Energy distributions from three-body decaying many-body resonances
R. Alvarez-Rodriguez, A.S. Jensen, D.V. Fedorov, H.O.U. Fynbo, E. Garrido
aa r X i v : . [ nu c l - t h ] J u l Energy distributions from three-body decaying many-body resonances
R. ´Alvarez-Rodr´ıguez, A.S. Jensen, D.V. Fedorov, H.O.U. Fynbo
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
E. Garrido
Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain (Dated: November 4, 2018)We compute energy distributions of three particles emerging from decaying many-body resonances.We reproduce the measured energy distributions from decays of two archetypal states chosen as thelowest 0 + and 1 + -resonances in C populated in β -decays. These states are dominated by sequential,through the Be ground state, and direct decays, respectively. These decay mechanisms are reflectedin the “dynamic” evolution from small, cluster or shell-model states, to large distances, where thecoordinate or momentum space continuum wavefunctions are accurately computed.
PACS numbers: 21.45.+v, 31.15.Ja, 25.70.Ef
Introduction.
Energy and momentum conservationguarantees that two particles, emerging from decay ofa given quantum state, appear with definite kinetic en-ergies inversely proportional to their masses. In three-body decays the available energy can be continuouslydistributed among the particles. Prominent classical ex-amples are α -emission and beta-decay, respectively. Sur-prisingly enough the decay of a many-body quantumsystem into three particles has not been well describedmicroscopically although discussed phenomenologicallyfor various systems. The process depends on the ini-tial state and the dynamic evolution or equivalently thedecay mechanism.This problem of three-body decay is common to sev-eral subfields of physics. The invention of Dalitz plotswas an early attempt to classify the decay mechanismsby use of intermediate two-body doorway states [1]. Theunderlying dynamics in particle physics may be describedas quark rearrangements. Similar decays occur in anni-hilation of a proton-antiproton pair from a Coulomb-likeorbit into three mesons [2]. In molecular physics an ex-ample is decay of excited states of the H -molecule intothree hydrogen atoms [3]. In nuclear physics there ex-ists a large number of three-body decaying systems ofdisparate structures and decay mechanisms, e.g. variousexcited states of He, Li, C, Ne. More and morehigh-quality experimental data become available in allsubfields [4, 5, 6, 7] and quantitatively accurate mod-els are needed to extract and understand the underlyingphysics.The purpose of the present letter is to compute theenergy distributions for three-body decaying excited nu-clear many-body resonances. We shall assume that theresonances are populated in β -decays and consequentlyonly an outgoing flux is present. For reactions an in-going flux is required to provide the population of thedecaying wavefunction. Such a generalization is eas-ily achieved by allowing initial conditions different fromthat of a resonance wavefunction. In all cases the ma-jor difficulty is to compute accurately the asymptoticlarge-distance three-body wavefunctions corresponding to genuine many-body resonances, which possibly dif-fer at small distances from cluster states formed by theemerging three particles [8].At least four problems must be solved to overcome thedifficulties, i.e. (i) the complex scaled resonance wave-functions must be accurately determined even thoughthey vanish exponentially at large distances, (ii) thewavefunctions must be traced as they “evolve dynami-cally” from relatively small to asymptotically large dis-tances, (iii) the Coulomb problem of coupling continuumstates at infinitely large distances must be solved, (iv)the mixture of two- and three-body asymptotics must beaccurately determined. Theoretical framework.
We use the hypersphericaladiabatic expansion method of the Faddeev equationscombined with complex scaling [9, 11]. The hyperradius ρ is the most important of the coordinates. For threeidentical particles of mass m α the definition is m N ρ = m α X i 92 MeV for 0 + and 1 + , respectively [21]. However, special treatment is required in the variationsfrom intermediate to large distances. The necessary ba-sis sizes become insurmountably large. Our solution isto compute an accurate wavefunction at intermediate,but relatively large, distances. This is achieved when aneven larger hyperradius, compensated by a larger basis,leads to the same observables derived from the wavefunc-tions. This stability condition is difficult to reach whenboth three-body background continuum states are pop-ulated simultaneously with resonances in one or more ofthe two-body subsystems. At sufficiently large distanceswe can precisely identify these structures as componentsin the complex scaled wavefunctions related to differentadiabatic potentials [10], e.g. sequential decay proceedsthrough a potential approaching the corresponding com-plex two-body energy E b [11], whereas no intermediatestructure is present for direct decay to the continuum.When the two-body intermediate states have largewidths the related radial wavefunction decreases quickly,because then the adiabatic couplings are large. These states then dissipate fast into the continuum described asdirect decay and the distinction becomes artificial. Thisprocess eventually happens for all sequential decays sincethe intermediate states are unstable. Classification intosequential and direct decay is related to the use of dif-ferent complete basis states, i.e. either two-body reso-nances and the third particle in the continuum or three-body continuum states. Interpretation as sequential ordirect is then meaningful when a few states in one basisare sufficient while many are needed in the other. Such adistinction between paths producing the same observableis not possible in quantum mechanics. Illuminating archetypes. Accurate measurements of α -particle energy distributions are available from decaysof 0 + and 1 + -resonances of C [6, 7, 12, 13]. The low-est 0 + -resonance is often described as a cluster state[14, 15, 16, 17, 18] whereas the 1 + -resonance in contrastis referred to as a shell-model state without any signifi-cant cluster structure [14, 15, 16, 17, 18]. Furthermore,the decay mechanisms are known to be different [6, 7].These cases are therefore ideally suited as illustrations ofthe present novel technique. E α /E α ,max P ( E α / E α , m a x ) ( a . u . ) n=1n=2 FIG. 2: The α -particle energy distribution for the 1 + reso-nance of C at 5.43 MeV above threshold at an excitation of12.71 MeV. The energy is measured in units of the maximumpossible, i.e. 2 × . / ρ = 70 , 100 fm. The thin curves arecontributions from separate adiabatic potentials. The mo-mentum space computation (described below) fall on top ofthe ρ = 100 fm curve (thick dashed). The histogram is theexperimental distribution [13]. In fig. 1 we show the lowest potentials where the attrac-tive pockets at small distance support the resonances andprovide the small distance boundary conditions. As thehyperradius increases beyond the barriers, the potentialsall decrease as 1 /ρ due to the Coulomb repulsion. Thestructure at large distances is necessarily of three-bodycharacter since this is the boundary condition imposed bythe measurement. In contrast, at small distances theseclusters overlap and the detailed description must use thenucleon degrees of freedom. The first adiabatic potentialcorresponds to the Be(0 + ) state and therefore associatedwith this sequential decay. We shall explore the, perhapssurprising, conjecture that the decay can be describedalmost entirely within the present cluster model. The + -resonance. With the potentials in fig. 1 weshow the computed energy distribution in fig. 2 for the1 + -resonance where sequential decay via the Be ground-state is forbidden. The asymptotic behavior is reachedalready for hyperradii larger than about 60 fm. The smallvariation of the distribution from 70 fm to 100 fm showthe convergence and the stability. Higher accuracy is ob-tained at these distances with a moderate basis size thanat larger distances where the basis quickly becomes insuf-ficient. Two interfering adiabatic potentials are necessaryto reach the impressive agreement with the measured dis-tributions. It is remarkable that the cluster model pro-vides this accuracy in spite of the fact that the initialdecaying state is a many-body resonance without anythree-body structure. ρ (fm) |f /f | |f /f | |f /f | E = 0.38 MeVE = 3.95 MeV|f | |f | FIG. 3: The four lowest radial wavefunctions as functionsof ρ for each of the two 0 + -resonances of C at 0 . 38 MeVand 3.95 MeV above threshold or at excitation energies of7 . 63 MeV and 11.2 MeV [13]. Ratios and the small-distancedominating wavefunctions are given for both states. To test the reliability we Fourier transformed the wave-function in two ways, i.e. first numerically with coordi-nates from ρ = 0 to 100 fm and secondly by use of the an-alytic solution obtained from the parametrized adiabaticpotentials which asymptotically are sums of 1 /ρ and 1 /ρ terms. The results are remarkably similar distributionsand the analytic result is in fact indistinguishable fromthe curve for ρ = 100 fm in fig. 2. Small deviations fromthe measurements could be due to two-body interactionswith resonance properties deviating slightly from the val-ues measured for Be. However, most of the differencesare more likely due to uncertainties arising from accep-tance of the detectors used in the experiment. It is amusing to estimate that the sequential decay viathe Be 2 + -resonance would produce a similar centralpeak of about twice the width. To reproduce the datastrong interference would then be necessary. Good phe-nomenological reproduction of the data is obtained byR-matrix theory where the smaller width is explaineddue to preferentially populating the low energy tail ofthe Be 2 + -resonance, and where effects of interferencealso play an important role [6]. E α (MeV) P ( E α ) ( a . u . ) direct (th)sequential (th)total (th)direct (exp)sequential (exp)total (exp) FIG. 4: The α -particle energy distribution for the second 0 + resonance of C at 4.3 MeV above threshold at an excita-tion of 11.2 MeV [7, 13]. The interactions in fig. 1 give anenergy of 3.95 MeV above threshold [21]. We use 2 . × . / ≈ . The + -resonances. The complex scaled radial wave-functions are shown in fig. 3 for the two lowest 0 + -resonances. The largest probability is found at smalldistance and they all vanish with increasing hyperra-dius. Their relative sizes are fairly insensitive to vari-ations of the hyperradius at large distances where theenergy distributions are determined. The first resonanceis described by the first adiabatic component for all dis-tances whereas the second resonance changes characterfrom small to large ρ from 4% to 75% of the first adiabaticcomponent. This component, which dominates for bothresonances at large distance, approaches the configura-tion of the 0 + -resonance in Be with the third α -particlefar away, i.e. s -waves between each pair of α -particlesfor each of the three Faddeev components. The resultis an energy distribution with characteristic features ofsequential three- α decay, i.e. a narrow high-energy peakand a distribution around one quarter of the maximumof the C resonance energy. Unfortunately these com-puted distributions are not accurate because the two-body asymptotic behavior in these cases are not reachedfor ρ less than 100 fm. However, the method provides theamount of sequential decay and we can substitute the in-accurate component by the known two-body asymptoticbehavior. The energy distribution from decay of the first0 + -resonance at 0 . 38 MeV is then seen from fig. 3 to bealmost entirely determined by the first potential whichmeans sequential decay. The direct decay is only about1% in agreement with the experimental upper limit [12].This energy distribution is then in complete agreementwith experimental data.The second 0 + -resonance is also dominated by the firstadiabatic potential at large distance. This is in strikingcontrast to the domination by the second potential atsmall distance. This is an example of the importance ofthe dynamical evolution from small to large distances.The result is about 75% sequential (first potential) and25% direct decay described by the other adiabatic poten-tials. In comparison with measurements complicationsarise for two reasons both related to the large width ofthe order 1 MeV. First, effects of energy-dependent feed-ing in the beta-decay populating the decaying state aresubstantial in the data [7, 13]. Higher beta-energies arerather strongly favored resulting in distributions movingtowards lower energies. Second, the experimental analy-sis is hampered by possible effects from other resonances.Their contributions are possibly not fully disentangled.The peak energy corresponding to the resonance po-sition is at about 2 . α -particle by us-ing the Breit-Wigner distribution defined with the mostprobable position at 2 . α -particles following from decay of Be are uniquely relatedby kinematic conditions resulting in a peak at lower en-ergy. The large width of the three-body decaying reso-nance smears out the latter distribution. Between thesetwo peaks appears the contribution of about 25% fromdirect decay described by the other adiabatic potentials.The inaccuracies in the computed distributions are firstthat deviations from the Breit-Wigner shape become im-portant for the large width of 1 MeV, and second thatthe fraction of sequential decay may be underestimatedby perhaps 10% due to missing higher partial waves. In any case, the shape of the sequential decay via the Be ground state is derived by precisely the same kine-matic conditions in both the computation and the ex-perimental analysis. The largest differences between the-ory and experiment is simulated by the shift of resonancepeak energy. The agreement is rather good and only pos-sible due to the computed decay mechanism of dynamicevolution with hyperradius. Summary and conclusions. We have computed theenergy distributions for three-body decaying many-bodyresonances. Combinations of short-range and repulsiveCoulomb interactions are allowed. We conjecture, andshow in specific cases, that the energy distributions ofthe decay fragments are insensitive to the short-distancemany-body structure, but accessible in a three-body clus-ter model. The resonance structures may be completelydifferent at small and large distances. This dynamic evo-lution is decisive for the decay mechanism. We separatecomponents with two- and three-body asymptotics cor-responding to sequential and direct decays. This distinc-tion is crucial to obtain accurate wavefunctions at largedistances. We test the method by comparing results fromcoordinate and momentum space.We illustrate by application to the archetypes of α -decaying 0 + and 1 + -states in C. The 1 + -resonance can-not be described as a three-body state but its decayproceeds directly into the three-body continuum. Thetwo 0 + -resonances both have substantial, but very dif-ferent, cluster components at small distances. However,they both decay preferentially through the same largedistance structure best described as the 0 + -resonance of Be. These sequential decays imply a total rearrange-ment of the second of these resonances from small tolarge distances. The accurately measured α -particle en-ergy distributions for all three resonances populated in β -decay are reproduced remarkably well. Thus the methodhas past very severe tests. It is reliable and with predic-tive powers. Acknowledgments. R.A.R. acknowledges support bya post-doctoral fellowship from Ministerio de Educaci´ony Ciencia (Spain). [1] R.H. Dalitz, Philosophical. mag. , 1068 (1953).[2] C. Amsler, Rev. Mod. Phys. , 1293 (1998).[3] U. Galster, U. M¨uller, H. Helm, Phys. Rev. Lett. ,073002 (2004).[4] U. Galster, F. Baumgartner, U. M¨uller, H. Helm, andM.Jungen, Phys. Rev. A72 , 062506 (2005).[5] B. Blank et al. , C. R. Physique , 521 (2003).[6] H.O.U. Fynbo et al., Phys. Rev. Lett. , 082502 (2003).[7] C. Aa. Diget et al., Nucl. Phys. A 760 , 3 (2005).[8] E. Garrido, D.V. Fedorov, H.O.U. Fynbo and A.S.Jensen, Nucl. Phys. A 781 , 387 (2007).[9] E. Nielsen, D.V. Fedorov, A.S. Jensen, and E. Garrido,Phys. Rep. , 373 (2001).[10] E. Garrido, D.V. Fedorov, A.S. Jensen and H.O.U.Fynbo, Nucl. Phys. A 766 , 74 (2005). [11] D.V. Fedorov, E. Garrido, and A.S. Jensen, Few-bodysystems, , 153 (2003).[12] M. Freer et al. , Phys. Rev. C49 , R1751 (1994)[13] C. Aa. Diget, PhD-thesis, IFA, Univ. of Aarhus, 2006.[14] Y. Kanada-En’yo, Phys. Rev. Lett. , 5291 (1998).[15] T. Neff and H. Feldmeier, Nucl. Phys. A 738 , 357 (2004).[16] P. Descouvemont, Nucl. Phys. A 709 , 275 (2002).[17] S.C. Pieper, Nucl. Phys. A 751 , 516 (2005).[18] P. Navratil, J.P. Vary, and B.R. Barrett, Phys. Rev. C62 , 054311 (2000).[19] D. V. Fedorov and A. S. Jensen, Phys. Lett. B389 , 631(1996).[20] S. Ali and A.R. Bodmer, Nucl. Phys. , 99 (1966).[21] R. ´Alvarez-Rodr´ıguez, E. Garrido, A.S. Jensen, D.V. Fe-dorov, and H.O.U. Fynbo, Eur. Phys. J. A 31 , 303, 303