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Potential barriers modified by coupling in the analysis of the nuclear fusion process
Nuclear Physics A 687 (2001) 385–404 www.elsevier.nl/locate/npe
Potential barriers modified by coupling in the analysis of the nuclear fusion process M.R. Spinella a , J.E. Testoni a,∗ , O. Dragún a , H.D. Marta b a Departamento de Física, Comisión Nacional de Energía Atómica, Avda. del Libertador 8250,
1429 Buenos Aires, Argentina b Facultad de Ingeniería, Universidad de la República, CC30, CP11000, Montevideo, Uruguay
Received 24 July 2000; revised 25 September 2000; accepted 8 November 2000
Abstract The behavior of several relevant physical quantities calculated on the basis of coupled-channel wavefunctions are used in the study of fusion barrier distributions. The introduction of potential barriers modified by coupling effects makes possible the decoupling of the wave equations. This procedure leads to a view of the fusion barrier distributions that does not require to consider channels related to states which are admixtures of the ordinary ones. The 16 O + 144 Sm system has been selected as a paradigmatic case in the comparison between the theoretical and experimental data. For the sake of simplicity, the elastic and just an inelastic reaction channel are taken into account. The characteristics of the barrier distributions are analyzed on the basis of the modified potential barriers and the spatial behavior of physical quantities such as fusion rates, sources or sinks, and the incoming and outgoing currents in the different channels. 2001 Elsevier Science B.V. All rights reserved. PACS: 25.70; 25.70.Bc; 25.70.Jj; 24.10.Eq Keywords: N UCLEAR REACTIONS : Coupled-channel formalism; Modified potential barriers; 16 O + 144 Sm; Ecm = 55–70 MeV; Fusion excitation functions; Fusion barrier distributions
1. Introduction Several years ago, it was pointed out that the opening of channels in nuclear collisions leading to fusion processes induces structure-dependent effects in the fusion excitation functions [1]. These effects become apparent as a consequence of the presence of different potential barriers that must be surpassed by quasiparticles described by combinations of the involved ordinary physical states [2]. The concept of fusion barrier distribution expressed by the second derivative respect to the energy of the fusion excitation function times * Corresponding author.
E-mail address: [email protected] (J.E. Testoni). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 5 8 2 - 0
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this energy has become an efficient instrument in the interpretation of measured data [3]. Experimental evidence confirmed the validity of the theoretical developments [4,5]. The extensively used CCFUS code and its different modified versions [6] reveal themselves as powerful tools in the analysis of fusion barrier distributions. Such formalisms are, however, somewhat schematic and include important approximations. More exact codes [7] have confirmed that structures in the above mentioned second derivative are indeed expressed by the coupled-channel formalism. But, it may be remarked that these codes, due to their complexity, are not practical tools in order to analyze the underlying mechanisms involved in the nuclear processes in consideration. This work intends to obtain, in a simple case, an insight of the mechanisms that explain how, when coupling is present, the perturbations in the excitation functions leading to a distribution of fusion barriers take place. The 16 O + 144 Sm system was chosen to perform the present study because good measurements of the fusion excitation function have been reported and good descriptions of the barrier distributions have been obtained by means of the above mentioned simple codes [5,8]. An extensive analysis of this system in which the validity of the two-channel model, and the importance of anharmonic effects of the phonon excitations are discussed, has been recently reported [9,10]. Taking into account the main aim of the present work, only the excitation of the one-phonon 3− state of the target nucleus is considered in the calculations. The inclusion, in a two-channel approach, of that dominant state, which produces a well-defined pair of peaks in both, the experimental and the theoretical barrier distributions, is an adequate option for the purposes of this study.
2. The formalism In the present coupled-channel analysis of the fusion process behavior in the 16 O + 144 Sm system, only the elastic scattering channel and the inelastic one due to the excitation of the octupolar 3− (1.81 MeV) vibrational state of the target will be considered. In the next subsections, the basic coupled-channel formalism and the procedure proposed in order to analyze the reaction mechanisms which explains the fusion barrier distributions considered in the study are presented. An incident plane wave in the elastic channel of unitary amplitude is always assumed. 2.1. Basic coupled-channel formulae The radial wavefunctions gn (r) that describe the collision process are found by solving the following set of coupled equations: 2 2 ( + 1) h¯ d 2 − + k (r) gn (r) = Vn,n (r)gn (r), (1) − V C n 2 2 2µ dr r n
where r is the coordinate of the relative motion between the projectile and the target, n characterizes the elastic or the inelastic channel, represents the orbital angularmomentum quantum number of the relative motion, µ is the reduced mass and kn is
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the asymptotic wave number of the channel. The type of analysis that is intended in the present study indicates the isocentrifugal approximation [3] as an appropriate choice; all the channels with the same total angular momentum are considered to have the same centrifugal potential. The diagonal matrix elements Vn,n (r), for both, the elastic and the inelastic channels, are represented by the same conventional Woods–Saxon shaped nuclear optical potential given by VN (r) =
W0 V0 +i , 1 + exp[(r − RV )/aV ] 1 + exp[(r − RW )/aW ]
(2)
where V0 and W0 are the strengths of the real and imaginary components of the optical potential, the nuclear radii Ri are related to the radial parameters r0i and to the projectile 1/3 1/3 and target mass numbers Ap and At by the expressions Ri = r0i (Ap + At ), i = V , W ; aV and aW being the respective diffuseness parameters. Improved descriptions of the Coulomb interaction in the nuclear inner region have been recently developed [11], however, this region is not sensitive to decide the fusion cross section at low energies. Thus, in which follows, we have adopted the conventional potentials corresponding to a spherical and uniform charge distribution. In this way, the Coulomb potential VC (r), is given by Z Z e2 /r for r RtC , (3) VC (r) = p t 2 2 2 Zp Zt e /2RtC 3 − r /RtC for r < RtC , where Zp and Zt are the atomic numbers of the projectile and target, respectively, e is the 1/3 electron charge and RtC = r0C At is the radius of the uniformly charged sphere. The symmetrical nondiagonal formfactors Vnn (r) determine the flux of particles between channels. In order to facilitate comparisons with the results obtained in other studies, the formfactor given in Ref. [12] have been adopted; the coupling strength for the excitation to the collective state is calculated assuming that the deformation of the nuclear matter and that of the charge are similar: λ dVNR 3ZpZt e2 RtC β −R + for r RtC , √ tV dr 2λ + 1 r λ+1 4π (4) Vnn (r) = dVNR 3ZpZt e2 r λ β + −RtV for r < RtC , √ λ+1 dr 2λ + 1 RtC 4π where VNR is the real part of the nuclear potential VN , λ the multipolarity of the transition, 1/3 and β the deformation parameter; the target nuclear radius being RtV = r0V At . In solving Eq. (1) the following boundary conditions are required: (i) regularity of the wavefunctions gn (r) at the origin, (ii) a Coulomb wave behavior in the asymptotic region and (iii) an incoming wave only in the elastic channel. 2.2. Modified potential barriers In order to analyze the behavior of the fusion barrier distributions, potential barriers modified by coupling effects [13] are defined as
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h¯ 2 ( + 1) + VC (r) + 2µ r2
n
Vn,n (r)gn (r) . gn (r)
(5)
mod ] must be compared with the conventional The real parts of the modified barriers [Vn potential barriers,
Vcon (r) =
h¯ 2 ( + 1) + VC (r) + VN (r) . 2µ r2
(6)
This formal transformation allows us to deal with uncoupled wave equations for the elastic and the inelastic channel: 2 2 h¯ d 2 mod + kn − Vn (r) gn (r) = 0. (7) 2µ dr 2 mod (r) determines the In each of these uncoupled equations a modified barrier Vn behavior of the wavefunction and, in consequence, the relevant physical quantities, i.e. spatial distributions of fusion rates, interchannel sources and sinks, and currents, which are useful tools in order to explain characteristics of the fusion barrier distribution. The real part of the modified barrier, mod h¯ 2 ( + 1) n Vnn (r)gn (r) + V (r) +
Vn (r) = , (8) C 2µ r2 gn (r)
substitutes, in the uncoupled equations, the conventional Coulomb barrier. In the imaginary part of the modified barrier, mod cpl (9) Vn (r) = VN (r) + Vn (r), two terms must be distinguished: the usual optical absorptive potential which is assumed as the unique responsible of the fusion process, [VN (r)] (see Eq. (2)) and the component, negative or positive, Vnn (r)gn (r) cpl Vn (r) = , n = n , (10) gn (r) related to the interchannel transitions induced by the coupling effects. Notice that, in the case of the inelastic channel, the second term in Eq. (9) accounts for the whole source of particles in the channel; while, in the case of the elastic channel, the source represented by an asymptotic incoming wave is also present. Eqs. (1) and (7) are two ways in which the same set of wave equations may be expressed; the latter, formally presented as describing “uncoupled” channels, facilitates, as will be discussed later, the interpretation of the reaction mechanisms that induce the behavior of the fusion barrier distribution. In particular, the wavefunctions gn (r) are the same in both equations. In the analysis reported in the following sections gn (r) are calculated by solving the coupled system and, then, these wavefunctions are applied in the evaluation of the modified barriers.
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2.3. Physical quantities used in the analysis In the next sections several physical quantities, which are specified below, will be used to analyze the fusion barrier distributions. In each case, a development in partial waves is performed. The radial distribution of the fusion rate in the channel n, depending on the incident energy E, is defined as 2 2 F Rn (r) = − VN (r) gn (r) , (11) F Rn (r) = h¯ fusion rate meaning the number of fusions per unity of time taking place in a shell of unitary width and radius r. The fusion excitation function σ f (E) may be expressed as σnf (E), (12) σ f (E) = n
where the excitation functions corresponding to each channel are given as follows: 1 µ f f σn (E) = σn (E) = F Rn (r)dr, ρ h¯ kn
(13)
where ρ is the probability density of the incident wave. The fusion barrier distribution is defined as an addition of partial barrier distributions Bn (E): f B(E) = Bn (E) = d2 (Eσn (E))/dE 2 . (14) n
n
The inelastic reaction cross section is determined by the S-matrix elements Sin, [13]: π kin σinR = 2 (2 + 1)|Sin, |2 . (15) kel kel
The knowledge of the incoming and outgoing currents in the channels as a function of the radial coordinate r is an useful instrument in the analysis of the behavior of the fusion barrier distributions from the point of view of the modified barriers. Except near the classical turning points, the partial wavefunctions can be divided into an incoming component and an outgoing one: R R (r)r + On (r) exp ikn (r)r , (16) gn (r) = In (r) exp −ikn R being the real part with amplitudes In and On slowly variant complex functions of r, kn of the complex wave number kn . It has been verified that the expressions
CI n (r) ∼
R hk ¯ n In (r)2 µ
and COn (r) ∼
R h¯ kn On (r)2 µ
(17)
give approximate descriptions of the incoming and the outgoing currents in each channel, useful in order to perform the analysis of Section 3. Notice that in Eq. (16) the wavefunctions conserve their asymptotic Coulomb wave behavior.
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The partial radial distribution of the source (or sink) in a given channel may be expressed as Sn (r) =
2 2 cpl Vn (r) gn (r) , h¯
(18)
the quantity Sn (r) has the same dimensions than the fusion rate F Rn (r). The source in the elastic channel Sel, has the same magnitude and opposite sign than that which corresponds to the inelastic one Sin, . 3. Analysis of the fusion barrier distribution in the 16 O + 144 Sm system The present analysis of the fusion barrier distribution in the 16 O + 144 Sm system over the center-of-mass energy range Ecm = 55–70 MeV will be made following the next steps: (a) determination of parameters that give a good description of the experimental fusion excitation function in terms of the formalism described in Section 2, (b) study of the buildup of the successive -contributions to the barrier distribution and selection of a representative one, (c) discussion of the behavior of specific excitation functions and barrier distributions on the basis of the modified potential barriers, sources, fusion rates, and the incoming and outgoing currents in the channels. The fusion excitation function obtained with the present formalism using the parameters of Table 1 and the experimental data reported in Ref. [5] are shown in Fig. 1. A set of real optical parameters close to that given in Ref. [12] has been adopted. The imaginary parameters were obtained by fitting the experimental fusion excitation function. In Fig. 2, the experimental barrier distribution, also reported in Ref. [5], is compared with that corresponding to the present calculation. As a reference, results obtained with the same optical parameters but neglecting deformation (i.e. β = 0) are shown in these figures. The near-barrier enhancement of the fusion cross sections and the splitting of the barrier distribution as consequences of the coupling effects can be observed in the figures. The agreements obtained between experimental and theoretical data allows us to undertake the proposed analysis of the mechanisms underlying the fusion barrier distribution using the method described above and the parameters of Table 1. Fig. 3a shows a typical partial fusion excitation function σf , i.e. the contribution corresponding to = 9 in Eq. (14), adopted as a representative term for the present analysis; in which follows, for the sake of simplicity, the subscripts indicating = 9 will be omitted. The increase from zero to the maximum of the fusion cross section as a function of Table 1 Optical and deformation parameters V0 (MeV)
r0V (fm)
aV (fm)
W0 (MeV)
r0W (fm)
aW (fm)
r0C (fm)
β
−105.1
1.1
0.765
−15
0.8
0.5
1.15
0.205
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Fig. 1. The experimental fusion excitation function for the 16 O + 144 Sm system reported in Ref. [5] (solid circles) and the fit obtained with the present formalism (solid line). The theoretical excitation function for the same optical parameters but β = 0 is also shown (long-dashed line).
Fig. 2. The fusion barrier distribution for the 16 O + 144 Sm system derived from the experimental fusion excitation function reported in Ref. [5] (solid circles) and theoretical calculation obtained with the parameters of Table 1 (solid line). The distribution for the same optical parameters but β = 0 is also shown (long-dashed line).
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Fig. 3. (a) The theoretical partial fusion excitation functions for = 9 (solid line). The elastic (dot-dashed line) and the inelastic (dashed line) contributions to the total excitation function are also plotted. (b) The fusion barrier distribution corresponding to the excitations functions shown in (a).
the energy occurs in two steps. For higher energies the partial excitation function decreases as 1/Ecm . Each step gives rise to a positive peak followed by a negative one in the barrier distribution plotted in Fig. 3b (solid line). As can be seen there, the pattern with two main oscillations is determined by the typical curvatures associated with the successive steps in the excitation function. In Fig. 3 are also plotted the contributions of each channel to the excitation function, as well as their corresponding barrier distributions. The two step behavior is present in σelf , however, the peak-like shaped contribution σinf around Ecm = 63 MeV enhances this behavior in the total excitation function. At Ecm = 66–67 MeV the fusion cross section in the elastic channel reaches a maximum while that in the inelastic one shows a minimum. These facts lead to the following consequences in the barrier distribution. In the first oscillation, as the energy increases, the contributions of both channels are coherent, while, in the second, the partial barrier distributions interfere determining the shape and position of this oscillation. At higher energies the elastic contribution to fusion is systematically transferred to the inelastic one. In this energy region, the curvatures of the excitation functions are smooth and with different signs. This occurs in such a way that the curvature of the total excitation function is null as corresponds to the classical 1/Ecm behavior. As a reference, the = 9 partial excitation function and the corresponding barrier distribution for the case without deformation are shown in Fig. 4. There, it can be observed the well known behavior of the excitation function for this case; that is, the increase from zero to the maximum of the cross section as a function of the energy in just one step. The midpoint of this step takes place close to the energy of the maximum of the Coulomb barrier, EC = 62.1 MeV. A single step gives rise to an unique oscillation composed by a positive peak followed by a negative one for crescent energies in the barrier distribution.
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Fig. 4. (a) The fusion excitation function when only the elastic channel is opened (i.e. β = 0). (b) The corresponding barrier distribution.
When the coupling represented by the deformation parameter β increases, these peaks suffer a progressive splitting leading to the pattern shown in Fig. 3b. It can be observed that the amplitude of the positive and negative peaks in the partial barrier distributions, as well as their energy positions, increases with the relative angular momentum . However, when the partial contributions are summed up, an interference process leads to the double positive peaked pattern in the fusion barrier distribution displayed by both, theoretical and experimental results. Fig. 5 illustrates how this process of interference occurs. In Fig. 5a the partial contributions to the barrier distribution from = 0 to = 40 are plotted, together with the total barrier distribution shown in Fig. 2. Notice in the figure the presence of two envelopes, each one related to one of the two final main peaks of the barrier distribution. Fig. 5b shows how the final distribution is being successively approached when the maximum -value in the sum, max , increases. The term in the sum corresponding to = 9 is highlighted in Fig. 5a; its characteristics suggest that this particular relative angular momentum constitutes a good choice in order to analyze the basic behavior of the barrier distributions and their relations with different significant physical quantities. In Fig. 6, partial excitation functions for the following processes are plotted together in order to perform comparisons: fusion σ f , fusion through the elastic channel σelf , fusion through the inelastic channel σinf , inelastic scattering σinR , and fusion without deformation f . The fusion cross section, σ f = σ f + σ f , is composed by the contributions of the σβ=0 el in elastic and the inelastic channels. The hindrance of the elastic contribution to fusion, in the energy region where the inelastic scattering is an open channel — see peak in σinR around Ecm = 63 MeV — is not fully compensated by the inelastic contribution, giving rise to the double peaked pattern in the barrier distribution. Notice that the fusion threshold in the coupled case is shifted toward lower energies in comparison with that of the β = 0 case;
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Fig. 5. (a) The barrier distributions for different orbital angular momenta (from = 1 to 40) and the total barrier distribution (solid line). The contribution for = 9 is highlighted. (b) The sum of partial distributions from = 0 up to max = 11 (dot-dashed line), 20 (dashed line), 30 (short–long-dashed line) and 40 (solid line).
this behavior characterizes the near-barrier enhancement of the fusion cross section. As a complement of the excitation functions of Fig. 6, asymptotic outgoing currents are plotted in Fig. 7. Two main features can be pointed out, both occurring in the energy region where the inelastic scattering channel is open: (a) between Ecm = 59 and 63 MeV the total
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Fig. 6. Comparison between several relevant excitation functions.
Fig. 7. Asymptotic outgoing currents in the elastic (dot-dashed line) and inelastic (dashed line) in the coupled case. The total outgoing currents for this case (solid line) and the β=0 one (long-dashed line) are also shown.
outgoing current is hindered as a correlate of the near-barrier fusion enhancement; (b) in the plateau-like region of the excitation function, i.e. between Ecm = 63 and 67 MeV, that current is kept in a high level in comparison with that corresponding to the β = 0 case — a fact related with the perturbations in the excitation functions that induce the multiplicity of barriers.
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The behavior, as a function of the energy, of the cross sections and currents will be analyzed taking into account some relevant quantities: the conventional Coulomb barrier V con , the real modified barriers [Vnmod ], the source of particles in the inelastic channel Sin , the total fusion rate F R = F Rel + F Rin , and the incoming CI n and outgoing COn currents in the channels. Representative energy regions have been selected in order to perform the analysis. It must be taken into account that the wavefunctions, fusion rates, sources and currents in the figures correspond to the uncoupled equations as well as to the coupled ones (see Eqs. (1) and (7)). It is useful, however, to perform the present study from the point of view of the uncoupled equations governed by the modified potential barriers. At incident sub-Coulomb energies, the conventional Coulomb barrier V con , as well as the real elastic and inelastic modified barriers [Vnmod ] are higher than the pertinent channel energies, Eel or Ein . The pattern, here, is one of almost complete subbarrier reflection, the source and the inelastic outgoing currents are essentially null because wave functions are negligible in the region where the formfactors are important, the elastic outgoing current assumes a purely reflective character and the coupling effects are not significant. At Ecm = 61.7 MeV, barely below the Coulomb barrier energy (see Fig. 8a), the energy Eel is just in between the tops of V con and [Velmod ]; that is, from the point of view of the conventional Coulomb barrier, the problem would be a subbarrier one, but this is not the case if [Velmod ] is the potential used in the uncoupled equations because the channel energy exceeds its top. This fact explains why, in the coupled case, there is a shift, toward lower energies, of the critical point in which the first step of the excitation function takes place (see Fig. 6); in other words, it explains the enhancement of the fusion cross section at near-barrier energies. The relation between total incoming (negative) and outgoing (positive) currents and fusion rates are shown in Fig. 8b for the cases with (β = 0) and without (β = 0) deformation. There, it can be seen that the fusion rate increases while, consistently, the reflected current decreases when the inelastic channel is open. Remark that, in absence of internal sources, underbarrier currents are null (see long-dashed lines in the figure). The incoming and outgoing components of the currents in each channel when deformation is present, as well as the source in the inelastic channel are shown in Fig. 8c. It is illustrative to observe how the source injects in the inelastic channel, simultaneously, an incoming and an outgoing wave. The incoming one fades-out feeding the fusion channel, while the outgoing one contributes to the inelastic scattering. As will be pointed out later, the balance between this processes is decisive in order to explain the generation of a distribution of fusion barriers. In Fig. 9a the gaps between the corresponding channel energies and the tops of the elastic (dot-dashed line), and inelastic (dashed line) modified real barriers and the conventional Coulomb barrier (long-dashed line) are plotted. For the elastic channel, three energy regions may be discerned: well below the midpoint of the window where the inelastic scattering channel is open, the gap for β = 0 is greater than that for β = 0; the inverse situation takes place well above this midpoint; over the window a transition occurs. This behavior suggests that the real elastic modified barrier [Velmod ] might be responsible for
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Fig. 8. Radial distributions of different quantities for Ecm = 61.7 MeV. (a) The real modified barriers for the elastic [Velmod ] (dot-dashed line) and the inelastic [Vinmod ] (dashed line) channels. The corresponding channel energies are shown. The conventional Coulomb barrier V con is also plotted (long-dashed line). (b) The total incoming (negative) and outgoing (positive) currents in the coupled case, i.e. elastic plus inelastic contributions (solid lines), and the corresponding currents for the case with β = 0 (long-dashed lines). Fusion rates for the coupled case (dot-dot-dashed line) and the β = 0 case (short–long-dashed line) are also displayed. (c) The incoming (negative) and outgoing (positive) elastic (dot-dashed lines) and inelastic (dashed lines) currents. The source in the inelastic channel is also shown (short–long-dashed line).
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Fig. 9. (a) The energy gap between the corresponding channel energies and the tops of the elastic real modified barrier (dot-dashed line), the inelastic real modified barrier (dashed line), and the conventional Coulomb barrier (long-dashed line). (b) Comparison between the fusion excitation function given by including in the calculation the complete modified barrier (solid line) and that obtained by using only the real elastic modified barrier (short–long-dashed line).
the two characteristic influences of the nuclear structure in the fusion process: the nearbarrier enhancement and the barrier distribution. In relation with the former item, the fact that, in the pertinent energy region, the β = 0 gap is greater than the β = 0 one means a favored transmission through the barrier leading to the enhancement of the fusion cross section due to coupling effects. In relation with the second item, it may be argued that the transition toward a less important gap on the inelastic scattering window can explain the hindrance in the excitation function, as well as the associated inflection that produces the curvatures responsible for the splitting of peaks in the barrier distribution. The fusion excitation function predicted by the calculations when only the real modified barrier is present in the elastic channel — i.e. the unique source in the elastic channel is the incident plane wave — is shown in Fig. 9b (short–long-dashed line). Two conclusions can be inferred if the result of this calculation is compared with the excitation function σ f (solid line): the mere real modified barrier explains the near-barrier enhancement, but, the justification of the barrier distribution behavior is only partially achieved. The interplay between the source and the modified barrier in the inelastic channel must be analyzed in order to clarify why the plateau-like behavior in the fusion excitation function takes place. This interplay is illustrated in Fig. 10 where the position of the source in the inelastic channel (short–long-dashed lines) respect to the top of the real inelastic modified barrier (see vertical solid lines) is put in evidence. The outgoing currents in the channel (dashed lines) are also plotted. The progressive transit of the source from the outer side to the inner side of the barrier is correlated with the amount of outgoing current. Between Ecm = 63 and 69 MeV, as can be concluded if the inelastic gap is compared with
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Fig. 10. The interplay between the source distribution (short–long-dashed lines) and the position of the top of the real modified barrier in the inelastic channel (vertical solid lines) in the energy region where fusion barriers take place. The outgoing current in the inelastic channel is also shown (dashed lines).
the gap for the β = 0 case in Fig. 9a, the coupling induces a progressive increment of the reflective character of the inelastic modified barrier; consequently, the passage of current through the barrier, in both directions, is hindered. In the cases with Ecm = 63 and 65 MeV the barrier and the source positions permit a significant amount of inelastic scattering. In particular, as can be seen in Fig. 11, at Ecm = 65 MeV, the corresponding gaps are 0.31 MeV for the inelastic case and 2.87 MeV for the β = 0 one. This fact explains why the high transmission in the last of this situations implies a negligible reflection of the impinging wave in the entrance channel which contributes almost exclusively to fusion. Conversely, in the coupled situation, a significant outgoing current is generated in the excited channel by the portion of the source outside the reflective inelastic barrier. This fact hinders the fusion process and, correlatively, increases the scattering one (see Figs. 7, 10a and 10b). Fig. 11 gives, for Ecm = 65 MeV, the equivalent information given in Fig. 8 for Ecm = 61.7 MeV. This figure is illustrative in order to verify properties described above: the reflective character of the real inelastic modified barrier in comparison with the transparent character of the conventional Coulomb barrier in the elastic channel (see Fig. 11a); the difference between the amount of inelastic scattering
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Fig. 11. Same as Fig. 8 for Ecm = 65 MeV.
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Fig. 12. Same as Fig. 8 for Ecm = 78 MeV. Notice that in (b) currents and fusion rates for both, the coupled and the β = 0 cases, are almost coincident.
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— i.e. outgoing currents — in the cases with and without deformation; and its correlative difference in the fusion rates that reveals the fusion hindrance induced by coupling (see Fig. 11b). Another interesting feature can be observed in Fig. 11c, the formfactor in the periphery of the nucleus transforms almost completely the elastic incoming flux into inelastic incoming flux at around 9 fm. However, immediately, the source in the inelastic channel becomes a sink, producing a reflux into the elastic channel that, at this energy, constitutes the main contribution to fusion. This effect has a maximum around Ecm = 66–67 MeV (see Fig. 6). Figs. 10c and 10d indicate how when, at Ecm = 67 and 69 MeV, the source is placed in the inner radial region respect to the inelastic barrier, the scattering in the excited channel is negligible. At higher energies, the processes with and without deformation assume common behaviors as can be observed in Fig. 12, devoted to Ecm = 78 MeV. It is to be noticed that, for energies beyond Ecm = 67 MeV, progressively, the above mentioned sink in the inelastic source disappear; this means that the main contribution to fusion will be provided by the inelastic channel. The source that fed this channel is confined by the real inelastic modified barrier in the nuclear interior as is indicated in the figure.
4. Summary The main purpose of the present work is to explore an alternative point of view in the analysis of the underlying reaction mechanisms that determine the behavior of the fusion barrier distributions. In this way, a better comprehension of the two main manifestations of the interplay between these reaction mechanisms and structure in nuclear collisions, i.e. near-barrier enhancement of the fusion cross section and fusion barrier distributions, can be achieved. The formal decoupling of the wave equations and the correlative definition of energydependent modified potential barriers allows us to study the behavior of some relevant physical quantities, i.e. fusion rates, interchannel sources and incoming and outgoing currents, in the simpler context of uncoupled channels. The dependence of the cross sections as functions of the energy is analyzed considering the behavior of those physical quantities as functions of the radial coordinate. As an example of the application of these procedures the fusion barrier distribution in the 16 O + 144 Sm system is studied. The selection of this extensively addressed system, which presents a dominant 3− excitation, permits to restrict the problem to a two-channel one that shows well separated peaks in the barrier distribution. By fitting the experimental data a convenient set of parameters is achieved. The importance of the partial-wave developments is emphasized. In particular, it is pointed out how the buildup of the successive contributions, with different -values, to the barrier distribution leads to a peculiar interference process; while it removes the negative peaks of the partial contributions, it produces the final pattern which exhibits only two
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positive peaks. This analysis leads to the selection of a representative term between the -contributions to the barrier distribution, i.e. that for = 9, in order to perform the study. Along the partial excitation functions, several characteristic energy regions have been chosen to be analyzed under the light of the spatial behavior of the above mentioned physical quantities. Below the threshold of the fusion process, a dominant reflective picture inhibits coupling effects. Around the threshold, the difference between the conventional Coulomb barrier and the real part of the elastic modified barrier explains the near-barrier enhancement of the fusion cross section. The inflection in the excitation function that originates the splitting effect in the barrier distribution seems to be due to the following two facts: (a) the passage of the real modified barrier top in the elastic channel from below to above the conventional Coulomb barrier top, and (b) a local interplay between the source and the real inelastic modified barrier that enhances the inelastic scattering; both acting in detriment of fusion. At energies well above those for which the inelastic scattering is important, again, the coupling effects do not affect the fusion cross section and, consequently, there are not fusion barriers. The model predicts interesting transitions between channels that determines which channel feeds the fusion processes at different energies. The present approach provides an explanation of enhancement and multiplicity of barriers in the fusion process based in energy-dependent potential barriers modified by coupling effects acting in independent wave equations; it constitutes an alternative and complementary interpretation of that based in a decoupling which introduces mixed quantum states.
Acknowledgements One of the authors (M.R.S.) is a postdoctoral fellowship of the Consejo de Investigaciones Científicas y Técnicas (CONICET), Argentina, and J.E.T. is member of the Carrera del Investigador Científico, CONICET. H.D.M. acknowledge financial support from PEDECIBA and CSIC (UDELAR), Uruguay, and Agencia Nacional de Promoción Científica y Tecnológica, Argentina, Pict 03-00000-01357.
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A.M. Stefanini, D. Ackermann, L. Corradi, D.R. Napoli, C. Petrache, P. Spolaore, P. Bednarczyk, H.Q. Zhang, S. Beghini, G. Montagnoli, L. Mueller, F. Scarlassara, G.F. Segato, F. Soramel, N. Rowley, Phys. Rev. Lett. 74 (1995) 864; A.M. Stefanini, D. Ackermann, L. Corradi, H.J. He, C. Montagnoli, S. Beghini, F. Scarlassara, G.F. Segato, Phys. Rev. C 52 (1995) R1727. C.R. Morton, M. Dasgupta, D.J. Hinde, J.R. Leigh, R.C. Lemmon, J.P. Lestone, J.C. Mein, J.O. Newton, H. Timmers, N. Rowley, A.T. Kruppa, Phys. Rev. Lett. 72 (1994) 4074; J.R. Leigh, M. Dasgupta, D.J. Hinde, J.C. Mein, C.R. Morton, R.C. Lemmon, J.P. Lestone, J.O. Newton, H. Timmers, J.X. Wei, N. Rowley, Phys. Rev. C 52 (1995) 3151. C.H. Dasso, S. Landowne, Comput. Phys. Commun. 46 (1987) 187; J.O. Fernández-Niello, C.H. Dasso, S. Landowne, Comput. Phys. Commun. 54 (1989) 409; M. Dasgupta, A. Navin, Y.K. Agarwal, C.V.K. Baba, H.C. Jain, M.L. Jhingan, A. Roy, Nucl. Phys. A 539 (1992) 351. I.J. Thompson, Comput. Phys. Rep. 7 (1988) 167. M. Dasgupta, D.J. Hinde, J.R. Leigh, K. Hagino, Nucl. Phys. A 630 (1998) 78c. K. Hagino, N. Takigawa, S. Kuyucak, Phys. Rev. Lett. 79 (1997) 2943. K. Hagino, S. Kuyucak, N. Takigawa, Phys. Rev. C 57 (1998) 1349. N. Takigawa, T. Rumin, N. Ihara, Phys. Rev. C 61 (2000) 044607. K. Hagino, N. Takigawa, M. Dasgupta, D.J. Hinde, J.R. Leigh, Phys. Rev. C 55 (1997) 276; C.R. Morton, Ph.D. thesis, The Australian National University, 1995. J.E. Testoni, O. Dragún, H. Massmann, M.R. Spinella, Nucl. Phys. A 669 (2000) 173.
Potential barriers modified by coupling in the analysis of the nuclear fusion process M.R. Spinella a , J.E. Testoni a,∗ , O. Dragún a , H.D. Marta b a Departamento de Física, Comisión Nacional de Energía Atómica, Avda. del Libertador 8250,
1429 Buenos Aires, Argentina b Facultad de Ingeniería, Universidad de la República, CC30, CP11000, Montevideo, Uruguay
Received 24 July 2000; revised 25 September 2000; accepted 8 November 2000
Abstract The behavior of several relevant physical quantities calculated on the basis of coupled-channel wavefunctions are used in the study of fusion barrier distributions. The introduction of potential barriers modified by coupling effects makes possible the decoupling of the wave equations. This procedure leads to a view of the fusion barrier distributions that does not require to consider channels related to states which are admixtures of the ordinary ones. The 16 O + 144 Sm system has been selected as a paradigmatic case in the comparison between the theoretical and experimental data. For the sake of simplicity, the elastic and just an inelastic reaction channel are taken into account. The characteristics of the barrier distributions are analyzed on the basis of the modified potential barriers and the spatial behavior of physical quantities such as fusion rates, sources or sinks, and the incoming and outgoing currents in the different channels. 2001 Elsevier Science B.V. All rights reserved. PACS: 25.70; 25.70.Bc; 25.70.Jj; 24.10.Eq Keywords: N UCLEAR REACTIONS : Coupled-channel formalism; Modified potential barriers; 16 O + 144 Sm; Ecm = 55–70 MeV; Fusion excitation functions; Fusion barrier distributions
1. Introduction Several years ago, it was pointed out that the opening of channels in nuclear collisions leading to fusion processes induces structure-dependent effects in the fusion excitation functions [1]. These effects become apparent as a consequence of the presence of different potential barriers that must be surpassed by quasiparticles described by combinations of the involved ordinary physical states [2]. The concept of fusion barrier distribution expressed by the second derivative respect to the energy of the fusion excitation function times * Corresponding author.
E-mail address: [email protected] (J.E. Testoni). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 5 8 2 - 0
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this energy has become an efficient instrument in the interpretation of measured data [3]. Experimental evidence confirmed the validity of the theoretical developments [4,5]. The extensively used CCFUS code and its different modified versions [6] reveal themselves as powerful tools in the analysis of fusion barrier distributions. Such formalisms are, however, somewhat schematic and include important approximations. More exact codes [7] have confirmed that structures in the above mentioned second derivative are indeed expressed by the coupled-channel formalism. But, it may be remarked that these codes, due to their complexity, are not practical tools in order to analyze the underlying mechanisms involved in the nuclear processes in consideration. This work intends to obtain, in a simple case, an insight of the mechanisms that explain how, when coupling is present, the perturbations in the excitation functions leading to a distribution of fusion barriers take place. The 16 O + 144 Sm system was chosen to perform the present study because good measurements of the fusion excitation function have been reported and good descriptions of the barrier distributions have been obtained by means of the above mentioned simple codes [5,8]. An extensive analysis of this system in which the validity of the two-channel model, and the importance of anharmonic effects of the phonon excitations are discussed, has been recently reported [9,10]. Taking into account the main aim of the present work, only the excitation of the one-phonon 3− state of the target nucleus is considered in the calculations. The inclusion, in a two-channel approach, of that dominant state, which produces a well-defined pair of peaks in both, the experimental and the theoretical barrier distributions, is an adequate option for the purposes of this study.
2. The formalism In the present coupled-channel analysis of the fusion process behavior in the 16 O + 144 Sm system, only the elastic scattering channel and the inelastic one due to the excitation of the octupolar 3− (1.81 MeV) vibrational state of the target will be considered. In the next subsections, the basic coupled-channel formalism and the procedure proposed in order to analyze the reaction mechanisms which explains the fusion barrier distributions considered in the study are presented. An incident plane wave in the elastic channel of unitary amplitude is always assumed. 2.1. Basic coupled-channel formulae The radial wavefunctions gn (r) that describe the collision process are found by solving the following set of coupled equations: 2 2 ( + 1) h¯ d 2 − + k (r) gn (r) = Vn,n (r)gn (r), (1) − V C n 2 2 2µ dr r n
where r is the coordinate of the relative motion between the projectile and the target, n characterizes the elastic or the inelastic channel, represents the orbital angularmomentum quantum number of the relative motion, µ is the reduced mass and kn is
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the asymptotic wave number of the channel. The type of analysis that is intended in the present study indicates the isocentrifugal approximation [3] as an appropriate choice; all the channels with the same total angular momentum are considered to have the same centrifugal potential. The diagonal matrix elements Vn,n (r), for both, the elastic and the inelastic channels, are represented by the same conventional Woods–Saxon shaped nuclear optical potential given by VN (r) =
W0 V0 +i , 1 + exp[(r − RV )/aV ] 1 + exp[(r − RW )/aW ]
(2)
where V0 and W0 are the strengths of the real and imaginary components of the optical potential, the nuclear radii Ri are related to the radial parameters r0i and to the projectile 1/3 1/3 and target mass numbers Ap and At by the expressions Ri = r0i (Ap + At ), i = V , W ; aV and aW being the respective diffuseness parameters. Improved descriptions of the Coulomb interaction in the nuclear inner region have been recently developed [11], however, this region is not sensitive to decide the fusion cross section at low energies. Thus, in which follows, we have adopted the conventional potentials corresponding to a spherical and uniform charge distribution. In this way, the Coulomb potential VC (r), is given by Z Z e2 /r for r RtC , (3) VC (r) = p t 2 2 2 Zp Zt e /2RtC 3 − r /RtC for r < RtC , where Zp and Zt are the atomic numbers of the projectile and target, respectively, e is the 1/3 electron charge and RtC = r0C At is the radius of the uniformly charged sphere. The symmetrical nondiagonal formfactors Vnn (r) determine the flux of particles between channels. In order to facilitate comparisons with the results obtained in other studies, the formfactor given in Ref. [12] have been adopted; the coupling strength for the excitation to the collective state is calculated assuming that the deformation of the nuclear matter and that of the charge are similar: λ dVNR 3ZpZt e2 RtC β −R + for r RtC , √ tV dr 2λ + 1 r λ+1 4π (4) Vnn (r) = dVNR 3ZpZt e2 r λ β + −RtV for r < RtC , √ λ+1 dr 2λ + 1 RtC 4π where VNR is the real part of the nuclear potential VN , λ the multipolarity of the transition, 1/3 and β the deformation parameter; the target nuclear radius being RtV = r0V At . In solving Eq. (1) the following boundary conditions are required: (i) regularity of the wavefunctions gn (r) at the origin, (ii) a Coulomb wave behavior in the asymptotic region and (iii) an incoming wave only in the elastic channel. 2.2. Modified potential barriers In order to analyze the behavior of the fusion barrier distributions, potential barriers modified by coupling effects [13] are defined as
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h¯ 2 ( + 1) + VC (r) + 2µ r2
n
Vn,n (r)gn (r) . gn (r)
(5)
mod ] must be compared with the conventional The real parts of the modified barriers [Vn potential barriers,
Vcon (r) =
h¯ 2 ( + 1) + VC (r) + VN (r) . 2µ r2
(6)
This formal transformation allows us to deal with uncoupled wave equations for the elastic and the inelastic channel: 2 2 h¯ d 2 mod + kn − Vn (r) gn (r) = 0. (7) 2µ dr 2 mod (r) determines the In each of these uncoupled equations a modified barrier Vn behavior of the wavefunction and, in consequence, the relevant physical quantities, i.e. spatial distributions of fusion rates, interchannel sources and sinks, and currents, which are useful tools in order to explain characteristics of the fusion barrier distribution. The real part of the modified barrier, mod h¯ 2 ( + 1) n Vnn (r)gn (r) + V (r) +
Vn (r) = , (8) C 2µ r2 gn (r)
substitutes, in the uncoupled equations, the conventional Coulomb barrier. In the imaginary part of the modified barrier, mod cpl (9) Vn (r) = VN (r) + Vn (r), two terms must be distinguished: the usual optical absorptive potential which is assumed as the unique responsible of the fusion process, [VN (r)] (see Eq. (2)) and the component, negative or positive, Vnn (r)gn (r) cpl Vn (r) = , n = n , (10) gn (r) related to the interchannel transitions induced by the coupling effects. Notice that, in the case of the inelastic channel, the second term in Eq. (9) accounts for the whole source of particles in the channel; while, in the case of the elastic channel, the source represented by an asymptotic incoming wave is also present. Eqs. (1) and (7) are two ways in which the same set of wave equations may be expressed; the latter, formally presented as describing “uncoupled” channels, facilitates, as will be discussed later, the interpretation of the reaction mechanisms that induce the behavior of the fusion barrier distribution. In particular, the wavefunctions gn (r) are the same in both equations. In the analysis reported in the following sections gn (r) are calculated by solving the coupled system and, then, these wavefunctions are applied in the evaluation of the modified barriers.
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2.3. Physical quantities used in the analysis In the next sections several physical quantities, which are specified below, will be used to analyze the fusion barrier distributions. In each case, a development in partial waves is performed. The radial distribution of the fusion rate in the channel n, depending on the incident energy E, is defined as 2 2 F Rn (r) = − VN (r) gn (r) , (11) F Rn (r) = h¯ fusion rate meaning the number of fusions per unity of time taking place in a shell of unitary width and radius r. The fusion excitation function σ f (E) may be expressed as σnf (E), (12) σ f (E) = n
where the excitation functions corresponding to each channel are given as follows: 1 µ f f σn (E) = σn (E) = F Rn (r)dr, ρ h¯ kn
(13)
where ρ is the probability density of the incident wave. The fusion barrier distribution is defined as an addition of partial barrier distributions Bn (E): f B(E) = Bn (E) = d2 (Eσn (E))/dE 2 . (14) n
n
The inelastic reaction cross section is determined by the S-matrix elements Sin, [13]: π kin σinR = 2 (2 + 1)|Sin, |2 . (15) kel kel
The knowledge of the incoming and outgoing currents in the channels as a function of the radial coordinate r is an useful instrument in the analysis of the behavior of the fusion barrier distributions from the point of view of the modified barriers. Except near the classical turning points, the partial wavefunctions can be divided into an incoming component and an outgoing one: R R (r)r + On (r) exp ikn (r)r , (16) gn (r) = In (r) exp −ikn R being the real part with amplitudes In and On slowly variant complex functions of r, kn of the complex wave number kn . It has been verified that the expressions
CI n (r) ∼
R hk ¯ n In (r)2 µ
and COn (r) ∼
R h¯ kn On (r)2 µ
(17)
give approximate descriptions of the incoming and the outgoing currents in each channel, useful in order to perform the analysis of Section 3. Notice that in Eq. (16) the wavefunctions conserve their asymptotic Coulomb wave behavior.
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The partial radial distribution of the source (or sink) in a given channel may be expressed as Sn (r) =
2 2 cpl Vn (r) gn (r) , h¯
(18)
the quantity Sn (r) has the same dimensions than the fusion rate F Rn (r). The source in the elastic channel Sel, has the same magnitude and opposite sign than that which corresponds to the inelastic one Sin, . 3. Analysis of the fusion barrier distribution in the 16 O + 144 Sm system The present analysis of the fusion barrier distribution in the 16 O + 144 Sm system over the center-of-mass energy range Ecm = 55–70 MeV will be made following the next steps: (a) determination of parameters that give a good description of the experimental fusion excitation function in terms of the formalism described in Section 2, (b) study of the buildup of the successive -contributions to the barrier distribution and selection of a representative one, (c) discussion of the behavior of specific excitation functions and barrier distributions on the basis of the modified potential barriers, sources, fusion rates, and the incoming and outgoing currents in the channels. The fusion excitation function obtained with the present formalism using the parameters of Table 1 and the experimental data reported in Ref. [5] are shown in Fig. 1. A set of real optical parameters close to that given in Ref. [12] has been adopted. The imaginary parameters were obtained by fitting the experimental fusion excitation function. In Fig. 2, the experimental barrier distribution, also reported in Ref. [5], is compared with that corresponding to the present calculation. As a reference, results obtained with the same optical parameters but neglecting deformation (i.e. β = 0) are shown in these figures. The near-barrier enhancement of the fusion cross sections and the splitting of the barrier distribution as consequences of the coupling effects can be observed in the figures. The agreements obtained between experimental and theoretical data allows us to undertake the proposed analysis of the mechanisms underlying the fusion barrier distribution using the method described above and the parameters of Table 1. Fig. 3a shows a typical partial fusion excitation function σf , i.e. the contribution corresponding to = 9 in Eq. (14), adopted as a representative term for the present analysis; in which follows, for the sake of simplicity, the subscripts indicating = 9 will be omitted. The increase from zero to the maximum of the fusion cross section as a function of Table 1 Optical and deformation parameters V0 (MeV)
r0V (fm)
aV (fm)
W0 (MeV)
r0W (fm)
aW (fm)
r0C (fm)
β
−105.1
1.1
0.765
−15
0.8
0.5
1.15
0.205
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Fig. 1. The experimental fusion excitation function for the 16 O + 144 Sm system reported in Ref. [5] (solid circles) and the fit obtained with the present formalism (solid line). The theoretical excitation function for the same optical parameters but β = 0 is also shown (long-dashed line).
Fig. 2. The fusion barrier distribution for the 16 O + 144 Sm system derived from the experimental fusion excitation function reported in Ref. [5] (solid circles) and theoretical calculation obtained with the parameters of Table 1 (solid line). The distribution for the same optical parameters but β = 0 is also shown (long-dashed line).
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Fig. 3. (a) The theoretical partial fusion excitation functions for = 9 (solid line). The elastic (dot-dashed line) and the inelastic (dashed line) contributions to the total excitation function are also plotted. (b) The fusion barrier distribution corresponding to the excitations functions shown in (a).
the energy occurs in two steps. For higher energies the partial excitation function decreases as 1/Ecm . Each step gives rise to a positive peak followed by a negative one in the barrier distribution plotted in Fig. 3b (solid line). As can be seen there, the pattern with two main oscillations is determined by the typical curvatures associated with the successive steps in the excitation function. In Fig. 3 are also plotted the contributions of each channel to the excitation function, as well as their corresponding barrier distributions. The two step behavior is present in σelf , however, the peak-like shaped contribution σinf around Ecm = 63 MeV enhances this behavior in the total excitation function. At Ecm = 66–67 MeV the fusion cross section in the elastic channel reaches a maximum while that in the inelastic one shows a minimum. These facts lead to the following consequences in the barrier distribution. In the first oscillation, as the energy increases, the contributions of both channels are coherent, while, in the second, the partial barrier distributions interfere determining the shape and position of this oscillation. At higher energies the elastic contribution to fusion is systematically transferred to the inelastic one. In this energy region, the curvatures of the excitation functions are smooth and with different signs. This occurs in such a way that the curvature of the total excitation function is null as corresponds to the classical 1/Ecm behavior. As a reference, the = 9 partial excitation function and the corresponding barrier distribution for the case without deformation are shown in Fig. 4. There, it can be observed the well known behavior of the excitation function for this case; that is, the increase from zero to the maximum of the cross section as a function of the energy in just one step. The midpoint of this step takes place close to the energy of the maximum of the Coulomb barrier, EC = 62.1 MeV. A single step gives rise to an unique oscillation composed by a positive peak followed by a negative one for crescent energies in the barrier distribution.
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Fig. 4. (a) The fusion excitation function when only the elastic channel is opened (i.e. β = 0). (b) The corresponding barrier distribution.
When the coupling represented by the deformation parameter β increases, these peaks suffer a progressive splitting leading to the pattern shown in Fig. 3b. It can be observed that the amplitude of the positive and negative peaks in the partial barrier distributions, as well as their energy positions, increases with the relative angular momentum . However, when the partial contributions are summed up, an interference process leads to the double positive peaked pattern in the fusion barrier distribution displayed by both, theoretical and experimental results. Fig. 5 illustrates how this process of interference occurs. In Fig. 5a the partial contributions to the barrier distribution from = 0 to = 40 are plotted, together with the total barrier distribution shown in Fig. 2. Notice in the figure the presence of two envelopes, each one related to one of the two final main peaks of the barrier distribution. Fig. 5b shows how the final distribution is being successively approached when the maximum -value in the sum, max , increases. The term in the sum corresponding to = 9 is highlighted in Fig. 5a; its characteristics suggest that this particular relative angular momentum constitutes a good choice in order to analyze the basic behavior of the barrier distributions and their relations with different significant physical quantities. In Fig. 6, partial excitation functions for the following processes are plotted together in order to perform comparisons: fusion σ f , fusion through the elastic channel σelf , fusion through the inelastic channel σinf , inelastic scattering σinR , and fusion without deformation f . The fusion cross section, σ f = σ f + σ f , is composed by the contributions of the σβ=0 el in elastic and the inelastic channels. The hindrance of the elastic contribution to fusion, in the energy region where the inelastic scattering is an open channel — see peak in σinR around Ecm = 63 MeV — is not fully compensated by the inelastic contribution, giving rise to the double peaked pattern in the barrier distribution. Notice that the fusion threshold in the coupled case is shifted toward lower energies in comparison with that of the β = 0 case;
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Fig. 5. (a) The barrier distributions for different orbital angular momenta (from = 1 to 40) and the total barrier distribution (solid line). The contribution for = 9 is highlighted. (b) The sum of partial distributions from = 0 up to max = 11 (dot-dashed line), 20 (dashed line), 30 (short–long-dashed line) and 40 (solid line).
this behavior characterizes the near-barrier enhancement of the fusion cross section. As a complement of the excitation functions of Fig. 6, asymptotic outgoing currents are plotted in Fig. 7. Two main features can be pointed out, both occurring in the energy region where the inelastic scattering channel is open: (a) between Ecm = 59 and 63 MeV the total
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Fig. 6. Comparison between several relevant excitation functions.
Fig. 7. Asymptotic outgoing currents in the elastic (dot-dashed line) and inelastic (dashed line) in the coupled case. The total outgoing currents for this case (solid line) and the β=0 one (long-dashed line) are also shown.
outgoing current is hindered as a correlate of the near-barrier fusion enhancement; (b) in the plateau-like region of the excitation function, i.e. between Ecm = 63 and 67 MeV, that current is kept in a high level in comparison with that corresponding to the β = 0 case — a fact related with the perturbations in the excitation functions that induce the multiplicity of barriers.
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The behavior, as a function of the energy, of the cross sections and currents will be analyzed taking into account some relevant quantities: the conventional Coulomb barrier V con , the real modified barriers [Vnmod ], the source of particles in the inelastic channel Sin , the total fusion rate F R = F Rel + F Rin , and the incoming CI n and outgoing COn currents in the channels. Representative energy regions have been selected in order to perform the analysis. It must be taken into account that the wavefunctions, fusion rates, sources and currents in the figures correspond to the uncoupled equations as well as to the coupled ones (see Eqs. (1) and (7)). It is useful, however, to perform the present study from the point of view of the uncoupled equations governed by the modified potential barriers. At incident sub-Coulomb energies, the conventional Coulomb barrier V con , as well as the real elastic and inelastic modified barriers [Vnmod ] are higher than the pertinent channel energies, Eel or Ein . The pattern, here, is one of almost complete subbarrier reflection, the source and the inelastic outgoing currents are essentially null because wave functions are negligible in the region where the formfactors are important, the elastic outgoing current assumes a purely reflective character and the coupling effects are not significant. At Ecm = 61.7 MeV, barely below the Coulomb barrier energy (see Fig. 8a), the energy Eel is just in between the tops of V con and [Velmod ]; that is, from the point of view of the conventional Coulomb barrier, the problem would be a subbarrier one, but this is not the case if [Velmod ] is the potential used in the uncoupled equations because the channel energy exceeds its top. This fact explains why, in the coupled case, there is a shift, toward lower energies, of the critical point in which the first step of the excitation function takes place (see Fig. 6); in other words, it explains the enhancement of the fusion cross section at near-barrier energies. The relation between total incoming (negative) and outgoing (positive) currents and fusion rates are shown in Fig. 8b for the cases with (β = 0) and without (β = 0) deformation. There, it can be seen that the fusion rate increases while, consistently, the reflected current decreases when the inelastic channel is open. Remark that, in absence of internal sources, underbarrier currents are null (see long-dashed lines in the figure). The incoming and outgoing components of the currents in each channel when deformation is present, as well as the source in the inelastic channel are shown in Fig. 8c. It is illustrative to observe how the source injects in the inelastic channel, simultaneously, an incoming and an outgoing wave. The incoming one fades-out feeding the fusion channel, while the outgoing one contributes to the inelastic scattering. As will be pointed out later, the balance between this processes is decisive in order to explain the generation of a distribution of fusion barriers. In Fig. 9a the gaps between the corresponding channel energies and the tops of the elastic (dot-dashed line), and inelastic (dashed line) modified real barriers and the conventional Coulomb barrier (long-dashed line) are plotted. For the elastic channel, three energy regions may be discerned: well below the midpoint of the window where the inelastic scattering channel is open, the gap for β = 0 is greater than that for β = 0; the inverse situation takes place well above this midpoint; over the window a transition occurs. This behavior suggests that the real elastic modified barrier [Velmod ] might be responsible for
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Fig. 8. Radial distributions of different quantities for Ecm = 61.7 MeV. (a) The real modified barriers for the elastic [Velmod ] (dot-dashed line) and the inelastic [Vinmod ] (dashed line) channels. The corresponding channel energies are shown. The conventional Coulomb barrier V con is also plotted (long-dashed line). (b) The total incoming (negative) and outgoing (positive) currents in the coupled case, i.e. elastic plus inelastic contributions (solid lines), and the corresponding currents for the case with β = 0 (long-dashed lines). Fusion rates for the coupled case (dot-dot-dashed line) and the β = 0 case (short–long-dashed line) are also displayed. (c) The incoming (negative) and outgoing (positive) elastic (dot-dashed lines) and inelastic (dashed lines) currents. The source in the inelastic channel is also shown (short–long-dashed line).
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Fig. 9. (a) The energy gap between the corresponding channel energies and the tops of the elastic real modified barrier (dot-dashed line), the inelastic real modified barrier (dashed line), and the conventional Coulomb barrier (long-dashed line). (b) Comparison between the fusion excitation function given by including in the calculation the complete modified barrier (solid line) and that obtained by using only the real elastic modified barrier (short–long-dashed line).
the two characteristic influences of the nuclear structure in the fusion process: the nearbarrier enhancement and the barrier distribution. In relation with the former item, the fact that, in the pertinent energy region, the β = 0 gap is greater than the β = 0 one means a favored transmission through the barrier leading to the enhancement of the fusion cross section due to coupling effects. In relation with the second item, it may be argued that the transition toward a less important gap on the inelastic scattering window can explain the hindrance in the excitation function, as well as the associated inflection that produces the curvatures responsible for the splitting of peaks in the barrier distribution. The fusion excitation function predicted by the calculations when only the real modified barrier is present in the elastic channel — i.e. the unique source in the elastic channel is the incident plane wave — is shown in Fig. 9b (short–long-dashed line). Two conclusions can be inferred if the result of this calculation is compared with the excitation function σ f (solid line): the mere real modified barrier explains the near-barrier enhancement, but, the justification of the barrier distribution behavior is only partially achieved. The interplay between the source and the modified barrier in the inelastic channel must be analyzed in order to clarify why the plateau-like behavior in the fusion excitation function takes place. This interplay is illustrated in Fig. 10 where the position of the source in the inelastic channel (short–long-dashed lines) respect to the top of the real inelastic modified barrier (see vertical solid lines) is put in evidence. The outgoing currents in the channel (dashed lines) are also plotted. The progressive transit of the source from the outer side to the inner side of the barrier is correlated with the amount of outgoing current. Between Ecm = 63 and 69 MeV, as can be concluded if the inelastic gap is compared with
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Fig. 10. The interplay between the source distribution (short–long-dashed lines) and the position of the top of the real modified barrier in the inelastic channel (vertical solid lines) in the energy region where fusion barriers take place. The outgoing current in the inelastic channel is also shown (dashed lines).
the gap for the β = 0 case in Fig. 9a, the coupling induces a progressive increment of the reflective character of the inelastic modified barrier; consequently, the passage of current through the barrier, in both directions, is hindered. In the cases with Ecm = 63 and 65 MeV the barrier and the source positions permit a significant amount of inelastic scattering. In particular, as can be seen in Fig. 11, at Ecm = 65 MeV, the corresponding gaps are 0.31 MeV for the inelastic case and 2.87 MeV for the β = 0 one. This fact explains why the high transmission in the last of this situations implies a negligible reflection of the impinging wave in the entrance channel which contributes almost exclusively to fusion. Conversely, in the coupled situation, a significant outgoing current is generated in the excited channel by the portion of the source outside the reflective inelastic barrier. This fact hinders the fusion process and, correlatively, increases the scattering one (see Figs. 7, 10a and 10b). Fig. 11 gives, for Ecm = 65 MeV, the equivalent information given in Fig. 8 for Ecm = 61.7 MeV. This figure is illustrative in order to verify properties described above: the reflective character of the real inelastic modified barrier in comparison with the transparent character of the conventional Coulomb barrier in the elastic channel (see Fig. 11a); the difference between the amount of inelastic scattering
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Fig. 11. Same as Fig. 8 for Ecm = 65 MeV.
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Fig. 12. Same as Fig. 8 for Ecm = 78 MeV. Notice that in (b) currents and fusion rates for both, the coupled and the β = 0 cases, are almost coincident.
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— i.e. outgoing currents — in the cases with and without deformation; and its correlative difference in the fusion rates that reveals the fusion hindrance induced by coupling (see Fig. 11b). Another interesting feature can be observed in Fig. 11c, the formfactor in the periphery of the nucleus transforms almost completely the elastic incoming flux into inelastic incoming flux at around 9 fm. However, immediately, the source in the inelastic channel becomes a sink, producing a reflux into the elastic channel that, at this energy, constitutes the main contribution to fusion. This effect has a maximum around Ecm = 66–67 MeV (see Fig. 6). Figs. 10c and 10d indicate how when, at Ecm = 67 and 69 MeV, the source is placed in the inner radial region respect to the inelastic barrier, the scattering in the excited channel is negligible. At higher energies, the processes with and without deformation assume common behaviors as can be observed in Fig. 12, devoted to Ecm = 78 MeV. It is to be noticed that, for energies beyond Ecm = 67 MeV, progressively, the above mentioned sink in the inelastic source disappear; this means that the main contribution to fusion will be provided by the inelastic channel. The source that fed this channel is confined by the real inelastic modified barrier in the nuclear interior as is indicated in the figure.
4. Summary The main purpose of the present work is to explore an alternative point of view in the analysis of the underlying reaction mechanisms that determine the behavior of the fusion barrier distributions. In this way, a better comprehension of the two main manifestations of the interplay between these reaction mechanisms and structure in nuclear collisions, i.e. near-barrier enhancement of the fusion cross section and fusion barrier distributions, can be achieved. The formal decoupling of the wave equations and the correlative definition of energydependent modified potential barriers allows us to study the behavior of some relevant physical quantities, i.e. fusion rates, interchannel sources and incoming and outgoing currents, in the simpler context of uncoupled channels. The dependence of the cross sections as functions of the energy is analyzed considering the behavior of those physical quantities as functions of the radial coordinate. As an example of the application of these procedures the fusion barrier distribution in the 16 O + 144 Sm system is studied. The selection of this extensively addressed system, which presents a dominant 3− excitation, permits to restrict the problem to a two-channel one that shows well separated peaks in the barrier distribution. By fitting the experimental data a convenient set of parameters is achieved. The importance of the partial-wave developments is emphasized. In particular, it is pointed out how the buildup of the successive contributions, with different -values, to the barrier distribution leads to a peculiar interference process; while it removes the negative peaks of the partial contributions, it produces the final pattern which exhibits only two
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positive peaks. This analysis leads to the selection of a representative term between the -contributions to the barrier distribution, i.e. that for = 9, in order to perform the study. Along the partial excitation functions, several characteristic energy regions have been chosen to be analyzed under the light of the spatial behavior of the above mentioned physical quantities. Below the threshold of the fusion process, a dominant reflective picture inhibits coupling effects. Around the threshold, the difference between the conventional Coulomb barrier and the real part of the elastic modified barrier explains the near-barrier enhancement of the fusion cross section. The inflection in the excitation function that originates the splitting effect in the barrier distribution seems to be due to the following two facts: (a) the passage of the real modified barrier top in the elastic channel from below to above the conventional Coulomb barrier top, and (b) a local interplay between the source and the real inelastic modified barrier that enhances the inelastic scattering; both acting in detriment of fusion. At energies well above those for which the inelastic scattering is important, again, the coupling effects do not affect the fusion cross section and, consequently, there are not fusion barriers. The model predicts interesting transitions between channels that determines which channel feeds the fusion processes at different energies. The present approach provides an explanation of enhancement and multiplicity of barriers in the fusion process based in energy-dependent potential barriers modified by coupling effects acting in independent wave equations; it constitutes an alternative and complementary interpretation of that based in a decoupling which introduces mixed quantum states.
Acknowledgements One of the authors (M.R.S.) is a postdoctoral fellowship of the Consejo de Investigaciones Científicas y Técnicas (CONICET), Argentina, and J.E.T. is member of the Carrera del Investigador Científico, CONICET. H.D.M. acknowledge financial support from PEDECIBA and CSIC (UDELAR), Uruguay, and Agencia Nacional de Promoción Científica y Tecnológica, Argentina, Pict 03-00000-01357.
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