Extraction of Astrophysical Cross Sections in the Trojan-Horse Method

Transcription

1 Few-Body Systems 29, 75±93 (2000) Extraction of Astrophysica Cross Sections in the Trojan-Horse Method S. Type and H. H. Woter Sektion Physik, Universitat Munchen, Am Couomwa 1, D Garching, Germany Astract. The Trojan-horse method has een proposed to extract S-matrix eements of a two-ody reaction at astrophysica energies from a reated reaction with three partices in the na state. This shoud e usefu in cases where the direct measurement of the two-ody reaction at the necessary ow energies is experimentay dif cut. The formaism of the Trojan-horse method for nucear reactions is deveoped in detai from asic scattering theory incuding spin degrees of freedom of the nucei and we specify the necessary approximations. The energy dependence of the three-ody reaction is determined y characteristic functions that represent the theoretica ingredients for the method. In a pane-wave Born approximation of the T-matrix the differentia cross section assumes a simpe structure. 1 Introduction The Trojan-horse method (THM) [1] has een suggested as an indirect method in order to determine cross sections of charged-partice reactions reevant to nucear astrophysics. Ideay, reaction cross sections which serve as an input to various astrophysica modes, as primordia nuceosynthesis or stear evoution, shoud e measured directy in the aoratory. However, for the reevant ow energies reaction rates ecome very sma ecause of Couom repusion of the interacting partices, and an experimenta determination is very dif cut or impossie. In order to circumvent this proem aternative methods have een proposed, where the considered reaction is not studied directy, ut a cosey reated process is investigated which can e measured experimentay. An exampe is the Couom dissociation method for the determination of radiative capture cross sections which has een successfuy used for severa reactions in recent years [2±4]. In genera, the reation to the astrophysica reaction is estaished with the hep of nucear reaction theories, however, approximations are necessary. These may depend on the reaction investigated giving the possiiity to seect the appropriate method. Because of Dedicated to Prof. A. Weiguny on the occasion of his retirement E-mai address: stefan.type@physik.uni-muenchen.de

2 76 S. Type and H. H. Woter these approximations one cannot expect to extract asoute cross sections for reactions of astrophysica interest, however, the Trojan-horse method wi give reiae information on their energy dependence. This is important, ecause at higher energies the asoute cross section is usuay we known from direct experiments and with the information from the Trojan-horse method it can e reiay extrapoated to the reevant energies. In particuar, direct measurements can suffer from the effect of eectron screening at very sma energies, which is not understood suf cienty we theoreticay. In contrast, eectron screening does not affect cross sections extracted in the Trojan-horse method. The method is thus aso of interest in understanding the eectron screening effect. Let us consider a nucear reaction A x! C c 1:1 with given Q-vaue. At ow reative energies in the initia channe the Couom repusion of the charged nucei A and x wi ead to a strong reduction of the cross section. In the Trojan-horse method the nucear custer x is hidden inside a projectie a ˆ x y attaching it to a nuceus therefore the name of the method and the reaction A a! C c 1:2 with three partices in the na channe is studied (Fig. 1). The reative energy in the A a channe is chosen aove the Couom arrier, thus there is no Couom reduction of the three-ody cross section. However, reactions etween A and x can sti e induced at sma reative energies due to the Fermi motion of x inside a which can compensate, at east partiay, the arge reative momentum in the A a system. The nuceus is thus assumed to e a mere spectator to the reevant process. The method, in principe, is very exie, since one is not restricted to ineastic processes ut aso the eastic A x! A x scattering can e investigated. The method can aso e empoyed where the partice c is a photon and C is a ound state of the A x system. In genera, various reaction mechanisms can ead to a na state with partices ; c, and C. Besides the process, where nuceus is emitted and ony x and A interact, one coud aso consider the formation of a compound nuceus D from A a and a susequent decay with some ux into the C c channe. In case of eastic A x scattering the na A x state can e reached y a reakup of a into x in the nucear pus Couom ed of A. These other mechanisms have to e distinguished experimentay from the process where the reaction A x! C c is not affected y the presence of partice. Fig. 1. Diagram for the three-ody reakup reaction in the quasi-free mechanism

3 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 77 The THM has aready een appied in some cases [5±8] in order to identify the quasi-free reaction mechanism and to test the method. The experimenta resuts were anayzed essentiay within the pane-wave-impuse approximation (PWIA), where nuceus is assumed to e unaffected y the coision. The PWIA was originay deveoped for the description of -partice knockout reactions in order to deduce momentum distriutions of the -custer inside nucei ike 6 Li or 7 Li [9]. In PWIA the reakup cross section is written as [9] d 3 ˆ KF j a k f d N B de Cc d Cc d j2 B d : 1:3!Cc It is a product of three factors: (i) a kinematica factor KF, (ii) a momentum distriution of the nuceus a at the B na momentum, and (iii) an off-she twoody A x! C c cross section, which is repaced in the quasi-free approximation y an appropriate on-she two-ody cross section. In the THM one is interested in the atter quantity. Athough this factorization is quite appeaing some questions remain aout the approximations invoved. This concerns the repacement of the compete three-ody T-operator y the two-ody T-operator, i.e., the impuse approximation [9], and the reation etween the off-she and the on-she two-ody cross section. In the appication of Eq. (1.3) d N =d has een interpreted as the nucear cross section without Couom arrier effects and the astrophysicay reevant two-ody cross section d=d has een written as d d ˆ G dn d ; 1:4 where G is supposed to account for the penetration factor for the dominant partia wave through the Couom arrier, which was taken in a simpe semicassica approximation. Such empirica procedures aso have to e justi ed. The reation etween the cross section of two-ody reaction (1.1) and the cross section of three-ody reaction (1.2) can e cari ed within the theory of direct nucear reactions as wi e shown in the foowing. In the origina suggestion of the Trojan-horse method [1] the reation was estaished more or ess quaitativey with emphasis on the energy dependence of the cross sections. The empoyed approximations were not speci ed in detai. Here, they wi e stated more expicity. They can e tested experimentay as we as theoreticay. Besides the fu formuation for the three-ody cross section in distorted-wave Born approximation (DWBA) we deduce a pane-wave Born approximation (PWBA) which resemes in structure the resut of the PWIA, ut is more genera. We treat oth eastic and ineastic processes for the two-ody reaction. Our work is organized as foows: In Sect. 2 we wi introduce the reevant quantities for the description of reaction (1.2) and express the cross section with three partices in the na state with the hep of the T-matrix of the process. The foowing section is devoted to the approximations for the T-matrix in the post-form distorted wave description. In Sect. 4 the reation of the cross sections for reaction (1.1) and reaction (1.2) is deduced with the hep of the asymptotic form of the twoody scattering wave function. In Sect. 5 the T-matrix for the three-ody reakup is cacuated in pane-wave Born approximation (PWBA) and compared to the PWIA. Finay, we cose with a summary. For transparency we deveop the

4 78 S. Type and H. H. Woter formaism-disregarding spin. The genera expressions incuding spin are given in an Appendix. 2 The Three-Body Breakup Reaction A a! C c We wi denote the mass, the spatia coordinate and the momentum of a nuceus i y m i ; r i, and p i ˆ hk i ˆ m i _r i, respectivey. Furthermore we introduce reative coordinates r ij ˆ r i r j 2:1 etween nucei i and j, and the conjugated momenta p ij ˆ hk ij ˆ ij _r ij ˆ mj p i m i p j ; 2:2 m i m j where ij ˆ m i m j = m i m j is the corresponding reduced mass. For a three-ody system with partices ; c, and C we have the tota energy E tot ˆ m C m c m E Cc E Cc P2 2M with the tota momentum P ˆ P i p i and tota mass M ˆ Pi m i. 2:3 E ij ˆ p2 ij 2:4 2 ij is the kinetic energy of reative motion etween partices i and j. The reative motion of the three partices is competey speci ed y the two Jacoi momenta p Cc and p Cc. In case of the two-ody system A a we have E tot ˆ m A m a E Aa P2 2:5 2M with ony one reevant kinetic energy E Aa. For simpicity we wi assume P ˆ 0in the foowing. In genera, the cross section for reaction (1.2) is given y [10, 11] d ˆ 2 Aa d 3 p f Cc d 3 p f B h p i Aa 2h 3 2h 3 jt k f Cc ; k f B ; ki Aa j2 E f Cc E f B Ei Aa Q ; 2:6 where B stands for the system C c. Quantities in the initia and na channes are given superscripts i and f, respectivey. The -function with the Q-vaue Q ˆ m A m a m C m c m 2:7 of the reaction guarantees energy conservation in the scattering process. The T- matrix contains a information aout the dynamics and its cacuation is the principe proem for the description of the reaction. With p f Cc dp f Cc ˆ CcdE f Cc we nd the tripe differentia cross section d 3 de f Cc d Cc d B ˆ Aa B Cc 2 5 h 6 f k B k f Cc kaa i jt k f Cc ; k f B ; ki Aa j2 ; 2:8

5 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 79 which depends on the na reative energy E f Cc in system B and the directions of oth reative momenta ^p f Cc and ^p f B for the given initia momentum pi Aa. Here and in the foowing we have negected possie spins of the participating nucei in order to simpify the equations. The reevant formuas with incusion of spin degrees of freedom wi e given in the Appendix. 3 Approximations to the Three-Body T-Matrix The exact T-matrix is given in terms of the interaction etween the partices and the exact soution of the scattering proem for the tota wave function. Of course, this is not known in the genera case and we have to empoy approximations which shoud retain the reevant physics of the reaction process. We assume that the tota interaction in our system is given y a sum of two-ody potentias V. Then the potentia etween two nucei V ij ˆ X 3:1 2i 2j depends not ony on the reative coordinate r ij ut aso on the interna coordinates of the nuceons inside the nucei i and j, respectivey. On the other hand, we can write the kinetic energy operator for the reative motion as T ij ˆ p2 ij 3:2 2 ij with the momentum operator of the reevant Jacoi momentum p ij. We de ne the interna Hamitonian of a nuceus i to e h i (incuding the rest mass) which depends ony on interna coordinates with the soution i of the corresponding Schrodinger equation. Considering the different partitions into nucei in the initia and na channes we have for the tota Hamitonian the expressions in the initia state H ˆ h A h a T Aa V Aa ; h a ˆ h x h T x V x 3:3 and in the na state H ˆ h B h T B V B ; h B ˆ h C h c T Cc V Cc : 3:4 The exact T-matrix eement is now given in the post formuation y [10, 11] T k f Cc ; k f B ; ki Aa ˆh f 0 B B jv B j Aa i; 3:5 where Aa is the fu soution of the scattering proem with a pane wave in the initia channe A a and outgoing spherica waves in a channes with or without rearrangement. The outgoing pane wave f 0 B ˆexp ik f B r B 3:6 appears on the eft-hand side. The wave function B for the system C c wi e speci ed in Sect. 4, depending on whether one considers a ound state or a scattering state. The T-matrix eement is transformed with the hep of the Ge-Mann± Goderger reation into a sum of two contriutions. For this we spit the tota V

6 80 S. Type and H. H. Woter Hamitonian H into two parts H ˆ H i V Aa U Aa ; H i ˆ h A h a T Aa U Aa ; 3:7 introducing an (optica) potentia U Aa depending ony on the reative coordinate r Aa. The scattering proem for the Hamitonian H i can e soved exacty as the potentia U Aa ony aows eastic Aa scattering without any excitation of interna degrees of freedom of the coiding nucei. The reative motion is given y a distorted wave T Aa U Aa Aa ˆEAa i Aa : 3:8 Simiary, in case of the na state we have the decomposition H ˆ H f V B U B ; H f ˆ h B h T B U B 3:9 and the distorted wave T B U B B ˆE f B B : 3:10 The Ge-Mann±Goderger reation [10] now eads to the resut T k f Cc ; k f B ; ki Aa ˆh B B jv Aa V B U B j 0 Aa A a i h B B jv B U B j Aa i; 3:11 which is sti exact. Note that in the na channe we have to insert the wave function B which is the time-reversed of B. The origina Ge-Mann± Goderger reation [12] was derived for identica partitions of the Hamitonian in the initia and na channes whereas here we appy the more genera resut for different partitions [10, 11]. Next we oserve that the rst contriution to the T-matrix eement vanishes exacty. By using Eqs. (3.3) and (3.4) we nd V Aa V B U B ˆ H h A h a T Aa H h B h T B U B ˆ h B h T B U B h A h a T Aa : 3:12 Since the wave functions in the matrix eement are soutions of the Schrodinger equations h B h T B U y B Š B B ˆ E tot B B 3:13 and h A h a T Aa Š 0 Aa A a ˆ E tot 0 Aa A a 3:14 with tota energy E tot the rst matrix eement vanishes. Then the T-matrix reevant for our cacuation is sti exacty given ony y the second contriution T k f Cc ; k f B ; ki Aa ˆh B B jv B U B j Aa i: 3:15 This expression sti contains the fu soution of the scattering proem on the right side of the T-matrix. As a rst approximation, we appy the distorted-wave Born approximation y repacing the fu soution Aa y the distorted wave Aa A a T DWBA k f Cc ; k f B ; ki Aa ˆh B B jv B U B j Aa A a i: 3:16

7 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 81 The next approximation is introduced as usua y repacing the potentia V B U B ˆ V C V c U B ˆ V A V x U B V x ; 3:17 where we assume that the transferred nuceus x is sma and that the optica potentia U B ts eastic scattering of y nuceus B. Negecting the difference V A U B is expected to e of the same order as the uncertainties of DWBA. This approximation for the interaction potentia has een extensivey discussed, e.g., in ref. [13] where it was appied to deuteron stripping. In a pane-wave approximation of the T-matrix the potentia U B is not introduced and the aove approximation means to negect the interaction etween nuceus A and the spectator. This just corresponds to the impuse approximation for the three-ody reaction. Finay the reevant T-matrix of the three-ody reakup reaction assumes the form T DWBA k f Cc ; k f B ; ki Aa ˆh B B jv x j Aa A a i: 3:18 4 Reation Between the Two-Body and the Three-Body Cross Sections We now formuate the reation etween the cross section for the two-ody reaction (1.1) and the three-ody reakup reaction (1.2). The T-matrix eement (3.18) for the three-ody reaction is cacuated with the wave function B for the system B in the na state. In our case, this is the fu scattering wave function of the unound B ˆ C c system, and thus the three-ody T-matrix eement aso contains the information on the C c! A x reaction, therefore giving the connection etween the two-ody and three-ody cross sections. In the foowing formuae we wi negect effects of antisymmetrization. 4.1 The Two-Body Scattering Cross Section The two-ody differentia cross section is given as d! Cc ˆv f Cc d Cc v i j f! Cc j 2 4:1 with the appropriate scattering ampitude f. The veocities are determined from the corresponding momenta p f Cc ˆ Ccv f Cc and pi ˆ v i. The scattering ampitude is derived from the fu scattering wave function ; k i with the system A x in the initia channe. Its asymptotic form is given y (we ony consider twoody na states) ; k! 4 X X ; k r k i i Y m ^r Y m r ^k i : 4:2 m The sum runs over a possie na channes, where is the corresponding wave function which depends on a interna variaes of the two partices in the na state. The radia wave function s ; k r ˆexp i i Š v i 2i v f S! u ; k r u ; k r Š 4:3

8 82 S. Type and H. H. Woter can e expressed y the (nucear) S-matrix eement S! and inear cominations of the reguar and irreguar Couom wave functions (F and G ), where we have introduced the notation u ; k r ˆG ; k r if ; k r! exp i k r n 2k r 2 : 4:4 The Couom phase shift for the partia wave depends on the Sommerfed parameter ˆ Z 1 Z 2 e 2 4:5 hv with the charge numers Z i of the two nucei i in channe and their reative veocity v. The asymptotic form of the radia wave function (4.3) wi e of importance for estaishing the reation etween the cross sections for the two- and three-ody reactions. Assuming that the initia momentum p i is directed aong the z-axis we have r Y m ^k i ˆ 2 1 m0 4:6 4 and for the tota scattering ampitude (incuding Couom scattering) p s v i X p f! Cc ˆ i 2 1Y 0 ^r Cc k i v f Cc exp i f Cc Šg S! Cc Cc Š: 4:7 Finay, the tota reaction cross section for an ineastic reaction Cc 6ˆ! Cc ˆ X 2 1 js k i! Cc j 2 4:8 2 is otained y integrating Eq. (4.1) over a possie directions ^r Cc. It is ovious that the S-matrix contains a the essentia information of the scattering process. At very sma energies E i in the initia state the energy dependence of the S-matrix is dominated y the Couom arrier and decreases ike exp i. In astrophysica appications one therefore de nes the S-factor S E i ˆ! Cc Ei exp 2i 4:9 which shows a much weaker variation with energy as compared to the cross section. 4.2 The Three-Body Scattering Cross Section In case of the three-ody reakup the tripe differentia cross section is given y Eq. (2.8). We now use the approximation (3.18) for the T-matrix. Assuming that the optica potentias U Aa and U B depend ony on r Aa and r B, respectivey, we write T k f Cc ; k f B ; ki Aa ˆh B; k f B Cc; k f Cc jv x j Aa; k i Aa A a i: 4:10

9 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 83 We have repaced the wave function B in Eq. (3.18) y the exact two-ody scattering wave function Cc; k f Cc which asymptoticay ehaves as a pane wave in the Cc channe and ingoing spherica waves in a other channes. It can e otained from Eq. (4.2) y appying the time-inversion operator and repacing y Cc. Then, its asymptotic form is given y Cc; k f Cc!4 X X ; q r k f i Y m ^r Y m r ^k f Cc 4:11 Cc with radia wave functions s ; k r ˆi exp i f Cc Š 2 v f Cc v S Cc! u ; k r Cc u ; k r Š 4:12 for a possie partitions. The expression (4.10) can e cacuated in DWBA y repacing the exact scattering wave Cc; k f Cc y a distorted wave. But then we do not otain an expicit reation to the two-ody cross section. We therefore introduce the essentia, so-caed surface approximation [14]. From the form of the T-matrix eement (4.10) we can deduce which channe gives the most important contriution. The potentia V x ony acts on the ound-state wave function a of the x system and does not depend on r Aa. The distorted wave Aa; k i Aa wi ecome very sma for sma r Aa since the optica potentia U Aa is strongy asorptive there. For sma momenta p i Aa the Couom arrier in the initia channe wi additionay reduce the ampitude of the wave function at sma r Aa. On the other hand, asymptoticay, the channes A a and C c ecome orthogona. Therefore, there wi e a sustantia contriution to the T-matrix eement ony in the surface region of nuceus A, i.e., for r Aa cose to the radius of A. Thus we can expect that the product V x Aa; k i Aa A a projects onto contriutions mainy from the A x structure and ony itte onto the C c and other partitions. We concude that in the wave function Cc; k f Cc the channe with the A x fragmentation is seected in the matrix eement and further, that we can use the asymptotic form (4.11) of the wave function, since the sustantia contriutions to the matrix eement arise from not too sma r. Using this surface approximation [14] we repace the fu wave function y its asymptotic form (in the A x channe) Cc; k f Cc! s ˆ 4 X k f exp i f Cc Š v f Cc Cc m v f i Y m ^r A x Y m ^k f Cc " # S ; k f Cc! r ; k f r Cc 4:13 r r with the radia wave functions ; k f r ˆ r R i 2 u f ; k f r ; 4:14

10 84 S. Type and H. H. Woter where we introduced a cut-off at a suitae radius R in the radia wave functions to eiminate contriutions from the interior incuding the divergence from the irreguar Couom function at sma r. The momentum in the channe and the corresponding veocity v f are determined y m A m x p f 2 ˆ m C m c p f Cc 2 ; 4: Cc i.e., the Cc! S-matrix is taken on-she. The surface approximation can e tested in an actua cacuation, e.g., y a comparison of the resuts otained with a distorted wave approximation for the radia wave function without the radia cut-off and with the asymptotic wave function with proper radia cut-off. In case of incusive deuterium reakup reactions the resuts of a fu cacuation and of the surface approximation were compared and found to e in satisfactory agreement [15]. For an ineastic process with Cc 6ˆ the T-matrix nay assumes the form s T k f Cc ; k f B ; ki Aa ˆ k f Cc k f Cc X m exp i f Cc ŠS Cc! Y m ^k f Cc t m k f ; k f B ; ki Aa with the reduced T-matrix eement 4:16 t m k f ; k f B ; ki Aa ˆ4i h B; k f B ; k f r k f r Y m ^r A x jv x j Aa; k i Aa A a i: In case of the eastic process with Cc ˆ we nd T k f ; k f B ; ˆX ki Aa exp i f ŠY m ^k f m 4:17 S! t m k f ; k f B ; ki Aa t m k f ; k f B ; ki Aa Š: 4:18 The reduced T-matrix eements t, Eq. (4.17), are the essentia ingredients in the THM, since they are needed to extract the two-ody S-matrix S Cc! from the measured three-ody cross section. They have the form of a DWBA T-matrix for the transfer reaction A x a! unound, i.e., for the transfer of x to an unound state in A x with reative momentum p f, Eq. (4.15), mediated y the interaction V x. Here p f is the momentum for the astrophysica energy E f,at which the A x! C c cross section is to e investigated. Thus the reduced T- matrix is the ampitude for preparing the -system at the corresponding energy. Its structure wi ecome cearer in the PWBA in the next section. As seen in Eqs. (4.16) and (4.18) there is no simpe reationship etween the two-ody and three-ody cross sections in genera. The S-matrix eements in Eq. (4.16) are for the inverse process C c! A x, ut oth are reated y unitarity.

11 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 85 In genera, one has interference etween different partia waves in the tripe differentia cross section (2.8), and one has to perform anguar correations etween the different fragments to extract the partia-wave S-matrix eements. The reduced T-matrix eements, depending on three momenta, have to e suppied from theoretica cacuations to the THM. In principe they can e evauated in DWBA, even though they invove six-dimensiona integration, i.e., are of the exact niterange type. The reduced T-matrix eements t are not mode-independent. They depend on assumptions on the distorted waves, the radia cut-offs, the ound-state wave function a, and the V x residua interaction. Thus the THM wi not aow to determine the two-ody S-matrix asoutey. However, it wi give reiay the energy dependence, which is just the important information in the extrapoation to astrophysica energies. 5 Pane-Wave Approximation In order to see the structure of the aove expressions and aso for a comparison to the iterature, we now use the pane-wave approximation. The cacuation of the reduced T-matrix eements (4.17) simpi es consideray, when we repace the distorted waves B; k f B and Aa; k i Aa y pane waves exp ik f B r B and exp ik i Aa r Aa, respectivey. Notice that in this approximation the wave function for the system B sti contains the fu effect of the Couom potentia. We Fourier transform the product of the potentia V x and the ound-state wave function a oth depending on r x d 3 q V x r x a r x ˆ 2 3 W q exp iq r x x ; 5:1 where we have introduced the momentum ampitude W q. A zero-range approximation for the product V x a woud correspond to a constant W. For the reduced T-matrix eement we nd t m Using k f Cc ; k f B ; ki Aa d 3 q ˆ W q d 3 r B d 3 r ; k f r k f r Y m ^r exp i k i Aa r Aa q r x k f B r BŠ : r x ˆ r B r Aa ˆ m A m A m x r ; m m x m r B we can write for the exponentia m x M m A m x m x m r ; exp i k i Aa r Aa q r x k f B r BŠ ˆ exp i q B r B q r Š 5:2 5:3 5:4 5:5

12 86 S. Type and H. H. Woter with the momenta m q B ˆ k i Aa m x m q k f B ; 5:6 m x M q ˆ m A m x m x m ki Aa m A q: 5:7 m A m x Integrating over r B yieds a -function and the q integration ecomes trivia. We otain t m k f Cc ; k f B ; ki Aa ˆ 4W Q B d 3 r ; k f r k f r Y m ^r exp iq Aa r 5:8 with Q B ˆ k f B m k i Aa m x m ; 5:9 Q Aa ˆ k i Aa m A k f B m A m : 5:10 x Here Q B is the difference etween the momentum of in the na channe and the fraction of the initia momentum in the projectie a. Thus it is the recoi of. Simiary Q Aa is the recoi of A. In PWBA the reduced T-matrix is the product of the Fourier transforms of the ound x wave function at Q B with that of the scattering state at Q Aa. Introducing a partia-wave expansion of the pane wave and performing the anguar integration for r we otain t m k f Cc ; k f B ; ki Aa ˆ 4 3 W k f B m k i Aa m x m X s X i m 00 0 m j 00 0 m 0 m 0 j 00 m R 00 0 ; ki Aa ; k f B ; k f Y 00 m 00 ^k i Aa Y 0 m 0 ^k f B 5:11 with the radia integra R 00 0 ; ki Aa ; k f B ; k f 1 ˆ k f 1 dr r j 00 kaa i r j 0 k f B r ; k f r 5:12 0 and the areviation ˆ ma : 5:13 m A m x Due to the parity Cesch-Gordan coef cient 0 0 0j 00 0 some cominations of 00 ; 0, and wi give no contriution to the reduced T-matrix eements. The radia integra over continuum wave functions can convenienty e cacuated with the methods given in ref. [16] y passing to the compex pane for the variae r with a suitae decomposition of the integrand. For sma momenta p f the contriution of the

13 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 87 irreguar Couom wave function in wi dominate and we nd with ref. [17] the very ow energy ehaviour of the radia integra R 00 0 ; ki Aa ; k f B ; k f / i exp f q 2Q k f 1 dr r j 00 kaa i r j 0 k f B r p p 2Q r K Q r 5:14 R with the (constant) inverse Bohr ength Q ˆ f k f ˆ ZAZ x e 2 h 2 : 5:15 In Eq. (5.14) ony the factor in front of the integra depends on k f. For ow energies the exponentia factor increases strongy canceing the decrease of the two-ody S-matrix eements in the three-ody T-matrix eement (4.16). Finay, in case of the ineastic two-ody reaction the resut for the tota threeody T-matrix ecomes s T k f Cc ; k f B ; ki Aa ˆ k f Cc k f W k f B m k i Aa m Cc x m X 2 1 exp i f Cc Š S Cc! X 00 0 R 00 0 ; ki Aa ; k f B ; k f X00 0 ^k f Cc ; ^k f B ; ^k i Aa 5:16 with the anguar distriution function s X 00 0 ^k f Cc ; ^k f B ; ^k i Aa ˆ 4 5=2 i j X m 0 m 0 j 00 m 00 Y 00 m 00 ^k i Aa Y 0 m 0 ^k f B Y m ^k f Cc : m 00m0 5:17 In the eastic case we nd m T k f ; k f B ; ki Aa ˆ W k f B m X k i Aa 2 1 exp i f m x m Š X X 00 0 ^k f ; ^k f B ; ^k i Aa 00 0 S! R 00 0 ; ki Aa ; k f B ; k f R 00 0 ; ki Aa ; k f B ; k f Š: 5:18 The essentia feature of the THM is the occurrence of the factor exp f in the ow-energy approximation (5.14) of the radia integras R 00 0 which compensates the ow-energy suppression of the S-matrix eement S Cc! from the Couom arrier in the channe. Therefore we have no suppression of the cross section in the three-ody reaction [1]. Of course, the energy dependence derived here in the

14 88 S. Type and H. H. Woter PWBA aso appies to the more genera case with distorted waves in the T-matrix eement. Rewriting the pane-wave approximation for the T-matrix T k f Cc ; k f B ; ki Aa ˆW k f B m k i Aa m x m we otain for the three-ody cross section d 3 de f Cc d Cc d B ˆ Aa f B Cc kb k f Cc 2 5 W k f h 6 B m k i Aa m x m k i Aa ~T k f Cc ; k f B ; ki Aa ; 5:19 2 j~t k f Cc ; k f B ; ki Aa j2 ; 5:20 a resut which resemes the PWIA (1.3) in structure. We aso have a product of three factors: (i) a kinematica factor, (ii) a momentum distriution, and (iii) an expression ike a cross section. But there are cear differences as compared to PWIA. The momentum distriution jwj 2 is not simpy the Fourier transform of the ground-state wave function of nuceus a ut the Fourier transform of the product of this wave function with the interaction V x. In case of a nuceus a with an ˆ 0 ground-state wave function for the x reative motion, e.g., 6 Li, jwj 2 wi peak at zero momentum. This means k f B m k i Aa m x m ; i.e., the na momentum of the spectator is just the corresponding mass fraction of the initia momentum of projectie a. Therefore it remains constant during the reaction as assumed in the impuse approximation. In the PWIA the ast factor is directy the two-ody cross section of the A x! C c reaction [9]. Here it is a function, where the S-matrix eements of this reaction enter. This function contains the off-she effects for the two-ody cross section and goes eyond the quasi-free approximation. In particuar it contains an expicit expression for the arrier penetration factors through the radia integras R. Athough we have used the pane-wave approximation for the Aa and B reative motion the Couom effects in the C c scattering are sti fuy incuded. 6 Summary We speci ed approximations invoved in the Trojan-horse method in order to determine the cross section of a two-ody reaction from a reated process with three odies in the na state. The fu expression for the tripe differentia cross section of a three-ody reakup reaction has een derived in a post-form distorted-wave Born description. The reation to the two-ody cross section of interest can e estaished via the S-matrix eements if the surface approximation is appied. In a further panewave Born approximation of the T-matrix eement the three-ody cross section assumes a form simiar to the pane-wave impuse approximation. In our formuation the fu effect of Couom penetration in the reevant two-ody process is incuded as we as off-she effects in the reakup. The Trojan-horse method has

15 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 89 aready een shown to e usefu in the appication with the simpi ed, semiempirica PWIA expressions of Eq. (1.3) [6±8]. For an appication of the Trojanhorse method to a speci c reaction of astrophysica interest various cominations of spectator nuceus and projectie energy with different three-ody na states can e seected. The actua choice depends strongy on the experimenta possiiities for the detection of the partices in the na state. In particuar, the range in reative energy of the two-ody reaction which is accessie in the experiment is connected to projectie energy and the momentum of the spectator in the na state. It shoud e chosen to e cose to the maximum of the momentum distriution so that the quasi-free reaction mechanism dominates. Thus, the appication of the method depends strongy on the experimenta reaization. A Trojan-horse experiment under suitay chosen conditions shoud e performed and the more detaied formuation of the THM given here shoud e appied and discussed in that context. A comparison of the extracted cross section for a two-ody reaction with data from a direct measurement woud e usefu to assess the appicaiity of the Trojan-horse method for nucear astrophysics. Acknowedgement. The authors woud ike to thank M. Aiotta, G. Baur, C. Spitaeri, and A. Tumino for interesting and fruitfu discussions. S.T. appreciates the warm hospitaity during his stays in Catania. This work was supported in part y grant LMWoT from GSI, Darmstadt, and y INFN- LNS, Catania. Appendix Considering partices with spin eads to more compex expressions for the cross sections, wave functions, and matrix eements. In the foowing we give the most important formuae, where we denote the spin of a nuceus j y s j and s jk is the channe spin of the system j k with projection jk. For cross sections we usuay have to sum over a na states and over a initia states in a reaction. In case of a two-ody reaction we cacuate the differentia cross section d! Cc d Cc 1 X X f Cc ˆ j f ; s ;! Cc; s Cc ; Cc j 2 A:1 2s A 1 2s x 1 v s ax s Cc Cc v i from the scattering ampitude f which can e derived from the fu scattering wave function ; k i ; s ; with the system A x in the initia channe. Its asymptotic form reads ; k ; s ;! 4 k i X X s JM X J s s ; k r r Y s JM ;^r Z s JM ^k i A:2 with the functions and Y s JM ;^r ˆX msjjm i Y m ^r s Z s m JM ^k ˆX msjjm Y m ^k ; m A:3 A:4

16 90 S. Type and H. H. Woter which descrie the anguar dependence. The radia wave function J s s ; k r s ˆ exp i i Š v i 2i v f S J s s! u ; k r ss u ; k r Š A:5 contains the (nucear) S-matrix eement S J s s! which depends on the tota anguar momentum J and on the orita anguar momenta and spins in the initia and na channes. The tota scattering ampitude (incuding Couom scattering) is now given y f ; s ;! Cc; s Cc ; Cc p s v i ˆ i k i v f Cc X X p 2 1Y Ccm Cc ^r Cc X Cc m Cc s Cc Cc jjm 0s jjm Ccm Cc JM exp i Cc f Cc Šg S J Ccs Cc s! Cc Cc Cc scc s Š; A:6 and the tota reaction cross section for an ineastic reaction Cc 6ˆ reads 1 X! Cc ˆ 2J 1 X X js 2s A 1 2s x 1 k i 2 J Cc s Cc s! Cc j 2 : A:7 J s s Cc Cc In case of the three-ody reakup the tripe differentia cross section incuding spin degrees of freedom is given y d 3 de f Cc d Cc d B ˆ Aa B Cc 2 5 h 6 f kb k f Cc kaa i 1 X X jt k f Cc 2s A 1 2s a 1 ; k f B ; s B; B ; k i Aa ; s Aa; Aa j 2 ; s Aa Aa s B B where the T-matrix depends on the spins and their projections T k f Cc ; k f B ; s B; B ; k i Aa ; s Aa; Aa ˆ X X s Cc Cc s js B B s A A s a a js Aa Aa s Cc A a h B; k f B Cc; k f Cc ; s Cc Cc s jv x j Aa; k i Aa A s A A a s a a i: Cc X The asymptotic scattering wave function in the C c channe can e written Cc; k f Cc ; s Cc; Cc! s ˆ 4 X X X k f exp i Cc f Cc Š v f Cc Cc s JM Cc v f Y s JM ;^r Z Ccs Cc Cc JM ^k f Cc " S J s Cc s Cc Cc! ; k f r Cc Cc ; k f r # ss Cc r r A:8 A:9 A:10 with the radia wave functions ; k f r ˆ r R i 2 u f ; k f r : A:11

17 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method 91 In an ineastic process with Cc 6ˆ the T-matrix eement assumes the form T k f Cc ; k f B ; s B; B ; k i Aa ; s Aa; Aa s k f X X X ˆ Cc k f exp i Cc f Cc Š SJ s Ccs Cc Cc! X s Cc Cc s js B B Cc J s Cc s Cc Cc X M Z CcsCcCc JM ^k f Cc t JM k f ; k f B ; ; s ; ; k i Aa ; s Aa; Aa A:12 with the reduced T-matrix eement t JM k f ; k f B ; ; s ; ; k i Aa ; s Aa; Aa ˆ 4 X s A A s a a js Aa Aa A a h B; k f B ; k f k f r A s A A a s a a i: r Y s JM ;^r s jv x j Aa; k i Aa In case of the eastic process with Cc ˆ we nd T k f ; k f B ; s B; B ; k i Aa ; s Aa; Aa ˆ X X X exp i 0 f Š X J s s 0 0 s 0 0 s js B B X Z 0 s0 0 JM ^k f 0 M S J s! t JM 0 s0 k f ; k f B ; ; s ; ; k i Aa ; s Aa; Aa t JM k f ; k f B ; ; s ; ; k i Aa ; s Aa; Aa Š: In the pane-wave approximation we introduce the Fourier transform of the product V x r x a s a a ; r x d 3 k ˆ s x x s 0 js a a x s x x s 0 A:13 A:14 A: W k exp ik r x X x 0 and otain for the reduced T-matrix t JM k f Cc ; k f B ; ; s ; Cc ; s ; ; k i Aa ; s Aa; Aa p p ˆ s x s A s Aa s s x s A s 2s 1 2s a 1 s Aa s s a X m s jjm s s js Aa Aa W k f B m k i Aa m m x m X s X 2 i j0 m 0 m 0 jm m 0 m 0 R 0 ; kaa i ; k f B ; k f Y m ^k i Aa Y 0 m 0 ^k f B A:16 after performing the anguar agera. Finay, in case of the ineastic two-ody reaction the resut for the tota three-ody T-matrix ecomes T k f Cc ; k f B ; s B; B ; k i Aa ; s Aa; Aa s k f ˆ Cc k f W k f B m k i Aa m Cc x m X 2J 1 X X exp i Cc f Cc Š SJ s Ccs Cc Cc! J Cc s Cc s X 0 R 0 ; q i Aa ; q f B ; q f XJ0 s Cc s Cc ^k f Cc ; ^k f B ; s B; B ; ^k i Aa ; s Aa; Aa A:17

18 92 S. Type and H. H. Woter with the anguar distriution function X J0 s Ccs Cc ^k f Cc ; ^k f B ; s B; B ; ^k i Aa ; s Aa; Aa ˆ 4 5=2 i j0 1 s x s A s Aa s s Cc s Cc r s p p 2 2s 1 2s a s Aa 1 2s B Cc s x s A s X >< s s s Aa >= 2K 1 s Cc J Cc s Aa s s a K >: >; s B K X s B B Kj m s Aa Aa Kj Cc m Cc X m m Cc X mm 0 m 0 m 0 jm Y m ^k i Aa Y 0 m 0 ^k f B Y Cc m Cc ^k f Cc : A:18 In the eastic case we have T k f ; k f B ; s B; B ; k i Aa ; s Aa; Aa ˆ W k f B m X k i Aa 2J 1 X m x m 0 X 0 J X s s 0 X J0 s 0 s0 ^k f ; ^k f B ; s B; B ; ^k i Aa ; s Aa; Aa exp i 0 f Š S J s! R 0 s0 0 ; q i Aa ; q f B ; q f R 0 ; q i Aa ; q f B ; q f Š: A:19 The pane-wave approximation for the fu T-matrix can again e factorized T k f Cc ; k f B ; s B; B ; k i Aa ; s Aa; Aa ˆ W k f B m k i Aa ~T k f Cc m x m ; k f B ; s B; B ; k i Aa ; s Aa; Aa A:20 with the resut d 3 de f Cc d ˆ Aa f B Cc kb k f Cc Cc d B 2 5 h 6 kaa i W k f B m 2 k i Aa m x m 1 X X j~t k f Cc 2s A 1 2s a 1 ; k f B ; s B; B ; k i Aa ; s Aa; Aa j 2 s Aa Aa s B B for the cross section. A:21 References 1. Baur, G.: Phys. Lett. B178, 135 (1986) 2. Baur, G., Bertuani, C. A., Ree, H.: Nuc. Phys. A458, 188 (1986) 3. Baur, G., Ree, H.: J. Phys. G20, 1 (1994) 4. Baur, G., Ree, H.: Ann. Rev. Nuc. Part. Sci. 46, 321 (1996) 5. Spitaeri, C., Aiotta, M., Figuera, P., Lattuada, M., Pizzone, R. G., Romano, S., Tumino, A., Rofs, C., Giaanea, L., Strieder, F., Cheruini, S., Musumarra, A., Mijanic, D., Type, S., Woter, H. H.: Eur. Phys. J. A7, 181 (2000) 6. Spitaeri, C., Aiotta, A., Cheruini, S., Lattuada, M., Mijanic, D., Romano, S., Soic, N., Zadro, M., Zappaa, R. A.: Phys. Rev. C60, (1999) 7. Cavi, G., Cheruini, S., Lattuada, M., Romano, S., Spitaeri, C., Aiotta, M., Rizzari, G., Sciuto, M., Zappaa, R. A., Kondratyev, V. N., Mijanic, D., Zadro, M., Baur, G., Goryunov, O. Yu., Shvedov, A. A.: Nuc. Phys. A621, 139c (1997)

19 Extraction of Astrophysica Cross Sections in the Trojan-Horse Method Cheruini, S., Kondratyev, V. N., Lattuada, M., Spitaeri, C., Mijanic, D., Zadro, M., Baur, G.: Ap. J. 457, 855 (1996) 9. Jain, M., Roos, P. G., Pugh, H. G., Homgren, H. D.: Nuc. Phys. A153, 49 (1970) 10. Joachain, C. J.: Quantum Coision Theory, Vo. 2, p Amsterdam: North-Hoand Newton, R. G.: Scattering Theory of Partices and Waves, p New York: Springer Ge-Mann, M., Goderger, M. L.: Phys. Rev. 91, 398 (1953) 13. Austern, N.: Direct Nucear Reaction Theories, p New York: Wiey Baur, G., Rose, F., Trautmann, D., Shyam, R.: Phys. Rep. 111, 333 (1984) 15. Kasano, A., Ichimura, M.: Phys. Lett. 115B, 81 (1982) 16. Vincent, C. M., Fortune, H. T.: Phys. Rev. C2, 782 (1970) 17. Aramowitz, M., Stegun, I. A.: Handook of Mathematica Functions. New York: Dover 1970 Received August 31, 1999; revised June 14, 2000; accepted for puication June 30, 2000