Cluster structure of light nuclei

We review recent studies of the cluster structure of light nuclei within the framework of the algebraic cluster model (ACM) for nuclei composed of k alpha-particles and within the framework of the cluster shell model (CSM) for nuclei composed of k alpha-particles plus x additional nucleons. The calculations, based on symmetry considerations and thus for the most part given in analytic form, are compared with experiments in light cluster nuclei. The comparison shows evidence for Z_2, D_{3h} and T_d symmetry in the even-even nuclei 8Be (k=2), 12C (k=3) and 16O (k=4), respectively, and for the associated double groups Z'_2 and D'_{3h} in the odd nuclei 9Be, 9B (k=2, x=1) and 13C (k=3, x=1), respectively.


Introduction
The cluster structure of light nuclei has a long history dating back to the 1930's with early studies of α-cluster models by Wheeler [1] and Hafstad and Teller [2], followed by Dennison [3] and Kameny [4]. Soon afterwards, the connection between the cluster model and the shell-model was investigated by Wildermuth and Kallenopoulos [5]. In 1965, Brink [6,7] suggested specific geometric in which these isotopes are seen as 8 Be plus neutrons and protons. In very recent years, a description of kα + x nuclei has been suggested in terms of the Cluster Shell Model (CSM) [61,62] which builds on the algebraic description of kα nuclei in terms of the ACM [63]. In the second part of this paper, we review the CSM as applied to kα + x nuclei with k = 2, 3, 4 and x = 1.
We note in this connection that the Cluster Shell Model (CSM) takes fully into account the Pauli principle, as discussed in Section 8. Individual nucleons are placed in the single-particle orbitals described in Section 7 according to the Pauli principle. The treatment of the Pauli principle in CSM is thus identical to that in the Nilsson model [64], in the Brink model [6] and the LCAO method [46,47]. The question of the Pauli principle in the Brink model is also discussed in [15,16] and in the molecular model with Z 2 symmetry dumbbell in [46,47].
We emphasize here the point that in the CSM one is able to take into account the Pauli principle not only for the dumbbell configuration (two-center shell model) but also for the triangular configuration (three-center shell model) and for the tetrahedral configuraton (four-center shell model). The latter two have not been discussed before within the context of nuclear physics.
The figures of Section 7 also show the occurrence of "magic" numbers for protons and neutrons at 4 for the dumbbell configuration, at 6 for the triangular configuration and at 8 for the tetrahedral configuration. The stability of the kα nuclei is thus inherent in the approach described here and justifies the ACM. A remaining question, however, is to what extent the reduction from the spherical shell model to the cluster model induces two-or higher order terms in the effective α-α interaction. The algebraic approach when written in terms of coordinates and momenta corresponds to an effective α-α interaction of the Morse type [65], as discussed in [66]. The Morse type interaction has a "hard core" which effectively mimics the Pauli principle, since it does not allow for two α-particles to get close and overlap strongly. Also, as shown in the figures in Section 2 where the matter density is plotted, the situation encountered in light nuclei is that in which the α-particles are in a close-packing situation, that is they just touch but do not overlap. Taking into account effectively the Pauli principle in the α-α interaction within the framework of the algebraic method ACM is therefore a good approximation to the full microscopic approach in terms of nucleons.
The aim of this paper is to present all formulas and calculations to compare with experimental data and, as a result, to show evidence for the occurrence of the geometric symmetries Z 2 , D 3h , T d and Z ′ 2 , D ′ 3h in the structure of kα and kα + 1 nuclei.
The ACM model reviewed in the first part of this article is purely phenomenological, as the collective model of Bohr and Mottelson and does not attempt a microscopic description in terms of nucleon-nucleon interactions, but rather exploits symmetry considerations to derive most of the observables in explicit analytic form that can be compared with experiment. Conversely, the CSM reviewed in the second part is a microscopic model that makes use of a symmetry-adapted basis, the cluster basis with Z 2 , D 3h and T d symmetry instead of the spherical basis.
One important question is the extent to which these symmetries emerge from microscopic calculations. Extensive calculations have been done for 8 Be and 9 Be (Z 2 and Z ′ 2 symmetry) within the framework of microscopic approaches mentioned in the paragraphs above. For these nuclei, microscopic approaches appear to produce cluster features correctly, although effective charges still need to be introduced in the analysis of electromagnetic transition rates in shell model based calculations [32]. A detailed comparison between the algebraic approach and microscopic approaches for 8 Be and 9 Be is given in [62]. Also for 12 C and 13 C (D 3h and D ′ 3h symmetry) extensive calculations exist, especially for 12 C. The AMD and FDM microscopic calculations produce results in quantitative agreement with the symmetry. Also, lattice EFT produces results in 12 C and 16 O which support D 3h and T d symmetry. It would be of great interest to understand whether the cluster structure of 12 C and 13 C emerges from ab initio calculations, such as the no-core shell model (NCCI) [31][32][33] for which calculations are planned. The results presented here, based on purely symmetry concepts, provide benchmarks for microscopic studies of cluster structure of q q q q q q q q q q q 16  light nuclei.

The algebraic cluster model
The algebraic cluster model is based on the algebraic theory of molecules introduced in 1981 [67] and reviewed in [66]. It amounts to a bosonic quantization of the Jacobi variables according to the general quantization scheme for problems with ν degrees of freedom in terms of the Lie algebra U (ν + 1) [68]. For kα structures, the number of degrees of freedom, after removing the center-of-mass motion, is ν = 3(k − 1), leading to the Lie algebra of U (3k − 2).
The ACM is a model which describes the relative motion of a cluster system.
We start by introducing the relative Jacobi coordinates for a k-body system (see  k Nucleus U (3k − 2) Discrete symmetry Jacobi variables 2 8 Be U (4) Z 2 ρ together with their conjugate momenta. Here r i represent the coordinates of the constituent particles. The relevant Jacobi coordinates are ρ for k = 2, ρ and λ for k = 3, and ρ, λ and η for k = 4 (see Table 1). The ACM uses the method of bosonic quantization which consists in quantizing the Jacobi coordinates and momenta with vector boson operators and adding an additional scalar boson under the constraint that the total number of bosons N is conserved. An explicit construction of the algebra and derivation of analytic formulas for energy levels, electromagnetic transition rates, matter and charge densities and associated form factors in electron scattering has been completed for cases k = 2 [67], k = 3 [38,39] and k = 4 [40,41]. It is summarized in Table 1 and results will be reviewed in the following subsections.

Classification of states
The discrete symmetry of clusters imposes conditions on the allowed quantum states. The mathematical method for determining the allowed states (i.e.

Symmetry adapter
Group G Transposition Cyclic permuation constructing representations of the discrete group G) is by means of the use of so-called symmetry adapter operators. For cases k = 2, 3, 4 and identical constituents, one can exploit the isomorphism of the discrete point group with the permutation group S k . The associated symmetry adapter operators are the transposition P (12) and the cyclic permutation P (12 · · · k), see Table 2. All other permutations can be expressed in terms of these elementary ones [69].
For the harmonic oscillator there exists a procedure for the explicit construction of states with good permutation symmetry [69]. However, in the application to the ACM the number of oscillator quanta may be large (up to 10) and moreover in general the oscillator shells are mixed. Therefore, a general procedure was developed in which the wave functions with good permutation symmetry |ψ t are generated numerically by diagonalizing S k invariant interactions. Subsequently, the permutation symmetry t of a given wave function is determined by examining its transformation properties under the transposition P (12) and the cyclic permutation P (12 · · · k). This procedure is explained in more detail in Appendix A.
Representations can be labeled either by S k or by the isomorphic discrete group G, as shown in Table 3. Here the representations of S k are labelled by the Young tableaux, while those of G are labelled by the standard notation used in molecular physics [70,71].
In application to α-cluster nuclei, like 8 Be, 12 C and 16 O, in which the constuent parts are identical, the eigenstates of the Hamiltonian should transform Group G S k Label G Label Degeneracy A Singly [11] B Singly

Dumbbell configuration
An algebraic description of this configuration is given by the algebra of U (4) [67]. This algebra is constructed with boson creation operators b † ρ,m with m = 0, ±1 and s † , altogether denoted by c † α with α = 1, . . . , 4, and annihilation operators b ρ,m , s. Here b † ρ,m and b ρ,m are the quantization of the Jacobi variable ρ (see Fig. 2) and its conjugate momentum, and s † , s is an auxiliary scalar boson. The bilinear products G αβ = c † α c β with α, β = 1, . . . , 4 of creation and annihilation operators generate the Lie algebra of U (4). Specifically these are whereb ρ,m = (−1) 1−m b ρ,−m ands = s. We consider here rotations and vibrations of the dumbbell configuration. States can be classified by a vibrational quantum number v = 0, 1, 2, . . . , and a rotational quantum number L and its projection M as |v, L, M . In the case in which the two constituents are identical (two α-particles) the dumbbell has Z 2 ∼ S 2 ∼ P symmetry. All vibrational states v have symmetry A under Z 2 since the two particles are identical (Fig. 2).
The angular momentum content of each vibrational band is L P = 0 + , 2 + , 4 + , . . ., where the parity P has been added, although here it is not an independent quantum number, P = (−) L .
We consider here rotations and vibrations of an equilateral triangular configuration. States can be classified as where t denotes the representations of D 3h , and K the projection of the angular momentum L on the symmetry axis. In the case in which the three constituents are identical (three α-particles), the triangular configuration of Fig. 3 Figure 3: Jacobi variables ρ, λ for an equilateral triangle configuration.  symmetry. This imposes some conditions on the allowed values of K and L. In 2 ) label the vibrational states with v 1 = 0, 1, . . ., v 2 = 0, 1, . . ., and ℓ 2 = v 2 , v 2 − 2, . . . , 1 or 0 for v 2 odd or even. The fundamental vibrations of a triangular configuration are shown in Fig. 4.
We consider here rotations and vibrations of a tetrahedral configuration.
States can be classified as q q q q q q q q q q q q q q q q q q where (v 1 , v ℓ2 2 , v ℓ3 3 ) denote the vibrational quantum numbers and t labels the representations of T d . In the case in which the four constituents are identical (four α-particles), the tetrahedral configuration of Fig. 5 has T d symmetry.
This imposes some conditions on the allowed values of L which depend on t. A derivation of the allowed values is given in [41]. For the ground state, t = A, and for the fundamental vibrations with t = A, E and F of Fig. 6, it can be summarized as follows t = A L P = 0 + , 3 − , 4 + , 6 ± , . . . , t = E L P = 2 ± , 4 ± , 5 ± , 6 ± , . . . ,

Energy formulas
Energy levels in ACM can be obtained by diagonalizing the Hamiltonian.
Computer programs have been written for all three cases, U (4), U (7) and U (10) [72]. These programs can deal with all situations encountered in two-, three-and four-body problems, including both soft and rigid situations. For applications here, we consider only rigid situations and write down analytic formulas that can be used to analyze experimental data.

Dumbbell configuration
The algebraic Hamiltonian describing roto-vibrations of a dumbbell configuration (diatomic molecule) is given in Eqs. (2.108) and (2.112) of [66]. Written explicitly in terms of boson operators, it has the form with m = 0, ±1. In this case, the Hamiltonian has a dynamic symmetry U (4) ⊃ SO(4) ⊃ SO(3) ⊃ SO (2). The eigenvalues of H can be written in explicit analytic form as Here N is the so-called vibron number, that is the number of bosons that characterizes the irreducible representations of U (4). The vibrational quantum number v takes the values for N odd or even, and the rotational quantum number L takes the integer values For identical constituents, i.e. Z 2 symmetry, only even values of L are allowed.
In the large N limit, one obtains the semiclassical formula for the energy levels of a dumbbell configuration where ω is the vibrational energy and B the inertial parameter B =h 2 /2I. A schematic spectrum of a rotating and vibrating dumbbell is shown in Fig. 7.

Equilateral triangle configuration
The situation here is more complicated than for the dumbbell configuration, since there is no dynamic symmetry corresponding to the rotation and vibration of a rigid symmetric top. The explicit form of the Hamiltonian is [39]  In a generic situation, this Hamiltonian needs to be diagonalized in the space of given vibron number N . However, in the limit N → ∞, one can write down a semiclassical formula which describes the energy levels of a symmetric top [39]. States are classified as in Eq. (5) and the values of K and L for (v 1 , v ℓ2=0,1 2 ) are given in Eqs. (6) and (7). In Fig. 8 we show a schematic rotational-vibrational spectrum of a triangular configuration.

Tetrahedral configuration
Also in this case there is no dynamic symmetry corresponding to the rotation and vibration of a spherical top. The explicit form of the Hamiltonian describing the vibrations is [40,41] The Hamiltonian describing rotations can be written as where L and I denote the angular momentum in coordinate space and index space, respectively, the explicit form of which is Again, in a generic situation, this Hamiltonian needs to be diagonalized in the space of given vibron number N . For N → ∞, one can write down a semiclassical formula [41] E which describes the energy levels of a spherical top. States are classified as in Eq. (9) and the values of t and L P for the ground state band and the fundamental vibrations are given in Eq. (10). In Fig. 9 we show a schematic rotationalvibrational spectrum of a tetrahedral configuration.

Form factors and transition probabilities
The transition form factors are the matrix elements of k i=1 exp(i q · r i ), where q is the momentum transfer and r i is the location of the α-particles. To do this calculation in the ACM, one first converts the transition operator to algebraic form and then calculates the form factors The transition probabilities B(EL) can be extracted from the form factors in the long wavelength limit where Ze is the total electric charge of the cluster.

Dumbbell configuration
Choosing the z-axis along the direction of the momentum transfer and using the fact that the two particles are identical, it is sufficient to consider the matrix elements of exp(iqr 2z ). After converting to Jacobi coordinates and integrating over the center-of-mass coordinate one has exp(−iqρ z ). The matrix elements of this operator can be obtained algebraically by making the replacement where β represents the scale of the coordinate and X D is given by the reduced matrix element of the dipole operator of Eq. (12). Explicit evaluation in the large N limit gives with where j L is the spherical Bessel function, and P L the Legendre polynomial.
From these, one can obtain the B(EL) value

Equilateral triangle configuration
Choosing again the z-axis along the direction of the momentum transfer and using the fact that the three particles are identical, it is sufficient to consider the matrix elements of exp(−iq 2/3λ z ). By making the replacement one can obtain the form factors for N → ∞ in explicit form as with

Cluster densities
All results in Sect. 2.3 are for point-like constituents, with density This situation is not realistic, since the constituent α-particles are not pointlike. Assuming a Gaussian form of the density of the α-particle, one has the more realistic cluster density Here α = 0.56 fm −2 describes the form factor of the α-particle [73]. For the density of Eq. (38), the form factors become which represents the convolution of the form factor of the cluster with that of the α-particle. B(EL) values, however, remain the same as in Sect. 2.3 since in the long-wavelength limit q → 0, the exponential factor exp(−q 2 /4α) → 1. The density of Eq. (38) can be visualized by making an expansion into multipoles.
By placing the particles at a distance β from the center of mass with spherical coordinates (β, θ i , φ i ) we then have [61] ρ( r) = α π

Dumbbell configuration
For Z 2 symmetry, the origin is chosen in the center of mass, and the angles of the two particles are given by ( This configuration has axial symmetry. In the multipole expansion, only µ = 0 and λ = even = 0, 2, . . . , remain. The charge and matter densities of the dumbbell configuration are shown in Fig. 10 for β = 0, 2 and 4 fm. The density describes the entire range from united constituent particles (β = 0) to separated constituent particles (β → ∞). Note that the density describes also break-up into two fragments, as shown in the panel on the right-hand side of Fig. 10. The color scale is in fm −3 . Reproduced from [61] with permission.

Equilateral triangle configuration
For the D 3h symmetry of an equilateral triangle, the angles of the particles are given by ( where κ = 1, 2, . . . , and µ ≤ λ. For this configuration, the remaining multipoles are λ = 0, 2, 3, 4, . . . , corresponding to the fact that the density is invariant under D 3h transformations and thus belongs to the symmetric representation of D 3h [39]. The charge (and matter) densities of a triangular configuration are shown in Fig. 11 for different values of β.

Moments of inertia and radii
From the density Eq. (40) one can calculate the moments of inertia and radii.
The three components of the moment of inertia are given by and radii by

Dumbbell configuration
Introducing the appropriate normalization, one has where A = 4k = 8, corresponding to a prolate top, and

Equilateral triangle configuration
In this case, one has [39] where A = 4k = 12, corresponding to an oblate top, and For the tetrahedral configuration, all three moments of inertia are the same where A = 4k = 16, corresponding to a spherical top, and

Evidence for cluster structures
The ACM provides a simple way to analyze experimental data, thus determining whether or not the symmetries Z 2 , D 3h , and T d appear in the spectra of 8 Be, 12 C and 16 O.

Dumbbell configuration
Energy levels for this configuration can be analyzed with Eq. (16). A comparison with data in 8 Be [74] is shown in Fig. 13. The occurrence of a rotational band in the experimental spectrum is clearly seen in Fig. 14, where the energy of the states is shown as a function of L(L + 1). No evidence for the vibrational bands is reported in [74], although Barker [75,76] suggested in the 1960's one such a band at E ∼ 6 MeV, in accordance to similar vibrational bands observed in 12 C and 16 O. The non-observation of the vibrational band in 8 Be may be due to its expected large width.
From the value of B =h 2 /2I extracted from the experimental energy dif-

Equilateral triangle configuration
Recent experiments [24] have confirmed the occurrence of D 3h symmetry in  Reproduced from [24] with permission.
here the occurrence of not only rotational bands with angular momentum content expected from D 3h symmetry, but also the occurrence of the fundamental vibrations of the triangle of Fig. 8 (v 1 , v ℓ2 2 ) = (1, 0 0 ) and (0, 1 1 ) with symmetry A and E, respectively.

Tetrahedral configuration
The occurrence of T d symmetry in 16 O was emphasized by Robson [13,14] in the 1970s and more recently revisited in [40,41]. Energy levels have been analyzed with Eq. (22). A comparison with data is shown in Figs. 17 and 18.

Dumbbell configuration
The nucleus 8 Be is unstable and therefore form factors in electron scattering cannot be measured. The value of β for this configuration is estimated from the moment of inertia to be β = 1.82 ± 0.04 fm, as given in Section 3.1.1.

Equilateral triangle configuration
Form factors in 12 C have been extensively investigated. In the rigid case, only states in the ground state band are excited with form factors given by Eq. (30) and no excitation of the vibrational bands occurs. Since experimentally excitation of these bands occurs, although with a small strength, one needs in this case to do a calculation with the general algebraic Hamiltonian, Eq. (17) [39]. The resulting form factors are shown in Fig. 19, where they are compared with experimental data. The value of β is determined from the first minimum of the elastic form factor to be β = 1.74 ± 0.04 fm [39] with an estimated error of 2 %. The experimental form factors in Fig. 19 compare well with the theoretical form factors, except for the transition form factor |F (0 + 1 → 0 + 2 )| 2 whose shape is correctly given but whose magnitude is smaller by a factor of 10. This discrepancy is the subject of current investigations [77] and seems to indicate that the structure of the 0 + 2 Hoyle state may be somewhat softer than calculated by the rigid oblate configuration.
Form factors in 16 O have also been extensively investigated. In the rigid case, the form factors are given by Eq. (34). Since also here excitation of the vibrational bands occurs, one needs to do a full calculation [41].

Electromagnetic transition rates
Electromagnetic transition rates and B(EL) values can be analyzed by making use of Eqs. (28), (32) and (36). A comparison with data for 8 Be, 12 C and 16 O is shown in Table 4. In this table, the experimental value for 8 Be was estimated using the Greens function Monte Carlo method (GFMC) [78]. The value of β = 1.82 fm for 8 Be is obtained from the moment of inertia, and for 12 C and 16 O from the first minimum in the elastic form factor, see Sect. 3.2.

Non-cluster states
The cluster model assumes that kα nuclei are composed of α-particles. However, these in turn are composed of two protons and two neutrons. At some excitation energy in the nucleus, the α-particle structure may break. In some cases non-cluster states can be clearly identified, since some states are forbidden by the discrete symmetry. Specifically, for Z 2 symmetry ( 8 Be) L P = 1 + states    Above these energies, cluster states co-exist with non-cluster states.

Softness and higher-order corrections
The situations described by the energy formulas in Eqs. (16), (18) and (22) correspond to rigid configurations. As mentioned in previous sections, soft (nonrigid) situations can be described by diagonalizing the full algebraic Hamiltonian. However, one can also write, for comparison with experimental data, simpler analytic expressions for non-rigid situations.
An analytic formula including anharmonic corrections and vibration-rotation interaction is

Equilateral triangle configuration
In this case an analytic expression is [24]

Tetrahedral configuration
The effect of anharmonicities here can be written as

Other geometric configurations
Within the ACM it is possible to provide analytic formulas for energies and electromagnetic transition rates for all possible geometric configurations and, most importantly, by diagonalizing the full algebraic Hamiltonian, it is possible to study non-rigid situation intermediate between two or more geometric situations and thus study the transitions between these, called ground-state phase transitions [81]. Some possible geometric configurations for three and configuration. An alternative interpretation was given by Brink [6,7], in which the excited states have a different geometric configuration as the ground state.
Specifically, in 12 C the ground state was suggested to be an equilateral triangle (D 3h symmetry) and the excited state to be linear (C ∞v symmetry) [82] or bent (C 2v symmetry) as obtained in lattice EFT calculations [35]. Similarly in 16 O, the ground state was suggested to be a regular tetrahedron (T d symmetry) and the excited state to be a square (D 4h symmetry) [36]. This situation, in which the symmetry of the state changes as a function of excitation energy is called an excited-state quantum-phase transitions (ESQPT) [83].
Work is currently underway to see whether experimental data support Brink's hypothesis [6,7] or lattice EFT calculations [34] or rather the oblate structure of the previous sections for the excited 0 + 2 state (Hoyle state) of 12 C. Within the ACM, the transition from bent to linear can be studied by adding to the Hamilonian of Eq. (17) which describes triangular configurations, a term ǫn [39,77]. Bent to linear transitions have been extensively investigated in molecular physics by making use of the algebraic approach described here [84,85]. the motion of a single-particle in the potential, V ( r), obtained by convoluting the density with the nucleon-alpha interaction v( r − r ′ ), Several forms of the nucleon-alpha interaction have been considered. By taking a Volkov-type Gaussian interaction [86], one obtains a potential V ( r) with the same dependence on r, θ, φ as in the density of Eq. (40), but with a different value of the parameter α. The basic form of the potential has been assumed to be where f λ (r) = e −α(r 2 +β 2 ) 4πi λ (2αβr) .
In addition, the odd-particle experiences also a spin-orbit interaction. Since V ( r) is not spherically symmetric, one must take for V so ( r) the symmetrized From Eq. (56), one haŝ Finally, if the odd particle is a proton one must add the Coulomb interaction between the odd particle and the cluster, given by The total single-particle Hamiltonian is then the sum of kinetic, nuclear spinindependent, nuclear spin-orbit, and Coulomb terms The single-particle energies, ǫ Ω , and the intrinsic wave functions, χ Ω , are then

Dumbbell configuration
The energy levels of a neutron in a potential with Z 2 symmetry are given in Fig. 24. At β = 0 the single-particle levels are those of the spherical shell model and can be labelled accordingly. As β increases the spherical levels split.
However, since the potential has axial symmetry, the projection of the angular momentum on the symmetry z-axis, K, is a good quantum number. All levels are doubly degenerate with ±K. The values of K contained in each j level are  Fig. 24, levels are labelled by K and the parity P = (−) l of the spherical level nlj from which they originate. These quantum numbers are conserved in the correlation diagram of Fig. 24. Alternatively the energy levels can be labelled by the molecular notation nσK, nπK, nδK, . . . and a g (gerade), u (ungerade) label. Here n = 1, 2, . . . denotes the 1 st , 2 nd , . . . , state, σ, π, δ, . . . denotes the projection of the orbital angular momentum L on the z-axis in spectroscopic notation 0 ≡ σ, 1 ≡ π, 2 ≡ δ, . . ., K the total projection including spin, and g, u the parity, g ≡ +, u ≡ −. which goes to zero at r → ∞, while in [60] a harmonic oscillator potential is used that goes to infinity at r → ∞.

Equilateral triangle configuration
The energy levels of a neutron in a potential with D 3h symmetry are shown in Fig. 26. At β = 0 the levels are again those of the spherical shell-model. As β increases, the levels split. The resolution of the representations D P j of SU (2) with angular momentum j and parity P into representations of the double group D ′ 3h which describes fermions is a complicated group-theoretical problem. It was solved by Koster [89] for applications to crystal physics and by Herzberg [70] for applications to molecular physics. The solution is given in Table 6, where we have used a notation more appropriate to nuclear physics [90]. Various notations are used for representations of D ′ 3h , whose conversion is E notation is the nuclear physics notation [90], the second is that of Herzberg [70], the third is that of Koster [89] and the fourth is that of Hamermesh [91].
The representations E 1/2 and E 3/2 can be further decomposed into with n = 1, 2, 3, . . . , and K > 0. The angular momenta are given by J = K, K + 1, K + 2, . . .. As a result, the rotational sequences for each one of the irreps of D ′ 3h are given by (see also Table 6) . .

Tetrahedral configuration
The energy levels of a neutron in a potential with T d symmetry are shown in Fig. 27. The resolution of single-particles levels j P into representations of T ′ d is given in Table 7. In this table, the notation of [89] is used as well as the notation appropriate to nuclear physics. The conversion between our notation  and that of others is E where the second notation is that of Herzberg [70], the third is that of Koster [89] and the fourth is that of Hamermesh [91].
In this case the projection of the angular momentum K is not a good quantum number. From Table 7

Energy formulas
We consider here the rotational and vibrational spectra of rigid configurations. Only the dumbbell and equilateral triangle configurations have been analyzed so far.
The rotational spectra of a dumbbell configuration can be analyzed with the energy formula [62] E rot (Ω, where J = K, K + 1, K + 2, . . .. The energy levels depend on the inertial parameter A Ω =h 2 /2I, where I is the moment of inertia, and on the so-called decoupling parameter a Ω [92] a where the expansion coefficients are given by Eq. (62) and Ω is restricted to (66) is identical to that used in the collective model which describes the rigid motion of an ellipsoidal shape [92]. The moment of inertia I in odd nuclei can be obtained by adding the contribution of the odd particles I n to that of the cluster I c where I c is given in Eq. (44). The assumption here is that the odd particle is dragged along in a rigid fashion. The odd particle contribution to the three components of the moment of inertia can be calculated as where m is the nucleon mass, and χ Ω is the intrinsic wave function of Eq. (62).
The vibrational spectra of a dumbbell configuration plus additional particles can be analyzed with the formula where the zero-point energy has been removed. There is in this case only one vibrational quantum number v Ω = 0, 1, . . . , as in Eq. (16).

Equilateral triangle configuration
An expression similar to Eq. (66) applies to the rotational energy levels of an equilateral triangle configuration. The rotational formula is where ε Ω is the intrinsic energy, A Ω =h 2 /2I the inertial parameter, b Ω a Coriolis term, and a Ω the decoupling parameter. The latter term applies only to representations Ω = E where again the zero-point energy has been removed. There are here three vibrational quantum numbers and therefore the situation is more complex than in the case of a dumbbell configuration. In Fig. 28, the expected vibrational levels of a triangular configuration are shown.

Electromagnetic transition probabilities
Electromagnetic transition probabilities and moments can be calculated in the same way as in the collective model. The wave functions are factorized as a product of the intrinsic wave functions, χ Ω , obtained as in Sect. 7, the vibrational functions of the cluster ψ vib which depend on the vibrational quantum numbers v i and a rotational part which is that of the symmetric top, where Ω, K labels the intrinsic state, J the angular momentum, and M and K its projection on the z-axis and the symmetry axis, respectively. Eq. (73) is valid when K is a good quantum number, as is the case for the dumbbell and equilateral triangle configuration.
The electric and magnetic multipole operators in the laboratory frame are written as a sum of single-particle and cluster contributions The matrix elements of the operators in Eq. (74) can be calculated in the stan-dard way and the transition probabilities, defined as thus obtained as The two terms in Eq. (76) come from the symmetrization of the wave function in Eq. (73), and The second term in Eq. (76) contributes only in the case λ ≥ K + K ′ .
Similarly, the electric multipole moments are defined in the usual fashion as Electromagnetic transition rates and moments in odd nuclei have contributions from both the single particle and the cluster, Eq. (74). The single-particle contribution is written in the standard form [93,94] T el,sp where e eff is the effective charge center-of-mass corrected [94] and the g-factors are given by The cluster contribution depends on the vibrational quantum numbers v i and on the charge and magnetization distribution. The electric cluster contribution can be evaluated using the algebraic cluster model (ACM) described in Sect. 2, and it depends on the configuration.

Equilateral triangle configuration
For the equilateral triangle configuration where c λ is given by Eq. (31).
The magnetic cluster contribution is rather difficult to evaluate. Since the cluster is composed of spin-less α-particles has been taken in all calculations performed so far.

Form factors in electron scatterng
Form factors in electron scattering can also be split into a single-particle and collective cluster contribution, The single-particle contribution F sp gives rise to longitudinal electric, transverse magnetic and transverse electric form factors. These contributions were derived in the laboratory frame by De Forest and Walecka [95]. They were converted to the intrinsic frame in [62], where explicit expressions are given. Since the cluster contribution is composed of spin-less α-particles, it is assumed that the cluster contribution F c applies only to the longitudinal form factors.

Dumbbell configuration
For the dumbbell configuration where λ = even = 0, 2, . . .. Here α and β are the parameters of the cluster density of Eq. (41). For odd multipolarities it has a more complicated dependence on β, as discussed in [62].

Evidence for cluster structure in odd nuclei
The CSM provides a simple way to analyze cluster structures in kα+x nuclei, In the following subsections, however, we analyze only cluster structures in nuclei with Z 2 symmetry plus one particle, 9 Be and 9 B, and in nuclei with D 3h symmetry plus one particle, 13 C. The study of nuclei with T d symmetry plus one particle, 17 O and 17 F, and of nuclei with Z 2 , D 3h and T d symmetry plus two particles, 10 Be, 14 C and 18 O, is planned for future investigations. We note here that while the case of the two-center shell model (Z 2 symmetry) has been extensively investigated [87,88], the three-and four-center shell model has not been studied within the context of nuclear physics.

Dumbbell configuration
The energy spectrum of 9 Be is shown in Fig. 29 where it is compared with the experimental spectrum. Three rotational bands have been observed with K P = 3/2 − , 1/2 − and 1/2 + which can be assigned to the representations Ω = [1π u 3/2], [1π u 1/2], [2σ g 1/2], respectively. It is convenient to visualize the three bands by plotting the energies of each level as a function of J(J + 1) as shown in Fig. 30. It is seen that the band with K P = 1/2 + has a large decoupling parameter. The ACM appears to describe the energy levels well, including the large decoupling of the K P = 1/2 + . Table 8 shows a comparison between the experimental inertia and decoupling parameters, A Ω and a Ω . The agreement is remarkable in view of the fact that there are no free parameters that have been adjusted, the value of β having been fixed from the moment of inertia of 8 Be.
The same situation occurs for the nucleus 9 B. In Fig. 31 the spectrum of 9 B is shown in comparison with CSM. The Coulomb displacement energies between states in 9 Be and 9 B are calculated well as shown in Table 9 [62]. Rotational

Equilateral triangle configuration
The rotational bands of 13 C are shown in Fig. 33  4π and quadrupole moment Inserting the value of β as determined from the moment of inertia of 8 Be we obtain the results of Table X. The agreement between theory and experiment is excellent and provides the strongest argument for the cluster structure of 9 Be seen as 8 Be + n.
Magnetic transitions within a rotational band are determined by the single- and quadrupole moment where G sp 1,0 is given by Eq. (77). Inserting g n s = −3.8256 gives the results of Table 10. The magnetic moment is well reproduced while B(M 1) is a factor of ∼ 2 smaller than the experimental value. (The units of B(M 1) are those used in electron scattering [96].)

Equilateral triangle configuration
Some calculations are available for electromagnetic transition rates in 13 C. Table 11 shows results for electric transitions in the ground state band, repre- 3h . One should note the large B(E3) value for the transition 5/2 + → 1/2 − .    Table 10. The same problem appears in large shell model calculations as reported in [96]. The disagreement may be due to an inconsistency between experiments measuring the magnetic moments and those extracting the form factors from electron scattering.

Equilateral triangle configuration
Only some longitudinal electric form factors have been calculated so far in 13 C. Fig. 36 shows the results for E2 form factors for the ground state band.
Particularly noteworthy is the fact that the two form factors in Fig. 36 are expected to be identical in the D ′ 3h symmetry, and indeed appear to be so.
where ε Ω , A Ω , b Ω and a Ω have the same meaning as in Eq. (71), and η Ω is a stretching parameter. Similarly, the vibrational energy needs to be modified to for a dumbbell configuration, and for a triangular configuration. The values of x ij,Ω are the anharmonicities.

Summary and conclusions
In this article, the cluster structure of light nuclei has been reviewed. In the first part, cluster structures in kα nuclei with k = 2 ( 8 Be), k = 3 ( 12 C) and k = 4 ( 16 O) have been analyzed in terms of the algebraic cluster model (ACM). The advantage of this model is that it produces explicit analytic results for energies, electromagnetic transition rates and form factors in electron scattering. Evidence for a cluster dumbbell configuration in 8 Be, an equilateral triangle configuration in 12 C and a tetrahedral configuration in 16 O has been presented. This evidence confirms early suggestions [6,7,12] for the occurrence of these configurations in 8 Be, 12 C and 16 O. The ACM makes use of algebraic methods adapted to the symmetry of the structure which is Z 2 (dumbbell), D 3h (equilateral triangle) and T d (tetrahedron). These symmetries are exploited to obtain the analytic results that are used to analyze experimental data.
In the second part, cluster structures of kα + x nuclei are analyzed in terms of a cluster shell model (CSM). The advantage of this model is that single particle levels in cluster potentials with arbitrary discrete symmetry can be easily calculated. The three cases of single-particle levels in cluster potentials with discrete symmetry Z 2 , D 3h , T d are shown explicitly. Here again the use of symmetry considerations plays an important role, particularly in the classification of states through the use of the double groups Z ′ 2 , D ′ 3h , T ′ d . Evidence for cluster structures in the odd nuclei 9 Be, 9 B (k = 2, x = 1) and 13 C (k = 3, x = 1) is presented. This evidence demonstrates that cluster structures survive the addition of one nucleon, and confirms early suggestions [57][58][59][60] that spectra of 8 Be and 9 B can be well described as 8 Be plus one particle.
We emphasize that in the ACM and the CSM most results can be obtained in terms of a single parameter, β, which represents the distance from the center of mass of the α particles to the center of mass of the nucleus. The value of this parameter is ∼ 2 fm for all nuclei described in this review. This is an astonishing result which supports the "simplicity in complexity" program advocated by the authors.
In this program of investigation of cluster structures in light nuclei what remains to be done is: (1) in even nuclei the study of 20 Ne (k = 5), 24 Mg (k = 6) and 28 Si (k = 7) suggested in [6,7] to have bi-pyramidal (k = 5), octahedral or bi-pyramidal (k = 6) and stacked triangular body-centered (k = 7) structure with symmetry D 3h , O h or D 2h , D 3h or D 3v , respectively; (2) in odd nuclei, the study of 17 O, 17 F (k = 4, x = 1) and, most importantly, the study of kα + x nuclei with x > 1, especially 10 Be, 10 B and 11 Be suggested in [57,58] to have a dumbbell configuration plus x = 2 and x = 3 particles. The case of 10 Be and 11 Be is particularly timely since many experimental studies of these nuclei have been done in recent times. A preliminary calculation of 10 Be within the framework of the CSM plus residual interactions has been done, which appears to indicate that cluster structures even survive the addition of two nucleons.
Most importantly, the review presented here in which most results are given in explicit analytic form, provides benchmarks for microscopic studies of cluster structures in light nuclei.

Appendix A. Permutation symmetry
For k identical clusters, the Hamiltonian has to be invariant under their permutation. Therefore, the eigenstates can be classified according to the representations of the permutation group S k . The permutation symmetry of k objects is determined by the transposition P (12) and the cyclic permutation P (12 · · · k) (see Table 2). All other permutations can be expressed in terms of these elementary ones [69]. In this appendix we review the construction of eigenfunctions of the ACM Hamiltonian with definite permutation symmetry for cluster composed of k = 2, 3 and 4 α-particles, and clarify the notation used in Tables 2 and 3 [63].

Appendix A.1. Dumbbell configuration
For the permutation of two objects there are two different symmetry classes characterized by the Young tableaux [2] and [11]. Due to the isomorphism S 2 ∼ Z 2 , the three symmetry classes can also be labeled by the irreducible representations of the point group Z 2 as [2] ∼ A and [11] ∼ B ( Table 3).
The permutation symmetry can be determined by considering the transposition P (12) In the ACM, the transformation properties under S 2 ∼ Z 2 follow from those of the building blocks. Algebraically, the transposition can be expressed as  (Table 3).
In this case, the permutation symmetry can be determined by considering the transposition P (12) and the cyclic permutation P (123). The transformation properties of the three different symmetry classes under P (12) and P (123) are given by and P (123) In the ACM, the transformation properties under S 3 ∼ D 3 follow from those of the building blocks. Algebraically, the transposition and cyclic permutation can be expressed in terms of the generators b † i b j ≡ m b † i,m b j,m that act in index space (i, j = ρ, λ). The transposition is given by and θ = arctan √ 3. The scalar boson, s † , transforms as the symmetric representation [3] ∼ A 1 , whereas the two vector Jacobi bosons, b † ρ and b † λ , transform as the two components of the mixed symmetry representation [21] ∼ E.

Appendix A.3. Tetrahedral configuration
For the permutation of four objects there are five different symmetry classes characterized by the Young tableaux [4], [ Table 3).
The transformation properties of the five different symmetry classes under the transposition P (12) and the cyclic permutation P (1234) are given by and P (1234) In the ACM, the transformation properties under S 4 ∼ T d follow from those of the building blocks. The transposition is given by The discrete symmetry t of a given wave function can be determined by evaluating the matrix elements ψ t | P (12) |ψ t = ψ t | U tr |ψ t = ±1 , ψ t | P (1234) |ψ t = ψ t | U cycl |ψ t , (A. 16) and comparing with Eqs. (A.11) and (A.12).