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Multistep processes in charge-exchange reactions
Nuclear Physics A 697 (2002) 171–182 www.elsevier.com/locate/npe
Multistep processes in charge-exchange reactions P. Demetriou a,∗ , A. Marcinkowski b , B. Maria´nski b a Institute of Nuclear Physics, NCSR “Demokritos”, GR-15310 Athens, Greece b The Andrzej Soltan Institute for Nuclear Studies, 00-681 Warsaw, Hoza 69, Poland
Received 29 June 2001; accepted 18 July 2001
Abstract Cross sections for the charge-exchange 65 Cu(p, n)65 Zn reaction at the incident energy of 27 MeV and the 100 Mo(p, n)100 Tc reaction at the incident energy of 26 MeV have been calculated using the multistep direct reaction theory of Feshbach, Kerman and Koonin. The theory was modified to include the non-DWBA matrix elements and the isovector collective vibrations according to the prescription of Marcinkowski and Maria´nski. The results show enhanced contributions from two-, three- and four-step direct reactions in agreement with experiment. 2002 Elsevier Science B.V. All rights reserved. PACS: 25.40.Kv; 24.60.Gv Keywords: N UCLEAR REACTIONS 65 Cu(p, xn), E = 27 MeV, 100 Mo(p, xn), E = 26 MeV; Multistep reaction cross sections
1. Introduction The quantum-mechanical theories developed to describe multistep processes distinguish between multistep compound (MSC) and multistep direct (MSD) reactions [1–3]. The MSD reactions became a subject of a controversy due to the biorthogonality of the distorted waves involved in the DWBA description of the subsequent reaction stages apart from the last stage that is a normal DWBA transition [4]. The biorthogonally conjugated wave functions result in non-DWBA matrix elements that can be expressed in terms of the normal ones including an inverse elastic S-matrix enhancing factor [5]. It has been argued that averaging over energy in the continuum removes these enhancing factors and hence permits the use of the DWBA matrix elements in practical calculations [6,7]. Thus most of previous analyses of inelastic scattering and charge-exchange reactions at low energies used normal DWBA matrix elements concluding that nucleon emission can be described * Corresponding author.
E-mail address: [email protected] (P. Demetriou). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 4 3 - X
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as one-step direct (1SD) reaction mainly. On the other hand, it was found that the limits imposed by the energy weighted sum rules (EWSR’s) on transitions of different angular momentum transfer reduce the 1SD cross sections considerably [8,9]. It was also shown that in neutron scattering this reduction can be well compensated by using the enhanced MSD cross sections obtained with the use of the non-DWBA matrix elements [10]. In this paper we calculate the MSD cross sections of the charge-exchange (p, n) reactions in the framework of the theory of Feshbach, Kerman and Koonin (FKK) [1]. The theory is modified to include the excitation of collective vibrations and the enhanced non-DWBA matrix elements according to the prescription of [8,11]. Given the fact that chargeexchange reactions involve only the weak isovector collective excitations it is expected that the multistep processes will be mainly due to incoherent excitation of the multiparticle–hole states. The calculations confirm this supposition and show the importance of different sequences of collisions of the leading continuum nucleon that contribute to the multistep reaction. When all the contributions are taken into account the measured cross sections and angular distributions for the 65 Cu(p, n)65 Zn reaction, at incident energy of 26.7 MeV [12], and for the 100 Mo(p, n)100 Tc reaction, at incident energy of 25.6 MeV [13], are well reproduced.
2. The model of the multistep direct reactions The cross section of the MSD reaction in the FKK theory is obtained by a convolution of 1SD cross sections [1]: 2 d σ dE dΩ MSD m1 E 1 mM−1 EM−1 m2 E 2 = dE1 dΩ1 dE2 dΩ2 . . . dEM−1 dΩM−1 2 2 2 2 (2π) h¯ (2π) h¯ (2π)2 h¯ 2 d2 σ d2 σ d2 σ × ... , (1) dEM dΩM 1SD dE2 dΩ2 1SD dE1 dΩ1 1SD where EM , ΩM and mM are the energy, solid angle and mass of the scattered nucleon after the Mth stage of the reaction. The matrix elements describing the successive 1SD transitions in Eq. (1) involve, except for the last one, the biorthogonally conjugated distorted waves. The corresponding non-DWBA matrix elements can be expressed in terms of the normal DWBA matrix elements including an inverse elastic scattering S-matrix term. Therefore the corresponding 1SD cross sections in (1) are given by the partial l-wave cross sections multiplied by modulus squared of the inverse elastic matrix elements Sl−2 . Eq. (1) has to be used repeatedly for all possible sequences of the M collisions of the leading continuum nucleon with a bound nucleon that contribute to the reaction considered. Bearing in mind the type of the continuum nucleon before and after the collision we consider the scattering (nn) or (pp) and the charge-exchange (pn) or (np) reaction stages. These stages form the following MSD sequences that contribute to the (p, n) reaction considered in this paper:
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1SD: 2SD: 3SD:
(pn), ppS −2 , pn , pnS −2 , nn , pnS −2 , nnS −2 , nn , ppS −2 , pnS −2 , nn , pnS −2 , npS −2 , pn ,
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ppS −2 , ppS −2 , pn ,
etc.
Here we have used the abbreviation (nnS −2 ) = l=λ σl (n, n)Sl−2 . Thus in order to obtain, e.g., a 3SD cross section the calculation of the r.h.s. of Eq. (1) has to be repeated four times. Even more striking is the fact that to obtain the MSD cross sections of the charge-exchange reaction, e.g., of the 65 Cu(p, n) reaction, one has to calculate in addition the 1SD cross sections in all the remaining reaction channels: 65 Cu(p, p ), 65 Zn(n, n ) and 65 Zn(n, p). This problem was first mentioned in Ref. [14] but until then, in calculations of (p, n) reactions it had been assumed that the charge-exchange stage of the reaction takes place at first, leading to the (pn, nn, nn, . . . ) sequence [15], or an unphysical sequence (pn, pn, pn, . . . ) had been employed. The elastic scattering matrix elements, |Sl |2 = 1 − Tl , are related to the partial wave transmission coefficients Tl [5] of the optical potentials for neutrons [16] and protons [17], respectively. The Tl ’s may approach the value 1 at the positions of the single-particle resonances [18] giving rise to divergences in the Sl−2 factors. These anomalies were removed by assuming an average behaviour of the corresponding Tl ’s. The 1SD cross sections entering Eq. (1) were derived by imposing the restrictions due to the EWSR’s on transitions of different multipolarity or angular momentum transfer. These restrictions split the 1SD cross sections into two terms: (i) the one due to the collective vibrations of multipolarity not exceeding λ = 4 and (ii) the other due to incoherent excitation of the ph-pairs of angular momentum transfer l > 4 [8]. From the cross section formula in [8] we obtain the 1SD cross sections specified for the charge-exchange (p, n) and (n, p) reactions: 2 1 dσ d σ 2 β = f [h¯ ωλ,1 , Γ ] dU dΩ 1SD 2 λ,1 dΩ λ λ4 dσ 1pπ 1hν 2 + (2 + 1)P × gπ gν U × R1,1 () × Vπν , (2) dΩ >4
and for nucleon scattering: 2 d σ dU dΩ 1SD 1 dσ 2 (1 + 3δ0,τ )βλ,τ = f [h¯ ωλ,τ , Γ ] 4 dΩ λ λ4,τ =0,1
+
>4
dσ 1pν 1hν 2 2 (2 + 1)P × R1,1 () × gν 2 δν,x U Vνν + gν 2 δπ,x U Vπν dΩ
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dσ 1pπ 1hπ 2 2 2 2 , + (2 + 1)P × R1,1 () × gπ δπ,x U Vππ + gπ δν,x U Vπν dΩ >4
(3) where x = π for proton scattering and x = ν for neutron scattering. The second and third terms in (3) describe the incoherent excitation of the neutron and proton particle–hole pairs separately. The macroscopic DWBA cross section (dσ/dΩ)λ of the collective model has to be calculated for the reaction of interest: (p, n) or (n, p) in (2), and (p, p ) or (n, n ) in (3). The macroscopic form factors are obtained by deforming the phenomenological complex optical potential. The sum over λ involves summation over known low-energy one-phonon vibrations of multipolarity λ, deformation parameters βλ,τ and energy hω ¯ λ,τ as well as over the giant resonances (GR) in the continuum. The sum over τ involves the isovector τ = 1 vibrations in (2), and both isoscalar τ = 0 and isovector τ = 1 in (3). In the latter case the summation is limited practically to the giant dipole resonance. Only the stronger electric s = 0 excitations are included in (2) and (3). f is the energy distribution function assumed to be gaussian for the low-energy collective levels and lorentzian for the GR’s. The microscopic DWBA cross sections (dσ/dΩ)l are calculated again for the reaction of interest using either (2) or (3), appropriately. They include the two-particle form factor with real effective interaction of Yukawa form of 1 fm range. The effective nucleon–nucleon interaction strength is assumed to be Vπ,π = Vν,ν = 12.7 MeV and Vπ,ν = Vν,π = 43.1 MeV [19]. The term P = 1/2 is the parity distribution and R1,1 (l) p+h p+h is the spin distribution of the density, ω1,1 (U ) = gπ gν U [20], of the 1p1h states. Here gπ = Z/13 and gν = N/13 are the single-particle state densities for protons and neutrons, respectively. The spin cutoff parameter in R1,1 (l) is given by the relation σ 2 = 0.56 A2/3 [21].
3. Results and discussion 3.1. The inelastic scattering reaction stages The macroscopic cross sections of the collective model were calculated according to the − first r.h.s. term of Eq. (3) for the isoscalar 2+ 1 and 31 one-phonon states in the even–even 64 100 Mo. The weak coupling model was used for the product nuclei core-nuclei Zn and 65 Cu and 100 Tc of the reactions studied. In addition the giant dipole, quadrupole and the low-energy octupole (LEOR) resonances were taken into account. The excitation energies of the GR’s were assumed to be 77 A−1/3 , 62 A−1/3 and 38 A−1/3 [22], respectively. The effective deformation parameters for the GR in the continuum were obtained by depleting the EWSR’s of respective multipolarity by the strengths of the low-energy one-phonon states just mentioned. Only 30% depletion of the total octupole strength was assumed [22]. The incoherent ph cross sections in Eq. (3) are summed over the transferred orbital angular momenta 4 < l < 12. Each partial cross section observes the limit of the EWSR. The microscopic DWBA cross sections (dσ/dΩ)l in (3) were calculated and averaged
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over the final shell–model neutron and proton particle–hole pairs separately (the second and third term in (3)). The distorted waves for neutrons and protons were calculated with the optical potentials of [16,17]. Both the macroscopic collective cross sections and the incoherent ph ones were calculated with the DWUCK-4 code [23]. The spectroscopic amplitude (2jh + 1)1/2 was taken into account in the microscopic option of DWUCK-4. 3.2. The charge-exchange reaction stages For the charge-exchange reaction stages (pn) and (np) the collective cross sections could not be calculated exactly according to the first term in (2). This is because the strength parameters βλ,1 for the weak isovector transitions to low-energy excited states have never been determined experimentally. Therefore, bearing in mind that the isovector GR’s, apart from the giant dipole one, have large widths (of 10 MeV and more [22]) and that the low energy levels, except for the isobaric analog state, do not concentrate considerable collective strength, we have approximated the collective cross section term by the ph cross sections of the second term in (2) reduced according to the limits imposed by the EWSR’s [8] on transitions with orbital angular momentum transfer l 4. A comparison of the relative contributions of the various sequences of the leading continuum nucleons that add to the 2SD, 3SD and 4SD cross sections for the 65 Cu(p, n)65 Zn reaction in Figs. 1–3 shows that the largest contributions are due to the occurrence of neutron inelastic scattering in the sequence of collisions. This is not surprising given that neutron scattering not only involves a significant number of incoherent ph excitations but also the strong isoscalar collective excitations. The presence of proton inelastic scattering also
Fig. 1. The calculated 2SD cross section for the 65 Cu(p, n)65 Zn reaction at incident energy of 27 MeV (thick solid line). The contributions of the two sequences of reaction stages are shown as thin lines.
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Fig. 2. The same as in Fig. 1 but for the 3SD reaction (thick solid line). The contributions of the four sequences of the reaction stages are shown as thin lines.
Fig. 3. The same as in Fig. 1 but for the 4SD reaction (thick solid line). The contributions of the eight sequences of reaction stages are shown as thin lines.
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gives rise to larger cross sections compared to the charge-exchange reactions, although the Coulomb barrier leads to cross sections smaller than those of neutron inelastic scattering. The cross sections for all sequences of the 65 Cu(p, n)65 Zn reaction stages integrated over energy and angle are included in Table 1. The same but for the 100 Mo(p, n)100 Tc reaction are given in Table 2. From Tables 1 and 2 it is evident that none of the single sequences of reaction stages, not even the (pn, nn, . . . ) one, is able to describe the corresponding MSD contribution to the (p, n) reaction satisfactorily. However, there are always two sequences, namely the (pn, nn) and (pp, pn) for the 2SD, (pn, nn, nn) and (pp, pn, nn) for the 3SD, (pn, nn, nn, nn) and (pp, pn, nn, nn) for the 4SD reaction, which provide more than 85% of the corresponding MSD cross section. The 1SD cross section together with the summed 2SD, 3SD and 4SD cross sections, are compared with the inclusive neutron emission spectrum of the 65 Cu(p, xn)65Zn reaction measured at an incident energy of 26.7 MeV, in Fig. 4. The same but for Table 1 The sequences of the leading particle reaction stages contributing to the 65 Cu(p, n)65 Zn reaction at 27 MeV σ (1SD)
σ (2SD)
σ (3SD)
σ (4SD)
(pn) 74 mb (pnS −2 , nn) 28 mb (pnS −2 , nnS −2 , nn) 9.8 mb (ppS −2 , pn) 10 mb (ppS −2 , pnS −2 , nn) 3.0 mb (ppS −2 , ppS −2 , pn) 0.6 mb (pnS −2 , npS −2 , pn) 0.2 mb
Total 74 mb
38 mb
(pnS −2 , nnS −2 , nnS −2 , nn) 3.15 mb (ppS −2 , pnS −2 , nnS −2 , nn) 0.9 mb (ppS −2 , ppS −2 , pnS −2 , nn) 0.16 mb (pnS −2 , npS −2 , pnS −2 , nn) 0.05 mb (pnS −2 , nnS −2 , npS −2 , pn) 0.04 mb (ppS −2 , ppS −2 , ppS −2 , pn) 0.02 mb (ppS −2 , pnS −2 , npS −2 , pn) 0.007 mb (pnS −2 , npS −2 , ppS −2 , pn) 0.004 mb 13.6 mb 4.3 mb
Table 2 The sequences of the leading particle reaction stages contributing to the 100 Mo(p, n)100 Tc reaction at 26 MeV σ (1SD)
σ (2SD)
σ (3SD)
σ (4SD)
(pn) 74 mb (pnS −2 , nn) 46 mb (pnS −2 , nnS −2 , nn) 26.5 mb (pnS −2 , nnS −2 , nnS −2 , nn) 14.5 mb (ppS −2 , pn) 9 mb (ppS −2 , pnS −2 , nn) 4.9 mb (ppS −2 , pnS −2 , nnS −2 , nn) 2.5 mb (ppS −2 , ppS −2 , pn) 3.3 mb (ppS −2 , ppS −2 , pnS −2 , nn) 1.8 mb (pn, S −2 , npS −2 , pn) 0.07 mb (ppS −2 , ppS −2 , ppS −2 , pn) 0.75 mb (pnS −2 , nnS −2 , npS −2 , pn) 0.03 mb (pnS −2 , npS −2 , pnS −2 , nn) 0.03 mb (pnS −2 , npS −2 , ppS −2 , pn) 0.01 mb (ppS −2 , pnS −2 , npS −2 , pn) 0.004 mb Total 74 mb 55 mb 34.8mb 20.0 mb
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Fig. 4. Comparison of the calculated cross sections with the spectrum of neutrons from the 65 Cu(p, n)65 Zn reaction measured at incident energy of 26.7 MeV [12]. The thick line is the sum of all contributions shown. The labels CN1, CN2, CN3 denote the primary, secondary and tertiary neutrons evaporated from the compound nucleus, respectively. CPN1 and CPN2 denote secondary and tertiary neutrons preceded by a proton and MSC labels the sum of contributions from the three steps of preequilibrium compound reaction. The 1SD, 2SD, 3SD and 4SD cross sections are described in the text. The peak at the outgoing energy of about 17 MeV in the experimental data corresponds to the excitation of the isobaric analog state.
Fig. 5. The same as in Fig. 4 but for the 100 Mo(p, n)100 Tc reaction measured at incident energy of 25.6 MeV [13]. The peak at the outgoing energy of about 13 MeV in the experimental data corresponds to the excitation of the isobaric analog state.
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the 100 Mo(p, xn)100Tc reaction measured at 25.6 MeV is shown in Fig. 5. Overall, a satisfactory agreement between theory and experiment can be found in both cases. The cross sections for multiple emission of low-energy neutrons from the compound nucleus was calculated according to the theory of Hauser–Feshbach and the preequilibrium compound contribution allowing for gradual absorption of the incident flux [24] was calculated according to the multistep compound reaction theory of FKK [25,26]. The radial overlap integral of the single-particle wave functions in the MSC cross section was calculated with constant wave functions within the nucleus. The unbound wave function was modified in order to approximate the result of the microscopic calculations [27]. Thus, the cross sections with the microscopic wave function are 1/4 of those with constant unbound wave function. In Figs. 6 and 7 the calculated angular distributions are compared with the ones measured for the two studied reactions. The theory once again is able to reproduce the experimental angular distributions satisfactorily, apart from the forward-angles where a systematic
Fig. 6. Comparison of the calculated double-differential cross sections with the angular distributions of neutrons from the 65 Cu(p, n)65 Zn reaction measured at incident energy of 26.7 MeV [12]. The outgoing neutron energies are indicated in the plots. The thick lines are sums of all the contributions shown. The labels CN1, CN2, CPN1, and MSC denote the same as in Fig. 4.
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Fig. 7. The same as in Fig. 6 but for the 100 Mo(p, n)100 Tc reaction measured at incident energy 25.6 MeV [13].
underprediction is observed at all outgoing energies. Up to three reaction steps contribute to the 65 Cu(p, n)65 Zn reaction at all outgoing energies, apart from the highest ones. The 4SD cross section contributes to about 3% of the total cross section and therefore can be neglected. For the 100 Mo(p, n)100 Tc reaction the four-step reactions become important and the large MSD contributions are mainly due to the large cross section for inelastic neutron scattering on 100 Mo.
4. Conclusions The contribution of multistep processes in the charge-exchange reactions 65 Cu(p, n)65 Zn and 100 Mo(p, n)100 Tc at about 26 MeV were investigated. Enhanced MSD reaction cross sections were obtained with the use of non-DWBA matrix elements. All the sequences of collisions of the continuum particle were taken into account in Eq. (1). The results show that up to three-step processes contribute mainly to the 65 Cu(p, n)65 Zn reaction while four-step processes are also important for the 100 Mo(p, n)100 Tc reaction. It was found that
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the largest contributions are due to the occurrence of neutron scattering in the sequence of collisions. Thus, at each multistep reaction considered the following two sequences: (pn, nn) and (pp, pn), (pn, nn, nn) and (pp, pn, nn), (pn, nn, nn, nn) and (pp, pn, nn, nn), etc., contribute to more than 85% of the corresponding MSD cross sections and are therefore, together with the 1SD cross section, sufficient to provide a good description of the (p, n) total cross section.
Acknowledgements This work was supported by the Greek–Polish bilateral agreement on scientific collaboration No-028/2001-2002, 3326/R01/R02. P.D. wishes to thank the Andrzej Soltan Institute for Nuclear Studies for the hospitality during her stay.
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[25] A. Marcinkowski, J. Rapaport, R.W. Finlay, C.E. Brient, M. Herman, M.B. Chadwick, Nucl. Phys. A 561 (1993) 387; A. Marcinkowski, D. Kielan, Nucl. Phys. A 578 (1994) 168. [26] G. Arbanas, M.B. Chadwick, F.S. Dietrich, A. Kerman, Phys. Rev. C 51 (1995) R1078. [27] T. Kawano, Phys. Rev. C 59 (1999) 865.
Multistep processes in charge-exchange reactions P. Demetriou a,∗ , A. Marcinkowski b , B. Maria´nski b a Institute of Nuclear Physics, NCSR “Demokritos”, GR-15310 Athens, Greece b The Andrzej Soltan Institute for Nuclear Studies, 00-681 Warsaw, Hoza 69, Poland
Received 29 June 2001; accepted 18 July 2001
Abstract Cross sections for the charge-exchange 65 Cu(p, n)65 Zn reaction at the incident energy of 27 MeV and the 100 Mo(p, n)100 Tc reaction at the incident energy of 26 MeV have been calculated using the multistep direct reaction theory of Feshbach, Kerman and Koonin. The theory was modified to include the non-DWBA matrix elements and the isovector collective vibrations according to the prescription of Marcinkowski and Maria´nski. The results show enhanced contributions from two-, three- and four-step direct reactions in agreement with experiment. 2002 Elsevier Science B.V. All rights reserved. PACS: 25.40.Kv; 24.60.Gv Keywords: N UCLEAR REACTIONS 65 Cu(p, xn), E = 27 MeV, 100 Mo(p, xn), E = 26 MeV; Multistep reaction cross sections
1. Introduction The quantum-mechanical theories developed to describe multistep processes distinguish between multistep compound (MSC) and multistep direct (MSD) reactions [1–3]. The MSD reactions became a subject of a controversy due to the biorthogonality of the distorted waves involved in the DWBA description of the subsequent reaction stages apart from the last stage that is a normal DWBA transition [4]. The biorthogonally conjugated wave functions result in non-DWBA matrix elements that can be expressed in terms of the normal ones including an inverse elastic S-matrix enhancing factor [5]. It has been argued that averaging over energy in the continuum removes these enhancing factors and hence permits the use of the DWBA matrix elements in practical calculations [6,7]. Thus most of previous analyses of inelastic scattering and charge-exchange reactions at low energies used normal DWBA matrix elements concluding that nucleon emission can be described * Corresponding author.
E-mail address: [email protected] (P. Demetriou). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 4 3 - X
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P. Demetriou et al. / Nuclear Physics A 697 (2002) 171–182
as one-step direct (1SD) reaction mainly. On the other hand, it was found that the limits imposed by the energy weighted sum rules (EWSR’s) on transitions of different angular momentum transfer reduce the 1SD cross sections considerably [8,9]. It was also shown that in neutron scattering this reduction can be well compensated by using the enhanced MSD cross sections obtained with the use of the non-DWBA matrix elements [10]. In this paper we calculate the MSD cross sections of the charge-exchange (p, n) reactions in the framework of the theory of Feshbach, Kerman and Koonin (FKK) [1]. The theory is modified to include the excitation of collective vibrations and the enhanced non-DWBA matrix elements according to the prescription of [8,11]. Given the fact that chargeexchange reactions involve only the weak isovector collective excitations it is expected that the multistep processes will be mainly due to incoherent excitation of the multiparticle–hole states. The calculations confirm this supposition and show the importance of different sequences of collisions of the leading continuum nucleon that contribute to the multistep reaction. When all the contributions are taken into account the measured cross sections and angular distributions for the 65 Cu(p, n)65 Zn reaction, at incident energy of 26.7 MeV [12], and for the 100 Mo(p, n)100 Tc reaction, at incident energy of 25.6 MeV [13], are well reproduced.
2. The model of the multistep direct reactions The cross section of the MSD reaction in the FKK theory is obtained by a convolution of 1SD cross sections [1]: 2 d σ dE dΩ MSD m1 E 1 mM−1 EM−1 m2 E 2 = dE1 dΩ1 dE2 dΩ2 . . . dEM−1 dΩM−1 2 2 2 2 (2π) h¯ (2π) h¯ (2π)2 h¯ 2 d2 σ d2 σ d2 σ × ... , (1) dEM dΩM 1SD dE2 dΩ2 1SD dE1 dΩ1 1SD where EM , ΩM and mM are the energy, solid angle and mass of the scattered nucleon after the Mth stage of the reaction. The matrix elements describing the successive 1SD transitions in Eq. (1) involve, except for the last one, the biorthogonally conjugated distorted waves. The corresponding non-DWBA matrix elements can be expressed in terms of the normal DWBA matrix elements including an inverse elastic scattering S-matrix term. Therefore the corresponding 1SD cross sections in (1) are given by the partial l-wave cross sections multiplied by modulus squared of the inverse elastic matrix elements Sl−2 . Eq. (1) has to be used repeatedly for all possible sequences of the M collisions of the leading continuum nucleon with a bound nucleon that contribute to the reaction considered. Bearing in mind the type of the continuum nucleon before and after the collision we consider the scattering (nn) or (pp) and the charge-exchange (pn) or (np) reaction stages. These stages form the following MSD sequences that contribute to the (p, n) reaction considered in this paper:
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1SD: 2SD: 3SD:
(pn), ppS −2 , pn , pnS −2 , nn , pnS −2 , nnS −2 , nn , ppS −2 , pnS −2 , nn , pnS −2 , npS −2 , pn ,
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ppS −2 , ppS −2 , pn ,
etc.
Here we have used the abbreviation (nnS −2 ) = l=λ σl (n, n)Sl−2 . Thus in order to obtain, e.g., a 3SD cross section the calculation of the r.h.s. of Eq. (1) has to be repeated four times. Even more striking is the fact that to obtain the MSD cross sections of the charge-exchange reaction, e.g., of the 65 Cu(p, n) reaction, one has to calculate in addition the 1SD cross sections in all the remaining reaction channels: 65 Cu(p, p ), 65 Zn(n, n ) and 65 Zn(n, p). This problem was first mentioned in Ref. [14] but until then, in calculations of (p, n) reactions it had been assumed that the charge-exchange stage of the reaction takes place at first, leading to the (pn, nn, nn, . . . ) sequence [15], or an unphysical sequence (pn, pn, pn, . . . ) had been employed. The elastic scattering matrix elements, |Sl |2 = 1 − Tl , are related to the partial wave transmission coefficients Tl [5] of the optical potentials for neutrons [16] and protons [17], respectively. The Tl ’s may approach the value 1 at the positions of the single-particle resonances [18] giving rise to divergences in the Sl−2 factors. These anomalies were removed by assuming an average behaviour of the corresponding Tl ’s. The 1SD cross sections entering Eq. (1) were derived by imposing the restrictions due to the EWSR’s on transitions of different multipolarity or angular momentum transfer. These restrictions split the 1SD cross sections into two terms: (i) the one due to the collective vibrations of multipolarity not exceeding λ = 4 and (ii) the other due to incoherent excitation of the ph-pairs of angular momentum transfer l > 4 [8]. From the cross section formula in [8] we obtain the 1SD cross sections specified for the charge-exchange (p, n) and (n, p) reactions: 2 1 dσ d σ 2 β = f [h¯ ωλ,1 , Γ ] dU dΩ 1SD 2 λ,1 dΩ λ λ4 dσ 1pπ 1hν 2 + (2 + 1)P × gπ gν U × R1,1 () × Vπν , (2) dΩ >4
and for nucleon scattering: 2 d σ dU dΩ 1SD 1 dσ 2 (1 + 3δ0,τ )βλ,τ = f [h¯ ωλ,τ , Γ ] 4 dΩ λ λ4,τ =0,1
+
>4
dσ 1pν 1hν 2 2 (2 + 1)P × R1,1 () × gν 2 δν,x U Vνν + gν 2 δπ,x U Vπν dΩ
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dσ 1pπ 1hπ 2 2 2 2 , + (2 + 1)P × R1,1 () × gπ δπ,x U Vππ + gπ δν,x U Vπν dΩ >4
(3) where x = π for proton scattering and x = ν for neutron scattering. The second and third terms in (3) describe the incoherent excitation of the neutron and proton particle–hole pairs separately. The macroscopic DWBA cross section (dσ/dΩ)λ of the collective model has to be calculated for the reaction of interest: (p, n) or (n, p) in (2), and (p, p ) or (n, n ) in (3). The macroscopic form factors are obtained by deforming the phenomenological complex optical potential. The sum over λ involves summation over known low-energy one-phonon vibrations of multipolarity λ, deformation parameters βλ,τ and energy hω ¯ λ,τ as well as over the giant resonances (GR) in the continuum. The sum over τ involves the isovector τ = 1 vibrations in (2), and both isoscalar τ = 0 and isovector τ = 1 in (3). In the latter case the summation is limited practically to the giant dipole resonance. Only the stronger electric s = 0 excitations are included in (2) and (3). f is the energy distribution function assumed to be gaussian for the low-energy collective levels and lorentzian for the GR’s. The microscopic DWBA cross sections (dσ/dΩ)l are calculated again for the reaction of interest using either (2) or (3), appropriately. They include the two-particle form factor with real effective interaction of Yukawa form of 1 fm range. The effective nucleon–nucleon interaction strength is assumed to be Vπ,π = Vν,ν = 12.7 MeV and Vπ,ν = Vν,π = 43.1 MeV [19]. The term P = 1/2 is the parity distribution and R1,1 (l) p+h p+h is the spin distribution of the density, ω1,1 (U ) = gπ gν U [20], of the 1p1h states. Here gπ = Z/13 and gν = N/13 are the single-particle state densities for protons and neutrons, respectively. The spin cutoff parameter in R1,1 (l) is given by the relation σ 2 = 0.56 A2/3 [21].
3. Results and discussion 3.1. The inelastic scattering reaction stages The macroscopic cross sections of the collective model were calculated according to the − first r.h.s. term of Eq. (3) for the isoscalar 2+ 1 and 31 one-phonon states in the even–even 64 100 Mo. The weak coupling model was used for the product nuclei core-nuclei Zn and 65 Cu and 100 Tc of the reactions studied. In addition the giant dipole, quadrupole and the low-energy octupole (LEOR) resonances were taken into account. The excitation energies of the GR’s were assumed to be 77 A−1/3 , 62 A−1/3 and 38 A−1/3 [22], respectively. The effective deformation parameters for the GR in the continuum were obtained by depleting the EWSR’s of respective multipolarity by the strengths of the low-energy one-phonon states just mentioned. Only 30% depletion of the total octupole strength was assumed [22]. The incoherent ph cross sections in Eq. (3) are summed over the transferred orbital angular momenta 4 < l < 12. Each partial cross section observes the limit of the EWSR. The microscopic DWBA cross sections (dσ/dΩ)l in (3) were calculated and averaged
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over the final shell–model neutron and proton particle–hole pairs separately (the second and third term in (3)). The distorted waves for neutrons and protons were calculated with the optical potentials of [16,17]. Both the macroscopic collective cross sections and the incoherent ph ones were calculated with the DWUCK-4 code [23]. The spectroscopic amplitude (2jh + 1)1/2 was taken into account in the microscopic option of DWUCK-4. 3.2. The charge-exchange reaction stages For the charge-exchange reaction stages (pn) and (np) the collective cross sections could not be calculated exactly according to the first term in (2). This is because the strength parameters βλ,1 for the weak isovector transitions to low-energy excited states have never been determined experimentally. Therefore, bearing in mind that the isovector GR’s, apart from the giant dipole one, have large widths (of 10 MeV and more [22]) and that the low energy levels, except for the isobaric analog state, do not concentrate considerable collective strength, we have approximated the collective cross section term by the ph cross sections of the second term in (2) reduced according to the limits imposed by the EWSR’s [8] on transitions with orbital angular momentum transfer l 4. A comparison of the relative contributions of the various sequences of the leading continuum nucleons that add to the 2SD, 3SD and 4SD cross sections for the 65 Cu(p, n)65 Zn reaction in Figs. 1–3 shows that the largest contributions are due to the occurrence of neutron inelastic scattering in the sequence of collisions. This is not surprising given that neutron scattering not only involves a significant number of incoherent ph excitations but also the strong isoscalar collective excitations. The presence of proton inelastic scattering also
Fig. 1. The calculated 2SD cross section for the 65 Cu(p, n)65 Zn reaction at incident energy of 27 MeV (thick solid line). The contributions of the two sequences of reaction stages are shown as thin lines.
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Fig. 2. The same as in Fig. 1 but for the 3SD reaction (thick solid line). The contributions of the four sequences of the reaction stages are shown as thin lines.
Fig. 3. The same as in Fig. 1 but for the 4SD reaction (thick solid line). The contributions of the eight sequences of reaction stages are shown as thin lines.
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gives rise to larger cross sections compared to the charge-exchange reactions, although the Coulomb barrier leads to cross sections smaller than those of neutron inelastic scattering. The cross sections for all sequences of the 65 Cu(p, n)65 Zn reaction stages integrated over energy and angle are included in Table 1. The same but for the 100 Mo(p, n)100 Tc reaction are given in Table 2. From Tables 1 and 2 it is evident that none of the single sequences of reaction stages, not even the (pn, nn, . . . ) one, is able to describe the corresponding MSD contribution to the (p, n) reaction satisfactorily. However, there are always two sequences, namely the (pn, nn) and (pp, pn) for the 2SD, (pn, nn, nn) and (pp, pn, nn) for the 3SD, (pn, nn, nn, nn) and (pp, pn, nn, nn) for the 4SD reaction, which provide more than 85% of the corresponding MSD cross section. The 1SD cross section together with the summed 2SD, 3SD and 4SD cross sections, are compared with the inclusive neutron emission spectrum of the 65 Cu(p, xn)65Zn reaction measured at an incident energy of 26.7 MeV, in Fig. 4. The same but for Table 1 The sequences of the leading particle reaction stages contributing to the 65 Cu(p, n)65 Zn reaction at 27 MeV σ (1SD)
σ (2SD)
σ (3SD)
σ (4SD)
(pn) 74 mb (pnS −2 , nn) 28 mb (pnS −2 , nnS −2 , nn) 9.8 mb (ppS −2 , pn) 10 mb (ppS −2 , pnS −2 , nn) 3.0 mb (ppS −2 , ppS −2 , pn) 0.6 mb (pnS −2 , npS −2 , pn) 0.2 mb
Total 74 mb
38 mb
(pnS −2 , nnS −2 , nnS −2 , nn) 3.15 mb (ppS −2 , pnS −2 , nnS −2 , nn) 0.9 mb (ppS −2 , ppS −2 , pnS −2 , nn) 0.16 mb (pnS −2 , npS −2 , pnS −2 , nn) 0.05 mb (pnS −2 , nnS −2 , npS −2 , pn) 0.04 mb (ppS −2 , ppS −2 , ppS −2 , pn) 0.02 mb (ppS −2 , pnS −2 , npS −2 , pn) 0.007 mb (pnS −2 , npS −2 , ppS −2 , pn) 0.004 mb 13.6 mb 4.3 mb
Table 2 The sequences of the leading particle reaction stages contributing to the 100 Mo(p, n)100 Tc reaction at 26 MeV σ (1SD)
σ (2SD)
σ (3SD)
σ (4SD)
(pn) 74 mb (pnS −2 , nn) 46 mb (pnS −2 , nnS −2 , nn) 26.5 mb (pnS −2 , nnS −2 , nnS −2 , nn) 14.5 mb (ppS −2 , pn) 9 mb (ppS −2 , pnS −2 , nn) 4.9 mb (ppS −2 , pnS −2 , nnS −2 , nn) 2.5 mb (ppS −2 , ppS −2 , pn) 3.3 mb (ppS −2 , ppS −2 , pnS −2 , nn) 1.8 mb (pn, S −2 , npS −2 , pn) 0.07 mb (ppS −2 , ppS −2 , ppS −2 , pn) 0.75 mb (pnS −2 , nnS −2 , npS −2 , pn) 0.03 mb (pnS −2 , npS −2 , pnS −2 , nn) 0.03 mb (pnS −2 , npS −2 , ppS −2 , pn) 0.01 mb (ppS −2 , pnS −2 , npS −2 , pn) 0.004 mb Total 74 mb 55 mb 34.8mb 20.0 mb
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Fig. 4. Comparison of the calculated cross sections with the spectrum of neutrons from the 65 Cu(p, n)65 Zn reaction measured at incident energy of 26.7 MeV [12]. The thick line is the sum of all contributions shown. The labels CN1, CN2, CN3 denote the primary, secondary and tertiary neutrons evaporated from the compound nucleus, respectively. CPN1 and CPN2 denote secondary and tertiary neutrons preceded by a proton and MSC labels the sum of contributions from the three steps of preequilibrium compound reaction. The 1SD, 2SD, 3SD and 4SD cross sections are described in the text. The peak at the outgoing energy of about 17 MeV in the experimental data corresponds to the excitation of the isobaric analog state.
Fig. 5. The same as in Fig. 4 but for the 100 Mo(p, n)100 Tc reaction measured at incident energy of 25.6 MeV [13]. The peak at the outgoing energy of about 13 MeV in the experimental data corresponds to the excitation of the isobaric analog state.
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the 100 Mo(p, xn)100Tc reaction measured at 25.6 MeV is shown in Fig. 5. Overall, a satisfactory agreement between theory and experiment can be found in both cases. The cross sections for multiple emission of low-energy neutrons from the compound nucleus was calculated according to the theory of Hauser–Feshbach and the preequilibrium compound contribution allowing for gradual absorption of the incident flux [24] was calculated according to the multistep compound reaction theory of FKK [25,26]. The radial overlap integral of the single-particle wave functions in the MSC cross section was calculated with constant wave functions within the nucleus. The unbound wave function was modified in order to approximate the result of the microscopic calculations [27]. Thus, the cross sections with the microscopic wave function are 1/4 of those with constant unbound wave function. In Figs. 6 and 7 the calculated angular distributions are compared with the ones measured for the two studied reactions. The theory once again is able to reproduce the experimental angular distributions satisfactorily, apart from the forward-angles where a systematic
Fig. 6. Comparison of the calculated double-differential cross sections with the angular distributions of neutrons from the 65 Cu(p, n)65 Zn reaction measured at incident energy of 26.7 MeV [12]. The outgoing neutron energies are indicated in the plots. The thick lines are sums of all the contributions shown. The labels CN1, CN2, CPN1, and MSC denote the same as in Fig. 4.
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Fig. 7. The same as in Fig. 6 but for the 100 Mo(p, n)100 Tc reaction measured at incident energy 25.6 MeV [13].
underprediction is observed at all outgoing energies. Up to three reaction steps contribute to the 65 Cu(p, n)65 Zn reaction at all outgoing energies, apart from the highest ones. The 4SD cross section contributes to about 3% of the total cross section and therefore can be neglected. For the 100 Mo(p, n)100 Tc reaction the four-step reactions become important and the large MSD contributions are mainly due to the large cross section for inelastic neutron scattering on 100 Mo.
4. Conclusions The contribution of multistep processes in the charge-exchange reactions 65 Cu(p, n)65 Zn and 100 Mo(p, n)100 Tc at about 26 MeV were investigated. Enhanced MSD reaction cross sections were obtained with the use of non-DWBA matrix elements. All the sequences of collisions of the continuum particle were taken into account in Eq. (1). The results show that up to three-step processes contribute mainly to the 65 Cu(p, n)65 Zn reaction while four-step processes are also important for the 100 Mo(p, n)100 Tc reaction. It was found that
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the largest contributions are due to the occurrence of neutron scattering in the sequence of collisions. Thus, at each multistep reaction considered the following two sequences: (pn, nn) and (pp, pn), (pn, nn, nn) and (pp, pn, nn), (pn, nn, nn, nn) and (pp, pn, nn, nn), etc., contribute to more than 85% of the corresponding MSD cross sections and are therefore, together with the 1SD cross section, sufficient to provide a good description of the (p, n) total cross section.
Acknowledgements This work was supported by the Greek–Polish bilateral agreement on scientific collaboration No-028/2001-2002, 3326/R01/R02. P.D. wishes to thank the Andrzej Soltan Institute for Nuclear Studies for the hospitality during her stay.
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