β decay of deformed r-process nuclei near A 80 and A 160 ...engelj/papers/sefmms.pdfPHYSICAL REVIEW C 94, 055802 (2016) β decay of deformed r-process nuclei near A = 80 and A = 160,

PHYSICAL REVIEW C 94, 055802 (2016)

β decay of deformed r-process nuclei near A = 80 and A = 160, including odd-A and odd-oddnuclei, with the Skyrme finite-amplitude method

T. Shafer* and J. Engel†Department of Physics and Astronomy, CB 3255, University of North Carolina, Chapel Hill, North Carolina 27599-3255

C. Fröhlich and G. C. McLaughlinDepartment of Physics, North Carolina State University, Box 8202, Raleigh, North Carolina 27695

M. MumpowerDepartment of Physics, University of Notre Dame, Notre Dame, Indiana 46556

and Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544

R. SurmanDepartment of Physics, University of Notre Dame, Notre Dame, Indiana 46556

(Received 22 June 2016; revised manuscript received 3 September 2016; published 7 November 2016)

After identifying the nuclei in the A � 80 and A � 160 regions for which β-decay rates have the greatesteffect on weak and main r-process abundance patterns, we apply the finite-amplitude method (FAM) withSkyrme energy-density functionals (EDFs) to calculate β-decay half-lives of those nuclei in the quasiparticlerandom-phase approximation (QRPA). We use the equal filling approximation to extend our implementation ofthe charge-changing FAM, which incorporates pairing correlations and allows axially symmetric deformation,to odd-A and odd-odd nuclei. Repeated calculations with A � 160 nuclei and multiple EDFs show a spread of1.9–3.3 in β-decay half-lives, with differences in calculated Q values playing an important role. We compare ourresults with those of previous work and investigate their implications for r-process simulations.

DOI: 10.1103/PhysRevC.94.055802

I. INTRODUCTION

The solar abundances of nuclei heavier than iron, on theneutron-rich side of stability, have traditionally been attributedto rapid neutron-capture, or r-process, nucleosynthesis [1].The three largest abundance peaks in the solar pattern, atA ∼ 80, 130, and 195, are associated with the closed neutronshells at N = 50, 82, and 126, suggesting that astrophysicalconditions of increasing neutron richness are responsible foreach. A smaller fourth abundance peak in the rare-earthelements (A ∼ 160) is also formed in neutron-rich environ-ments. Observational data from meteorites and metal-poorhalo stars confirm the separate origins for 70 � A � 120(“weak”) and A > 120 (“main”) r-process nuclei and providehints of the nature of the r-process astrophysical site, thoughthe exact site (or sites) has not yet been definitively pinneddown [2].

In principle the r-process sites can be identified by com-paring simulations of prospective astrophysical environmentswith observational data from the solar system and other stars(see, e.g., Ref. [3]). The precision of r-process abundancepredictions, however, is limited by our incomplete knowledgeof properties—such as masses, reaction rates, and decaylifetimes—of nuclei on the neutron-rich side of stability [4].It is particularly important that we better determine decaylifetimes, because r-process nuclei are built up via a sequence

*[email protected]†[email protected]

of captures and β decays. Thus, β-decay lifetimes determinethe relative abundances of the nuclei along the r-process path[1,2,5] and the overall time scale for neutron capture [6,7].At late times, as nuclei move back from the r-process pathtoward stability and the last remaining neutrons are captured,the lifetimes determine the shape of the final abundance pattern[5,8]. Finally, for a weak r process β-decay rates controlthe amount of material that remains trapped in the A ∼ 80peak and the amount that moves to higher mass numbers[9], i.e., they determine where the weak r process terminates.For all these reasons, an accurate picture of the β decay ofneutron-rich nuclei is crucial for the accuracy of r-processsimulations.

Although many β-decay lifetimes have been measured (see,e.g., Refs. [9–13]), most of the nuclei populated during the rprocess remain out of reach. Simulations must therefore rely oncalculated lifetimes. The most widely used sets of theoreticalrates are from gross theory [14–16] and from an application ofthe quasiparticle random-phase approximation (QRPA) withina macroscopic-microscopic framework [6,17] that employsgross theory for first-forbidden transitions. Here we use a fullymicroscopic Skyrme QRPA, implemented through the proton-neutron finite-amplitude method (pnFAM) [18] and extendedto treat odd-A and odd-odd nuclei (hereafter “odd” nuclei)in the equal-filling approximation (EFA) [19]. We can nowuse arbitrary Skyrme energy-density functionals (EDFs) toself-consistently compute β-decay rates of both even-even andodd axially symmetric nuclei, including contributions of bothallowed (Jπ = 1+) and first-forbidden (Jπ = 0−,1−, or 2−)transitions.

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https://doi.org/10.1103/PhysRevC.94.055802

T. SHAFER et al. PHYSICAL REVIEW C 94, 055802 (2016)

We evaluate lifetimes for key r-process nuclei in twohighly populated regions of the abundance pattern: the largemaximum at A ∼ 80 and the smaller rare-earth peak at A ∼160. In a main r process, the rare earth peak forms in a differentway than do the large peaks at A ∼ 130 and 195, both of whichoriginate from long-lived “waiting points” near closed neutronshells at N = 82 and 126. The rare-earth peak, by contrast,forms during the late stages of the r process, as β decay,neutron capture, and photodissociation all compete with oneanother and the r-process path moves toward stability [20,21].The A ∼ 160 abundance peak is thus useful for studying themain r-process environment [22,23]. The A ∼ 80 region is notso clearly related to the main r process. In fact, nuclei with70 � A � 120 can be created in a variety of nucleosyntheticprocesses, ranging from the neutron-rich weak r process tothe proton-rich νp process [24–26] (see also Refs. [27,28]).Untangling the various contributions to these elements requiresrigorous abundance pattern predictions, which in turn requirea better knowledge of still unmeasured β-decay half-lives.

In this paper, we aim to study and improve r-processabundance predictions for both weak r-process nuclei andthe rare-earth elements by identifying and recalculating keyβ-decay rates. We begin in Sec. II by reviewing the pnFAMand then discussing our extension to odd nuclei. In Sec. III,guided by r-process sensitivity studies, we calculate β-decayrates separately for the important isotopes in the two massregions (after optimizing the Skyrme EDF separately for eachregion). We also examine the effect on β-decay half-livesof varying the Skyrme EDF in rare-earth nuclei. Finally, inSec. IV we discuss the impact of our β-decay rates on r-processabundances. Section V is a conclusion.

II. NUCLEAR STRUCTURE

A. The proton-neutron finite-amplitude method

The finite-amplitude method (FAM) is an efficient way tocalculate strength distributions in the random-phase approxi-mation (RPA) or the QRPA. Nakatsukasa et al. first introducedthe FAM to calculate the RPA response of deformed nuclei[29] with Skyrme EDFs, and the method was rapidly extendedto include pairing correlations in Skyrme QRPA, both forspherical [30] and axially deformed nuclei [31], and to includesimilar correlations in the relativistic QRPA [32]. Reference[18] applied the same ideas to charge-changing transitions, inparticular those involved in β decay; the resulting method iscalled the pnFAM. Like the FAM implemented in Ref. [31],the pnFAM computes strength functions for transitions thatchange the K quantum number by arbitrary (integer) amountsin spherical or deformed superfluid nuclei.

The first work with the pnFAM focused on the impact oftensor terms in Skyrme EDFs [18]. More recently, the authorsof Ref. [33] used the method to constrain the time-odd part ofthe Skyrme EDF and compute a β-decay table that includesthe half-lives of 1387 even-even nuclei. We leave most detailsof the pnFAM itself to these references, but repeat the mainpoints here in anticipation of the extension to odd nuclei inSec. II B.

QRPA strength functions are related to the linear time-dependent response of the Hartree-Fock-Bogoliubov (HFB)mean field (see, e.g., Refs. [34,35] for a discussion). The staticHFB equation can be written as

[H0,R0] = 0, (1)where (e.g., for protons or neutrons)

R0 =(

ρ0 κ0

−κ∗0 1 − ρ∗0

), H0 =

(h0 �0

−�∗0 −h∗0

). (2)

In Eq. (2), R0 is the generalized static density (the subscript0 indicates a static quantity), built from the single-particledensity ρ0 and the pairing tensor κ0 (see, e.g., Ref. [34]), andH0 is the static generalized mean field, built from the staticmean field h0 and the static pairing field �0. The generalizedmean field H0 depends on both ρ0 and κ0 and is usually writtenH0[R0]. The matricesR0 andH0 are diagonalized by a unitaryBogoliubov transformation,

W =(

U V ∗

V U ∗

), (3)

which connects the set of single-particle states (created by c†k)in which the problem is formulated to a set of quasiparticlestates (created by α†μ):(

c

c†

)=

(U V ∗

V U ∗

)(α

α†

). (4)

Thus, the transformed generalized density and mean field,

R0 ≡ W†R0W, H0 ≡ W†H0W, (5)are in the quasiparticle basis and have the diagonal form,

R0 =(

0 0

0 1

), H0 =

(E 0

0 −E)

. (6)

In the pnFAM we solve the small-amplitude time-dependent HFB (TDHFB) equation,

iṘ(t) = [H[R(t)] + F(t), R(t)], (7)where F(t) is a time-dependent external field that changesneutrons into protons or vice versa. Equation (7) determinesthe oscillation of the generalized density around the staticsolution R0 of Eq. (1); for external fields proportional to a smallparameter η, a first-order expansion R(t) ≈ R0 + ηδR(t) issufficient to describe the behavior of the nucleus. It leads tothe linear-response equation:

iδṘ(t) = [H0, δR(t)] + [δH(t) + F(t), R0]. (8)Here δH(t) and δR(t) are the first-order changes in thegeneralized mean field and density.

If the perturbing field oscillates at a frequency ω, theresulting generalized density can be written in the form:

δR(t) = δR(ω)e−iωt + δR†(ω)eiωt , (9)with

δR(ω) ≡(

0 X(ω)

−Y (ω) 0)

, (10)

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β DECAY OF DEFORMED r-PROCESS NUCLEI . . . PHYSICAL REVIEW C 94, 055802 (2016)

where the requirement that R(t) remain projective (R2 = R)forces the diagonal blocks to be zero [32]. The time-dependentgeneralized Hamiltonian also oscillates harmonically, with

δH(ω) =(

δH 11(ω) δH 20(ω)

−δH 02(ω) −δH 11(ω)

). (11)

The block superscripts refer to the number of quasiparticlescreated and destroyed by the corresponding block Hamiltonian(cf. Refs. [30,32]); the 20 and 02 blocks are made up of termsproportional to α†α† and αα , respectively, and the 11 and 11blocks of terms proportional to α†α and αα†.

Putting everything together in Eq. (8) [including theoscillating external field F(t), which we have not written outexplicitly here], and evaluating the commutators, one obtainsthe pnFAM equations [18]:

Xπν(Eπ + Eν − ω) + δH 20πν(ω) = −F 20πν, (12a)Yπν(Eπ + Eν + ω) + δH 02πν(ω) = −F 02πν, (12b)

where π and ν label proton and neutron states, and Eπ and Eνare single-quasiparticle energies. Equation (12) can be put intomatrix-QRPA form [30], but it is more easily solved directly(through iteration) [18,29,30]. The FAM transition strength isthen just given by S(F ; ω) = tr F†δR(pn)(ω) [18,29,30].

B. The equal-filling approximation and the linearresponse of odd nuclei

Our pnFAM code and HFBTHO, the HFB code on whichit is based, require time-reversal-symmetric nuclear states[36]. To apply the FAM to odd nuclei, the ground statesof which break time-reversal symmetry, we use the EFA, a“phenomenological” approximation, in the words of Ref. [19],in which the interaction between the odd nucleon and the coreare captured at least partially without breaking time-reversalsymmetry.

In odd-nucleus density-functional theory, the ground stateis typically represented in leading order by a one-quasiparticleexcitation of an even-even core, |�〉 = α†|�〉. This state,however, produces the time-reversal-breaking single-particleand pairing densities [19,37],

ρkk′ = (V ∗V T )kk′ + UkU ∗k′ − V ∗kVk′, (13a)κkk′ = (V ∗UT )kk′ + UkV ∗k′ − V ∗kUk′. (13b)

The EFA replaces the densities in Eq. (13) with new onesthat average contributions from the state α†|�〉 and its time-reversed partner α†

̄|�〉:

ρEFAkk′ = (V ∗V T )kk′ + 12 (UkU ∗k′ + Uk̄U ∗k′̄−V ∗kVk′ − V ∗k̄Vk′̄), (14a)

κEFAkk′ = (V ∗UT )kk′ + 12 (UkV ∗k′ + Uk̄V ∗k′̄−V ∗kUk′ − V ∗k̄Uk′̄). (14b)

Here we have assumed that the state of the even-even core|�〉 is time-reversal even. The odd-A HFB calculation thenproceeds as usual with ρ → ρEFA and κ → κEFA [19].

The EFA appears to be an excellent approximation to thefull HFB solution for odd-A nuclei. Reference [37] containscalculations of odd-proton excitation energies in rare-earthnuclei, in both the EFA and the less restrictive blockingapproximation. The EFA reproduces the full one-quasiparticleenergies to within a few hundred keV. The approximation wasgiven a theoretical foundation in Ref. [19], which showed thatρEFA and κEFA can be obtained rigorously by abandoning theusual product form of the HFB solution and instead describingthe nucleus as a mixed state. From this point of view, thenucleus is not represented by a single state vector |�〉but rather by a statistical ensemble with a density operatorD̂ ≡ exp K̂ [38]:

D̂ = |�〉〈�| +∑

μ

α†μ|�〉pμ〈�|αμ

+ 12!

∑μν

α†μα†ν |�〉pμpν〈�|αναμ + · · · . (15)

In Eq. (15), pμ is the probability that the excitation α†μ|�〉 iscontained in the ensemble. Expectation values are traces withD̂ in Fock space (we use ‘Tr’ for these traces and ‘tr’ for theusual trace of a matrix),

〈A〉 = Tr[D̂Â]/ Tr[D̂], (16)so that, e.g., the particle density is

ρkk′ = Tr[D̂c†k′ck]/ Tr[D̂]. (17)An ensemble like the above is familiar from finite-

temperature HFB [39], where the quasiparticle occupationsare statistical and determined during the HFB minimization.Reference [19] shows that the EFA emerges from a specificnonthermal choice of the ensemble probabilities:

pμ ={

1, μ ∈ [,̄]0, otherwise.

(18)

With these values of pμ, one finds that for an arbitrary one-body operator Ô,

〈Ô〉o-e = 12 (〈�|αÔα†|�〉 + 〈�|ᾱÔα†

̄|�〉), (19)

and the trace in Eq. (17) produces ρEFA (14a). The formalismmay be applied in a straightforward way to odd-odd nucleias well by constructing an ensemble from the proton (π ) andneutron (ν) orbitals π,̄π ,ν , and ̄ν . Then one finds that

〈Ô〉o-o = 12 (〈�|αν απ Ôα†

π

α†

ν

|�〉+ 〈�|α

̄να

̄πÔα

†

̄π

α†

̄ν

|�〉). (20)The statistical interpretation of the EFA allows us to extend

the pnFAM, which is an approximate time-dependent HFB,to odd nuclei. The thermal QRPA, described in Refs. [40–44], generalizes Eq. (8) to a statistical density operator D̂ =exp K̂ and a thermal ensemble; here we do the same with thenonthermal ensemble in Eq. (18).

Of the matrices that enter the TDHFB equations (7), onlythe generalized density R is fundamentally altered in theEFA; the external field is unaffected and the ground-stateHamiltonian matrix H0 assumes its usual form [19]. But the

055802-3


replacement 〈�|Â|�〉 by Tr[D̂Â]/ Tr[D̂] has implications forboth the static density R0 and the time-dependent perturbationδR(t). In the usual HFB, the definition of the generalizeddensity in the quasiparticle basis [34,35],

R =(

〈α†α〉〈αα〉〈α†α†〉〈αα†〉

), (21)

leads to the form of R0 in Eq. (6). In the EFA ensemble,however, the expectation values 〈α†α〉 and 〈αα†〉 are [19]

〈α†ναμ〉 = δμνfμ, (22a)〈ανα†μ〉 = δμν(1 − fμ), (22b)

leading to an R0 with the more general form,

REFA0 =(

f 0

0 1 − f)

. (23)

The matrix f is diagonal, with factors fμ related to the pμ(18) and taking on the values,

fμ ={

12 , μ ∈ [,̄],0, otherwise.

(24)

The use of an ensemble also changes the way we calculatethe response δR(t). Following Ref. [40], we consider the evo-lution of the density operator under a unitary transformationU (t) = exp[iηŜ(t)]. The operator Ŝ(t) is undetermined, butHermitian. To first order in η, the ensemble evolves as

D̂(t) � [1 + iηŜ(t)]D̂(0)[1 − iηŜ(t)]≡ D̂(0) + ηδD̂(t),

(25)

with δD̂(t) = −i[D̂, Ŝ(t)]. The cyclic invariance of the trace[19] guarantees that Tr[δD̂(t)] = 0. The time evolution of D̂(t)determines the evolution of δR(t), e.g., via

〈α†α〉 → Tr[D̂(t)α†α]/ Tr[D̂(0)]= 〈α†α〉 − iη〈[Ŝ(t), α†α]〉, (26)

so that δR is no longer block antidiagonal as in Eq. (10).Instead it has the form,

δR(t) ≡(

Pπν(t) Xπν(t)

−X∗πν(t) −P ∗πν(t)

), (27)

with P (t) and X(t) proportional to matrix elements of Ŝ(t):

Pπν(t) ≡ i(fν − fπ )S11πν(t), (28a)Xπν(t) ≡ i(1 − fν − fπ )S20πν(t). (28b)

[The matrices S11 and S20 arise from the quasiparticlerepresentation of the one-body operator Ŝ(t); see, e.g., theAppendix of Ref. [34].] When the external field is sinusoidal,we have

Pπν(t) = Pπν(ω)e−iωt + Q∗πν(ω)eiωt , (29a)Xπν(t) = Xπν(ω)e−iωt + Y ∗πν(ω)eiωt , (29b)

and, finally, the frequency-dependent perturbed density for anodd nucleus in the EFA is

δR(ω) =(

Pπν(ω) Xπν(ω)

−Yπν(ω) −Qπν(ω))

. (30)

The use of the EFA ensemble doubles the number of pnFAMequations, from two to four:

Xπν(ω)[(Eπ + Eν) − ω] = −(1 − fν − fπ )[δH 20πν(ω)+F 20πν

],

(31a)

Yπν(ω)[(Eπ + Eν) + ω] = −(1 − fν − fπ )[δH 02πν(ω)+F 02πν

],

(31b)

Pπν(ω)[(Eπ − Eν) − ω] = −(fν − fπ )[δH 11πν(ω) + F 11πν

],

(31c)

Qπν(ω)[(Eπ − Eν) + ω] = −(fν − fπ )[δH 11πν(ω) + F 11πν

].

(31d)

Equations (31) are coupled through the dependence of theHamiltonian matrix δH(ω) on the perturbed density δR(ω). Inaddition to the two “core” equations for X and Y , which aremodified from Eq. (12), the EFA pnFAM includes equations forthe matrices P (31c) and Q (31d), which describe transitionsof the odd quasiparticle(s).

The EFA pnFAM equations are actually no more difficult tosolve than the usual ones. Equation (31) contains the additionalmatrices labeled 11 and 11, but, because we solve for Hiteratively in the single-particle basis and then transform tothe quasiparticle basis [18], we multiply by the Bogoliubovmatrix W in Eq. (3) to obtain these additional matrices. A fewmore iterations may be needed to solve the linear responseequations, but that does not significantly increase computationtime.

We compute the strength function in odd nuclei in thesame way as in even ones, but because δR(ω) is not blockantidiagonal, the valence nucleon(s) affects S(F ; ω) explicitlythrough P and Q, as well as implicitly through X and Y :

S(F ; ω) =∑πν

[F 20∗πν Xπν(ω) + F 02∗πν Yπν(ω)

+F 11∗πν Pπν(ω) + F 11∗πν Qπν(ω)]. (32)

Equation (32) can be obtained directly from the EFA expec-tation value Tr[F̂ †D̂(t)]/ Tr[D̂(0)] (cf. Refs. [18,29,30]) byrequiring that δD̂(t) vary sinusoidally. With the EFA ensemble,S(F ; ω) is simply the average transition strength from theequally occupied odd-A ground states α†|�〉 and α†̄|�〉:

S(F ; ω) = 12 [S(F ; ω) + S̄(F ; ω)]. (33)[See Eq. (19).] The two EFA states include the polarization ofthe core because of the valence nucleon, at least partially [19].

In Fig. 1, we plot the total Gamow-Teller strength functionfor the proton-odd nucleus 71Ga. (In our EFA calculation,71Ga has a slight deformation β2 = −0.007; our methodsfor extracting laboratory-frame transition strength from adeformed intrinsic nuclear ensemble are presented in theappendix). The top panel compares the strength functions

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0 2 4 6 8 10

QRPA energy (MeV)

10–2

10–1

1

10

GT

stre

ngth

(MeV

–1)

(a)EFA

Even-even

0 2 4 6 8 10

Excitation energy (MeV)

10–3

10–2

10–1

1

10

GT

stre

ngth

(MeV

–1)

(b)Borzov et al. (1995)

FIG. 1. (Top panel) Gamow-Teller transition strength for 71Ga,computed with the EFA pnFAM (red, solid lines) and the even-evenpnFAM with the underlying HFB state constrained to have the correctodd average particle number, as in Ref. [45] (green, dashed lines).(Bottom panel) The same EFA pnFAM strength function as in the toppanel, plotted versus excitation energy Eex = EQRPA − E0(pn) [seeEqs. (34) and (38)], alongside the strength function from Ref. [46](blue, dotted line).

obtained with the EFA pnFAM and the even-even pnFAM,artificially constrained to obtain the correct odd particlenumber as suggested in Ref. [45]. The two calculations applythe same Skyrme energy-density functional (SV-min) withoutproton-neutron isoscalar pairing, but begin with distinct HFBcalculations. The EFA calculation clearly includes importantone-quasiparticle transition strength near EQRPA = 1 MeV thatis not present in the other calculation. The bottom panelcompares the EFA-pnFAM strength, this time as a function ofexcitation energy in the daughter nucleus [shifted downward inenergy by E0(pn) � 749 keV—see Eq. (38) and the discussionaround Eq. (34)] with a finite Fermi system calculation fromRef. [46]. Although the two calculations do not yield identicalstrength functions, they clearly mirror one another, and bothinclude low-energy one-quasiparticle strength.

Finally, the odd-A formalism of Ref. [47], used by theauthors of Ref. [17], is an approximate version of ours.We would recover similar expressions to those in Ref. [47]by substituting a separable Gamow-Teller interaction for the

Skyrme interaction and dropping terms beyond leading orderin Pπν and Qπν .

C. Application to β decay in deformed nuclei

Reference [18] discusses the calculation of β-decay ratesfrom pnFAM strength functions at length, so we make onlya few important points here. First, we treat the quenchingof Gamow-Teller strength by using an effective value gA =−1.0 for the axial-vector coupling constant, in both allowedand first-forbidden β-decay transitions. This renormalizationis slightly different from that in Ref. [33], where only Gamow-Teller transitions were quenched. Second, we apply the Q-value approximation of Ref. [7]:

Qβ = �Mn−H + λn − λp − E0(pn). (34)Here �Mn−H is the neutron-hydrogen mass difference, the λqare Fermi energies, and E0(pn) is the energy of the lowest two-quasiparticle state for even-even nuclei, or the smallest one-quasiparticle transition energy for odd nuclei. We approximateE0(pn) with one-quasiparticle energies from the HFB solution;this choice affects only Qβ , not the size of the QRPA energywindow, which is determined as in Ref. [18]:

EmaxQRPA = Qβ + E0(pn) = λn − λp + �Mn−H . (35)Our procedure for going from the intrinsic frame to thelaboratory frame, generalized to include the odd-A EFAensemble, is described in the Appendix.

III. HALF-LIFE CALCULATIONS

A. Identification of important nuclei

To identify the most important β-decay rates for weakand rare-earth r-process nucleosynthesis, we turn to two setsof nucleosynthesis sensitivity studies. Reference [4] reviewssensitivity studies for main r processes; our studies hereproceed as described in Refs. [5,48,49] and Sec. 5.2 of thereview, Ref. [4].

The rare-earth peak is formed in a main r process, sofor our first set of studies we begin with several choices ofastrophysical conditions that produce a good match to thesolar r-process pattern for A � 120. These conditions includehot and cold parametrized winds, similar to those that mayoccur in core-collapse supernovae or accretion disk outflows,along with mildly heated neutron star merger ejecta. Werun a baseline simulation for each astrophysical trajectory(i.e., condition) chosen, and then repeat it with individualβ-decay rates changed by a small factor K . Individual β-decayhalf-lives in the rare-earth region tend to produce local changesto the final abundance pattern that influence the size, shape,and location of the rare-earth peak. Thus we compare the finalabundances with those of the baseline simulation by using alocal metric flocal, defined as

flocal(Z,N ) = 100 ×180∑

A=150|YK (A) − Yb(A)|, (36)

where Yb is the final baseline isotopic abundance, and YK isthe final abundance in a simulation in which the β-decay rate

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FIG. 2. Influential β-decay rates in the rare-earth region forhot, cold, and merger r-process conditions. The hot conditions areparametrized as in Ref. [50] with entropy s/k = 200, dynamical timescale τdyn = 80 ms, and initial electron fraction Ye = 0.3; the coldconditions are parametrized as in Ref. [51] with s/k = 150,τdyn =20 ms, and Ye = 0.3; and the merger conditions are from a simulationof A. Bauswain and H.-Th. Janka, similar to that of Ref. [52]. Weperformed two sensitivity studies for each trajectory, looking at theresults of increases and decreases to the rates by a factor of K = 5.In order of lightest to darkest, the shades are white (flocal = 0), lightblue (0.1 < flocal � 0.5), medium blue (0.5 < flocal � 1.0), dark blue(1 < flocal � 5), and darkest blue (flocal > 5).

of the nucleus with Z protons and N neutrons is multiplied bya factor K . Results for six studies, in which individual β-decayrates were changed by a factor of K = 5, with hot, cold, andneutron star merger r-process conditions appear in Fig. 2. Thelargest impacts on the final abundances occur near the peak(A ∼ 160), although which nuclei are most sensitive dependsa little on the astrophysical conditions chosen.

In the second set of studies, focused on the A ∼ 80 peak, westart with a baseline weak r-process simulation that producesan abundance pattern with a good match to the solar patternfor 70 < A < 110, as identified in Ref. [49]. Here we chooseconditions qualitatively similar to those found in the outflowsof neutron star or neutron star-black hole accretion disks[53,54], which are attractive candidate sites for the weak rprocess. We run a sensitivity study as described above, varyingeach β-decay lifetime in turn by a factor of K = 10 andcomparing the result to the baseline pattern. Unlike in therare-earth region, where the influence of an individual β-decayrate is primarily confined to the surrounding nuclei, rates inthe peak regions can produce global changes to the pattern [5]and can influence how far the process proceeds in A. Thus,in this study we compare each pattern to the baseline with a

FIG. 3. Influential β-decay rates in the A ∼ 80 region for weakr-process conditions, parametrized as in Ref. [50] with entropy perbaryon s/k = 10, dynamic time scale τ = 200 ms, and startingelectron fraction Ye = 0.3. The shaded boxes show the globalsensitivity measures Fglobal resulting from β-decay rate increases ofa factor of K = 10. Stability is indicated by crosses; the β-decayrates of nuclei to the right of stability and to the left of the solid grayline have all been measured and so are not included in the sensitivityanalysis.

global sensitivity measure Fglobal:

Fglobal = 100 ×∑A

|XK (A) − Xb(A)|, (37)

where Xb(A) and XK (A) are the final mass fractions of thebaseline simulation and the simulation with the β-decay ratechanged, respectively. Figure 3 shows representative results.The pattern of most influential β-decay lifetimes is similar tothat identified for a main r process [5]: The important nucleitend to be even-N isotopes along either the r-process path orthe decay pathways of the most abundant nuclei.

We select isotopic chains with the highest sensitivitymeasures, according to Figs. 2 and 3, and carefully recalculatetheir β-decay half-lives. The selected chains encompass 70nuclei in the rare-earth region and 45 nuclei in the A ∼ 80region.1

B. Selection and adjustment of Skyrme EDFs

Our density-dependent nucleon-nucleon interactions arederived from Skyrme EDFs. References [56–58] contain com-prehensive reviews of the properties of Skyrme functionals;Refs. [18,33] contain discussions of the most important termsof the EDF for β decay. In our calculations, we largely applythe Skyrme EDF “as is,” but adjust a few important parametersthat affect ground-state properties and β-decay rates. Amongthese are the proton and neutron like-particle pairing strengthsVp and Vn; the spin-isospin coupling constant Cs10; and the

1Measured half-lives of 76,77Co and 80,81Cu were recently reportedin Ref. [55]. We still include these nuclei in our calculations.

055802-6


proton-neutron isoscalar pairing strength V0. We tune theseparameters separately for each mass region; the couplingconstants that multiply the remaining “time-odd” terms of theSkyrme EDF are set either to values determined by local gaugeinvariance [58] or to zero. Though our pnFAM code is able tohandle tensor interactions [18], none of the EDFs we use herehave nonzero tensor couplings CFt or C

∇st (in the notation of

Refs. [18,59]).

1. Multiple Skyrme EDFs for the rare-earth elements

The pnFAM’s efficiency significantly reduces the compu-tational effort in β-decay calculations. The smaller computa-tional cost makes repeated calculations feasible and allows usto examine the extent to which β-decay predictions depend onthe choice of Skyrme EDF. Here we use four very differentSkyrme functionals: SkO′ [60], SV-min [61], UNEDF1-HFB[62], and SLy5 [63]. SkO′ was already applied to the β decayof spherical nuclei [7]; it was also chosen for the recent globalcalculations of Ref. [33]. SV-min and UNEDF1-HFB are morerecent; the latter is a re-fit of the UNEDF1 parametrization[64], without Lipkin-Nogami pairing. SLy5 tends to yieldless-collective Gamow-Teller strength than some other Skyrmeparametrizations [65].

We do most of our calculations in a 16-shell harmonicoscillator basis, a choice that further reduces computationaltime from that associated with the 20-shell basis applied in theUNEDF parametrizations of Refs. [64,66,67]. Because UNEDF1-HFB was constructed with HFBTHO in a 20-shell basis, however,we use this larger basis for that particular functional. Wedetermine the nuclear deformation by starting from three trialshapes (spherical, prolate, and oblate) and selecting the mostbound result after the HFB energy and deformation have beendetermined self-consistently. We obtain the mean-field groundstates of odd nuclei within the EFA, beginning from a referenceeven-even solution and then computing odd-A solutions for alist of blocking candidates reported by HFBTHO. For odd-oddnuclei, we try all Np × Nn proton-neutron configurations totake into account as many odd-odd trial states as are practical.Again, we select the most-bound quasiparticle vacuum fromamong these candidates.

Returning to the functionals themselves, to adjust thepairing strengths and coupling constant Cs10, we start fromthe published parametrizations.2 Then we fix the like-particlepairing strengths Vp and Vn by comparing the average HFBpairing gap to the experimental odd-even staggering (OES) ofnuclear binding energies for the small set of test nuclei listedin Table I.3 Following the procedure in Refs. [64,66,67], we

2With a few exceptions, we use the same nucleon mass for protonand neutrons, unlike Ref. [61], which originally determined SV-min,and we employ the SLy5 parametrization written into HFBTHO, whichdiffers from that published in Ref. [63]. The HFBTHO values are thesame as those of Ref. [68], but t0 = −2483.45 MeV fm3 instead of−2488.345 MeV fm3.

3The UNEDF1-HFB pairing strengths were originally fit simultane-ously with the rest of the functional (with HFBTHO), so we do notreadjust the UNEDF1-HFB pairing strengths.

TABLE I. OES indicators �̃(3) for the even-even nuclei used tofit the pairing strengths Vp and Vn.

Z N �̃(3)p (MeV) �̃(3)n (MeV)

52 84 0.79096 ± 0.00464 0.75491 ± 0.002454 86 0.90975 ± 0.01527 0.87276 ± 0.002256 90 0.92059 ± 0.01093 0.92025 ± 0.020458 90 0.99503 ± 0.00975 0.97777 ± 0.009260 92 0.68605 ± 0.01176 0.77895 ± 0.029862 94 0.57543 ± 0.02887 0.67368 ± 0.004262 96 0.55867 ± 0.05021 0.58183 ± 0.004964 96 0.57608 ± 0.00276 0.67969 ± 0.001866 98 0.53795 ± 0.00277 0.67866 ± 0.001668 100 0.55392 ± 0.00312 0.64734 ± 0.001768 102 0.50391 ± 0.03646 0.60222 ± 0.002170 104 0.52725 ± 0.00300 0.53483 ± 0.001772 106 0.62796 ± 0.00168 0.63470 ± 0.001672 108 0.62486 ± 0.00388 0.57799 ± 0.002274 110 0.55784 ± 0.00199 0.66483 ± 0.000874 112 0.60795 ± 0.01224 0.70165 ± 0.001374 114 0.67773 ± 0.03607 0.79595 ± 0.022776 116 0.78248 ± 0.01110 0.83218 ± 0.002078 118 0.75364 ± 0.00128 0.88139 ± 0.0009

adjust the HFB pairing gap to match the indicator (e.g., for neu-trons) �̃(3)n (Z,N ) = 12 [�(3)n (Z,N + 1) + �(3)n (Z,N − 1)] foreven-even nuclei. We obtain the usual three-point indicators�(3) [69,70] from mass excesses in the 2012 Atomic MassEvaluation [71,72]—after using the prescription of Ref. [72]to remove the electron binding [66] from the atomic bindingenergies [73]. After finding pairing strengths that correspond toone-σ uncertainties in �̃(3)(treating asymmetric uncertaintiesas in Ref. [74]), we find best-fit values Vp and Vn for our sampleset of nuclei. For all EDFs except SV-min, we choose mixedvolume-surface pairing, with α = 0.5 as in Ref. [18]. SV-min’spairing piece was originally fixed along with the rest of thefunctional, but in the HF+BCS framework. We therefore refitthe pairing strengths to better represent ground-state propertieswith HFBTHO, keeping the coefficient that specifies densitydependence at its value of α = 0.75618 from Ref. [61].

Next, we determine an appropriate value for Cs10 bycomparing the excitation energy,

Eex = EQRPA − E0(pn), (38)of the Gamow-Teller giant resonance (GTR) to an experi-mentally measured value in a nearby nucleus. This constantCs10 is the same one we adjusted to GTR data in the past[18], following the work of Ref. [59]; Ref. [33] recentlyshowed that it is the only particle-hole constant that is trulyimportant for β decay. In these A � 160 nuclei, we use theresonance associated with the doubly magic nucleus 208Pb,with Eex = 15.6 ± 0.2 MeV in the odd-odd daughter 208Bi[75], to fix it. We also use the deformed rare-earth nucleus150Nd (Eex � 15.25 MeV in 150Pm [76]), to fix an alternativevalue Cs10 in SkO

′, calling the resulting functional SkO′-Nd.The two fits result in values of Cs10 that differ by nearly20%. Figure 4 compares the Gamow-Teller strength functionsproduced by the two values in 150Pm. Not only are the

055802-7


0 5 10 15 20

Excitation energy (MeV)

0

5

10

15

20

25

Gam

ow-T

elle

rst

reng

th(M

eV–

1)

Cs10(SkO ) = 128.0 MeV fm3

Cs10(SkO -Nd) = 102.0 MeV fm3

FIG. 4. Gamow-Teller strength functions in 150Pm for SkO′ (reddashed line, with Cs10 fit to the GTR energy in

208Pb) and SkO′-Nd(purple solid line, with Cs10 fit to the GTR energy in

150Nd). Thevertical line marks the measured GTR energy in 150Pm.

resonances at different places, but there is also a big differencein the strength functions at the low energies that are importantfor β decay.

To adjust the T = 0 pairing, we select short-lived even-evenisotopes with Z = 54, 56, 58, 60, 62, and 64 with β-decayrates that have been measured reasonably precisely, accordingto Ref. [77]; the 18 nuclei we use are listed in Table II. Foreach nucleus, we attempt to find a pairing strength V0 thatreproduces the measured half-life. If a calculated half-life istoo short, even when V0 = 0, we remove the nucleus fromconsideration; This prevents our fit from being influenced by

TABLE II. Isotopes used to fit the proton-neutron isoscalarpairing to experimental half-lives from Ref. [77]. Labels a–e in the“Excluded?” column note which isotopes were excluded from the fitsfor the functionals (a) SkO′, (b) SkO′-Nd, (c) SV-min, (d) SLy5, and(e) UNEDF1-HFB, as discussed in the text.

Z N Element T1/2(expt.) Excluded?

54 88 142Xe 1.23 b d e54 90 144Xe 0.388 b d e54 92 146Xe 0.146 d e56 88 144Ba 11.5 a b d e56 90 146Ba 2.22 b d e56 92 148Ba 0.612 e58 90 148Ce 56 b c d e58 92 150Ce 4 d e58 94 152Ce 1.4 d e60 92 152Nd 684 a b c d e60 94 154Nd 25.9 b c d e60 96 156Nd 5.06 b d e62 96 158Sm 318 a b c d e62 98 160Sm 9.6 e62 100 162Sm 2.4 e64 98 162Gd 504 b e64 100 164Gd 45 e64 102 166Gd 4.8 e

TABLE III. Summary of proton-neutron T = 0 pairing fit, in-cluding the amount of test data in each fit (N ) and the resultingpairing strength (V0).

EDF N V0

SkO′ 15/18 − 320.0SV-min 14/18 − 370.0SkO′-Nd 9/18 − 300.0SLy5 6/18 − 240.0UNEDF1-HFB 0/18 − 0.0

especially long-lived or sensitive isotopes. After determiningan approximate V0 = 0 for each nucleus (where possible), wecompute the average of these values, weighing fast decaysmore than slow ones (because the very neutron-rich r-processnuclei are short-lived), with weight factors,

wi = 1log10

[T

expt1/2 (i)/35 ms

] . (39)The fit is fairly insensitive to the weighting half-life T0 =35 ms; with T0 = 25 ms the fit values of V0 change by �2%.

Table III lists the values for V0 that we end up with andthe number of nuclei incorporated into the fit for each EDF.We find that none of the EDFs predict long-enough half-livesto fix V0 = 0 for the entire set of test nuclei; SkO′ (15 of18) and SV-min (14) come the closest, while SLy5 (only 6)and UNEDF1-HFB (zero) come less close and are thus poorlyconstrained by β decay. (We discuss SkO′-Nd momentarily.)One cannot really have confidence in fits (SLy5, UNEDF1-HFB)that take into account less than half of the available data, butFig. 5 provides at least a partial explanation. It compares ourcalculated Q values (34) to measured values [72] and thoseof the finite-range droplet model in Ref. [6]. Our Q valuesare almost uniformly larger than experiment (those of Ref. [6]are generally smaller), and those of SLy5 and UNEDF1-HFB aremuch larger. Because the β-decay rate is roughly proportionalto Q5 [78], a Q value that is too large will lead to an artificially

142 Xe

144 Xe

146 Xe

144 Ba

146 Ba

148 Ba

148 C

e15

0 Ce15

2 Nd15

4 Nd15

6 Nd15

8 Sm16

0 Sm16

2 Gd

–1.0

–0.5

0.0

0.5

1.0

1.5

2.0

Qβ

(cal

c)-

Qβ

(exp

t)(M

eV)

FIG. 5. The difference between calculated and experimentalβ-decay Q values, with SkO′ (circles), SV-min (diamonds), SLy5(squares), and UNEDF1-HFB (triangles). Q values of Ref. [6] (crosses)also appear.

055802-8


1 102 10410–4

10–2

1

102

104 SkO

1 102 104

SkO -Nd

1 102 104

SV-min

1 102 10410–4

10–2

1

102

104 SLy5

1 102 104

Experimental half-life (s)

Experimental half-life (s)

T1/

2(c

alc)

/T1/

2(e

xpt)

T1/

2(c

alc)

/T1/

2(e

xpt)

FIG. 6. Performance of fit functionals in even-even rare-earthnuclei. Filled symbols mark nuclei used to fit the T = 0 pairing. Theexperimental data are from Ref. [77].

short half-life. The T = 0 pairing only makes the half-livesshorter.

The Q-value fitting difficulties, however, do not manifestthemselves in actual half-life predictions as much as theymight, even with SLy5 and UNEDF1-HFB. Figure 6 compares ourcalculations to experimental measurements in the 18 test nucleiused to fit V0 (listed in Table II) and an additional 18 rare-earthnuclei (listed in Table IV). Our results display the same pattern

TABLE IV. The 18 even-even rare-earth nuclei in Fig. 6 that arenot included in the EDF fitting. Experimental half-lives, in seconds,are from Ref. [77].

Z N Element T1/2(expt.)

50 84 134Sn 1.0550 86 136Sn 0.2552 82 134Te 250852 84 136Te 17.6352 86 138Te 1.454 84 138Xe 844.854 86 140Xe 13.656 86 142Ba 63656 94 150Ba 0.358 90 146Ce 811.262 90 156Sm 3384066 102 168Dy 52268 106 174Er 19270 108 178Yb 444070 110 180Yb 14472 112 184Hf 1483272 114 186Hf 15674 116 190W 1800

TABLE V. A � 80 nuclei used to fit the like-particle pairingstrengths. We again use data from Ref. [72] to compute theexperimental indicators �̃(3). We set �̃(3) = 0 for nuclei with Z = 28or N = 50.

Z N �̃(3)p (MeV) �̃(3)n (MeV)

24 32 1.26354 1.0163426 38 1.17387 1.2926930 44 1.01199 1.4143332 46 1.13235 1.2483532 48 0.99635 1.1777934 52 1.17100 0.7896836 54 1.15773 0.8399036 56 1.18243 0.9064438 58 1.11437 0.9306338 60 0.99089 0.8559142 62 0.99786 0.9507728 38 0.00000 1.2097532 50 0.95788 0.0000028 50 0.00000 0.00000

as many others (e.g., Refs. [17,45,79]), reproducing half-livesof short-lived nuclei better than those of longer-lived ones. TheQ-value errors discussed previously show up as systematicbiases in our half-life predictions (particularly with SLy5 andUNEDF1-HFB, for which half-lives of long-lived nuclei areartificially reduced). The shortest-lived nuclei, however, arenot so poorly represented even with SLy5 and UNEDF1-HFB;these half-lives are still systematically short but by less than afactor of about two (the shaded region in Fig. 6 covers a factorof 5 relative to measured values). The systematic problemsin the two functionals based on SkO′ are barely noticeable.SV-min, which performs the best overall, is somewhere in themiddle. Because all the functionals do well with the short-livedisotopes, we use them all for our rare-earth calculations. Thelifetimes we get with UNEDF1-HFB serve as lower bounds onour predictions.

Finally, although SkO′-Nd, the SkO′ variant that reproducesthe GTR in 150Nd, fails in 9 of the 18 nuclei used for fitting,its predictions do not differ significantly from those of SkO′.Figure 4 shows increased low-energy Gamow-Teller transitionstrength as Cs10 is reduced to reproduce the rare-earth GTR.This increased low-lying strength reduces β-decay half-livesso that a smaller pairing strength is required (see Table III),with no loss of quality.

2. Weak r-process elements

To calculate the half-lives of A � 80 nuclei, we electto apply the functional from the previous section that bestreproduces measured half-lives. Both Fig. 6 and the metrics inRef. [17] point to SV-min as the best EDF. We adjust SV-minfor these lighter nuclei in much the same way as discussedin the previous section. We fit the like-particle pairing tocalculated OES values (see Table V), this time employing thefitting software POUNDerS [80] to search for the best values ofVp and Vn. We provide POUNDerS with the weighted residuals,

Xi = wi[�̄i(Vp,Vn) − �̃(3)i

], (40)

055802-9


TABLE VI. Open-shell (top) and semimagic (bottom) even-evenA � 80 nuclei whose half-lives are used to adjust the proton-neutronisoscalar pairing. Experimental half-lives are from the ENSDF [82].

Z N Isotope T1/2 (s)

22 38 60Ti 0.02224 40 64Cr 0.04326 44 70Fe 0.09430 52 82Zn 0.22832 52 84Ge 0.95434 54 88Se 1.53036 60 96Kr 0.080

28 46 74Ni 0.68028 48 76Ni 0.23830 50 80Zn 0.54032 50 82Ge 4.560

where �̄i is the average HFB pairing gap for the ith nucleusand the weight factor wi is 1 for nonmagic nuclei and 10for magic ones. The search yields Vp = −361.0 MeV fm3 andVn = −320.9 MeV fm3. We adjust the coupling constant Cs10to the GTR in a lighter doubly magic nucleus, 48Ca (Eex =10.6 MeV in 48Sc [81]).

Finally, as before, we adjust the T = 0 pairing strength toreproduce measured half-lives of even-even nuclei, now withA � 80 (see Table VI). In this region of the isotopic chart thefit is complicated by the presence of both proton and neutronclosed shells (see Fig. 3), so we include Z = 28 and N = 50semimagic nuclei in the fit. Figure 7 shows the impact of theT = 0 pairing on half-lives and in particular in the differencein the effect between nonmagic nuclei (solid lines) andsemimagic nuclei (dashed lines). We search for distinct valuesof V0 for these two cases, finding V0(nm) = −353.0 MeV fm3for nonmagic nuclei and V0(sm) = −549.0 MeV fm3 is for ourset of semimagic nuclei.

The top panel of Fig. 8 shows the results of these adjust-ments, comparing calculated β-decay half-lives of even-even(left panel) and singly odd (right) nuclei with measured values

0 100 200 300 400 500 600

–V0 (MeV fm3)

10–2

10–1

1

10

102

T1/

2(c

alc)

/T1/

2(e

xpt)

FIG. 7. Impact of T = 0 pairing on β-decay half-lives, bothfor nonmagic (solid lines) and semimagic (dashed lines) nuclei.The shaded region marks agreement between our calculation andmeasured half-lives [82] to within a factor of two.

10–4

10–2

1

102

104 This work )b()a(

10–2 1 102 104 10610–4

10–2

1

102

104 rellöM et al. (2003) (c)

10–2 1 102 104 106

Experimental half-life (s)T

1/2

(cal

c)/T

1/2

(exp

t)

(d)

FIG. 8. Performance of our SV-min calculations (top panels)and the FRDM calculations of Ref. [17] (bottom panels) for nucleiwith 22 < Z < 36 and T1/2 � 1 day. Left panels show the ratio ofcalculated to measured half-lives in even-even nuclei; right panelsshow the ratio in odd-A nuclei. Filled circles in the upper-left paneldenote even-even nuclei used to fit the T = 0 proton-neutron pairing.The shaded horizontal band marks agreement to within a factor offive, and the white background on the left side marks nuclei withmeasured half-lives that are shorter than 1 s.

[82], for nuclei with 22 < Z < 36 and T1/2 < 1 day. Thebottom panel shows the same comparison for the finite-rangedroplet model (FRDM) calculation of Ref. [17]. The two setsof results are comparable, but those of Ref. [17] have a clearbias in even-even nuclei. A metric defined in Ref. [17],

ri = log10[

T calc1/2 (i)

Texpt

1/2 (i)

], (41)

captures the bias. The mean and RMS deviation of 10ri , calledM10r and �

10r , quantify the deviation between calculation and

experiment: A value M10r = 2 would signify that calculationsproduce half-lives that are too long by a factor of two, onaverage. Our calculated rates yield M10r = 1.32 in even-evennuclei while those of Ref. [17] give M10r = 3.55. For thestandard deviation, our rates yield �10r = 5.14, vs 7.50 forthose of Ref. [17]. Thus, we indeed do measurably better ineven-even nuclei. Our results in odd-A nuclei are worse thanthose of Ref. [17], however; we obtain M10r = 2.71 (vs 0.95)and �10r = 11.61 vs (6.46). The two sets of calculations arecomparable for short-lived odd-A nuclei, however: We getM10r = 1.11 vs 0.96 and �10r = 2.48 vs 2.21 for isotopes withT1/2 � 1 s.

C. Results near A = 160Guided by the sensitivity studies in Fig. 2, we identify 70

rare-earth nuclei, all even even or proton odd, with rates that

055802-10


10–2

10–1

1

10

102

103

Z = 58

(a)

Z = 60

(b)

90 98 106 114

10–2

10–1

1

10

102

103

Z = 58

(c)

90 98 106 114

Z = 60

(d)

Neutron number

Hal

f-lif

e(s

)

FIG. 9. (Top panels) Half-lives for nuclei in the Ce (Z = 58, left)and Nd (Z = 60, right) isotopic chains, calculated with the SkyrmeEDFs described in the text. Symbols correspond to the same EDFs asin Fig. 6. (Bottom panels) Calculated half-lives of Ref. [17] (×), [33](+), and [83] (�), measured half-lives [82] (circles), and the range ofhalf-lives reported in this paper (shaded region).

strongly affect r-process abundances near A = 160. (Neutron-odd nuclei do not significantly affect the r process becausethey quickly capture neutrons to form even-N isotopes [1].)Figures 9 and 10 present new calculated half-lives in fourisotopic chains, with even-Z isotopes in Fig. 9 and odd-Zisotopes in Fig. 10. The top panels of these figures includepredictions with all five adjusted Skyrme EDFs, and thebottom panels compare our calculations to measured values[82] and the results of previous QRPA calculations [17,33,83]where they are available. Our calculations span the (narrow)range of predicted half-lives in these isotopic chains, withSLy5 and UNEDF1-HFB predicting the shortest half-lives forthe most neutron-rich isotopes, as one could expect fromthe analysis of Sec. III B 1. While the UNEDF1-HFB half-livesare uniformly short, however, those of SLy5 actually arethe longest predictions (and the closest to measured values)for nuclei nearer to stability. The variability of our half-lifepredictions in Figs. 9 and 10 is typical of all 70 nuclei in thecalculation. The longest and shortest calculated half-lives forany nucleus in our set differ by a factor ranging from about 1.9to 3.3, and this interval does not depend strongly on whethera nucleus has an even or odd number of protons.

The results of Refs. [17,33,83] are actually fairly similar toours, spanning roughly the full range of our predicted values inFig. 9. (Neither Ref. [33] nor [83] report half-lives for odd-Znuclei and so cannot be a part of Fig. 10.) The half-lives ofRef. [33] are close to our own SkO′ half-lives, a result thatis unsurprising given that the EDF in that paper is a modifiedversion of SkO′ (and that we use the same pnFAM code). The

10–3

10–2

10–1 Z = 55

(a)

Z = 57

(b)

102 104 106 108 110 112

10–3

10–2

10–1 Z = 55

(c)

106 110 114 118

Z = 57

(d)

Neutron numberH

alf-

life

(s)

FIG. 10. (Top panels) Half-lives for nuclei in the Cs (Z = 55, left)and La (Z = 57, right) isotopic chains, calculated with the SkyrmeEDFs described in the text. Symbols correspond to the same EDFs asin Figs. 6 and 9. (Bottom panels) Calculated half-lives from Ref. [17](×) superimposed upon the range of half-lives reported in this paper(shaded region).

half-lives of Ref. [17] lie, for the most part, right in the middleof our predictions and follow those of SV-min fairly closely(the Q values are similar). Finally, Fang’s recent calculations[83] yield relatively short half-lives, shorter than even thoseof UNEDF1-HFB most of the time. Still, the band of predictedhalf-lives is relatively narrow among these three calculationseven in the most neutron-rich nuclei. Reference [33] points outthat despite their differences, most global QRPA calculationsproduce comparable half-lives. Our results in both A � 80 andA � 160 nuclei support this observation.

Figures 9 and 10 (as well as Fig. 11, discussed momentarily)show that the overall pattern of β decays in the A � 160 region

95 100 105 110 115 120

Neutron number

48

50

52

54

56

58

60

62

64

66

Pro

ton

num

ber

0%

10%

20%

30%

40%

50%+

FIG. 11. Impact of first-forbidden β transitions in rare-earthnuclei that are important for the r-process nuclei, with the SkyrmeEDF SV-min.

055802-11


does not depend much on whether an element has an even orodd number of protons. As mentioned above, the variabilityin predictions is approximately equal for these two classes ofnuclei, and we see in Fig. 10 that the calculations of Ref. [17]continue to lie toward the middle of our predictions. We alsofind that our calculations for both even-even and odd-Z nucleipredict smoothly decreasing in half-lives (on a logarithmicscale) with increasing neutron number. In this respect ourcalculations differ slightly from those of Ref. [17], the resultsof which are more variable (a fact most easily seen in Fig. 10).

Finally, we have examined the impact of first-forbidden βdecay on half-lives of rare-earth nuclei. Figure 11 shows thatin heavier nuclei forbidden decay makes up between 10% and40% of the total decay rate. The percentage generally increaseswith A.

D. Results near A = 80Following the weak r-process sensitivity study in Fig. 3,

we present new half-lives for 45 A � 80 nuclei in Fig. 12,comparing our results to those of Refs. [17,33]. Not surpris-ingly, in light of Fig. 8, our calculated half-lives (circles) areoften slightly shorter than those of Ref. [17] (crosses). Theyare also similar to those of Ref. [33], which used the samepnFAM code for even-even nuclei. We have also comparedour A � 80 half-lives to those of the QRPA calculations inRef. [79], finding very similar results for the few isotopicchains discussed both here and there.

One interesting feature of our calculation is that the half-lives of 85,86Zn, 89Ge, and (to a lesser extent) 90As are longcompared to those of Ref. [17]. The top panel of Fig. 13, whichplots our calculated quadrupole deformation β2 for A � 80nuclei, suggests these longer half-lives are at least partially

44 48 52 56 60 64

Neutron number

24

26

28

30

32

34

Pro

ton

num

ber

Quadrupole deformation β2

(a)

44 48 52 56 60 64

Neutron number

24

26

28

30

32

34

Pro

ton

num

ber

Forbidden contribution (%)

(b)

–0.2

–0.1

+0.0

+0.1

+0.2

0%

10%

20%

30%

40%

50%+

FIG. 13. (Top panel) Quadrupole deformation β2 of the 45A � 80 nuclei whose half-lives we calculate and display in Fig. 12.(Bottom panel) Contribution of first-forbidden β decay (0,1,2−

transitions) to the decay rate of the same 45 nuclei, plotted as apercentage.

from changes in ground-state deformation. The Zn isotopesswitch from being slightly prolate to oblate near 86Zn (N =56), while Ge and As isotopes do the same near N = 57. Ourcalculations find 89Ge to be spherical and situated between two

44 46 4810–3

10–2

10–1

1

Hal

f-lif

e(s

)

Z = 24

46 48 50

Z = 25

46 48 50

Z = 26

49 50

Z = 27

51 52

Z = 29

54 5610–3

10–2

10–1

1

Hal

f-lif

e(s

)

Z = 30

55 57

Z = 31

54 56 58 60

Z = 32

56 58 60 62

Z = 33

58 60 62 64

Z = 34

Neutron number

FIG. 12. Computed half-lives in A � 80 β decay (red circles), along with those of Refs. [33] (orange squares) and [17] (blue crosses).

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isotopes with β2 ∼ ±0.17. The authors of Ref. [17] appear toforce 83−90Ge to be spherical.

The bottom panel of Fig. 13 shows the impact of first-forbidden β decay in this mass region; together with thetop panel it connects negative-parity transitions with ground-state deformation. As discussed in Ref. [84], first-forbiddencontributions to β-decay rates are small, except in the oblatetransitional nuclei discussed above. These results largely agreewith those of the recent global calculation in Ref. [33].

We also find that while the most deformed nuclei decayalmost entirely via allowed transitions, spherical isotopes showa large scatter in the contribution of forbidden decay. Morethan 80% of the 89Ge decay rate is driven by first-forbiddentransitions. This analysis may bear on the large first-forbiddencontributions near N = 50 and Z = 28 reported by Ref. [45],which restricted nuclei to spherical shapes. Reference [79],which considers the effect of deformation on Gamow-Tellerstrength functions, suggests that it is important near this massregion.

Our calculations near A = 80 include even-even, odd-even, even-odd, and odd-odd nuclei. Though odd pnFAMcalculations are no more computationally difficult than evenones, a few odd half-lives are probably a little less reliablethan their even counterparts for two (related) reasons. First,we have found that our calculations produce negative contri-butions to decay rates in some first-forbidden channels of afew nuclei. We simply remove these negative contributions,with the following results: The total β-decay rate of 77Cochanges by only 0.1%, 80Cu by 160%, 81Cu by 17%, and97Se by 6%. Second, we have found that the lowest-energy(unperturbed) charge-changing single-quasiparticle transitionlies at an energy E < 0 for a few odd nuclei. In other words,there are a few odd-proton nuclei for which Eν − Eπ < 0 and afew odd-neutron nuclei for which Eπ − Eν < 0. The situationoccasionally allows negative QRPA energies. Our techniquefor calculating the decay rate then fails because of the form ofthe pnFAM strength function [18]: For every state at E = �ωthat has β− strength, the pnFAM generates a state with β+strength equal to the negative of the β− strength at E = −�ω.As a result, if there are states with negative energy and nonzeroβ− strength, then when we apply the residue theorem to obtainintegrated strength [85] we include (the negative of) spuriousβ+ contributions. Because that strength is at low energy, itis strongly weighted by β-decay phase space. The stronglyweighted spurious strength might help to explain some of ournegative forbidden decay contributions. While we feel it im-portant to note these issues, Fig. 13 suggests that their impact islimited. Indeed, first-forbidden decay contributes about 14% ofthe total decay rate on average in this mass region. If we removethe four largest contributions (89Ge and the three Zn isotopes—notable outliers), first-forbidden decays contribute less than10% to the total rate on average. The spurious contributions ina few nuclei are interesting, then, but not critical.

IV. CONSEQUENCES FOR THE R PROCESS

In rare-earth nuclei, our calculation produces rates that areeither consistently larger than or smaller than (depending onthe functional) those of, e.g., Ref. [17]. We now use these new

FIG. 14. The effect of our new β-decay rates on final r-processabundances. The same trajectories are used as in Fig. 2: (a) hot,(b) cold, and (c) nsm. Black circles mark solar abundances.

rates in simulations of the r process. The resulting rare-earthabundances are shown in Fig. 14. Generally, calculations thatpredict low rates build up the rare-earth peak in both hotand cold r-process trajectories, and calculations that predicthigher rates (SLy5 and UNEDF1-HFB) reduce the peak. Ourneutron star merger calculation (bottom panel of Fig. 14)demonstrates a different effect: Longer-lived nuclei broadenthe rare-earth peak, and shorter-lived nuclei narrow it. For allthree trajectories the change in abundances is fairly localized,with the effects caused by our most reliable parametrizations(SkO′, SkO′-Nd, and SV-min) modifying abundances byfactors, roughly, of two to four.

Near A = 80, our calculations produce small changes inweak r-process abundances from those obtained with theβ-decay rates of Ref. [17] and larger changes from those withcompilations. Figure 15 shows the baseline weak r-processcalculation from the sensitivity study of Fig. 3 (blue line),where the β-decay lifetimes are taken from the REACLIBdatabase [86]4 everywhere. We compare this abundancepattern to those produced when the β-decay rates for the setof 45 nuclei calculated in this work are replaced with our rates(red), those of Ref. [17] (purple), and those from Ref. [45](teal). Although the differences in abundance produced byour rates and those of Ref. [17] are fairly small, differencesproduced by ours and those of Ref. [45] or REACLIB arenoticeable, with the widely used REACLIB rates producingthe most divergent results. It appears that many β-decay ratesnear A = 80 in the REACLIB database come from a mucholder QRPA calculation [87] that differs significantly from the

4https://groups.nscl.msu.edu/jina/reaclib/db/

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FIG. 15. Impact of our β-decay rates near A = 80 on weakr-process abundances, with rates from this work (red solid line),Ref. [17] (purple short dashes), Ref. [45] (light blue dot dashes), andthe REACLIB database [86] (dark blue long dashes). Black crossesmark solar abundances.

more modern calculations, especially in lighter nuclei. In someinstances, our rates differ from those listed in REACLIB by afactor of seven.

Even though many of our calculated rates are higher thanthose of Ref. [17] (Fig. 12), their impact on a weak r-processabundance pattern is not a uniform speeding-up of the passageof material through this region, as appears to be the case for amain r process (see, e.g., Ref. [7]). β-decay rates can influencehow much neutron capture occurs in the A ∼ 80 peak regionand, consequently, how many neutrons remain for captureelsewhere [9]. Higher rates do not necessarily lead to a morerobust weak r process; in fact, the opposite is more usually thecase, because more capture in the peak region generally leadsto fewer neutrons available for capture above the peak. Thiseffect is illustrated in Fig. 16, which compares the abundancepattern for the baseline weak r-process simulation fromSec. III A with those obtained by using subsets of our newlycalculated rates in the same simulation. Consider first theinfluence of the rates of the iron isotopes (green line in Fig. 16),particularly 76Fe. This N = 50 closed shell nucleus lies on ther-process path, below the N = 50 nucleus closest to stabilityalong the path 78Ni. Thus an increase to the β-decay rate of76Fe over the baseline causes more material to move throughthe iron isotopic chain and reach the long waiting point at78Ni. The abundances near the A ∼ 80 peak increase and thoseabove the peak region decrease because the neutrons used toshift material from the very abundant 76Fe to 78Ni are no longeravailable for capture elsewhere. Changes to the β-decay ratesof nuclei just above the N = 50 closed shell, however, can havea quite different effect on the pattern. The germanium isotopes,particularly 86Ge and 88Ge, are just above the N = 50 closedshell, so increases to their β-decay rates from the baseline willmove material out of those isotopes to higher A (orange line inFig. 16). Thus, abundances above the peak increase and morematerial makes it to the next closed shell, N = 82. In the end,the two very different effects partly cancel one another so thatour rates do not change abundances significantly compared tothose obtained with the rates of Ref. [17].

FIG. 16. (a) Abundance pattern for the baseline trajectory de-scribed in the text, using the rates of Ref. [17] rates for all of the keynuclei identified in Sec. II (black dashed line), compared to resultsof simulations with the same astrophysical trajectory and with newrates for 68−72Cr only (light green line), new rates for 72−76Fe only(green line), new rates for 86−92Ge only (yellow line), and new ratesfor 89−95As only (orange line). (b) Percent difference between theabundances produced by the baseline simulation (black line) and thesimulations with the new rates (colored lines).

Given that modern QRPA calculations appear to haveconverged to roughly a factor of two or so, we use MonteCarlo variations to investigate the influence of this amountof uncertainty in all the β-decay rates required for r-processsimulations. We start with astrophysical trajectories that seemtypical for three types of main r-process environments (hotwind, cold wind, and merger). Then, for each Monte Carlostep, we vary all of the β-decay rates by factors sampled froma log-normal probability distribution,

p(x) = 1x√

2πσexp

[− (μ − ln(x))

2

2σ 2

], (42)

where μ is the mean, and σ is the standard deviation of theunderlying normal distribution and x is a random variable.We take μ = 0 and σ = ln(1.4) which yields a spread inrandom rate factors corresponding roughly to the factor of twouncertainty in modern QRPA calculations (see, e.g., Ref. [33]).For each set of rate factors generated with the log-normal distri-bution the r-process simulation is then rerun. Figure 17 showsthe resulting final r-process abundance pattern variances for10 000 such steps. In each case, though some abundancepattern features stand out as clear matches or mismatches tothe solar pattern, the widths of the main peaks and the sizeand shape of the rare-earth peak are not clearly defined. Thereal uncertainty in β-decay rates is larger than a factor of twobecause all QRPA calculations miss what could be importantlow-lying correlations. Thus, more work is needed, whether itbe theoretical refinement or advances in experimental reach.

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FIG. 17. Ranges of abundance patterns produced from MonteCarlo sampling of all β-decay lifetimes assuming roughly a factorof two rate uncertainty from a log-normal distribution in three mainr-process trajectories: hot (a), cold (b), and merger (c) as in Ref. [88].Points are solar residuals from Ref. [2].

V. CONCLUSIONS

We have adapted the proton-neutron finite-amplitudemethod (pnFAM) to calculate the linear response of odd-A andodd-odd nuclei, as well as the even-even nuclei for which it wasoriginally developed, by extending the method to the equal-filling approximation (EFA). The fast pnFAM can now be usedto compute strength functions and β-decay rates in all nuclei.

After optimizing the nuclear interaction to best representhalf-lives in each mass region separately, we have calculatednew half-lives for 70 rare-earth nuclei and 45 nuclei nearA = 80. Our calculated half-lives are broadly similar to thoseobtained in the global calculations of Möller et al. [17] aswell as to those of more recent work. As a result, r-processabundances derived from our calculated half-lives are similarto those computed with the standard rates of Ref. [17]. Ourcalculations support the conclusions of Ref. [33], which com-pared multiple QRPA β-decay calculations and found that theyall had similar predictions. Still, the comparison of r-processpredictions with much older ones in the REACLIB databaseand the discussion of uncertainty in Sec. IV demonstrate theneed for continued work on nuclear β decay.

ACKNOWLEDGMENTS

T.S. gratefully acknowledges many helpful conversationswith M. T. Mustonen, clarifying notes on SV-min from P.-G.

Reinhardt, a useful HFB fitting program from N. Schunck, anddiscussions with D. L. Fang. This work was supported by theUS Department of Energy through the Topical Collaborationin Nuclear Science “Neutrinos and Nucleosynthesis in Hot andDense Matter,” under Contract No. DE-SC0004142; throughEarly Career Award Grant No. SC0010263 (C.F.); and underindividual Contracts No. DE-FG02-97ER41019 (J.E.), No.DE-FG02-02ER41216 (G.C.M.), and No. DE-SC0013039(R.S.). M.M. was supported by the National Science Foun-dation through the Joint Institute for Nuclear AstrophysicsGrants No. PHY0822648 and No. PHY1419765 and underthe auspices of the National Nuclear Security Administrationof the US Department of Energy at Los Alamos NationalLaboratory under Contract No. DE-AC52-06NA25396. Wecarried out some of our calculations in the Extreme Science andEngineering Discovery Environment (XSEDE) [89], which issupported by National Science Foundation Grant No. ACI-1053575, and with HPC resources provided by the TexasAdvanced Computing Center (TACC) at The University ofTexas at Austin.

APPENDIX: RESTORING ANGULARMOMENTUM SYMMETRY

Deformed intrinsic states like those we generate in HFBTHOrequire angular-momentum projection. Here we use the rotormodel [90,91], which is equivalent to projection in the limit ofmany nucleons or rigid deformation [34]. Even-even nuclei,with K = 0 ground states (K is the intrinsic z componentof the angular momentum) are particularly simple rotors.Their “laboratory-frame” reduced matrix elements are justproportional to full intrinsic ones [91],

〈J K||ÔJ ||0 0〉 = �K〈K|ÔJK |0〉intr, (A1)

where �K = 1 for K = 0 and �K =√

2 for K > 0. Thecorresponding transition strength to an excited state |J K〉 is

B(ÔJ ; 0 0 → J K) = �2K |〈K|ÔJK |0〉intr|2. (A2)

FAM strength functions are essentially composed of squaredmatrix elements [29,85], so we simply multiply K > 0strength functions by �2K = 2.

The situation is more complicated in odd nuclei, whichK = 0 ground-state angular momenta. The transformationbetween laboratory and intrinsic frames, corresponding toEq. (A1), includes an additional term involving the time-reversed intrinsic state |Ki〉 [91]:

〈Jf Kf ||Ôλ||Ji Ki〉 =√

2Ji + 1[( Ji Ki ; λ Ki−Kf |Jf Kf )×〈Kf |Ôλ,Kf −Ki |Ki〉intr+ (Ji −Ki ; λ Ki + Kf |Jf Kf )×〈Kf |Ôλ,Kf +Ki |Ki〉intr]. (A3)

If we neglect rotational energies in comparison with intrinsicenergies, we can sum over Ji that appear in Eq. (A3) to

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obtain [90]

B(Ôλ; Ji Ki → Kf )

= 12Ji + 1

Ji+λ∑Jf =|Ji−λ|

|〈Jf Kf ||Ôλ||Ji Ki〉|2

= |〈Kf |Ôλ,Ki−Kf |Ki〉intr|2

+ |〈Kf |Ôλ,Ki+Kf |Ki〉intr|2. (A4)

The EFA pnFAM, by preserving time-reversal symmetryand providing the combined transition strength from auxiliarystates α†|�〉 and α†̄|�〉, directly yields the terms in Eq. (A4).For example, the Gamow-Teller decay of a nucleus withJi = Ki = 3/2 involves, according to Eq. (A4), three intrinsic

transitions:〈32

∣∣ÔK=0∣∣ 32 〉, 〈 52 ∣∣ÔK=1∣∣ 32 〉, 〈 12 ∣∣ÔK=−1∣∣ 32 〉.The third matrix element is equivalent because of time-reversalsymmetry to 〈− 12 |ÔK=1|− 32 〉, up to a phase. Thus, a half-lifecalculation to each band requires a K = 0 transition and apair of K = 1 transitions from states | 32 〉 and |− 32 〉. These arethe auxiliary states that make up the EFA-pnFAM strengthfunction. As a result, we obtain the total strength to a band inan odd nucleus the same way as in an even one: Stotal(F ; ω) =SK=0(F ; ω) + 2SK=1(F ; ω) for Gamow-Teller transitions. Thecalculation of forbidden strength is similar. The factor of twofor K > 0 strength cancels the factors of 1/2 that appear inthe EFA strength function in Eq. (33). K = 0 transitions donot need this factor because, e.g., K = 3/2 → K = 3/2 andK = −3/2 → K = −3/2 transitions are equivalent and cometogether in the strength function.

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