Generalizing the calculable R-matrix theory and eigenvector continuation to the incoming wave boundary condition
GGeneralizing the calculable R -matrix theory and eigenvector continuation to theincoming wave boundary condition Dong Bai ∗ and Zhongzhou Ren
1, 2, † School of Physics Science and Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Advanced Micro-Structure Materials, Ministry of Education, Shanghai 200092, China
The calculable R -matrix theory has been formulated successfully for regular boundary conditionswith vanishing radial wave functions at the coordinate origins [P. Descouvemont and D. Baye, Rept.Prog. Phys. , 036301 (2010)]. We generalize the calculable R -matrix theory to the incoming waveboundary condition (IWBC), which is widely used in theoretical studies of low-energy heavy-ionfusion reactions to simulate the strong absorption of incoming flux inside the Coulomb barriers. Thegeneralized calculable R -matrix theory also provides a natural starting point to extend eigenvectorcontinuation (EC) [D. Frame et al. , Phys. Rev. Lett. , 032501 (2018)] to fusion observables.The N + C fusion reaction is taken as an example to validate these new theoretical tools. Bothlocal and nonlocal potentials are considered in numerical calculations. Our generalizations of thecalculable R -matrix theory and EC are found to work well for IWBC. I. INTRODUCTION
The calculable R -matrix theory provides a powerfulframework to solve the Schr¨odinger equations in regu-lar boundary conditions [1–4], where radial wave func-tions vanish at the coordinate origins and can take dif-ferent asymptotic forms in different problems. The cal-culable R -matrix theory divides the configuration spaceinto internal and external regions by the channel radius.Compared to the internal region, physics in the exter-nal region gets simplified thanks to the negligibility ofshort-range interactions therein, and external wave func-tions are known explicitly up to a few coefficients. TheBloch operator is often adopted to match internal andexternal wave functions continuously at the channel ra-dius [5]. The resultant Bloch-Schr¨odinger equations arethen solved by, e.g., the variational method. In nuclearphysics, the calculable R -matrix theory has been usedsuccessfully to study bound states, resonant states, elas-tic/inelastic scatterings, transfer reactions, breakup re-actions, and fusion reactions. See Refs. [6–13] for somerecent works.The incoming wave boundary condition (IWBC) playsa fundamental role in modern theoretical studies of low-energy heavy-ion fusion reactions [14–16]. Consider theSchr¨odinger equation (cid:20) − µ d d r + L ( L + 1)2 µr + V C ( r ) − E (cid:21) χ L ( r )= − V N ( r ) χ L ( r ) , (cid:82) d r (cid:48) W ( L ) N ( r, r (cid:48) ) χ L ( r (cid:48) ) , (1)with L being the orbital angular momentum, µ being thetwo-body reduced mass, E > ∗ [email protected] † Corresponding author: [email protected] in the center-of-mass (CM) frame, V C ( r ) = Z P Z T e /r being the Coulomb potential, V N ( r ) and W ( L ) N ( r, r (cid:48) ) be-ing the local and nonlocal nuclear potentials, and χ L ( r )being the radial wave function. IWBC is given by χ L ( r ) ∼ T L ( E ) exp (cid:20) − i (cid:90) rr abs d r (cid:48) k L ( r (cid:48) ) (cid:21) , ≤ r ≤ r abs , (2)= H ( − ) L ( η, kr ) − S L ( E ) H (+) L ( η, kr ) , r ≥ r c . (3)Here, r abs is the absorption radius inside the Coulombbarrier, r c is the channel radius chosen to be so largethat the nuclear interaction and antisymmetrization be-tween the target and the projectile become negligible inthe external region, k = √ µE is the relative momentumat r → ∞ , k L ( r ) is the relative momentum at r ≤ r abs , η is the Sommerfeld parameter, H ( ∓ ) L ( η, kr ) are the incom-ing/outgoing Coulomb-Hankel functions, and S L ( E ) and T L ( E ) are the S - and transmission matrix elements. Therelative momentum k L ( r ) could be estimated by k L ( r )= (cid:114) µ (cid:104) E − L ( L +1)2 µr − V N ( r ) − V C ( r ) (cid:105) for the local poten-tial and k L ( r ) = (cid:114) µ (cid:104) E − L ( L +1)2 µr − W LE N ( r ) − V C ( r ) (cid:105) forthe nonlocal potential, with W LE N being some local equiv-alence of the nonlocal potential (see Section III). IWBCis different from regular boundary conditions which im-pose χ L ( r ) = 0 at r = 0. It is widely used in nuclearfusion problems to simulate the strong absorption of in-coming flux inside the Coulomb barriers and has becomeone of the standard ansatzes to calculate fusion observ-ables. Compared with the regular-boundary-conditionapproach to nuclear fusion reactions, IWBC is often re-garded as more predictive in the sense that no extra imag-inary optical potential is needed.In this work, the generalized calculable R -matrix the-ory is proposed to solve the Schr¨odinger equations inIWBC. Besides the academic interest to enlarge the ap-plicable scope of the calculable R -matrix theory, it pro-vides a unified framework for both local and nonlocalpotentials in nuclear fusion studies. Nonlocal potentials a r X i v : . [ nu c l - t h ] J a n r c r abs r V N ( r )+ V C ( r ) V N ( r )+ V C ( r ) V C ( r ) χ L int ( r ) χ L ext ( r ) ∼ T L ( E ) e - i ∫ r abs r ⅆ r ' k L ( r ' ) χ L ext ( r ) = H L (-) ( η ,kr )- S L ( E ) H L (+) ( η ,kr ) Internal Region External RegionAbsorption Region
FIG. 1. The trisection of the configuration space in the generalized calculable R -matrix theory for IWBC. In Absorption andInternal Regions, both the nuclear and Coulomb potentials are sizable, while in External Region, only the Coulomb potential issizable. The nuclear potential is taken to be the local potential V N ( r ) for simplicity. The trisection of the configuration spaceworks similarly for nonlocal potentials. have important applications in nuclear physics datingback to Perey and Buck in the 1960s [17] and get revivedin recent years [18–26]. Nucleus-nucleus nuclear interac-tions are intrinsically nonlocal in coordinate space thanksto the antisymmetrization effect and model-space trun-cations [27, 28]. Moreover, modern realistic interactionsof nucleons based on chiral effective field theory are gen-erally nonlocal [29–31], which might also contribute tononlocality in nucleus-nucleus potentials. Nonlocal po-tentials have been adopted by some authors to study low-energy heavy-ion fusion reactions. Refs. [32–35] studyfusion problems with nonlocal potentials in the frame-work of the WKB approximation. Refs. [36, 37] solvethe Schr¨odinger equations with nonlocal complex opti-cal potentials in regular boundary conditions. As faras we know, there is few publication on how to solvethe Schr¨odinger equations with nonlocal potentials andIWBC exactly. The popular implementation of IWBCin the CCFULL code [14] is based on the modified Nu-merov method [38] and deals with local potentials only.It is not easy to extend the method to nonlocal potentials.Our generalized R -matrix theory fills this gap and givesa valuable opportunity to study the impacts of nonlocalpotentials in nuclear fusion reactions.The generalized calculable R -matrix theory also pro- vides a natural starting point to extend eigenvector con-tinuation (EC) [39], a variational emulator for bound-state observables, to fusion observables. EC is character-ized by choosing basis functions from the Hamiltonianeigenstates at selected training points in the parameterspace. It has been shown to be a reliable and efficienttool in uncertainty quantification and global sensitiv-ity analysis [39–44], where model evaluations, sometimescomputationally expensive, have to be carried out for alarge number of times at different points in the parame-ter space. An on-going direction of EC is to extend themethod to reaction observables. Recently, some progresshas been made for scattering observables via the Kohnvariational principle [45]. Generally speaking, the calcu-lable R-matrix theory and its generalizations provide acomprehensive roadmap to extend EC from bound-stateobservables to other structural and reaction observables,not limited to fusion observables. Here, we just focus onfusion observables for concreteness.The rest parts are organized as follows. In Section II,our generalizations of the calculable R -matrix theory andEC are presented. In Section III, numerical reliability ofthese methods is examined in detail by studying the N+ C fusion reaction. Conclusions are given in Section IV.The natural units (cid:126) = c = 1 are adopted in this work. II. THEORETICAL FORMALISMA. The Generalized Calculable R -Matrix Theory We divide the configuration space into three parts, “Absorption Region” [0 , r abs ], “Internal Region” [ r abs , r c ], and“External Region” [ r c , ∞ ). See Fig. 1 for an illustration. In Absorption and Internal Regions, both the nuclearpotential V N ( r ) and the Coulomb potential V C ( r ) are sizable, while in External Region the nuclear potential becomesnegligible and only the Coulomb potential predominates. External wave functions in Absorption and External Regionsare given by IWBC in Eqs. (2)-(3), which should be matched continuously to internal wave functions in Internal Region.In the calculable R -matrix theory, the Schr¨odinger equation in Eq. (1) is promoted to the Bloch-Schr¨odinger equation (cid:20) − µ d d r + L ( L + 1)2 µr + L ( r c ) − L ( r abs ) + V C ( r ) − E (cid:21) χ int L ( r ) = [ L ( r c ) − L ( r abs )] χ ext L ( r ) − V N ( r ) χ int L ( r ) , (cid:82) r c r abs d r (cid:48) W ( L ) N ( r, r (cid:48) ) χ int L ( r (cid:48) ) , (4)with L ( R ) = µ δ ( r − R ) dd r being the Bloch operator. It is easy to verify that − µ d d r + L ( r c ) − L ( r abs ) is hermitianin Internal Region [ r abs , r c ]. Besides, the continuity of wave functions imposes χ int L ( r abs ) = χ ext L ( r abs ) and χ int L ( r c ) = χ ext L ( r c ). The Bloch-Schr¨odinger equation in Eq. (4) could be solved by the variational method with χ int L ( r ) = (cid:80) Nn =1 c n ( E ) ϕ n ( r ). The transmission matrix element T L ( E ) and the S -matrix element S L ( E ) are obtained by solvingthe linear equations N (cid:88) n =1 C mn ( E ) c n ( E ) = 12 µ (cid:110) kϕ m ( r c ) (cid:104) H ( − ) (cid:48) L ( η, kr c ) − S L ( E ) H (+) (cid:48) L ( η, kr c ) (cid:105) + iT L ( E ) k L ( r abs ) ϕ m ( r abs ) (cid:111) , (5) N (cid:88) n =1 ϕ n ( r abs ) c n ( E ) = T L ( E ) , (6) N (cid:88) n =1 ϕ n ( r c ) c n ( E ) = H ( − ) L ( η, kr c ) − S L ( E ) H (+) L ( η, kr c ) , (7)with C mn ( E ) = (cid:18) ϕ m (cid:12)(cid:12)(cid:12)(cid:12) − µ d d r + L ( L + 1)2 µr + L ( r c ) − L ( r abs ) + V C ( r ) − E (cid:12)(cid:12)(cid:12)(cid:12) ϕ n (cid:19) + ( ϕ m | V N ( r ) | ϕ n ) , (cid:16) ϕ m (cid:12)(cid:12)(cid:12) W ( L ) N ( r, r (cid:48) ) (cid:12)(cid:12)(cid:12) ϕ n (cid:17) . (8)Here, the round bracket denotes the inner product over Internal Region [ r abs , r c ], i.e., ( φ |O| ψ ) = (cid:82) r c r abs d r φ ( r ) O ( r ) ψ ( r ).In the calculable R -matrix theory, it is often convenient to use the Lagrange functions { L Ni ( r ) } [46] as varia-tional basis functions (see Appendix A). Noticeably, the Lagrange functions used here are a bit different fromthe ordinary Lagrange functions L Ni ( r ) ∝ r − r abs defined in the same interval [ r abs , r c ]. The latter always satisfy L Ni ( r ) = ( − N + i r − r abs ∆ rx i (cid:112) x i (1 − x i )∆ r P N [(2 r − r abs − r c ) / ∆ r ] r − x i ∆ r − r abs , with P N ( x ) being the Legendre polynomial of order N and∆ r = r c − r abs . This means that any finite combination of the ordinary Lagrange functions becomes exactly zero atthe absorption radius and violates IWBC by construction. As a result, the matching between the wave functions inAbsorption and Internal Regions cannot be handled properly with the ordinary Lagrange functions. In comparison,the Lagrange functions defined in Appendix A give nonzero values at the absorption radius and thus are better suitedto matching the wave functions in Absorption and Internal Regions. The numerical calculations in Section III showexplicitly that the Lagrange functions defined in Appendix A are suitable for our purpose. There are other choices forthe variational basis functions. For example, we have checked that the Gaussian basis functions ϕ n ( r ) = exp( − ν n r )could be used as the basis functions as well. However, the corresponding matrix elements have to be calculated bynumerical integration, except in some specific cases. This makes the Gaussian basis functions numerically less friendlythan the Lagrange functions. As shown in Appendix A, all the relevant matrix elements could be calculated analyti-cally with the Lagrange functions, which is an important advantage in numerical calculations. The above theoreticalformalism might turn out to be a bit similar to the R -matrix propagation method [47], but it has different theoreticalmotivations and application scenarios. With the transmission matrix element T L ( E ), the fusion cross section is givenby σ fus ( E ) = L max (cid:88) L =0 σ L ( E ) = πk L max (cid:88) L =0 (2 L + 1) P L ( E ) , (9) P L ( E ) = k L ( r abs ) k | T L ( E ) | , (10)with L max being the maximal partial-wave angular momentum taken into account. The resultant uncertainties arereferred to as truncation errors. B. The Generalized Eigenvector Continuation
The Bloch-Schr¨odinger equation in Eq. (4) could berewritten schematically as H E ( α ) χ int L ( α ) = L χ ext L ( α ) . (11) Here, H E ( α ) and L stand for the operators acting onthe internal and external wave functions, with α being ExpLocalNonlocal3 4 5 6 7 8 9 1010 - ( MeV ) σ f u s ( m b ) ( a ) N = = = - - - ( MeV ) | R e l a t i v e E rr o r s | ( b ) Local N = = = - - - -
10 E ( MeV ) | R e l a t i v e E rr o r s | ( c ) Nonlocal
FIG. 2. (a) Theoretical fusion cross sections of the N + C fusion reaction at different reaction energies in the CM framegiven by the generalized calculable R -matrix theory for the local potential in Eq. (13) (solid red line) and the nonlocal potentialin Eq. (14) (dashed pink line). The parameters are given by V = 60 MeV, R = 1 . A / T + A / P ) fm, a = 0 . N ≥
20 and N ≥
100 Lagrange functions are used in the numerical calculations for local and nonlocal potentials,respectively. The dark red points are the experimental data from Ref. [49]. (b) Absolute values of the relative errors of thefusion cross sections given by the generalized calculable R -matrix theory with the local potential. The benchmark values tocalculate relative errors are given by the generalized calculable R -matrix theory with N = 200 Lagrange functions. (c) Absolutevalues of the relative errors of the fusion cross sections given by the generalized calculable R -matrix theory with the nonlocalpotential. N = 400 Lagrange functions are used to calculate the benchmark values. the model parameters. The subscript “ E ” stresses thatthe operator H E ( α ) depends on the reaction energy E .Let { χ int L ( α tr i ) } be the exact internal wave functions ofthe Bloch-Schr¨odinger equations at the training points { α tr i } in the parameter space. Following the philosophyof EC, we construct the variational emulator χ int L ( α te (cid:12) )for the test point α te (cid:12) χ int L ( α te (cid:12) ) = N EC (cid:88) i =1 c i χ int L ( α tr i ) . (12)The coefficients { c i } and fusion observables are obtainedby solving Eqs. (5)-(7) with the EC basis functions { χ int L ( α tr i ) } . III. NUMERICAL RESULTS
We take the N + C fusion reaction as a proof-of-concept example to test our generalizations of the calcu-lable R -matrix theory and EC. Both local and nonlocalpotentials are used to describe nuclear interactions be-tween N and C. The local nuclear potential V N ( r ) is taken to be the Woods-Saxon form V N ( r ) = − V r − R ) /a ] , (13)while the nonlocal nuclear potential W N ( r , r (cid:48) ) is takento be the Perey-Buck form [17] W N ( r , r (cid:48) ) = V N [( r + r (cid:48) ) /
2] exp[ − ( r − r (cid:48) ) /β ] π / β . (14)Here, β = β /A red denotes the range of nonlocality [48],with β = 0 .
85 fm being the range of nonlocality fromneutron scattering [17] and A red being the reduced massnumber in fusion problems. Expanding W N ( r , r (cid:48) ) in par-tial waves, we have W ( L ) N ( r, r (cid:48) ) = V N [( r + r (cid:48) ) / × rr (cid:48) exp[ − ( r + r (cid:48) ) /β ] π / β i L (cid:18) β rr (cid:48) (cid:19) . (15) i L ( z ) is the modified spherical Bessel function of the firstkind. The relative momentum k L ( r ) in Eq. (2) is es-timated by k L ( r ) = (cid:114) µ (cid:104) E − L ( L +1)2 µr − V N ( r ) − V C ( r ) (cid:105) ( MeV ) | R e l a t i v e E rr o r s |
40 6080 100 120 140160 180200220 240 ( a ) N EC = - - - E ( MeV ) | R e l a t i v e E rr o r s |
40 60 80 100120 140 160180200 220240 ( b ) N EC = FIG. 3. Absolute values of the relative errors of the fusion cross sections in the N + C fusion reaction given by thegeneralized EC. The EC basis functions are taken to be the exact internal wave functions at (a) N EC = 6 training points V = 45 , , · · · , ,
145 MeV and (b) N EC = 10 training points V = 45 , , · · · , ,
135 MeV. The test points are taken tobe V = 40 , , · · · , ,
240 MeV. X is the abbreviation of V = X MeV. N =
20 N =
50 N = =
200 N =
400 N =
50 100 500 10000.11101001000 r c ( fm ) σ f u s ( m b ) ( a ) Generalized R - Matrix N EC = N EC =
50 100 500 100010 - - - - - r c ( fm ) | R e l a t i v e E rr o r s | ( b ) Generalized EC
FIG. 4. Theoretical results for the local potential in Eqs. (13) and (16) at different channel radii between 20 fm and 1000 fm:(a) the fusion cross sections given by the generalized R -matrix theory with N = 20 , , , , ,
800 Lagrange functionsand (b) absolute values of the relative errors of fusion cross sections given by the generalized EC with N EC = 6 ,
10 EC basisfunctions at the same training points as Fig. 3. and k L ( r ) = (cid:114) µ (cid:104) E − L ( L +1)2 µr − W LE N ( r ) − V C ( r ) (cid:105) for lo-cal and nonlocal potentials, respectively. W LE N ( r ) =exp (cid:110) − µβ (cid:2) E − V C ( r ) − W LE N ( r ) (cid:3)(cid:111) V N ( r ) is the localequivalence of the Perey-Buck potential W N ( r , r (cid:48) ) [17].In numerical calculations, we take the absorption radiusto be r abs = A / P + A / T for all the partial waves. This isslightly different from the CCFULL’s convention where r abs is taken to be the local minimum of the total two-body potential (including the centrifugal potential) insidethe Coulomb barrier. We have verified explicitly thatthese two choices give fusion cross sections numericallyclose to each other. The maximal partial-wave angularmomentum is taken to be L max = 10. Such a truncationgives rise to the truncation errors in theoretical results.It is important to distinguish them from numerical er-rors from solving the Bloch-Schr¨odinger equations. Ournumerical codes are written by using arbitrary-precisionarithmetic. They inevitably become less efficient than numerical codes written in double precision. But, theyallow us to handle the so-called “ill-conditioned” matricesin a more straightforward way, without introducing ex-tra nuggets by hand to regularizing these matrices [45].It thus helps us better understand our new theoreticaltools.We first calculate the fusion cross sections of the N + C fusion reaction by using the generalized calculable R -matrix theory on a Lagrange mesh. We take V = 60 MeV , R = 1 . A / T + A / P ) fm , a = 0 . , (16)for both the local and nonlocal potentials. The channelradius is taken to be r c = 20 fm unless otherwise men-tioned. In Fig. 2(a), the fusion cross sections are pre-sented for both the local and nonlocal potentials, alongwith experimental data from Ref. [49]. For the local andnonlocal potentials, we take the numbers of Lagrangefunctions to be N ≥
20 and N ≥ β ∼ .
13 fm used in the nonlocal potential, whichsuppresses nonlocality significantly and makes the nonlo-cal potential nearly diagonal in r - r (cid:48) space. Such a “sin-gular” nonlocal potential might be less interesting fromthe physical viewpoint. But, it turns out to be a techni-cal challenge for numerical calculations and provides anideal playground to test the robustness of the general-ized calculable R -matrix theory, as well as our numericalimplementations in arbitrary precision. It is found thatmore than 100 Lagrange functions are needed for numeri-cal convergence in the case of the nonlocal potential, andour arbitrary-precision codes give the reliable numericalresults. We also test our numerical codes for mild non-local potentials with larger ranges of nonlocality. It isfound that much fewer Lagrange functions are needed formoderate precision goals in these cases. In Figs. 2(b) and2(c), we calculate relative errors = (theoretical results − benchmark results) / benchmark results of the generalizedcalculable R -matrix theory for different numbers of La-grange functions. For the local potential, it is foundthat the relative errors of the N = 20 , ,
100 resultsare around 0 .
01, 10 − , and 10 − , respectively, with thebenchmark results taken at N = 200. Similarly, for thenonlocal potential, it is found that the relative errors ofthe N = 100 , ,
300 results are around 0 .
01, 10 − , and10 − , respectively, with the benchmark results taken at N = 400. These results show that the generalized calcu-lable R -matrix theory generally gets convergent quicklywith respect to the increasing numbers of Lagrange func-tions. We would like to emphasize that the relative errorshere are the numerical errors from solving the Bloch-Schr¨odinger equations and do not include the truncationerrors from the partial-wave truncations. The latter arefound to be around 10 − -10 − for E ∈ [3 ,
10] MeV.We then explore the parameter space of the N + Cfusion reaction with the generalized EC. EC and its gen-eralizations have been shown to be useful for quantifyingtheoretical uncertainties and analyzing global sensitiv-ity in bound-state and scattering problems. Our gen-eralized EC aims to extend the method to low-energyheavy-ion fusion reactions. The local potentials in theWoods-Saxon form are adopted to validate the general-ized EC. Numerical calculations with the nonlocal po-tentials in the Perey-Buck form are similar but moretime-consuming. We treat V as the free parameterand take its value between 40 MeV and 240 MeV. Twosets of training points are examined, i.e., Training Set(a) with six training points at V = 45 , , · · · , , V = 45 , , · · · , ,
135 MeV. In Fig. 3, we plot theabsolute values of the relative errors of the fusion crosssections given by the generalized EC at the test points V = 40 , , · · · , ,
240 MeV. In Fig. 3(a), the rela-tive errors of the fusion cross sections are found to be ∼ − -0 . V = 60 , · · · , ,
120 MeV are muchsmaller than those at the extrapolating points V =40 , , · · · , ,
240 MeV. In other words, the general-ized EC shows better performance for interpolation thanextrapolation. In Fig. 3(b), we increase the EC basissize from N EC = 6 to N EC = 10. The relative errorsget decreased systematically for all the test points andare found to be around 10 − to 10 − . Once again, thegeneralized EC gives better results for the interpolatingpoints.We also calculate the fusion cross sections at differ-ent channel radii. The local potential in Eqs. (13) and(16) is used. For concreteness, we take the reaction en-ergy E = 5 MeV and the channel radius r c between 20fm and 1000 fm. A channel radius as large as 1000 fmis certainly not necessary for the N + C fusion re-action. But it gives a valuable chance to explore theadvantages and limitations of our generalizations of thecalculable R -matrix theory and EC. Also, there are im-portant cases where channel radii as large as ∼ R -matrix theorywith N = 20 , , , , ,
800 Lagrange functions atthe channel radii r c ∈ [20 , r c = 20 , , N EC = 6 ,
10 training points asFig. 3. The EC basis functions { χ int L ( α tr i ) } are taken tobe the exact solutions of the Bloch-Schr¨odinger equationsat each channel radius, and the relative errors of fusioncross sections from the generalized EC are found to beabout 2 × − and 2 × − for N EC = 6 and 10, respec-tively, which remain stable over the whole channel-radiusinterval [20 , R -matrix theory. This contributes to the higher numer-ical efficiency of the generalized EC in scanning the pa-rameter space than the generalized calculable R -matrixtheory, especially at large channel radii. Also, it is in-teresting to note that in our cases the relations betweenthe internal wave functions and the EC basis functions(i.e., Eq. (12)) learned at the small channel radii remainvalid approximately at the large channel radii. Explicitcalculations show that { c i } at different channel radii areindeed numerically close to each other. In other words,the generalized EC works out universal relations amongthe internal wave functions at different points in the pa-rameter space. These relations remain valid even whenthe channel radius is enlarged from 20 fm to 1000 fmand the internal region is inflated by a factor of morethan 50 along the radial direction. This is drasticallydifferent from the generalized R -matrix theory + the La-grange functions, where the relations between the inter-nal wave functions and the Lagrange functions are byno means universal and change significantly at differentchannel radii. IV. CONCLUSIONS
IWBC assumes incoming-wave profiles near the co-ordinate origins. It is widely used in theoretical stud-ies of low-energy heavy-ion fusion reactions to simulatethe strong absorption inside the Coulomb barriers. Inthis work, we generalize the calculable R -matrix theoryand EC to IWBC. The calculable R -matrix theory hasbeen formulated for regular boundary conditions and ap-plied successfully to study various structural and reactionproblems in nuclear physics. On the other hand, EC isa variational emulator to calculate physical observablesat different points in the parameter space. It has beenworked out for bound-state and scattering observables.In the generalized calculable R -matrix theory, we divideconfiguration space into three regions by the absorptionradius r abs and the channel radius r c and solve the Bloch-Schr¨odinger equations for internal wave functions on the Lagrange meshes to extract fusion observables. The gen-eralized R -matrix theory then provides a natural startingpoint to generalize EC to fusion observables. As a proofof concept, we use our generalizations of the calculable R -matrix theory and EC to study the N + C fusionreaction. Both local and non-local potentials are used inthe numerical calculations. We check the numerical reli-ability of our generalizations of the calculable R -matrixtheory and EC systematically. Both of them are foundto work well for IWBC. ACKNOWLEDGMENTS
This work is supported by the National Natural Sci-ence Foundation of China (Grants No. 12035011, No.11905103, No. 11947211, No. 11535004, No. 11975167,No. 11761161001, No. 11565010, No. 11961141003, andNo. 12022517), by the National Key R&D Programof China (Contracts No. 2018YFA0404403 and No.2016YFE0129300), by the Science and Technology De-velopment Fund of Macau (Grants No. 0048/2020/A1and No. 008/2017/AFJ), by the Fundamental Re-search Funds for the Central Universities (Grant No.22120200101), and by the China Postdoctoral Sci-ence Foundation (Grants No. 2020T130478 and No.2019M660095).
Appendix A: Lagrange Functions
Let { x i , λ i } be the abscissae and weights associated with the Gauss quadrature on the interval [0 ,
1] [46, 54], P N (2 x i −
1) = 0 , λ i = 14 x i (1 − x i )[ P (cid:48) N (2 x i − , (A1)with i = 1 , · · · , N . Here, P N ( x ) is the Legendre polynomial of order N and P (cid:48) N ( x ) ≡ d P N ( x ) / d x . The Lagrangefunctions { L Ni ( r ) } are given by L Ni ( r ) = ( − N + i (cid:112) x i (1 − x i )∆ r P N [(2 r − r abs − r c ) / ∆ r ] r − x i ∆ r − r abs , (A2)with ∆ r = r c − r abs . It is straightforward to show that L Ni ( r abs + x j ∆ r ) = (∆ rλ i ) − / δ ij . The relevant matrixelements of the Lagrange functions are given as follows: (cid:0) L Ni | L Nj (cid:1) ≡ (cid:90) r c r abs d r L Ni ( r ) L Nj ( r ) = δ ij , (A3) (cid:0) L Ni | V ( r ) | L Nj (cid:1) ≡ (cid:90) r c r abs d r L Ni ( r ) V ( r ) L Nj ( r ) = δ ij V ( r abs + x i ∆ r ) , (A4) (cid:0) L Ni | W ( r, r (cid:48) ) | L Nj (cid:1) ≡ (cid:90) r c r abs d r (cid:90) r c r abs d r (cid:48) L Ni ( r ) W ( r, r (cid:48) ) L Nj ( r (cid:48) )= ∆ r (cid:112) λ i λ j W ( r abs + x i ∆ r, r abs + x j ∆ r ) , (A5) (cid:18) L Ni (cid:12)(cid:12)(cid:12)(cid:12) d d r (cid:12)(cid:12)(cid:12)(cid:12) L Nj (cid:19) ≡ (cid:90) r c r abs d r L Ni ( r ) L Nj (cid:48)(cid:48) ( r )= ( N + N +6 ) ( x i − x i +23∆ r ( x i − x i , i = j, ( − i + j √ ( x i − x j − x j [ x i (4 x i − x j − x j ]∆ r ( x i − x / i ( x i − x j ) , i (cid:54) = j, (A6) (cid:18) L Ni (cid:12)(cid:12)(cid:12)(cid:12) δ ( r − r abs ) dd r (cid:12)(cid:12)(cid:12)(cid:12) L Nj (cid:19) ≡ (cid:90) r c r abs d r L Ni ( r ) L Nj (cid:48) ( r ) δ ( r − r abs )= ( − i + j +1 (cid:112) ( x i − x i ( x j − x j [ N ( N + 1) x j − r x i x j , (A7) (cid:18) L Ni (cid:12)(cid:12)(cid:12)(cid:12) δ ( r − r c ) dd r (cid:12)(cid:12)(cid:12)(cid:12) L Nj (cid:19) ≡ (cid:90) r c r abs d r L Ni ( r ) L Nj (cid:48) ( r ) δ ( r − r c )= ( − i + j +1 x i [ N ( N + 1) ( x j −
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