The geometric interpretation of the semimicroscopic algebraic cluster model and the role of the Pauli principle

We study the properties of the Semimicroscopic Algebraic Cluster Model (SACM) under phase transitions. We show that the SACM can undergo first and second order phase transitions and a critical line appears. When the Pauli principle is not observed, the critical line appears, but other non-physical properties arise. At the end the meaning of the SO(4) dynamical limit is discussed.


Introduction
In this contribution the structure of phase transition within the SACM is discussed. The same will be done when the Pauli exclusion principle is not taken into account. Some physical interpretation of the SO(4) limit is also given. The discussion of phase transitions is enormously simplified when a geometrical mapping is applied. A semi-classical potential is obtained which depends on variables related to physical observables as the distance between two clusters. Phase transitions then manifest themselves by certain structural changes in the potential.
In section 2 the SACM is shortly reviewed, the Hamiltonian is defined and the method of geometrical mapping is explained. The semi-classical potential is obtained, which is used to investigate the phase transitions of the model. Changes are discussed when the Pauli exclusion principle is not observed. At the end some remarks are made concerning the physical meaning of the SO(4) dynamical limit. In section 3 conclusions are drawn.

The SACM and the geometrical mapping
We shall shortly resume the main features of the SACM [1,2]. The relative degrees of freedom are described by the oscillator boson operators π † m and π m , with m = 0, ±1. To that the scalar boson operators σ † and σ are also added: these have no physical meaning but are needed in order to introduce a cutoff in the theory. All combinations of any creation with any annihilation operator form the U R (4) group, where the R refers to the relative motion. The total number of bosons N = n π + n σ is kept constant, i.e., the number of π-bosons ranges from 0 to N . The individual clusters are described via the SU (3)-model of Elliott [3]. In this way both the relative motion and the orbital structure of the clusters is characterized by an SU (3) irreducible representation (irrep) (λ, µ).
The model space is constructed as follows. First, one determines the result of the product (λ 1 , µ 1 ) ⊗ (λ 2 , µ 2 ) ⊗ (n π , 0), which gives a sum of total SU (3) irreps with a certain multiplicity. Second, all possible excitations of the nucleons within the shell model are constructed, for nhω excitation, with the center of mass removed. This list of shell model irreps, which by construction observe the Pauli exclusion principle is compared to the former list. Only those irreps are included in the SACM model space, which appear also in the shell model space. This method automatically reproduces the Wildermuth condition, which states that there is a minimal number of π-bosons (n 0 ) needed as a necessary condition to satisfy the Pauli exclusion principle. In this manner, the Pauli exclusion principle is observed. Since the model space is constructed microscopically, one can study in this framework the overlap of the cluster configurations with the states of the shell and collective models, e.g. the clusterization of the superdeformed (SD) and hyperdeformed (HD) states. It is remarkable that the predictions of the model seem to be justified by the experimental observation in some cases, like for the SD state of 28 Si [4] and for the HD state of the 36 Ar [5].
The Hamiltonian is phenomenological, indicated by "semi" in the acronym SACM. We consider the sum of Hamiltonians, each describing a particular dynamical symmetry limit. Due to lack of space we shall present here the Hamiltonian taking into account only the SU (3) and SO(4) dynamical symmetries. The first one is known as the vibrational limit, while the second one as the deformed limit, though, as will be shown below, within the SACM the potential is already deformed by construction.
The Hamiltonian has the form where x is the parameter ranging from 0 to 1, determining the relative weight of each symmetry.
In their most general form, the individual components are (2) The SU (3) Hamiltonian contains a third-order interaction. Because the interaction proportional, for example, to the C 2 (n π , 0) interaction is given by n π (n π + 3), this term dominates for large n π and destroys the mean field parthωn π . In models with a fixed number of n π this is avoided by limiting the model space to fewhω excitations. However, we include mixing of n π and this requires to test convergence for increasing N . This type of Hamiltonian was applied successfully in several studies, of which we cite here only a few [6,7].
The semi-classical potential is obtained by choosing first a trial state | α and then calculating the expectation value of the Hamiltonian with respect to this normalized trial state, i.e., V (α) = α | H | α . Here α is a short hand notation for the variables of the state.
As a trial state we use where, for convenience, we redefined the total number of relative oscillator quanta as (N + n 0 ), while the γ 1 parameter has to be set equal to 1 after the differentiation, and the normalization factor N N,n 0 is given in [8]. Already here we obtain an essential difference with respect to the case when the Pauli principle is not observed. As shown in [8], the α variable is proportional to √ n 0 and also proportional to the separation of the clusters. Thus the so-called vibrational limit (α = 0) describes already a deformed system. Nevertheless, we shall keep the notation as it is. A special limit arises when n 0 is set to zero, i.e. no Pauli exclusion principle is observed. Models with this property we call PACM (Phenomenologic Algebraic Cluster Model).
The resulting potential has a surprisingly simple structure. The main part depends on only three parameters and is given bỹ The expressions for F pq α 2 as well as for the coefficients A (x, y), B (x, y) and C (x, y) can be found in Ref. [9]. They are functions of all parameters of the model. It is important to note that although the Hamiltonians depends on many parameters, the semi-classical potential depends effectively only on three of them. Thus the discussion of phase transition properties is reduced to the variation of three parameters only! The same potential could, in principle, be obtained in the PACM too, by paying the price of the introduction of non-polynomial interactions. This is similar to simulating Fermi repulsion in nucleus-nucleus collisions with a potential that is repulsive for small distances. Phase transition is investigated, following the steps: i) The minima of the potential are determined numerically. There will appear only two possible minima, a deformed (α 2 > 0) and a spherical one, the latter corresponding to α 1 = 0. The point α 1 = 0 is always an extremum.
ii) The three parameters are varied and those points are determined, where the two minima coincide. These points, which form a surface in the space of the parameters, are at the phase transition boundary. iii) The corresponding α variables are substituted into the potential. For the spherical minimum α 1 = 0 is always zero, which results in an expression for the potential which is identical zero, and therefore the derivatives of it with respect to the parameters is always zero. iv) The derivatives with respect to the parameters are determined and compared at the point of phase transition. Due to the structure of the potential a simple rule could be derived [9,10]: If α 2 at the point of phase transition is greater than zero, then the first derivatives in the two potential minima are different and the transition is, therefore, of first-order. However, when α 2 tends to zero approaching the point of phase transition, then the phase transition is of second order. The new feature is that from a given point on, no phase transition appears and a critical line is obtained. As a conclusion, the SACM permits first-and second-order phase transitions. The situation is illustrated in fig. 1. In the right panel α 2 is plotted versus A and C (B is determined via the condition that only points at the surface of phase transition are considered). The area where α 2 is zero represents the surface of second-order phase transition, while the other area corresponds to the surface of a first-order phase transition. In the left panel this surface is plotted in the space of the parameters. Note the appearance of a dashed line (the solid line separates the first-from the second-order phase transitions). Beyond this line no phase transition appears and this represents a critical line.
The same considerations were applied, i.e. in the PACM, where the minimal number of π-oscillation quanta is zero. Though the Hamiltonian is the same, the semi-classical potential changes significantly. The important part has the form with β 2 = α 2 1+α 2 . The expression for the coefficients A (x, y) and B (x, y), can be found in Ref. [9]. The points of phase transitions now form a line. First-and second-order phase transition appear but not a critical point. The first-order phase transition appears due to the inclusion of a third-order interaction in the SU (3) Hamiltonian, otherwise only second-order phase transitions are permitted.
In fig. 2 the space of phase transitions is plotted. There are only two relevant parameters A and B. The explanation is given in the figure caption. The horizontal axis is B, while the vertical axis corresponds to A. In Region I and Region II two minima exist, one spherical and one deformed. In Region I the global minimum is the deformed one, while in Region II it is the spherical minimum. In Region III only a spherical minimum exists and in Region IV the only minimum is a deformed one.
Finally, we would like to point out severe problems for states at low energy, when the Pauli exclusion principle is not observed. As an example we consider the 16 O+α → 20 Ne system. In the SU (3) limit the ground state would be formed by a 0 + state in the (0,0) SU (3)-irrep, a 2 + state in (2,0), a 4 + state in (4,0), etc. Even when one adds the interaction term L 2 , which gives a rotational L(L + 1) structure, simulating a rotational band, this band is not rotational because each state has a different intrinsic structure. Using the SO(4) dynamical symmetry limit gives a quite good agreement to experiment (!), however, the resulting states jump between low n π and hight n π , near the cut-off N , making the result dubious. Another problem appears in the geometrical limit: the interaction π † ·π † − σ † 2 (π ·π) − (σ) 2 tends to a potential of the type af (α)α 2 − bN , where f (α) is a simple expression of the order of one and a, b depend on N and n 0 (which is 0 for the PACM and > 0 for the SACM). Removing the cut-off (N → ∞) the minimum of the deformed potential tends to infinity too, being more equivalent to a dissociation limit. This problem arises in both the SACM and PACM, shedding doubt on the physical meaning of the SO(4) dynamical symmetry as a deformed limit.

Conclusions
In this contribution we showed that the SACM permits a very rich structure of phase transitions, including first and second order. A critical line also appears. When the Pauli exclusion principle is not observed, and a Hamiltonian of up to third order interactions is applied, then first-and second-order phase transitions still appear, but not a critical line. The results of the SACM could, in principle, be also reproduced within the PACM, but only paying the price of using a very complicated non-polynomial interaction that simulates the Pauli-exclusion principle. We also showed that neglecting the Pauli exclusion principle leads to dubious behavior for the states at low energy, even though the experimental spectrum is well reproduced. This puts some doubt on the argument that when the experiment is reproduced the model is fine. More has to be included, such as the structure of the states and the fulfillment of basic principles, as the Pauli exclusion principle is. These deficiencies may, of course, manifest themselves in failing to reproduce further experimental observables not discussed in the model. Finally, we discussed the physical interpretation of the SO(4) dynamical symmetry limit, which rather describes a dissociation of the cluster system than a deformed limit.