11institutetext: Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México, Circuito Exterior, C.U., A.P. 70-543, 04510 México D.F.,
Mexico 22institutetext: Frankfurt Institute for Advanced Studies, Johann
Wolfgang Goethe Universität, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main,
Germany

# A Semimicroscopic Algebraic Cluster Model for Heavy Nuclei

I: One heavy and one light cluster

Peter O. Hess 1122 Leonardo J. Chávez-Nuñez 11

(Received: date / Revised version: date)

###### Abstract

An extension of the Semimicroscopic Algebraic Cluster Model (SACM) is
proposed, based on the pseudo-$SU(3)$ model ($\widetilde{SU}(3)$). The
Hamiltonian and the spectroscopic factor operator of the model are presented
and a procedure of constructing the model space. Because a huge number of
$SU(3)$ irreducible representations (irrep) appear, one has to be careful in
designing a practical, consistent path to reduce the Hilbert space. The
concept of forbiddenness, taking into account excitations of the clusters, is
introduced and applied. The applications are to two systems with a low
forbiddenness, namely to 236U $\rightarrow$ 210Pb + 26Ne and 224Ra
$\rightarrow$ 210Pb + 14C, and to 236U $\rightarrow$ 146Xe + 90Sr, which
appears in the fission of ${}^{236}U$, which requires a large forbiddenness.
Energies, electromagnetic transitions and spectroscopic factors are
calculated.

###### pacs:

21.00nuclear structure and 21.60.Csshell model and 21.60.Gxcluster model

##  1 Introduction

The Semimicroscopic Algebraic Cluster Model (SACM) was proposed in cseh-letter
; cseh-levai-anph  for light nuclei (see, for example, sacm-appl1 ), with the
intend as an alternative to also describe nuclear molecules, i.e., an alegraic
version of their geometrical description scheid1995 ; hess-1984 . A first
attempt to extend it to heavy nuclei is published in cseh-algora , using the
$\widetilde{SU}(3)$ (pseudo-$SU(3)$) model hecht ; arima . While the heavy
cluster is treated within the $\widetilde{SU}(3)$ the light cluster is
described by the $SU(3)$ standard model. In cseh-algora ; cseh-scheid , the
separation of nucleons into the unique and normal orbitals in the united
nucleus, compared to the ones in each cluster, is not well defined:
Distinction is made when a cluster is light, which is then treated within the
standard shell model, or both are heavy. In different words: Each cluster and
the united nucleus are not treated in the same mean field.

Since then, many applications of the SACM have been studied and further
attempts to extend it to heavy nuclei: For example, a construction of the
effective $SU(3)$ irreducible representations (irreps) for heavy nuclei
hunyadi , using the Nilsson model ring , and a study of preferences in
radioactive decays and/or fission sacm-fission1 ; sacm-fission2 ; sacm-
fission3 . In hess-86 , the spectra of $\alpha$ cluster nuclei, of significant
interest in astrophysics related to the production of heavy elements, and
their spectroscopic factors were calculated. In phase-I ; phase-II  the first
steps in investigating phase transitions were taken and more recently david  a
complete description of phase transitions using the catastrophe theory gilmore
was published. In renorm  the renormalization of the coherent state parameters
is investigated, used for the geometric mapping.

More recently, the $\widetilde{SU}(3)$ model was applied to the shell like
quarteting in heavy nuclei cseh2020 , another method to restrict effectively
the shell model space on physical grounds. The quarteting model was first
proposed in arima-quart ; danos-quart  and applied in cseh-quart  within the
SACM for light nuclei. However, the proton and the neutron part are coupled
directly to a total $SU(3)$ irreducible representation (irrep) without taking
into account the preference for certain couplings, leading as a result to too
many irreps at low energy. This is also a problem of the model we present in
this contribution and a path on how to tackle it is proposed. The quarteting
model was compared to another procedure, called the proxy-$SU(3)$ bonatsos2017
, showing comparable results. All methods have in common to exploit a
symmetry, showing up in the single particle spectrum, and using it to
effectively cut the model space to a manageable size. Here we will propose an
equivalent manner to describe heavy nuclei, using the $\widetilde{SU}(3)$
model and a proposal on how to restrict the model space to the most important
contributions.

A further prove of the success of the SACM is the introduction of the multi-
channel symmetry multi , were different clusterizations were connected via the
same Hamiltonian, thus, reducing the complexity of the model and delivering
more insight into the structure of cluster systems.

In hunyadi  a quite powerful method was presented on how to treat heavy
nuclei, it only delivers the ground state, or the first super- and hyper-
deformed states cseh2006 . Therefore, it is of interest to look for
alternative procedures in order to deal also with excited states.
Unfortunately, we cannot list all contributions of the SACM, but the ones
cited clearly demonstrate the success of the SACM.

The $\widetilde{SU}(3)$ was proposed in hecht ; arima  for its application to
heavy nuclei, where it is observed that when the intruder states, called
unique orbitals, with $j=\eta+\frac{1}{2}$ in each shell ($\eta$ is the shell
number), are excluded, the remaining orbitals, called normal orbitals, are
grouped, just by counting, into shells of what is denominated as the
$\widetilde{SU}(3)$ symmetry. The normal orbital in the $\eta$-shell are
renamed (see next section) such that they correspond to a pseudo-shell of
${\tilde{\eta}}=\eta-1$. Within the Nilsson model asymptotic states also show
a degeneration in orbitals denoted by the asymptotic quantum numbers of the
Nilsson model $\Omega\left[\eta\eta_{z}\Lambda\right]$ ring ; plb1994 .

That the nuclear force exhibits such an unexpected symmetry is today well
understood, parting from microscopic field theoretic models of the nuclear
interaction fieldsu3  and mapping to the effective nuclear interaction. The
complete Hilbert space of the shell model is a direct product of a state
described within a $\widetilde{SU}(3)$, in the same manner as the $SU(3)$
model of Elliott for light nuclei, and a state describing the nucleons in the
intruder orbitals. The nucleons in the intruder orbitals are assumed to play
only the role of an observer: Nucleons in the unique orbitals play a passive
role in the dynamics due to their opposite parity and their coupling to spin-
zero pairs, contributing only to the binding energy. The pairing energy of
nucleons in the unique orbitals is big, due to the large $j$ ring , compared
to the nucleons in the normal orbitals. Thus, nucleons in the unique orbitals
only contribute at high energies, e.g., in the back-bending effect and related
phenomena draayer-book . Inter-shell excitations are considered only within
the normal states, for the same reasons. The contribution of the nucleons in
the unique orbitals are treated via well defined effective charges for
electromagnetic transitions NPA1994 . The effective charges do not depend on
parameters, but only on the total number of nucleons $A$, ${\tilde{A}}$ and
protons $Z$, ${\tilde{Z}}$, where the symbols with the tilde refer to numbers
in the normal orbitals. In conclusion, the restriction to the
$\widetilde{SU}(3)$ is well justified for states at low energy. Though, the
$\widetilde{SU}(3)$ has its limits, as not including independent dynamical
effects of nucleons in the unique orbitals, it is still useful for
investigating an extension of the SACM to heavy nuclei.

Already within the SACM for light nuclei, new problems arise: When the
lightest cluster is too large and assuming both clusters in their ground
state, the corresponding $SU(3)$ irrep of the united nucleus cannot be
reached. The main reason is that for increasing number of nucleons of the
light cluster, the number of quanta in the relative motion becomes too large
and the coupling of the relative 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 smirnov  and the authors
introduced the concept of forbiddenness. The main idea is as follows: When the
two clusters are in their ground state and all missing quanta are put into the
relative motion (the Wildermuth condition wildermuth ) and the ground state of
the united nucleus can be reached, everything is fine. No excitations are then
allowed within a cluster, because these are included in the excitation of the
relative motion. However, when the ground state cannot be reached, one or the
two clusters are allowed to be excited, subtracting quanta from the relative
motion, up to the point when for the first time the ground state is reached.
The excited state of the cluster ($C_{k}^{*}$) is then fixed and the rest of
the procedure is identical as in the SACM for light nuclei. The minimal number
required for the excitation of the clusters is called the forbiddenness.

In smirnov  the importance of this concept and its consequences was proven and
applied quite successfully in the prediction of preferences of fission
fragments and fusion of light and heavy systems. The larger the forbiddenness,
the more suppressed is the reaction channel. To summarize: For low lying
states and a large light cluster, the cluster system has to be excited!
Reconsidering the definition of forbiddenness from an alternative angle, we
were able to obtain a simple formula on the minimal number of excitation
quanta huitz-2015  needed, which will be resumed further below.

Both models, the $\widetilde{SU}(3)$ and the SACM extended to heavy nuclei,
will be explained in section 2. The formalism includes a restricted model
Hamiltonian, the calculation of spectra, electromagnetic transition rates and
the determination of spectroscopic factors. In particular, in Section 3 we
will demonstrate the importance of the concept of the forbiddenness. In
Section 3 the model is applied to several heavy cluster systems. Two systems
only require a low number of forbiddenness, while the third system exhibits a
large forbiddenness and demonstrates its importance. In section 4 conclusions
are drawn.

##  2 The SACM and its extension to heavy nuclei

The approach to $\widetilde{SU}(3)$ is as follows: In each harmonic oscillator
shell $\eta$ the orbital belonging to the largest spin $j=\eta+\frac{1}{2}$ is
removed from consideration as an active orbital, it is considered rather as a
spectator. To the remaining orbitals the redefinition

$\displaystyle j~{}=~{}l\pm\frac{1}{2}$ $\displaystyle\rightarrow$
$\displaystyle{\tilde{l}}~{}=~{}l\mp\frac{1}{2}~{},~{}\eta~{}\rightarrow~{}{\tilde{\eta}}~{}=~{}\eta-1~{}~{}~{},$
(1)

is applied, where ${\tilde{l}}$ denotes the pseudo-orbital angular momentum
and ${\tilde{\eta}}$ the pseudo-shell number. With this redefinition alone it
is easily verified that each shell $\widetilde{\eta}$ has the same content as
the corresponding shell in the standard $SU(3)$ model.

For large deformations, the Nilsson states for axial symmetric nuclei are
classified by their asymptotic quantum numbers ring

$\displaystyle\Omega\left[\eta\eta_{z}\Lambda\right]~{}~{}~{},$ (2)

where $\eta_{z}$ is the oscillation number in the $z$-direction, $\Lambda$ is
the projection of the orbital angular momentum onto the same axis and $\Omega$
= $\Lambda\pm\frac{1}{2}$.

Excluding the intruder levels, the reassignment of the orbitals is

$\displaystyle\Omega~{}=~{}\Lambda\pm\frac{1}{2}$ $\displaystyle\rightarrow$
$\displaystyle\widetilde{\Lambda}~{}=~{}\Lambda\mp\frac{1}{2}~{}~{}~{}.$ (3)

Inspecting the Nilsson diagrams ring , those orbitals with the same quantum
numbers $\left[\widetilde{\eta}\widetilde{\eta_{z}}\widetilde{\Lambda}\right]$
are degenerate, which implies a very small pseudo-spin-orbit interaction and
as a consequence an approximate symmetry. In addition, the content of the
$\widetilde{\eta}$ shell corresponds to the same one in the standard shell
model. Thus, the shell model for light nuclei can be directly extended to
heavy nuclei, using the $\widetilde{SU}(3)$ model instead of the $SU(3)$
model.

The basis in the Hilbert space is a direct product of the $\widetilde{SU}(3)$
states and the ones describing the unique (intruder) orbitals. As mentioned in
the introduction, their contribution to the nuclear dynamics is taken into
account by a well defined scaling factor (effective charge).

For example, the quadrupole effective charge is given by NPA1994 :

$\displaystyle e_{eff}(E2)$ $\displaystyle=$
$\displaystyle\left(\frac{Z}{\widetilde{A}}\right)^{2}\left(\frac{A}{{\widetilde{A}}}\right)^{\frac{4}{3}}~{}~{}~{},$
(4)

where $A$ and ${\widetilde{A}}$ are the total number of nucleons and the
number of nucleons in the normal orbitals, respectively. $Z$ is the total
number of protons.

Restricting the dynamics only to the nucleons in the normal orbitals was
justified in the introduction, recognizing that at large excitation energies,
where backbending effects play a role, the model has to be modified by adding
the contributions of the dynamics in the unique orbitals. The effectiveness of
$\widetilde{SU}(3)$ was demonstrated in NPA1994 , for the case of the pseudo-
symplectic model of the nucleus pseudo-sympl  and many other applications of
the $\widetilde{SU}(3)$ model (see, for example, cseh-quart ).

###  2.1 Construction of the model space in light nuclei

In the SACM for light nuclei, the $SU(3)$ irreps are determined using the
following path: Each cluster is represented by an irrep
$\left(\lambda_{k},\mu_{k}\right)$ ($k=1,2$) in their ground state. Adding the
number of oscillation quanta contained in each cluster and comparing them with
the number of oscillation quanta of the united nucleus results in a mismatch:
The number of oscillation quanta of the united nucleus is larger than the sum
of both clusters. Wildermuth wildermuth  showed that the necessary condition
to satisfy the Pauli exclusion principle is to add the missing quanta into the
relative motion, introducing a minimal number of relative oscillation quanta
$n_{0}$. This is known as the Wildermuth condition. However, there are still
irreps which are not allowed by the Pauli-exclusion principle.

An elegant solution to it, avoiding cumbersome explicit antisymmetrization of
the wave function, was proposed in cseh-letter ; cseh-levai-anph , the
original publication of the SACM: The coupling of the cluster irreps with the
one of the relative motion generates a list of $SU(3)$ irreps, i.e.,

$\displaystyle\left(\lambda_{1},\mu_{1}\right)\otimes\left(\lambda_{2},\mu_{2}\right)\otimes\left(n_{\pi},0\right)$
$\displaystyle=$
$\displaystyle\sum_{m_{\lambda,\mu}}m_{\lambda,\mu}\left(\lambda,\mu\right)~{}~{}~{},$
(5)

where $n_{\pi}$ is the number of relative oscillation quanta ($\pi$ gives
reference to the relative $\pi$-bosons of spin 1), limited from below by
$n_{0}$, and $m_{\lambda,\mu}$ is the multiplicity of
$\left(\lambda,\mu\right)$. 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 model space can be called microscopic.

In such a manner, the basis states are described by the ket state

$\displaystyle\mid(\lambda_{1},\mu_{1})(\lambda_{2},\mu_{2});\rho_{C}(\lambda_{C},\mu_{C})(n_{\pi},0);\rho(\lambda,\mu)\kappa
LM\rangle~{}~{}~{},$ (6)

where $\rho_{C}$, $\rho$ and $\kappa$ are multiplicity labels. The cluster
irreps are coupled first to $(\lambda_{C},\mu_{C})$, then with $(n_{\pi}0)$ to
the final $SU(3)$ irrep $(\lambda,\mu)$. The advantage of using the ket-
formalism is the absence to the need of a coordinate space description, it
suffices to get the quantum numbers describing Pauli allowed cluster states.
Of course, the disadvantage is that no explicit space distribution of the
clusters are depicted.

This representation of the model space is of advantage, because it involves
only $SU(3)$ groups, which refer to the shell model space of the clusters and
the complete nucleus.

The word Semi in the name of SACM appears due to the phenomenological
character of the Hamiltonian, which is a sum of terms associated to the single
particle energy, quadrupole-quadrupole interactions, angular momentum
operators, etc. Note, that this is an additional ingredient and the
construction of the cluster space is independent of it, which can be used in
any microscopic model.

Finally, we expose the path on how to deduce the $\widetilde{SU}(3)$ shell
model space: Each shell $\eta$ has $\frac{1}{2}(\eta+1)(\eta+2)$ orbital
degrees of freedom. Taking into account the two spin degrees of freedom, the
group-chain classifying the states within the shell $\eta$ is given by

$\displaystyle U((\eta+1)(\eta+2))$
$\displaystyle\supset~{}~{}~{}U(\frac{1}{2}(\eta+1)(\eta+2))\otimes U_{S}(2)$
$\displaystyle\left[1^{N}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\widetilde{\left[h\right]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left[h\right]$
$\displaystyle U(\frac{1}{2}(\eta+1)(\eta+2))$
$\displaystyle\supset~{}~{}~{}SU(3)$ $\displaystyle\left[h\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\lambda,\mu)~{}~{}~{},$ (7)

where $[h]$ is a short hand notation for $\left[h_{1},h_{2}\right]$ and the
tilde denotes the conjugate Young diagram where rows and columns are
interchanged. (7) gives the relation of the spin-part (denoted by the index
$S$) and the orbital part, such that the complete state is anti-symmetric
($\left[1^{N}\right]$, with $N$ as the number nucleons in the shell $\eta$).
In the case of light nuclei, instead of the spin group $SU_{S}(2)$ the spin-
isospin group $SU_{ST}(4)$ appears. In contrast, in heavy nuclei the protons
and neutron have to be considered within different shells, which is the reason
why (7) is used. For the reduction to $SU(3)$ programs are available bahri .

The reduction scheme (7) is applied to each shell, which contains nucleons.
Each $\Delta n_{\pi}\hbar\omega$ excitations ($\Delta n_{\pi}=0,1,...$)
corresponds to a particular distribution of the nucleons within the shells.
The $SU(3)$ content of each shell is multiplied with all others, resulting in
a preliminary list. Finally, the center of mass motion is removed from the
$\Delta n_{\pi}\hbar\omega$ excitation, by multiplying the result for
$0\hbar\omega$ with $(\Delta n_{\pi},0)$, the $1\hbar\omega$ result by
$(\Delta n_{\pi}-1,0)$, etc. and subtracting all the results from the before
obtained large list. This is a standard procedure, which for heavy nuclei is
applied separately to the proton and neutron part.

###  2.2 Concepts for the extension to heavy nuclei

When dealing with heavy nuclei one encounters an exploding number of states
($SU(3)$ irreps). Thus, one not only has to find a consistent path on how to
combine the cluster irreps with the relative motion to the final irreps, but
also a way to restrict further the irreps according to their importance to
contribute at low energy. This has to be done twice, namely for protons and
neutrons, and combining them leads to even more irreps. Thus, the method to
restrict the Hilbert space is very essential.

For the extension of the SACM to heavy nuclei requires explanations of some
facts, concepts and assumptions:

  * •

All nucleons move in the same mean field of the parent nucleus. This is
obvious, because even when clusters are defined as consisting of a subset of
$A_{k}$ ($k=1,2$) nucleons, these nucleons are still part of the parent
nucleus and not free clusters. All nucleons are part of the same shell model,
characterized by the same oscillator energy $\hbar\omega$ of the united
nucleus. As a consequence, a light cluster, cannot be the same within the
parent nucleus, as it is as a free nucleus. For example, for a free and
independent cluster the $\hbar\omega$ value is already different. The
identification of the clusters is only via their content in their number of
protons and neutrons.

In addition, in $\widetilde{SU}(3)$ each cluster has nucleons in the normal
and unique orbitals and only the ones in the normal orbitals are counted.
Thus, a cluster changes to a pseudo-cluster with $\widetilde{A}_{k}$ nucleons
and $\widetilde{Z}_{k}$ protons in the normal orbitals ($k=1,2$).

  * •

Because protons and neutrons are in different shells, the construction of the
model space is applied separately to them, as done in any shell model
application for heavy nuclei ring . The proton and neutron part move in the
same mean field, i.e., with the same $\hbar\omega$.

  * •

The nucleons are filled into the Nilsson orbitals at the deformation of the
united nucleus. The deformation values for a nucleus is retrieved from the
tables nix-tables . This does not suggest that we are working in different
shell models, it is just a fact that protons and neutrons are filling
different kind of orbitals, which is obvious by inspecting the Nilsson
diagrams for protons and neutrons ring . For both, however, the deformation
value is the same.

  * •

The nucleons of the heavy cluster are filled in first and then the ones of the
light clusters. This involves the assumption that the light cluster is small,
compared to the heavier one, and is preformed within the united nucleus, a
phenomenological picture used in radioactive decays.

This step will provide us with a heavy and light cluster, for each one with a
certain number of nucleons in the normal orbitals, denoted by
$\widetilde{A}_{k}$ ($k=1,2$).

  * •

In a final step, the proton and neutron part are combined, proposing a
particular selection, whose origin is the demand that the proton and neutron
fluid are aligned. In draayer1  this is discussed and in more detail in NPA576
: In the nuclear shell model the aligned irrep, i.e.
$(\lambda_{p},\mu_{p})\otimes(\lambda_{n},\mu_{n})$ $\rightarrow$
$(\lambda_{p}+\lambda_{n},\mu_{p}+\mu_{n})$ ($p$ for protons and $n$ for
neutrons), corresponds to aligned principle axes of the two (proton-neutron)
rotors. All other irreps are non-aligned proton-neutron rotors, which lie at
higher energy, corresponding to scissors mode, or isovector resonances
eisenberg . Thus, the restriction to aligned proton-neutron irreps is a pretty
good approximation, taking effectively into account the interaction, which
tends to align the proton and neutron fluid.

###  2.3 Construction of the model space and further restrictions

The construction of the ${\widetilde{S}U}(3)$ model space for heavy nuclei is
in line with the one for light nuclei. First some notational definitions: An
$\alpha=p,n$ denotes the type of the nucleons (protons or neutrons),
$(\widetilde{\lambda}_{k\alpha},\widetilde{\mu}_{k\alpha})$ ($k=1,2$) refers
to cluster number $k$ and for each type of nucleon. The
$(\widetilde{\lambda}_{\alpha C},\widetilde{\mu}_{\alpha_{C}})$ is called the
cluster irrep for the $\alpha$-type nucleons and it is the result of the
coupling

$\displaystyle(\widetilde{\lambda}_{1\alpha},\widetilde{\mu}_{1\alpha})\otimes(\widetilde{\lambda}_{2\alpha},\widetilde{\mu}_{2\alpha})$
$\displaystyle=$ $\displaystyle\sum_{\widetilde{\lambda}_{\alpha
C},\widetilde{\mu}_{\alpha_{C}}}m_{\widetilde{\lambda}_{\alpha
C},\widetilde{\mu}_{\alpha_{C}}}(\widetilde{\lambda}_{\alpha
C},\widetilde{\mu}_{\alpha_{C}})~{}~{}~{},$ (8)

being $m_{\widetilde{\lambda}_{\alpha C},\widetilde{\mu}_{\alpha_{C}}}$ the
multiplicity of $(\widetilde{\lambda}_{\alpha
C},\widetilde{\mu}_{\alpha_{C}})$.

Afterward, each $(\widetilde{\lambda}_{\alpha
C},\widetilde{\mu}_{\alpha_{C}})$ is coupled with the relative motion irrep
$(\widetilde{n}_{\alpha\pi},0)$, i.e.,

$\displaystyle(\widetilde{\lambda}_{\alpha
C},\widetilde{\mu}_{\alpha_{C}})\otimes(\widetilde{n}_{\alpha\pi},0)$
$\displaystyle=$
$\displaystyle\sum_{\widetilde{\lambda}_{\alpha},\widetilde{\mu}_{\alpha}}m_{\widetilde{\lambda}_{\alpha},\widetilde{\mu}_{\alpha}}(\widetilde{\lambda}_{\alpha},\widetilde{\mu}_{\alpha})~{}~{}~{}.$
(9)

The minimal number of ${\widetilde{n}}_{\alpha\pi}$ is determined in the same
manner as in the SACM for light nuclei: It is the difference of the number of
$\pi$-oscillation quanta in the united nucleus (only counting those in the
normal orbitals) and the sum of oscillation quanta of the two clusters, for
protons and neutrons separately.

This leads to a large list of irreps
$(\widetilde{\lambda}_{\alpha},\widetilde{\mu}_{\alpha})$ for each sector of
nucleons. This list is compared to the one obtained by the shell model. A
program, which does this automatically, can be send on request.

For a two cluster system, the path explained is resumed in the following group
chains:

$\displaystyle\widetilde{SU}_{1\alpha}(3)\otimes\widetilde{SU}_{2\alpha}(3)\otimes\widetilde{SU}_{R\alpha}(3)\supset$
$\displaystyle(\lambda_{1\alpha},\mu_{1\alpha})~{}~{}(\lambda_{2\alpha},\mu_{1\alpha})~{}~{}(\widetilde{n}_{\alpha\pi},0)$
$\displaystyle\widetilde{SU}_{\alpha
C}(3)\otimes\widetilde{SU}_{R\alpha}(3)\supset\widetilde{SU}(3)$
$\displaystyle~{}~{}(\lambda_{\alpha C},\mu_{\alpha
C})~{}~{}~{}(\widetilde{n}_{\alpha\pi},0)~{}~{}~{}~{}~{}~{}(\widetilde{\lambda},\widetilde{\mu})~{}~{}~{},$
(10)

where $R$ refers to the relative motion part. Below each group the
corresponding quantum numbers are listed. The $\widetilde{SU}(3)$ can be
further reduced to the angular momentum group.

Up to here, the procedure is in accordance to the SACM for light nuclei cseh-
letter , with the difference that due to the breaking of isospin symmetry the
protons and neutrons are treated separately. In order to obtain the final list
of Pauli allowed total irreps, the one of protons is multiplied with the one
of neutrons, i.e.,

$\displaystyle(\widetilde{\lambda}_{p},\widetilde{\mu}_{p})\otimes(\widetilde{\lambda}_{n},\widetilde{\mu}_{n})$
$\displaystyle=$
$\displaystyle\sum_{\widetilde{\lambda},\widetilde{\mu}}m_{\widetilde{\lambda},\widetilde{\mu}}(\widetilde{\lambda},\widetilde{\mu})~{}~{}~{},$
(11)

which is represented by the following group chain

$\displaystyle\widetilde{SU}_{p}(3)\otimes\widetilde{SU}_{n}(3)$
$\displaystyle\supset$ $\displaystyle\widetilde{SU}(3)~{}~{}~{}.$ (12)

To reduce further the large list, obtained in (11), we select only the irreps
corresponding to a linear coupling, namely

$\displaystyle(\lambda_{p},\mu_{p})\otimes(\lambda_{n},\mu_{n})$
$\displaystyle\rightarrow$
$\displaystyle(\lambda_{p}+\lambda_{n},\mu_{p}+\mu_{n})~{}~{}~{}.$ (13)

The justification is the same as mentioned further above, i.e., all other
irreps correspond to scissors modes at large high energy draayer1 .

###  2.4 The concept of forbiddenness

As for large clusters in light nuclei, for heavy clusters in heavy nuclei an
additional problem arises:

Usually, the clusters are in their ground state and all shell excitations are
dealt via the radial excitation. This is because an excitation of the clusters
is equal to the excitation in the radial motion.

However, as mentioned above, for the light cluster being sufficiently large
the clusters have to be excited, in order to connect to the ground state irrep
of the parent nucleus. The required number of excitation quanta are subtracted
from the relative motion. The cluster system has to be excited by a minimal
number of shell excitation $n_{C}$ = $n_{pC}+n_{nC}$, such that for the first
time the ground state can be reached in the coupling of the cluster irreps and
the relative motion, containing now less number of quanta. Further excitations
have to be added to the relative motion with the same argument as given in the
last paragraph. In smirnov  this concept was denoted forbiddenenness with
important consequences for the explanation of the preferences of fission. This
concept is barely known but applied with success within the SACM sacm-fission1
; sacm-fission2 ; sacm-fission3 . In the examples presented in section 3 we
will demonstrate the importance of the forbiddenness.

An easy to apply formula to determine the forbiddenness $n_{\alpha C}$ is
given in huitz-2015  (again for protons and neutrons separately, with
$\alpha=p$ or $n$, and all variables are denoted by a tilde in order to stress
that we work within the $\widetilde{SU}(3)$ language):

$\displaystyle{\widetilde{n}}_{\alpha C}=$ $\displaystyle{\rm
max}\left[0,\frac{1}{3}\left\\{{\widetilde{n}}_{\alpha
0}-({\widetilde{\lambda}}_{\alpha}-{\widetilde{\mu}}_{\alpha})-(2{\widetilde{\lambda}}_{\alpha
C}+{\widetilde{\mu}}_{\alpha C})\right\\}\right]$ $\displaystyle+{\rm
max}\left[0,\frac{1}{3}\left\\{{\widetilde{n}}_{\alpha
0}-({\widetilde{\lambda}}_{\alpha}+2{\widetilde{\mu}}_{\alpha})+({\widetilde{\lambda}}_{\alpha
C}-{\widetilde{\mu}}_{\alpha C})\right\\}\right]~{}~{}~{},$ (14)

where $\widetilde{n}_{\alpha 0}$ is the minimal number of relative oscillation
quanta for $\alpha$ (p or n) as required by the Wildermuth condition. The
forbiddenness can be zero, as is the case for most cluster systems of light
nuclei. The $({\widetilde{\lambda}}_{\alpha C},{\widetilde{\mu}}_{\alpha C})$
denotes the cluster irrep (to which the two clusters are coupled and there may
appear several) and
$({\widetilde{\lambda}}_{\alpha},{\widetilde{\mu}}_{\alpha})$ is the final
$SU(3)$ irrep of the united nucleus. For later use, we define
$\left({\widetilde{\lambda}}_{\alpha 0},{\widetilde{\mu}}_{\alpha 0}\right)$
as the difference of the excited cluster irrep
$\left({\widetilde{\lambda}}^{e}_{\alpha C},{\widetilde{\mu}}^{e}_{\alpha
C}\right)$ to the non-excited one $\left({\widetilde{\lambda}}_{\alpha
C},{\widetilde{\mu}}_{\alpha C}\right)$ (the letter $e$ stands for excited),
via

$\displaystyle\left({\widetilde{\lambda}}^{e}_{\alpha
C},{\widetilde{\mu}}^{e}_{\alpha C}\right)$ $\displaystyle=$
$\displaystyle\left({\widetilde{\lambda}}_{\alpha
C}+{\widetilde{\lambda}}_{\alpha 0},{\widetilde{\mu}}_{\alpha
C}+{\widetilde{\mu}}_{\alpha 0}\right)~{}~{}~{}.$ (15)

Eq. (14) can be interpreted as follows: The first term in (14) indicates, that
in order to minimize ${\widetilde{n}}_{\alpha C}$, we have to maximize
$({\widetilde{\lambda}}_{\alpha C}+2{\widetilde{\mu}}_{\alpha C})$. The second
term suggests to minimize the difference $\left({\widetilde{\lambda}}_{\alpha
C}-{\widetilde{\mu}}_{\alpha C}\right)$. The condition of a maximal
$({\widetilde{\lambda}}_{\alpha C}+2{\widetilde{\mu}}_{\alpha C})$ and a
minimal $\left({\widetilde{\lambda}}_{\alpha C}-{\widetilde{\mu}}_{\alpha
C}\right)$ implies a large compact and oblate configuration of the two-cluster
system. Information on the relative orientation of the clusters is obtained in
comparing the distribution of oscillation quanta for each cluster in the
different spatial directions NPA576 ; orient .

These conditions are achieved, determining the whole product of
$({\widetilde{\lambda}}_{1\alpha},{\widetilde{\mu}}_{1\alpha})\otimes({\widetilde{\lambda}}_{2\alpha},{\widetilde{\mu}}_{2\alpha})$,
using (8) and searching for the irrep that corresponds to a large compact
structure. The ${\widetilde{n}}_{0}={\widetilde{n}}_{p0}+{\widetilde{n}}_{n0}$
is the total minimal number of relative excitation quanta and
${\widetilde{n}}_{C}={\widetilde{n}}_{pC}+{\widetilde{n}}_{nC}$ is the total
forbiddenness.

The $\widetilde{n}_{\alpha C}$ are transferred to the clusters, where the
result does not depend on how the distribution is done. For example, when only
one cluster is excited, first the irrep of the number of one less valence
nucleons is determined and then coupled with the nucleon in the higher shell,
with $n_{\alpha C}$ quanta above the valence shell. The important criterion is
that at the end one finds a combination of irreps which couple to the final
irrep in the united cluster. This is a direct but rather cumbersome procedure
and how to do it will be illustrated in the applications.

More restrictions have to be implemented, in order to avoid a too large, not
manageable list of irreps:

  * •

When coupling the cluster irreps, only those $\left(\lambda_{\alpha
C},\mu_{\alpha C}\right)$ irreps in the product are considered which couple to
the ground state of the united nucleus, for protons and neutron separately.

  * •

Of those, only the ones are considered with the largest eigenvalues of the
second order Casimir operator of ${\widetilde{S}U}(3)$, often the irrep with
the largest eigenvalue suffices. The main argument is that in deformed nuclei,
with a significant quadrupole-quadrupole interaction, only these selected
irreps contribute significantly at low energy.

  * •

In coupling protons and neutrons together, only the linear coupling is
considered, for reasons explained before.

  * •

The final list is, in general, still too large and one has to restrict to the
first few irreps with the larges eigenvalues of the second order Casimir
operator, an argument valid for dominant quadrupole-quadrupole interaction.

  * •

Convergence criterium: Convergence due to these restrictions have to be
verified by adding and/or restricting some irreps. If no sensible change is
observed, then the cut-off procedure is set.

This is a consistent procedure and provides a basis for the extension to heavy
nuclei: The sum of nucleons in the normal orbitals, of the two cluster, will
be always equal to the ones in the united nucleus, which is for example
ignored in cseh-algora . All procedures known from SACM can now be applied in
the same manner.

For completeness, we mention a different definition for the forbiddenness: In
order to be able to deal with heavy systems, the alternative definition was
proposed in cseh-algora ; cseh-scheid , where the relation of an irrep to its
deformation was exploited rowe1 . The criterion applied is as follows: If an
irrep of the list has a similar deformation as the one in the shell model
allowed irrep, it implies a less forbidden state than irreps with a larger
difference in the irreps. The reason why it works is that the deformation can
be related to $SU(3)$ irreps rowe1 ; TE  and comparing deformations is
equivalent to comparing the dimension of these irreps.

Therefore, the definition of forbiddenness used in cseh-scheid ; cseh-algora
is

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{1}{1+min\left[\sqrt{\Delta n_{1}^{2}+\Delta
n_{2}^{2}+\Delta n_{3}^{2}}\right]}~{}~{}~{},$ (16)

where $\Delta n_{i}=\mid n_{i}-n_{i,\xi}\mid$ and in contrast to $S$, as
defined in cseh-scheid ; cseh-algora , we use $F$ because the letter $S$ will
be used later exclusively for the spectroscopic factor. The index $i$ refers
to the spatial direction of the oscillation and $\xi$ to the several cluster
irreps allowed by the Pauli-exclusion principle. The $n_{i}$ is the number of
oscillation quanta in direction $i$. When all $\Delta n_{i}$ are zero then
$F=1$ and the irrep is allowed. If at least one of those numbers is different
from zero the irrep is partially forbidden and when $F=0$ it is completely
forbidden.

###  2.5 The structure of the Hamiltonian and the quadrupole transition
operator

The most general algebraic Hamiltonian has the same structure as for light
nuclei, save that the operators (number operator, quadrupole operator, etc.)
are substituted by their pseudo counter parts. How this mapping is achieved,
is explained in detail in draayer1 , where an operator $O$ in $SU(3)$ is
mapped to its counterpart $O$ $\approx$
$\kappa{\mbox{\boldmath$\widetilde{O}$}}$ and $\kappa$ has a simple
approximation, namely $\kappa=\frac{\eta+\frac{3}{2}}{\eta+\frac{1}{2}}$,
close to 1. The factor can be assimilated into the parameters of the SACM
Hamiltonian.

We restrict to the $SU(3)$ limit, which turns out to be sufficient, due to the
large deformation of the systems considered. This, however, will result in
zero $B(E2)$ transition rates between states belonging to different
${\widetilde{S}U}(3)$ irreps.

A simplified model Hamiltonian is selected, which has the following structure:

$H$ $\displaystyle=$
$\displaystyle\hbar\omega\mbox{\boldmath$\widetilde{n}$}_{\pi}+(a_{2}-a_{5}\Delta\mbox{\boldmath$\widetilde{n}$}_{\pi})\mathit{\widetilde{{\mbox{\boldmath$C$}}}}_{2}\left(\widetilde{\lambda},\widetilde{\mu}\right)$
(17)
$\displaystyle+t_{3}\left[\mathit{\widetilde{{\mbox{\boldmath$C$}}}}_{2}\left(\widetilde{\lambda},\widetilde{\mu}\right)\right]^{2}+t_{1}\mathit{\widetilde{{\mbox{\boldmath$C$}}}}_{3}\left(\widetilde{\lambda},\widetilde{\mu}\right)$
$\displaystyle+\left(a_{3}+a_{Lnp}\Delta\widetilde{{\mbox{\boldmath$n$}}}_{\pi}\right){\mbox{\boldmath$\widetilde{L}$}}^{2}+t_{2}{\mbox{\boldmath$\widetilde{K}$}}^{2}$

where
$\Delta\widetilde{{\mbox{\boldmath$n$}}}_{\pi}=\widetilde{{\mbox{\boldmath$n$}}}_{\pi}-({\widetilde{n}}_{0}-{\widetilde{n}}_{C})$,
$({\widetilde{n}}_{0}-{\widetilde{n}}_{C})$ being the total minimal number of
quanta required by the Pauli principle and the possible effects of the
forbiddeness are taken into account by ${\widetilde{n}}_{C}$. The moment of
inertia, contained in the factor of the total angular momentum operator,
$\widetilde{{\mbox{\boldmath$L$}}}^{2}$, may depend on the excitation in
$\tilde{n}_{\pi}$ (excited states change their deformation, corresponding to a
variable momentum of inertia). The ${\mbox{\boldmath$K$}}^{2}$ term serves to
split the degeneracy in ${\widetilde{\mbox{\boldmath$L$}}}$ within the same
$\widetilde{SU}(3)$ irrep. The first term of the $\widetilde{SU}(3)$
Hamiltonian, $\hbar\omega\mbox{\boldmath$\widetilde{n}$}_{\pi}$, contains the
linear invariant operator of the $\widetilde{U}_{R}(3)$ subgroup defining the
mean field, and the $\hbar\omega$ is fixed via $(45A^{-1/3}-25A^{-2/3})$ MeV
for light nuclei hw , which can also be used for heavy nuclei. For heavy
nuclei $\hbar\omega=41A^{-\frac{1}{3}}$ MeV is more common. The $A$ is the
mass number of the real nucleus and not the number
${\tilde{A}}=\widetilde{A}_{1}+\widetilde{A}_{2}$ of nucleons in the normal
orbitals for the united nucleus.

The
$\widetilde{{\mbox{\boldmath$C$}}}_{2}\left({\widetilde{\lambda}},{\widetilde{\mu}}\right)$
is the second order Casimir-invariant of the coupled $\widetilde{SU}(3)$
group, having contributions both from the internal cluster part and from the
relative motion. The
$\widetilde{{\mbox{\boldmath$C$}}}_{2}\left({\widetilde{\lambda}},{\widetilde{\mu}}\right)$
is given by:

$\displaystyle\mbox{\boldmath$\widetilde{C}$}_{2}(\widetilde{\lambda},\widetilde{\mu})$
$\displaystyle=$ $\displaystyle
2\mbox{\boldmath$\widetilde{Q}$}^{2}+\frac{3}{4}\mbox{\boldmath$\widetilde{L}$}^{2},$
$\displaystyle\rightarrow$
$\displaystyle\left(\widetilde{\lambda}^{2}+\widetilde{\lambda}\widetilde{\mu}+\widetilde{\mu}^{2}+3\widetilde{\lambda}+3\widetilde{\mu}\right),$
$\widetilde{Q}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\widetilde{Q}$}_{C}+\mbox{\boldmath$\widetilde{Q}$}_{R},$
$\widetilde{L}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\widetilde{L}$}_{C}+\mbox{\boldmath$\widetilde{L}$}_{R},$
(18)

where $\widetilde{Q}$ and $\widetilde{L}$ are the total quadrupole and angular
momentum operators, respectively, and $R$ refers to the relative motion. The
eigenvalue of
$\mbox{\boldmath$\widetilde{C}$}_{2}(\widetilde{\lambda},\widetilde{\mu})$ is
also indicated. The relations of the quadrupole and angular momentum operators
to the $\widetilde{C}^{(1,1)}_{2m}$ generators of the $\widetilde{SU}(3)$
group, expressed in terms of $\widetilde{SU}(3)$-coupled $\pi$-boson creation
and annihilation operators, are escher :

$\displaystyle\mbox{\boldmath$\widetilde{Q}$}_{k,2m}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\widetilde{C}^{(1,1)}_{k2m},$
$\displaystyle\mbox{\boldmath$\widetilde{L}$}_{k1m}$ $\displaystyle=$
$\displaystyle\widetilde{C}^{(1,1)}_{k1m},$
$\displaystyle\mbox{\boldmath$\widetilde{C}$}^{(1,1)}_{lm}$ $\displaystyle=$
$\displaystyle\sqrt{2}\left[\mbox{\boldmath$\pi$}^{\dagger}\otimes\mbox{\boldmath$\pi$}\right]^{(1,1)}_{lm}.$
(19)

The quadrupole electromagnetic transition operator is defined as

$\displaystyle\mbox{\boldmath$T$}_{m}^{(E2)}$ $\displaystyle=$
$\displaystyle\sum_{\gamma}e_{\gamma}^{(2)}\mbox{\boldmath$Q$}^{(2)}_{\gamma,m}~{}~{}~{},$
(20)

where $e_{\gamma}^{(2)}$ is the effective charge of the contribution to the
quadrupole operator, coming from the cluster $\gamma$ = $C_{1}$, $C_{2}$ and
from the relative motion $R$. The effective charges are determined as
explained in great detail in fraser-2012 .

###  2.6 Spectroscopic factors

A successful parametrization of the spectroscopic factor, within the SACM for
light nuclei, is given in specfac-draayer :

$\displaystyle S=$ $\displaystyle e^{{\cal A}+Bn_{\pi}+C{\cal
C}_{2}(\lambda_{1},\mu_{1})+D{\cal C}_{2}(\lambda_{2},\mu_{2})+E{\cal
C}_{2}(\lambda_{c},\mu_{c})}$ $\displaystyle\times e^{F{\cal
C}_{2}(\lambda,\mu)+G{\cal C}_{3}(\lambda,\mu)+H\Delta n_{\pi}}$
$\displaystyle\mid\langle(\lambda_{1},\mu_{1})\kappa_{1}L_{1},(\lambda_{2},\mu_{2})\kappa_{2}L_{2}\mid\mid(\lambda_{C},\mu_{C})\kappa_{C}L_{C}\rangle_{\varrho_{C}}$
$\displaystyle\cdot\langle(\lambda_{C},\mu_{C})\kappa_{C}L_{C},(n_{\pi},0)1l\mid\mid(\lambda,\mu)\kappa
L\rangle_{1}\mid^{2}~{}~{}~{},$ (21)

where the $\varrho$-numbers refer to multiplicities in the coupling to $SU(3)$
irreps and the $\kappa$’s to the multiplicities to the reduction to $SO(3)$.
The parameters were adjusted to theoretically exactly calculated spectroscopic
factors within the p- and sd-shell, using the $SU(3)$ shell model draayer2 ,
with an excellent coincidence. For the good agreement, the factor depending on
the $SU(3)$-isoscalar factors turns out to be crucial.

For heavy nuclei, spectroscopic factors are poorly or not at all known
experimentally. Therefore, we have to propose a simplified manageable ansatz,
compared to (21), including the forbiddenness and, if possible, parameter
free.

In what follows, we will propose an expression for the spectroscopic factor
which is motivated by the one used for nuclei in the p- and sd-shell. As in
(21) the expression is divided into two factors, the first one is an
exponential factor and the second one of pure geometrical origin specfac-
draayer , an expression in terms of coupling coefficients. The second factor
is maintained, because it refers to coupling of $SU(3)$ irreps only.

The first exponential factor deserves more explanation: As argued in specfac-
draayer , this term is the result of the relative part of the wave-function,
which for zero angular momentum is proportional to $e^{-aR^{2}}\sim
e^{-a\frac{\hbar}{\mu\omega}n_{\pi}}$, where $R$ is the relative distance of
the two clusters (though, an $e^{-aR}$ ansatz would be more appropriate, but
the clusters are defined in the harmonic oscillator picture and we stay to it
for consistency) and $a$ has units of $\rm{fm}^{-2}$. The $\mu$ is the reduced
mass. Let us restrict to the minimum value $n_{0}$ of $n_{\pi}$. Using the
relation of $r_{0}=\sqrt{\frac{\hbar}{\mu\omega}n_{0}}$ geom , where $r_{0}$
is the average, minimal distance between the clusters, and taking into account
that for this case $R=r_{0}$, we obtain $e^{-\mid B\mid n_{0}}$, with $\mid
B\mid=a\frac{\hbar}{\mu\omega}$ and $B<0$. When the wave function acquires the
value $e^{-1}$ it results in the relation $\mid B\mid=\frac{1}{n_{0}}$. For
the nuclei in the sd-shell, the adjustment of the parameters was done for
cases with $n_{0}=8$, which corresponds according to this estimation to
$B\approx-0.13$. This has to be compared to the value $-0.36$ as obtained in
the fit in specfac-draayer , i.e., it is only an approximation which gives at
least the correct order. The most important part of (21) is the factor
depending on the $SU(3)$ isoscalar factors, which are responsible for the
relative changes, while the influence of the exponential factor is not
dominant for the relative numerical values of the spectroscopic factors.
Furthermore, when ratios of spectroscopic factors are used, the exponential
contribution cancels for states in the $0\hbar\omega$ shell. Also when the
$(n_{0}-nc)$ is large, the corrections for $\Delta n_{\pi}$ of the order of
one will be negligible. We do not see any possibility to estimate the
parameter $\cal{A}$ in the exponential factor, which represents a
normalization. The other terms in the exponential factor represent corrections
to the inter-cluster distance, because they correspond to deformation effects
and the parameters in front turned out to be consistently small, thus, they
can for the moment be neglected.

In light of the above estimation and discussion, for heavy nuclei we propose a
similar expression as in (21), but due to the non-availability of a sufficient
number of spectroscopic factor values (or none at all) for heavy nuclei, we
propose the following simplified expression:

$\displaystyle S=$ $\displaystyle e^{{\tilde{\cal
A}}+{\tilde{B}}({\tilde{n}}_{0}-{\tilde{n}}_{C}+\Delta{\tilde{n}}_{\pi})}$
$\displaystyle\mid\langle({\tilde{\lambda}}_{1},{\tilde{\mu}}_{1}){\tilde{\kappa}}_{1}{\tilde{L}}_{1},({\tilde{\lambda}}_{2},{\tilde{\mu}}_{2}){\tilde{\kappa}}_{2}{\tilde{L}}_{2}\mid\mid({\tilde{\lambda}}_{C}+{\widetilde{\lambda}}_{0},{\tilde{\mu}}_{C}+{\widetilde{\mu}}_{0}){\tilde{\kappa}}_{C}{\tilde{L}}_{C}\rangle_{\varrho_{C}}$
$\displaystyle\cdot\langle({\tilde{\lambda}}_{C}+{\widetilde{\lambda}}_{0},{\tilde{\mu}}_{C}+{\widetilde{\mu}}_{0}){\tilde{\kappa}}_{C}{\tilde{L}}_{C},({\tilde{n}}_{\pi},0)1{\tilde{l}}\mid\mid({\tilde{\lambda}},{\tilde{\mu}}){\tilde{\kappa}}{\tilde{L}}\rangle_{1}\mid^{2}~{}~{}~{}.$
(22)

The $n_{\pi}$ in the exponential factor was substituted by
$[(\tilde{n}_{0}-\tilde{n}_{C})+\Delta n_{\pi}]$. The
$({\tilde{n}}_{0}-{\tilde{n}}_{C})$ is the number of relative oscillation
quanta in $0\hbar\omega$ ($\tilde{n}_{C}$ was added to the excitation of the
clusters).

The parameter is estimated as
${\tilde{B}}=-\frac{1}{{(\tilde{n}}_{0}-\tilde{n}_{C})}$. Because we can not
determine the parameter ${\tilde{\cal A}}$, as a consequence only
spectroscopic factors divided by the exponential factor $e^{{\widetilde{A}}}$
are listed. An additional dependence on $\tilde{n}_{C}$ is contained in the
product of reduced coupling coefficients, with the appearance of
${\widetilde{\lambda}}_{0}$ and ${\widetilde{\mu}}_{0}$ (see the definition in
(15)). ${\widetilde{n}}_{0}$ and ${\widetilde{n}}_{C}$ refer to the values
within the parent nucleus. In general, in (22) further parameters can be
added, as for light nuclei, which have though to be adjusted.

With this choice, the spectroscopic factor is parameter free, except for an
overall normalization, which does not play a role when only ratios are of
interest.

##  3 Applications

In this section we apply the pseudo-SACM to three sample systems. The first is
${}^{236}_{92}$U144 $\rightarrow$ ${}^{210}_{82}$Pb128+${}^{26}_{10}$Ne16, the
second one is ${}^{224}_{88}$Ra136 $\rightarrow$
${}^{210}_{82}$Pb128+${}^{14}_{6}$C8 and the third one is ${}^{236}_{92}$U144
$\rightarrow$ ${}^{146}_{54}$Xe92+${}^{90}_{38}$Sr52, which appears in the
fission channel of 236U. In the first two cases, the forbiddeness is small.
They serve to illustrate how to add the quanta to the clusters and how to
determine the final cluster irrep. The importance of the forbiddeness is
significant in the third case and it will be much harder to determine the
cluster irreps. This sequence also shows that the larger the lighter cluster
is, the larger the forbiddenness becomes.

For illustrative reasons, only the $\widetilde{SU}(3)$ dynamical symmetry
limit is considered, i.e., the united nucleus must be well deformed. The
mixing of $SU(3)$ irreps will be investigated in a future publication. The
examples also serve to illustrate the applicability of the model and on how to
deduce the quantum numbers. However, some transitions will be forbidden due to
$SU(3)$ selection rules.

Very useful is the Table 1, where the number of nucleons in a given shell
$\eta$ is related to the one of the pseudo-shell number $\widetilde{\eta}$. In
the last columns the accumulated number of oscillation quanta within the
$\widetilde{SU}(3)$ shell model is listed.

$\eta$ | No. of nucleons | $\widetilde{\eta}$ | No. of nucleons | acum. No. of quanta
---|---|---|---|---
0 | 2 | - | - | -
1 | 6 | 0 | 2 | 0
2 | 12 | 1 | 6 | 6
3 | 20 | 2 | 12 | 30
4 | 30 | 3 | 20 | 90
5 | 42 | 4 | 30 | 210
6 | 56 | 5 | 42 | 420
Table 1:  Table, listing the number of particles in the shell number $\eta$
and the pseudo-shell number $\widetilde{\eta}$, including the spin degree of
freedom. The last column lists the accumulated number of quanta of the
$\widetilde{SU}(3)$, when each shell is full up to the valence one. The final
total number of quanta is obtained, by adding to the number of quanta, reached
up to the last closed shell, the number of quanta of the nucleons in the
valence shell.

In this section, we explain trough the examples in great detail the
calculations which lead us to the cluster irreps and with the relative motion
to the ground state irreps in the proton and neutron part. This requires a lot
of cumbersome counting and determination of irreps, especially when relative
oscillation quanta from the forbiddenness are added to the clusters. We
apologize for that and ask the reader to be patient. If she (he) is not
interested in the details, she (he) just can skip this part and jump to the
final numbers of the cluster irreps.

###  3.1 ${}^{236}_{92}$U144 $\rightarrow$
${}^{210}_{82}$Pb128+${}^{26}_{10}$Ne16

$\widetilde{\eta}$ | 0 | 1 | 2 | 3 | 4 |  | $N$
---|---|---|---|---|---|---|---
${}^{128}_{46}Pd_{82}$ | 2 | 6 | 12 | 20 | 6 | 2(0)+6+2(12)) |
|  |  |  |  |  | +3(20)+4(6 | 114
${}^{112}_{40}Zr_{72}$ | 2 | 6 | 12 | 20 | 0 | 2(0)+6+2(12) |
|  |  |  |  |  | +3(20) | 90
${}^{14}_{6}C_{8}$ | 2 | 4 | 0 | 0 | 0 | 2(0)+4 | 4
Table 2: An example on how to determine the number of oscillation quanta ($N$) in the system derived from 236U $\rightarrow$ 210Pb+26Ne, with the help of Table 1. The same is applied for the two other sample cases, where Table 1 is very helpful for counting the number of relative excitation quanta. When one cluster is excited by $\widetilde{n}_{C}$ quanta, this number has still to be added.  Parameter | system a | system b | system c
---|---|---|---
$\hbar\omega$ | $6.63$ | $6.73$ | $6.63$
$a_{2}$ | $-0.020841$ | $-0.0082194$ | $-0.023251$
$a_{3}$ | $0.0087042$ | $0.0099524$ | $0.0067124$
$t_{1}$ | $6.698\times 10^{-5}$ | $-0.00061005$ | $-2.74\times 10^{-5}$
$t_{2}$ | 0.40 | $0.2205$ | $0.0364275$
$a_{5}$ | -0.0081127 | $0.1$ | $0.0098151$
$a_{L}$ | $-0.0012576$ | $0.0025050$ | $0.00073054$
$a_{L_{n_{p}}}$ | $6.55\times 10^{-3}$ | $0.00025837$ | $-0.00018069$
$t_{3}$ | $-1.420\times 10^{-7}$ | $1.10\times 10^{-5}$ | $5.10\times 10^{-7}$
Table 3: Non-zero parameter values for system a: 236U $\rightarrow$ 210Pb +
26Ne; system b: 224Ra $\rightarrow$ 210Pb + 14C and system c: 236U
$\rightarrow$ 148Xe + 90Sr.

Figure 1:  Spectrum of 236U, described by the clusterization 210Pb+26Ne. Only
states up to angular momentum 6 are depicted. The theoretical spectrum (left
panel) is compared to experiment (right panel). Below each rotational band the
content of the number of $\pi$ bosons ($n_{\pi}$) and the
${\widetilde{S}U}(3)$ irrep is indicated.

As explained above, the protons and neutrons are treated separately and the
nucleons in each sector are filled into the Nilsson diagram from below, at the
deformation value $\epsilon_{2}=0.200$ ($\beta_{2}=0.215$) nix-tables . The
$\hbar\omega=6.63$ MeV.

For 236U, the united nucleus, we obtain 46 protons in the normal orbitals and
the valence shell is $\widetilde{\eta}_{p}=4$ with 6 valence protons. The
ground state $\widetilde{SU}(3)$ irrep for the proton part is
$(\tilde{\lambda},\tilde{\mu})_{p}=(18,0)_{p}$, while for the neutrons there
are 82 particles in normal orbitals with 12 in the $\widetilde{\eta}_{n}=5$
valence shell, giving the ground state irrep
$(\tilde{\lambda},\tilde{\mu})_{n}=(36,0)_{n}$. These two irreps can be
coupled to the total one for 236U, namely
$(\tilde{\lambda},\tilde{\mu})=(54,0)$. The determination of the ground state
irrep is necessary for the evaluation of the forbiddenness (see (14)).

These considerations have to be repeated for the two clusters involved, with
210Pb being the largest cluster and 14C the lightest one. For 210Pb, filling
the protons into the Nilsson diagram, at the same deformation as for the
united nucleus, we obtain 40 protons in normal orbitals, where the valence
shell is ${\widetilde{\eta}}=3$ and closed, thus the corresponding irrep is
$(0,0)^{\rm Pb}_{p}$. For the neutrons one has 72 in normal orbitals with 2
neutrons in the ${\widetilde{\eta}}_{n}=5$ pseudo-shell. The corresponding
irrep is $(10,0)^{\rm Pb}_{n}$.

$J_{k}^{P}$ |  236U $E_{{\rm exp}}$ [MeV] |  224Ra $E_{{\rm exp}}$ [MeV]
---|---|---
$0_{2}^{+}$ | 0.919 | 0.916
$2_{1}^{+}$ | 0.045 | 0.084
$2_{2}^{+}$ | 0.958 | -
$2_{3}^{+}$ | 0.960 | 0.993
$3_{1}^{+}$ | 1.002 | -
$4_{1}^{+}$ | 0.150 | 0.251
$6_{1}^{+}$ | 0.210 | 0.479
$1_{1}^{-}$ | 0.688 | 0.216
$1_{2}^{-}$ | 0.967 | -
$3_{1}^{-}$ | 0.744 | 0.290
$5_{1}^{-}$ | 0.848 | 0.433
$J_{i}^{P}\rightarrow J_{f}^{P}$ |  236: $B(E2)$ [WU] |  224Ra: $B(E2)$ [WU]
$2_{1}\rightarrow 0_{1}$ | 250. | 99.
$4_{1}\rightarrow 2_{1}$ | 357. | 144
Table 4:  Experimental data used in the fit of the parameters of the model Hamiltonian. The second column lists the data used for 236U and the third column for 224Ra. If no data are mentioned (dashed sign), the experimental value is not used in the fit.  $J_{k}^{P}$ |  236U-1 (th) |  224Ra (th) |  236U-2
---|---|---|---
$0_{1}^{+}$ | 0.0 | 0.0 | $0.00403$
$0_{2}^{+}$ | 0.0 | 0.0 | $0.0000356$
$2_{1}^{+}$ | 0.0 | 0.0 | $0.00384$
$2_{2}^{+}$ | 0.0 | 0.0 | $4.79\times 10^{-7}$
$4_{1}^{+}$ | 0.0 | 0.0 | $0.00346$
$4_{2}^{+}$ | 0.0 | 0.0 | $7.00\times 10^{-6}$
$1_{1}^{-}$ | 0.0 | 0.0 | $1.04\times 10^{-5}$
$2_{1}^{-}$ | 0.0 | 0.0 | $1.04\times 10^{-5}$
$3_{1}^{-}$ | 0.0 | 0.0 | $6.03\times 10^{-5}$
Table 5:  Some spectroscopic factors of low lying states, divided by
$e^{\widetilde{A}}$. In the first column the state quantum numbers are
tabulated. The values of the spectroscopic factor for 236U containing the
210Pb cluster, 224Ra and 236U containing the 146Xe cluster are in the second,
third and fourth column, respectively. Only values which are greater than
$10^{-8}$ are listed. As seen, only the last system shows significant
deviations from zero.

Using the numbers just deduced, the system ${}^{236}_{92}$U $\rightarrow$
${}^{210}_{82}$Pb+${}^{26}_{10}$Ne can be viewed within the
$\widetilde{SU}(3)$ description as a ${}^{128}_{46}\widetilde{\rm{Pd}}$
$\rightarrow$ ${}^{112}_{40}\widetilde{\rm{Zr}}$ \+
${}^{16}_{6}\widetilde{\rm{C}}$ cluster system, obtained by counting the
protons and neutrons in the normal orbitals. Of course, these so-called
pseudo-nuclei are only schematic in nature.

The Tables 1 and 2 serve to illustrate on how to determine the total number of
quanta in each cluster and the united nucleus, for the particular case
considered. For the other cases treated in this manuscript, it will be
similar.

The minimal number of quanta, which have to be added in the proton part, is 20
corresponding to a $(20,0)_{pR}$ irrep in the relative part. For the neutron
part, this number is 40, i.e., an irrep $(40,0)_{nR}$.

In the next step, the proton part of the clusters are coupled with the
relative part of the proton section of the united nucleus. The same is done
for the neutrons. For the proton part, the product
$(0,0)_{p}\otimes(0,2)_{p}\otimes(20,0)_{pR}$ (the index $pR$ refers to the
relative motion of the protons) contains the proton irrep $(18,0)_{p}$ of the
united nucleus, thus, the forbiddenness for the proton part is zero. The
situation is different for the neutron part: The product
$(10,0)_{n}\otimes(4,0)_{n}\otimes(40,0)_{nR}$ does not contain (36,0), which
is the irrep in the united nucleus. This indicates that the clusters have to
be excited, thus, the forbiddenness is different from zero. Using the formula
(14) we obtain a forbiddenness of $\widetilde{n}_{C}=2$. The excitation of the
clusters is achieved, changing as one possibility the irrep of 26Ne from
$(4,0)_{n}$ to $(6,0)_{n}$. The relative part is now reduced by two quanta,
leaving $(38,0)_{nR}$ (the index $nR$ refers to the relative motion of the
neutrons). With this change, the product
$(10,0)_{n}\otimes(6,0)_{n}\otimes(38,0)_{nR}$ now contains the dominant irrep
for neutrons in 236U. This is verified by Eq. (14). Distributing the
excitation quanta in a different manner 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 parameters obtained are listed in Table 3, second column. With these
parameters, the spectrum calculated is depicted in Figure 1. The calculated
B(E2)-transition values are listed in Table 6, second (theory) and third
(experiment) column.

As can be noted, the agreement to experiment is good and shows the
effectiveness of the pseudo-SACM to describe the collective structure of heavy
nuclei.

Next, we calculated some spectroscopic factors, listed in Table 5, second
column. The Equation (22) was used with the approximation of the parameter
$\widetilde{B}$ as $\left(-\frac{1}{{\tilde{n}}_{0}-n_{C}}\right)$. The total
number of relative oscillation quanta for the system under study is
${\tilde{n}}_{0}=60$ and ${\widetilde{n}}_{C}=2$, thus,
$\widetilde{B}\approx-0.0172$ and the exponential factor in (22) acquires the
form
$e^{\widetilde{A}-0.0172({\tilde{n}}_{0}-n_{c}+\Delta{\tilde{n}}_{\pi})}$
$\approx$
$(0.983)^{({\tilde{n}}_{0}-n_{c}+\Delta{\tilde{n}}_{\pi})}e^{\widetilde{A}}$.

The factor $e^{\widetilde{A}}$ is unknown and, as explained before, the
spectroscopic factors values can be found in Table 5, divided by
$e^{{\widetilde{A}}}$. As observed, the spectroscopic factors to
$\Delta{\tilde{n}}_{\pi}=1$ are suppressed, with a value smaller than
$10^{-8}$ considered to be zero.

###  3.2 ${}^{224}_{88}$Ra136 $\rightarrow$ ${}^{210}_{82}$Pb128 \+
${}^{14}_{6}$C8

$J_{i}^{P_{i}}\rightarrow J_{f}^{P_{f}}$ | a | b | c |  $U$-exp |  $Ra$-exp
---|---|---|---|---|---
$2^{+}_{1}\rightarrow 0^{+}_{1}$ | $251$ | $99$ | $250$ | $250\pm 10$ | $99\pm 3$
$2^{+}_{2}\rightarrow 0^{+}_{1}$ | $0$ | $0.321$ | $0$ | - |
$4^{+}_{1}\rightarrow 2^{+}_{1}$ | $357$ | $141$ | $357$ | $357\pm 23$ | $141\pm 7$
$4^{+}_{2}\rightarrow 2^{+}_{1}$ | $0$ | $0.0312$ | $0$ | - | -
$4^{-}_{1}\rightarrow 2^{-}_{1}$ | $309$ | $120$ | $3.554$ | - | -
$4^{-}_{2}\rightarrow 2^{-}_{1}$ | $0.$ | $0.00169$ | $337$ | - | -
$6^{+}_{1}\rightarrow 4^{+}_{1}$ | $391$ | $155$ | $390$ | $385\pm 22$ | $157\pm 13$
$2^{+}_{1}\rightarrow 0^{+}_{2}$ | $0$ | $0.115$ | $0$ | - | -
$2^{+}_{2}\rightarrow 0^{+}_{2}$ | $269$ | $142$ | $16.13$ | - | -
$4^{+}_{1}\rightarrow 2^{+}_{2}$ | $0$ | $0.00472$ | $0$ | - | -
$4^{+}_{2}\rightarrow 2^{+}_{2}$ | $383$ | $58.94$ | $21$ | - | -
$4^{-}_{1}\rightarrow 2^{-}_{2}$ | $0$ | $0.974$ | $0$ | - | -
$4^{-}_{2}\rightarrow 2^{-}_{2}$ | $368$ | $6.25\times 10^{-6}$ | $0$ | - |
Table 6: Theoretical $B(E2)$-transition values of the system a: 236U
$\rightarrow$ 210Pb + 26Ne; system b: 224Ra $\rightarrow$ 210Pb + 14C and
system c: 224U $\rightarrow$ 146Xe + 90Sr. The last two columns list the
experimental values for 236U and 224Ra, respectively. The unit is in WU and
the theoretical values are compared to available experimental data brook .
Note that in 238U most of the inter-band transitions are zero. This is due to
the fact that the ground state irrep is (54,0) and that we are working in the
$SU(3)$ limit, thus, there are no transitions between states in distinct
$SU(3)$ irrep. This changes for 224Ra, whose ground state irrep in the $SU(3)$
limit is (48,4), which contains several $K$ bands ($K=0,2,4$) and allows now a
transition between states from different bands at low energy.

Figure 2:  Spectrum of 224Ra, described by the clusterization 210Pb+14C. Only
states up to angular momentum 6 are depicted The theoretical spectrum (left
panel) is compared to experiment (right panel).

As in the former section, the protons and neutrons are treated separately,
where the nucleons are filled into the Nilsson diagram from below, at the
deformation value $\epsilon_{2}=0.150$ ($\beta_{2})=0.164$) nix-tables . The
$\hbar\omega=6.73$ MeV.

For 224Ra, the united nucleus, we obtain 46 protons in the normal orbitals and
the valence shell is ${\widetilde{\eta}}_{p}=4$ with 6 valence protons. The
$\widetilde{SU}(3)$ irrep is $(\tilde{\lambda},\tilde{\mu})_{p}=(18,0)^{\rm
Ra}_{p}$, while for the neutrons we have 80 normal particles with 10 in the
${\widetilde{\eta}}_{n}=5$ valence shell, giving
$(\tilde{\lambda},\tilde{\mu})^{\rm Ra}_{n}=(30,4)^{\rm Ra}_{n}$. These two
irreps can be coupled to the ground state irrep for 224Ra, namely
$(\tilde{\lambda},\tilde{\mu})=(48,4)$.

The compilation of the valence shells, their nucleon content and the
corresponding irreps for 236U can be found in the former subsection.

The light cluster 14C is added on top of the heavy cluster. We have then 6
protons and 8 neutrons in normal orbitals, which gives $(0,2)^{\rm C}_{p}$
(two holes in the $\widetilde{p}$ shell) and $(0,0)^{\rm C}_{n}$ (closed
$\widetilde{p}$ shell).

Using the numbers just deduced, the system ${}^{224}_{88}$Ra136 $\rightarrow$
${}^{210}_{82}$Pb128 \+ ${}^{14}_{6}$C8 can be viewed within the
$\widetilde{SU}(3)$ description as a ${}^{126}_{46}\widetilde{\rm{Ru}}_{80}$
$\rightarrow$ ${}^{112}_{40}\widetilde{\rm{Zr}_{72}}$ \+
${}^{14}_{6}\widetilde{\rm{C}}_{8}$ cluster system. Of course, these so-called
pseudo-nuclei only serve for illustration. In what follows, we will continue
to use the notation for the real nuclei.

The minimal number of quanta which have to be added in the proton part of the
united nucleus is 20, which is the result of counting the difference of the
oscillation in the united pseudo-nucleus to the sum of the oscillation quanta
of the two pseudo-clusters. This corresponds to a $(20,0)_{Rp}$ irrep in the
relative part. For the neutron part the difference in the oscillation quanta
is 34 and, thus, the irrep of the relative motion is $(34,0)_{Rn}$

For the proton part, the product $\left[(0,0)^{\rm
Pb}_{p}\otimes(0,2)^{C}_{p}\right]$ $\otimes$ $(20,0)_{Rp}$ does lead to the
final ground state irrep $(18,0)^{\rm Ra}_{p}$ and, therefore, the
forbiddenness is zero. For the neutron part, however, this is no longer the
case: The product $\left[(0,0)^{\rm Pb}_{n}\otimes(0,0)^{\rm C}_{n}\right]$
$\otimes$ $(34,0)_{Rn}$ does not contain the ground stte irrep $(30,4)^{\rm
Ra}_{\nu}$. Applying the formula for the forbiddenness gives $n_{C}^{n}=2$.
These two quanta are subtracted from the relative motion, giving $(32,0)_{Rn}$
and are added to the Pb-cluster, as one possibility (any other leads to the
same final result). The irrep used is obtained by subtracting one neutron from
the $\widetilde{\eta}_{n}=5$ and exciting one neutron to the
$\widetilde{\eta}_{n}$ = (5+2,0) = (7,0). The neutrons in the valence shell
provide the irrep (5,0) (only one valence neutron left) and the product with
(7,0) contains the $(8,2)^{\rm Pb}_{n}$ irrep. The product with the relative
motion is sufficient, because the $C$ pseudo-nucleus has the scalar neutron
irrep $(0,0)^{\rm C}_{n}$. The product $(8,2)^{\rm Pb}_{n}\otimes(32,0)_{Rn}$
contains the ground state irrep $(30,4)^{\rm Ra}_{n}$.

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 parameters obtained are listed in Table 3, third column. With these
parameters, the spectrum calculated is depicted in Figure 2. The calculated
B(E2)-transition values are listed in Table 6, third (theory) and sixth
(experiment) column.

As can be noted, the agreement to experiment is good and shows also in this
example the effectiveness of the pseudo-SACM to describe the collective
structure of heavy nuclei.

Next, we calculated some spectroscopic factors for 224Ra, listed in Table 5,
third column. The total number of relative oscillation quanta for the system
under study is ${\tilde{n}}_{0}=40$ ($n_{C}=2$), thus
$\widetilde{B}\approx-0.026$ and the exponential factor in (22) acquires the
form

$\displaystyle
e^{\widetilde{A}-0.026({\tilde{n}}_{0}-n_{c}+\Delta{\tilde{n}}_{\pi})}$
$\displaystyle\approx$
$\displaystyle(0.974)^{({\tilde{n}}_{0}-n_{c}+\Delta{\tilde{n}}_{\pi})}e^{\widetilde{A}}~{}~{}~{}.$
(23)

The factor $e^{\widetilde{A}}$ is unknown and thus in Table 5, the
spectroscopic factors were divided by $e^{{\widetilde{A}}}$.

###  3.3 ${}^{236}_{92}$U144 $\rightarrow$
${}^{146}_{54}$Xe92+${}^{90}_{38}$Sr52

Figure 3:  Spectrum of 236U, described by the clusterization 146Xe+90Sr. Only
states up to angular momentum are depicted. The theoretical spectrum (left
panel) is compared to experiment (right panel).

Consulting the table of nix-tables  the deformation of 236U is
$\epsilon_{2}=0.200$ ($\beta_{2}=0.215$), further $\hbar\omega=6.63$MeV. The
number of normal and unique protons and neutrons is given in subsections 3.1,
therefore, we will resume the numbers for 146Xe and 90Sr, only.

Counting for 146Xe the number of protons in the normal orbitals, we obtain 4
in the pseudo-shell $\widetilde{\eta}_{p}$=3. Taking into account the lower
shells, we obtain for the total number of protons in normal orbitals
$\widetilde{Z}$=24. As the leading irrep, obtained in the reduction of
$U(\frac{1}{2}\left(\widetilde{\eta}_{p}+1\right)\left(\widetilde{\eta_{p}}+2\right))$
$\supset$ $\widetilde{SU}(3)$ we have $(8,2)_{p}$. For the neutron part, there
are 8 neutrons in the $\widetilde{\eta}_{n}$ = 4 shell, corresponding to
$(18,4)_{n}$ as the leading irrep. The total number of normal neutrons is 48.

The lighter cluster, 90Sr, is put on top of the 146Xe cluster. Counting the
difference of the normal protons and neutron in the 236U nucleus to the 146Xe
nucleus, we obtain 22 normal protons and 34 normal neutrons for 90Sr.
Distributing them in the $\widetilde{SU}(3)$ shell model, we have 2 valence
protons in the $\widetilde{\eta}_{\pi}$=3 shell and 14 neutrons in the same
pseudo-shell, which correspond to 6 holes. The leading irreps are respectively
$(6,0)_{p}$ and $(0,12)_{n}$.

Counting the difference in the oscillation quanta in the proton sector between
the 236U nucleus and the sum of the two clusters, we obtain 36 quanta.
However, when we couple the two cluster irreps and then with the relative
motion irrep $(36,0)_{R}$, the $(18,0)^{U}_{p}$ irrep of 236U cannot be
reached, which is a clear indication that the proton part requires the use of
the forbiddenness. Using the formula for the forbiddenness we obtain
$n^{p}_{C}=10$, demonstrating the importance of the concept of forbiddenness.
The 10 oscillation quanta are to be added to the clusters, choosing a path
which is easier to follow: As one possibility (all others lead to the same
result) we distribute 6 to the large and 4 to the light cluster (such that in
each an even number is added). For the proton part, in 146Xe the irrep of one
proton less in the valence shell is (7,1) (instead of the former (8,2)). One
proton is excited to $\widetilde{\eta}_{p}$ = (3+6,0) = (9,0). The product of
$(7,1)\otimes(9,0)$ contains the irrep $(14,2)^{\rm Xe}$. In Sr the irrep with
one nucleon less in the valence shell $\widetilde{\eta}_{p}=3$ is (3,0). One
nucleon is excited the $\widetilde{\eta}_{p}$ = 3+4 =7, i.e., with the irrep
(7,0). In the product of $(3,0)\otimes(7,0)$ the irrep $(10,0)^{\rm Sr}$
appears. In the final coupling with the radial irrep $(36-10,0)_{pR}$ =
$(26,0)_{pR}$ we obtain $(14,2)^{\rm Xe}_{p}\otimes(10,0)^{\rm Sr}_{p}$, which
contains (4,12) and coupled with $(26,0)_{pR}$ leads to the final proton irrep
$(18,0)^{U}_{p}$ and, thus, we reached our objective for the proton part

The same procedure has to be applied for the neutron part. Counting the
difference in the oscillation quanta in the neutron sector between the 236U
nucleus and the sum of the two clusters, we obtain 76 quanta. However, when we
couple the two cluster irreps and then with the relative motion irrep
$(76,0)_{nR}$, we cannot reach the $(36,0)^{U}_{n}$ irrep of 236U, which is a
clear indication that the neutron part requires the use of the forbiddenness.
Using the formula for the forbiddenness we obtain $n^{\nu}_{C}=48$, again
showing that the concept of forbiddenness is very important. The 48
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 $n_{C}^{\nu}$ = 48 between the clusters. This is the reason why
we now follow this path. Before, in Sr there were 6 holes, now there will be
7, because one neutron is excited to the $\widetilde{\eta}_{n}$ = 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, $(2,11)\otimes(51,0)$,
a large list is obtained were we choose the most compact irrep, namely (38,2).
The reason to choose the most compact irrep is that this alone leads in the
product with the relative motion to the smallest irreps possible. Therefore,
we consider the product $\left[(38,2)\otimes(18,4)^{\rm Xe}_{n}\right]$
$\otimes$ $(28,0)_{nR}$ $\rightarrow$, which leads to the possible combination
of $(22,14)\otimes(28,0)_{nR}$, where (22,14) appears in the product
$(38,2)\otimes(18,4)$. The (22,14) was chosen, because taking the product with
the relative irrep $(28,0)_{\nu R}$ it contains the final irrep $(36,0)^{\rm
U}_{\nu}$.

The cluster irreps for Xe and Sr are obtained by couple linearly for each
cluster the proton and neutron irrep. Using the above results of the
compilations, we get $(48,2)^{\rm Sr}$ and $(32,6)^{\rm Xe}$, once coupled
contains the irrep (30,30), which coupled again with the relative motion
(54,0), where 54 is the sum of the relative motions in the proton (26) and the
neutron part (28). The final result contains the ground state irrep of 236U,
namely (54,0).

Counting only the normal nucleons, the system ${}^{236}_{92}$U144
$\rightarrow$ ${}^{146}_{46}$Xe92 \+ ${}^{90}_{22}$Sr52 can be viewed as the
system ${}^{128}_{68}$$\widetilde{{\rm Pd}}$82 $\rightarrow$
${}^{72}_{24}$$\widetilde{{\rm Cr}}$48 \+ ${}^{56}_{22}$$\widetilde{\rm
Ti}$34. The agreement to experiment is satisfactorily, as can be see by
consulting Figure 3 for the spectrum and Table 6 for the transition values.
Comparing to the 236U $\rightarrow$ 210Pb \+ 26Ne, the obtained spectrum has a
similar agreement to experiment, with some shifts in the band heads and
changes in the moment of inertia.

For the calculation of the spectroscopic factor, the parameter $\widetilde{B}$
= $-\frac{1}{\widetilde{n}_{0}-n_{C}}$, where $(\widetilde{n}_{0}-n_{C})$ =
-0.0185 are the remaining total relative oscillation quanta after subtracting
the forbiddenness. The spectroscopic factors are listed in the last column in
Table 5.

##  4 Conclusions

We have presented an extension of the Semimicroscopic Algebraic Cluster Model
(SACM), valid for light nuclei, to the pseudo-SACM, for heavy nuclei, limiting
to the $\widetilde{SU}(3)$ dynamical symmetry limit. Though, there exist
earlier attempts to extend the SACM to heavy nuclei, we found it necessary to
construct a model, which enables us to determine the complete spectrum and
circumvent some conceptual and practical problems of the former approaches and
to deliver a consistent procedure, as working in the same mean field of the
parent nucleus and simultaneously conserving 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 they are coupled together. The protons and
neutrons are distributed within the normal and unique orbitals in such a
manner that the sum of normal nucleons of the clusters is the same as in the
parent nucleus. The construction of the model space is in complete analogy to
the SACM.

As examples, we considered 236U $\rightarrow$ 210Pb+26Ne, 224Ra $\rightarrow$
210Pb+14C and 236U $\rightarrow$ 146Xe+90Sr. We demonstrated that the model is
able to describe the spectrum and electromagnetic transition probabilities.
Spectroscopic factors were also calculated, without further fitting and they
can be considered as a prediction of the model. With this, we demonstrated the
usefulness of the pseudo-SACM for treating heavy nuclei. A more systematic
study of several nuclei is planned in the future.

The restriction to the $\widetilde{SU}(3)$ dynamical symmetry limit has to be
relaxed in future applications, including the other dynamical symmetries as
$SO(4)$. One has to study the extension of $\widetilde{SU}(3)$ too, including
the active participation of nucleons in the unique orbitals. Also the study of
phase transitions is of interest, requiring the use of the geometrical mapping
geom  of the SACM.

## Acknowledgments

We acknowledge financial support form DGAPA-PAPIIT (IN100421).

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