Dynamically Generated Secondary Rainbow

• Dynamically generated secondary rainbow is a secondary refractive caustic created when channel coupling induces an extra stationary point in the deflection function.
• The process involves coupling to collective excitations that modify the real interaction potential, generating a new Airy sequence in the elastic channel.
• This phenomenon resolves scattering anomalies in systems like 12C+12C, refining deep refractive potential models and linking high-energy oscillatory patterns to low-energy molecular dynamics.

A dynamically generated secondary rainbow is a secondary refractive caustic that is absent in a static single-channel description and appears only when dynamical effects create an additional stationary point of the deflection function. In heavy-ion scattering, the term denotes a second nuclear rainbow produced by channel coupling to collective excitations, which induces a dynamical polarization potential and an additional Airy sequence in the elastic channel. The concept has become central to the interpretation of refractive scattering in systems such as $^{16}$O+$^{12}$C, $^{13}$C+$^{12}$C, and especially $^{12}$C+$^{12}$C, where it resolves the long-standing anomaly in the $90^\circ$ excitation function and establishes the existence of a fourth Airy elephant (Ohkubo et al., 26 Jul 2025, Ohkubo et al., 2014).

1. Definition and physical basis

Nuclear rainbow scattering is the refractive, farside-dominated component of elastic heavy-ion scattering generated by a deep, attractive mean-field potential between the colliding nuclei. In the semiclassical picture, the deflection function $\Theta(l)$ has a single extremum when refraction is strong, and rays clustered near that extremum form Newton’s zero-order, or primary, nuclear rainbow. Unlike the meteorological rainbow, the nuclear rainbow is produced by refraction alone and does not require internal reflection.

A dynamically generated secondary rainbow is a distinct phenomenon. It is not a higher-order reflection effect and is not present in the single-channel mean-field picture. Instead, it arises when channel coupling to low-lying collective states modifies the elastic interaction through a dynamical polarization potential. That induced potential changes the refractive landscape, produces an additional extremum in the deflection function, and generates a second Airy pattern on the dark side of the primary bow. This mechanism was first demonstrated for $^{16}$O+$^{12}$C, then confirmed in $^{12}$0C+$^{12}$1C, and later established in $^{12}$2C+$^{12}$3C (Ohkubo et al., 2014, Ohkubo et al., 26 Jul 2025).

Airy structures are the oscillatory minima and maxima associated with rainbow scattering. Near a rainbow angle, the scattering amplitude is described by an Airy function, and the ordered minima are labeled $^{12}$4, $^{12}$5, and so forth. In the $^{12}$6C+$^{12}$7C literature, the gross humps in the $^{12}$8 excitation function are called “Airy elephants”; they are separated by the energies at which Airy minima cross $^{12}$9 (Ohkubo et al., 26 Jul 2025).

2. Semiclassical and coupled-channel description

The elastic scattering amplitude and differential cross section are written as

$^{13}$0

For identical spin-0 bosons such as $^{13}$1C+$^{13}$2C, the observable cross section must be symmetrized,

$^{13}$3

and this symmetry interference becomes decisive near $^{13}$4.

The semiclassical rainbow condition is expressed through the deflection function,

$^{13}$5

or, in impact-parameter form, through the stationary points of $^{13}$6. Near a rainbow angle $^{13}$7, the amplitude is described by a uniform Airy approximation,

$^{13}$8

so the zeros of $^{13}$9 generate the ordered Airy minima. Nearside–farside decomposition shows that the rainbow region is dominated by the farside component, confirming its refractive origin (Ohkubo et al., 26 Jul 2025, Ohkubo et al., 18 Sep 2025).

Microscopically, the real interaction is constructed with an extended double-folding formalism,

$^{12}$0

and, in coupled channels,

$^{12}$1

The coupled-channel radial equations then determine the elastic-channel $^{12}$2-matrix and thereby the modified deflection function (Ohkubo et al., 26 Jul 2025, Ohkubo et al., 18 Sep 2025).

3. Microscopic origin of the secondary bow

In the $^{12}$3C+$^{12}$4C analyses, the real folded interaction is based on microscopic $^{12}$5C densities and transition densities from the three-$^{12}$6-cluster resonating group method of Kamimura, combined with the density-dependent finite-range interaction DDM3Y-FR. Coulomb folding is included analogously, while absorption is represented by a Woods–Saxon volume-type imaginary potential. Reported real renormalizations include $^{12}$7 at $^{12}$8 MeV, $^{12}$9 at $^{12}$0 MeV, and $^{12}$1 at $^{12}$2 MeV; reported imaginary strengths range from $^{12}$3 to $^{12}$4 MeV with $^{12}$5 fm and $^{12}$6 fm (Ohkubo et al., 26 Jul 2025, Ohkubo et al., 18 Sep 2025).

The six-channel $^{12}$7C+$^{12}$8C scheme includes

$^{12}$9

Mutual excitation of the $^{12}$0 state and orientation effects are treated explicitly. The secondary bow is generated because these couplings induce a dynamical polarization potential with energy- and radius-dependent structure. In the single-channel problem, $^{12}$1 has one extremum; with coupling, the induced potential can create an additional extremum at larger scattering angles, producing a second refractive trajectory and a second Airy system. In $^{12}$2C+$^{12}$3C, the dominant driver is coupling to the strong $^{12}$4 state, and two-channel coupling $^{12}$5 is already sufficient to generate the secondary bow, while mutual excitation shifts $^{12}$6 and $^{12}$7 toward the full six-channel result (Ohkubo et al., 18 Sep 2025).

This mechanism is structurally different from any explanation based on renormalizing a static potential. The coupled-channel studies in $^{12}$8O+$^{12}$9C and the inversion analysis of the resulting elastic $90^\circ$0-matrix show that the local dynamical polarization potential has a strongly radius-dependent real part and a nontrivial imaginary part that cannot be reproduced by uniform renormalization. A plausible implication is that the secondary rainbow is best understood as a coupling-induced refractive catastrophe rather than as a perturbative displacement of the primary bow (Mackintosh et al., 2015).

4. The $90^\circ$1C+$90^\circ$2C anomaly and the fourth Airy elephant

The $90^\circ$3C+$90^\circ$4C system posed a historical puzzle because the primary $90^\circ$5 minimum crosses $90^\circ$6 at $90^\circ$7 MeV, or $90^\circ$8 MeV for the symmetric system. This is much lower than the corresponding $90^\circ$9 crossings in $\Theta(l)$0O+$\Theta(l)$1C, where $\Theta(l)$2 MeV, and in $\Theta(l)$3O+$\Theta(l)$4O, where $\Theta(l)$5 MeV. Earlier deep-potential rainbow analyses established the primary rainbow but left the $\Theta(l)$6 excitation-function discrepancy unresolved; Demyanova and collaborators concluded from precise data at $\Theta(l)$7 MeV that no Airy minimum crosses $\Theta(l)$8 above $\Theta(l)$9 MeV if only the primary rainbow is considered (Ohkubo et al., 26 Jul 2025).

The resolution is that the last relevant $^{16}$0 crossing in $^{16}$1C+$^{16}$2C is not the primary $^{16}$3, but the secondary $^{16}$4, which crosses $^{16}$5 at $^{16}$6 MeV, or $^{16}$7 MeV. This restores the inter-system systematics near $^{16}$8 MeV and establishes a fourth Airy elephant between the primary $^{16}$9 and the secondary $^{12}$0 (Ohkubo et al., 26 Jul 2025).

Crossing at $^{12}$1	Approx. $^{12}$2	Structural role
$^{12}$3	$^{12}$4 MeV	first primary boundary
$^{12}$5	$^{12}$6 MeV	second primary boundary
$^{12}$7	$^{12}$8 MeV	third primary boundary
$^{12}$9	$^{12}$00 MeV	last primary boundary
$^{12}$01	$^{12}$02 MeV	secondary boundary creating the fourth elephant

In this scheme, there are four gross humps in the $^{12}$03 excitation function: between $^{12}$04 and $^{12}$05, $^{12}$06 and $^{12}$07, $^{12}$08 and $^{12}$09, and finally between $^{12}$10 and $^{12}$11. The highest-order primary minimum $^{12}$12 fades at lower energies and does not cross $^{12}$13. The recognition of the dynamically generated secondary rainbow therefore resolves a decades-long interpretive problem and refines the extraction of the deep refractive potential that also underpins quasi-molecular structure and molecular resonances in the $^{12}$14Mg compound system (Ohkubo et al., 26 Jul 2025).

5. Angular distributions, ripples, and threshold behavior

The coupled-channel extended double-folding calculations for $^{12}$15C+$^{12}$16C reproduce the angular distributions in the $^{12}$17–$^{12}$18 MeV laboratory-energy range and identify both primary and secondary Airy minima. At $^{12}$19 MeV, the primary minimum is at $^{12}$20 and the secondary minimum at $^{12}$21. At $^{12}$22 MeV, the corresponding values are $^{12}$23 and $^{12}$24. The large-angle fall-off beyond roughly $^{12}$25 at $^{12}$26 MeV is the dark side of the primary rainbow, and the secondary bow develops within that region (Ohkubo et al., 18 Sep 2025).

In the symmetric $^{12}$27C+$^{12}$28C system, the secondary bow is not observed as a smooth bright bump. Because the observable amplitude is $^{12}$29, bosonic symmetrization generates strong interference near $^{12}$30, breaking up the bright side of the secondary bow into ripples superimposed on the Airy structure. The unsymmetrized calculation reveals the secondary bow more clearly, while the symmetrized calculation reproduces the experimentally observed non-monotonic large-angle behavior. The ripples are therefore not simply nearside–farside interference; they arise from symmetry interference acting on a refractive farside amplitude (Ohkubo et al., 18 Sep 2025).

The energy evolution is equally diagnostic. With the six-channel EDF potential, no secondary bow appears at $^{12}$31 MeV, where $^{12}$32 and the large-angle distribution remains a fall-off. By $^{12}$33 MeV, the fall-off halts and a plateau forms near $^{12}$34. By $^{12}$35 MeV, a clear $^{12}$36 and $^{12}$37 appear. The secondary bow persists to $^{12}$38 MeV, where $^{12}$39, $^{12}$40, and $^{12}$41. The threshold behavior around $^{12}$42 MeV is explained by the fact that inelastic $^{12}$43 cross sections are very strong and can exceed elastic cross sections in the rainbow region; as the primary fall-off is pushed forward with energy, coupling-driven large-angle strength becomes comparatively dominant (Ohkubo et al., 18 Sep 2025).

6. Comparative nuclear systematics

The first clear evidence for a dynamically generated secondary nuclear rainbow came from $^{12}$44O+$^{12}$45C elastic scattering. Measurements at $^{12}$46 MeV extended the angular coverage to about $^{12}$47 and found an Airy minimum near $^{12}$48, far larger than the $^{12}$49 expected from the established global potential. Coupled-channel EDF calculations with the $^{12}$50C $^{12}$51 and $^{12}$52 states reproduced the large-angle minimum and showed, at $^{12}$53 MeV, multiple extrema in the deflection function: $^{12}$54, $^{12}$55, and $^{12}$56. Coupling to the $^{12}$57C $^{12}$58 state was essential, the $^{12}$59 state alone had negligible effect, and removing the imaginary potential left $^{12}$60 intact, demonstrating that the secondary bow is generated by the real coupled interaction rather than by absorption (Ohkubo et al., 2014).

The same mechanism was then established in $^{12}$61C+$^{12}$62C at $^{12}$63 MeV. There, a distinct minimum at $^{12}$64 could not be reproduced by an uncoupled folding potential but emerged when coupling to the $^{12}$65C $^{12}$66 state was included. The analysis further showed that the quadrupole $^{12}$67 reorientation term of the $^{12}$68 channel is essential for the appearance of the secondary bow, because removing that term eliminates the extra Airy minimum even when the $^{12}$69 transition coupling is retained. The same framework predicts $^{12}$70 near $^{12}$71 at $^{12}$72 MeV and near $^{12}$73 at $^{12}$74 MeV, with a sharper forward secondary minimum by $^{12}$75 MeV (Ohkubo et al., 2015).

The $^{12}$76C+$^{12}$77C case is therefore not an isolated anomaly but the symmetric realization of a broader heavy-ion refractive systematics. What makes it distinctive is that the secondary $^{12}$78 actually crosses $^{12}$79, whereas in $^{12}$80O+$^{12}$81C the secondary bow is observed at angles smaller than $^{12}$82. That difference is what allows the dynamically generated secondary rainbow to resolve the Airy-elephant problem uniquely in the $^{12}$83C+$^{12}$84C system (Ohkubo et al., 26 Jul 2025).

7. Analogues beyond heavy-ion scattering

A mathematically related use of the concept appears in wave scattering by a Schwarzschild black hole surrounded by a thin spherical shell of matter. In that system, the shell makes the classical deflection function $^{12}$85 non-monotonic, producing both a local maximum and a local minimum. These two stationary points generate two fold caustics, interpreted as primary and secondary rainbows. For $^{12}$86 and $^{12}$87, the reported extrema occur at $^{12}$88 and $^{12}$89; as $^{12}$90 increases, the two stationary points approach each other and eventually disappear, restoring the monotonic behavior of a bare Schwarzschild spacetime. The associated wave-scattering signal exhibits Airy-type peaks and supernumerary oscillations near the rainbow angles (Leite et al., 2019).

A different analogue is provided by geometric-optics studies of rainbows in Venusian clouds composed of aqueous sulfuric acid droplets. There the secondary bow is the ordinary $^{12}$91 reflection rainbow, but its angular position is strongly modulated by the refractive index $^{12}$92 of the solution. At $^{12}$93 nm, increasing the acid mass fraction from $^{12}$94 to $^{12}$95 shifts the primary radius from $^{12}$96 to $^{12}$97, the secondary radius from $^{12}$98 to $^{12}$99, and Alexander’s dark band from $^{13}$00 to $^{13}$01. This suggests a broader cross-disciplinary pattern: a secondary rainbow becomes “dynamic” whenever changes in geometry, medium, or coupling create or strongly shift an additional caustic, even though the nuclear case remains distinctive because its secondary bow is generated by channel coupling rather than by internal reflection (Zaikin, 24 Feb 2026).

Across these settings, the common mathematical signature is the appearance of an additional stationary point in a deflection function and the associated Airy structure. In nuclear physics, that signature has direct spectroscopic and structural significance: it constrains the interior nucleus–nucleus interaction, exposes the dynamical role of collective excitations, and links high-energy refractive scattering to low-energy quasi-molecular dynamics.