Geometrical symmetries of nuclear systems: and symmetries in light nuclei
The role of discrete (or point-group) symmetries in -cluster nuclei is discussed in the framework of the algebraic cluster model which describes the relative motion of the -particles. Particular attention is paid to the discrete symmetry of the geometric arrangement of the -particles, and the consequences for the structure of the corresponding rotational bands. The method is applied to study cluster states in the nuclei C and O. The observed level sequences can be understood in a simple way as a consequence of the underlying discrete symmetry that characterizes the geometrical configuration of the -particles, i.e. an equilateral triangle with symmetry for C, and a tetrahedron with symmetry for O. The structure of rotational bands provides a fingerprint of the underlying geometrical configuration of -particles.
pacs:21.60.Fw, 21.60.Gx, 21.10.-k, 27.20.+n
Keywords: Alpha-cluster nuclei, geometrical symmetries, algebraic cluster model, energy spectrum, electromagnetic form factors, values
The concept of symmetries has played an important role in nuclear structure physics, both continuous and discrete symmetries. Examples of continuous symmetries are isospin symmetry , Wignerâs combined spin-isospin symmetry , the (generalized) seniority scheme [3, 4], the Elliott model  and the interacting boson model . In addition to providing simple analytic solutions which can be used to analyze and interpret the structure of nuclei, symmetries play a crucial role in establishing the connection between different models of nuclear structure, such as the spherical shell model of Goeppert-Mayer  and Jensen , the geometric collective model of Bohr and Mottelson  and the interacting boson model of Arima and Iachello . In particular, the Elliott model provides a link between the spherical shell model and the quadrupole deformation of the geometric collective model, and the dynamical symmetries of the interacting boson model correspond to the harmonic vibrator, axial rotor and -unstable rotor limits of the geometric collective model. An recent review of the relation between symmetries of the spherical shell model and quadrupole and octupole deformations of the collective model can be found in Ref. .
On the other hand, discrete symmetries have been used in the context of collective models to characterize the intrinsic shape of the nucleus, such as axial symmetry for quadrupole deformations , and tetrahedral  and octahedral [11, 12] symmetries for deformations of higher multipoles. A different application is found in the context of -particle clustering in light nuclei to describe the geometric configuration of the particles. Early work on -cluster models goes back to the 1930’s with studies by Wheeler , and Hafstad and Teller , followed by later work by Brink [15, 16] and Robson . Recently, there has been a lot of renewed interest in the structure of -cluster nuclei, especially for the nucleus C . The measurement of new rotational excitations of the ground state [19, 20] and the Hoyle state [21, 22, 23, 24] has stimulated a large theoretical effort to understand the structure of C ranging from studies based on the semi-microscopic algebraic cluster model , antisymmetrized molecular dynamics , fermionic molecular dynamics , BEC-like cluster model , ab initio no-core shell model , lattice effective field theory , no-core symplectic model , and the algebraic cluster model (ACM) [20, 32, 33]. A recent review on the structure of C can be found in Ref. .
It is the aim of this contribution to discuss the ACM for two-, three- and four-body clusters, and study possible applications in -cluster nuclei, like Be, C and O. In these applications, it is important to take into account the permutation symmetry between the identical clusters. The manuscript is organized as follows. In Section 2 the radiation problem of various configurations of identical charged particles is discussed at the classical level. In Sections 3 and 4 some general properties of the ACM are presented which are relevant for two-, three- and four-body identical clusters, such as the structure of the Hamiltonian, permutation and geometrical symmetries, the classical limit, electromagnetic couplings, dynamical symmetries and shape-phase transitions. In the next three sections, the ACM for two-, three- and four-body clusters is developed in more detail. Particular attention is paid to the cases of the axial rotor, the oblate top and the spherical top, and their respective point group symmetries. Even though these special solutions do not correspond to a dynamical symmetry of the Hamiltonian, approximate formula for energies, form factors and electromagnetic transition rates can be derived in a semi-classical mean-field analysis. Finally, I discuss applications of the ACM to the cluster states in the nuclei C and O.
2 Classical treatment
In order to appreciate the effect of the geometric configuration of a number of identical charges (without magnetic moment, as in the case for the -particle model) on the multipole radiation, first consider the classical radiation problem for a charge distribution with multipole moments
From these multipole moments, the transition probability per unit time can be calculated as 
For a point-like charge distribution of identical charges with coordinates () the charge distribution is given by
The corresponding values are given by
where denotes the relative angle between the vectors and
For two-body clusters (), the coordinates of the two particles are taken with respect to the center of mass: and (see Fig. 1). The corresponding values are
which vanish for the odd multipoles, and are equal to
for the even multipoles.
Next consider the case of three identical particles at the vertices of an equilateral triangle (point-group symmetry ) (see Fig. 1). The origin is placed at the center of mass so that the distance from the center is the same for all three particles . The spherical coordinates of the three particles can be taken as , and for and , respectively, such that the relative angles for all . For this configuration, the multipole radiation is given by
The dipole radiation vanishes because of the spatial symmetry of the charge distribution.
Finally, consider the case of four identical particles at the vertices of a tetrahedron (point-group symmetry ) (see Fig. 1). Again, the origin is placed at the center of mass so that the distance from the center is the same for all four particles . The spherical coordinates of the three particles can be taken as , , and , respectively, such that the relative angles satisfies for all . For this configuration, the multipole radiation is given by
In this case, the multipoles with , and vanish as a consequence of the tetrahedral symmetry of the configuration of four particles.
3 Algebraic cluster model
Algebraic models have found useful applications both in many-body and in few-body systems. Algebraic methods are based on the general criterion to introduce a spectrum generating algebra for a bound-state problem with degrees of freedom. Well-known examples are the interacting boson model for the quadrupole degrees of freedom in collective nuclei , and the vibron model for the dipole degrees of freedom in diatomic molecules .
In this section, I briefly review the basic ingredients of the algebraic cluster model (ACM) which was introduced to describe the relative motion of cluster systems . The relevant degrees of freedom of a system of -body clusters are given by the relative Jacobi coordinates
and their conjugate momenta, . Here denotes the position vector of the -th cluster. The building blocks of the ACM consist of a vector boson for each relative coordinate and conjugate momentum
with and , and a scalar boson, , . In the ACM, cluster states are described in terms of a system of interacting bosons with angular momentum and parity (dipole or vector bosons) and (monopole or scalar bosons). The components of the vector bosons together with the scalar boson span a -dimensional space with group structure . The many-body states are classified according to the totally symmetric irreducible representation of , where represents the total number of bosons .
In summary, the ACM is an algebraic treatment of cluster states which is based on the spectrum generating algebra of where denotes the number of clusters. For two-body clusters (), it reduces to the vibron model which was introduced originally in molecular physics , but which has also found applications in nuclear physics , and hadronic physics (mesons) . The ACM for three-body clusters () was introduced to describe the relative motion of three-quark configurations in baryons , with applications in molecular physics  and nuclear physics [20, 32] as well. More recently, the ACM was extended to four-body clusters () to describe the cluster states of the nucleus O in terms of the relative motion of four-alpha particles [33, 40].
3.1 Geometrical symmetries
In application to -cluster nuclei, like Be, C and O, in which the constuent parts are identical, the eigenstates of the Hamiltonian should transform according to the symmetric representations of the permutation group ( for identical objects). The permutation symmetry of objects is determined by the transposition and the cyclic permutation . All other permutations can be expressed in terms of these elementary ones . Algebraically, the transposition and cyclic permutation can be expressed in terms of the generators that act in index space (), The scalar boson, , transforms as the symmetric representation characterized by Young tableau , whereas the Jacobi vector bosons, with , transform as the components of the mixed symmetry representation . Next, one can use the multiplication rules for to to construct physical operators with the appropriate symmetry properties. For example, for the bilinear products of the Jacobi bosons, one finds
for . For three clusters () only the first three terms are present, and for two clusters () only the first term. The isomorphism between the permutation group and the symmetries of a regular simplex in dimensions  will be used in the next sections to establish the connection with the and point groups for cluster configurations where the clusters are located at the vertices of an equilateral triangle and a regular tetrahedron, respectively.
As a result, for a system of identical clusters the most general one- and two-body Hamiltonian that conserves the total number of bosons, angular momentum and parity, and is invariant under the permutation group , is given by
with and . The , , , and terms in Eq. (10lmnp) are scalars under . The restrictions imposed by the permutation symmetry on the coefficients appearing in the last term, will be discussed in more detail in the next sections for the cases of two-, three- and four-body clusters.
In general, the matrix elements of the Hamiltonian of Eq. (10lmnp) are calculated in a set of coupled harmonic oscillator basis states characterized by the total number of bosons , angular momentum and parity . Although for the harmonic oscillator there exists a procedure for the explicit construction of states with good permutation symmetry , in the application to the ACM the number of oscillator quanta may be large (up to 10) and moreover the oscillator shells are mixed by the term. Therefore, a general procedure was developed in which the wave functions with good permutation symmetry are generated numerically by diagonalizing invariant interactions. Subsequently, the permutation symmetry of a given wave function is determined by examining its transformation properties under the transposition and the cyclic permutation .
In general, geometric properties of algebraic models such as the interacting boson model , the vibron model  and the algebraic cluster model  can be studied with time-dependent mean-field approximations. The mean-field equations can be derived by minimizing the action [43, 44]
Here represents an intrinsic or coherent state as a variational wave function in terms of a condensate of deformed bosons which depend on geometric variables [45, 46, 47]. For the -body ACM, these coherent states are given by
where the condensate boson is parametrized in terms of complex variables
The variational principle gives Hamilton’s equations of motion
where and represent canonical coordinates and momenta, and denotes the classical limit of the Hamiltonian which is defined as the expectation value of the normally ordered Hamiltonian in the coherent state of Eq. (10lmnr) divided by the total number of bosons
Bound states correspond to periodic classical trajectories , with period that satisfy a Bohr-Sommerfeld type quantization rule 
The energy associated with a periodic classical orbital is independent of time and is given by .
3.4 Electromagnetic couplings
The transition form factor for the excitation of discrete nuclear levels is defined as 
In the long wavelength limit, only one multipole contributes (with ). After summing the square of the transition form factor over final and averaging over initial magnetic substates, one obtains
As a consequence, the reduced transition probabilities can be extracted from the transition form factors in the long wavelength limit
For the point-like charge distribution of Eq. (4) one finds for the transition form factor
This result was obtained by first using the permutation symmetry of the initial and final wave functions, and next carrying out a transformation to center-of-mass and relative Jacobi coordinates, and integrating over the center-of-mass coordinate. The reduced transition probabilities can be obtained in the long wavelength limit by
In order to calculate transition form factors and transition probabilities in the algebraic cluster model one has to express the transition operator in terms of an algebraic operator. The matrix elements can be obtained algebraically by making the replacement
where represents the scale of the coordinate and is a normalization factor which is related the reduced matrix element of the dipole operator. The replacement in Eq. (10lmnaf) comes from the fact that in the large limit, the dipole operator
corresponds to the Jacobi coordinate .
where is the angular momentum in polar coordinates of the -th oscillator
The square root factor appearing in Eq. (10lmnah) is due to the presence of the scalar boson in the dipole operator, and is a consequence of the finiteness of the model space of the ACM.
In summary, the transition form factors can be expressed in the ACM in terms of the matrix elements
In general, the transition form factors cannot be obtained in closed analytic form, but have to be evaluated numerically.
The results discussed so far, are for point-like constituent particles with a charge distribution given by Eq. (4). Next, let’s consider the case in which the constituent particles have a finite size. With the application to -cluster nuclei in mind, it is reasonable to assume a Gaussian form
As a consequence, all form factors are multiplied by an exponential factor . The charge radius can be obtained from the slope of the elastic form factor in the origin
Form factors and values only depend on the parameters and . The coefficient can be determined from the first minimum in the elastic form factor, and the charge radius can be used to fix the value of .
4 Dynamical symmetries
In general, the ACM Hamiltonian has to diagonalized numerically in order to obtain the energy eigenvalues and corresponding eigenvectors. It is of general interest to study limiting cases of the Hamiltonian of Eq. (10lmnp), in which the energy spectra, electromagnetic transition rates and form factors can be obtained in closed form. These special solutions provide benchmarks in which energy spectra and other spectroscopic properties can be interpreted in a clear and transparent way. These special cases correspond to so-called dynamical symmetries which arise when the Hamiltonian has a certain group structure , and it can be expressed in terms of Casimir invariants of a chain of subgroups of only. The eigenstates can then be classified uniquely according to the irreducible representations of and its subgroups. The energy eigenvalues are given by the expectation values of the Casimir operators.
The algebraic cluster model has a rich algebraic structure, which includes both continuous and discrete symmetries. The ACM Hamiltonian for the -body problem has the group structure . In this section, I discuss the two dynamical symmetries which are related to the group lattice
These dynamical symmetries are limiting cases of the ACM, and are called the and limit, respectively. In the following, I discuss these special solutions for any number of clusters , and will show that, by studying the classical limit, they can be interpreted as the harmonic oscillator and the deformed oscillator in dimensions .
4.1 Harmonic oscillator
The first dynamical symmetry corresponds to the group chain
The basis states are classified by the quantum numbers , and , which characterize the irreducible representations of , and , respectively. Here is the total number of bosons, and denotes the number of oscillator quanta . The energy levels are organized into oscillator shells , and are further labeled by with or for odd or even. The parity of the levels is given by .
Here, I consider the one-body Hamiltonian
The corresponding spectrum of the multiplets is shown in the left panel of Fig. 2.
The classical limit of this Hamiltonian is, according to Eq. (10lmnu), given by its coherent state expectation value
where is the angular momentum of the -th oscillator expressed in polar coordinates
It is convenient to make a change of variables from the coordinates to hyperspherical coordinates, the hyperradius and the angles
together with their conjugate momenta, and . In the hyperspherical coordinates, the classical limit of the limit is given by
Here denotes the generalized angular momentum for rotations in dimensions
with and . It is the classical limit of the Casimir operator and is a constant of the motion. Therefore, one can first apply the requantization conditions to the coordinates and momenta contained in , which yields that be replaced by . The difference from the exact result is typical for the semi-classical approximation. The quantization condition in the phase space is given by
The integral can be solved exactly to obtain
which is identical to the exact result of Eq. (10lmnap) with . This semi-classical analysis shows the correspondence between the limit and the -dimensional spherical oscillator.
In , the ACM form factors in the limit were derived in closed analytic form. As an example, the elastic form factor is given by
where is given by with . The exponential form shown on the right-hand side is obtained in the large limit which is taken such that remains finite.
4.2 Deformed oscillator
For the (an)harmonic oscillator, the number of oscillator quanta is a good quantum number. When in Eq. (10lmnp), the oscillator shells with are mixed, and the eigenfunctions are spread over many different oscillator shells. A dynamical symmetry that involves the mixing between oscillator shells, is provided by the reduction
The basis states are classified by the quantum numbers , and , where characterizes the irreducible representations of . and have the same meaning as in the limit. In this case, the energy levels are organized into bands labeled by with or for odd or even, respectively, The rotational excitations are denoted by with .
Let’s consider a Hamiltonian of the form
where is the number operator
The difference between the Casimir operators of and corresponds to a dipole-dipole interaction (see Eq. (10lmnag)). The energy spectrum is obtained from the expectation values of the Casimir operators as
The energy spectrum is shown in the right panel of Fig. 2. Although the size of the model space, and hence the total number of states, is the same as for the harmonic oscillator, the ordering and classification of the states is different. In the limit all states are vibrational, whereas the limit gives rise to a rotation-vibration spectrum, where the vibrations are labeled by and the rotations by .
The classical limit of is given by Eq. (10lmnu)
Also in this case, the generalized angular momentum is a constant of the motion, and hence can be requantized first. The remaining quantization condition in the phase space
can be solved exactly to obtain
In the large limit, this expression reduces to the exact one of Eq. (10lmnasba), if one associates the vibrational quantum number with
To leading order in , the vibrational frequency coincides. In conclusion, this semi-classical analysis shows that the limit corresponds to a deformed oscillator in dimensions.
Also in the limit the ACM form factors can be derived in closed analytic form . In this case, the elastic form factor is given in terms of a Gegenbauer polynomial which in the large limit reduces to a spherical (cylindrical) Bessel function for even (odd)