Pocket resonances in low-energy antineutrons reactions with nuclei
PPocket resonances in low-energy antineutrons reactions with nuclei
Teck-Ghee Lee ∗ Department of Physics, Auburn University, Auburn, Alabama 36849, USA.
Orhan Bayrak † Department of Physics, Akdeniz University, Antalya, Turkey
Cheuk-Yin Wong ‡ Physics Division, Oak Ridge National Laboratory d , Oak Ridge, Tennessee 37831, USA. Upon investigating whether the variation of the antineutron-nucleus annihilation cross-sectionsat very low energies satisfy Bethe-Landau’s power law of σ ann ( p ) ∝ /p α behavior as a functionof the antineutron momentum p , we uncover unexpected regular oscillatory structures in the lowantineutron energy region from 0.001 to 10 MeV, with small amplitudes and narrow periodicityin the logarithm of the antineutron energies, for large- A nuclei such as Pb and Ag. Subsequentsemiclassical analyses of the S matrices reveal that these oscillations are pocket resonances thatarise from quasi-bound states inside the pocket and the interference between the waves reflectinginside the optical potential pockets with those from beyond the potential barriers, implicit in thenuclear Ramsauer effect. They are the continuation of bound states in the continuum. Experimentalobservations of these pocket resonances will provide vital information on the properties of the opticalmodel potentials and the nature of the antineutron annihilation process. d This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Departmentof Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the USgovernment retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript,or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored researchin accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan), Oak Ridge, Tennessee 37831,USA ∗ [email protected]; [email protected] † [email protected] ‡ [email protected] a r X i v : . [ nu c l - t h ] F e b I. INTRODUCTION
Matter-antimatter asymmetry in the Universe is one of the greatest mysteries of modern physics [1]. To unravel thismystery, there has been a great deal of experimental and theoretical investigations on matter-antimatter interactions.So far, most of the obtained information centers around antiproton-nucleus (¯ p A) reactions and structures [2–14]. Thecorresponding information on antineutron-nucleus (¯ n A) reaction [15], on the other hand, remains comparatively sparseand limited with the most recent work from the OBELIX’s collaboration [16, 17]. Nevertheless, the ¯ nA annihilationis essential in the process of quantifying signals from n → ¯ n oscillations in matter [18–21], and the significant theconnection between the ¯ nA interaction-potential and the n → ¯ n oscillations rates have also been examined theoretically[22–24].It has been generally expected that in the s -wave limit the ¯ nA and ¯ pA annihilation cross-sections are to obeyBethe-Landau’s power-law, σ ¯ nA, ¯ pA ann ( p ) ∝ /p α as a function of the antineutron momentum p , with α = 1 for ¯ nA and α = 2 for ¯ pA [25]. But recent experimental data (see Fig. 5 of Ref.[3]) revealed that the annihilation cross sectionsof ¯ n , and ¯ p , on C nuclei at low energies, appear to follow a similar trend with only minor differences. Much curiousabout the puzzle, we recently introduced a new momentum-dependent optical potential to investigate the behavior ofthe annihilation cross-sections for the ¯ n , and the ¯ p , on C, Al, Fe, Cu, Ag, Sn, and Pb nuclei in the momentum range50 −
500 MeV/ c [26], via the ECIS code [27]. The calculated results were concordant with the OBELIX’s annihilationcross-section data [16, 17]. However, between 40 and 100 MeV/ c , the optical model calculations indicated that α ≈ / nA and α ≈ . pA , leading us to conclude our low-energy annihilation reaction in question was yet to reachthe s -wave limit. optical model: n_Pbsemiclassical: n_Pboptical model: n_Agoptical model: n_C -3 -2 -1 Energy [MeV] (a) annihilation(b) elastic C r o ss s ec ti on s [ b a r n s ] n_Pb n_C n_Ag n_Pbn_Agn_C FIG. 1: Energy variation of the (a) ¯ nA annihilation and (b) ¯ nA elastic cross sections obtained from the opticalmodel calculations for different nuclei. The semiclassical results for ¯ n Pb reactions are shown as the dotted-curve.In the course of investigating the energy dependence of the cross-sections at even lower energies, we uncover, toour surprise, unexpected regular oscillatory structures with small amplitudes and narrow periods (in the logarithm ofthe energy) in the annihilation and elastic cross-sections, in the region from 0.001 MeV −
10 MeV, as shown in Fig.1.Such oscillations are absent for small nuclei and gain in strength as the nuclear radius increases. Its amplitude islarger for the elastic scattering than for the annihilation process. The dependency on the size of the nucleus revealsthat it occurs when many partial-wave L values are involved in its interference as many more partial waves canbe accommodated when the size of the nucleus increase. Such behavior, undoubtedly, contradicts Bethe-Landau’spower-law. It also reminiscences of wave-interference.Furthermore, the predicted oscillations appear somewhat different from that of neutron-induced total reaction crosssections for Pb, Cd, and Ho nuclei, which arises from the Ramsauer’s resonances [28–30]. They are also unlike thosepresent in the high-energy C+ C fusion, which is due to successive addition of contributions from even values ofpartial waves to the identical-particle fusion cross-section, with increasing energies [31–33].To understand the nature of these oscillations, we shall use a semiclassical S matrix method as it is capable ofproviding a clear intuitive picture to guide our understanding of the reaction process. Since the oscillations show upmore clearly in the ¯ n Pb reaction, we shall focus on this system. We wish to show that the cross-section oscillationsare physical and constitute pocket resonances. Such resonance phenomena are general features of potential scatteringin atomic, molecular, and heavy-ion collisions. A thorough analysis of the reaction process is therefore needed tounderstand the origin of such resonances.
II. SEMICLASSICAL ANALYSES OF THE ANNIHILATION OSCILLATIONS AND ELASTICOSCILLATIONS
Semiclassical descriptions of quantum effects in a potential scattering problem have been discussed in great detailsby Ford and Wheeler [34] using the Wentzel-Kramers-Brillouin (WKB) approximation, and later consolidated byother pioneering works [35–39]. However, we shall follow a collection of WKB works for complex-potential scatteringby Brink, Takigawa, Lee, Marty, and Ohkubo [37–39] as they deemed proper for the present analyses. r [fm] -40-30-20-10010203040 R e ( V e ff ) [ M e V ] E r r r L=02 4 6 8 10L=12
FIG. 2: Real part of the ¯ n Pb interaction potential Re( V eff ( r, E )) at a sample energy E = 5.3 MeV, where only thepotential curves for even orbital angular momentum L are displayed. The r , , denote three turning points forRe( V eff ( r, E )).We adapt the same momentum-dependent optical potential used in our earlier work [26] in the present semiclassicaltheory. We consider the concept of the interference between a barrier wave reflected at the potential barrier andmodified by the tunneling effect, and an internal wave which penetrates the barrier, into the potential pocket, andreemerges through the barrier out to r → ∞ .According to [37], implicitly for a given E and a partial wave L , the total phase function δ = δ ( E, L ) is related tothe WKB phase function δ WKB , the action phase angle δ ij between the turning points r i and r j , tunneling phase φ ,tunneling phase angle ξ and tunneling coefficient w by δ = δ WKB − φ + tan − { w ( ξ ) tan( δ − φ ) } , (1) δ WKB = π L + 12 ) − kr + (cid:114) µ (cid:126) (cid:90) ∞ r ( K − k ) dr, (2) δ ij = (cid:114) µ (cid:126) (cid:90) r j r i Kdr, (3) φ = 12 (cid:18) ξ ln( ξ/e ) − arg Γ( 12 + iξ ) (cid:19) , (4) ξ = − iπ δ , (5) w ( ξ ) = (cid:112) (1 + e − πξ ) − (cid:112) (1 + e − πξ ) + 1 , (6)where K = ( E − V eff ( r, E )) / , k = E / , E is the center-of-mass energy and µ is the reduced mass of the collision-pair.The turning points r , , are the roots of E − V eff ( r, E ) = 0, as displayed in Fig.(2). In our case, V eff ( r, E ) is theLanger’s modified interaction potential, V eff ( r, E ) = − ( V ( E ) + iW ( E ))1 + exp { ( r − r o ) /a o } + (cid:126) ( L + 1 / µr , (7)which is complex and the values for the parameters r o , a o and volume terms V and W are listed in Table 1, 2 and 3in [26]. To keep the analyses simple, we consider only the nuclear volume terms in the present semiclassical theory.Figure 2 gives the real part of the potential Re( V eff ( r, E )) as a function of internuclear distance r and different L at a sample energy E = 5 . n Pb interaction. With the nuclear potential that has a negative imaginarypotential W , the turning points r , are below the real axis of the complex-plane and r lies above, and the V eff ( r, E )for a given E must be an analytic function of r . The integrands in the action integrals are multiple-valued involvingbranch cuts. The integration paths in the complex plane need to be chosen so that the action angles δ ij and ξ arepositive, as discussed in [37].Upon finding the oscillating behavior from the optical model calculation which is rather unusual behavior, we havecarried out the above semiclassical calculations for comparison to analyze the underlying origin of the oscillations.We find that the semiclassical calculation using the above formulation not only reproduces the result of the opticalmodel but fits the oscillatory structures reasonably well in energy region 0.1 −
10 MeV as shown in Fig. 1. At muchlower energies and L = 0, the semiclassical cross-section is expected to deviate from that of the optical model becausethe de Broglie wavelengths at such low energies are significantly larger than the width of the nuclear potential.As the semiclassical calculations contains an explicit contributions from physical processes which can illuminatethe interplay between partial waves of various origins, our task is to carry out a detailed study of the partial waveanalysis to pinpoint the precise origins of such oscillations. Accordingly, we focus on the interference between thebarrier waves and the internal waves and split the total S matrix element into the barrier wave contribution S B , andthe internal wave contribution S I : S = e iδ = S B + S I , (8) S B = e iδ WKB
N , (9) S I = (cid:20) e iδ WKB N (cid:21) e − πξ (cid:20) e iδ N (cid:21) (cid:34) e iδ N + (cid:18) e iδ N (cid:19) + (cid:18) e iδ N (cid:19) + · · · (cid:35) = (cid:20) e iδ WKB N (cid:21) e − πξ (cid:20) e iδ N (cid:21) (cid:20)
11 + e iδ /N (cid:21) (10)where N = √ π exp( − πξ + iξ ln( ξ/e )) / Γ( + iξ ) is the barrier-penetration factor. The factor 1 / (1+ e iδ /N ) in the S I term is from Pad´e approximation, where e iδ /N is the amplitude of penetration-weighted internal-wave. Thus, thefactor e iδ / ( N + e iδ ) essentially describes the sum of amplitudes of multiple reflections of internal-wave inside thepotential pocket [37, 38]. And knowing these S matrices, the annihilation cross section: σ ann = πk (cid:80) L (2 L +1)(1 −| S | )and elastic cross section: σ el = πk (cid:80) L (2 L + 1) | − S | can be readily evaluated. σ a nn ( L ) [ b a r n s ] σ B I a nn ( L ) [ b a r n s ] -3 -2 -1 Energy [MeV] -808 σ i n t a nn ( L ) [ b a r n s ] -100 L=1 L=2 L=3 L=4 L=5 L=6L=1 L=2 L=3 L=4 L=5
L=6
L=1 L=2 L=3 L=4 L=5 σ ann (L) = σ BIann (L) + σ intann (L) L=0 L=0L=0 (a) (b) (c)(d) σ intann σ ann σ BIann σ BIann (L) σ intann (L) n_Pb C r o ss s ec ti on s [ b a r n s ] σ ann = σ BIann + σ intann σ ann (L) σ ann FIG. 3: Semiclassical annihilation cross sections for ¯ n Pb as a function of energy. (a) is from Eq.(11), (b) is from thesum of Eq.(12) and Eq.(13), (c) is from Eq.(12) and (d) is from Eq.(13) . Partial wave analysis of the annihilation oscillations in terms of the interference of the barrier and the internalwaves gives [39] σ ann = σ BIann + σ intann = ∞ (cid:88) L =0 ( σ BIann ( L ) + σ intann ( L )) , (11) σ BIann ( L ) = πk (2 L + 1)(1 − | S B | − | S I | ) , (12) σ intann ( L ) = πk (2 L + 1)( − { S B S ∗ I } ) , (13)where σ int ( L ) constitutes the interference between the internal and barrier waves, and σ BI ( L ) is and the remainingnon-interference term.In Fig. 3(a), we use Eq.(11) to separate the total annihilation cross section σ ann into the non-interference part, σ BIann ,and the interference part, σ intann . The non-interference σ BIann produces the familiar 1 /p -like dependence with decreasingenergy without any oscillations. The interference part σ intann , oscillates, giving rise to a total sum σ ann that oscillateson the smooth 1 /p background in Fig. 3(a).Fig. 3(b) shows the break-up of the total annihilation cross section σ ann into different partial waves contributions, σ ann ( L ). Except for the L = 0 partial-wave curve that resembles the 1 /p -like behavior, the rest, on the other hand,display a double peak with the largest peak at the lower energy end. In the low-energy region in Fig. 3(c) andFig. 3(d), the σ BIann ( L ) given by Eq.(12) produces the broad peak of the cross-section, whereas the interference term σ intann ( L ) given by Eq.(13) produces the finer oscillations. For a particular partial wave L , the sum of σ BIann ( L ) and σ intann ( L ) resulted in the observed double peaks in σ ann ( L ). As E increases, these double-peak structures graduallyturn to a single-peak where the contribution from higher L becomes dominant. This is because, at high energies, fora given high partial-wave, σ BIann ( L ) with broad maximum overtakes the less relevant, finer oscillatory σ intann ( L ).Focusing on the non-interference part from different partial waves L , Fig. 3(c) shows a plot of σ BIann ( L ) given byEq.(12). As expected, the L = 0 curve faithfully follows the 1 /p dependence which seems to imply the difference( | S I | − | S B | ) is small in comparison to the first term of 1. In contrast, the L > | S I | and | S B | in Eq.(12); theinterference between the part of the ¯ n wave passing through the nucleus with the part of the wave which has gonearound the nucleus, as pointed out in Peterson [29]. Note that σ BIann ( L ) contributes the most to the broad maximumfeature in σ ann ( L ) of Fig.3(b).Now, focusing on the interference part from different partial waves L , Fig. 3(d) indicates that the oscillations inthe annihilation reaction are attributed to the barrier wave interfering with the internal wave. Inspection of Eq.(9),(10) and (13) reveal that the oscillations are governed by [ − S B S ∗ I )] in which S B S ∗ I can be factorized into S B S ∗ I = (cid:20) e iδ WKB N (cid:21) (cid:20) e i ( δ WKB + δ ) N (cid:21) ∗ (cid:124) (cid:123)(cid:122) (cid:125) barrier factor (cid:20) e iδ /N e iδ /N (cid:21) ∗ (cid:124) (cid:123)(cid:122) (cid:125) pocket factor (14)and is depended upon a combination of complex phases δ , δ and δ WKB , which are functions of E and L . Toisolate the precise origin of the oscillations, we must consider Fig.4 that shows the energy dependence of the realpart of the pocket and the barrier factors as separate elements of the whole Eq.(14) term for various partial waves.For incident energy greater than zero and below the barrier, in the r < r < r tunneling region, we find from ournumerical calculations that δ are predominantly imaginary, resulting exp(2 iδ ) ≈ exp( − πξ ). We also find thatthe contribution from the difference between δ WKB and δ ∗ WKB of the r > r outer most region are predominantlyimaginary as well, rendering exp(2 i ( δ WKB − δ ∗ WKB )) ≈ exp( − η ). We recast Eq.(13) into σ intann ( L ) ∼ πk (2 L +1) | S B || S I | e − πξ + η ) [ − δ )] . (15)We find consequently that the the maxima of the annihilation cross sections σ intann ( L ) with L = 1 in Fig. 3(d) as“annihilation resonances” are located at the minima of the real part of the pocket factor of Eq. (14) in Fig. 4(a) and -0.8-0.6-0.4-0.200.2 R e ( P o c k e t f ac t o r) L=0L=1L=2L=3L=4L=5 R e ( B a rr i e r f ac t o r) L=0 L=1L=2L=3L=4L=5 -0.2-0.100.10.2 R e { S B S * I } -3 -2 -1 Energy [MeV] R e ( δ ) (d) Re( δ ) L=5L=4 (a) Re(Pocket factor)
L=0 (b) Re(Barrier factor)(c) Re{S B S* I } L=3L=2L=1L=0L=1L=2L=3L=4L=5 π π π FIG. 4: Energy dependence of (a) the real part of the pocket factor, (b) the real part of the barrier factor, (c)Re( S B S ∗ I ) of Eq.(14) and (d) Re( δ ) for various partial waves.in Re( S B S ∗ I ) in Fig. 4(c), as marked by the vertical dashed lines in Fig. 4(c) and 4(d). Though the barrier factor ofFig. 4(b) is seen to increase with increasing energy, it reduces the oscillation-amplitude of the pocket factor of Fig.4(a) without varying the final positions of maximum and minimum of the oscillations as shown in Fig. 4(c). Theminimum of the pocket factor, according to Eq.(15), must occur whenever cos(2 δ ) = − δ = (2 n + 1) π ) andthe minima are indeed located at δ ≈ π , 6.5 π and 5.5 π , respectively, as indicated by horizontal lines in Fig.4(d). The quantity δ varies by one π unit between resonances with ∂δ /∂n ≈ π . We further examine and findthat wherever the horizontal lines cross the δ curves of Fig. 4(d), every cross-point matches every single minimumof Re { S B S ∗ I } in Fig. 4(c) and a maximum σ intann ( L ) in Fig. 3(d). This means that the condition for an annihilationresonance at a cross section maximum, at the energy E , is δ ( E, L ) ≈ ( n L + 12 ) π, (16)corresponding to the quantization condition for a quasi-bound state inside the pocket, where n L is the number ofbound and quasi-bound states below E for the angular momentum L . As δ ( E, L ) is the action integral for a pathwithin the confining potential V eff ( r, E )), the “pocket resonance” is in fact a continuation of the bound states of thepotential into the continuum. While the locations of these pocket resonances for the present problem are specific tothe optical potential in question, the general condition at which the annihilation resonance occurs suggests that thecondition of Eq. (16) is quite general in nature. σ e l ( L ) [ b a r n s ] σ B I e l ( L ) [ b a r n s ] -8-4048 σ i n t e l ( L ) [ b a r n s ] L=1
L=2
L=3L=4L=5L=0 L=6 (d) σ intel (L)(e) Re( δ WKB + δ ) L=2 -3 -2 -1 Energy [MeV] R e ( δ W K B + δ ) L=1 L=2L=3 L=5 L=6L=1 L=2L=3L=4L=5L=6L=4L=0L=0 C r o ss s ec ti on [ b a r n s ] σ BIel x10 -4 σ intel x10 -4 σ el = σ BIel + σ intel σ el (L) = σ BIel (L)+ σ intel (L) (c) σ BIel (L) n_Pb (a) σ el (b) σ el (L) π π π π π FIG. 5: Semiclassical elastic cross sections for ¯ n Pb as a function of energy. (a) is from Eq.(17), (b) is from the sumof Eq.(18) and Eq.(19), (c) is from Eq.(18) and (d) is from Eq.(19).We now examine the elastic oscillations with σ el = σ BI + σ int = ∞ (cid:88) L =0 ( σ BIel ( L ) + σ intel ( L )) , (17) σ BIel ( L ) = πk (2 L + 1)(1 + | S I | + | S B | ) , (18) σ intel ( L ) = πk (2 L + 1)(2Re { S ∗ I S B − S ∗ I − S B } ) . (19)The quantity σ BIel and its partial wave decomposition in Fig.5(a) and 5(c) show the all σ BIel ( L ) are monotonicallydecreasing functions of the energy. There is a significant cancellation of σ BIel by σ intel where σ intel has an oscillatingbehavior, resulting in an oscillating total elastic cross section σ el as a function of energy as shown in Fig. 5(a).Fig.5(b) shows the break-up of the total elastic σ el into different partial waves contributions σ el ( L ). Here we noticeonly the L = 0 partial wave has three, narrow, sinusoidal oscillations with decreasing amplitudes. Those of L > σ el near 0.05 MeVseems to correspond to the first maximum of L = 0 curve at the same energy. The subsequent, second maximum of σ el that is near 0.2 MeV, corresponds to the contributions from the L = 1 curve and the second maximum of L =0 curve. Beyond the second maximum, each consecutive maximum now appears to associate with a specific partialwave of L = 2, 3, 4, and so forth.In Fig.5(c) we show the curves for various partial-wave non-interference cross-sections σ BIel ( L ) from Eq.(18). Theyall obey Bethe-Landau’s power law in proper order as L increases and unlike the diffraction pattern depicted earlierin the case of annihilation. Such behavior can be understood from Eq.(18) and qualitatively interpreted as there is“no” path-length difference between | S I | and | S B | .Focusing on the interference part of the contributions to different partial waves, Fig.5(d) exhibits the rapid oscillationfunctions of σ el ( L ) as a function of E for different partial waves. The oscillation is similar to the wave-interference inthe annihilation cross section. Although Eq.(19) is more complicated than Eq.(13) for the annihilation, we have foundfrom our numerical calculations that the [ − S I )] term dominates in the evaluation of σ intel ( L ) over the energyrange. Furthermore, the maximum of σ el ( L ) as an “elastic resonance” for various partial waves is located at the sameenergy as the minimum of Re( S I ). The phase of S I , accordingly to Eq. (10), is given predominantly by ( δ WKB + δ ).Numerical calculations of ( δ WKB + δ ) indicate that the difference of the phases between elastic resonances is π , i.e., ∂ ( δ WKB + δ ) /∂n ≈ π . Figures 5(d) and 5(e) show that the condition for the elastic resonances at the energy E arising from the pocket is ( δ WKB + δ )( E, L ) ≈ ( n L + 12 ) π, (20)at which cos[2( δ WKB + δ )( E, L )] = −
1, substantiating the quantization condition for the elastic resonance similar tothat for the annihilation resonance in Eq. (16), but with the addition of δ WKB to δ . And because of this additional δ WKB the location of the elastic resonance energy E is slightly shifted from the annihilation resonance energy for thesame n and L .Another interesting feature besides the elastic oscillations is the resonance energies shifting toward lower energieswith an increasing nuclear radius or mass. This is because for any quantum scattering by an effective potential suchas Eq.(7); if the pocket is sufficiently deep, resonances will occur whenever a whole number of wavelengths can befitted into the pocket at energies below the barrier. Consequently, as the nuclear radius increases, the wavelengthsof the wave function inside the pocket must correspondingly increase, and the energy E of the wave function mustcorrespondingly decrease to maintain the resonant condition. Note that the opposite behavior is also possible wherethe resonance shifts toward higher energies with increasing nuclear radius. And such behavior was exemplified in theneutron total cross-section data for Cu, Cd, and Pb [40] and Ho [30] nuclei from 2 MeV to 125 MeV. The cause ofthe broad resonance and its shifts to higher energies have been explained by Peterson [29] and by Wong in [30] usingthe concept of Ramsauer’s interference [28].As the incident energy further decreases below 0.03 MeV, we see the optical model predicted annihilation cross-sections for C, Ag, and Pb nuclei merge into one single curve, whatever their mass. Parametrizing these curves with σ ¯ nA ann ( p ) ∝ /p α ¯ N in this energy region (i.e., 1.0 ≤ p ≤ c ), we found α ¯ N = ∂ ln( σ ¯ nA ann ) /∂ ln( p ) for C, Ag andPb nuclei to be ∼ ∼ ∼ s -wave predictionof α ¯ n = 1. On the elastic scattering at energies below 0.03 MeV, we notice the scattering becomes isotropic andenergy-independent. Consequently, the “ s -wave” cross-section is purely geometrical effect and a constant, which canbe described by a black-nucleus model.Lastly, we find from additional calculations that as the imaginary potential W increases, the amplitude of thepocket resonance oscillation decreases. And we further note that the annihilation and elastic curves for W = 0.028MeV are essentially identical to those of W = 2.8 MeV. III. CONCLUSIONS
The optical-model calculations show oscillatory structures in ¯ nA annihilation and elastic cross-sections in the low ¯ n energy range 0.001 −
10 MeV. The cross-sections oscillate as a function of log( E ) with small amplitudes and narrowperiods for massive nuclei. This unexpected behavior contradicts the generally expected Bethe-Landau’s power-lawin the s -wave limit.We, therefore, used a semiclassical S -matrix method to obtain further insight to the nature and the precise origin ofthe oscillations. We found two main contributions to the structures in the reaction cross sections: (1) For annihilationand elastic scattering, the oscillations are attributed to the interference between the internal and barrier waves, whiletheir smooth, 1 /p , backgrounds are mainly due to the non-interference term. (2) Delve deeper into the interferenceterm for both reactions, we identified the maxima of the cross-sections as resonances occurred within the potentialpocket. The condition for an annihilation resonance is the quantization rule: δ ( E, L ) ≈ ( n L + 1 / π , and for anelastic resonance: ( δ WKB + δ )( E, L ) ≈ ( n L +1 / π where n L is the number of bound and quasi-bound L -states in thepocket below E . So in conclusion, the existence of resonances is connected to the potential pocket and barrier that,and of course, depends sensitively on the depth and the radial dimension of the optical potential and its magnitude ofthe absorptive potential. It is for this reason that the pockets may not exists for lighter nuclei for which these pocketresonances do not appear. From this aspect, the occurrence of the pocket resonances may be a sensitive indicator ofthe depth of the potential. Pocket resonances may also not exist if the imaginary potential is large. Experimentalobservations of these pocket resonances will provide valuable information on the properties of the optical modelpotentials and the nature of the ¯ n -annihilation process for different nuclei. Though here we focused on ¯ nA reactions,similar work on the potential pocket analysis can also be extended to C+ C nuclei [41] which is an importantreactions in nuclear astrophysics.
ACKNOWLEDGMENT
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