Signatures of $\alpha$-clustering in $^{12}$C and $^{13}$C

We study the cluster structure of $^{12}$C and $^{13}$C in the framework of the cluster shell model. Simple relations are derived for ratios of longitudinal form factors as well as transition probabilities. It is shown that the available experimental data for $^{12}$C and $^{13}$C can be well described by a triangular structure with ${\cal D}_{3h}$ and ${\cal D}'_{3h}$ symmetry, respectively.

An interesting question concerns to what extent the cluster structure survives under the addition of extra nucleons.The aim of this article is to address this question for the case of 12 C and 13 C.The splitting of single-particle levels in the deformed field generated by a cluster of the three α-particles with D 3h triangular symmetry was discussed in the framework of the Cluster Shell Model (CSM) [31].A study of both the rotation-vibration spectra and the electromagnetic transition rates showed that the 13 C can be considered as a system of three α-particles in a triangular configuration plus an additional neutron moving in the deformed field generated by the cluster characterized by D ′ 3h symmetry [32].
The main idea of this article is to make a comparison between electromagnetic properties of 12 C and 13 C and to establish explicit relations between Coulomb form factors and transition rates in both nuclei.These relations are used to address the question to what extent the α-cluster structure as observed in 12 C is still present in the neighboring nucleus 13 C.
Hereto we present a simultaneous study of longitudinal  13 C in the framework of an algebraic approach to clustering (ACM and CSM) which emphasizes the symmetry structure of the cluster configurations.We derive explicit expressions between longitudinal form factors and transition probabilities in 13 C which are similar to the Alaga rules in the collective geometric model.
Moreover, it will be shown that since the Coulomb form factors are dominated by the collective (or cluster) part, the q dependence of the longitudinal form factors in 12 C and 13 C is expected to be very similar.
2. Odd-cluster nucleus: 13 C The cluster shell model was introduced [31,33,32,28] to describe nuclei composed of k α-particles plus additional nucleons, denoted as kα + x nuclei.The CSM combines cluster and single-particle degrees of freedom, and is very similar in spirit as the Nilsson model [34], but in the CSM the odd nucleon moves in the deformed field generated by the (collective) cluster degrees of freedom.For the case of 13 C the total Hamiltonian is given by where H int is the intrinsic single-particle CSM Hamiltonian [31,32] i.e. the sum of the kinetic energy, a central potential obtained by convoluting the density with the interaction between the α-particle and the nucleon, a spin-orbit interaction and, for an odd proton, a Coulomb potential.Here ⃗ r i = (r i , θ i , ϕ i ) denote the coordinates of the α-particles with respect to the center-of-mass.For the case of a triangular configuration the relative distance of the three α-particles to the center-of-mass is the same r i = β, and the coordinates are given by (β, π 2 , 0), (β, π 2 , 2π 3 ) and (β, π 2 , 4π 3 ).The CSM makes use of a symmetry-adapted basis for D 3h symmetry instead of a spherical basis [32,35].For the case of triangular symmetry the eigenstates of the CSM Hamiltonian of Eq. ( 2) can be classified according to the doubly degenerate spinor representations of the double point group D ′ 3h [36]: or, in the notation of Ref. [32], as 1/2 and E 3/2 , respectively.The label γ was introduced to distinguish between the two components of each one of the doublets [35].The Hamiltonian of the CSM is solved in the intrinsic, or body-fixed, system.The single-particle energies and the intrinsic wave functions are obtained by diagonalizing the H in the harmonic oscillator basis The rotational states are labeled by the angular momentum I, parity P and its projection K on the symmetry axis, |I P K⟩.The allowed values of K P for each one of the spinor representations are given by [32,35] + , . . .[32].The complete wave function is given by 2 [35] Ωγ; where ψ v is the vibrational wave function, ϕ Ωγ the intrinsic wave function and D I M K (ω) the rotational wave function.The wave function is invariant under the transformation S −1 i S e where the operator S is the product of a rotation about π followed by a parity transformation [35].The operator S i acts on the intrinsic wave function and S e on the rotational wave function.

Longitudinal form factors
For a system consisting of a cluster of three αparticles in a triangular geometry and a single nucleon the charge distribution is taken be the sum of a Gaussian-like distribution for the three α-particles and a point-like distribution for the extra nucleon with Here (Ze) c is the electric charge of the 3α core nucleus and ẽ denotes the effective charge of the extra nucleon.
The longitudinal or Coulomb form factor F Cλ is related to the matrix element of the Fourier transform of the charge distribution summed over final and averaged over initial states with (Ze) odd denotes the electric charge of the odd nucleus.
For the vibrationally elastic case with v = v ′ = 0 the collective part is given by The single-particle part is obtained in the CSM by For longitudinal form factors which are diagonal in the intrinsic states, the contribution of the term H sp vanishes identically.As a consequence, ratios of these diagonal longitudinal form factors do not depend on the momentum transfer q, and are given simply by ratios of Clebsch-Gordan coefficients.These relations are valid for both the collective and the single-particle part, and are similar, but not identical, to the wellknown Alaga rules in the collective geometric model [37].For excitations from the ground state of 13 C with , K 1 relative to the final state with I 2 = λ + 1 2 , K 2 the ratio is given by The explicit results are shown in the last column of Table 1.

Even-cluster nucleus: 12 C
The even-cluster nucleus 12 C was described successfully in the Algebraic Cluster Model (ACM) as a triangular configuration of three α-particles [27].In particular, the L P = 5 − was predicted more than a decade before its measurement [3].The relevant point group symmetry is D 3h .The ground state band in 12 C has and consists of several rotational bands labeled by K P = 0 + , 3 − , 6 + , . .., where K is the projection of the angular momentum L on the symmetry axis.The angular momentum content is given by L P = 0 + , 2 + , 4 + , . . .for K P = 0 + and L P = K P , (K + 1) P , . . .for K ̸ = 0.The longitudinal (or Coulomb) form factor F Cλ for vibrationally elastic excitations from the ground state with A ′ 1 ⊃ A 1 symmetry is given by [27] F Cλ (q; 0 + , 0 → L P , K) where the coefficients c 2 LK can be obtained from Eq. ( 12) as Table 1: Relation between the collective contribution to the Coulomb form factors for excitations from the ground state in 12 C with L P = 0 + , K = 0 and the ground state in 13 C with The transition probabilities are obtained in the long wavelength limit as The classical result of multipole radiation for a system of three α particles located at the vertices of an equilateral triangle given in Eq. (2.21) of Ref. [27] corresponds quantummechanically to a sum over all states that can be excited from the ground state.The explicit connection is a result of the equality

Comparison between 12 C and 13 C
In this section, we discuss the Coulomb form factors and the electromagnetic transition probabilities for 12 C and 13 C.

Longitudinal form factors
There is a close relation between the Coulomb form factors and transition probabilities in the nuclei 12 C and 13 C. Fig. 1 shows the non-vanishing Coulomb form factors for vibrationally elastic excitations (v = v ′ = 0 and Ω ′ γ ′ = Ωγ) from the ground state in both nuclei.
For a given multipole the relative ratios for Coulomb form factors to excited states in 13 C are given by Eq. ( 14).In 12 C only a single state is excited for multipoles C2, C3 and C4.Even though in 13 C the strength is fragmented, it is only fragmented over a limited number of excited states, two states for C2 and C4 and 3 states for C3.
In  13 C all have the same q dependence, and the summed strength for the excitation from the ground state to excited states is the same in both nuclei where K ′ = 1 2 for K = 0 and K ′ = K ± 1 2 for K ̸ = 0.The final states in 12 C have parity P = (−1) λ and those in 13 In Fig. 2  A special case of Eq. 19 can be seen in Fig. 3 in which we compare the summed C2 strength in both nuclei.
Whereas in 12 C only the L P = 2 + state is excited, in 13 C the strength is fragmented among the 3 2 − 1 and 1 states.The experimental data in Figs. 2 and 3 show indeed that the q dependence is very similar for all three form factors, and hence also for the summed strength, in agreement with the CSM results in Eq. 19.

Transition probabilities
The transition probabilities B(Eλ) can be obtained from the Coulomb form factors in the long wavelength limit.Therefore, the B(Eλ ↑) values for excitations from the ground state of 12 C and 13 C satisfy the same The experimental data are taken from [40,41] and [44].
In Table 2 we  than that of the ground state [47].In the present approach this would imply a state-dependent value of β.
A slightly higher value of β for the L P = 3 − state by 10 % would augment the B(E3) value which depends on β 6 by a factor of 1.8, right up to the experimental value.Moreover, Eq. (15) shows that a larger value of β moves the maximum of the C3 form factor to a smaller q value thus improving the agreement with ex- perimental data.Since the main idea of this article was to establish explicit relations between Coulomb form factors and transition rates in 12 C and 13 C in order to study the extent in which the α-cluster structure as observed in 12 C is still present in the neighboring nucleus 13 C, we have used a single value of β.

Summary and conclusions
We investigated to what extent the cluster structure in 12 C survives under the addition of an extra neutron.Hereto we presented a simultaneous analysis of longitudinal form factors and transition probabilities in 12 C and 13 C in the framework of the ACM and the CSM.Since the contribution of the single-particle part is small with respect to that of the collective (or cluster) part, the q dependence of the form factors is expected to be very similar in 12 C and 13 C which is confirmed by the available experimental data for the elastic C0 and inelastic C2 form factors.It was shown that, whereas for a given multipole only a single state in 12 C is excited, in 13 C the strength is fragmented, but only over a few states.Moreover, the summed strength for excitations from the ground state is the same in both nuclei.
It is tempting to use a similar analysis to help identify the analog of the Hoyle state in 13 C.In general, the search for analogs of the Hoyle state in neighboring nuclei of 12 C is a topic of lot of interest, see Refs.[48,49] and references therein.In Fig. 4 we compare the longitudinal form factor of the Hoyle state with that of the 1 2 − 2 state at 8.86 MeV in 13 C.The q dependence is very similar, although there is some discrepancy in the absolute value.The theoretical curve corresponds to a numerical calculation in the ACM [27].The possible assignment of the 1 2 − 2 state as the analog of the Hoyle state is in agreement with previous analyses of energy systematics [32,50], charge radii [51] and spin-orbit splitting in 13 C and 13 N [52].
In conclusion, an analysis of the Coulomb form factors and transition probabilities shows evidence for the persistence of the cluster structure of three α-particles in a triangular configuration in the neighboring nucleus 13 C.   [40,41,53] and [44].The line shows calculations taken from [27], where only the core structure is considered.
p p p p p p p p p p p p p p p p p p p
we show a comparison of the elastic form factors in 12 C (left) and 13 C (right).The coefficients α and β are determined from the elastic form factors to be α = 0.56 1/fm 2 and β = 1.74 fm for 12 C, and α = 0.58 1/fm 2 and β = 1.71 fm for 13 C.The effective charge of the odd neutron is taken as the center-of-mass corrected value, ẽ = (−) λ Ze/A λ [33].The similarity between the C0 and C2 form factors in 12 C and 13 C is a consequence of the cluster structure in both nuclei.Since for these states the Coulomb form factors are dominated by the collective (or cluster) part, the q dependence of the longitudinal form factors in 12 C and 13 C is very similar.

2 − 1 and 3 2 − 1
present a comparison between calculated B(Eλ) values and quadrupole moments and experimental data.There is good agreement between the calculations and the experimental data.The quadrupole transitions from the 5 states to the ground state have very similar values as is predicted in the CSM.It is important to note that in the present calculation all states are characterized by the same value of β, the relative distance of the α-particles with respect to the center of mass.The value of β was determined by the zero of the elastic Coulomb form factor, i.e. by ground state properties.It was shown by Yamada et al. that the nuclear radius for excited states is larger

Table 1 ,
we show the relation between the Coulomb form factors in 12 C and 13 C.The longitudinal or Coulomb form factors are dominated by the collective part.Since the contribution of the single-particle part is of the order of a few percent only, it is neglected in the table.For a given multipole the Cλ Coulomb form factors in 12 C and