Inclusive breakup reaction of a two-cluster projectile on a two-fragment target: A genuine four-body problem
aa r X i v : . [ nu c l - t h ] N ov Inclusive breakup reaction of a two-cluster projectile on a two-fragmenttarget: A genuine four-body problem
M. S. Hussein , , , C. A. Bertulani , B. V. Carlson , and T. Frederico Instituto de Estudos Avanc¸ados, Universidade de S˜ao Paulo, Caixa Postal 72012, 05508-970 S˜ao Paulo, SP, Brazil Instituto de F´ısica, Universidade de S˜ao Paulo, Caixa Postal 66318, 05314-970 S˜ao Paulo, SP, Brazil Departamento de F´ısica, Instituto Tecnol´ogico de Aerona´utica, DCTA, 12.228-900 S˜ao Jos´e dos Campos, SP, Brazil Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429-3011, USA
Abstract.
We develop a four-body model for the inclusive breakup of two-fragment halo projectiles collidingwith two-fragment targets. In the case of a short lived projectiles, such as halo nuclei, on a deuteron target, themodel allows the extraction of the neutron capture cross section of such projectiles. We supply examples.
The ongoing research on the reaction of radioactive nuclei has supplied us with invaluable information about thestructure of nuclei near the drip line. Further, they produced important information on capture reactions and otherdirect reactions needed to fill the gaps in the chain of reactions in the r and s processes in astrophysics. The neutroncapture reactions referred to above involve capture by stable nuclei. Neutron capture reactions on radioactivenuclei, especially near the drip nuclei are not available. A possible way to obtain these cross sections is throughindirect hybrid reactions. One such method is the Surrogate Method [1]. So far this method was mostly used toobtain neutron capture cross section of fast neutrons by actinide nuclei for use in research in fast breeder reactors.Recently, the Surrogate Method was proposed to obtain the neutron capture cross section of radioactive nuclei[2]. A recent review gives an account of the (d, p) inclusive breakup reaction which is the basis of the SurrogateMethod [3]. The theory employed for this is the Inclusive Nonelastic Breakup (INEB) Reaction theory [4,6,5,7,8].In this contribution we report on recent work that extends the application of the INEB to the case of capture bya radioactive target or projectile. In our approach we consider first the three-body case of a non-cluster projectileinteracting with a two-cluster target, such as the deuteron. In this case the reaction is a neutron pickup. Through themeasurement of the inclusive proton spectrum one is able to extract the neutron capture cross section. This crosssection is not the free capture cross section as several factors come into play owing to the fact that the neutron isbound in the deuteron. The second case we consider is the four-body one involving three-cluster projectile and no-cluster target [11]. In this contribution we propose an extension of the theory of [11] to the case of a two-fragmentprojectile on a two-fragment target. One such reaction involves the one proton halo nucleus B, B + d → p + Bor p + ( Be + d). So the inclusive proton spectrum will exhibit two groups a low proton energy one associatedwith the incomplete fusion Be + d and a higher proton energy group connected with the capture reaction. We alsoconsider the one-neutron halo projectiles on the deuteron target, Be + d and the C + d. Our work reported hereshould be useful to assess the applicability of the INEB theory to isotopes such as
Xe whose lifetime is 9.8hours, which is a notorious nuclear reactor poison as its thermal neutron capture cross section is huge, 2.5 × barns. Several other nuclei exhibit very large thermal neutron capture cross sections [12], whose explanation wasattempted in [13]. Our aim is to use Xe as a benchmark to test the inclusive proton spectrum in a reaction of thetype d + Xe → p + Xe.
We will consider the scattering of a radioactive projectile with a two-cluster target (deuteron). Let us take
Xe asan example. Its life time is 9.6 hours (very long). The reaction we want to describe is
Xe + d → p + Xe. Apickup reaction. The spectrum of the protons is measured, and the theory for this inclusive reaction is available. Thequantity which is extracted from the measurement and the analysis is the total reaction cross section n + Xe → Xe, the neutron capture reaction. The cross section is given by the Austern or Hussein-McVoy (HM) expression d σ p dE p dΩ p = ρ p ( E p )ˆ σ nR , (1)here ˆ σ nR is the medium-modified total reaction cross section of the process n + A, ˆ σ nR = ˆ σ R ( n + A ) . (2)More explicitly, the Inclusive Nonelastic Breakup theory gives for the reaction a + A → b + (x + A) d σ INEB b dE b dΩ b = ˆ σ xR ρ b ( E b ) , (3)where ˆ σ xR is the total reaction cross section of the interacting fragment, x , and ρ b ( E b ) ≡ d k b (2 π ) dE b dΩ b = µ b k b (2 π ) ¯ h (4)is the density of state of the observed, spectator fragment, b . The reaction cross section ˆ σ xR is given by [18] ˆ σ xR = − k x E x h ˆ ρ x ( r x ) | W x ( r x ) | ˆ ρ x ( r x ) i , (5)where W x is the imaginary part of the complex optical potential, U x , of the interacting fragment, x , in the field ofthe target, A. The source function ˆ ρ x ( r x ) is the overlap of the distorted wave of the interacting fragment, x , andthe total wave function of the incident channel. In the DWBA limit of the this latter wave function and using thepost form of the interaction, V xb , the source function in the HM approach [7], is just ˆ ρ x ( r x ) = ( χ ( − ) b | χ (+) a Φ a > ( r x ) . (6)In the IAV theory [4], based on the post form of the interaction, V xb , the source function contains a Green’s functionreferring to the propagation of x , ˆ ρ x ( r x ) = 1 E x − U x + iε ( χ ( − ) b | V xb | χ (+) a Φ a i . (7)The cross section in Eq.(5), according to the Hussein-McVoy model [7], can be decomposed into partial wavesgiving E x k x ˆ σ xR = Z d r x | ˆ S b ( r x ) | W ( r x ) | χ (+) x ( r x ) | , (8)where ˆ S b ( r x ) ≡ Z d r b h χ ( − ) b | χ (+) b i ( r b ) Φ a ( r b , r x ) , (9)and Φ a ( r b , r x ) is the internal wave function of the projectile which carries the observed spectator fragment, b. Theabove formalism has recently been employed to calculate the (d, p) inclusive proton spectrum in (d, p) reactions[19,20,21,22,23,24].In applying the above formalism to the reaction involving the deuteron as a projectile and Xe as the target,or vice versa, one is reminded once again of the lifetime of the latter, 9.8 hours. So there is the practical questionwhich of these two reactions is feasible. In any case the final result in either case is the medium-modified totalreaction cross section of the system n +
Xe. The capture cross section is the difference between this crosssection and the contributions of other direct reactions, such as inelastic excitation of
Xe. In passing we remindthe reader once again that in free space the thermal neutron capture cross sections of several nuclei is abnormallylarge [12,13].
Recently we have developed the theory of INEB involving a three-fragment projectiles, a = b + x + x , suchas Be = He + He + n and Borromean nuclei such as Li = Li + n + n, The cross section for this four-bodyprocess, b + x + x + A , where b is the observed spectator fragment and x and x are the interacting participantsfragments, is d σ INEBb dE b dΩ b = ρ b ( E b ) σ BR , (10) BR = k a E a (cid:20) E x k x σ x R + E x k x σ x R + E CM ( x , x )( k x + k x ) σ BR (cid:21) , (11)where, the form of the reaction or fusion cross section as derived in [18] is used, σ x R = k x E x h ˆ ρ x ,x | W x | ˆ ρ x ,x i , (12) σ x R = k x E x h ˆ ρ x ,x | W x | ˆ ρ x ,x i , (13)and, σ BR = ( k x + k x ) E CM ( x , x ) h ˆ ρ x ,x | W B | ˆ ρ x ,x i , (14)is a three -body, x + x + A , reaction cross section. The energies of the different fragments are defined through thebeam energy, since the projectiles we are considering are weakly bound and thus the binding energy is marginallyimportant in deciding the energies of the three fragments. Thus, e.g., E x ,Lab = E a,Lab ( M x /M a ) , where by M a and M x we mean the mass numbers of the projectile and fragment x , respectively. The three-body sourcefunction, ˆ ρ x ,x , is a generalisation of the two-body source function in Eqs. (6,7), ˆ ρ x ,x ( r x , r x ) = ( χ ( − ) b ( r b ) | χ (+) a ( r b , r x , r x ) Φ a ( r b , r x , r x ) i . (15)The cross sections σ x R and σ x R are the reaction cross sections of x + A and x + A individually, while theother fragments, x and x respectively, are scattered and not observed. E x k x σ x R = Z d r x d r x | ˆ S b ( r x , r x ) | | χ (+) x ( r x ) | W ( r x ) | χ (+) x ( r x ) | , (16) E x k x σ x R = Z d r x d r x | ˆ S b ( r x , r x ) | | χ (+) x ( r x ) | W ( r x ) | χ (+) x ( r x ) | . (17) In the following we treat another four-body breakup problem: the case of two-fragment projectile and two-fragmenttarget. Both projectile and target can break into their two fragments. This is a genuine four-body scattering problem.In principle the formalism of Ref. [11] can be applied after several modifications. Thus the target is a = b + x ,and the projectile is A = x + B. Thus the inclusive spectrum of b will contain breakup of the projectile with x interacting with the target a, x + a, and the breakup of the target with x interacting with the projectile, x +A. In principle this process is a complicated four-body reaction. Here, however we take a simpler approach andtreat the process as a two three-body problems. As such we have the breakup of the projectile without affectingthe target and the breakup of the target without affecting the projectile. In the calculation of the inclusive non-elastic breakup, one would obtain two distinct groups of detected spectator fragments, one related to the target andthe other to the projectile. This method would be valuable in the case of a projectile being an exotic, neutron orproton-rich nucleus.In the following we consider the reaction B + d, which leads to p + (n + B) → p + B, and p + ( Be + d).We remind the reader that B is a one proton halo with a halo separation energy of 0.137 MeV. The first reactionresults in the neutron capture by a one-proton halo nucleus, while the second reaction results in the incompletefusion of the core of this halo nucleus with the deuteron target. The inclusive non-elastic proton spectrum can bewritten as (denoting the proton originating from the radioactive projectile by p and that from the deuteron targetbreakup by p ) d σ p dE p dΩ p = ρ ( E p )ˆ σ R ( n + B ) + ρ ( E p )ˆ σ R ( d + Be ) + · · · (18)The first term on the RHS of the above equation contains the neutron capture cross section of the halo nucleus andwould be concentrated at higher proton energy (the proton separation energy of the deuteron is 2.22 MeV) in itsspectrum, while the second term corresponds to the incomplete fusion, Be + d, which involves the emission of thehalo proton in B and the collision of its core Be with the deuteron. This process should dominate the low energypart of the inclusive proton spectrum.In the case of a one-neutron halo projectile such as Be or C, with halo neutron separation energies, E s = 0.501 MeV and E s = 0.530 MeV, respectively, the same type of reaction will results in an inclusive protonpectrum which should exhibit a now low energy peak related to to the target deuteron breakup at 2.22 MeV, andhigh energy and weaker peak connected with removing a proton from the tightly bound cores, Be, C. d σ p dE p dΩ p = ρ ( E p )ˆ σ R ( n + Be) + ρ ( E p )ˆ σ R ( d + Be) + · · · (19) d σ p dE p dΩ p = ρ ( E p )ˆ σ R ( n + C) + ρ ( E p )ˆ σ R ( d + B) + · · · (20)The cross sections, ˆ σ R ( n + Be), ˆ σ R ( n + C), ˆ σ R ( d + Be), ˆ σ R ( d + B), are given by expressions similar to Eq.(8). One needs the S-matrix elements, ˆ S p ( r p ) and ˆ S p ( r p ) in order to evaluate the above cross sections. Thesematrix elements can be evaluated once appropriate optical potentials for protons on deuteron and on the differenthalo projectiles are given. Further, optical potentials for the projectile target systems are needed, as well as thosefor the generation of the participant fragment distorted waves. These are n + Be, n + C, d + Be, d + B.For the proton halo nucleus B, we need similar ingredients: ˆ S p ( r p ) for p + d elastic scattering and ˆ S p ( r p ) for p + B. Similarly one needs the d + B optical potential and the n + B and d + Be optical potentials. Thesepotentials in principle are known from elastic scattering data.Once the incomplete fusion cross sections are calculated from fusion theory [26], the neutron capture crosssections can be obtained from the general form of the breakup cross sections, Eqs. (18, 19, 20). Thus the InclusiveNon-Elastic Breakup is a potentially powerful method to extract the neutron capture cross section of short-livedradioactive nuclei.The density of states of the observed proton in Eq.s (18), (19) and (20) are given by: ρ p ( E p ) = m p k p (2 π ) ¯ h (21)In B + d, due to the low value of the halo proton separation energy, of E s , in the inclusive nonelastic breakupreaction, we expect a low energy peak in the inclusive proton spectrum connected with the incomplete fusion d + Be, and a higher energy peak connected with the neutron capture n + B reaction.In Be + d, with E s = 0.5 MeV, we expect a lower energy peak associated with the neutron capture n + Be and a much higher energy peak connected with the incomplete fusion d + Li. The higher energy peak isconnected to the proton emitted from the core, Be with a separation energy of E s = 5 MeV. Similarly, for C +d: a low energy peak n + C with a higher energy peak d + B. Here we present an outline of the derivation of Eqs.(18, 19, 20). We take the projectile A to be a bound system oftwo fragments, x and B, and the target a as similarly composed of a bound system of two fragments, x and b. Inthis derivation we follow the works of [4,8,11].We invoke the spectator model in the sense that the observed fragment is only optically scattered from theprojectile or target. Thus we take the Hamiltonian to be H = K x + K x + K b + K a + V x x + V x A + V x b + V x b + U x A + U bA . (22)The steps to be followed to obtain Eqs. (18), Eq.(19) and Eq. (20) are lengthy but rest on a generalization of thecase of three-fragment projectile breakup formalison of [11], and will be reported elsewhere [27]. Acknowledgements.
This work was partly supported by the US-NSF and by the Brazilian agencies, Fundac¸˜aode Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP), the Conselho Nacional de Desenvolvimento Cient´ıfico eTecnol´ogico (CNPq) and INCT-FNA project 464898/2014-5. CAB acknowledges a Visiting Professor support fromFAPESP and MSH acknowledges a Senior Visiting Professorship granted by the Coordenac¸˜ao de Aperfeic¸oamentode Pessoal de N´ıvel Superior (CAPES), through the CAPES/ITA-PVS program. CAB also acknowledges supportby the U.S. NSF Grant No. 1415656 and the U.S. DOE Grant No. DE-FG02-08ER41533.
References
1. J. E. Escher, J. T. Burke, F. S. Dietrich, N. D. Scielzo, I. J. Thompson, and W. Younes, Rev. Mod. Phys. , 353 (2012).. J. E. Escher, J. T. Burke, R. O. Hughes, N. D. Scielzo, R. J. Casperson, S. Ota, H. I. Park, A. Saastamoinen, and T. J. Ross,Phys. Rev. Lett. 121, 052501 (2018).3. G. Potel, G. Perdikakis, B. V. Carlson, M. C. Atkinson, P. Capel, W. H. Dickhoff, J. E. Escher, M. S. Hussein, J. Lei, W.Li, A. O. Macchiavelli, A. M. Moro, F. M. Nunes, S. D. Pain, J. Rotureau, Eur.Phys.J.A , 178 (2017)4. M. Ichimura, N. Austern, and C. M. Vincent, Phys. Rev. C , 431 (1985)5. A. Kasano and M. Ichimura, Phys. Lett. B , 81 (1982).6. T. Udagawa and T. Tamura, Phys. Rev. C , 1348 (1981).7. M. S. Hussein and K. W. McVoy, Nucl. Phys. A , 124 (1985).8. N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher, and M. Yahiro, Phys. Rep. , 125 (1987).9. L. F. Canto and M. S. Hussein, Scattering Theory of Molecules, Atoms and Nuclei , World Scientific 2013.10. C. A. Bertulani, M. S. Hussein and S. Typel, Phys. Lett. B ,217 (2018).11. B. V. Carlson, T. Frederico and M. S. Hussein, Phys. Lett. B , 53 (2017).12. S. F. Mughabghab, “Thermal Neutron Capture Cross Sections Resonance Integrals and G-Factors”, Int. Atomic. EnergyAgency, INDC(NDS)-440 (2003).13. B. V. Carlson, M. S. Hussein and A. K. Kerman, Act. Phys.Polonica B
491 (2016).14. C. Spitaleri et al., Phys. of Atomic Nuclei , 1725 (2011).15. A. Tumino et al., Few Body Systems, , 745 (2012).16. R.G. Pizzone, et al., Eur. Phys. J. , 00034 (2015).17. M. S. Hussein, Two-step nuclear reactions: The Surrogate Method, the Trojan Horse Method and their common founda-tions , EPJA, , 110 (2017).18. M. S. Hussein, Phys. Rev. C
0, 1962 (1984).19. G. Potel, F. M. Nunes, and I. J. Thompson, Phys. Rev. C , 034611 (2015).20. Q. Ducasse et al., Phys. Rev. C , 024614 (2016).21. ]J. Lei and A. M. Moro, C , 061602(R) (2015).22. J. Lei and A. M. Moro, Phys. Rev. C , 044616 (2015)23. B.V. Carlson, R. Capote, M. Sin, Few-Body Syst.
7, 307 (2016).24. J. Lei, A.M. Moro, Phys. Rev. C , 044605 (201725. S. F. Mughabghab, “Thermal Neutron Capture Cross Sections Resonance Integrals and G-Factors”, Int. Atomic. EnergyAgency, INDC(NDS)-440 (2003).26. See, e.g., L. F. Canto, P. R. S. Gomes, R. Donangelo, and M. S. Hussein, Phys. Rep.424