High-spin molecular resonances in 12C + 12C

Resonances observed in the C + C collisions are studied with a molecular model. At high spins J = 10–18, a stable dinuclear configuration is found to be an equator-equator touching one. Firstly, normal modes have been solved around the equilibrium, with spin J and K-quantum number being specified for rotation of the whole system. Secondly, with respect to large centrifugal energy, Coriolis coupling has been diagonalized among low-lying 11 states of normal-mode excitations, which brings K-mixing. The analyses of decay widths and excitation functions have been done. The molecular ground state exhibits alignments of the orbital angular momentum and the C spins, while some of the molecular excited states exhibit disalignments with small widths. Those results are surprisingly in good agreement with the experiments, which will light up a new physical picture of the highspin C + C resonances.


Introduction
Well above the Coulomb barrier of the 12 C+ 12 C system, series of resonances have been found with high spins over 10h, which exhibit prominent peaks in the elastic and inelastic 2 + channels [1]. Band Crossing Model (BCM), based on the double resonance mechanism, has successfully explained resonance mechanism with the aligned configurations of the orbital angular momentum and the spins of the excited states of 12 C nuclei [2].
On the other hand, for high-spin resonances observed in 24 Mg + 24 Mg and 28 Si + 28 Si, we have developed a new molecular model [3,4]. The model has an advantage to describe various geometrical configurations of two constituent deformed nuclei with the method of normal mode around the stable energy minimum under strong nucleus-nucleus interaction, which gives intuitive understanding on nuclear structure of the molecular resonances. For the oblate-oblate systems such as 28 Si − 28 Si and 12 C − 12 C, the stable configurations with axial asymmetry exhibit a series of low-lying excited states with K = 2 or 4, in which spins of the constituent nuclei incline to the molecular z -axis. Thus, those states are expected to have an interesting property of spin disalignments with the orbital angular momentum in contrast to the alignments in BCM. How to resolve an apparent contradiction?
In rotating systems, Coriolis coupling plays an important role in optimization of the centrifugal energy. Actually, a diagonalization of the coupling has brought a result that the molecular ground state with the K = 1 mixed configuration has aligned components similar to those of BCM [5]. In addition, a chain coupling of Coriolis is expected to connect the K = 2 and K = 4 states with the elastic channel, which gives rise to intermediate structures comparable to the observation.

Dynamical aspect and normal modes
Assuming a constant deformation and axial symmetry of the constituent nuclei, we have seven degrees of freedom (q i ) = (θ 1 , θ 2 , θ 3 , R, α, β 1 , β 2 ), as illustrated in fig. 1. The relative vector (R, θ 2 , θ 1 ) of two 12 C nuclei defines the molecular z -axis. Orientations of the symmetry axes of 12 C as internal degrees of freedom are described with Euler angles (α i , β i ) referring to the : Dinuclear molecular coordinates; seven degrees of freedom of 12 C + 12 C (oblate-oblate system) are described with (q i ) = (θ 1 , θ 2 , θ 3 , R, α, β 1 , β 2 ).  molecular axes. With α 1 and α 2 , θ 3 = (α 1 + α 2 )/2 of the total K-rotation around z -axis and α = (α 1 − α 2 )/2 of twisting motion are defined. The kinetic energy is given classically in terms of angular velocities at first, and then the quantization is done to describe the whole rotating system. The resultant kinetic energy is classified into three parts, aŝ T =T rot +T int +T C , whereT rot denotes the rotational energy with the total angular momentum operatorĴ ,T int the energy of internal motions, and finallyT C the Coriolis coupling. The nucleus-nucleus interaction is given by a folding potential with nucleon-nucleon interaction DDM3Y. Expecting a stable configuration, we describe the interaction potential by the internal degrees of freedom (R, α, β 1 , β 2 ), which specify geometrical configurations. Consistently with the coordinate system, at first we introduce a rotationvibration type wave function specified with K-quantum number as basis one, where χ K describes the internal motions. In order to know dynamical aspects of multidimensional internal motions, we calculate the effective potential for given J and K, (1) where T rot denotes a multidimensional centrifugal potential with specified J and K, reduced from the rotational kinetic energy operatorT rot . In fig. 2, a multidimensional energy surface is displayed for J = 14. A local minimum is obtained around β = 90 • and R = 5.7 fm, which is called as equator-equator (E-E) configuration. The barrier height is 4.8 MeV located at R = 7.5 fm.
H coupl and H(α) for the twisting motion have been diagonalized. In fig. 3, the spectrum of the normal modes with J = 12 is displayed, classified with quantum numbers K = 0-4. The axially asymmetric shape of the whole composite system allows states with K = 2m, m being integer, in low excitation. With 1hω β excitation around the equilibrium, the system has states with K = 2m + 1. The molecular quantum numbers also indicate rotational motions of two constituent 12 C nuclei by {n 1 , μ 1 = (K + ν)/2} and {n 2 , μ 2 = (K − ν)/2}, respectively. A set of numbers I i = n i + |μ i | is related to the set of spins of 12 C nuclei, like a channel assignment. In fig. 3, single 2 + and mutual 2 + assignments are depicted with connected lines.

Role of Coriolis coupling
With the normal-mode hamiltonian, the total hamiltonian is written as H = H 0 +T C + (higher order). (2) Generally in the rotating system, we have Coriolis couplingT C , which can be described with a simple product −J · (S 1 + S 2 )/μR 2 of the total angular momentum operator J and spins of 12 C S i . Note that they refer to the molecular axes as designated with prime. In the model, it is given with many terms; J is consisted with differential operators of (θ i ), which are combined into lowering and raising operators of K, J ± = J x ± iJ y , respectively. As for spins S i , they are described usually with differential operators of Euler angles β i , i.e., given with those of the internal variables (α, β 1 , β 2 ). We consider the Coriolis terms as residual coupling, which causes Kmixing with ΔK = 1. The mixing induces aligned components. Actually, in the molecular ground state, a (K = 1, ν = 1) mixed component appears to bring strong alignments in the single 2 + channel [5]. In the rightmost column of fig. 3, levels of the K-mixed solutions caused by the Coriolis coupling are displayed for J = 12.

Molecular spectrum and excitation functions
We have found the stable E-E configuration for high spins from J = 10 to J = 18. For lower spins than J = 10, stable configurations are not found in the E-E one, but belly-belly one, etc. in closer contact of 12 C nuclei. In fig. 4, the molecular bands with the E-E structure are shown. Of course, there are many possible molecular bands for lower spins, which may correspond to those observed around the Coulomb barrier region.  We have obtained many molecular excited states with small widths for the same spin, the number of which are comparable with the experimental observations. To obtain such a wide variety of the excited states, we introduced 11 bases of the molecular-mode states with low excitation energies, corresponding to single 2 + and mutual 2 + excitations. Important are low-lying states with high μ i excitations, such as ν = 4 or K = 4, which contains components of spin disalignments due to rotations of constituent 12 C nuclei around the molecular z -axis. Surprisingly, widths of those states are not large due to very small amounts of the elastic components. The elastic components are almost exhausted in the molecular ground state and the first excited state.
In fig. 5 [6]. This spin assignment is very important, because spins of all three resonances are suggested to be J = 12 by Cormier [1]. In the present results, the molecular ground state with J = 14 appears at 19.9 MeV in good agreement with the Ledoux's data. As for J = 12, the ground state is obtained at 14 MeV, while the second and third excited states appear around 19 MeV (see fig. 5). Thus three close-by resonances obtained around E cm = 19 MeV are in good correspondence with three intermediate peaks on the gross peak observed in the single 2 + channel. It is noted that the excitation functions of both single and mutual 2 + excitations are in good agreement with experiment [1].

Conclusions
Dinuclear molecular model has been applied to high-spin resonances of the 12 C+ 12 C system. The model is capable to describe various angular momentum couplings, by the role of the Coriolis coupling. The resultant molecular ground state has its own characteristics as a token of aligned configuration, consistent with BCM, while in the excited states high K-rotational states appear relatively low in energy as non-aligned configurations. The latter multi-excited states are expected to weakly couple to the elastic channel and to be observed as resonances with small intermediate widths. As a result, characteristic features of excitation functions in the single 2 + and mutual 2 + channels, i.e., the prominent gross peaks topped with the intermediate structures have been very well reproduced over a wide energy range. This gives rise to a novel visage to the known molecular resonances. Experiments to explore nature of those resonance states are strongly called for.