Folding model analysis of the elastic and inelastic scattering of K + from 12 C

Optical potentials for the scattering of K + from the 12 C nucleus are calculated using the folding model. Angular distributions of the elastic and inelastic scattering differential cross sections at 635, 715, and 800 MeV/ c are successfully described using these potentials. Good ﬁts with data are obtained without modifying any of the potential parameters. The extracted deformation parameters and reaction and total cross sections are also considered.


Formalism
On the basis of the Watanabe superposition model [5], the optical potential of the 12 C nucleus has been investigated in terms of the α-particle optical potential, using the 3α-cluster structure to describe the 12 C nucleus: wherer is the separation vector between the centers of the two scattered particles, V 1 , V 2 , and V 3 are the optical potentials of the alpha clusters constituting 12 C and φ(R, ρ) is the internal wave function of 12 C. The internal position vectorsR andρ are defined by the position vectors of the three alpha particles constituting 12 C asR Neglecting the internal structure of the α clusters in 12 C, the internal wave function reduces to [6] |φ(R, ρ)| 2 = 24 √ 3γ 3 π 3 e −γ (4ρ 2 +3R 2 ) . ( The form of V 12 C Watanabe was obtained by Taylor expansions of V 1 (|r + 1 3ρ − 1 2R |), V 2 (|r + 1 3ρ + 1 2R |), and V 3 (|r − 2 3ρ |) aboutR =ρ = 0, up to second order inR andρ. Such expansions reduce V 12 C Watanabe of expression (1) to the simple form [7] V 12 C with where the alpha-particle potential V α is chosen to be consistent with the superposition model; V α is taken as having a phenomenological Woods-Saxon shape for both the real and imaginary parts of the potential: , and a v are, respectively, the depth, radius, and diffuseness of the real part of the potential and W, R w (= r w A 1/3 ), and a w are the corresponding quantities for the imaginary part, A = 4. The six parameters V o , r v , a v , W I , r w , and a w of the WS potential are taken to be the average values given in Table 2 in Ref. [2]; these values are V o = 28 MeV, W I = 55 MeV, r v = r w = 1.6 fm, and a v = a w = 0.7 fm. On the other hand, since φ(R, ρ) of expression (2) depends explicitly on the spatial coordinates R and ρ, φ(R, ρ) is symmetric with respect to the exchange of the spatial coordinates of the alpha clusters constituting 12 C. This allows Eq. (1) to be written as Taking the z axis along the direction of the vectorr and putting (2) into (5), one finds where μ = cos θ =r ·ρ r ρ , and θ is the angle between the vectorsr andρ. The potentials given by Eqs. (3) and (6) depend on the parameter γ of the internal wave function of 12 C. This parameter may be adjusted to give the root mean square (RMS) radius of the ground state of the 12 C nucleus. Adopting a definition for the RMS radius of the 12 C nucleus and neglecting the antisymmetrization between the three constituent alpha clusters of 12 C [8], one may write where r 2 is the mean square radius, the operator O op is given by andR i is the position vector of the ith nucleon of the 12 C nucleus. As a first approximation one neglects the internal structure of the constituent α clusters, hence the operator O op reduces to The substitution of (2) and (9) into (7) yields One may include the internal structure of the α particle by adding a term representing the RMS radius of the α particle where r 2 1/2 exp = 2.37 fm [9] and the experimental RMS radius of the α particle, r 2 1/2 α exp = 1.43 fm [10]. Neglecting the internal structure of the α particle (Eq. (10)), we get γ = 0.029 67 fm −2 . On the other hand, if we include the structure of the α particle (Eq. (11)), we get γ = 0.046 67 fm −2 . Using Eq. (11), Kermode's value of γ = 0.0884 fm −2 [6] gives an RMS radius for 12 C equal to 2.116 fm.
In the present work, we adopt the value γ = 0.046 67 fm −2 in order to calculate potentials (3) and (6). The potentials obtained are used to analyze the elastic scattering process.
Inelastic-scattering measurements are usually analyzed using a deformed optical model potential [11]. This provides a transition potential whose radial dependence is where V (r ) is the optical potential found to fit the corresponding elastic scattering (Eqs. (3) and (6)), the deformation length δ l determines the strength of the interaction, and l is the multipolarity; it denotes the corresponding deformation length for the transition to the 2 + and 3 − states in 12 C.

Results and discussion
Based upon the Watanabe superposition model, both expressions (3) and (6) fold the alpha-particle optical potential into the internal wave function of 12 C. Expression (3) is an approximate relation for V12 C in terms of V α , while expression (6) is an exact one. The potential V α used in expressions (3) and (6) is taken, as mentioned in the previous section, from the average potential parameters listed in Table 2 in Ref. [2]. They are kept fixed during the calculations. The resulting real and imaginary optical potentials for K + scattered from 12 C from expressions (3) and (6) are shown in Fig. 1. As shown in this figure, the real parts of both potentials (left panel), calculated using γ = 0.04667 fm −2 , have very similar shapes. The full folding potential, however, seems to be slightly shallower than the other one over the total plotted radial range (R = 0-9 fm). Similar behavior is noticed for the imaginary parts of both potentials (right panel). Very similar results are also found considering the other values of the γ parameter, 0.02967 and 0.0884 fm −2 . So, there is no need to plot the potentials derived using these values. Hence we can note that the Taylor expansion used here is an appropriate simplification to calculate the optical potentials of the K + -12 C system. The potentials of expression (3) and (6) are used to generate the angular distribution of differential cross sections for the elastic and inelastic scattering of K + from 12 C using the value of the parameter γ , which equals 0.04667 fm −2 . An analysis of the experimental data of K + elastically and inelastically scattered from 12 C at different momenta has been carried out using the computing program CHUCK3 [4], fed with the values obtained using expressions (3) and (6). The radial integrations have been carried out to a maximum radius of 40 fm in steps of 0.1 fm to account properly for Coulomb excitations. The calculations have been done by individually coupling each state to the ground state for all cases under consideration. The Coulomb potential used here is due to a uniformly charged sphere of radius R C = r 0C A 1/3 fm, where r 0C = 1.  corrections are negligible here due to the large binding energy of 12 C against three-alpha break-up. It may also be useful to mention here that the considered experimental data were previously analyzed using the nuclear matter densities and a Kisslinger local potential for positive kaons elastically and inelastically scattered from 6 Li and 12 C at different energies by simply taking into account relativistic kinematics [15]. It could be noted that the present calculations fit the data as well as these previous calculations. It is noted, however, that the comparison with the data is made over a limited angular range (≤50 • ); additional measurements at angles larger than 50 • are required in order to investigate the reality of the considered potentials. Therefore, the calculated differential cross sections depend consecutively on the parameter γ such that the cross sections increase as γ increases, and the fitting with the experimental data when γ = 0.04667 fm −2 is more reasonable than that for the other two values of γ at all energies considered here. The diffractive maxima and minima increase as the beam energy increases.
Since inelastic scattering in the collective model is driven by the derivative of the optical potential, these calculations are carried out for inelastic scattering to the 2 + and 3 − states of 12 C. The present calculations are made for K + at 635, 715, and 800 MeV/c using the average potential parameters mentioned above. The CCBA fits to the inelastic angular distribution for the excitation of the 2 + and 3 − states in 12 C are shown in Figs. 3 and 4. The present values of deformation length δ l are adjusted to obtain reasonable agreement with the data. To calculate the deformation parameter from the deformation length, we use the relation where β l is the deformation parameter and r o = 1.6 fm. The resulting deformation parameters for both the real and imaginary potentials are listed in Table 1. As shown in Figs. 3 and 4, the inelastic scattering data for the 2 + and 3 − collective states in 12 C are well reproduced by the deformed potentials using the same parameters employed in the elastic scattering analysis. The obtained deformation parameters β l agree well with the corresponding values extracted by others [13,16,17]. The Watanabe superposition model is thus a good alternative formalism to other sophisticated formalisms. The CHUCK3 code, using either of the two forms of potential considered here, also calculates the reaction and total cross sections, σ R and σ T , respectively, of K + scattering from 12 C at 488, 531, 656, and 714 MeV/c kaon laboratory momenta. These energies are chosen because there are corresponding calculations of σ R and σ T . Figure 5 shows the predicted values of σ R and σ T from the present work compared with those estimated by Friedman et al. [18]. The values of σ R and σ T predicted by the two potentials are in good agreement with those of Ref. [18]. However, it is noticed that, for all considered energies, the predictions of the exact expression (6) are the nearest to the corresponding cross sections estimated in Ref. [18] for both the σ R and σ T results. 6

Conclusions
Considering the 3α-cluster structure of the 12 C nucleus, the K + -12 C nuclear potential is derived using the Watanabe superposition model. Two procedures in the calculation are adopted. Firstly, we use the Taylor expansion approximation and, secondly, an exact calculation is performed using the folding model formalism. Almost identical potentials are obtained using both the considered procedures. Angular distributions of the differential cross section for K + elastically and inelastically scattered from 12 C at 635, 715, and 800 MeV/c kaon laboratory momenta are calculated, using both the extracted real and imaginary potentials. Both potentials produce a good reproduction of the data. However, the calculation of the second folding procedure is more successful in describing the data than that based upon the approximated Taylor expansion, particularly for the inelastic scattering. The extracted reaction and total reaction cross sections, as well as the deformation parameters for both considered excited states, are quite consistent with the corresponding values deduced in previous studies using more sophisticated calculations. Finally, it is worth concluding that the 3α representation of the 12 C nucleus seems to be a fruitful tool for constructing single folding optical potentials to perform successful analysis of scattering reactions involving the 12 C nucleus.