Asymptotic normalization coefficient method for two-proton radiative capture
L.V. Grigorenko, Yu.L. Parfenova, N.B. Shulgina, M.V. Zhukov
aa r X i v : . [ nu c l - t h ] S e p Asymptotic normalization coefficient method for two-proton radiative capture
L.V. Grigorenko a , b , c , ∗ , Yu.L. Parfenova a , N.B. Shulgina c , d , M.V. Zhukov e a Flerov Laboratory of Nuclear Reactions, JINR, 141980 Dubna, Russia b National Research Nuclear University “MEPhI”, Kashirskoye shosse 31, 115409 Moscow, Russia c National Research Centre “Kurchatov Institute”, Kurchatov sq. 1, 123182 Moscow, Russia d Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia e Department of Physics, Chalmers University of Technology, S-41296 G¨oteborg, Sweden abstract
The method of asymptotic normalization coefficients is a standard approach for studies of two-body non-resonant radiative captureprocesses in nuclear astrophysics. This method suggests a fully analytical description of the radiative capture cross section in the low-energy region of the astrophysical interest. We demonstrate how this method can be generalized to the case of three-body 2 p radiativecaptures. It was found that an essential feature of this process is the highly correlated nature of the capture. This reflects the complexityof three-body Coulomb continuum problem. Radiative capture O+ p + p → Ne+ γ is considered as an illustration. Keywords: asymptotic normalization coefficient method; two-proton nonresonant radiative capture; E1 strength function; three-bodyhyperspherical harmonic method.
Date:
September 29, 2020.
1. Introduction
In the asymptotic normalization coefficient (ANC) approachthe nuclear wave function (WF) is characterized only by thebehavior of its asymptotics. This asymptotics is defined interms of the modified Bessel function of the second kind K inneutral case ψ gs ( r → ∞ ) = C p qr/π K l +1 / ( qr ) ∼ C exp[ − qr ] , or in terms of the Whittaker function W in Coulombic case ψ gs ( r → ∞ ) = C W − η,l +1 / (2 qr ) ∼ C (2 kr ) − η exp[ − qr ] , where η = Z Z e M/q is the Sommerfeld parameter. Thusthe asymptotics and, hence, the related observables are definedjust by two parameters: the g.s. binding energy E b = q / (2 M )and the 2-body ANC value C .Such an approximation is valid for highly peripheral pro-cesses. The nonresonant radiative capture reactions at as-trophysical energies are the main subject of interest here [1, 2, 3, 4, 5]. An asymptotic normalization coefficient charac-terizes the virtual decay of a nucleus into clusters and, there-fore, it is equivalent to coupling constant in particle physics[ 6]. For that reason the ANC formalism naturally providesa framework for deriving the low-energy astrophysical infor-mation from peripheral reactions, such as direct transfer reac-tions, at intermediate energies (the so-called “Trojan horse” ∗ Corresponding author. E-mail address: [email protected] (L.V.Grigorenko) method [ 7, 8, 9]). From the short list of references above,it can be seen that the ANC study is quite active and has anumber of controversial unresolved issues.For the network nucleosynthesis calculations in a thermal-ized stellar environment it is necessary to determine the as-trophysical radiative capture rates h σ part ,γ v i . The two-body resonant radiative captures h σ part ,γ v i ( T ) ∝ T n/ exp (cid:20) − E r kT (cid:21) Γ γ Γ part Γ tot , (1)can be related to experimentally observable quantities [ 10,11, 12]: resonance position E r , gamma Γ γ and Γ part particlewidths ( n = 1 for two-body and n = 2 for three-body cap-tures).The situation is much more complicated for nonresonant radiative capture rates. The direct measurements of the low-energy capture cross sections could be extremely difficult fortwo-body processes. However, for the three-body capturerates the direct measurements of the corresponding capturecross sections are not possible at all. Therefore, experimen-tal approaches to three-body processes include studies of thephoto and Coulomb dissociation, which are reciprocal pro-cesses for radiative captures. However, the “extrapolation” ofthree-body cross sections from experimentally accessible en-ergies to the low energies, important for astrophysics, mayrequire tedious theoretical calculations. This is because rela-tively simple “standard” quasiclassical sequential formalism [10, 11] may not work in essentially quantum mechanical cases[ 12, 13, 14, 15].The 2 n and 2 p astrophysical captures are becoming impor-tant at extreme conditions in which density and temperatureare so high that triple collisions are possible. However, thetemperature should not be too high to avoid the inverse pho-todisintegration process. For the 2 n captures the followingpossible astrophysical sites are investigated: (i) the neutrino-heated hot bubble between the nascent neutron star and theoverlying stellar mantle of a type-II supernova, (ii) the shockejection of neutronized material via supernovae, (iii) the merg-ing neutron stars. The 2 p captures may be important for ex-plosive hydrogen burning in novae and X-ray bursts.The 2 n and 2 p nonresonant radiative capture rates havebeen investigated in a series of papers Ref. [ 13, 14, 15] by theexamples of the He+ n + n → He+ γ and O+ p + p → Ne+ γ transitions. These works also required the development ofexactly solvable approximations to understand underlyingphysics of the process and achieve the accuracy needed forastrophysical calculations [ 16, 17, 18]. Some of the universalphysical aspects observed in the papers mentioned above havemotivated the search for simple analytic models. The follow-ing qualitative aspects of the low-energy E1 strength function(SF) behavior were emphasized in [ 13, 14, 15] for 2 n and2 p captures: (i) sensitivity to the g.s. binding energy E b ; (ii)sensitivity to the asymptotic weights of configurations deter-mining the transition; (iii) importance of one of near-thresholdresonances in the two-body subsystems (virtual state in n - n channel in the neutral case and lowest resonance in the core- p channel in the Coulombic case), which effect on SF is foundto be crucial even at asymptotically low three-body energies.Points (i) and (ii) are the obvious motivation for ANC-like de-velopments; point (iii) represents important and problematicdifference from the two-body case.This work to some extent summarizes this line of researchsuggesting analytical framework for two-nucleon astrophysicalcapture processes. We demonstrate that it is possible to gener-alize the two-body ANC2 method to the ANC3 method in thesituation of three-body radiative captures. While for the 2 n capture the practical applicability of ANC3 method remainsquestionable, for the 2 p captures it is established beyond anydoubt. In this work we provide compact fully analytical frame-work for the processes, which previously could be consideredonly in bulky numerical three-body calculations.
2. ANC3 in the hyperspherical harmonics (HH) ap-proximation
The HH formalism for calculations of the E1 SF is providedin details in Ref. [ 19] and here we just give a sketch. Assumethat the bound and continuum wave functions (WF) can bedescribed in a three-cluster core+ N + N approach by solvingthe three-body Schr¨odinger equation ( ˆ H − E T )Ψ J gs = 0 , ˆ H = ˆ T + V N N ( r ) + V cN ( r ) + V cN ( r ) + V ( ρ ) , (2)where E T is the energy relative to the three-cluster breakupthreshold. See Fig. 1 for definition of coordinates used in thiswork. The pairwise interactions V ij are motivated by spectraof the subsystems, while V is phenomenological three-bodypotential used for fine-tuning of the three-body resonance en-ergies. In the hyperspherical harmonics method this equation r r r Jacobi "T"Jacobi "Y" corecore N N core n n p p (c)(b) V ( non r e s ) y ( Y ) V (res)n n (X) V c N ( r ) V N N (r ) Three-body Hamiltonian core + N + N V c N ( r ) V ( non r e s ) y ( Y ) Simplified Hamiltonian core + p + p V (r e s ) c p ( X ) Simplified Hamiltonian core + n + n (a) Fig. 1. Coordinate systems and potential sets for “hyperspherical har-monics” HH and “simplified Hamiltonian” SH approaches to ANC3. (a)The complete 3-body Hamiltonian is applied both to core+ p + p andcore+ n + n systems. (b) For core+ p + p system the dynamical domina-tion of lowest resonance in the core- p subsystem motivates the use ofsimplified Hamiltonian in the “Y” Jacobi system. (c) For core+ n + n sys-tem the dynamical domination of the n - n final state interaction motivatesthe use of simplified Hamiltonian in the “T” Jacobi system. is reduced to a set of coupled differential equationsΨ J gs ( ρ, Ω ) = ρ − / X Kγ χ Kγ ( ρ ) J JKγ (Ω ) , (cid:20) d dρ − L ( L + 1) ρ + 2 M ( E T − V Kγ,Kγ ( ρ )) (cid:21) χ JKγ ( ρ )= X K ′ γ ′ = Kγ M V K ′ γ ′ ,Kγ ( ρ ) χ JK ′ γ ′ ( ρ ) , (3) ρ = ( A A r + A A r + A A r ) / ( A + A + A ) , depending on the collective coordinate — hyperradius ρ . The“scaling” mass M is taken as an average nucleon mass in thesystem and J JKγ (Ω ) is the hyperspherical harmonic with thedefinite total spin J . The three-body potentials are defined as V K ′ γ ′ ,Kγ ( ρ ) = hJ JK ′ γ ′ | P i>j V ij ( r ij ) |J JKγ i . The effective orbital momentum L = K + 3 / K = 0.The continuum three-body problem is solved using the sameEq. (3) set but for continuum WF χ JK f γ f ,K ′ f γ ′ f ( κ ρ ) (squarematrix of solutions) diagonalizing S-matrix. Hypermomentum κ is defined as κ = √ M E T . In the no-Coulomb case the WFis constructed by diagonalizing the 3 → χ JKγ,K ′ γ ′ ( κ ρ ) = exp( iδ Kγ,K ′ γ ′ ) sin( κ ρ − ( K + 2) π/ δ Kγ,K ′ γ ′ ) , S Kγ,K ′ γ ′ = exp(2 iδ Kγ,K ′ γ ′ ) , in analogy with the two-body case. This WF contains planethree-body wave and outgoing waves. The formulation ofthe boundary conditions becomes problematic in the Coulombcase and methods with only outgoing waves (including the SHmodel introduced later in Section 3) is a preferable choice. Thedetails of the method and its applications are well explainedin the literature [ 20, 21, 12, 22, 23, 19] and we will not dwellon that too much.The form of hyperspherical equations (3) immediately pro-vides the vision for the low-energy behavior of observables inE1 continuum since the only K = 1 component with the lowestcentrifugal barrier is important in the E T → N + N states are induced by the following operator O E1 ,m = e X i =1 , Z i r i Y m (ˆ r i ) = p / (4 π ) D m , -6 -5 -4 -3 -2 -1 He WF He asympt. Ne WF Ne asympt. Z hh by Eq. (16) Ne asympt. Z hh fitted K = () (f m ) (fm) R a ti o o f K = () t o a s y m p t o ti c Fig. 2. Left axis: ground state WF components with K = 0 for He (solidcurve) and Ne (dashed curve), matched to asymptotic Bessel (thickgray curves) and Whittaker (short dashed and dotted curves) functions.For He ANC3 value is C = 0 . − / . For Ne ANC3 values are C = 13400 fm − / with Z hh = 27 . C = 5958 fm − / with fitted Z hh = 26 .
14. The lines on the right axis show the ratio of Ne WF to Whittaker functions with mentioned Z hh and C . where D = P i =1 , eZ i r i is the dipole operator, and O E1 ,m = Z eff ρ cos( θ ρ ) Y m (ˆ y ) , (4) Z = (cid:26) Z ( Z − A ) (cid:27) e ( A + A ) A ( A + A + A ) . (5)The upper value in curly braces is for core+ n + n and the lowerone is for core+ p + p three-body systems, taking into accountthe c.m. relation r + r = − A r for the three-body system.For historical reasons the astrophysical E1 nonresonant ra-diative capture rate is expressed via the SF of the reciprocalE1 dissociation , see Eq. (47). The E1 dissociation SF in theHH approach is dB E dE T = X J f G fi r M E T X K f γ f | M J f K f γ f | . (6)where J i is total spin of bound state, J f is total spin ofcontinuum state, G fi = (2 J f + 1) / (2 J i + 1) is a statisticalfactor, and the E1 matrix element is M J f K f γ f = Z eff X K ′ f γ ′ f X K i γ i M a M hh Z dρ p /π × χ J f K f γ f ,K ′ f γ ′ f ( κ ρ ) ρ χ J i K i γ i ( ρ ) ,M a = (cid:10) J f γ ′ f k Y (ˆ y ) k J i γ i (cid:11) , M hh = (cid:10) K ′ f γ ′ f | cos( θ ρ ) | K i γ i (cid:11) . For example, the reduced angular momentum matrix element M a = 1 / √ π for J i = 0 → J f = 1 transition and the hyper-angular matrix element M hh = 1 / √ K i = 0 → K f = 1transition. For the three-body plane-wave case the solution matrix isdiagonal and expressed in terms of cylindrical Bessel functions χ J f K f γ f ,K ′ f γ ′ f ( κ ρ ) = r π δ K f ,K ′ f δ γ f ,γ ′ f r π κ ρ J K f +2 ( κ ρ ) , with asymptotics for small κ ρ √ κ ρ J K f +2 ( κ ρ ) ∼ ( κ ρ ) K f +5 / / [( K f + 2)! 2 K f +2 ] . (7) This expression can be used to separate the leading term ofthe low-energy dependence of the matrix element, labeled forsimplicity only by the values of K for the initial and final states M K f K i ( E T ) = p /π κ K f +5 / Z eff M a M hh I K f K i ( E T ) , (8) I K f K i ( E T ) = 1 κ K f +2 Z dρ ρ / J K f +2 ( κ ρ ) χ J i K i γ i ( ρ ) . (9)where the overlap integral I K f K i tends to a constant at E T → K i = 0 to the lowest E1 continuum compo-nent with K f = 1: dB E dE T = 1 π G fi Z M M (2 M ) E T I ( E T ) . (10)Now we replace the bound state WF χ in Eq. (9) by its long-range asymptotics expressed in terms of the three-body ANCvalue C and cylindrical Bessel functions Kχ K =0 ( ρ ) → C p κρ/π K ( κρ ) , (11)where the g.s. hypermoment κ = √ M E b is defined via thebinding energy E b . This approximation is valid in a broadrange of ρ values, see Fig. 2. The He WF is taken from [19, 15]. The overlap integral now has simple analytical form I ( E T ) = 4 C / [(2 M E b ) / (1 + E T /E b ) ] . (12)It can be found that the ANC3 approximation of the overlapvalue (12) deviates within very reasonable ∼
7% limits fromthe directly calculated by Eq. (9) in a broad energy range( E T . He case
The E1 SF and the astrophysical capture rate for the α + n + n → He+ γ was recently studied in Refs. [ 15, 19].It can be found that Eq. (10) is not sufficient in this case fortwo reasons:(i) In the p -shell He nucleus not only the K i = 0 → K f = 1transition is important, but also K i = 2 → K f = 1. Theasymptotics of the K i = 2 WF component falls off much fasterthan that of the component K i = 0. However, the weight ofthe K i = 2 WF component corresponding to [ p ] configura-tion is much larger ( ∼ K i = 0WF component ( ∼ s ] config-uration. So, finally their contributions to the low-energy MEare comparable.(ii) It was shown in [ 15, 19] that the low-energy part of theE1 SF is highly sensitive to the final state n - n interaction (anincrease in SF when the n - n interaction is taken into accountis a factor of 8). The paper [ 15] is devoted to the study of thiseffect in the dynamic dineutron model. We do not currentlysee a method to consider this effect analytically.Applicability of the approximation (10) to the other casesof 2 n capture should be considered separately. Let us consider the transition to the single K f = 1 contin-uum final state. The low-energy behavior of continuum singlechannel WF in the Coulomb case is provided by the regularat the origin Coulomb WF χ K f ( κ ρ ) → p /π F K f +3 / ( η hh , κ ρ ) . (13)The suitable asymptotics of the Coulomb WFs are F l ( η, kr ) = (2 l + 1)! C l ( η )(2 η ) l +1 p βr I l +1 (2 p βr ) , (14) G l ( η, kr ) = 2(2 η ) l (2 l + 1)! C l ( η ) p βr K l +1 (2 p βr ) , (15) C l ( η ) = 2 l exp[ − πη/ | Γ[ l + iη + 1] | / Γ[2( l + 1)] , (16) C ′ l ( η ) = √ π (2 η ) l +1 / (2 l + 1)! exp[ − πη ] , β = ηk , (17) D l ( η, k ) = (2 l + 1)! C ′ l ( η )(2 η ) l +1 p β = √ πk exp[ − πη ] , (18)where I and K are modified Bessel functions. Approximation(17) for the Coulomb coefficient (16) works for η ≫ l .In the ANC3 approximation the g.s. WF χ can be replacedby its long-range asymptotics χ K i ( ρ ) → C W − η gs ,K i +2 (2 κρ ) . (19)This asymptotics is valid when all three particles are well sepa-rated. We will find out later that at least the core- p distances,which contribute E1 SF, are simultaneously large, see Fig. 5(b). The Sommerfeld parameters η for continuum and boundstates are η hh = β hh / κ , β hh = Z hh e M , η gs = Z hh e M/κ . (20)The effective charges of isolated hyperspherical channels canbe defined as Z ( Kl x l y )hh = ρ (cid:28) Kl x l y (cid:12)(cid:12)(cid:12)(cid:12) Z Z r + Z Z r + Z Z r (cid:12)(cid:12)(cid:12)(cid:12) Kl x l y (cid:29) . (21)For the Ne case the K = 0 and K = 1 effective charges are Z (000)hh = 27 . , Z (101)hh = 27 . . (22)Fig. 2 shows that the substitution Eq. (19) works well in avery broad range of radii (the Ne g.s. WF is from Ref. [ 24]).The effective charge in Eq. (22) obtained for K i = 0 is veryreasonable. However, slightly different effective charge value Z (000)hh = 26 .
14 is required for an almost perfect match to theasymptotics. This is a clear indication of coupled-channel dy-namics in this case. It is actually a nontrivial fact that all thecomplexity of this dynamics reduces to a simple renormaliza-tion of effective charges.Using Eqs. (14) and (18) we can factorize the E M K f K i = p /π D K f +3 / ( η hh , κ ) M a M hh Z eff I ( c ) K f K i ( E T ) ,I ( c ) K f K i ( E T ) = Z dρ F K f +3 / ( η hh , κ ρ ) D K f +3 / ( η hh , κ ) ρ / χ J i K i γ i ( ρ ) , (23)where the overlap integral I ( c ) K f K i weakly depends on the energyand in the limit E T → I ( c ) K f K i = Z dρ I l +1 (2 p β hh ρ ) ρ / χ J i K i γ i ( ρ ) . (24)The overlaps (23) for K i = 0 → K f = 1 transition are shownin Fig. 3. It can be found that in the ANC3 approximation theEq. (19) is quite accurate: in this case the overlap increasesjust less than 6% compared the calculation with the real g.s.WF. It is also seen that the use of simple energy-independent I (cc)10 , real g.s. WF I (cc)10 , Whittaker tail I (c)10 , real g.s. WF I (c)10 , Whittaker tail B, real g.s. WF -3 -2 -1 I (cc)10 I ( c ) (f m ) , I ( cc ) a nd B (f m ) E T (MeV) all scaled by x10 -4 ~ (1+E T /E cy ) (1+E T /E b ) Fig. 3. Overlap integrals for HH I ( c )10 ( E T ) [Eq. (23)], SH I ( cc )10 ( ε , E T )[Eq. (31)], and B ( ε , E T ) [Eq. (44)]. The solid gray curve shows analyticalapproximation Eq. (38) for I ( cc )10 . overlap Eq. (24) instead of (23) gives almost perfect resultbelow 10 keV and is reasonable below 100 keV. For the E dB E dE T = G fi Z M M M I ( c )210 ( E T ) exp[ − πη hh ] . (25)The energy dependence of the derived expression at E T → − πη hh ]. The SF calculationresults are shown in Fig. 4. They strongly disagree with cal-culation results from Refs. [ 13] and [ 14]. The modificationof the “effective continuum charge” Z (101)hh from Eq. (22) doesnot save the situation since the energy dependence of the SFin Eq. (25) and that of the SF in [ 13, 14] are too different.We demonstrate in the next section that the Eq. (25) is actu-ally incorrect. However, the derivations of this section are stillimportant for our further discussion.
3. ANC3 in the simplified Hamiltonian (SH) approx-imation
The approximation is based on the usage of a simplifiedthree-body Hamiltonian for the E1 continuum instead of thereal oneˆ H → ˆ H ′ = ˆ T + V cN ( X ) + V y ( Y ) , (26)where X ≡ r is the Jacobi vector in the “Y” Jacobi system,while Y corresponds to the second Jacobi vector, see Fig. 1.Such a Hamiltonian is quite reliable since the nuclear interac-tion with a proton in a non-natural parity state is weak. Themodel was used for nonresonant astrophysical rate calculationsin Ne in Ref. [ 13] and in He in Ref. [ 15]. A thorough checkof the model is given in Ref. [ 16], and the detailed descriptionof the formalism for complicated angular momentum couplingsin Ref. [ 14].To obtain the E1 dissociation strength function in this ap-proximation we solve the inhomogeneous Schr¨odinger equation( ˆ H ′ − E T )Ψ J f M f (+) M i m = O E1 ,m Ψ J i M i gs , for WF Ψ (+) with pure outgoing wave boundary conditions.The transition operator Eq. (4) dependent on r can be rewrit-ten in X and Y coordinates using relation: r = X A / ( A + A ) − Y A / ( A + A + A ) . (27) -35 -30 -25 -20 -15 Ne O+p+p(a)
HH I (c) (E T ) SH I (cc) ( ,E T ) I (cc) SH I (cc) ( ,E T ) SH I (cc) ( ,E T ) + + resonance correction d B E / d E T ( e f m / M e V ) ~ -13 -11 -9 -7 d B E / d E T ( e f m / M e V ) [Grigorenko, 2006] Three-body HH [Parfenova, 2018] SH, one component, decayvia F 0 at E r = 535 keV E T (MeV) (b) Turnover to sequential capture mechanism E r Fig. 4. The E1 strength functions for Ne → O+ p + p transition. Solidblack curve corresponds to E1 SF obtained with ANC3 HH method of Eq.(25). Green dash-dotted curve corresponds to simple energy-independentapproximation Eq. (43) I ( cc ) ( ε, E T ) → ˜ I ( cc ) . Red dashed and blue dottedcurves corresponds to ANC3 SH method of Eq. (33) without and withthe resonance correction Eq. (43), respectively. Thick gray curve andthin magenta solid curves correspond to SFs from Refs. [ 13] and [ 14],respectively. Since the factorized form of the Hamiltonian Eq. (26) allows asemi-analytical expression for the three-body Green’s function,a rather simple expression for the SF can be obtained dB E dE T = G fi π E T Z dε M x M y k x k y | A ( E x , E y ) | , (28) E x = εE T , E y = (1 − ε ) E T , k x,y = p M x,y E x,y , where ε is the energy distribution parameter. The amplitude A is defined as A ( E x , E y ) = Z dXdY f l x ( k x X ) f l y ( k y Y ) Φ( X, Y ) . (29)where the “source function” Φ is defined by the E1 operatoracting on Ψ gs . The WFs f l x and f l y are eigenfunctions of sub-Hamiltonians depending on X and Y Jacobi coordinates inS-matrix representation with asymptotics f l ( kr ) = e iδ l [ F l ( η, kr ) cos( δ l ) + G l ( η, kr ) sin( δ l )] . (30)Eq. (29) is given in a simplified form, neglecting angularmomentum couplings, more details can be found in [ 14]. Weskip this part of the formalism in this work. The calculationsof the E1 strength function in the SH approximation withoutfinal state interactions in X and Y channels for the 2 n captureare equivalent to calculations in the HH approximation. So,we skip no-Coulomb case and proceed to the 2 p capture. With good accuracy, one can calculate the amplitude onlyfor the Y coordinate and then double the result. This is notdifficult to prove, but tedious, so we do not provide a proofhere. The amplitude A for the Y coordinate [see Eq. (27)]from the transition operator Eq. (4) with extracted by Eqs.(14) and (18) low-energy dependence is written in terms ofthe overlap integral I ( cc ) as A ( E x , E y ) = M a D l x ( η x , k x ) D l y ( η y , k y ) I ( cc ) l x l y ( ε, E T ) ,I ( cc ) l x l y ( ε, E T ) = Z dXdY F l x ( k x X ) D l x ( η x , k x ) F l y ( k y Y ) D l y ( η y , k y ) Y ψ gs ( X, Y ) , (31) η x = Z Z e M x /k x , η y = ( Z + Z ) Z e M y /k y . The asymptotic form of this overlap, independent of energy, is˜ I ( cc ) l x l y = Z dXdY I l x +1 (2 p β x X ) I l y +1 (2 p β y Y ) × √ XY ψ gs ( X, Y ) . (32)The WF ψ gs and the integrand of Eq. (31) on the { X, Y } plane are shown in Figs. 5 (a) and (b). Their comparisonillustrates the extreme peripheral character of the low-energyE1 transition: the WF maximum is at a distance of ∼ ∼
60 fm.The E1 SF with antisymmetry between nucleons taken intoaccount is dB E dE T = G fi M M x M y Z − A ) e ( A + A ) ( A + A + A ) A I ε ( E T ) , (33) I ε ( E T ) = E T Z dεI ( cc )210 ( ε, E T ) exp[ − π ( η x + η y )] . (34)The Coulomb exponent in I ε has a very sharp energy depen-dence, see Fig. 6. The energy dependence of I ( cc ) is shownin Fig. 5 (c): it is quite flat for ε ≈ ε . Thus, I ( cc ) can beevaluated at the peak ε = ε and the ε integration can beperformed by the saddle point method: I ε ( E T ) = I ( cc )210 ( ε , E T ) E T exp( − πη sh ) √ R ε η sh , (35) η sh = Z sh e M/ κ , Z sh = ( b x + b y ) / , (36) ε = b x / ( b x + b y ) , R ε = ( b x + b y ) / (4 b x b y ) ,b x = [ Z Z M x /M ] / , b y = [( Z + Z ) Z M y /M ] / . For the Ne → O+ p + p transition ε = 0 . , Z sh = 23 . , (37)The accuracy of the saddle point integration is ∼
2% and ∼ I ( cc )10 ( ε , E T ) = ˜ I ( cc )10 p E T /E cy (1 + E T /E b ) , E cy = 2 M y β y − ε , (38)Eq. (38) contains additional Coulomb correction for l y = 1motion in Y coordinate (with E cy = 3 .
67 MeV) and we use itlater for astrophysical rate derivation. (b)
X (fm) Y (f m ) X (fm) Y (f m ) (a) -3 -2 -1 0-3-2-10 (c) Log(E x ) L og ( E y ) X (fm) Y (f m ) Fig. 5. (a) The WF ψ gs component with K = 0, normalized to unity. (b) Integrand of Eq. (32). (c) I ( cc ) l x l y ( ε , E T ) / ˜ I ( cc ) l x l y on logarithmic axes. (d)Integrand of Eq. (44). = 0.48 Phase vol. E T (MeV) 1.0 0.1 0.01 0.001 d B E ( E T , ) / d ( a r b . un it s ) = E core-p /E T = E x /E T Fig. 6. Energy distribution between core and one of the protons fordifferent decay energies E T , governed by the exp[ − π ( η x + η y )] term inEq. (34). All distributions are normalized to unity value at peak. The results of the SF calculation in the SH approximationare shown in Fig. 4 by the red dashed curve. Now there is nosignificant disagreement for E T → p emission/capture In the HH Eq. (25) and SH Eq. (33) approximations we getSF expressions with the low-energy asymptotics dB E dE T ∝ exp( − πη hh ) , dB E dE T ∝ E / T exp( − πη sh ) , (39)which are qualitatively different. There are two main points.(i) Effective charge, entering the Coulomb exponent is signif-icantly lower in SH case, see Eq. (37) compared to Eq. (22):27.41 vs. 23.282. (ii) There is an additional power dependenceon energy E / T which, evidently, cannot be compensated, forexample, by some modification of the effective charge. Whatis the source of qualitative difference between Eqs. (25) and(33)?The answer is actually provided in Fig. 6: in the SH ap-proach the emission of two protons is highly correlated process,which produces the narrow bell-shaped ε distributions. In theapproximation K f = 1 used in Eq. (25) the momentum distri-bution is described by the phase space (thick gray curve in Fig. 6). In the correlated calculation Eq. (33) this distribution isdrastically modified by the three-body Coulombic effect. Themomentum distribution which “shrinks” to the proximity ofthe ε value allows an easier penetration, which is reflected alsoin the smaller effective charge (37) in the Coulomb exponentin Eq. (39).So, what is wrong with Eq. (25)? Formally the transitionby the dipole operator from K i = 0 g.s. occurs only to K f = 1continuum, as we assumed. This means that the substitutionof Eq. (13) is incorrect. This substitution is based on the as-sumption that for E T → K f = 1 and, hence, the smallest centrifugal barrier,contributes to the penetrability. Now it is clear that for thethree-body continuum Coulomb problem this “evident” argu-ment is incorrect. Within the complete HH couple-channelproblem the K f = 1 channel should be affected by an infinitesum of the other channels in such a way that their cumulativeeffect does not vanish even in the limit E T → p radioactivity process. It was predicted by Goldansky in hispioneering work on 2 p radioactivity [ 25]: in the Coulomb-correlated emission of two protons the energies of the protonstend to be equal in the limit of infinitely strong Coulomb in-teraction in the core+ p channel. This effect for two-protonradioactivity and resonant “true” two-proton emission is nowwell studied experimentally and understood in details in the-oretical calculations [ 23, 26]. It is proved that the approxi-mations like Eq. (33) represent well the underlying physics ofthe phenomenon. It was shown in calculations of [ 13, 14] (see Figs. 3 and4-5 in these works) that the resonant state in the core- p sub-system with “natural parity” quantum numbers significantlyaffects both the profile of the E1 strength function in a widerange of energies and the asymptotic behavior at low E T val-ues. To evaluate the influence of resonance on the asymptoticsanalytically, let us consider the two-body resonant scatteringWF in the quasistationary approximation : f l ( kr ) = e iδ l F l ( kr ) cos( δ l ) + p v Γ( E ) / E r − E − i Γ( E ) / ψ l ( r ) . (40)This expression can be easily connected with the asymptoticsEq. (30) by using the resonant R-matrix formulas:tan( δ l ) = Γ( E ) E r − E → e iδ l sin( δ l ) = Γ( E ) / E r − E + i Γ( E ) / . (41)The ˜ ψ l ( r ) is so-called quasistationary WF, defined at resonantenergy E r by the irregular Coulomb WF boundary conditionand normalized to unity in the “internal region” r < r c :˜ ψ l ( r c ) E = E r ∝ G l ( k r r c ) , Z r c dr | ˜ ψ l ( r ) | = 1 . (42)The low-energy behavior of the overlap integrals Eq. (31)with the resonant continuum WF (40) in X coordinate is then I ( cc ) ′ l x l y ( ε , E T ) = I ( cc ) l x l y ( ε , E T ) + B ( ε , E T )1 − ε E T /E r , (43) B ( ε , E T ) = B c Z dXdY ˜ ψ l x ( X ) F l y ( η y , k y Y ) D l y ( η y , k y ) Y ψ gs ( X, Y ) , (44) B c = θ x [4 M x r c K l x +1 (2 p β x r c ) E r ] − . Here we use the R-matrix width definitionΓ( E ) = θ M r c P l ( E, r c ) , P l ( E, r c ) = kr c F l ( kr c ) + G l ( kr c ) , (45)which is simplified in the low-energy region using Eq. (15).The integrand of Eq. (44) is shown in Fig. 5 (d) and ithas quite peripheral character compared to the g.s. WF Fig.5 (a). The “resonance correction function” B is shown inFig. 3 demonstrating very weak dependence on energy. Itis evaluated with function ˜ ψ l x ( X ) approximated by HultenAnsatz with rms X value 3.5 fm. Parameters θ x = 1 and r c = 3 . F 0 − ground state at E r = 535keV. So, in the whole energy range of interest we can approx-imate B as˜ B = B ( ε , E T → . (46)The blue dotted curve in Fig. 4 shows nice agreement of the“resonance corrected” E1 SF with complete three-body calcu-lations up to ∼
600 keV. At this energy the two-body reso-nance well enters the “energy window” for three-body capture E r < E T and turnover to sequential capture mechanism istaking place. The nonresonant radiative capture rate is expressed via theSF of E1 dissociation Eq. (6) as h σ p,γ v i = (cid:18) P n A n A A A (cid:19) / (cid:18) πmkT (cid:19) J f + 12(2 J i + 1) I E ( T ) ,I E ( T ) = Z dE π E γ dB E ( E ) dE exp (cid:20) − EkT (cid:21) , (47)where J i and J f are spins of the O and Ne g.s., respectively[ 14].The energy dependence of Eq. (33) is too complex to allow adirect analytical calculation of the astrophysical capture rate.However, using Eqs. (35), (38), and (46), the main analyticalterms can be obtained by the saddle point calculation near theGamow peak energy E G : I E ( T ) ∝ Z dE T ( E b + E T ) I ε ( E T ) exp (cid:20) − E T kT (cid:21) = 2 πE E / γ √ R ε × E G /E cy E G /E b (cid:18) ˜ I ( cc )10 + (1 + E G /E b ) − ε E G /E r ˜ B (cid:19) exp (cid:20) − γ / ( kT ) / (cid:21) ,E G = ( γkT ) / , γ = πZ sh e p M/ , πη sh = γ/ p E T . (48) -2 -1 ~ v / ( v [ P a rf e nov a , ] ) T (GK)
Numerical I (cc) ( ,E T ) I (cc) Numerical I (cc) ( ,E T ) + B(E T )/(E r 0 E T ) Analytical I (cc) + B/(E r 0 E T ) ~ ~ T /E cy (1+E T /E b ) Fig. 7. Ratio of the rates for astrophysical nonresonant three-body cap-ture reaction O+ p + p → Ne+ γ obtained in this work and in the paper[ 14] (see the thin magenta solid line SF in Fig. 4). Blue dotted and solidorange curve show the results of numerical rate calculation by Eq. (47)and analytical result by Eq. (48) for the resonance-corrected I ( cc ) ′ ( ε, E T )from Eq. (43). Green dash-dotted curve shows the rate for simple energy-independent SF I ( cc ) ( ε, E T ) → ˜ I ( cc ) from Eq. (32). The Gamow peak energy can be found as { . , . , } MeVat { . , . , } GK. Comparison of the rates calculated in amodel of the paper Ref. [ 14] and in this work is given in Fig.7. It can be seen that even the very crude energy-independentapproximation Eq. (32) I ( cc ) ( ε, E T ) → ˜ I ( cc ) for T < . T < . T < . nonresonant p capture rate is h σ p,γ v i ∝ C T / exp[ − ( T eff /T ) / ] , kT eff ≈
193 MeV . Analogous dependence for the resonant rate is (e.g. Ref. [ 12]) h σ p,γ v i ∝ Γ p T − exp[ − ( E r /kT )] , E r = 0 .
355 MeV , where E r is the lowest state decaying via 2 p emission withΓ p (for Ne this is the first excited 3 / − state). Here itcan be found that nonresonant capture dominates in the low-temperature limit, see also discussion in Ref. [ 13].So, we have obtained a compact analytical expression for the2 p capture rate, which depends only on the global parametersof the system under consideration ( C , E b , E r , Z sh ) and twouniversal overlaps ( ˜ I ( cc ) and ˜ B ) calculated at E T →
4. General note on three-body Coulomb continuumproblem
The three-body Coulomb continuum problem is a famouslong-term conundrum of theoretical and mathematical physics.Complexity of this problem is defined by the possible presenceof the Coulomb correlations, bound and resonant states inthe two-body subsystems. As a result, no compact analyticalform of the asymptotics is known for the three-body Coulombcontinuum problem. There is known approximate asymp-totic solution of this problem (so-called “Redmond-Merkuriev”asymptotics) [ 27, 28], which is valid in four regions: in one re-gion all three particles are far from each other r ∼ r ∼ r and there are three regions where different pairs of three parti-cles are close r ij ≪ r jk ∼ r ki . There were fruitful applications core p p k y k x c o s ( k ) d W / d d c o s ( k ) ( a r b . un it s ) = E x / E T (b) (a) c o s ( k ) = ( k x , k y ) / ( k x k y ) = E x /E T p -p core-p core-p Fig. 8. (a) Regions in the kinematical plane { ε, cos( θ k ) } where effects ofthe two-body Coulomb repulsion are dominating. Different colors corre-spond to different pairs of particles, where r ij ≪ r jk ∼ r ki . Solid colorcorresponds to E T ∼ E T ∼ −
100 keV. The dotted magenta curve illustrates theregion of p - p repulsion dominance at some hypothetical extremely smallenergy E T ≪ Ne(g.s.) → O+ p + p with E T = 1 .
466 MeV, very well reproducing thedata [ 26]. of this asymptotics, e.g. to atomic problems [ 29, 30]. Thereis a wide range of works dedicated to improvement of thisasymptotics [ 31, 32, 33, 34]. One of modern trends is notto struggle with analytical problems of this asymptotics, butto use powerful computing and propagate numerical solutionsto distances where uncertainties of the asymptotics does notplay a practical role. However, we do not think that this isa completely satisfactory approach, which should replace theanalytical developments.In this work we deal with a limited subset of the three-bodyCoulomb continuum problem: only repulsive Coulomb interac-tions and no bound states in the subsystems. Specific featureof our problem is that the three-body energies are extremelysmall and the solution residue in the kinematical region, wherethe contributions of two two-body Coulomb asymptotics (core- p and core- p channels) overlap, see Fig. 8 (a). This justifiesamplitude factorization and consequent analytical calculationsof Section 3.1. However, reliability of this approximation isbased on the fact that p - p Coulomb interaction is quite smallcompared to core- p interaction. It can be found from Fig. 8 (b)that a minor part of the kinematical space is affected by the p - p repulsion for energies 1 < E T < E T → p - p Coulomb interaction will become important in the wholekinematical plane { ε, cos( θ k ) } and the low-energy asymptotics,which we deduced in this work, will be broken.
5. Conclusion
In this paper, we provide a formalism for a complete an-alytical description of low-energy three-body 2 p nonresonantradiative capture processes. The developed approach is a gen-eralization of the ANC method, which has proven itself wellfor two-body nonresonant radiative captures. The ordinary(two-body) ANC2 method demonstrates the sensitivity of thelow-energy E1 strength function, important for astrophysics,to only two parameters: the binding energy E b and the ANC2value C . For the three-body ANC3 method one more param-eter should be employed: the energy E r of the lowest to thethreshold “natural parity” two-body resonance with appropri-ate quantum numbers. An interesting formal result is related to the problem of thethree-body Coulomb interaction in the continuum. We demon-strate that ANC3 method developed completely in three-bodyhyperspherical harmonics representation (named “HH approx-imation”) is not valid in the Coulomb case as it gives incor-rect low-energy asymptotics of SF and hence incorrect low-temperature asymptotics of the astrophysical rate. The rea-son for this is the highly correlated nature of the 2 p capture.The correct asymptotics can be obtained using the Coulomb-correlated SH approximation based on a simplified three-bodyHamiltonian. The latter approximation also allows to deter-mine the correction related to the low-lying two-body resonantstate in the core-nucleon channel.The two-dimensional overlap integrals involved in the ANC3approximation in the correlated Coulomb case are rather com-plicated compared to those in the ANC2 case. However, theircalculation is a task that is incomparably simpler than anycomplete three-body calculation. The whole formal frameworkis compact and elegant and requires only two overlap calcula-tions: ˜ I ( cc ) and ˜ B . Thus, we find that ANC3 approximation ina three-body case is a valuable development providing robusttool for estimates of the three-body nonresonant capture ratesin the low-temperature ( T . . − Acknowledgments. — LVG, YLP, and NBS were supportedin part by the Russian Science Foundation grant No. 17-12-01367.
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