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ISSN 1063-7788, Physics of Atomic Nuclei, 2007, Vol. 70, No. 5, pp. 848–854. c Pleiades Publishing, Ltd., 2007. Original Russian Text c L.I. Galanina, N.S. Zelenskaja, A.K. Morzabaev, 2007, published in Yadernaya Fizika, 2007, Vol. 70, No. 5, pp. 882–888. NUCLEI Theory Role of Various Mechanisms in the Formation of a 12C Nucleus in the Reaction 13C(3He, α)12C L. I. Galanina1)* , N. S. Zelenskaja1), and A. K. Morzabaev2) Received March 21, 2006; in final form, October 5, 2006 Abstract—The contributions of various mechanisms to the production of the final nucleus 12 C in the reaction 13 C(3He, α)12 C are estimated. These are neutron stripping and the transfer of the heavy cluster 9Be in the pole approximation or via a sequential transfer of virtual 8Be and a neutron. It is shown that the sequential mechanism of heavy-cluster transfer must be taken into account in order to describe correctly experimental data over the whole angular range. PACS numbers: 24.10.-i DOI: 10.1134/S1063778807050067 In this study, we perform a comprehensive analysis of 1. INTRODUCTION Investigation into mechanisms of nuclear reactions on light nuclei remains one of the urgent challenges in intermediate-energy nuclear physics. In particular, it is of interest to study processes for which it is assumed that they are governed by mechanisms more complex than the one-step transfer of a nucleon 13 3 12 or a group of nucleons. The reaction C( He, α) C + + (0 ; 2 , 4.443 MeV) is one of such processes. In [1, 2], where spin-tensor components of the density 12 matrix for the findal-state nucleus C + in the 2 state at 4.443 MeV were reconstructed at projectile energies of about 7 to 10 MeV per nucleon on the basis of correlation measurements, it was shown that the neutron-pickup mechanism is insuﬃcient for describing the reaction in question. It turned out that the components of rank k = 4 are quite sizable, but they must vanish in the case of the neutron-pickup mechanism, where the angularmomentum transfer is l = 1 [3]. In analyzing the reaction in question, Lebedev et al. [2] used the coupled-channel method, which takes into account 12 the reaction 13C(3He, α)12 C (0+ ; 2+, 4.443 MeV) for the first time, taking into account not only the one-step mechanisms of neutron pickup and the exchange of the heavy cluster 9Be but also the mechanism of the sequential transfer of virtual 8Be and a virtual neutron. The two-step mechanism allowing for the delay of interaction (box diagram) corresponds to the second order of perturbation theory. The role of this mechanism is determined by the structural features of the nuclei involved in the reaction. The ensuing exposition is organized as follows. The general formulas used to calculate the amplitude 13 3 12 + + of the reaction C( He, α) C (0 ; 2 , 4.443 MeV) for various mechanisms are presented in Section 2. In Section 3, we discuss the results of our calculations for the diﬀerential cross section with respect to angles and for the spin-tensor components of the density 12 matrix for the final nucleus C and compare these results with experimental data. Intermediate formulas used to calculate the matrix element for the two-step mechanism corresponding to the box diagram are presented in the Appendix. the collective nature of the C nucleus and, hence, the possibility of multistep (but not diﬀerent-time) nucleon exchange in the formation of this nucleus, whereby they were able to relax the selection rule in the angular-momentum transfer l. Their calculations permitted qualitatively describing experimental data for the forward hemisphere but did not lead to perfect agreement with experimental data. 1) Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, 119992 Russia 2) Gumilev National Eurasian University, Astana, Kazakhstan * E-mail: [email protected] 2. FORMULAS FOR CALCULATING THE PROPERTIES OF THE REACTION 13 3 C( He, α)12C (0+ ; 2+, 4.443 MeV) WITH ALLOWANCE FOR VARIOUS MECHANISMS Figure 1 shows the diagrams illustrating various 13 3 12 + + mechanisms of the reaction C( He, α) C (0 ; 2 , 4.443 MeV). The first of them (Fig. 1a) corresponds to the direct neutron-pickup process, in which case 848 ROLE OF VARIOUS MECHANISMS IN THE FORMATION OF A 12C NUCLEUS (a) 12 (b) α ë α 12 9 n 13 3 ë α He 849 (c) 8 Be ë ë 12ë α 12 n Be 3 ë 13 3 He13ë He Fig. 1. Diagrams illustrating various mechanisms of the reaction 13 C(3 He, α)12 C: (a) neutron-pickup process, (b) exchange of the heavy cluster 9Be in the pole approximation, and (c) two-step transfer of virtual 8 Be and a virtual neutron. l = 1. The last two diagrams illustrate the exchange of 9 the heavy cluster Be either in the pole approx-imation (Fig. 1b) or via the two-step mechanism according to 8 which virtual Be and a virtual neutron are transferred sequentially in time (box diagram in Fig. 1c). There is the problem of assessing the degree to which it is of importance to include the mechanism taking into account delay in the calculation of the matrix 13 3 12 element for the reaction C( He, α) C. The energy diﬀerence ∆E between virtual transferred par-ticles is known to be the main criterion of the realiza-tion of mechanisms taking into account delay that are represented by the box diagram [4]. In our case, ∆E is great both because of the high binding energy ε in the n + 3 ∗ He → α vertex and because of a wide band ∆E of levels 8 of the virtual nucleus Be that are allowed by selection 8 rules. Therefore, the wave functions for the Be cluster and the neutron overlap only slightly, with result that the probability of their transfer to the target nucleus at diﬀerent instants is great. These arguments give reasons to assume a priori that the contributions of the one-step pole mechanism and the mechanism of sequential 13 3 12 transfer in the reaction C( He, α) C are commensurate. The matrix elements for the one-step transfer of particle C (pole diagrams in Figs. 1a and 1b) can be calculated in the distorted-wave Born approximation for a finite interaction range (DWBA–FIR) [3]. We denote the “elementary” and the composite particle in the entrance (exit) channel by x (y) and A = y + C (B = x + C ), respectively. The DWBA–FIR matrix element in terms of the angularmomentum transfers J1 and J2 and the orbital-angularmomentum trans-fer l is then given by M (1) DWBA 2 +2 (2J1 + 1)(2J2 + 1)(−1) Jx Jy −M1−J1 = J1 J2 l involved in the reaction (i = x, A, y, B), Λ1 and Λ2 are the orbital angular momenta of cluster relative motion in the decay vertices for nuclei B and A, βΛ1 Λ2l (θy ) are terms that contain the entire body of information on the reaction mechanism and which depend on the emission angle θy of final particles, and the structural factors Θ Λ1 Λ2 l β (θ )Θ Λ1Λ2 l y J1 J2 _J1 − M1J2M2|lm_, Λ1 Λ2 PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007 Λ1Λ2 l are related to the J1J2 _ _ B→x+C and reduced decay widths of the nuclei (ΘΛ1 A→x+C Θ ). Λ2 The kinematical factors βΛ1 Λ2l (θy ) are determined by the double integral of the invariant form factor F Lx Ly x y (r , r ) and the partial distorted waves Λ1 Λ2 l χLx (kx rx) and χLy (ky ry ) of the initial and final reaction channels with respect to radial variables. The expression for the matrix element M♦ under the assumption of the mechanism of two-step sequential transfer (Fig. 1c) is obtained as the convolution of the matrix elements of the two pole mechanisms that arise upon cutting the complete box diagram. In general, this expression contains additional integrals featuring a variable upper limit and involves additional coherent summation over the angular mo-menta of the intermediate system [4]. For the reac-tion being considered, it is impossible to simplify the general formulas for calculating the relevant matrix elements [5] since the main condition under which simplifications are ∗ possible, ∆E _ ε, does not hold here. Thus, the matrix element M♦ for the two-step 9 mechanism of the transfer of the Be cluster is obtained as the convolution of the amplitudes for the two mechanisms (neutron pickup and the exchange of the 8 × _Jx MxJ1M1|JB MB __Jy My J2M2|JA MA_ × where Ji (Mi ) are the total spins (their projections) of the nuclei heavy cluster Be) with the respective Green’s function. Two independent coordinate spaces arise upon cutting the complete box diagram in Fig. 1c. Figure 2 shows the corresponding pole diagrams. Expanding 850 GALANINA et al. (a) (b) y=α α ëρ B= σ 12 8 n 3 x = He 13 A= ë 12ë 12 ë Be σ α ρ Fig. 2. Pole mechanisms arising upon cutting the complete box diagram. l1 l2 l the Green’s function, we obtain M♦ = −2µρσ ikρσ Λ × _J1 − M1J2M2|lml _θ J1 J2 ∞2 r r drr 0 jΛ 2 _ _ dr M1 M2 _2 _ 0 h 2 × L x j l2 r drM1 Λ M2 Λ , r dr _lml Ly − ml |Lx0_ 0 r + Lx +Ly Lx Ly Λ (2) ∞ i × hΛ l1 Λ Ly l P (θ )I , Ly |ml | y Lx Ly Λ ll l is expressed in where the spectroscopic factor θ 1 2 0 J1J2 where µρσ and kρσ are, respectively, the reduced mass of the virtual system ρ + σ and the momentum of its relative motion, zΛ terms of the DWBA–FIR structural factors as √ l1l2 l θ zΛ and M1 (M2 ) corresponds to the matrix elements of the bottom (top) half of the com-plete box diagram featuring the interaction Vi (Vf ) for the virtual state characterized by an orbital angular moment Λ and a cylindrical function J1J2 = (2l + 1)(2l + 1) 1 (−1)2Jρ +2Jσ +J12 +J22−J2+l+l2+l1 × J11J12 J21 J22 (2J11 + 1)(2J12 + 1)(2J21 + 1)(2J22 + 1) × J describing the relative motion of the particles involved, zΛ (kρσ Λ r) (zΛ = jΛ , h ). Specifically, we have M zΛ x 1 M Λ11 Λ12 l1 _ 3 =d r dΩ zΛ =d r z (k r )Y ∗ (Ω Λρσ Λµ r i Θ 11 J12 J J21J22 The matrix elements for the top and bottom halves of the complete box diagram are calculated within the DWBA–FIR scheme. The corresponding expres-sions are presented in the Appendix. In order to complete the calculation of the matrix element for the process featuring the delay of the interaction, it is necessary to multiply the z z expressions for the matrix elements M1 Λ and M2 Λ and to perform summation over the projections of the intermediate angular momenta in the products of the Clebsch– Gordan coeﬃcients involved. As a result, the matrix element M♦ for the mechanism corresponding to the box diagram with unidirectional arrows is calculated by the formula (−1)2Jx +2Jy −J1−M1 (3) × _Jx MxJ1M1|JB MB __Jy My J2M2|JA MA_ J2 l 22 2 The multidimensional integral ILxLy Λ has the form √ I = ( 1) r dr χ × x Λ kxky x x r _ _ rdr r dr 0 (5) 0 _ (k r ) F Lx Λ Lx x ∞ 4 2π − Lx Ly Λ _ )J (k r ) (r , r x Λ11 Λ12 l1 Λρσ ΛLy ×r dr χ (k r )F y y Ly y y (r, r )h (k r) y Λ21 Λ22 l2 ∞ Λ ρσ r _ _ r dr + 0 r dr χ × x x rdr 0 _ (r , r (k r ) F Lx Λ Lx x x Λ11 Λ12 l1 _ )h (k r ) x Λρσ ΛLy (2J1 + 1)(2J2 + 1) J1J2lml J l1 l 12 × U (Jx J11JB J21 : Jσ J1)U (Jy J22JAJ12 : Jρ J2). f ♦ 21 J1 ry dΩrzΛ (kρσ r)YΛµ(Ωr)Vf Φ∗ (ry ). M J 11 x 3 2 = Λ21Λ22 l2 × ΘJ )V Φ (r ), i (4) 2l + 1 2 r dr χ (k r )F y y Ly y y Λ21 Λ22 l2 (r, ry )jΛ (kρσ r). The spin-tensor components of the density matrix for the final nucleus B produced in the binary reaction × PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007 2 ROLE OF VARIOUS MECHANISMS IN THE FORMATION OF A 1 C NUCLEUS Structural factors for various mechanisms of the reaction 13 3 12 + 851 + C( He, α) C(0 , 2 ) Reaction mechanism J1 J2 Λ1 l Λ2 ΘΛ1Λ2 l J 1J 2 13 Neutron pickup 13 Neutron pickup 12 C→n+ 12 C→n+ + C(2 ) + C(0 ) 8 12 + 12 + Exchange of the heavy cluster Be: α + C(2 ) = α + C(2 ) 8 12 + 12 9 12 + 13 + Exchange of the heavy cluster Be: α + C(0 ) = α + C(0 ) 9 12 + 13 9 Exchange of the heavy cluster Be for C(0 ): C → Be + α with unpolarized particles are given by √ µ µ k xA yB x ρkκ(JB ) = 4π 2 J (1) B y −M − × 2JB + 1 k (2Jx + 1)(2JA + 1) _ JM B _B J M B B − B | 3/2 0 1 1 −0.3424714 1/2 1/2 0 1 1 −0.2635027 2 2 0 2 0 2 0 0 0.20966 2 2 1 0.11483 2 2 4 0.07739 0 2 0 2 0 0 0.47602 0.33028 4 4 0 0.20129 0 Exchange of the heavy cluster Be for C(2 ): C → Be + α 9 1/2 0 0.06425 3 3 1 1 3 3 0 2 1 1 0.05603 5 1 5 2 3 −0.07866 1 1 1 1 0 2 1 1 −0.10461 0.06735 0.18604 mechanism of heavy-cluster stripping and the twostep sequential-transfer mechanism corresponding to (6) the box diagram. It should be noted that no additional normalization factors were introduced in calculating kκ the matrix element for neutron pickup and exchange _ processes. MB MB M (M )M ∗ (M _ ). × if B if 3. CALCULATION OF THE DIFFERENTIAL CROSS SECTION FOR THE REACTION B MxMA My In this case, the normalization factors is chosen _ in such a way that ρ00(JB ) = SpρJB (MB MB ) = dσ/dΩ. Expressions (1) and (3)–(6) make it possible to obtain spin-tensor components of the density 13C(3He, α)12 C AND OF SPIN-TENSOR COMPONENTS OF THE DENSITY MATRIX FOR THE PRODUCT 12C(2+) NUCLEUS AND COMPARISON OF THE RESULTS WITH EXPERIMENTAL DATA matrix for the 12C nucleus produced in the reaction The table summarizes the maximum values that we calculated by formulas (4), (A.4), and (A.6) for pickup mechanism and the mechanism of the ex- the structural factors, assuming various mechanisms 9 13 3 12 + + change of the heavy cluster Be are realized in the of the reaction C( He, α) C (0 , 2 ), such as the neutron-pickup process 13C → n + 12C (0+, 2+), the pole approximation and via the sequential transfer of 13C(3He, α)12 C(0+, 2+), for which the neutron- 8 virtual Be and a virtual neutron, so that n Mif = M + M 9Be ; M 9 Be = M DWBA 9 Be DWBA +M. ♦ exchange of the heavy cluster 13 9 9 Be via the process C → Be + α, and the exchange of the heavy cluster 8Be via the process 12C(0+ , 2+) → 8Be + α. From The matrix elements for neutron pickup and the the table, one can see that these terms are on the same exchange of the heavy cluster 9 Be are summed in- 12 order of magnitude for each state of the final nucleus coherently, since they are associated with diﬀerent C; therefore, we can expect that the contributions of all three mechanisms to the reaction cross section fore, there is no interference between them. In turn, will be sizable. the matrix element for 9Be exchange is determined Figure 3 shows the calculated diﬀerential cross 13 3 12 by the coherent sum of the amplitudes for the pole sections for the reaction C( He, α) C induced by cluster configurations of the initial nucleus; there- PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007 852 GALANINA et al. dσ/dΩ, mb/sr 10 0 10 –1 10 –2 10 0 10 –1 10 –2 12 C (2 ) + 12 C (0 ) + –3 10 –4 10 60 20 100 140 160 θα(c.m.), deg Fig. 3. Diﬀerential cross sections for the reaction 13 C(3 He, α)12 C occurring at EHe = 18 MeV and leading to the formation of a 12 C nucleus in the ground (0+ ) and an excited (2+ ) state according to calculations assuming various reaction mechanisms: (dotted curve) results for the neutron-stripping mechanism, (dashed curve) results for the mechanism involving the transfer of the heavy cluster 9Be within the DWBA–FIR scheme, (dash-dotted curve) results for the transfer of the heavy cluster 9Be with allowance for both the pole and the box diagram, and (solid curve) sum of all of these contributions. The displayed experimental data were borrowed from [1]. 18-MeV projectile helium ions that is considered for the cases where the final nucleus is produced in the + + ground (0 ) and an excited (2 ) state and under the assumption of various reaction mechanisms. It is obvious from this figure that the pole mechanism of neutron stripping is dominant in the forward hemiArb. units 0.2 0.1 0 – 0.1 ⟨T10⟩ ⟨T20⟩ 0.1 0 action C( He, α) C (0 , 2 ). The spin-tensor components of the density matrix make it possible to estimate the orientation of 12 + various multipole moments of the product C (2 ) – 0.1 ⟨T40⟩ 0.1 0 – 0.1 – 0.2 sphere, but that the exchange mechanism of heavycluster transfer determines the cross section in the backward hemisphere. The contribution of the twostep mechanism allowing for delay is significant at all emission angles of alpha particles. Moreover, one can 12 see that, in the case of ground-state C, this mechanism increases the cross section for the exchange process in the forward hemisphere. This makes it possible to improve considerably agreement between the theoretical cross section and experimental data [1] over the whole angular range. In other words, it is necessary to take into account the two-step mechanism of heavy-cluster transfer in order to describe correctly the experimental cross sections for the re13 3 12 + + nucleus with respect to its symmetry axis. In the xyz coordinate frame chosen in such a way that x||kB and z ⊥ 20 60 100 140 160 θα(c.m.), deg Fig. 4. Polarization tensors of the 2+ excited state of the product nucleus 12 C under the assumption of vari-ous reaction mechanisms. The notation for the curves is identical to that in Fig. 3. [kx × ky ], the mean values of the polarization tensors 1 _Tkκ_ = ρ kκ 2JB + 1 ρ00 characterize the spin orientation along the quantization axis and in the plane orthogonal to the reaction plane [6]. The polarization tensors of lowest rank have a clear physical meaning: the tensor _T10_ is PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007 ROLE OF VARIOUS MECHANISMS IN THE FORMATION OF A 12C NUCLEUS proportional to the polarization vector and characterizes the induced polarization of the system in the plane orthogonal to the reaction plane. Apart from normalization, the tensors _T20_ and _T22_ coincide with, respectively, the longitudinal and the transverse component of the alignment tensor. Figure 4 presents the contributions of various reaction mechanisms to some of the polarization ten+ 12 sors for the 2 excited state of the C nucleus. From this figure, one can see that the pole mechanism of neutron stripping is important for the emergence of the induced polarization of the product nucleus. Po-larization tensors of higher rank are determined by the mechanism of heavy-cluster exchange, the influence of the sequentialtransfer mechanism being the most significant in the region of intermediate angles. APPENDIX CALCULATION OF THE MATRIX ELEMENTS FOR THE TOP AND BOTTOM PARTS OF THE BOX DIAGRAM In order to systematize the notation, all interme-diate momenta over which summation is performed are labeled with indices “1n” or simply “1” for the diagram in Fig. 2a and “2n” or simply “2” for the diagram in Fig. 2b. z Let us represent the matrix element M1 Λ ≡ M1 corresponding to the diagram in Fig. 2a in a form analogous to that in (1); that is, j 3 12 + = MΛ Thus, we have used here diﬀerent features of the 13 853 (A.1) (2J11 + 1)(2J12 + 1) 1 J11J12 l1 + reaction C( He, α) C (0 ; 2 , 4.443 MeV) (dif-ferential cross sections for elastic and inelastic scat-tering, spin-tensor components of the density matrix, induced vector and tensor polarizations) to demon-strate the role of various mechanisms in the produc-tion of the final nucleus. The results indicating the importance of taking into account the mechanism of sequential virtual-cluster transfer (box diagram) are the most interesting here. The contribution of this mechanism proved to be sizable even in calculating the cross section for elastic and inelastic scattering; moreover, the inclusion of the two-step mechanism even leads to qualitative changes in polarization prop-erties, especially in the region of intermediate emis-sion angles of alpha particles. This circumstance emphasizes once again that the mechanism of nuclear reactions involving alpha particles is nontrivial, radically diﬀering from the pole mechanisms of stripping or pickup. Moreover, the inclusion of exchange processes requires considering second-order two-step eﬀects in addition to the mechanism of direct heavycluster transfer. The contribution of these eﬀects is determined by the structural features of the nuclei involved in the reaction. In the reaction considered here, these structural features are such (see Introduction) that the role of all mechanisms examined in this study proved to be significant, so that only by combining these mechanisms were we able to describe 13 3 12 + The work was supported by a grant from the Presi-dent of the Russian Federation for Support of Leading Scientific Schools (no. 5365.2006.2) and a grant from the Scientific and Technological Program Universi-ties of Russia (no. 02.02.506). PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007 J11J12 Λ11 Λ12 × _JxMxJ11 M11|Jσ Mσ __Jρ Mρ J12M12|JA MA_ × _J11 − M11J12M12|l1m1_. The expression for the kinematical factor βΛ11 Λ12 l1 can be obtained by introducing the invariant form factor F _ Lx Λ (r , rx ) and performing integration with Λ11 Λ12 l1 respect to the angular variables. We have 1 4π β = µ kr Λ11 Λ12 l1 xx 2l1 + 2 (−1) √ (A.2) i _LxµxΛ − µ|l1 − m1_YL∗x µx (kx) Lx × Lx µx r dr χ × x (k r ) F Lx Λ Lx x x x _ (r , r _ )j (k r ). x Λ11 Λ12 l1 Λρσ Substituting (A.2) into (A.1), we find for the matrix element M jΛ that 1 j 4π MΛ= (A.3) ( 1)2Jx +2Jρ −M11 −J11 − kx J11 J12l1 1 Λ11 Λ12 l1 (2J + 1)(2J + 1)Θ × 11 12 J11J12 × _JxMxJ11 M11|Jσ Mσ __Jρ Mρ J12M12|JA MA_ × _J11 − M11J12M12|l1m1_ µ Lx (−1) i _LxµxΛ − µ|l1 − m1_YL∗x µx (kx) × ACKNOWLEDGMENTS ΘΛ11Λ12 l1 Λ12 l1 Λ11 + the properties of the reaction C( He, α) C (0 ; 2 , 4.443 MeV). β −M11 ( 1)2Jx +2Jρ −J11 ×− Lx µx r dr χ × x x (k r ) F Lx Λ Lx x x Λ11 Λ12 l1 _ (r , r _ )j (k r ). x Λρσ In calculating the spectroscopic factor, we consider that a neutron plays the role of the transferred particle C . The corresponding expression then takes the 854 GALANINA et al. form ement of the top half of the box diagram (Fig. 2b), which corresponds to the exchange of the heavy clusΛ11Λ12 l1 (2J11 + 1)(2J12 + 1)(2l1 + 1) = Θ J11J12 2Λ 11 8 ter +1 Be, is written as (A.4) M h Λ k ( 1)−Jx −2Jρ +JC +J12 +J11+Sσ +l1+Λ11 2 − × 4π = ρ J A C 11 Λ12 SC J12 L ΛJl S A (2J21 + 1)(2J22 + 1) J21 J22 J 12121 ×_ _ | i− y _ΛµLy µy |l2 − m2_YLy µy (ky ) L × J J J Ly µ y σ 11 x S S C Λ , σ r dr × 11 y Λ21 Λ22 l2 Θ L = (−1) yy ) h ( k r) . (r , r Λ21 Λ22 l2 y Λρσ , we consider that x, y ≤ α (Lx = Ly = 0, Jx = Sx, Jy = Sy ). We then have +L B y _ (k r )F ΛLy Ly J21 J22 In just the same way as in (A.3), the matrix el- Λ21 Λ22 l2 χ In the expression for the spectroscopic factor where Li and Si (i = A, B, C, σ, ρ) are the orbital angular momenta and spins of the nuclei involved. J21 J22 __ | A Θ JB MB Jy My J22 M22 Jρ Mρ × _J21 − M21J22M22|l1m1_ J Λ S ρ 11 × Θ Jσ Mσ J21M21 × ΘΛ Θ Λ 12 11 ρ J21 J22l2 − y × _ A→C+ρ _ σ→C+x S (A.5) 1)2Jσ +2Jy −M21 −J21 Λ21 Λ22 l2 × (2LA + 1)(2SA + 1)(2Lσ + 1)(2Sσ + 1)(2Jρ + 1) L ( +L A J C − S +Λ 21 − C _ B→x+C _ A→y+C +Λ 22 21 (A.6) Θ Θ Λ21 Λ22 LL SA LB SB LC SC × (2LB + 1)(2SB + 1)(2LA + 1)(2SA + 1)(2l2 + 1)(J21 + 1)(2J22 + 1) S L J × C B 21 J B Sx SB SC L J A 22 J A Sy SA Expressions (A.3)–(A.6) solve completely the problem of calculating the matrix elements for the top and bottom halves of the box diagram. REFERENCES 1. O. I. Vasil’eva et al., Izv. Akad. Nauk SSSR, Ser. Fiz. 48, 1959 (1984). 2. V. M. Lebedev, N. V. Orlova, and A. V. Spassky, Yad. Fiz. 62, 1546 (1999) [Phys. At. Nucl. 62, 1455 (1999)]. 3. N. S. Zelenskaya and I. B. Teplov, Features of Excited Nuclear States and Angular Correlations in LB l2 LA Λ L Λ 22 C 21 LB l2 LA J S J 22 . C 21 ´ Nuclear Reactions (Energoatomizdat, Moscow, 1995) [in Russian]. 4. N. S. Zelenskaya, Yad. Fiz. 13, 734 (1971) [Sov. J. Nucl. Phys. 13, 417 (1971)]. 5. L. I. Galanina and N. S. Zelenskaya, Izv. Akad. Nauk, Ser. Fiz. 69, 1741 (2005). 6. L. I. Galanina and N. S. Zelenskaya, Izv. Akad. Nauk, Ser. Fiz. 63, 47 (1999). Translated by E. Kozlovsky PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 5 2007