Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering
Nguyen Hoang Phuc, Dao T. Khoa, Nguyen Tri Toan Phuc, Do Cong Cuong
aarXiv:2102.07399v1 [nucl-th] 15 Feb 2021
Nguyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering
EPJ manuscript No. (will be inserted by the editor)
Suppression of the nuclear rainbow in the inelasticnucleus-nucleus scattering
Nguyen Hoang Phuc , Dao T. Khoa , Nguyen Tri Toan Phuc , , and Do Cong Cuong Institute for Nuclear Science and Technology, VINATOM179 Hoang Quoc Viet, Cau Giay, 100000 Hanoi, Vietnam Department of Nuclear Physics, University of Science, VNU-HCM227 Nguyen Van Cu, District 5, 700000 Ho Chi Minh City, VietnamReceived: date / Revised version: date
Abstract.
The nuclear rainbow observed in the elastic α -nucleus and light heavy-ion scattering is provento be due to the refraction of the scattering wave by a deep, attractive real optical potential. The nuclearrainbow pattern, established as a broad oscillation of the Airy minima in the elastic cross section, originatesfrom an interference of the refracted far-side scattering amplitudes. It is natural to expect a similar rainbowpattern also in the inelastic scattering of a nucleus-nucleus system that exhibits a pronounced rainbowpattern in the elastic channel. Although some feature of the nuclear rainbow in the inelastic nucleus-nucleus scattering was observed in experiment, the measured inelastic cross sections exhibit much weakerrainbow pattern, where the Airy oscillation is suppressed and smeared out. To investigate this effect, anovel method of the near-far decomposition of the inelastic scattering amplitude is proposed to explicitlyreveal the coupled partial-wave contributions to the inelastic cross section. Using the new decompositionmethod, our coupled channel analysis of the elastic and inelastic C+ C and O+ C scattering atthe refractive energies shows unambiguously that the suppression of the nuclear rainbow pattern in theinelastic scattering cross section is caused by a destructive interference of the partial waves of differentmultipoles. However, the inelastic scattering remains strongly refractive in these cases, where the far-sidescattering is dominant at medium and large angles like that observed in the elastic scattering.
PACS.
The optical model (OM) studies of elastic heavy-ion (HI)scattering usually shows a strong absorption that sup-presses the refractive (large-angle) scattering, and elas-tic HI scattering occurs mainly at the surface. However,with the nuclear rainbow pattern observed in the elasticscattering of some α -nucleus and light HI systems, the ab-sorption turns out to be much weaker and the refractive,far-side scattering becomes dominant at medium and largeangles [1,2,3]. Not only a fascinating semiclassical analogto the atmospheric rainbow, the nuclear rainbow also en-ables the determination of the real nucleus-nucleus opticalpotential (OP) with much less ambiguity, down to smallinternuclear distances [3]. The pattern of the nuclear rain-bow is usually characterized by a broad oscillation of theAiry minima [1,2,3] in the elastic cross section. The obser-vation of these minima, especially, the first Airy minimumA1 that is followed by a broad shoulder-like maximum, notonly facilitates the determination of the real OP but alsoprovides an useful probe of the cluster structure of lightnuclei [4,5]. Similarly to the atmospheric rainbow, the nuclear rain-bow can be interpreted as a pattern resulted from an in-terference of the two scattering amplitudes, as shown bythe barrier-wave/internal-wave (BI) or near-side/far-side(NF) decomposition of the elastic scattering amplitude.The BI method proposed by Brink and Takigawa [6] de-scribes elastic HI scattering in terms of the internal wavespenetrating through the potential barrier into the nuclearinterior, and barrier waves reflected from the barrier. Onthe other hand, the NF decomposition suggested by Fuller[7] splits the elastic scattering amplitude into the near-side (N) and far-side (F) components, corresponding tothe waves deflected to the near side and far side of thescattering center, respectively. These two interpretationsare complementary, and the broad Airy oscillation of thenuclear rainbow pattern is given by the interference of twofar-side amplitudes [3,8,9,10]. These are either the bar-rier f BF and internal f IF far-side amplitudes [6], or f F < and f F > far-side amplitudes with the orbital momenta L smaller or larger than a critical value L R associated withthe rainbow angle θ R [7]. guyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering 3 From such a pattern of the nuclear rainbow, a similarAiry structure is expected to be seen also in the inelasticscattering of a light HI or α -nucleus system that shows astrong rainbow pattern in the elastic scattering. In fact,some feature of the nuclear rainbow was observed in theinelastic light HI scattering [11,12,13,14,15,16,17,18,19,20], and some of these data were analyzed using the BI [14]and NF [17,18,19,20] decomposition methods. Althoughthese studies have confirmed the dominance of the far-side scattering at large angles, the Airy oscillation patterncould not be clearly identified in the inelastic cross section.Such suppression of the nuclear rainbow was assumed by -4 -3 -2 -1 -5 -4 -3 -2 -1 A1A1 d / d ( m b / s r) C+ C, E lab = 240 MeV c.m. (deg) O+ C, E lab = 200 MeV elastic2 + (4.44 MeV) Fig. 1.
Angular distributions of the elastic scattering andinelastic scattering to the 2 +1 state of C measured for the C+ C system at E lab = 240 MeV [21,22] (upper part) and O+ C system at E lab = 200 MeV [16,23] (lower part). A1indicates the location of the first Airy minimum established inthe extended OM analysis [24] of the elastic scattering data. Khoa et al. [13] to be due to the increased absorption inthe exit channel, with the target nucleus in an excitedstate. Dem’yanova et al. [18] suggested that this effect islinked to the shape of the inelastic form factor, which isdifferent from that of the attractive, deep real OP. We notethat the inelastic scattering to the 0 +2 state of C (Hoylestate) seems to exhibit the same rainbow pattern as thatseen in the elastic scattering cross section, and Hamada et al. [15] suggested that this is due to the extended 3 α -cluster structure of the Hoyle state.The weakening or suppression of the Airy oscillationin the inelastic nucleus-nucleus scattering remains so faran unsolved problem. As an illustration, we show in Fig. 1the elastic and inelastic scattering data measured for the C+ C system at E lab = 240 MeV [21,22] and for the O+ C system at E lab = 200 MeV [16,23]. These elastic scattering data were proven to be strongly refractive, withthe first Airy minimum A1 followed by a broad rainbowshoulder [24], especially, in the elastic O+ C scatteringdata at 200 MeV shown in the lower panel of Fig. 1. Whilethe cross section of the inelastic scattering to the 2 +1 stateof C target is quite strong, A1 disappears in the inelasticcross section and the Airy pattern is smeared out. Themeasured inelastic scattering cross section of the 2 +1 stateis even larger than the elastic scattering cross section atmedium and large angles, which confirms that the inelasticscattering channel remains strongly refractive. Thus, thedisappearance of the Airy pattern should not be due toan enhanced absorption in the exit 2 +1 channel.To explore this effect, we propose a compact methodfor the NF decomposition of the inelastic scattering am-plitude to determine explicitly all partial wave contribu-tions to the angular momentum transfer to the spin ofthe excited target. For this purpose, the NF decomposi-tion method by Fuller [7] is extended to split the inelas-tic scattering amplitude of the coupled partial waves intothe near-side and far-side components, so that the refrac-tion in the inelastic scattering channel is studied on equalfooting with that in the elastic channel, and the forma-tion of the nuclear rainbow therein. Given the prominentnuclear rainbow pattern observed in the elastic C+ Cand O+ C scattering at the refractive energies around10 ∼
20 MeV/nucleon, we apply the extended NF de-composition method to the inelastic scattering to the 2 +1 (4.44 MeV) state of C target in the coupled channel(CC) analysis of the inelastic scattering data measuredfor the C+ C system at E lab = 240 MeV [21,22], andthe O+ C system at E lab = 200 and 260 MeV [16,23]. We recall here briefly the coupled channel equations forthe elastic and inelastic nucleus-nucleus scattering. Thescattering wave function is obtained at each total angu-lar momentum J π of the nucleus-nucleus system from thesolution of the following CC equations [25] (cid:26) ~ µ β (cid:20) d dR + k − L ( L + 1) R (cid:21) − h β ( LI ) J | V | β ( LI ) J i (cid:27) × χ βJ ( k, R ) = X β ′ L ′ I ′ h β ( LI ) J | V | β ′ ( L ′ I ′ ) J i χ β ′ J ( k ′ , R ) , (1)where β and β ′ denote the entrance and exit channels,respectively; µ β is the reduced mass, ~ k = p µ β E β isthe center-of-mass (c.m.) momentum, and χ βJ ( k, R ) is thescattering wave function at the internuclear radius R . Thetotal angular momentum J is determined from the spin I ( I ′ ) and orbital momentum L ( L ′ ) of the entrance (exit)channel by the angular momentum addition J = L + I = L ′ + I ′ . In the present work, we focus on the scatteringof the two spinless nuclei ( I = 0) with a natural-parity Nguyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering excitation of the target. Then, | L ′ − I ′ | J = L L ′ + I ′ ,where I ′ is the spin of the excited target.The diagonal matrix element V ββ ( R ) of the projectile-target interaction in Eq. (1) is the nucleus-nucleus OP,and the nondiagonal matrix element V ββ ′ ( R ) is the tran-sition potential, which is also dubbed as the inelastic scat-tering form factor (FF). The OP and inelastic scatteringFF can be evaluated microscopically in the double-foldingmodel (DFM) using the ground-state (g.s.) and transitionnuclear densities, respectively, and an effective nucleon-nucleon (NN) interaction between the projectile- and tar-get nucleons (see more details in Ref. [25]). From the so-lution of the CC equations (1), we obtain the elastic scat-tering amplitude as f ( θ ) = f C ( θ )+ 12 ik X L (2 L +1) exp(2 iσ L )( S L − P L (cos θ ) , (2)where f C ( θ ) and σ L are the Rutherford scattering am-plitude and Coulomb phase shift, respectively; S L is thediagonal element of the elastic scattering S matrix, and P L (cos θ ) ≡ P LM =0 (cos θ ) is the Legendre function of thefirst kind. Within the CC formalism [26], the amplitudeof the inelastic scattering to an excited state of the targetwith spin I ′ and projection M I ′ is written explicitly as f M I ′ ( θ, φ ) = √ π ik X LL ′ √ L + 1 h L ′ − M I ′ I ′ M I ′ | L i× exp[ i ( σ L + σ ′ L ′ )] S ′ L ′ L Y L ′ − M I ′ ( θ, φ ) . (3)Here Y LM ( θ, φ ) is the spherical harmonics, the Coulombphase shift σ ′ L ′ is evaluated from the c.m. momentum k ′ inthe exit channel, and S ′ L ′ L is the element of the inelasticscattering S matrix. The orbital angular momenta in theentrance and exit channels are linked with spin I ′ of theexcited target by the triangular rule L ′ = L − I ′ , L − I ′ + 2 , ..., L + I ′ − , L + I ′ , (4)where the step of two angular-momentum units is impliedby the parity conservation. One can see from the expansion (3) that the coupled par-tial waves of different multipolarities can contribute co-herently to the inelastic scattering amplitude at the samescattering angle θ when I ′ = 0. By expressing the selectionrule (4) as L ′ = L + K , the inelastic scattering amplitude(3) can be written in terms of the K -subamplitudes al-lowed by the selection rule (4) f M I ′ ( θ, φ ) = I ′ X K = − I ′ f ( K ) M I ′ ( θ, φ ) . (5)Like the elastic amplitude (2), each K -subamplitude of f M I ′ ( θ, φ ) can be expanded over the orbital momenta of the entrance channel L as f ( K ) M I ′ ( θ ) = √ π ik X L √ L + 1 h ( L + K ) − M I ′ I ′ M I ′ | L i× exp[ i ( σ L + σ ′ L + K )] S ′ ( L + K ) L Y ( L + K ) − M I ′ ( θ, φ ) . (6)In terms of the inelastic scattering cross section, the con-tribution from each K -subamplitude is obtained at thegiven scattering angle as dσ K dΩ = X M I ′ (cid:12)(cid:12)(cid:12) f ( K ) M I ′ ( θ, φ ) (cid:12)(cid:12)(cid:12) , (7)so that the full cross section of the inelastic scattering toan excited state of the target with spin I ′ is dσdΩ = X M I ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ′ X K = − I ′ f ( K ) M I ′ ( θ, φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (8)Thus, for an excited state with spin I ′ = 0, the full inelas-tic scattering cross section is given by the interference ofthe K -subamplitudes with K = − I ′ , I ′ + 2 , ..., I ′ − , I ′ . As mentioned above, the NF decomposition method byFuller [7] is a very helpful tool to analyze the interferenceof the near-side and far-side scattering amplitudes in theelastic scattering [2,3]. Namely, the elastic scattering am-plitude is decomposed into the near-side ( f N ) and far-side( f F ) components as f ( θ ) = f N ( θ ) + f F ( θ ) = f NC ( θ ) + f FC ( θ )+ 12 ik X L (2 L + 1) exp(2 iσ L )( S L − × h ˜ Q ( − ) L (cos θ ) + ˜ Q (+) L (cos θ ) i , (9)where f N(F)C ( θ ) is the near-side (far-side) component of theRutherford scattering amplitude [7], the relative strengthof the near-side and far-side nuclear scattering is given by˜ Q ( − ) L (cos θ ) and ˜ Q (+) L (cos θ ), respectively,˜ Q ( ∓ ) L (cos θ ) = 12 (cid:20) P L (cos θ ) ± iπ Q L (cos θ ) (cid:21) , (10)where Q L (cos θ ) ≡ Q LM =0 (cos θ ) is the Legendre functionof the second kind. It is well established [2,3,27] that thenuclear rainbow pattern observed in the elastic α -nucleusand light HI scattering is determined entirely by the far-side component of the elastic amplitude (9). The nuclearrainbow pattern is a broad oscillation of the Airy minimaat medium and large scattering angles that results froman interference between f F < ( θ ) and f F > ( θ ) subamplitudesof the far-side component in (9), with L being smaller orlarger than a critical partial wave L R associated with therainbow angle θ R [27]. guyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering 5 It is natural to expect a similar rainbow pattern also inthe inelastic scattering cross section of a nucleus-nucleussystem that exhibits a pronounced nuclear rainbow in theelastic scattering channel. For this purpose, a NF decom-position of the inelastic scattering amplitude (3) shouldbe done for each projection M I ′ of the target spin I ′ ,with the contributions of all allowed K -subamplitudes (6)treated explicitly. However, such a detailed decompositionmethod is so far unavailable, and only some general dis-cussion on possible Airy structure of the inelastic scat-tering cross section was made [15,16] based on the Airypattern established in the elastic scattering cross section.We note here an early attempt to extend the NF decom-position method to the inelastic HI scattering by Deanand Rowley [28], where the near-side and far-side scatter-ing amplitudes obtained for each M I ′ magnetic substateof the target excitation with I ′ = 0 were shown to be not in phase. However, the refractive Airy pattern of thenuclear rainbow was not discussed at all in Ref. [28]. Toclose this gap of the scattering theory, we suggest in thepresent work a method of the NF decomposition of theinelastic scattering amplitude (3) to investigate explicitlythe Airy oscillation pattern in the inelastic scattering crosssection. Thus, the NF decomposition (9) is generalized todecompose the inelastic scattering amplitude (3) using theassociated Legendre functions as f M I ′ ( θ, φ ) = f N M I ′ ( θ, φ ) + f F M I ′ ( θ, φ )= √ π ik X LL ′ √ L + 1 h L ′ − M I ′ I ′ M I ′ | L i× A L ′ M I ′ exp[ i ( σ L + σ ′ L ′ )] exp( − iM I ′ φ ) × S ′ L ′ L h ˜ Q ( − ) L ′ − M I ′ (cos θ ) + ˜ Q (+) L ′ − M I ′ (cos θ ) i . (11)where ˜ Q ( ∓ ) LM (cos θ ) = 12 (cid:20) P LM (cos θ ) ± iπ Q LM (cos θ ) (cid:21) , and A LM = s L + 14 π ( L + M )!( L − M )! . Here P LM (cos θ ) and Q LM (cos θ ) are the associated Leg-endre functions of the first- and second kind, respectively.We note that the inelastic scattering FF includes both theCoulomb and nuclear contributions [25], and the inelasticCoulomb scattering amplitude is not treated separately asin the elastic scattering channel.Expressing L ′ = L + K in the generalized NF decom-position (11), we obtain explicitly the near-side and far-side components of each K -subamplitude of the inelasticscattering amplitude (5) as f ( K ) M I ′ ( θ, φ ) = f ( K )N ,M I ′ ( θ, φ ) + f ( K )F ,M I ′ ( θ, φ )= √ π ik X L √ L + 1 h ( L + K ) − M I ′ I ′ M I ′ | L i× A ( L + K ) M I ′ exp[ i ( σ L + σ ′ L + K − M I ′ φ )] S ′ ( L + K ) L × h ˜ Q ( − )( L + K ) − M I ′ (cos θ ) + ˜ Q (+)( L + K ) − M I ′ (cos θ ) i . (12) Thus, the generalized NF decomposition (12) allows us todetermine the near-side and far-side contributions fromeach K -subamplitude to the partial (7) and full inelasticcross section (8), and to study the formation of the nuclearrainbow in the inelastic nucleus-nucleus scattering in thesame manner as done for the elastic scattering using theFuller method (9). C+ C and O+ Cscattering at the refractive energies
The diagonal matrix element V ββ ( R ) of the projectile-target interaction in the CC equations (1) is determined bythe total optical potential U ( R ) [25]. The new version ofthe density dependent CDM3Y3 interaction with the rear-rangement term included [24] is used in the double-foldingcalculation of the real optical potential V ( R ). Becausethe nuclear rainbow is a subtle effect that can be observedonly when the absorption of the dinuclear system is weak,the imaginary OP in the flexible Woods-Saxon (WS) formis usually used for a proper identification of the rainbowpattern. Thus, we have U ( R ) = N R V ( R ) + iW ( R ) + V C ( R ) , where W ( R ) = − W V R − R V ) /a V ] . (13)The Coulomb potential V C ( R ) is obtained by folding thetwo uniform charge distributions with their mean-squaredradii chosen to be close to the measured charge radii ofthe two nuclei. The nuclear g.s. densities used in the DFMcalculation are taken as the Fermi distributions with pa-rameters chosen to reproduce the empirical matter radiiof the considered nuclei [29]. The renormalization N R ofthe real folded potential and the WS parameters (13) areadjusted in each case to the best CC description of theelastic scattering data, and a small deviation of N R fromunity validates the use of the folding model. The best-fitOP parameters used in the present CC study of the elasticand inelastic C+ C and O+ C scattering are givenin Table 1.The nondiagonal matrix element V ββ ′ ( R ) in the CCequations (1) is given by the inelastic form factor U I ′ ( R )that accounts for inelastic scattering to the target excitedstate with spin I ′ (see details in Ref. [25]), U I ′ ( R ) = N R V I ′ ( R ) − iδ I ′ ∂W ( R ) ∂R + V C ,I ′ ( R ) , (14)where the real nuclear V I ′ ( R ) and Coulomb V C ,I ′ ( R ) in-elastic form factors are calculated in the DFM using thenuclear transition densities of the excited states of Cobtained in the resonating group method (RGM) [30].The nuclear deformation lengths δ I ′ are determined by thecollective-model prescription using the measured B ( EI ′ )transition rates [31,32] of the considered excited statesof C. All the CC calculations have been done using thecode ECIS97 written by Jacques Raynal [33] that providesthe detailed output of the elastic and inelastic scattering
Nguyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering
Table 1.
The OP parameters (13) used in the CC calculation (1) of the elastic and inelastic C+ C and O+ C scattering. J R and J W are the volume integrals (per interacting nucleon pair) of the real and imaginary parts of the OP, respectively. E lab N R J R W V R V a V J W Data(MeV) (MeV fm ) (MeV) (fm) (fm) (MeV fm ) C+ C 240 1.067 336.0 19.29 5.743 0.595 117.5 [21,22] O+ C 200 0.936 300.4 13.32 6.150 0.502 72.06 [16,23] O+ C 260 0.930 291.8 18.50 5.756 0.550 83.92 [16,23] matrices necessary for the NF decomposition (9) and (12)of the corresponding scattering amplitudes. It should berecalled that the sequential iteration method implementedin the ECIS code was developed by Raynal to tackle theinelastic HI scattering with a strong Coulomb contribu-tion, focusing in particular on the scattering experimentsbeing carried out at GANIL at that time [34]. By usingthe recurrence relations for the Coulomb excitation inte-grals in the CC calculations [35], the ECIS integration ofthe coupled equations is highly stable and accurate up tovery large radii with sufficiently high number of partialwaves. For the C+ C and O+ C systems under thepresent study, the ECIS integration up to R max ≈
25 fmin steps of dR = 0 .
05 fm is needed to ensure the conver-gence of the calculated cross section, taking into accountup to 180 partial waves. At the considered energies, the CCresults obtained using the nonrelativistic and relativistickinematics are about the same.The elastic and inelastic C+ C scattering has beenwidely studied at energies ranging from the Coulomb bar-rier [10] up to about 200 MeV/nucleon [36]. While the elas-tic C+ C scattering at the barrier energies was shownto be of interest for nuclear astrophysics [37], the scatter-ing data measured for this system at the refractive energiesaround 20 MeV/nucleon [21,22] exhibit a nuclear rainbowpattern that enabled an unambiguous determination ofthe real OP down to small distances [3,24]. In particu-lar, the C+ C scattering data measured accurately at E lab = 240 MeV [21,22] are very important for our studybecause this energy was found optimal for the observa-tion of the first Airy minimum A1 of the nuclear rainbowin the elastic C+ C scattering [24]. As shown abovein Fig. 1, the data of the inelastic C+ C scattering tothe 2 +1 (4.44 MeV) state of C measured at this sameenergy [21,22] does not have any minimum that can beinterpreted as the remnant of A1, at angles near the loca-tion of A1 established in the elastic cross section. Anotherlight HI system that exhibits a prominent rainbow patternin the elastic scattering is O+ C [23]. Unlike C+ C,the O+ C system does not have the boson symmetry,and the angular evolution of the Airy pattern could be ob-served with the increasing energy. The strongest rainbowpattern, the deep A1 minimum followed by an exponen-tial fall-off of the rainbow shoulder, is well confirmed in theelastic O+ C scattering data measured at E lab = 200MeV [23]. The question why the inelastic O+ C scat-tering data measured at this same energy [16] does notshow a similar Airy pattern (see lower panel of Fig. 1) isso far unanswered. In the present work, we try to explain the suppression of the first Airy minimum in the inelastic C+ C and O+ C scattering cross sections at the en-ergies where A1 was clearly identified in the elastic crosssections measured for these systems. -5 -4 -3 -2 -1 -4 -3 -2 -1 A1 C+ C, E lab = 240 MeV
Elastic Far-side c.m. (deg) d / d ( m b / s r) d / d M o tt A1 K=-2 A1 K=2 A1 K=0
Inelastic (2 + ) Far-side K =2 K =0 K =-2 Fig. 2.
CC description (solid lines) of the elastic and inelas-tic (to the 2 +1 state of C) C+ C scattering at E lab = 240MeV, in comparison with the measured data [21,22]. The dash-dotted, dotted, and dash-dot-dotted lines are the partial inelas-tic cross sections (7) given by the subamplitudes with K = 2 , The CC results for the elastic and inelastic C+ Cscattering describe well the data as shown in Fig. 2. Thedominance of the far-side cross sections at medium andlarge angles indicates that both the elastic- and inelastic C+ C at the considered energies are strongly refrac-tive. The nuclear rainbow pattern is well established inthe elastic C+ C scattering, with the first Airy mini-mum A1 unambiguously identified [24] at the scatteringangle θ c . m . ≈ ◦ based on the NF decomposition (9) of guyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering 7 the elastic scattering amplitude. The data of the inelastic C+ C scattering to the 2 +1 state of C are reproducedreasonably by the CC calculation (1), but the Airy struc-ture seen in the elastic cross section is smeared out in theinelastic cross section. Given the dominance of the far-sidescattering at medium and large angles, the suppression ofA1 in the inelastic cross section is definitely not causedby the near-side/far-side interference, but more likely bya destructive interference of the far-side subamplitudes. -6 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 A1 O+ C, E lab = 200 MeV
Elastic Far-side c.m. (deg) d / d ( m b / s r) d / d R Inelastic (2 + ) Far-side K =2 K =0 K =-2A1 K=2 A1 K=0 A1 K=-2
Fig. 3.
The same as Fig. 2 but for the elastic and inelastic O+ C scattering at E lab = 200 MeV [16,23]. One can see in the partial-wave expansion of the in-elastic scattering amplitude (3) that the inelastic scatter-ing cross section at the given scattering angle containsthe contributions from the subamplitudes of different par-tial waves ( L ′ = L ) when the spin of the excited state isnonzero ( I ′ = 0). For the inelastic C+ C scattering tothe 2 +1 state of C shown in Fig. 2, each L -componentof the inelastic scattering amplitude is resulting from aninterference of the three K -subamplitudes with K = L ′ − L = − , ,
2. The partial inelastic scattering cross sections(7) given by the three K -subamplitudes (summed over allpartial waves L ) are shown separately in the lower panelof Fig. 2. By tracing the angular evolution of the corre-sponding far-side cross sections, we have identified the firstAiry minimum A1 in the partial inelastic C+ C crosssection with K = 0 at θ c . m . ≈ ◦ which is close to the lo-cations of A1 in the elastic C+ C cross section. Whilea slight remnant of A1 with K = 0 can still be seen in thecalculated inelastic scattering cross section (8), it is notobserved in the measured data. Rather weak rainbow pat-terns of the two partial inelastic C+ C cross sections -5 -4 -3 -2 -1 -4 -3 -2 -1 A1 O+ C, E lab = 260 MeV
Elastic Far-side c.m. (deg) d / d ( m b / s r) d / d R Inelastic (2 + ) Far-side K =2 K =0 K =-2A1 K=2 A1 K=0 A1 K=-2
Fig. 4.
The same as Fig. 2 but for the elastic and inelastic O+ C scattering at E lab = 260 MeV [16,23]. with K = 0 were found which are shifted in angles, withA1 located at θ c . m . ≈ ◦ and 52 ◦ in the partial inelasticcross section with K = 2 and K = −
2, respectively. Onecan see in the lower panel of Fig. 2 that the partial inelas-tic cross sections with K = ± K = 0 at medium and large angles, and no remnantsof A1 with K = ± O+ C scattering at E lab = 200MeV plotted in Fig. 3. While the prominent A1 minimumis located at θ c . m . ≈ ◦ in the elastic cross section, itseems to disappear in the inelastic O+ C scatteringcross section. Such an effect was found also in the re-sults of the earlier CC analysis of the inelastic O+ Cscattering [38] as well as those of a cluster folded modelstudy [39]. As discussed for the C+ C system, threedifferent Airy oscillation patterns can be seen in the par-tial inelastic O+ C cross sections given by the subam-plitudes with K = 2 , , −
2, with A1 located at θ c . m . ≈ ◦ , ◦ , and 84 ◦ , respectively. Again, the location of A1with K = 0 is quite close to the locations of A1 in theelastic O+ C cross section. The full inelastic scatter-ing cross section (8) includes the contributions from allallowed K -subamplitudes, and their out-of-phase interfer-ence smears out the individual A1 minima seen in thepartial inelastic cross sections (7). While a slight remnantof A1 with K = 0 is seen in the calculated inelastic crosssection (solid line in the lower panel of Fig. 3), it can-not be clearly resolved in the measured data. The sameCC results for the elastic and inelastic O+ C scat-
Nguyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering tering at E lab = 260 MeV are compared with the data[16,23] in Fig. 4. We found that the absorption becomesslightly stronger (see Table 1) with the increasing energy,and the Airy oscillation pattern is weakened and shiftedto smaller angles (with A1 in the elastic cross section lo-cated at θ c . m . ≈ ◦ ). The weaker Airy oscillation patternof each partial inelastic cross section can still be seen butthe remnant of A1 with K = 0 disappears in both thecalculated inelastic scattering cross section and measureddata (lower panel of Fig. 4). In conclusion, the resultsof our CC analysis shown in Figs. 2-4 explain naturallywhy the Airy oscillation pattern of the nuclear rainbow isstrongly suppressed in the measured data of the inelastic C+ C and O+ C scattering to the 2 +1 state of Cat the rainbow energies. -4 -3 -2 -1 -3 -2 -1 A3 A2 A1 d / d R Elastic Far-side O+ C, E lab = 200 MeV A2 K=2 A2 K=0 A2 K=-2
Inelastic (2 + ) Far-side K =2 K =0 K =-2 c.m. (deg) d / d ( m b / s r) A1 K=2 A1 K=0 A1 K=-2
Fig. 5.
The same CC results as those in Fig. 3 for the elasticand inelastic O+ C scattering at E lab = 200 MeV, obtainedwith a less absorptive OP (13) with W V → W V / It is well-known that the nuclear rainbow is formed bythe interference of the far-side scattering waves, refractedby the attractive real
OP [2,3]. That’s the reason whythe nuclear rainbow could be observed only when the ab-sorption of the scattering system is weak enough for thefar-side trajectories to survive at the medium and largescattering angles. In practice, the absorptive strength ofthe OP is often reduced to artificially enhance the far-side scattering amplitude for a proper identification of theAiry oscillation pattern [24]. The results of the CC calcula-tion of the elastic and inelastic O+ C scattering at 200MeV given by a less absorptive OP (with W V → W V / K -subamplitudes (6), but the locations ofthe Airy minima are shifted to the smaller angles when K = 2, and to the larger angles when K = −
2. It isvery essential to emphasize again that the Airy oscillationpattern in the partial inelastic cross section with K = 0remains about the same as that observed in the elasticscattering thanks to an in-phase interference of the par-tial waves with L ′ = L . When K = 0, the out-of-phaseinterference of the partial waves with L ′ = L smears outthe different Airy oscillation patterns in the full inelastic2 +1 scattering cross section. Because the partial inelasticcross section with K = 0 is substantially larger that thosewith K = ± K = 0 can be very wellseen in the full inelastic scattering cross section (solid linein the lower panel of Fig. 5) when a reduced absorption W V was used in the CC calculation. In fact, the broadrainbow shoulder following A1 with K = 0 is still visiblein the data measured at E lab = 200 MeV for the inelastic O+ C scattering to the 2 +1 state of C (see lower panelof Fig. 3).It becomes clear now that there is no unique Airy pat-tern of the nuclear rainbow in the full (far-side) cross sec-tion of the inelastic nucleus-nucleus scattering to an ex-cited nuclear state with nonzero spin. In such a case, onlythe Airy oscillation pattern of the partial inelastic crosssection (7) given separately by each K -subamplitude (6)can be determined in the same manner as done in thecase of elastic scattering. Thus, the detailed locations ofthe Airy minima A1, A2, and A3 in the inelastic O+ Cscattering cross section deduced visually by Ohkubo et al. from the calculated angular distribution [38] are not prop-erly founded.Although the inelastic O+ C scattering to the 0 +2 (Hoyle) and 3 − excited states (at E x ≈ .
65 and 9.64MeV, respectively) were not measured at E lab = 200 MeV,the CC prediction of the inelastic cross sections for thesestates should be of interest for the revelation of the nuclearrainbow pattern therein. The CC results for the inelastic O+ C scattering at 200 MeV obtained with the nucleartransition densities of the 0 +2 and 3 − states of C givenby the RGM [30] and the same OP as given in Table 1 areshown in Fig. 6. One can see that the refractive (far-side)scattering is also dominant at medium and large anglesin the inelastic scattering to the Hoyle (0 +2 ) state, with K ≡ L ′ ≡ L ) and dσ K /dΩ ≡ dσ/dΩ . In this case,spin of the excited state is zero and there is no interferenceof the scattering subamplitudes with different K . As aresult, the Airy pattern in the angular distribution of theinelastic scattering to the Hoyle state is determined with asingle ( K = 0) inelastic scattering amplitude, in the samemanner as done for the elastic O+ C scattering. Thus,the deep minimum of the inelastic 0 +2 cross section can beconfirmed as the first Airy minimum A1 which is locatedat about the same angle as A1 of the elastic cross section(see upper panels of Figs. 5 and 6). guyen Hoang Phuc et al.: Suppression of the nuclear rainbow in the inelastic nucleus-nucleus scattering 9 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 O+ C, E
Lab = 200 MeV A1 + (7.65 MeV) Far-side A1 K=-3 A1 K=1 A1 K=3 A1 K=-1 d / d ( m b / s r) c.m. (deg) - (9.64 MeV) Far-side K=3 K=1 K=-1 K=-3 Fig. 6.
The same CC results as those in Fig. 2 but for theinelastic O+ C scattering to the 0 +2 (upper part) and 3 − (lower part) excited states of O at E lab = 200 MeV. For the inelastic O+ C scattering to the 3 − state of C, there are four subamplitudes (6) with K = − L ′ = L − , K = − L ′ = L − , K = 1 ( L ′ = L + 1),and K = 3 ( L ′ = L + 3) with the corresponding A1 min-ima shifted away from each other by a few degrees in thescattering angle. Among these K -subamplitudes, The firstAiry minimum A1 of the partial inelastic cross sectionwith K = − O+ C scattering to the 2 +1 state of C. The present work explains why the Airy pattern of nu-clear rainbow is suppressed in the inelastic C+ C and O+ C scattering to the 2 +1 state of C at the refrac-tive energies, where a strong rainbow pattern has beenobserved in the elastic scattering. For this purpose, thenear-far decomposition method by Fuller is generalized todetermine the near-side and far-side components of theinelastic scattering amplitude for all partial wave contri-butions. Using the generalized NF decomposition method,our coupled channel analysis of the elastic and inelastic C+ C and O+ C scattering at the energies understudy shows unambiguously that the destructive interfer-ence of the inelastic partial waves of different multipoles suppresses the Airy oscillation pattern in the inelasticscattering cross section. Nevertheless, the inelastic scat-tering remains strongly refractive in these cases, with thedominant far-side scattering at medium and large scatter-ing angles.We conclude, therefore, that it is not possible to iden-tify uniquely the Airy pattern of the nuclear rainbow inthe angular distribution of the inelastic nucleus-nucleusscattering to an excited state with nonzero spin. Semi-classically, such a refractive mixing of the partial wavesof different multipoles is analogous to an optical prism re-fracting ray of light of different wave lengths. The onlyexception is the inelastic scattering to a monopole excita-tion which does not mix different multipoles in the inelas-tic scattering amplitude, and the Airy pattern in the in-elastic cross section can be determined consistently in thesame manner as done for the elastic scattering. In light ofthis result, an accurate measurement of the inelastic α orlight ion scattering to the 0 +2 excitation of the C targetshould be of interest for the future studies of the nuclearrainbow scattering as well as the α -cluster structure of theHoyle state [40,41].Last but not least, we gratefully notice that over theyears our nuclear scattering study has been relied on sev-eral versions of the coupled channel code ECIS writtenby Jacques Raynal. This state-of-the-art computer codeof nuclear scattering is still being actively used in thecommunity, and Jacques’ important contribution to thedevelopment of the nuclear physics research is stronglyappreciated by many of us. Acknowledgments
The present research was supported, in part, by the Na-tional Foundation for Science and Technology Develop-ment (NAFOSTED). The authors also thank Prof. M.Kamimura for his permission to use the RGM nuclear den-sities in the double-folding calculation.
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