Post-prior equivalence for transfer reactions with complex potentials
aa r X i v : . [ nu c l - t h ] N ov Post-prior equivalence for transfer reactions with complex potentials
Jin Lei ∗ and A. M. Moro † Institute of Nuclear and Particle Physics, and Department ofPhysics and Astronomy, Ohio University, Athens, Ohio 45701, USA Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain. (Dated: April 29, 2018)In this paper, we address the problem of the post-prior equivalence in the calculation of inclusivebreakup and transfer cross sections. For that, we employ the model proposed by Ichimura, Austern,and Vincent [Phys. Rev. C 32, 431 (1985)], conveniently generalized to include the part of thecross section corresponding the transfer to bound states. We pay particular attention to the casein which the unobserved particle is left in a bound state of the residual nucleus, in which casethe theory prescribes the use of a complex potential, responsible for the spreading width of thepopulated single-particle states. We see that the introduction of this complex potential gives rise toan additional term in the prior cross section formula, not present in the usual case of real bindingpotentials. The equivalence is numerically tested for reaction induced by deuterons.
I. INTRODUCTION
The post-prior equivalence of the transition amplitudefor direct nuclear reactions involving different rearrange-ment channels is a key result of nuclear reaction theory.This result provides two formally equivalent ways of ex-pressing the transition amplitude, depending on whetherthe main interaction appearing in the transition operatoris that based on the initial or final internal Hamiltonians.The result holds for the exact transition amplitude and,in the case of transfer reactions, also for the DWBA limit.In this latter case, the post and prior expressions are for-mally identical, differing only in the transition operator.This result has indeed been confirmed in practical cases.In the case of inclusive breakup reactions of the form a + A → B ∗ + b , where a = b + x and B ∗ is any A + x state,the problem has deserved attention in the past. Severalgroups proposed formulae for the calculation of inclusivecross sections using either the post or prior DWBA repre-sentations [1–4]. Ichimura, Austern and Vincent(IAV) [3]showed that the post and prior equivalence holds also forthese inclusive processes but, in this case, it involves anadditional term, not present in the usual transfer processbetween bound states. This terms arises from the non-orthogonality between the initial ( a + A ) and final ( b + B )partitions. Although some authors (see e.g. Li, Udagawaand Tamura [5]) regarded this term as nonphysical, in ourrecent work [6] we showed in practical cases that the in-clusion of this term is essential to preserve the post-priorequivalence and to reproduce correctly the experimentaldata.The calculations of Ref. [6] were restricted to unbound x + A states (i.e. E x >
0, where E x is the final relativeenergy between x and A ). However, the E x < B system and, hence, the process ∗ [email protected] † [email protected] a + A → B + b becomes a transfer reaction in the usualsense. In some models, such as the DWBA, the scatter-ing amplitude involves the overlap function between the A and B systems, i.e. h A | B i . Although these overlapsshould be in principle obtained from many-body wavefunctions of A and B , they are most commonly approx-imated the single-particle wave functions calculated in amean-field potential, with the correct quantum numbersand separation energy, and multiplied by a spectroscopicamplitude. The latter accounts for the fragmentation ofsingle-particle strength due to beyond mean-field correla-tions. If one is not interested in the population of specificfinal states, but just in their sum, one may incorporatethe effect of this fragmentation by means of a complex po-tential, whose imaginary part accounts for the spreadingwidth of the single-particle levels into these more com-plicated configurations. This is the case of the dispersiveoptical potential, first introduced by Mahaux and Sartor[7] and recently pursued by several groups (see Ref. [8]for a recent review). The use of this dispersive poten-tials permits a natural extension of the inclusive breakupmodels to negative energies [9, 10]. A recent work, usingthe IAV model in prior form [11], has shown that this pro-cedure leads to a smooth transition between the positiveand negative E x values and hence between the breakupand transfer regions. However, the relation between theprior and form formulations for the case of transfer reac-tions with complex binding potentials has not been estab-lished to our knowledge. In particular, it remains to clar-ify the importance of the non-orthogonality term in thiscase. Indeed, for real potentials, these results should leadto the well-known post-prior equivalence used in trans-fer reactions, and the non-orthogonality term should notcontribute in this case.Guided by these considerations, in this paper, we ad-dress the post-prior equivalence for transfer reactions ofthe form of a + A → b + B ∗ in presence of complex x + A potentials. For that purpose, we revisit and generalizethe IAV model which allows us to describe the breakup( E x >
0) and transfer ( E x <
0) regions in the same foot-ing. We will see that, in the extended version of the IAVmodel, the use of a complex U xA potential leads to dif-ferent formulas for post and prior representations. As apractical application of the derived formulas, we presentcalculations for the Ni( d , pX ) reaction at E = 80 MeV.The paper is organized as follows. In Sec. II we sum-marize the main formulas of the IAV model in post andprior forms, and outline its relation with the prior formUT model. We show how the IAV model can be naturallyextended to final bound states. In Sec. III, the formalismis applied to the Ni( d , pX ) reaction. Finally, in Sec. IVwe summarize the main results. II. THEORETICAL FRAMEWORK
In this section, we briefly summarize the IAV model, inits post and prior forms, and highlight its connection withthe UT model. Further details can be found in Ref. [3]as well as in our previous works [6, 12, 13]. We write theprocess under study as a (= b + x ) + A → b + B ∗ , (1)where the projectile a , composed of b + x , collides with thetarget A , emitting the ejectile b and leaving the residualsystem B ∗ (= x + A ) in any possible final state com-patible with energy and momentum conservation. Thisincludes x + A states with both positive and negativerelative energies.The IAV model, as well as the UT model, treats theparticle b as a spectator, meaning that its interactionwith the target nucleus is described with an optical po-tential U bA .Using the post-form IAV model in DWBA, the inclu-sive breakup differential cross section, as a function ofthe detected angle and energy of the fragment b , is givenby d σd Ω b E b (cid:12)(cid:12)(cid:12) post = 2 π ~ v a ρ ( E b ) X c |h χ ( − ) b Ψ c, ( − ) xA | V post | χ (+) a φ a φ A i| × δ ( E − E b − E c ) , (2)where v a is the velocity of the incoming particle a , V post ≡ V bx + U bA − U bB is the post-form transition oper-ator, ρ b ( E b ) = k b µ b / [(2 π ) ~ ] (with µ b the reduced massof b + B and k b their relative wave number), | φ a i and | φ A i are the projectile and target ground states, χ (+) a and χ ( − ) b are distorted waves describing the a − A and b − B relative motion by the optical potentials U aA and U bB , respectively, and Ψ c, ( − ) xA is any possible state of the x + A many body system, with c = 0 denoting the x and A ground states. Thus, the c = 0 term in Eq. (2) cor-responds to the processes in which the target remains inthe ground state after the breakup, usually called elasticbreakup (EBU), whereas the terms c = 0 correspond tothe so-called non-elastic breakup (NEB) contributions. The theory of IAV allows to perform the sum in a for-mal way, making use of the Feshbach projection formal-ism and the optical model reduction, and leading to thefollowing closed form for the NEB differential cross sec-tion: d σdE b d Ω b (cid:12)(cid:12)(cid:12)(cid:12) postIAV = − ~ v a ρ b ( E b ) h ψ post x | W x | ψ post x i , (3)where W x is the imaginary part of the optical potential U x , which describes x + A elastic scattering. ψ post x ( ~r x ) isa projected x -channel wave function which describes the x − A relative motion for a given outgoing momentum ~k b of the b particle, and obtained after projection onto the A ground state using the Feshbach formalism. It verifiesthe equation | ψ post x i = G x | ρ i post , (4)where G x = 1 / ( E x − H x + iǫ ), with H x = T x + U x , E x = E − E b and h ~r x | ρ post i = ( χ ( − ) b ~r x | V post | φ a χ (+) a i .Udagawa and Tamura [4] proposed a very similar formulafor the same problem, but making use of the prior-formrepresentation. The prior-form x − channel wave functionreads | ψ prior x i = G x | ρ i prior , (5)where h ~r x | ρ prior i = ( χ ( − ) b ~r x | V prior | φ a χ (+) a i . Using thischannel wave function, UT proposed the following prior-form inclusive breakup formula d σdE b d Ω b (cid:12)(cid:12)(cid:12)(cid:12) UT = − ~ v a ρ b ( E b ) h ψ prior x | W x | ψ prior x i , (6)which is analogous to the IAV formula, Eq. (3) .Despite their formal analogy, the UT and IAV expres-sions give rise to different predictions for NEB cross sec-tion. This discrepancy led to a long-standing disputebetween these two groups. The problem has been also re-examined recently [12, 14]. These works have concludedthat the UT formula is incomplete, and must be sup-plemented with additional terms, as we show below. Thecomparison of the IAV and UT models with experimentaldata supports this interpretation [12–17]In general, the NEB will contain also contributionscoming from the population of states below the breakupthreshold of the x + A system ( E x < et al. [11], who have provided an ef-ficient implementation of this idea. The key point is therealization that the Green function G x ( r x , r ′ x ) in Eqs. (4)and (5) can be expressed for both E x > E x < V post = V prior + ( V bx + T bx + U aA + T aA ) − ( U xA + T xA + U bB + T bB ) . (7)We consider a definite final state of the x − A system,denoted | φ n i and evaluate the difference h φ n | ψ post x i − h φ n | ψ prior x i = h φ n χ ( − ) b | ( V bx + T bx + U aA + T aA ) − ( U xA + T xA + U bB + T bB ) | χ (+) a φ a i E x − H x . (8)The first term in parenthesis can be replaced by E when acting on | χ (+) a φ a i . The second parenthesis, actingon h φ n χ ( − ) b | gives also the total energy E , provided H x is Hermitian (i.e. U x real). In that case, we have h φ n χ ( − ) b | V post | χ (+) a φ a i = h φ n χ ( − ) b | V prior | χ (+) a φ a i , (9)which corresponds to the well-known post-prior equiva-lence for transfer reactions.However, when U x is complex we can not perform thelast step. Instead, we may rewrite (8) as | ψ post x i − | ψ prior x i = ( χ ( − ) b | E x − H x | χ (+) a φ a i E x − H x = ( χ ( − ) b | χ (+) a φ a i , (10)The function | ψ NO x i = ( χ ( − ) b | χ (+) a φ a i is the so-called non-orthogonality (NO) overlap, also referred to in the liter-ature as Hussein-McVoy term. Upon replacement of thisrelation into Eq. (3) we finally get d σd Ω b E b (cid:12)(cid:12)(cid:12) IAV = d σd Ω b E b (cid:12)(cid:12)(cid:12) UT + d σd Ω b E b (cid:12)(cid:12)(cid:12) NO + d σd Ω b E b (cid:12)(cid:12)(cid:12) IN (11)where the first term is the UT prior-form formula of theNEB cross section, Eq. (6), d σdE b d Ω b (cid:12)(cid:12)(cid:12)(cid:12) NO = − ~ v a ρ b ( E b ) h ψ NO x | W x | ψ NO x i , (12)is the non-orthogonality term and d σdE b d Ω b (cid:12)(cid:12)(cid:12)(cid:12) IN = − ~ v a ρ b ( E b ) Re [ h ψ NO x | W x | ψ prior x i ] , (13)is the interference term.Equation (11) shows that the IAV post-form formulaand UT prior-form formula are not equivalent. The lat-ter needs to be supplemented with two additional terms,which stem from the non-orthogonality of the initial and final states. Recent calculations comparing these expres-sions with experimental data support the interpretationof IAV [6, 12].We also note that the relations (10) and (11) hold forboth E x < E x > x − A potential is real the contribution of these termsvanish in the DWBA expression. It is one of the pur-poses of this work to assess the validity of (11) in actualcalculations for complex U x .It is enlightening to consider the simple case in which W x is taken as a constant. In this case, if one insertsthe Green’s operator into Eq. (3), the double differentialcross section for transferring particle x to bound statesin post-form resultsd σ d E dΩ (cid:12)(cid:12)(cid:12) postIAV = − ~ v a ρ b ( E b ) h ρ | G † x W x G x | ρ i post (14)Using the explicit form of G x and introducing the iden-tity, P n | φ n ih φ n | , we getd σ d E dΩ (cid:12)(cid:12)(cid:12) postIAV = X n ω n dσ n d Ω (cid:12)(cid:12)(cid:12) post , (15)where we have introduced the notation ω n = − π Γ n ( E x − E n ) + Γ n , (16)with Γ n = h φ n | W x | φ n i = W x and dσ n d Ω (cid:12)(cid:12)(cid:12) post = 2 π ~ v a ρ b ( E b ) (cid:12)(cid:12) h φ n | ρ i post (cid:12)(cid:12) = 2 π ~ v a ρ b ( E b ) |h φ n χ ( − ) b | V post | χ (+) a φ a i| . (17)It should be noted that the single-differential cross sec-tion given by Eq. (17) is nothing but the DWBA crosssection of transfer cross sections. Equation (16) showsthat, for constant W x , the energy distribution of thedouble differential cross section follows a Breit-Wigner(Lorentzian) shape for the bound states E n <
0, andΓ n represents the spreading width of the single-particlelevels generated by the V x potential.Notice that, in the limit case W → ω n → δ ( E x − E n )and hence d σ/ d E dΩ | post is zero everywhere except atthe pole energies E x = E n . III. CALCULATIONS
We consider the reaction Ni( d , pX ) at E lab d =80 MeV. This reaction was also considered in our pre-vious work [12], where we compared with the inclusivebreakup data from Ref. [20] using the original IAV modeland considering only the E x > n - Niresidual system. Here, we extend these calculations tonegative energies ( E x < U x potential, and comparing thepost and prior results.In the present calculations, we consider for the n − p interaction the simple Gaussian form of Ref. [21]. Thedeuteron and proton distorted waves are generated withthe same optical potentials used in Ref. [12]. Theneutron- Ni potential is extrapolated to negative en-ergies by simply fixing its real and imaginary parts totheir values at E n = 1 MeV for E n ≤ U x ( E x < U x ( E x = 1 MeV). The bin proce-dure is used to average the distorted wave χ b over smallmomentum intervals to evaluate the post-form formula.Although this is not required for the prior-form formula,the same averaging procedure is adopted in that case forconsistency with the post-form results.In Fig. 1(a) we present the post and prior calculationsfor the angle-integrated differential cross section of theoutgoing protons as a function of their center-of-massenergy. The black thick solid line is the post-form cal-culation obtained with the IAV post-form model and thered thin solid line is the UT prior-form calculation. It isseen that there is a significant difference between thesetwo calculations, for both E n < E n > B system [17].The fact that the difference between the IAV and UTresults at negative neutron energies originates from theuse of a complex neutron potential is illustrated in panel(b), where the imaginary part of this potential is reducedby a factor of 10. As expected, for E n > E n <
20 40 60 80 100-505101520 d σ / d E p ( m b / M e V ) IAVUTNOINUT+NO+IN20 40 60 80 100 E pc.m. (MeV) d σ / d E p ( m b / M e V ) Ni(d,pX) @ 80 MeV(a)(b)
FIG. 1. (Color online) (a) Angle-integrated proton energyspectra Ni( d , pX ) at E d = 80 MeV. The thick solid line isthe post-form calculation (IAV model). The thin solid, dottedand dot-dashed lines are the UT, NO, IN terms contributingto the prior-form cross section and the dashed line is theirsum. The vertical line indicates the threshold ( E n = 0) en-ergy. (b) Same as panel (a), but with the imaginary part ofthe n - Ni reduced by a factor of 10. See text for details. most perfect agreement between the IAV and UT formu-las, which is the usual post-prior equivalence for transferreactions. We note also that, for this weak-absorptioncase, the differential cross sections displays marked peaksat the position of the bound states and resonances ofthe neutron- Ni potential. In particular, a very narrow ℓ = 4 resonance is found near the neutron threshold.Therefore, the role of the imaginary part is to increaseNEB cross section, but also to smear the contribution ofthe bound states and resonances. This is also apparentfrom Eq. (14) which, in the case of constant W x , predictsa Lorentzian shape with a width given by Γ = W x . Toconclude this section, we notice that, even in the limit ofsmall W x , the IAV and UT results differ for E n >
0. Inthis case, the addition of the NO and IN terms is essen-tial to restore the post-prior equivalence, as shown in ourprevious work [6].
IV. SUMMARY AND CONCLUSIONS
In summary, we have addressed the problem of thepost-prior equivalence in the calculation of inclusivetransfer reactions of the form A ( a, b ) B , were B is anybound state of the x + A system. For that, we have con-sidered the post-form inclusive breakup model proposedby Ichimura, Austern and Vincent (IAV) [1–3], conve-niently extended to negative (bound states) of the x + A system. We have also considered the prior-form model ofUdagawa and Tamura (UT) [4].We have shown that the equivalence between the post-form (IAV) and prior-form (UT) expressions holds onlyfor real x − A potentials. For complex interaction, thenon-orthogonality (NO) term is indispensable. Once thisterm is included, the post-prior equivalence is restored.To assess this equivalence at a numerical level, we haveperformed calculations for the Ni( d , pX ) reaction at 80MeV. We find that, when a complex potential is usedfor the x − A system, the IAV and UT results signif-icantly disagree, both for the unbound ( E x >
0) andbound ( E x <
0) regions. Inclusion of the NO term givesan excellent agreement between the post and prior crosssections. We have also verified that, as the imaginary part of U x is reduced, the UT result approaches the IAVone, thus recovering the well-known post-prior equiva-lence of the DWBA formula.We believe that the present results are relevant becausethey extend a fundamental property of the transition am-plitude, namely, the post-prior equivalence, to the caseof non-Hermitian binding potentials. In particular, theresults will be useful in the context of the exclusive orinclusive transfer reactions with dispersive optical poten-tials, currently under development. ACKNOWLEDGMENTS
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