Theory of the Trojan-Horse Method

Transcription

1 Progress of Theoretica Physics Suppement No. 154, Theory of the Trojan-Horse Method Gerhard Baur 1 and Stefan Type 2 1 Institut für Kernphysik, Forschungszentrum Jüich, D Jüich, Germany 2 Geseschaft für Schwerionenforschung (GSI), D Darmstadt, Germany The Trojan-Horse method is an indirect approach to determine the energy dependence of S factors of astrophysicay reevant two-body reactions. This is accompished by studying cosey reated three-body reactions under quasi-free scattering conditions. The basic theory of the Trojan-Horse method is deveoped starting from a post-form distorted wave Born approximation of the T-matrix eement. In the surface approximation the cross section of the three-body reaction can be reated to the S-matrix eements of the two-body reaction. The essentia feature of the Trojan-Horse method is the effective suppression of the Couomb barrier at ow energies for the astrophysica reaction eading to finite cross sections at the threshod of the two-body reaction. In a modified pane wave approximation the reation between the two-body and three-body cross sections becomes very transparent. Appications of the Trojan Horse Method are discussed. It is of specia interest that eectron screening corrections are negigibe due to the high projectie energy. 1. Introduction Many astrophysica modes depend heaviy on precise information about nucear reaction rates that are ideay measured directy in the aboratory. 1) However, cross sections of reactions with charged partices become very sma with decreasing energy due to the Couomb barrier and the astrophysicay reevant energy range cannot be reached in direct measurements except a few cases. Therefore, the cross section σ(e) at ow energies is obtained by extrapoating experimenta data at higher energies with the astrophysica S factor S(E) =σ(e) E exp(2πη), (1.1) where E is the c.m. energy and η = Z 1 Z 2 e 2 /( v) is the Sommerfed parameter depending on the charge numbers Z 1, Z 2 of the coiding nucei and their reative veocity v. The extrapoation process introduces uncertainties and important contributions to the cross section, ike resonances, can be missed. Additionay, a correction has to be appied to obtain the cross section for bare nucei because direct aboratory measurements are affected by eectron screening that enhances the measured cross sections. 2), 3) Independent information on ow-energy cross sections is vauabe for a quantitative description of eectron screening that is not yet competey understood. During the ast years, severa indirect methods have been deveoped to extract astrophysicay reevant cross sections from reated reactions at higher energies. For exampe, the Couomb dissociation method 4) and the method of asymptotic normaization coefficients (ANC) 5), 6) aow to extract information on ow-energy radiative capture reactions. For genera nucear reactions the Trojan-Horse method (THM) can be appied. In this approach the astrophysica two-body reaction is repaced by a suitaby chosen three-body reaction that is measured under specia kinematica Downoaded from by guest on 12 September 2018

2 334 G. Baur and S. Type conditions. The reation between the cross sections is estabished with the hep of reaction theory. Without doubt, the indirect process wi introduce some uncertainties, but vauabe information can be obtained on the astrophysica reaction. Additionay, the errors are independent of that of the direct measurement. Of course, firm concusions can be drawn from indirect experiments ony if the methods have been vaidated by studying we-known reactions and if the theoretica approximations are understood. 7) A simiarity between cross sections for two-body and cosey reated three-body reactions under certain kinematica conditions 8) ed to the introduction of the Trojan- Horse method, 9) 11) see aso Refs. 12) and 13). In this indirect approach a two-body reaction A + x C + c (1.2) that is reevant to nucear astrophysics is repaced by a reaction A + a C + c + b (1.3) with three partices in the fina states assuming that the Trojan Horse a is composed predominanty of custers x and b, i.e. a =(x + b). This reaction can be considered as a specia case of a transfer reaction to the continuum. The energy in the entrance channe of reaction (1.3) is chosen around or above the Couomb barrier and effects from eectron screening are negigibe. Nevertheess, under quasifree kinematica conditions very sma energies can be reached in reaction (1.2). The essentia feature of the THM is the suppression of the Couomb barrier in the two-body reaction. The cross section of the three-body reaction remains finite when the c.m. energy in the A + x system approaches zero. In 2 some genera aspects in the theoretica description of transfer reactions into the continuum are discussed. This eads to the formuation of the THM theory. In a modified pane-wave approximation the reation between the cross section of reactions (1.2) and (1.3) becomes very transparent. For detais we refer to Ref. 11). Appications of the THM are discussed in 3 where aso a summary and an outook are presented. 2. Theory 2.1. Transfer reactions into the continuum in post-form DWBA We assume a three-body mode where the target nuceus A interacts with a projectie a = b + x. The T-matrix eement for the eastic breakup reaction A + a A + x + b (2.1) is given in the post-form of the distorted-wave Born approximation (DWBA) as (see aso Eq. (10) of Ref. 12)) T = χ ( ) Bb ( k Bb )Ψ ( ) B ( k Ax )Φ b V xb χ (+) Aa ( k Aa )Φ A Φ a, (2.2) where B denotes the system A + x in the fina state. Φ a, Φ b,andφ A are the boundstate wave functions of a, b and A, respectivey, and V bx is the potentia between x Downoaded from by guest on 12 September 2018

3 Theory of the Trojan-Horse Method 335 and b. Theχ s are the scattering wave functions generated by the appropriate optica potentias. This expression for the T-matrix eement is quite difficut to evauate in genera. At high beam energies eikona methods 14) can be used to simpify it. For an intermediate mode see, e.g., Ref. 15). It contains some simpe imits, ike the Serber mode: see, e.g., Ref. 12). In the distorted waves of Eq. (2.2) the interaction of the target with the participant x as we as the spectator b is incuded to a orders in genera. It is of interest to treat aso the case where the subsystem B = A + x can go to other fina channes C + c. This reaction is sketched in Fig. 1 with the reevant momenta of the nucei. The theoretica description is especiay simpe when the surface approximation can be appied: due to Couomb repusion and/or strong absorption the wave function of the transferred partice dξ A Ψ ( ) B Φ A =4π m i f (r Ax )Y m (ˆr Ax )Y m (ˆk Cc ) (2.3) has ony to be known in the nucear exterior. The integration in Eq. (2.3) is over the nuceon variabes of A. In this case the overap integra is given in terms of the S-matrix eement of the C + c A + x reaction, which we denote by S,as f (r Ax )=δ AxCc j (k Ax r Ax )+ 1 max k Ax (S δ AxCc ) h (+) 2 m Cc k (k Ax r Ax ) (2.4) Cc for r Ax R with a cutoff radius R. Here we assume spiness partices for the sake of simpicity. For charged partices x the appropriate Couomb functions have to be used in pace of the Besse (Hanke) functions j (h (+) ). The vaidity of the surface approximation was checked by Kasano and Ichimura. 16) It was found to be quite good for the (d,p) reaction at E d = 26 MeV. Incusive breakup spectra were measured for many different systems and compared to theory. Agreement is generay good. 12) The theory of incusive breakup reactions was substantiay generaized in a series of papers by Ichimura, Austern and Vincent ( IAV ). We give two references, A ka kx = k a k b x kc C kc kb = k C + k c c Downoaded from by guest on 12 September 2018 a ka kb b Fig. 1. Momenta of the nucei in the Trojan-Horse reaction (1.3).

4 336 G. Baur and S. Type from where the fu story can be traced back. 17), 18) In this series of papers, aso many forma aspects have been deepy eucidated and the reation of post-form to prior-form DWBA (they give identica resuts) has aso been made very cear Cross section in modified pane wave approximation and THM The appearance of the S-matrix eement of the two-body reaction in (2.4) aows to estabish a reation between the cross sections of reaction (1.2) and (1.3), see Ref. 11) for detais. Repacing the distorted waves in Eq. (2.2) by pane waves and appying the surface approximation, the cross section for the three-body reaction d 2 σ = KF W ( de Cc dω Cc dω Q Bb ) 2 dσ TH (2.5) Bb dω factorizes into a kinematica factor KF = µ Aaµ Bb µ Cc (2π) 5 6 k Bb k Cc k Aa 16π 2 k Ax Q Aa v Cc v Ax, (2.6) a momentum distribution W 2 and the so-caed TH cross section dσ TH /dω (see Ref. 11) for the definition of reduces masses, momenta etc.). The momentum ampitude ( W ( Q Bb )= ε a + 2 Q 2 ) Bb exp(iq 2µ Bb r xb )Φ x Φ b Φ a (2.7) xb is reated to the wavefunction of the Trojan horse a with binding energy ε a (> 0) in momentum space. It depends on the momentum Q Bb = m b k Bb kaa. (2.8) m b + m x Negecting the Fermi motion of b inside the Trojan Horse the second term is the momentum of the incoming spectator b with respect to A, and Q Bb corresponds to the momentum transfer to the spectator b. The momentum distribution essentiay describes the Fermi motion of b and x inside the Trojan horse a. The TH cross section dσ TH dω = 1 4k 2 Cc (2 +1)P (ˆk Cc ˆQ Aa ) [ S J (+) ] 2 δ AxCc J ( ) (2.9) with Legendre poynomias P and the S-matrix eements S ooks very simiar to the cross section for the inverse of the astrophysica reaction (1.2) except for the TH integras J (±) (R, η Ax,k Ax,Q Aa )=k ax Q Aa dr r u ± (η Ax; k Ax r) j (Q Aa r) (2.10) R with the Couomb wave functions u ± = e σ (G ± if ). The TH integras depend on the cutoff radius R of the surface approximation, the c.m. momentum k Ax in the A + x reative motion and Q Aa = m A k Aa kbb, (2.11) m A + m x Downoaded from by guest on 12 September 2018

5 Theory of the Trojan-Horse Method 337 that reduces to k x for target mass m A. The properties of the TH integras are discussed extensivey in Ref. 11). The expression (2.5) resembes the form of the cross section in a pane-wave impuse approximation 19) that has been used in the past in order to extract information on the momentum distribution of nucei. However, ony the DWBA with the surface approximation expains the effective reduction of the Couomb barrier for sma c.m. energies in the A + x system Threshod behaviour of cross sections The energy dependence of the two-body cross section dσ dω = 1 2 4kAx 2 (2 +1)P (ˆk Cc ˆk Ax )S k 2 Ax exp( 2πη Ax) (2.12) for the ineastic reaction (1.2) is governed by the k 2 Ax factor and the energy dependence S exp( πη Ax ) of the reevant S-matrix eement. This motivates the introduction of the astrophysica S factor (1.1) for the extrapoation of experimenta data to ow energies. In the TH cross section (2.9) the factor k 2 Ax is repaced with k 2 (±) Cc and the TH integras J appear. Their energy dependence for sma k Ax is determined by k Ax vax exp(πη Ax ) from the contribution of the irreguar Couomb wave function. This eads to a k Ax dependence of the three-body cross section (2.5) according to d 3 σ de C dω C dω c k 2 Ax v 1 Ax exp( 2πη Ax)k 2 Axv Ax exp(2πη Ax ) = const (2.13) in the owest order of k Ax. As a resut the cross section does not vanish at the threshod but takes on a finite vaue. Aso in the case of neutron transfer, ike in a (d,p) stripping reaction, it is we known that the cross section is finite at the threshod E n =0. 12), 13) The reason is the same as in the case of charged partices: the momentum dependence of the S-matrix eement is canceed by the corresponding Trojan-Horse enhancement factor. In a simiar way, the threshod behaviour can be studied in the eastic breakup case. In this case we have three contributions, the pure Couomb, the nucear and the interference term, see Eqs. (70) (73) of Ref. 11). A three terms show the same threshod behaviour, the cross section behaves as k Ax exp( 2πη) cose to threshod. In contrast, the Couomb term dominates in the direct two-body eastic scattering of the A + x-system. The d + p p + p + n breakup reaction was studied recenty in the reevant kinematica region in Ref. 20) Kinematica conditions In most experiments so far nucei with a dominant s-wave contribution in their ground state have been empoyed as Trojan horses. Then, the momentum ampitude W ( Q Bb ) has a maximum at zero. Correspondingy, the equation Q Bb = 0 defines the so-caed quasi-free condition in the three-body phase space where the cross section for the quasi-free reaction reaches a maximum. From this condition the Downoaded from by guest on 12 September 2018

6 338 G. Baur and S. Type corresponding quasi-free c.m. energy E qf Ax = E Aa ( 1 µ Aa µ 2 ) bx µ Bb m 2 ε a (2.14) x in the initia channe of the two-body reaction (1.2) is derived from energy conservation. The reation between E qf Ax and E Aa is purey a kinematica consequence. It is obvious that even with a arge c.m. energy E Aa in the entrance channe of the three-body reaction (1.3) a sma energy E Ax can be reached. The width of the momentum ampitude W ( Q Bb ) determines the range of energies around E qf Ax that can be expored due to the Fermi motion of b and x inside the Trojan horse a. In an actua experiment a cutoff in the momentum Q Bb is chosen to seect the region where the quasi-free process dominates the cross section over a processes. 3. Appications of the Trojan-Horse method, summary and outook Severa reactions have been studied with the TH method recenty. 20), 22) 29) with 2 Hand 6 Li (= α+d) as typica Trojan Horses. These nucei aow to study the transfer of protons, neutrons, deuterons and α-partices, which covers most of the cases of astrophysica interest for the two-body reaction. In nucear astrophysics, transfer reactions (ike (d,p) or ( 3 He,d), or (Li,α)) are used to study resonant states. For exampe, in the 22 Na( 3 He,d) 23 Mg reaction states near the proton threshod were studied. 21) This is reevant for the hydrogen burning of 22 Na. In principe, aso the continuum can be studied. For exampe, the paraeism of (d,p) and (n,n) reactions has been beautifuy shown aready in 1971, see Ref. 8). The d+ 6 Li reaction was investigated in Ref. 28) in this indirect way. Another recent appication is given in Ref. 29) to the 7 Li(p,α) 4 He-reaction. An especiay interesting case woud be the indirect study of the 12 C(α, γ) 16 O reaction by means of a ( 7 Li,t) or ( 6 Li,d) reaction. Quite recenty 30) the sub-couomb α-transfer reaction ( 6 Li,d) and ( 7 Li,t) to the bound 2 + and 1 states in 16 O has been used to obtain information on the astrophysica S-factor. In this contribution, the basic theory of the Trojan-Horse method was reviewed starting from a distorted wave Born approximation of the T-matrix eement. The essentia surface approximation aows to find the reation between the cross section of the three-body reaction and the S-matrix eements of the astrophysicay reevant two-body reaction. In the modified pane wave approximation the reation between the three-body and two-body cross sections becomes very transparent. The threebody cross section is a product of a kinematica factor, a momentum distribution and a so-caed Trojan-Horse two-body cross section. The energy dependence of the appearing Trojan-Horse integras eads to a finite cross section of the three-body reaction at the threshod of the two-body reaction without the suppression by the Couomb barrier. This aows to extract the energy dependence of astrophysica cross sections from the three-body breakup reaction to very ow energies without the probems of eectron screening and extremey ow cross section. A comparison of resuts for S factors from direct and indirect experiments can improve the information on the eectron screening effect, see aso Ref. 31). However, dedicated Downoaded from by guest on 12 September 2018

7 Theory of the Trojan-Horse Method 339 Trojan-Horse experiments are necessary in order to achieve a precision comparabe to direct measurements. The vaidity of the Trojan-Horse method can be tested by comparing the cross sections extracted from the indirect experiment with resuts from direct measurements of we studied reactions. In principe it is possibe to assess systematic uncertainties of the Trojan-Horse method by studying various combinations of projectie energies, spectators in the Trojan Horse and scattering anges. Furthermore, different theoretica approximations can be compared, e.g. fu DWBA cacuations with and without the surface approximation and simper modified pane wave approximations. One may aso envisage appications of the Trojan-Horse method to exotic nucear beams. An unstabe projectie hits a Trojan-Horse target aowing to study specific reactions on exotic nucei. We mention the d( 56 Ni,p) 57 Ni reaction studied in inverse kinematics in Ref. 32). In this paper stripping to bound states was studied; extension to stripping into the continuum woud be of interest for this and other reactions of this type. A study of ow-energy eastic scattering with the Trojan-Horse method opens another appication which can ead to improved information reevant to the theoretica description of nucear reactions at ow energies. Acknowedgements We gratefuy acknowedge the hospitaity of Caudio Spitaeri and his group in Catania where many of the ideas were shaped. References 1) C. E. Rofs and W. S. Rodney, Caudrons in the Cosmos (University of Chicago Press, Chicago, 1988). 2) H. J. Assenbaum, K. Langanke and C. Rofs, Z. Phys. A 327 (1987), ) C. Rofs, Prog. Theor. Phys. Supp. No. 154 (2004), ) G. Baur and H. Rebe, J. of Phys. G 20 (1994), 1; Annu. Rev. Nuc. Part. Sci. 46 (1996), 321. G.Baur,C.A.BertuaniandH.Rebe,Nuc.Phys.A458 (1986), ) A. Azhari, V. Burjan, F. Carstoiu, C. A. Gagiardi, V. Kroha, A. M. Mukhamedzhanov, F. M. Nunes, X. Tang, L. Trache and R. E. Tribbe, Phys. Rev. C 63 (2001), ) H. M. Xu, C. A. Gagiardi, R. E. Tribbe, A. M. Mukhamedzhanov and N. K. Timofeyuk, Phys.Rev.Lett.73 (1994), ) S. Austin, nuc-th/ ) H. Fuchs, H. Homeyer, Th. Lorenz and H. Oescher, Phys. Lett. B 37 (1971), ) G. Baur, Phys. Lett. B 178 (1986), ) S. Type and H. H. Woter, Few-Body Systems 29 (2000), ) S. Type and G. Baur, Ann. of Phys. 305 (2003), ) G.Baur,F.Röse, D. Trautmann and R. Shyam, Phys. Rep. 111 (1984), ) G. Baur and D. Trautmann, Phys. Rep. 25C (1976), ) M. S. Hussein and K. McVoy, Nuc. Phys. A 445 (1985), ) A. Bonaccorso, Phys. Rev. C 60 (1999), ) A. Kasano and M. Ichimura, Phys. Lett. B 115 (1982), ) M.Ichimura,N.AusternandC.M.Vincent,Phys.Rev.C37 (1988), ) M. Ichimura, Theory of Incusive Breakup Reactions, invited tak at the Internationa Conference on Nucear Reaction Mechanism, January 3-9, 1989, Cacutta, India. 19) M. Jain, P. G. Roos, H. G. Pugh and H. D. Homgren, Nuc. Phys. A 153 (1970), 49. Downoaded from by guest on 12 September 2018

8 340 G. Baur and S. Type 20) M. G. Peegriti, PhD thesis, University of Catania (2000); Prog. Theor. Phys. Supp. No. 154 (2004), ) S. Schmidt et a., Nuc. Phys. A 591 (1995), ) A. Musumarra, R. G. Pizzone, S. Bagus, M. Bogovac, P. Figuera, M. Lattuada, M. Miin, D. Mijanic, M. G. Peegriti, D. Rendic, C. Rofs, N. Soic, C. Spitaeri, S. Type, H. H. Woter and M. Zadro, Phys. Rev. C 64 (2001), ) C.Spitaeri,S.Type,R.G.Pizzone,M.Aiotta,S.Bagus,M.Bogovac,S.Cherubini,P. Figuera, M. Lattuada, M.Miin, D. Mijanic, A. Musumarra, M. G. Peegriti, D. Rendic, C. Rofs, S. Romano, N. Soic, A. Tumino, H. H. Woter and M. Zadro, Phys. Rev. C 63 (2001), ) M. Lattuada, R. G. Pizzone, S. Type, P. Figuera, D. Mijanic, A. Musumarra, M. G. Peegriti, C. Rofs, C. Spitaeri and H. H. Woter, Astrophys. J. 562 (2001), ) M. Aiotta, C. Spitaeri, M. Lattuada, M. Musumarra,R.G.Pizzone,A.Tumino,C.Rofs and F. Strieder, Eur. Phys. J. A 9 (2000), ) G. Cavi, S. Cherubini, M. Lattuada, S. Romano, C. Spitaeri, M. Aiotta, G. Rizzari, M. Sciuto, R. A. Zappaa, V. N. Kondratyev, D. Mijanic, M. Zadro, G. Baur, O. Yu. Goryunov and A. A. Shvedov, Nuc. Phys. A 621 (1997), 139c. 27) C. Spitaeri, M. Aiotta, P. Figuera, M. Lattuada, R. G. Pizzone, S. Romano, A. Tumino, C. Rofs, L. Giaanea, F. Strieder, S. Cherubini, A. Musumarra, D. Mijanic, S. Type andh.h.woter,eur.phys.j.a7 (2000), ) S. Cherubini, V. N. Kondratyev, M. Lattuada, C. Spitaeri, D. Mijanic, M. Zadro and G. Baur, Astrophys. J. 457 (1996), ) C. Spitaeri, A. Aiotta, S. Cherubini, M. Lattuada, D. Mijanic, S. Romano, N. Soic, M. Zadro and R. A. Zappaa, Phys. Rev. C 60 (1999), ) C. R. Brune et a., Phys. Rev. Lett. 83 (1999), ) A. Tumino, Prog. Theor. Phys. Supp. No. 154 (2004), ) K. E. Rehm et a., Phys. Rev. Lett. 80 (1998), 676. Downoaded from by guest on 12 September 2018