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Title Author(s) Citation Issue Date Similarity between nuclear rainbow and meteorological rainbow: Evidence for nuclear ripples Ohkubo, S.; Hirabayashi, Y. Physical Review C, 89(6): 61601 2014-06-27 DOI Doc URL http://hdl.handle.net/2115/56861 Right ©2014 American Physical Society Type article Additional Information File Information PhysRevC.89.061601.pdf Instructions for use Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP RAPID COMMUNICATIONS PHYSICAL REVIEW C 89, 061601(R) (2014) Similarity between nuclear rainbow and meteorological rainbow: Evidence for nuclear ripples S. Ohkubo1,* and Y. Hirabayashi2 1 Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan 2 Information Initiative Center, Hokkaido University, Sapporo 060-0811, Japan (Received 14 May 2014; revised manuscript received 10 June 2014; published 27 June 2014) We present evidence for the nuclear ripples superimposed on the Airy structure of the nuclear rainbow, which is similar to the meteorological rainbow. The mechanism of the nuclear ripples is also similar to that of the meteorological rainbow, which is caused by the interference between the externally reflective waves and refractive waves. The nuclear ripple structure was confirmed by analyzing the elastic angular distribution in 16 O + 12 C rainbow scattering at EL = 115.9 MeV using the coupled channels method by taking account of coupling to the excited states of 12 C and 16 O with a double folding model derived from a density-dependent effective nucleon-nucleon force with realistic wave functions for 12 C and 16 O. The coupling to the excited states plays the role of creating the external reflection. DOI: 10.1103/PhysRevC.89.061601 PACS number(s): 25.70.Bc, 24.10.Eq, 42.25.Gy Descartes [1] and subsequently Newton [2] explained the rainbow in optics by reflection and refraction in the raindrops. Airy [3] understood the supernumerary rainbow by the wave nature of light. The mechanism of the meteorological rainbow was understood precisely only recently by Nussenzveig using the electromagnetic theory of light [4]. In analogy with the meteorological rainbow the nuclear rainbow was predicted theoretically [5] and observed in α particle scattering [6]. The rainbow has been observed also in other systems such as in atom-atom collisions, atom-molecule collisions [7], electron-molecule collisions [8], and atom scattering from crystal surfaces [9]. Although the mechanism of Newton’s zero-order (p = 1 in Fig. 1) nuclear rainbow [10], where only refraction is active, is very different from that of the meteorological rainbow (p = 2 in Fig. 1), a similar Airy structure has been observed. As shown in Fig. 2, the precise description of the meteorological rainbow given by solving Mie scattering shows the rapidly oscillating structure, the high-frequency ripple structure, superimposed on the Airy structure of the rainbow [4,11,12]. The ripple structure is not predicted by the semiclassical theory of the nuclear rainbow of Ref. [5] and no attention has been paid to its possible existence. Here we report for the first time evidence for the existence of the ripple structure in the observed nuclear rainbow and explain its mechanism. The Airy structure of nuclear rainbows has been studied extensively especially for heavy ion scattering such as 16 O + 16 O, 16 O + 12 C, and 12 C + 12 C [13,14]. For the most typical 16 O + 16 O system, similar to the typical α + 16 O and α + 40 Ca scattering [15,16], a global deep potential has been determined uniquely from the rainbow scattering. It reproduces the experimental data over a wide range of energies from negative energy to the incident energy EL = 1120 MeV—that is, the rainbows [17], prerainbows [18], molecular resonances and cluster structures with the superdeformed configuration [19]—in a unified way. Unfortunately the observed Airy structure in the angular distributions is obscured due to symmetrization of two identical bosons. In this respect rainbow scattering of the asymmetric 16 O + 12 C system is important and has been thoroughly investigated [20–25]. A global deep potential could describe well the rainbows in the high-energy region, prerainbows [10], molecular resonances, and cluster structures with the 16 O + 12 C configuration in the quasibound energy region in a unified way [26]. However at energies around EL = 100 MeV the global optical potential calculations [23] only reproduced the experimental angular distributions in a qualitative way at larger angles. Also, to reproduce the high-frequency oscillations, imaginary potentials—with a thin-skinned volume term and an extraordinary small diffuseness parameter around 0.1 fm accompanying a surface term peaked at a larger radius—were needed [23,25]. The purpose of this paper is to show that the high-frequency oscillations superimposed on the Airy structure are nothing but the ripple structure of the nuclear rainbow, and can be explained by fully taking account of coupling to the excited states of 12 C and 16 O by using the microscopic wave functions and the extended double folding model. The mechanism of the ripple structure and the role of coupling to the excited states is clarified, and the similarity between the macroscopic meteorological rainbow and the quantum nuclear rainbow, despite the difference of the underlying interactions, is discussed. We study 16 O + 12 C scattering with the coupled channels method using an extended double folding (EDF) model that describes all the diagonal and off-diagonal coupling potentials derived from the microscopic realistic wave functions for 12 C and 16 O using a density-dependent nucleon-nucleon force. The diagonal and coupling potentials for the 16 O + 12 C system are calculated using the EDF model without introducing a normalization factor: Vij,kl (R) = * O) 12 (r1 ) ρkl( C) (r2 ) ×vNN (E,ρ,r1 + R − r2 ) dr1 dr2 , [email protected] 0556-2813/2014/89(6)/061601(5) 16 ρij( 061601-1 (1) ©2014 American Physical Society RAPID COMMUNICATIONS PHYSICAL REVIEW C 89, 061601(R) (2014) σ σ S. OHKUBO AND Y. HIRABAYASHI FIG. 1. (Color online) Illustrative figure of path of an incident ray (i) in a spherical raindrop in geometrical optics. The rays p = 0, p = 1, and p = 2 correspond to reflection, refraction only, and the primary rainbow, respectively. 16 where ρij( O) (r) is the diagonal (i = j ) or transition (i = j ) nucleon density of 16 O taken from the microscopic α + 12 C cluster model wave functions calculated in the orthogonality condition model (OCM) in Ref. [27]. This model uses a realistic size parameter both for the α particle and 12 C, and is an extended version of the OCM α cluster model of Ref. [28], which reproduces almost all the energy levels well up to Ex ≈ 13 MeV and the electric transition probabilities in 16 O. We take into account the important transition densities available in Ref. [27], i.e., g.s. ↔ 3− (6.13 MeV) and 2+ (6.92 MeV) in addition to all the diagonal potentials. 12 ρkl( C) (r) represents the diagonal (k = l) or transition (k = l) nucleon density of 12 C calculated using the microscopic three-α cluster model in the resonating group method [29]. This model reproduces the structure of 12 C well, and the wave functions have been checked for many experimental data [29]. In the coupled channels calculations we take + into account the 0+ (4.44 MeV), and 3− 1 (0.0 MeV), 2 12 (9.64 MeV) states of C. The mutual excitation channels in which both 12 C and 16 O are excited simultaneously are not included. For the effective interaction vNN we use the DDM3Y-FR interaction [30], which takes into account the FIG. 2. (Color online) Cross sections for the primary rainbow in Fig. 1. The dashed line is by classical theory in optics and the dark region before the critical angle (θc = 138◦ ) is displayed by a black shade. The dotted and solid lines show calculations of Mie scattering [11] and in the Airy approximation [11] in the wave theory of light, respectively. θ FIG. 3. (Color online) The experimental cross sections (points) in 16 O + 12 C scattering at EL = 115.9 MeV [23] are compared with the calculations (solid line) using the EDF potential: (a) the coupled channels calculations with aW = 0.2 (blue line), (b) the single channel calculations with aW = 0.6, and (c) the single channel calculations using the extremely thin-skinned volume-type imaginary potential with aW = 0.1. For comparison, in (a) the coupled channels calculations with aW = 0.4 are displayed by the green line. The calculated cross sections (solid line) are decomposed into the farside (dashed line) and nearside (dotted line) components. finite-range nucleon exchange effect. An imaginary potential (nondeformed) is introduced phenomenologically to take into account the effect of absorption due to other channels. In Fig. 3(a) the angular distributions of elastic 16 O + 12 C scattering at EL = 115.9 MeV calculated using the coupled channels method (blue solid line) are compared with the experimental data. We found that the EDF potential works well without introducing a normalization factor. The volume integral per nucleon pair of the ground state diagonal part, JV = 317.7 MeV fm3 , is consistent with those used in other optical potential model calculations and belongs to the same global potential family found in the EL = 62−1503 MeV region [23,24,31]. The parameters used in the imaginary potential with a Woods-Saxon volume-type form factor displayed in Fig. 4 are WV = 14 MeV, RW = 5.6 fm, and aW = 0.20 fm with a volume integral per nucleon pair JW = 54.3 MeV fm3 , in the conventional notation. We see that the refractive farside scattering dominates at the intermediate and large angles. The calculation reproduces well the two 061601-2 RAPID COMMUNICATIONS SIMILARITY BETWEEN NUCLEAR RAINBOW AND . . . PHYSICAL REVIEW C 89, 061601(R) (2014) FIG. 4. (Color online) The real folding and Woods-Saxon volume-type imaginary potentials used in the coupled channels calculations [Fig. 3(a)] in 16 O + 12 C scattering are displayed by the red solid line and the blue solid line, respectively. The imaginary potentials used in the single calculations in Figs. 3(b) and 3(c) are displayed by the green dotted line and the black dashed line, respectively. broad Airy maxima in the angular range, θ = 60–90◦ (Airy maximum A2) and θ = 100–140◦ (Airy maximum A1) in the experimental angular distribution, which are brought about by the refracted farside component. Also the high-frequency oscillations superimposed on the two broad Airy maxima, A1 and A2, in the experimental data are reproduced well. We see that the high-frequency oscillations are brought about by the interference between the farside and nearside scattering components. The investigation of the contributions of each channel reveals that none is overwhelmingly dominant and that the contribution of the excited states of 16 O is as much as that of 12 C, which is quite different from the higher energy region around EL = 300 MeV where coupling to the 2+ state of 12 C contributed dominantly in creating the secondary bow in the classically forbidden dark side of the primary rainbow [32]. Also neither the extremely thin-skinned volume-type imaginary potential with aW =0.1 nor the surface imaginary potential peaked at a larger radius used in Refs. [23,25] were needed. In Fig. 3(b) the angular distributions calculated in the single channel calculation using the readjusted imaginary potential (displayed in Fig. 4), WV = 11.5 MeV, RW = 5.9 fm, and aW = 0.6 fm (JW = 52.8 MeV fm3 ), which is similar to Ref. [31], are shown. Although the farside scattering is dominant, similar to Fig. 3(a), and the gross behavior of the Airy structure of the experimental angular distribution is reproduced, the high-frequency oscillations are missing. By comparing Fig. 3(a) (blue solid line) and Fig. 3(b), we note that the channel coupling to the excited states of 16 O and 12 C contributes in generating the high-frequency oscillations, although the rather small aW = 0.2 is needed. In Fig. 3(b) we note that the nearside component is damped more than two order of magnitude compared with the farside component, and no interference between them occurs resulting no highfrequency oscillations. While the JW values are almost the same for Figs. 3(a) and 3(b), the nearside scattering is retained significantly in Fig. 3(a). This means that the channel coupling in Fig. 3(a) plays a role of increasing the nearside scattering component, i.e., reflection. This can be confirmed in Fig. 3(c) where the angular distributions calculated in the single channel using both the extremely thin-skinned volume-type (Woods-Saxon squared) imaginary potential with WV = 16 MeV, RW = 4.4 fm, and aW = 0.1 fm and the surface imaginary potential with WD = 7 MeV, RD = 6.1 fm, and aD = 0.46 fm (JW = 59.8 MeV fm3 ) (displayed in Fig. 4), which are similar to those in Refs. [23] and [25], are shown. We note that the values of JW are almost the same for the three cases (a), (b), and (c). The high-frequency oscillations superimposed on the Airy structure are recovered only by using this extremely thin-skinned imaginary potential. We see that the nearside component needed to bring about the high-frequency oscillations is significantly increased compared with Fig. 3(b). The sharper the diffuseness of the imaginary potential is, the more the nearside component is increased. This can be checked by decreasing the ratio aW /WV ; that is, if we increase the strength of the imaginary potential in Fig. 3(c) by 50% to WV = 24 MeV from the original 16 MeV, the magnitude of the calculated cross section and its farside component decrease as expected from the increase of absorption. However, the magnitude of the nearside component is increased. This means that for the nearside scattering the imaginary potential does not act as absorption but acts as “divergence,” i.e., increasing reflective waves under the small diffuseness aW = 0.1. Thus the increased nearside component is found to be reflective in origin. The same behavior is observed for the coupled channels calculations in Fig. 3(a) with aW = 0.2. We see in Fig. 3(a) that the calculated cross sections with a moderate smooth diffuseness aW = 0.4 (green line) show no high-frequency oscillations, which means that no reflective waves are created. We note that the magnitudes of the S matrix in Figs. 5(a) (blue line) and 5(a) (green line) are similar to those in Figs. 5(c) and 5(b), respectively. We show that the high-frequency oscillations superimposed on the Airy structure in Fig. 3(a) are nothing but the ripple structure of the nuclear rainbow. In Fig. 2 the ripple structure in the meteorological rainbow is generated by the interference between the p = 0 external direct reflection and the p = 2 refractive rainbow rays with one internal reflection [4]. However it has been considered that in the nuclear rainbow, which is caused by a Luneberg lens [10] of a nuclear potential (p = 1) with a smoothly diffused surface, direct reflection scarcely occurs. This is seen in Fig. 3(b) where no high-frequency oscillations appear in the calculations using the real and imaginary potentials with a smooth surface. On the other hand, we see in Fig. 3(c) that the high-frequency oscillations are created by the interference between the reflective nearside component caused by the sharp-edged imaginary potential and the farside component. Thus the high-frequency oscillations superimposed on the Airy structure are considered to be the nuclear ripples because they have the same physical origin as those of the meteorological rainbow. In Fig. 3(a) it is found that the coupling to the excited states also contributes in creating the reflective nearside waves that are caused by 061601-3 RAPID COMMUNICATIONS PHYSICAL REVIEW C 89, 061601(R) (2014) σ σ S. OHKUBO AND Y. HIRABAYASHI FIG. 5. (Color online) The moduli of the calculated S matrices in Figs. 3(a), 3(b), and 3(c). The lines are explained in the caption of Fig. 3. the sharp-edged imaginary potential in Fig. 3(c). The nearside scattering waves that are responsible for the generation of the high-frequency oscillations in Figs. 3(a) and 3(c) correspond to the externally reflected waves in Mie scattering of the meteorological rainbow. Now the physical meaning and the origin of the extremely thin-skinned volume-type imaginary potential needed in Refs. [18,25] are clear. They were needed to mimic the effect of the channel coupling to enhance the reflective waves. In the single channel calculations, the increase of the nearside component that corresponds to the externally reflective waves is only attained by using the extraordinary small diffuseness parameter aW = 0.1 for the imaginary potential. This necessarily accompanies introducing the additional surface imaginary potential at the large radius to preserve the net absorption, i.e., the volume integral of the imaginary potential. In the coupled channel calculations in Fig. 3(a), in fact, no surface imaginary potential at the large radius was needed. It is important to treat channel coupling to the excited states microscopically to avoid the unphysically sharp-edged volume-type imaginary potential and the surface imaginary potential at the large radius. How the ripple structure emerges on the Airy structure by the reflective waves is shown in Fig. 6 where the angular distributions—calculated by replacing the SL matrices with the orbital angular momentum L = 0 ∼ Lc generated by the potential of Fig. 3(b) by those generated by the potential of Fig. 3(c)—are displayed. With around Lc = 8 (corresponding impact parameter b = 2.1 fm) the ripple structure starts to emerge on the A1 Airy peak in the backward angles beyond θ ≈ 100◦ and with around Lc = 14 (b = 3.6 fm) they also θ FIG. 6. (Color online) The angular distributions in 16 O + 12 C scattering at EL =115.9 MeV calculated by replacing the SL with L = 0 ∼ Lc among the SL -matrices generated by the potential used for Fig. 3(b) by those generated by the potential used for Fig. 3(c) (solid line) and its farside (dashed line) and nearside (dotted line) components are compared with the experimental data (points) [23]. appear on the Airy peak A2 in the intermediate angles before θ ≈ 90◦ . For Lc = 14–17 (b = 3.6–4.3 fm), which corresponds to the radius of the sharp edge of the imaginary potential in Fig. 4 (black dashed line), the high-frequency oscillations in phase with the experimental data are reproduced. The impact parameters of these partial waves are significantly smaller than b = 5.9 fm of the grazing partial wave L = 24 for which |SL | = 0.5 (see Fig. 5). The channel coupling in Fig. 3(a) plays two roles; that is, enhancing the reflective waves and enhancing absorption at the surface, which makes it possible to use the relatively larger diffuseness parameter aW = 0.2 fm and no surface imaginary potential needed in Fig. 3(c). Finally we mention that the high-frequency oscillations are not due to the elastic transfer of the α particle [31]. In fact, we see in the detailed coupled reaction channels calculations of 16 O + 12 C scattering at EL = 115.9 MeV in Ref. [33] that the contribution of the elastic transfer is three orders of magnitude smaller than the experimental data. Also the present calculations take into account the one-nucleon exchange effect, which is suggested to prevail over other transfer reactions [33], 061601-4 RAPID COMMUNICATIONS SIMILARITY BETWEEN NUCLEAR RAINBOW AND . . . PHYSICAL REVIEW C 89, 061601(R) (2014) by using the effective interaction DDM3Y, in which the knock-on exchange effect is incorporated [14,30]. To summarize, we have calculated 16 O + 12 C scattering with the Airy structure at EL = 115.9 MeV using a coupled channels method with an extended double folding (EDF) potential that is derived by using the microscopic realistic wave functions for 12 C and 16 O by taking account of excited states of the 2+ (4.44 MeV) and 3− (9.64 MeV) states of 12 C and the 3− (6.13 MeV) and 2+ (6.92 MeV) states of 16 O. Our calculations reproduce the high-frequency oscillations superimposed on the Airy structure. It is found that the high-frequency oscillations are nothing but the nuclear ripples similar to those superimposed on the Airy structure in Mie scattering of the meteorological rainbow. The nuclear ripples are generated by the interference between the refractive waves and the externally reflected waves. The coupling to the excited states of 16 O and 12 C plays the role of creating external reflection. 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Part of this work was supported by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. 061601-5