [6,9], the separation of nucleons into the uniqueand normal orbitals in the united nucleus, compared tothe ones in each cluster, is not well defined: Distinction ismade when a cluster is light, which is then treated withinthe standard shell model, or both are heavy. In differentwords: Each cluster and the united nucleus are not treatedin the same mean field.Since then, many applications of the SACM have beenstudied and further attempts to extend it to heavy nu-clei: For example, a construction of the effective SU (3)irreducible representations (irreps) for heavy nuclei [10],using the Nilsson model [28], and a study of preferencesin radioactive decays and/or fission [12,13,14]. In [15], thespectra of α cluster nuclei, of significant interest in astro-physics related to the production of heavy elements, andtheir spectroscopic factors were calculated. In [16,17] thefirst steps in investigating phase transitions were takenand more recently [18] a complete description of phase transitions using the catastrophe theory [19] was pub-lished. In [20] the renormalization of the coherent stateparameters is investigated, used for the geometric map-ping.More recently, the g SU (3) model was applied to the shell like quarteting in heavy nuclei [21], another methodto restrict effectively the shell model space on physicalgrounds. The quarteting model was first proposed in [22,23] and applied in [24] within the SACM for light nuclei.However, the proton and the neutron part are coupled di-rectly to a total SU (3) irreducible representation (irrep)without taking into account the preference for certain cou-plings, leading as a result to too many irreps at low en-ergy. This is also a problem of the model we present in thiscontribution and a path on how to tackle it is proposed.The quarteting model was compared to another proce-dure, called the proxy - SU (3) [25], showing comparable re-sults. All methods have in common to exploit a symmetry,showing up in the single particle spectrum, and using itto effectively cut the model space to a manageable size.Here we will propose an equivalent manner to describeheavy nuclei, using the g SU (3) model and a proposal onhow to restrict the model space to the most importantcontributions.A further prove of the success of the SACM is the intro-duction of the multi-channel symmetry [26], were differentclusterizations were connected via the same Hamiltonian, Peter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei thus, reducing the complexity of the model and deliveringmore insight into the structure of cluster systems.In [10] a quite powerful method was presented on howto treat heavy nuclei, it only delivers the ground state , orthe first super- and hyper-deformed states [27]. Therefore,it is of interest to look for alternative procedures in orderto deal also with excited states. Unfortunately, we can-not list all contributions of the SACM, but the ones citedclearly demonstrate the success of the SACM.The g SU (3) was proposed in [7,8] for its application toheavy nuclei, where it is observed that when the intruderstates, called unique orbitals , with j = η + in each shell ( η is the shell number), are excluded, the remaining orbitals,called normal orbitals , are grouped, just by counting, intoshells of what is denominated as the g SU (3) symmetry.The normal orbital in the η -shell are renamed (see nextsection) such that they correspond to a pseudo-shell of˜ η = η −
1. Within the Nilsson model asymptotic states alsoshow a degeneration in orbitals denoted by the asymptoticquantum numbers of the Nilsson model Ω [ ηη z Λ ] [28,29].That the nuclear force exhibits such an unexpectedsymmetry is today well understood, parting from micro-scopic field theoretic models of the nuclear interaction [30]and mapping to the effective nuclear interaction. The com-plete Hilbert space of the shell model is a direct productof a state described within a g SU (3), in the same manneras the SU (3) model of Elliott for light nuclei, and a statedescribing the nucleons in the intruder orbitals. The nucle-ons in the intruder orbitals are assumed to play only therole of an observer: Nucleons in the unique orbitals play apassive role in the dynamics due to their opposite parityand their coupling to spin-zero pairs, contributing only tothe binding energy. The pairing energy of nucleons in theunique orbitals is big, due to the large j [28], comparedto the nucleons in the normal orbitals. Thus, nucleons inthe unique orbitals only contribute at high energies, e.g.,in the back-bending effect and related phenomena [31].Inter-shell excitations are considered only within the nor-mal states, for the same reasons. The contribution of thenucleons in the unique orbitals are treated via well de-fined effective charges for electromagnetic transitions [32].The effective charges do not depend on parameters, butonly on the total number of nucleons A , ˜ A and protons Z ,˜ Z , where the symbols with the tilde refer to numbers inthe normal orbitals. In conclusion, the restriction to the g SU (3) is well justified for states at low energy. Though,the g SU (3) has its limits, as not including independent dy-namical effects of nucleons in the unique orbitals, it is stilluseful for investigating an extension of the SACM to heavynuclei.Already within the SACM for light nuclei, new prob-lems arise: When the lightest cluster is too large and as-suming both clusters in their ground state, the correspond-ing SU (3) irrep of the united nucleus cannot be reached.The main reason is that for increasing number of nucleonsof the light cluster, the number of quanta in the relativemotion becomes too large and the coupling of the rela-tive motion irrep to the cluster irreps leads to large final SU (3) irreps, not coinciding wityh the ground state irrep of the united nucleus. This problem was recognized in [33]and the authors introduced the concept of forbiddenness .The main idea is as follows: When the two clusters arein their ground state and all missing quanta are put intothe relative motion (the Wildermuth condition [34]) andthe ground state of the united nucleus can be reached,everything is fine. No excitations are then allowed withina cluster, because these are included in the excitation ofthe relative motion. However, when the ground state can-not be reached, one or the two clusters are allowed tobe excited, subtracting quanta from the relative motion,up to the point when for the first time the ground stateis reached. The excited state of the cluster ( C ∗ k ) is thenfixed and the rest of the procedure is identical as in theSACM for light nuclei. The minimal number required forthe excitation of the clusters is called the forbiddenness .In [33] the importance of this concept and its conse-quences was proven and applied quite successfully in theprediction of preferences of fission fragments and fusionof light and heavy systems. The larger the forbiddenness ,the more suppressed is the reaction channel. To summa-rize: For low lying states and a large light cluster, thecluster system has to be excited! Reconsidering the defi-nition of forbiddenness from an alternative angle, we wereable to obtain a simple formula on the minimal numberof excitation quanta [35] needed, which will be resumedfurther below.Both models, the g SU (3) and the SACM extended toheavy nuclei, will be explained in section 2. The formalismincludes a restricted model Hamiltonian, the calculationof spectra, electromagnetic transition rates and the deter-mination of spectroscopic factors. In particular, in Section3 we will demonstrate the importance of the concept of the forbiddenness . In Section 3 the model is applied to severalheavy cluster systems. Two systems only require a lownumber of forbiddenness, while the third system exhibitsa large forbiddenness and demonstrates its importance. Insection 4 conclusions are drawn. The approach to g SU (3) is as follows: In each harmonicoscillator shell η the orbital belonging to the largest spin j = η + is removed from consideration as an activeorbital, it is considered rather as a spectator. To the re-maining orbitals the redefinition j = l ± → ˜ l = l ∓ , η → ˜ η = η − , (1)is applied, where ˜ l denotes the pseudo-orbital angular mo-mentum and ˜ η the pseudo-shell number. With this redef-inition alone it is easily verified that each shell e η has thesame content as the corresponding shell in the standard SU (3) model.For large deformations, the Nilsson states for axialsymmetric nuclei are classified by their asymptotic quan-tum numbers [28] eter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei 3 Ω [ ηη z Λ ] , (2)where η z is the oscillation number in the z -direction, Λ isthe projection of the orbital angular momentum onto thesame axis and Ω = Λ ± .Excluding the intruder levels, the reassignment of theorbitals is Ω = Λ ± → e Λ = Λ ∓ . (3)Inspecting the Nilsson diagrams [28], those orbitals with the same quantum numbers he η e η z e Λ i are degenerate, whichimplies a very small pseudo-spin-orbit interaction and asa consequence an approximate symmetry . In addition, thecontent of the e η shell corresponds to the same one in thestandard shell model. Thus, the shell model for light nucleican be directly extended to heavy nuclei, using the g SU (3)model instead of the SU (3) model.The basis in the Hilbert space is a direct product ofthe g SU (3) states and the ones describing the unique (in-truder) orbitals. As mentioned in the introduction, theircontribution to the nuclear dynamics is taken into accountby a well defined scaling factor (effective charge).For example, the quadrupole effective charge is givenby [32]: e eff ( E
2) = (cid:18) Z e A (cid:19) (cid:18) A e A (cid:19) , (4)where A and e A are the total number of nucleons and thenumber of nucleons in the normal orbitals, respectively. Z is the total number of protons.Restricting the dynamics only to the nucleons in thenormal orbitals was justified in the introduction, recogniz-ing that at large excitation energies, where backbendingeffects play a role, the model has to be modified by addingthe contributions of the dynamics in the unique orbitals.The effectiveness of g SU (3) was demonstrated in [32], forthe case of the pseudo-symplectic model of the nucleus [36]and many other applications of the g SU (3) model (see, forexample, [24]). In the SACM for light nuclei , the SU (3) irreps are de-termined using the following path: Each cluster is repre-sented by an irrep ( λ k , µ k ) ( k = 1 , in their ground state .Adding the number of oscillation quanta contained in eachcluster and comparing them with the number of oscilla-tion quanta of the united nucleus results in a mismatch:The number of oscillation quanta of the united nucleusis larger than the sum of both clusters. Wildermuth [34]showed that the necessary condition to satisfy the Pauli exclusion principle is to add the missing quanta into therelative motion, introducing a minimal number of relativeoscillation quanta n . This is known as the Wildermuthcondition . However, there are still irreps which are notallowed by the Pauli-exclusion principle.An elegant solution to it, avoiding cumbersome explicit antisymmetrization of the wave function, was proposed in[1,2], the original publication of the SACM: The couplingof the cluster irreps with the one of the relative motiongenerates a list of SU (3) irreps, i.e.,( λ , µ ) ⊗ ( λ , µ ) ⊗ ( n π ,
0) = X m λ,µ m λ,µ ( λ, µ ) , (5)where n π is the number of relative oscillation quanta ( π gives reference to the relative π -bosons of spin 1), limitedfrom below by n , and m λ,µ is the multiplicity of ( λ, µ ).This list of irreps is compared to the one of the shell model.Only those, which have a counterpart in the shell model,are included in the SACM model space. In this manner,the Pauli exclusion principle is observed and the modelspace can be called microscopic.In such a manner, the basis states are described by theket state | ( λ , µ )( λ , µ ); ρ C ( λ C , µ C )( n π , ρ ( λ, µ ) κLM i , (6)where ρ C , ρ and κ are multiplicity labels. The cluster ir-reps are coupled first to ( λ C , µ C ), then with ( n π
0) to thefinal SU (3) irrep ( λ, µ ). The advantage of using the ket-formalism is the absence to the need of a coordinate spacedescription, it suffices to get the quantum numbers de-scribing Pauli allowed cluster states. Of course, the disad-vantage is that no explicit space distribution of the clus-ters are depicted.This representation of the model space is of advantage,because it involves only SU (3) groups, which refer to theshell model space of the clusters and the complete nucleus.The word Semi in the name of SACM appears due tothe phenomenological character of the Hamiltonian, whichis a sum of terms associated to the single particle energy,quadrupole-quadrupole interactions, angular momentumoperators, etc. Note, that this is an additional ingredientand the construction of the cluster space is independentof it, which can be used in any microscopic model.Finally, we expose the path on how to deduce the g SU (3) shell model space: Each shell η has ( η + 1)( η + 2)orbital degrees of freedom. Taking into account the twospin degrees of freedom, the group-chain classifying thestates within the shell η is given by U (( η + 1)( η + 2)) ⊃ U ( ( η + 1)( η + 2)) ⊗ U S (2) (cid:2) N (cid:3) f [ h ] [ h ] U ( 12 ( η + 1)( η + 2)) ⊃ SU (3)[ h ] ( λ, µ ) , (7) Peter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei where [ h ] is a short hand notation for [ h , h ] and thetilde denotes the conjugate Young diagram where rowsand columns are interchanged. (7) gives the relation of thespin-part (denoted by the index S ) and the orbital part,such that the complete state is anti-symmetric ( (cid:2) N (cid:3) , with N as the number nucleons in the shell η ). In the case oflight nuclei, instead of the spin group SU S (2) the spin-isospin group SU ST (4) appears. In contrast, in heavy nu-clei the protons and neutron have to be considered withindifferent shells, which is the reason why (7) is used. Forthe reduction to SU (3) programs are available [37].The reduction scheme (7) is applied to each shell, whichcontains nucleons. Each ∆n π ~ ω excitations ( ∆n π = 0 , , ... )corresponds to a particular distribution of the nucleonswithin the shells. The SU (3) content of each shell is mul-tiplied with all others, resulting in a preliminary list. Fi-nally, the center of mass motion is removed from the ∆n π ~ ω excitation, by multiplying the result for 0 ~ ω with ( ∆n π , ~ ω result by ( ∆n π − , When dealing with heavy nuclei one encounters an ex-ploding number of states ( SU (3) irreps). Thus, one notonly has to find a consistent path on how to combine thecluster irreps with the relative motion to the final irreps,but also a way to restrict further the irreps according totheir importance to contribute at low energy. This has tobe done twice, namely for protons and neutrons, and com-bining them leads to even more irreps. Thus, the methodto restrict the Hilbert space is very essential.For the extension of the SACM to heavy nuclei requiresexplanations of some facts, concepts and assumptions: – All nucleons move in the same mean field of the parentnucleus. This is obvious, because even when clustersare defined as consisting of a subset of A k ( k = 1 , All nucleons are part of thesame shell model , characterized by the same oscillatorenergy ~ ω of the united nucleus. As a consequence,a light cluster, cannot be the same within the parentnucleus, as it is as a free nucleus. For example, for afree and independent cluster the ~ ω value is alreadydifferent. The identification of the clusters is only viatheir content in their number of protons and neutrons.In addition, in g SU (3) each cluster has nucleons in thenormal and unique orbitals and only the ones in thenormal orbitals are counted. Thus, a cluster changesto a pseudo-cluster with e A k nucleons and e Z k protonsin the normal orbitals ( k = 1 , – Because protons and neutrons are in different shells,the construction of the model space is applied sepa-rately to them, as done in any shell model applica-tion for heavy nuclei [28]. The proton and neutron partmove in the same mean field, i.e., with the same ~ ω . – The nucleons are filled into the Nilsson orbitals at thedeformation of the united nucleus . The deformationvalues for a nucleus is retrieved from the tables [38].This does not suggest that we are working in differentshell models, it is just a fact that protons and neutronsare filling different kind of orbitals, which is obvious byinspecting the Nilsson diagrams for protons and neu-trons [28]. For both, however, the deformation value isthe same . – The nucleons of the heavy cluster are filled in first andthen the ones of the light clusters. This involves theassumption that the light cluster is small, comparedto the heavier one, and is preformed within the unitednucleus, a phenomenological picture used in radioac-tive decays.This step will provide us with a heavy and light cluster,for each one with a certain number of nucleons in thenormal orbitals, denoted by e A k ( k = 1 , – In a final step, the proton and neutron part are com-bined, proposing a particular selection, whose originis the demand that the proton and neutron fluid arealigned. In [39] this is discussed and in more detail in[40]: In the nuclear shell model the aligned irrep, i.e.( λ p , µ p ) ⊗ ( λ n , µ n ) → ( λ p + λ n , µ p + µ n ) ( p for pro-tons and n for neutrons), corresponds to aligned prin-ciple axes of the two (proton-neutron) rotors. All otherirreps are non-aligned proton-neutron rotors, whichlie at higher energy, corresponding to scissors mode,or isovector resonances [41]. Thus, the restriction toaligned proton-neutron irreps is a pretty good approx-imation, taking effectively into account the interaction,which tends to align the proton and neutron fluid. The construction of the e SU (3) model space for heavy nu-clei is in line with the one for light nuclei. First somenotational definitions: An α = p, n denotes the type ofthe nucleons (protons or neutrons), ( e λ kα , e µ kα ) ( k = 1 , k and for each type of nucleon.The ( e λ αC , e µ α C ) is called the cluster irrep for the α -typenucleons and it is the result of the coupling( e λ α , e µ α ) ⊗ ( e λ α , e µ α ) = X e λ αC , e µ αC m e λ αC , e µ αC ( e λ αC , e µ α C ) , (8)being m e λ αC , e µ αC the multiplicity of ( e λ αC , e µ α C ).Afterward, each ( e λ αC , e µ α C ) is coupled with the relativemotion irrep ( e n απ , e λ αC , e µ α C ) ⊗ ( e n απ ,
0) = X e λ α , e µ α m e λ α , e µ α ( e λ α , e µ α ) . (9)The minimal number of e n απ is determined in the samemanner as in the SACM for light nuclei: It is the difference eter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei 5 of the number of π -oscillation quanta in the united nucleus(only counting those in the normal orbitals) and the sumof oscillation quanta of the two clusters, for protons andneutrons separately.This leads to a large list of irreps ( e λ α , e µ α ) for each sec-tor of nucleons. This list is compared to the one obtainedby the shell model. A program, which does this automat-ically, can be send on request.For a two cluster system, the path explained is resumedin the following group chains: g SU α (3) ⊗ g SU α (3) ⊗ g SU Rα (3) ⊃ ( λ α , µ α ) ( λ α , µ α ) ( e n απ , g SU αC (3) ⊗ g SU Rα (3) ⊃ g SU (3)( λ αC , µ αC ) ( e n απ ,
0) ( e λ, e µ ) , (10)where R refers to the relative motion part. Below eachgroup the corresponding quantum numbers are listed. The g SU (3) can be further reduced to the angular momentumgroup.Up to here, the procedure is in accordance to the SACMfor light nuclei [1], with the difference that due to thebreaking of isospin symmetry the protons and neutronsare treated separately. In order to obtain the final list ofPauli allowed total irreps, the one of protons is multipliedwith the one of neutrons, i.e.,( e λ p , e µ p ) ⊗ ( e λ n , e µ n ) = X e λ, e µ m e λ, e µ ( e λ, e µ ) , (11)which is represented by the following group chain g SU p (3) ⊗ g SU n (3) ⊃ g SU (3) . (12)To reduce further the large list, obtained in (11), we selectonly the irreps corresponding to a linear coupling, namely( λ p , µ p ) ⊗ ( λ n , µ n ) → ( λ p + λ n , µ p + µ n ) . (13)The justification is the same as mentioned further above,i.e., all other irreps correspond to scissors modes at largehigh energy [39]. As for large clusters in light nuclei, for heavy clusters inheavy nuclei an additional problem arises:Usually, the clusters are in their ground state and allshell excitations are dealt via the radial excitation. Thisis because an excitation of the clusters is equal to theexcitation in the radial motion.However, as mentioned above, for the light cluster be-ing sufficiently large the clusters have to be excited, inorder to connect to the ground state irrep of the parent nucleus. The required number of excitation quanta aresubtracted from the relative motion. The cluster systemhas to be excited by a minimal number of shell excitation n C = n pC + n nC , such that for the first time the groundstate can be reached in the coupling of the cluster irrepsand the relative motion, containing now less number ofquanta. Further excitations have to be added to the rela-tive motion with the same argument as given in the lastparagraph. In [33] this concept was denoted forbiddenen-ness with important consequences for the explanation ofthe preferences of fission. This concept is barely knownbut applied with success within the SACM [12,13,14]. Inthe examples presented in section 3 we will demonstratethe importance of the forbiddenness .An easy to apply formula to determine the forbidden-ness n αC is given in [35] (again for protons and neutronsseparately, with α = p or n , and all variables are denotedby a tilde in order to stress that we work within the g SU (3)language): e n αC =max h , ne n α − ( e λ α − e µ α ) − (2 e λ αC + e µ αC ) oi +max h , ne n α − ( e λ α + 2 e µ α ) + ( e λ αC − e µ αC ) oi , (14)where e n α is the minimal number of relative oscillationquanta for α (p or n) as required by the Wildermuth con-dition. The forbiddenness can be zero, as is the case formost cluster systems of light nuclei. The ( e λ αC , e µ αC ) de-notes the cluster irrep (to which the two clusters are cou-pled and there may appear several) and ( e λ α , e µ α ) is thefinal SU (3) irrep of the united nucleus. For later use, wedefine (cid:16)e λ α , e µ α (cid:17) as the difference of the excited clusterirrep (cid:16)e λ eαC , e µ eαC (cid:17) to the non-excited one (cid:16)e λ αC , e µ αC (cid:17) (theletter e stands for excited ), via (cid:16)e λ eαC , e µ eαC (cid:17) = (cid:16)e λ αC + e λ α , e µ αC + e µ α (cid:17) . (15)Eq. (14) can be interpreted as follows: The first termin (14) indicates, that in order to minimize e n αC , we haveto maximize ( e λ αC + 2 e µ αC ). The second term suggests tominimize the difference (cid:16)e λ αC − e µ αC (cid:17) . The condition ofa maximal ( e λ αC + 2 e µ αC ) and a minimal (cid:16)e λ αC − e µ αC (cid:17) implies a large compact and oblate configuration of thetwo-cluster system. Information on the relative orientationof the clusters is obtained in comparing the distribution ofoscillation quanta for each cluster in the different spatialdirections [40,42].These conditions are achieved, determining the wholeproduct of ( e λ α , e µ α ) ⊗ ( e λ α , e µ α ), using (8) and searchingfor the irrep that corresponds to a large compact struc-ture. The e n = e n p + e n n is the total minimal number Peter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei of relative excitation quanta and e n C = e n pC + e n nC is thetotal forbiddenness .The e n αC are transferred to the clusters, where the re-sult does not depend on how the distribution is done. Forexample, when only one cluster is excited, first the irrep ofthe number of one less valence nucleons is determined andthen coupled with the nucleon in the higher shell, with n αC quanta above the valence shell. The important crite-rion is that at the end one finds a combination of irrepswhich couple to the final irrep in the united cluster. Thisis a direct but rather cumbersome procedure and how todo it will be illustrated in the applications.More restrictions have to be implemented, in order toavoid a too large, not manageable list of irreps: – When coupling the cluster irreps, only those ( λ αC , µ αC )irreps in the product are considered which couple tothe ground state of the united nucleus, for protons andneutron separately. – Of those, only the ones are considered with the largesteigenvalues of the second order Casimir operator of e SU (3), often the irrep with the largest eigenvalue suf-fices. The main argument is that in deformed nuclei,with a significant quadrupole-quadrupole interaction,only these selected irreps contribute significantly atlow energy. – In coupling protons and neutrons together, only thelinear coupling is considered, for reasons explained be-fore. – The final list is, in general, still too large and one has torestrict to the first few irreps with the larges eigenval-ues of the second order Casimir operator, an argumentvalid for dominant quadrupole-quadrupole interaction. – Convergence criterium : Convergence due to these re-strictions have to be verified by adding and/or restrict-ing some irreps. If no sensible change is observed, thenthe cut-off procedure is set.This is a consistent procedure and provides a basis forthe extension to heavy nuclei: The sum of nucleons in thenormal orbitals, of the two cluster, will be always equalto the ones in the united nucleus, which is for exampleignored in [6]. All procedures known from SACM can nowbe applied in the same manner.For completeness, we mention a different definition forthe forbiddenness : In order to be able to deal with heavysystems, the alternative definition was proposed in [6,9],where the relation of an irrep to its deformation was ex-ploited [43]. The criterion applied is as follows: If an irrepof the list has a similar deformation as the one in the shellmodel allowed irrep, it implies a less forbidden state thanirreps with a larger difference in the irreps. The reason whyit works is that the deformation can be related to SU (3)irreps [43,44] and comparing deformations is equivalent tocomparing the dimension of these irreps.Therefore, the definition of forbiddenness used in [9,6]is F = 11 + min hp ∆n + ∆n + ∆n i , (16) where ∆n i = | n i − n i,ξ | and in contrast to S , as definedin [9,6], we use F because the letter S will be used laterexclusively for the spectroscopic factor. The index i refersto the spatial direction of the oscillation and ξ to the sev-eral cluster irreps allowed by the Pauli-exclusion principle.The n i is the number of oscillation quanta in direction i .When all ∆n i are zero then F = 1 and the irrep is al-lowed. If at least one of those numbers is different fromzero the irrep is partially forbidden and when F = 0 it iscompletely forbidden. The most general algebraic Hamiltonian has the samestructure as for light nuclei, save that the operators (num-ber operator, quadrupole operator, etc.) are substitutedby their pseudo counter parts. How this mapping is achieved,is explained in detail in [39], where an operator O in SU (3)is mapped to its counterpart O ≈ κ e O and κ has a sim-ple approximation, namely κ = η + η + , close to 1. The fac-tor can be assimilated into the parameters of the SACMHamiltonian.We restrict to the SU (3) limit, which turns out to besufficient, due to the large deformation of the systems con-sidered. This, however, will result in zero B ( E
2) transitionrates between states belonging to different e SU (3) irreps.A simplified model Hamiltonian is selected, which hasthe following structure: H = ~ ω e n π + ( a − a ∆ e n π ) e C (cid:16)e λ, e µ (cid:17) + t h e C (cid:16)e λ, e µ (cid:17)i + t e C (cid:16)e λ, e µ (cid:17) + ( a + a Lnp ∆ e n π ) e L + t f K (17)where ∆ e n π = e n π − ( e n − e n C ), ( e n − e n C ) being the totalminimal number of quanta required by the Pauli principleand the possible effects of the forbiddeness are taken intoaccount by e n C . The moment of inertia, contained in thefactor of the total angular momentum operator, e L , maydepend on the excitation in ˜ n π (excited states change theirdeformation, corresponding to a variable momentum ofinertia). The K term serves to split the degeneracy in e L within the same g SU (3) irrep. The first term of the g SU (3)Hamiltonian, ~ ω e n π , contains the linear invariant operatorof the e U R (3) subgroup defining the mean field, and the ~ ω is fixed via (45 A − / − A − / ) MeV for light nuclei[45], which can also be used for heavy nuclei. For heavynuclei ~ ω = 41 A − MeV is more common. The A is themass number of the real nucleus and not the number ˜ A = e A + e A of nucleons in the normal orbitals for the unitednucleus.The e C (cid:16)e λ, e µ (cid:17) is the second order Casimir-invariant ofthe coupled g SU (3) group, having contributions both from eter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei 7 the internal cluster part and from the relative motion. The e C (cid:16)e λ, e µ (cid:17) is given by: e C ( e λ, e µ ) = 2 e Q + 34 e L , → (cid:16)e λ + e λ e µ + e µ + 3 e λ + 3 e µ (cid:17) , e Q = e Q C + e Q R , e L = e L C + e L R , (18)where e Q and e L are the total quadrupole and angular mo-mentum operators, respectively, and R refers to the rela-tive motion. The eigenvalue of e C ( e λ, e µ ) is also indicated.The relations of the quadrupole and angular momentumoperators to the e C (1 , m generators of the g SU (3) group, ex-pressed in terms of g SU (3)-coupled π -boson creation andannihilation operators, are [46]: e Q k, m = 1 √ e C (1 , k m , e L k m = e C (1 , k m , e C (1 , lm = √ (cid:2) π † ⊗ π (cid:3) (1 , lm . (19)The quadrupole electromagnetic transition operator isdefined as T ( E m = X γ e (2) γ Q (2) γ,m , (20)where e (2) γ is the effective charge of the contribution to thequadrupole operator, coming from the cluster γ = C , C and from the relative motion R . The effective charges aredetermined as explained in great detail in [47]. A successful parametrization of the spectroscopic factor,within the SACM for light nuclei, is given in [48]: S = e A + Bn π + C C ( λ ,µ )+ D C ( λ ,µ )+ E C ( λ c ,µ c ) × e F C ( λ,µ )+ G C ( λ,µ )+ H∆n π | h ( λ , µ ) κ L , ( λ , µ ) κ L || ( λ C , µ C ) κ C L C i ̺ C ·h ( λ C , µ C ) κ C L C , ( n π , l || ( λ, µ ) κL i | , (21)where the ̺ -numbers refer to multiplicities in the couplingto SU (3) irreps and the κ ’s to the multiplicities to the re-duction to SO (3). The parameters were adjusted to the-oretically exactly calculated spectroscopic factors withinthe p- and sd-shell, using the SU (3) shell model [49], withan excellent coincidence. For the good agreement, the fac-tor depending on the SU (3)-isoscalar factors turns out tobe crucial. For heavy nuclei, spectroscopic factors are poorly ornot at all known experimentally. Therefore, we have topropose a simplified manageable ansatz, compared to (21),including the forbiddenness and, if possible, parameterfree .In what follows, we will propose an expression for thespectroscopic factor which is motivated by the one usedfor nuclei in the p- and sd-shell. As in (21) the expressionis divided into two factors, the first one is an exponentialfactor and the second one of pure geometrical origin [48],an expression in terms of coupling coefficients. The secondfactor is maintained, because it refers to coupling of SU (3)irreps only.The first exponential factor deserves more explanation:As argued in [48], this term is the result of the relative partof the wave-function, which for zero angular momentumis proportional to e − aR ∼ e − a ~ µω n π , where R is the rela-tive distance of the two clusters (though, an e − aR ansatzwould be more appropriate, but the clusters are defined inthe harmonic oscillator picture and we stay to it for con-sistency) and a has units of fm − . The µ is the reducedmass. Let us restrict to the minimum value n of n π . Usingthe relation of r = q ~ µω n [50], where r is the average,minimal distance between the clusters, and taking into ac-count that for this case R = r , we obtain e −| B | n , with | B | = a ~ µω and B <
0. When the wave function acquiresthe value e − it results in the relation | B | = n . For thenuclei in the sd-shell, the adjustment of the parameterswas done for cases with n = 8, which corresponds ac-cording to this estimation to B ≈ − .
13. This has to becompared to the value − .
36 as obtained in the fit in [48],i.e., it is only an approximation which gives at least thecorrect order. The most important part of (21) is the fac-tor depending on the SU (3) isoscalar factors, which areresponsible for the relative changes, while the influence ofthe exponential factor is not dominant for the relative nu-merical values of the spectroscopic factors. Furthermore,when ratios of spectroscopic factors are used, the expo-nential contribution cancels for states in the 0 ~ ω shell.Also when the ( n − nc ) is large, the corrections for ∆n π of the order of one will be negligible. We do not see anypossibility to estimate the parameter A in the exponen-tial factor, which represents a normalization. The otherterms in the exponential factor represent corrections tothe inter-cluster distance, because they correspond to de-formation effects and the parameters in front turned outto be consistently small, thus, they can for the momentbe neglected.In light of the above estimation and discussion, forheavy nuclei we propose a similar expression as in (21),but due to the non-availability of a sufficient number ofspectroscopic factor values (or none at all) for heavy nu-clei, we propose the following simplified expression: Peter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei S = e ˜ A + ˜ B (˜ n − ˜ n C + ∆ ˜ n π ) | h (˜ λ , ˜ µ )˜ κ ˜ L , (˜ λ , ˜ µ )˜ κ ˜ L || (˜ λ C + e λ , ˜ µ C + e µ )˜ κ C ˜ L C i ̺ C ·h (˜ λ C + e λ , ˜ µ C + e µ )˜ κ C ˜ L C , (˜ n π , l || (˜ λ, ˜ µ )˜ κ ˜ L i | . (22)The n π in the exponential factor was substituted by [(˜ n − ˜ n C ) + ∆n π ]. The (˜ n − ˜ n C ) is the number of relative os-cillation quanta in 0 ~ ω (˜ n C was added to the excitationof the clusters).The parameter is estimated as ˜ B = − n − ˜ n C ) . Becausewe can not determine the parameter ˜ A , as a consequenceonly spectroscopic factors divided by the exponential fac-tor e e A are listed. An additional dependence on ˜ n C is con-tained in the product of reduced coupling coefficients, withthe appearance of e λ and e µ (see the definition in (15)). e n and e n C refer to the values within the parent nucleus.In general, in (22) further parameters can be added, as forlight nuclei, which have though to be adjusted.With this choice, the spectroscopic factor is parameterfree , except for an overall normalization, which does notplay a role when only ratios are of interest. In this section we apply the pseudo-SACM to three samplesystems. The first is U → Pb + Ne , the sec-ond one is Ra → Pb + C and the third oneis U → Xe + Sr , which appears in the fissionchannel of U. In the first two cases, the forbiddeness is small. They serve to illustrate how to add the quantato the clusters and how to determine the final cluster ir-rep. The importance of the forbiddeness is significant inthe third case and it will be much harder to determine thecluster irreps. This sequence also shows that the larger thelighter cluster is, the larger the forbiddenness becomes.For illustrative reasons, only the g SU (3) dynamical sym-metry limit is considered, i.e., the united nucleus must bewell deformed. The mixing of SU (3) irreps will be inves-tigated in a future publication. The examples also serveto illustrate the applicability of the model and on how todeduce the quantum numbers. However, some transitionswill be forbidden due to SU (3) selection rules.Very useful is the Table 1, where the number of nucle-ons in a given shell η is related to the one of the pseudo-shell number e η . In the last columns the accumulated num-ber of oscillation quanta within the g SU (3) shell model islisted.In this section, we explain trough the examples in greatdetail the calculations which lead us to the cluster irrepsand with the relative motion to the ground state irreps inthe proton and neutron part. This requires a lot of cum-bersome counting and determination of irreps, especially η No. of nucleons e η No. of nucleons acum. No. of quanta0 2 - - -1 6 0 2 02 12 1 6 63 20 2 12 304 30 3 20 905 42 4 30 2106 56 5 42 420
Table 1.
Table, listing the number of particles in the shellnumber η and the pseudo-shell number e η , including the spindegree of freedom. The last column lists the accumulated num-ber of quanta of the g SU (3), when each shell is full up to thevalence one. The final total number of quanta is obtained, byadding to the number of quanta, reached up to the last closedshell, the number of quanta of the nucleons in the valence shell. Parameter system a system b system c ~ ω .
63 6 .
73 6 . a − . − . − . a . . . t . × − − . − . × − t . . a -0.0081127 0 . . a L − . . . a L np . × − . − . t − . × − . × − . × − Table 3.
Non-zero parameter values for system a: U → Pb + Ne; system b: Ra → Pb + C and system c: U → Xe + Sr. when relative oscillation quanta from the forbiddenness are added to the clusters. We apologize for that and askthe reader to be patient. If she (he) is not interested inthe details, she (he) just can skip this part and jump tothe final numbers of the cluster irreps. U → Pb + Ne e η N P d Zr C Table 2.
An example on how to determine the number ofoscillation quanta ( N ) in the system derived from U → Pb+ Ne, with the help of Table 1. The same is appliedfor the two other sample cases, where Table 1 is very helpfulfor counting the number of relative excitation quanta. Whenone cluster is excited by e n C quanta, this number has still tobe added.eter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei 9
0+ 0 +
2+ 2 + + + + + + + + + + − − − − − − − − − − − − (54,0) (58,2) (58,2) (56,1) (52,6) (53,4) E [ M e V ] Pb + Ne + + + + + + + + + + + − − − − − − − − − − E [ M e V ] Exp
Fig. 1.
Spectrum of
U, described by the clusterization
Pb+ Ne. Only states up to angular momentum 6 are de-picted. The theoretical spectrum (left panel) is compared toexperiment (right panel). Below each rotational band the con-tent of the number of π bosons ( n π ) and the e SU (3) irrep isindicated. As explained above, the protons and neutrons are treatedseparately and the nucleons in each sector are filled intothe Nilsson diagram from below, at the deformation value ǫ = 0 .
200 ( β = 0 . ~ ω = 6 .
63 MeV.For
U, the united nucleus, we obtain 46 protonsin the normal orbitals and the valence shell is e η p = 4with 6 valence protons. The ground state g SU (3) irrep forthe proton part is (˜ λ, ˜ µ ) p = (18 , p , while for the neu-trons there are 82 particles in normal orbitals with 12 inthe e η n = 5 valence shell, giving the ground state irrep(˜ λ, ˜ µ ) n = (36 , n . These two irreps can be coupled tothe total one for U, namely (˜ λ, ˜ µ ) = (54 , forbiddenness (see (14)).These considerations have to be repeated for the twoclusters involved, with Pb being the largest cluster and C the lightest one. For
Pb, filling the protons intothe Nilsson diagram, at the same deformation as for theunited nucleus, we obtain 40 protons in normal orbitals,where the valence shell is e η = 3 and closed, thus the cor-responding irrep is (0 , Pb p . For the neutrons one has 72in normal orbitals with 2 neutrons in the e η n = 5 pseudo-shell. The corresponding irrep is (10 , Pb n .Using the numbers just deduced, the system U → Pb+
Ne can be viewed within the g SU (3) descriptionas a f Pd → f Zr + e C cluster system, obtained bycounting the protons and neutrons in the normal orbitals.Of course, these so-called pseudo-nuclei are only schematicin nature. J Pk U E exp [MeV] Ra E exp [MeV]0 +2 +1 +2 +3 +1 +1 +1 − − − − J Pi → J Pf : B ( E
2) [WU]
Ra: B ( E
2) [WU]2 → → Table 4.
Experimental data used in the fit of the parametersof the model Hamiltonian. The second column lists the dataused for
U and the third column for
Ra. If no data arementioned (dashed sign), the experimental value is not used inthe fit. J Pk U-1 (th)
Ra (th)
U-20 +1 . +2 . +1 . +2 . × − +1 . +2 . × − − . × − − . × − − . × − Table 5.
Some spectroscopic factors of low lying states, di-vided by e e A . In the first column the state quantum numbersare tabulated. The values of the spectroscopic factor for Ucontaining the
Pb cluster,
Ra and
U containing the
Xe cluster are in the second, third and fourth column, re-spectively. Only values which are greater than 10 − are listed.As seen, only the last system shows significant deviations fromzero. The Tables 1 and 2 serve to illustrate on how to deter-mine the total number of quanta in each cluster and theunited nucleus, for the particular case considered. For theother cases treated in this manuscript, it will be similar.The minimal number of quanta, which have to be addedin the proton part, is 20 corresponding to a (20 , pR irrepin the relative part. For the neutron part, this number is40, i.e., an irrep (40 , nR .In the next step, the proton part of the clusters arecoupled with the relative part of the proton section of theunited nucleus. The same is done for the neutrons. Forthe proton part, the product (0 , p ⊗ (0 , p ⊗ (20 , pR (the index pR refers to the relative motion of the pro-tons) contains the proton irrep (18 , p of the united nu-cleus, thus, the forbiddenness for the proton part is zero.The situation is different for the neutron part: The prod-uct (10 , n ⊗ (4 , n ⊗ (40 , nR does not contain (36,0), which is the irrep in the united nucleus. This indicates thatthe clusters have to be excited, thus, the forbiddenness isdifferent from zero. Using the formula (14) we obtain a forbiddenness of e n C = 2. The excitation of the clustersis achieved, changing as one possibility the irrep of Nefrom (4 , n to (6 , n . The relative part is now reducedby two quanta, leaving (38 , nR (the index nR refers tothe relative motion of the neutrons). With this change,the product (10 , n ⊗ (6 , n ⊗ (38 , nR now contains thedominant irrep for neutrons in U. This is verified byEq. (14). Distributing the excitation quanta in a differentmanner leads to the same final result.Using the Hamiltonian in the SU (3)-dynamical limit,the coefficients are adjusted to the experimental data,listed in Table 4 in the second column. The optimal pa-rameters obtained are listed in Table 3, second column.With these parameters, the spectrum calculated is de-picted in Figure 1. The calculated B(E2)-transition valuesare listed in Table 6, second (theory) and third (experi-ment) column.As can be noted, the agreement to experiment is goodand shows the effectiveness of the pseudo-SACM to de-scribe the collective structure of heavy nuclei.Next, we calculated some spectroscopic factors, listedin Table 5, second column. The Equation (22) was usedwith the approximation of the parameter e B as (cid:16) − n − n C (cid:17) .The total number of relative oscillation quanta for thesystem under study is ˜ n = 60 and e n C = 2, thus, e B ≈− . e e A − . n − n c + ∆ ˜ n π ) ≈ (0 . (˜ n − n c + ∆ ˜ n π ) e e A .The factor e e A is unknown and, as explained before,the spectroscopic factors values can be found in Table 5,divided by e e A . As observed, the spectroscopic factors to ∆ ˜ n π = 1 are suppressed, with a value smaller than 10 − considered to be zero. Ra → Pb + C As in the former section, the protons and neutrons aretreated separately, where the nucleons are filled into theNilsson diagram from below, at the deformation value ǫ =0 .
150 ( β ) = 0 . ~ ω = 6 .
73 MeV.For
Ra, the united nucleus, we obtain 46 protons inthe normal orbitals and the valence shell is e η p = 4 with6 valence protons. The g SU (3) irrep is (˜ λ, ˜ µ ) p = (18 , Ra p ,while for the neutrons we have 80 normal particles with10 in the e η n = 5 valence shell, giving (˜ λ, ˜ µ ) Ra n = (30 , Ra n .These two irreps can be coupled to the ground state irrepfor Ra, namely (˜ λ, ˜ µ ) = (48 , U can be foundin the former subsection.The light cluster C is added on top of the heavycluster. We have then 6 protons and 8 neutrons in normalorbitals, which gives (0 , C p (two holes in the e p shell) and(0 , C n (closed e p shell). J P i i → J P f f a b c U -exp Ra -exp2 +1 → +1
251 99 250 250 ±
10 99 ± +2 → +1 .
321 0 -4 +1 → +1
357 141 357 357 ±
23 141 ± +2 → +1 . − → −
309 120 3 .
554 - -4 − → − . . +1 → +1
391 155 390 385 ±
22 157 ± +1 → +2 .
115 0 - -2 +2 → +2
269 142 16 .
13 - -4 +1 → +2 . +2 → +2
383 58 .
94 21 - -4 − → − .
974 0 - -4 − → −
368 6 . × − Table 6.
Theoretical B ( E U → Pb + Ne; system b: Ra → Pb + Cand system c: U → Xe + Sr. The last two columnslist the experimental values for
U and
Ra, respectively.The unit is in WU and the theoretical values are comparedto available experimental data [51]. Note that in
U most ofthe inter-band transitions are zero. This is due to the fact thatthe ground state irrep is (54,0) and that we are working inthe SU (3) limit, thus, there are no transitions between statesin distinct SU (3) irrep. This changes for Ra, whose groundstate irrep in the SU (3) limit is (48,4), which contains several K bands ( K = 0 , ,
4) and allows now a transition betweenstates from different bands at low energy. + + + + + + + + + + + + + − − − − − − (48,4) (48,4) (46,12) (52,6) (50,5) E [ M e V ] Pb + C−−−> Ra + + + + + + + + − − − E [ M e V ] Experimental
Fig. 2.
Spectrum of
Ra, described by the clusterization
Pb+ C. Only states up to angular momentum 6 are de-picted The theoretical spectrum (left panel) is compared toexperiment (right panel).eter O. Hess, Leonardo J. Ch´avez-Nu˜nez: A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei 11
Using the numbers just deduced, the system Ra → Pb + C can be viewed within the g SU (3) de-scription as a f Ru → g Zr + e C cluster system.Of course, these so-called pseudo-nuclei only serve for il-lustration. In what follows, we will continue to use thenotation for the real nuclei.The minimal number of quanta which have to be addedin the proton part of the united nucleus is 20, which isthe result of counting the difference of the oscillation inthe united pseudo-nucleus to the sum of the oscillationquanta of the two pseudo-clusters. This corresponds to a(20 , Rp irrep in the relative part. For the neutron partthe difference in the oscillation quanta is 34 and, thus, theirrep of the relative motion is (34 , Rn For the proton part, the product (cid:2) (0 , Pb p ⊗ (0 , Cp (cid:3) ⊗ (20 , Rp does lead to the final ground state irrep (18 , Ra p and, therefore, the forbiddenness is zero. For the neu-tron part, however, this is no longer the case: The prod-uct (cid:2) (0 , Pb n ⊗ (0 , C n (cid:3) ⊗ (34 , Rn does not contain theground stte irrep (30 , Ra ν . Applying the formula for the forbiddenness gives n nC = 2. These two quanta are sub-tracted from the relative motion, giving (32 , Rn and areadded to the Pb-cluster, as one possibility (any other leadsto the same final result). The irrep used is obtained bysubtracting one neutron from the e η n = 5 and exciting oneneutron to the e η n = (5+2,0) = (7,0). The neutrons in thevalence shell provide the irrep (5,0) (only one valence neu-tron left) and the product with (7,0) contains the (8 , Pb n irrep. The product with the relative motion is sufficient,because the C pseudo-nucleus has the scalar neutron ir-rep (0 , C n . The product (8 , Pb n ⊗ (32 , Rn contains theground state irrep (30 , Ra n .Using the Hamiltonian in the SU (3)-dynamical limit,the coefficients are adjusted to the experimental data,listed in Table 4, third column. The optimal parametersobtained are listed in Table 3, third column. With theseparameters, the spectrum calculated is depicted in Fig-ure 2. The calculated B(E2)-transition values are listed inTable 6, third (theory) and sixth (experiment) column.As can be noted, the agreement to experiment is goodand shows also in this example the effectiveness of thepseudo-SACM to describe the collective structure of heavynuclei.Next, we calculated some spectroscopic factors for Ra,listed in Table 5, third column. The total number of rela-tive oscillation quanta for the system under study is ˜ n =40 ( n C = 2), thus e B ≈ − .
026 and the exponential factorin (22) acquires the form e e A − . n − n c + ∆ ˜ n π ) ≈ (0 . (˜ n − n c + ∆ ˜ n π ) e e A . (23)The factor e e A is unknown and thus in Table 5, thespectroscopic factors were divided by e e A . U → Xe + Sr Consulting the table of [38] the deformation of
U is ǫ =0 .
200 ( β = 0 . ~ ω = 6 . + + + + + + + + + + + + + − − − − − − − − − − − − (54,0) (54,10) (54,10) (54,5) (54,5) (54,5) E [ M e V ] Xe + Sr + + + + + + + + + + + − − − − − − − − − − E [ M e V ] Exp
Fig. 3.
Spectrum of
U, described by the clusterization
Xe+ Sr. Only states up to angular momentum are de-picted. The theoretical spectrum (left panel) is compared toexperiment (right panel). of normal and unique protons and neutrons is given insubsections 3.1, therefore, we will resume the numbers for
Xe and Sr, only.Counting for
Xe the number of protons in the nor-mal orbitals, we obtain 4 in the pseudo-shell e η p =3. Takinginto account the lower shells, we obtain for the total num-ber of protons in normal orbitals e Z =24. As the leadingirrep, obtained in the reduction of U ( ( e η p + 1) ( e η p + 2)) ⊃ g SU (3) we have (8 , p . For the neutron part, there are8 neutrons in the e η n = 4 shell, corresponding to (18 , n as the leading irrep. The total number of normal neutronsis 48.The lighter cluster, Sr, is put on top of the
Xecluster. Counting the difference of the normal protons andneutron in the
U nucleus to the
Xe nucleus, we ob-tain 22 normal protons and 34 normal neutrons for Sr.Distributing them in the g SU (3) shell model, we have 2valence protons in the e η π =3 shell and 14 neutrons in thesame pseudo-shell, which correspond to 6 holes. The lead-ing irreps are respectively (6 , p and (0 , n .Counting the difference in the oscillation quanta in theproton sector between the U nucleus and the sum ofthe two clusters, we obtain 36 quanta. However, when wecouple the two cluster irreps and then with the relativemotion irrep (36 , R , the (18 , Up irrep of U cannot bereached, which is a clear indication that the proton partrequires the use of the forbiddenness . Using the formulafor the forbiddenness we obtain n pC = 10, demonstratingthe importance of the concept of forbiddenness . The 10oscillation quanta are to be added to the clusters, choos-ing a path which is easier to follow: As one possibility(all others lead to the same result) we distribute 6 to thelarge and 4 to the light cluster (such that in each an evennumber is added). For the proton part, in Xe the irrepof one proton less in the valence shell is (7,1) (instead of the former (8,2)). One proton is excited to e η p = (3+6,0)= (9,0). The product of (7 , ⊗ (9 ,
0) contains the irrep(14 , Xe . In Sr the irrep with one nucleon less in the va-lence shell e η p = 3 is (3,0). One nucleon is excited the e η p = 3+4 =7, i.e., with the irrep (7,0). In the product of(3 , ⊗ (7 ,
0) the irrep (10 , Sr appears. In the final cou-pling with the radial irrep (36 − , pR = (26 , pR weobtain (14 , Xe p ⊗ (10 , Sr p , which contains (4,12) and cou-pled with (26 , pR leads to the final proton irrep (18 , Up and, thus, we reached our objective for the proton partThe same procedure has to be applied for the neutronpart. Counting the difference in the oscillation quanta inthe neutron sector between the U nucleus and the sumof the two clusters, we obtain 76 quanta. However, whenwe couple the two cluster irreps and then with the relativemotion irrep (76 , nR , we cannot reach the (36 , Un irrepof U, which is a clear indication that the neutron partrequires the use of the forbiddenness . Using the formulafor the forbiddenness we obtain n νC = 48, again showingthat the concept of forbiddenness is very important. The48 oscillation quanta, have to be added to the clusters.This time, all the 48 quanta are added to the light cluster,because it does not matter how we distribute the n νC = 48between the clusters. This is the reason why we now followthis path. Before, in Sr there were 6 holes, now there willbe 7, because one neutron is excited to the e η n = 3+48 =51 and the irrep which carries the single neutron is (51,0).The valence shell, with 7 holes, provides the irrep (2,11).Coupling both, (2 , ⊗ (51 , (cid:2) (38 , ⊗ (18 , Xe n (cid:3) ⊗ (28 , nR → , which leadsto the possible combination of (22 , ⊗ (28 , nR , where(22,14) appears in the product (38 , ⊗ (18 , , νR it contains the final irrep (36 , U ν .The cluster irreps for Xe and Sr are obtained by couplelinearly for each cluster the proton and neutron irrep. Us-ing the above results of the compilations, we get (48 , Sr and (32 , Xe , once coupled contains the irrep (30,30),which coupled again with the relative motion (54,0), where54 is the sum of the relative motions in the proton (26)and the neutron part (28). The final result contains theground state irrep of U, namely (54,0).Counting only the normal nucleons, the system U → Xe + Sr can be viewed as the system f Pd → f Cr + f Ti . The agreement to experiment is sat-isfactorily, as can be see by consulting Figure 3 for thespectrum and Table 6 for the transition values. Compar-ing to the U → Pb + Ne, the obtained spectrumhas a similar agreement to experiment, with some shiftsin the band heads and changes in the moment of inertia.For the calculation of the spectroscopic factor, the pa-rameter e B = − e n − n C , where ( e n − n C ) = -0.0185 are theremaining total relative oscillation quanta after subtract-ing the forbiddenness . The spectroscopic factors are listedin the last column in Table 5. We have presented an extension of the
SemimicroscopicAlgebraic Cluster Model (SACM), valid for light nuclei, tothe pseudo-SACM , for heavy nuclei, limiting to the g SU (3)dynamical symmetry limit. Though, there exist earlier at-tempts to extend the SACM to heavy nuclei, we found itnecessary to construct a model, which enables us to de-termine the complete spectrum and circumvent some con-ceptual and practical problems of the former approachesand to deliver a consistent procedure, as working in thesame mean field of the parent nucleus and simultaneouslyconserving the simplicity of the SACM for light nuclei.Protons and neutrons have to be treated separately,because they occupy different shells. Only at the end theyare coupled together. The protons and neutrons are dis-tributed within the normal and unique orbitals in such amanner that the sum of normal nucleons of the clustersis the same as in the parent nucleus. The construction ofthe model space is in complete analogy to the SACM.As examples, we considered U → Pb+ Ne, Ra → Pb+ C and U → Xe+ Sr. We demonstratedthat the model is able to describe the spectrum and elec-tromagnetic transition probabilities. Spectroscopic factorswere also calculated, without further fitting and they canbe considered as a prediction of the model. With this,we demonstrated the usefulness of the pseudo-SACM fortreating heavy nuclei. A more systematic study of severalnuclei is planned in the future.The restriction to the g SU (3) dynamical symmetry limithas to be relaxed in future applications, including theother dynamical symmetries as SO (4). One has to studythe extension of g SU (3) too, including the active partici-pation of nucleons in the unique orbitals. Also the studyof phase transitions is of interest, requiring the use of thegeometrical mapping [50] of the SACM. Acknowledgments
We acknowledge financial support form DGAPA-PAPIIT(IN100421).
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