Shell-model half-lives for r-process waiting point nuclei including first-forbidden contributions

0.41 MB·English

by  Q. Zhi

Shell-model half-lives for r-process waiting point nuclei including first-forbidden
contributions
Q. Zhi,1,2,3 E. Caurier,4 J. J. Cuenca-Garc´ıa,2 K. Langanke,2,1,5 G. Mart´ınez-Pinedo,1,2 and K. Sieja4
1Institut fu¨r Kernphysik (Theoriezentrum), Technische Universita¨t Darmstadt,
Schlossgartenstraße 2, 64289 Darmstadt, Germany
2GSI Helmholtzzentrum fu¨r Schwerioneneforschung, Planckstraße 1, 64291 Darmstadt, Germany
3School of Physics and Electronic Science, Guizhou Normal University, 550001 Guiyang, PR China
4Universit´e de Strasbourg, IPHC, 23 rue du Loess,
67037 Strasbourg France CNRS, UMR7178, 67037 Strasbourg, France
5Frankfurt Institute of Advanced Studies, Ruth-Moufang Str. 1, 60438 Frankfurt, Germany
(Dated: January 23, 2013)
We have performed large-scale shell-model calculations of the half-lives and neutron-branching
3
probabilities of the r-process waiting point nuclei at the magic neutron numbers N = 50, 82,
1
and 126. The calculations include contributions from allowed Gamow-Teller and first-forbidden
0
transitions. Wefindgoodagreementwiththemeasuredhalf-livesfortheN =50nucleiwithcharge
2
numbers Z = 28–32 and for the N = 82 nuclei 129Ag and 130Cd. The contribution of forbidden

n transitions reduce the half-lives of the N = 126 waiting point nuclei significantly, while they have
a only a small effect on the half-lives of theN =50 and 82 r-process nuclei.
J

2 PACSnumbers: 21.60.Cs,23.40.-s,26.30.Hj
2

] I. INTRODUCTION the conditions of the astrophysical environment. If the
h
r-process has sufficient time for beta-flow equilibrium to
t
- establish, the relative elemental abundances are propor-
l Although the actualsite of the astrophysicalr-process
c tional to the beta half-life [6].
isstillnotknownwithcertainty,itiscommonlyaccepted
u
n thatoccursinanexplosiveenvironmentofrelativelyhigh Despite their importance only a few half-lives of wait-
[ temperatures (T 109 K) and very high neutron densi- ing points with magic neutron numbers N = 50 and 82
ties(>1020cm−3≈)[1–5]. Undersuchconditions,neutron are known experimentally [7–10], while no experimental
1
capturesaremuchfasterthancompetingbetadecaysand data exist yet for the N =126 waiting points. The situ-
v
ther-processpathinthenuclearchartproceedsthrougha ation is expected to improve in the near future with the
5
2 chainofextremelyneutronrichnucleiwithrelativelylow advent of new experimental facilities. For example, the
2 and approximately constant neutron separation energies beta-decayhalf-livesof38newneutron-richisotopesfrom
5 (S <3 MeV). Due to the relatively stronger binding of Krto Tc close to the r-processpath havebeenmeasured
n
1. nucle∼i with magic neutron numbers, the neutron separa- at the new RIBF facility at RIKEN [11]. Furthermore,
0 tion energies show discontinuities at the magic numbers researchersatGSIhavemeasuredhalf-livesofnucleiclose
3 N = 50, 82, and 126. As a consequence the r-process to N = 126 using a novel analysis method [12]. Despite
1 matter flow slows down when it reaches these neutron- this progress, the half-lives needed for r-process simula-
v: magic nuclei and has to wait for several beta decays tionshavemainlytorelyontheoreticalestimates. Asthe
i (which are also longer than for other nuclei on the r- Q-values involved are rather low, such calculations have
X
processpath)tooccuruntilfurtherneutroncapturesare traditionally been based on allowed, i.e. Gamow-Teller,
r possible carrying the mass flow to heavier nuclei. Thus transitions. Most of these studies used the Quasiparti-
a
matter is accumulated at these r-process waiting points cleRandomPhaseApproximation(QRPA)eitherontop
associated with the neutron numbers N = 50, 82, and of semi-empirical global models [13–15] or the Hartree-
126 leading to the well-known peaks in the observed r- Fock-Bogoliubovmethod[16]. Althoughthecalculations
process abundance distribution. giveafairaccountofthefewexperimentalhalf-lives,itis
The beta half-lives of the waiting points have at least wellknownthat these models underestimate the correla-
two important effects on the r-process dynamics and tionsamongnucleonswhichpulldowntheGamow-Teller
abundancedistributions. Atfirst,theymainlydetermine (GT)strengthtolowenergies. Thisshortcomingisover-
the time it takes the mass flow within the r-process to comewithintheinteractingshellmodelwhichindeedde-
transmute seed nuclei to heavy nuclei in the third peak scribesthe measuredhalf-lives ofr-processwaiting point
around A 200. Second, in the astrophysical environ- nuclei very well [17–19].
∼
mentthenuclearr-processtimescale(givenbythesumof It is expected that the appearance of intruder single
beta half-lives of nuclei in the r-process path) competes particle states with different parity may have influence
with some dynamical timescale of the environment, e.g. on the low-energy spectra of the r-process waiting point
the expansion timescale of the ejected matter. If the r- nuclei. Thus it is conceivable that first-forbidden tran-
processpathandhalf-liveswereknown,thereproduction sitions might contribute to the half-lives of these nuclei.
of the abundance distribution can be used to constrain A first attempt to estimate such forbidden contributions
2
has beentakenwithin the grosstheory [14]. This model, corrects the phase space integral for the Coulomb dis-
however, has been found as rather inaccurate when ap- tortion of the electron wave function near the nucleus.
pliedtoGamow-Tellertransitions. Morerecently,Borzov The partial decay rate is related to the partial half-life:
extended the QRPA studies based on the Fayans energy λ = ln2/t. The total decay rate is given by summing
functional to a consistenttreatment of allowedand first- over the partial decay rates to all possible final states.
forbiddencontributionstor-processhalf-lives[20]. While For allowed transitions, the shape factor does not de-
these calculations find that forbidden contributions give pend on the electron energy and for β− decay has the
only a small correction to the half-lives of the N = 50 form:
and N =82 waiting point nuclei, they result in a signif-
icant reduction of the N = 126 half-lives. This impor- C(W)=B(GT). (3)
tant finding has been our motivation to extend our shell
model calculations of waiting point half-lives to include The GT reduced transition probability is given by:
also first-forbidden transitions. We expect that correla-
tions among nucleons will not only affect the half-lives,
butareliabledescriptionofthe detailedallowedandfor- B(GT)= gA 2 hf|| kσktk−||ii2, (4)
bidden strength function is needed to estimate the prob- (cid:18)gV (cid:19) P2Ji+1
abilities for beta-delayed neutron emission rates which
where the matrix element is reduced with respect to the
are known to be important to describe the decay of the
spinoperatorσonly(Racahconvention[34])andthesum
r-process nuclei towards stability after freeze-out.
runsoverallnucleons. Fortheisospinloweringoperator,
We note that GT and higher multipole transitions are
relevanttodescribeneutrino-nucleusreactionswhichare we use the convention t−n = p. Finally, (gA/gV) =
1.2701(25)istheratioofweakaxialandvectorcoupling
important in many astrophysical sites [21, 22]. Tra-
−
constants.
ditionally, these reactions have been studied within
For first-forbidden (FF) transitions, the shape factor
the Random Phase Approximation [23, 24], includ-
is:
ing nuclei relevant to r-process nucleosynthesis [25–27].
In an interesting recent development, neutrino-induced
C(W)=k+kaW +kb/W +kcW2 (5)
reactions on light nuclei with relevance to neutrino-
nucleosynthesis[28,29]havebeencalculatedonthebasis
wherethe coefficients k,ka,kb,kc,dependonthe FF nu-
oftheshell-model,includingGTandfirst-forbiddentran-
clearmatrixelements,themaximumelectronenergy,W ,
sitions [30, 31]. 0
andthequantityξ =αZ/(2R)withRtheradiusofauni-
formly chargedsphere approximating the nuclear charge
distribution[35]. FollowingthetreatmentofBehrensand
II. SHELL MODEL AND β DECAY THEORY
Bu¨hring [36] they are given by:
In our half-life calculations, we consider allowed and
first-forbidden contributions. These are obtained using 1 (0) 1 4
thediagonalizationshellmodecodeNATHANdeveloped k = ζ2+ w2 + ζ2+ (x+u)2 µ γ u(x+u)
0 9 1 9 − 9 1 1
by Etienne Caurier [32, 33] to calculate the initial and (cid:20) (cid:21) (cid:20)
final nuclear states and the corresponding nuclear tran- 1 1 (1)
+ W2(2x+u)2 λ (2x u)2
sition matrix elements. Model spaces and residual inter- 18 0 − 18 2 −
actions will be discussed for the three different sets of (cid:21)
waiting point nuclei with N = 50,82 and 126 individu- + 1 z2(W2 λ ) (2),
ally. The partial half-life, t, for a transition between an 12 0 − 2
(cid:20) (cid:21)
initial (normally the ground state) and a final nuclear
state is related to the phase space factor by: 4 1 (1) 1 (2)
ka= uY W (4x2+5u2) z2W , (6)
0 0
−3 − 9 − 6
ft=K =6146 s. (1) (cid:20) (cid:21) (cid:20) (cid:21)
2
The phase factor has the form: kb= µ1γ1 [ζ0w](0)+[ζ1(x+u)](1) ,
3 −
n o
f = W0C(W)F(Z,W)(W2 1)1/2W(W W)2dW . kc= 1 8u2+(2x+u)2+λ (2x u)2 (1)
0 2
− − 18 −
Z1
(2) + 1(cid:2) z2(1+λ ) (2). (cid:3)
C(W) is the so called shape factor that depends on the 12 2
electron energy, W in units of the electron mass. W0 is (cid:2) (cid:3)
themaximumelectronenergy,alsoinelectronmassunits, with
that is givenby the difference in nuclearmassesbetween V =ξ′v+ξw′, ζ =V + 1wW ,
the initial and final nuclear states, W = Q/(m c2) = 0 3 0
0 e (7)
(M M )/m . F(Z,W) is the Fermi function that Y =ξ′y ξ(u′+x′), ζ =Y + 1(u x)W .
i − f e − 1 3 − 0
3
The numbers in parenthesis on the closing bracket de- cleon mass. The quantity I(1,1,1,1,r) appearing in
note the rank of the operators inside the bracket. The the primed matrix elements takes in accountthe nuclear
parameter γ is given by 1 (αZ)2. For the Coulomb charge distribution that can be approximated by a uni-
1
−
functions µ and λ we use the approximations µ 1 form spherical distribution [36]:
1 2 1
p ≈
and λ 1 [37].
2
≈
After anon-relativisticreduction,the matrix elements
1 r 2
can be related to the form-factor coefficients, A,VFKls, 1 0 r R,
defined in refs. [36, 37]. In the Condon and Shortley 3 (cid:20) − 5(cid:16)R(cid:17) (cid:21) ≤ ≤
phase convention [38] the matrix elements are: I(1,1,1,1,r)= 2 R 1 R 3 r R.
w ==−Rg A√F300h1f1|| krk C1k×σk 0tk−||ii, (8a) Based on the conserve"drve−ct5or(cid:18)crur(cid:19)ren#t theo≥ry and(1th0e)
− A √2J +1 assumption that the isospin is a good quantum number,
P (cid:2) i (cid:3) ′
the matrix element ξ y can be related to the x matrix
1
x= R VF0 element [39]:
−√3 110
= hf|| krkC1ktk−||ii, (8b) ξ′y =Eγx, (11)
− √2J +1
P i where energy E is defined as the energy difference be-
γ
2 tween the isobaric analog of initial state and the final
u= R AF0
− 3 111 state:
r
f r Ck σk 1tk i
= g √2h || k k 1 × −|| i, (8c)
A
− P √(cid:2)2Ji+1 (cid:3) Eγ =Eias(i) Ef =Q+∆EC (mnc2 mpc2), (12)
− − −
2
z = R AF0
√3 211 where mn and mp are the neutron and proton masses
and ∆E is the Coulomb displacement energy between
=2g hf|| krk C1k×σ 2tk−||ii, (8d) isobaricCanalog states that can be approximated by [40]:
A
√2J +1
P (cid:2) i (cid:3)
2
w′ = R AF0 (1,1,1,1) ∆E =1.4136(1)Z¯/A1/3 0.91338(11)MeV, (13)
−3 011 C −
= g √3hf|| k 23rkI(1,1,1,1,rk) C1k×σ 0tk−||i(i8,e) with Z¯ =(Zi+Zf)/2.
− A √2J +1 To compare the first-forbidden and Gamow-Teller
P i (cid:2) (cid:3)
transitions, we define the averagedshape factor:
2
x′ = R VF0 (1,1,1,1)
−3√3 110 C(W)=f/f , (14)
0
f 2r I(1,1,1,1,r )Cktk i
= h || k 3 k k 1 −|| i, (8f) where f takes the form of Eq. 2 and f is
− √2J +1 0
P i
2√2 W0
u′ = R AF0 (1,1,1,1) f = F(Z,W)(W2 1)1/2W(W W)2dW. (15)
−3√3 111 0 − 0−
Z1
f 2r I(1,1,1,1,r ) Ck σk 1tk i
= g √2h || k 3 k k 1 × −|(|8ig,)
A
− P √2Ji+1(cid:2) (cid:3) III. MODEL SPACES AND QUENCHING
ξ′v =AF0
000
= gA√3hf|| k σk×∇k 0tk−||ii, (8h) forWre-phroavceesspewrfaoirtminegdpboeintats-dbecaasyedhoanlf-lliavregec-aslccaulelatsihoenlsl
M P (cid:2)√2Ji+1(cid:3) model calculations. In particular we chose the following
ξ′y =VF0 model spaces and respective interactions.
101
= 1 hf|| k∇ktk−||ii, (8i) spaFnonretdhebyNth=e500fnuclei wanedha1vpe adoptoerdbiatsmfoordeplrsoptoances
−M √2J +1 7/2,5/2 3/2,1/2
P i andbythe0f ,1p ,and0g orbitsforneutrons.
5/2 3/2,1/2 9/2
where The single-particle energies and the residual interaction
are the ones adopted in [41] to study the shell evolution
4π
C = Y , (9) between 68Ni and 78Ni.
lm lm
2l+1
r Our shell model calculations for the N = 82 waiting
with Y the spherical harmonics. The weak axial cou- point nuclei follows the shell model studies presented in
lm
pling constant is g = 1.2701(25) and M is the nu- [19]. Fromthetwomodelspacesdefinedin[19]weadopt
A
−
4
theonebuiltona88Srcore. Thatisweexplicitlyconsider turbation theory, Warburton [43, 49] showed that the
the 1p proton orbit which is expected to be impor- quenching of first-forbidden transitions depends slightly
1/2
tant for the description of the negative parity states and on the initial and final single particle orbits. In partic-
hence the first-forbidden transitions. Our model space ular, Warburton found that transitions mediated by the
is then spanned by the 0g ,1d ,2s ,0h or- rank 0 operator (Eq. 8h) (called relativistic matrix ele-
7/2 3/2,5/2 1/2 11/2
bits outside the N = 50 core for neutrons, and the ment)appeartobeenhancedcomparedtotheotherfirst-
1p ,0g ,1d ,2s orbits for protons. This forbiddentransitions,duetomesonexchangeeffects[50].
1/2 9/2,7/2 3/2,5/2 1/2
model space avoids spurious center-of-mass excitations
by omitting the 0h orbit for protons and the 0g
11/2 9/2
orbit for neutrons. We adopt the residual interaction
given in [19] based on 88Sr core which gives a good ac- 100
count of the spectroscopy of nuclei in the neighborhood p.
of 132Sn. In particular, our calculation reproduces the x
E
excitation energy of the first 1+ state in 130In as well as )
the low-energy spectrum of 128Cd and of the r-process m
waiting point nucleus 130Cd [42]. 2 (f50
1/
The model space for the N = 126 waiting points has W)
beenspannedbythe0g ,1d ,0h and2s or- (
7/2 5/2,3/2 11/2 1/2 C
bits for protonsandthe 0h ,1f , 0i , 2p
9/2 7/2,5/2 13/2 3/2,1/2 0
orbits for neutrons. As interaction we use the effective
Kuo-HerlinginteractionKHH ofref.[43]whichhasbeen
e
constructed based on holes in a 208Pb core. It is the 0 50 100
same model space and effective interaction as has been 1/2
C(W) (fm) Theo.
used in a previous calculation of the half-lives, which,
however, has only considered pure GT transitions [21]. FIG. 1. Comparison of calculated first-forbidden average
shapefactors,obtainedforthebestfitvaluesofthequenching
These are mainly connected to neutron 0h to proton
9/2 factors (Eq. (16)), with experimental data [51–53].
0h transitions. However, it is expected that first-
11/2
forbidden transitions can compete, mainly via neutron
To treat the quenching of the Gamow-Teller and first-
0i to proton 0h transitions. Full diagonalization
13/2 11/2
forbidden transitions in our shell model calculations we
in this model space exceeds current computer capabili-
haveassumedthatthequenchingfactorsarethesamefor
ties. Hence, we performed truncated calculations follow-
allnuclei. FollowingthefindingsofWarburton[49,50]we
ing a generalize seniority scheme that allows for config-
havefurthermoreassumedthatthequenchingfactorsfor
urations with maximum seniority 8, i.e. we consider a
theoperatorsofrank0,1,and2contributingtothefirst-
maximum of 4 non-J = 0 pairs, for even even nuclei.
forbidden transitions as defined in Eq. (8) can be differ-
For odd-even nuclei, the number of broken pairs had to
be limited to three, while for 199Ta no limitation has ent. To determine these individual quenching factors we
have performed shell model calculations for experimen-
been enforced. We expect our model spaces to be large
tally known beta-decays of nuclei in the vicinity of the
enough to give a reasonable account for the low-lying
magic neutron numbers N =82 and N =126. Here, we
Gamow-Teller and first-forbidden transitions. Neverthe-
haveadoptedthesimilarsetoffirst-forbiddentransitions
lessthemodelspacesaretoorestrictedtorecoverthefull
ofnucleiintheleadregionaschoseninthestudyofWar-
Gamow-Teller and first-forbidden strengths built on the
burton [50], supplemented by the decays of the ground
groundorisomericstates. Themissedstrength,however,
state of 205Au (N =126) and the (1/2)− isomeric states
residesmainlyoutsidetheQ windowanddoeshencenot
β
in 131In (N = 82) and 129In (N = 80) which are both
affect our half-life calculations.
known to decay by first-forbidden transitions. By per-
Althoughtheshellmodelusuallygivesagoodaccount
forming a least-squares fit to the experimental data we
of the relative strength distributions, it overestimates
obtained the following quenching factors for the various
the total strength. For Gamow-Teller transitions this
matrix elements defined in Eq. (8):
shortcoming can be corrected for by replacing the bare
Gamow-Telleroperator by an effective operator GTeff = q(ξ′v)=1.266, q(w)=q(w′)=0.66,
q σt. The quenching factor q has been found to be ap- q(x)=q(x′)=0.51, q(u)=q(u′)=0.38, (16)
proximately constant over the nuclear chart [44–46]. In q(z)=0.42.
practice,usingq 0.7hasbeenshowntogiveagoodre-
≈
production of the absolute Gamow-Teller distributions. The calculated half-lives and the corresponding average
There is evidence that also the absolute first-forbidden shapefactorsaresummarizedinTableI.Fig. 1compares
transition strength is overestimated within shell model the experimental and calculated shape factors.
approaches. Ejiri and collaborators [47, 48] related this Upon a closer inspection there is quite a good agree-
fact to core polarization effects and suggested the intro- ment between our shell model half-lives for the isomeric
duction of a constant hindrance factor. Based on per- states in the In isotopes with the experimental values.
5
torq =0.66wereproducethemeasuredhalf-lifeusing
GT
1/2
TABLE I. Comparison of calculated logf0t and (cid:16)C(W)(cid:17) both GT and first-forbidden transitions. The latter con-
tribute about 13% to the half-life and hence are small,
forfirst-forbiddentransitionswithexperimentaldata[51–53].
butnotnegligible. Ourfactorq isonlyslightlysmaller
GT
Transition log f0t (cid:16)C(W)(cid:17)1/2 than the customary quenching value of 0.7.
All half-lives presented in the next sections for the r-
Initial Final the exp the exp
131In(1−) 131Sn(3+) 5.32 ≈5.1 65.8 85.4 processwaitingpointnucleihavebeenobtainedusingthe
2 2 quenching factors for Gamow-Teller and first-forbidden
131Sn(21+) 5.74 6.5 41.1 17.1 transitions derived above.
129In(12−) 129Sn(23+) 5.57 5.9(3) 49.8 34(12) Before we present our results we have to discuss an-
129Sn(1+) 5.80 5.5(1) 38.1 54(6) other potential shortcoming of our calculations of first-
2
205Hg(1−) 205Tl(1+) 5.37 5.257(11) 62.3 71.3(9) forbidden transitions and how we will handle it. A com-
2 21
205Tl(1+) 6.77 7.03(25) 12.5 9(3) pletely converged calculation of the first-forbidden tran-
22
205Tl(3+) 7.26 6.51(21) 18.9 17(4) sition strength in our chosen model space is prohibited
205Tl(321+) 6.32 7.61(22) 10.3 5(1) due to computational limitations. We have derived the
22 strengthwithinthe Lanczosscheme using100iterations.
205Tl(5+) 8.16 8.70(21) 1.91 1.3(3)
2 As a consequence the lowest states are converged and
206Hg(0+) 206Tl(0−) 5.42 5.41(6) 59.2 60(4) correspond to physical states, while the Lanczos states
206Tl(1−) 5.18 5.24(10) 77.6 73(8) at higher excitation energies are unphysical and repre-
1
206Tl(1−) 5.68 5.67 43.6 44.3 sentstrengthper energyinterval. Furthermore,theiren-
2
207Tl(1+) 207Pb(1−) 5.14 5.108(6) 81.7 84.5(6) ergypositions depend onthe sumrule (pivot) state used
2 2
207Pb(3−) 6.18 6.157(22) 24.7 25.3(6) for the calculation of the strength function. For exam-
2 ple, starting from the pivot state of the x operator one
206Tl(0−) 206Pb(0+) 5.42 5.1775(13) 52.4 78.0(1)
1 obtains different energy positions for the non-converged
206Pb(0+) 5.18 5.99(6) 32.4 31(2)
2 states than starting from the sum rule state of the u
206Pb(2+) 5.68 8.60(3) 1.87 1.52(5) operator. Asweneedtocomputesuperpositionsofoper-
205Au(3+) 205Hg(1−) 6.79 5.79(9) 12.1 39(4) atorslikeu+x,whereboththemagnitudeandthephase
2 21
205Hg(3−) 7.33 6.43(11) 6.5 18(2) of the individual operators matter, we have followed the
21
205Hg(5−) 5.82 6.37(12) 37.3 20(3) same procedure as is used in shell-model calculations of
21
double-betadecays[32]. Hencewestartwithanarbitrary
sum rule state that canbe any linear combinationof op-
erators of the same rank. During the Lanczos iteration
For the nuclei in the vicinity of N = 126 we find, how- procedure we compute the overlaps with the individual
ever, a noticeably larger scatter between calculation and operators. This iteration procedure is stopped when at
data. Satisfyingly there are no systematic deviations. least 80% of the total strength for each individual oper-
With the exception of the two 205Au decays, where our ator is recovered.
calculation overestimates (to the 5/2− state in 205Hg) To illustrate this point, we have performed calcula-
−
or underestimates (to the 1/2 state) the average shape tionsfor199Tainthemodelspacedefinedaboveandhave
factor roughly a factor 9, we generally find agreementof used two different linear combinations of rank 1 opera-
our calculated C(¯W) with data within a factor of 4. As tors,ξ′y ξ(x+u)and(x+u),aspivotsforthe Lanczos
alreadyobservedwhile determining the quenchingfactor calculatio−ns of the operators x and u. For the combina-
forGTtransitionsinshellmodelcalculations[54]thede- tion ξ′y ξ(x+u) we obtain 92% and 80% of the total
scription of a decay between specific states is noticeably strength−fortheoperatorsxandu,respectively,whilefor
more sensitive to nuclear structure effects than global thecombinationx+u werecover85%and95%after100
quantities like half-lives or total strengths. Hence we ex- iterations. We stress that the effect which these short-
pectthatourprescriptionofquenchingforfirst-forbidden comingshaveonthefirst-forbiddenhalf-livesismildened
transitions yields a fair description of the N =126 half- as the contributions of the low-lying states, which are
lives. convergedin our Lanczos scheme, are strongly enhanced
As already stressed by Warburton [50] the relativistic by the phase space energy dependence.
matrix element (Eq. 8h) is enhanced compared to the To quantify the potential uncertainty in our first-
otherfirst-forbiddentransitions. We confirmthis finding forbidden half-lives, we have performed again calcula-
′
asthe value for q(ξ v) is noticeablylargerthanthe other tionsfor199Tainthemodelspaceasdefinedabove,how-
quenching factors. ever, allowing only one proton pair to be broken in our
Having determined the quenching of first-forbidden seniority scheme. This truncated space allows for the
transitions,weadjustthequenchingoftheGamow-Teller calculation of a fully converged first-forbidden strength
transition to the half-life of 130Cd which is expected distribution in the Q window. Fig. 2 compares the
β
to decay dominantly by Gamow-Teller. This is indeed partial decay rates to the various states in the daugh-
borne out in our calculation. Using the quenching fac- ter nucleus obtained in the fully converged calculation
6

and 716 ms when calculating the rank 1 operators from
′
thelinearcombinationsξ y ξ(x+u)and(x+u),respec-
/ / / −
tively,whilethepartialhalf-lifeintheconvergedstudyis
0.2 y- (u+x)
651 ms agreeing within 10 percent with the two approx-
imate calculations. In the following we will calculate the

contributions from the first-forbidden rank 1 operators
′
usingaLanczosschemewiththepivotstateξ y ξ(x+u)
−
and 100 iterations.
(a)
)
1
- s 0.0
(
s 0.2 IV. HALF-LIVES OF THE N =50 WAITING
e POINT NUCLEI
x+u
t
a
y r
a
c
e 20
D FRDM
(b)
SM
0.0 Audi-Wapstra
0.2 Full V) 15
e
M
(

Q
10
(c)
0.0 5
0 1 2 3 4 5 6 7
FIG. 2. Partial decay rates for 199Ta calculated from first- 24 26 28 30 32

forbidden rank 1 transitions only. The top (a) and medium
Charge number Z
(b) panel shows the decay rates obtained by using the lin-
ear combinations ξ′y−ξ(x+u) and (x+u), respectively, as FIG. 3. (Color online) Comparison of Qβ values of the N =
50 isotones between experimental data [55] and theoretical
pivotelementstocalculatethecontributionsofrank1opera-
resultsfromFRDM[14]andthepresentshellmodelapproach.
torswithinaLanczosschemewith100iterations. Thebottom
panel(c)showsthepartialdecayrateusingaLanczosscheme
with300iterationsinwhichthestateswithintheQβ window To calculate half-lives, a good description of the tran-
areconverged. Theseshellmodelcalculations havebeenper-
sition matrix elements and also of the Q values is re-
formedinatruncatedmodelspacecomparedtootherstudies β
quired. AsisdemonstratedinFig. 3ourshellmodelcal-
of the N =126 isotones.
culation reproduces the Q -values as given in the Audi-
β
Wapstra compilation well [55]. Hence we will use the
shell model Q values in the following calculation of the
β
(with 300 Lanczos iterations) with those where the rank half-lives and β-delayed neutron emission probabilities
1 contributions were derived using the linear combina- for the N =50 waiting point nuclei.
′
tions ξ y ξ(x+u) and (x+u) as pivots for the Lanc- As is shown in Table II and in Fig. 4 the shell model
−
zos scheme with 100 iterations. As expected, the lowest half-lives agree quite well with the data, although they
Lanczos states are converged and hence the calculated overestimate the ones of 82Ge and 79Cu by about 50%.
strength is the same for either choice of pivot combina- Nevertheless the agreement is significantly better than
tions. TheLanczosschemewith100iterationsisnotsuf- obtainedbasedontheglobalFRDMandETFSImodels.
ficienttoconvergethestatesforexcitationenergieslarger The HFB results [16], which are restricted to the decay
thanabout2.5MeV.Theyrepresentunphysicalstatesas of even-even nuclei, are very similar to the shell model
discussed above. Obviously further iterations lead to a results, except for the half-life of the double-magic nu-
stronger fragmentation of the strength in a small energy cleus 78Ni. Here only the shell model reproduces the
interval around the unphysical states. The energy inter- measured value [7], while all other models predict a sig-
val is small enough that this redistribution of strength nificantly longer half-life. This underlines again the fact
due to different phase space weighting has negligible ef- thatmany-bodyconfigurationmixingisneededtorepro-
fect on the half-life. But also the interference contribu- duce the cross-gap correlations in double-magic nuclei.
tions lead to rather mild differences between the trun- Similar results have been found in studies of the isotope
cated and the converged calculations. We find a partial shifts in calcium [57] or the M1 strength distributions in
half-lifeduetorank1first-forbiddenoperatorsof655ms argon isotopes [58].
7

TABLE II. Comparison of the present shell model half-lives 100
and the ones of reference [21]. All half-lives are in s.
FRDM+QRPA
DF3+QRPA
Nucleus Half-Life (s)
SM
Expt. present shell model (ref [21])
82Ge 4.55±0.05 6.90 2.057
81Ga 1.217±0.005 1.03 0.577 50
/
80Zn 0.545±0.016 0.53 0.432 FF
79Cu 0.188±0.025 0.27 0.222
78Ni 0.11+0.1 0.15 0.127
−0.06
77Co 0.016 0.016
76Fe 8.26×10−3 7.82×10−3
0
75Mn 3.66×10−3 3.52×10−3
74Cr 2.23×10−3 2.07×10−3 24 26 28 30 32
Charge Number Z
FIG. 5. (Color online) Percentage of the contributions from
first-forbidden transitions to the half-lives of the N= 50 iso-
2
10 tonesfromFRDM+QRPA[14],DF3+QRPA[20,56]andthe
present shell model.
FRDM+QRPA
1 DF3+QRPA
10
HFB+QRPA
SM
transitions stay low. For these nuclei they are nearly ex-
) 0 Exp.
s 10 clusivelydue to contributionsfromthe rank2 operators.
(
T 1/2 For the nuclei with Z < 28 the proton f7/2 orbital is
-1 not fully occupied and first-forbidden transitions due to
10
the rank 1 operators are possible. They are noticeably
largerthoseoftherank2operators. Hencethetotalfirst-
-2
10 forbidden strength is significantly larger for nuclei with
Z <28 than for the nuclei with Z 28. Furthermore it
≥
-3 increases with decreasing charge number due to the de-
10
populationofthe protonf orbitalinthe daughternu-
24 26 28 30 32 7/2
cleus. However, also Gamow-Teller transitions from the
Charge number Z
neutron f orbital into the proton f orbital become
5/2 7/2
FIG.4. (Coloronline)Comparisonofhalf-livesoftheN =50
unblocked. Hence reducing the chargenumber, increases
isotones between experimental data and theoretical results
both the GT and first-forbidden transitions due to de-
from FRDM+QRPA [14], HFB+QRPA [16], DF3+QRPA
creasing Pauli-blocking of the dominant transitions into
[20, 56] and thepresent shell model approach.
the f orbital. However, the relative decrease for the
7/2
GT half-lives with decreasing charge number is stronger
thanforthefirst-forbiddentransitions. Thisisrelatedto
The contribution of first-forbidden transitions to the
phasespace. Examplesofdifferentialdecayratesasfunc-
N = 50 half-lives is shown in Fig. 5. For the decay of tionofexcitationenergiesfornuclei74Cr,77Co,78Niand
the nucleiwith Z 28 the probabilityis very small(less 82Ge are shown in Figs. 6. From these figures we note
≥
than5%). However,first-forbiddentransitionscontribute
that the first-forbidden transitions are dominantly pro-
about 25% to the 77Co decay, while they are smaller,
ceeding to states in the daughter at low excitation ener-
but still sizable for the decay of the nuclei with charge
gies(usuallyupto2-2.5MeV)fortheN =50nucleiwith
numbers Z =24–26.
Z < 28, while the GT transitions go to states with ex-
To understand this behavior, we note that first- citation energies of order 5-7 MeV, simply reflecting the
forbiddencontributionsarerelatedtothetransitionfrom factthat it is energeticallymore favorableto havea f
5/2
a g9/2 neutron orbital to a f7/2 proton orbital for the neutron hole and a closed g9/2 shell, than having a hole
rank 1 operators and to a f5/2 proton orbital for rank in the g9/2 orbital. As the energy gain in the transitions
2 operators. (There are no contributions from rank 0 is smallerfor the GT transitions,they aremoresensitive
operators in our model space.) In the simple Indepen- to the increase of the Q value with decreasing charge
β
dent Particle Model the f7/2 level gets completely occu- number. This explains why the relative contribution of
pied for 78Ni and consequently this transition is Pauli- first-forbidden transitions decreases with reduced charge
blocked for N = 50 nuclei with Z 28. In the shell number below the double-magic 78Ni. Above 78Ni the
≥
model, the blocking is partially removed by configura- GT transitions proceed to daughter states at relatively
tion mixing, but the importance of the first-forbidden low excitation energies. (Fig. 6 shows the differential
8

150 6
74 (a) 78 (b)
Cr GT Ni GT
1 ) FF FF
- s
(100 Q 4 Q

e
t
a
r

y
a
c 50 2
e
D
0 0
0 2 4 6 8 1 0 12 14 16 18 20 0 2 4 6 8 10 12
0.10
30 77Co (c) GT 82Ge (d) GT
1 ) FF FF
- s
( Q Q

e 20
t
a 0.05
r

y
a
c
e 10
D
0 0.00
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12
E (MeV) E (MeV)
FIG. 6. (Color online) Partial decay rates including GT and FF transitions for the N = 50 isotones 74Cr (a), 78Ni (b), 77Co
(c), and 82Ge (d).
decay rates for 82Ge as an example.) As a consequence V. HALF-LIVES OF THE N =82 WAITING
firstforbiddentransitions,duetotheirsmallertransition POINT NUCLEI
matrix elements, cannot compete with GT transitions.
The present interaction and model space, based on a
88Sr core, are not the same as used in Ref. [19] to cal-
culate Q values and Gamow-Teller strength functions.
β
However, we stress that the present shell model calcula-
tiongivesverysimilarresults to the onesofRef. [19]. In
We have used the shell model neutron separation en- particular we reproduce the experimentally available Qβ
ergies to calculate the β-delayed neutron emission prob- values very well, as is shown in Fig. 8. This figure also
abilities, which are shown in Fig. 7. As demonstrated shows that the agreement of the Qβ values obtained in
aboveGTtransitionsdominatethedecaysoftheN =50 othermodelsisusuallynotasgoodasbytheshellmodel
r-process waiting points. As for the nuclei with Z 29 results. We will in the following use the shell model Qβ
these transitions connect mainly to daughter state≤s at values for the calculation of the half-lives.
excitation energies above the respective neutron thresh- The 1/2− isomer in 131In corresponds approximately
old, the decay is accompanied by neutron emission with to a 132Sn configurationwith a hole in the 1p orbital.
1/2
a high probability. The probability comes close to 100% Our calculation reproduces the energy of the isomer at
forthedecaysof55Mnand54Crassociatedwiththevery 0.302 MeV. (This quantity was one of the experimental
lowneutronseparationenergiesinthedaughters. Forthe ingredients to which the interaction has been adjusted.)
nuclei with Z > 29, the GT transitions reside at lower The calculated half-lives for the N =82 waiting point
excitationenergiesinthe respectivedaughters,while the nuclei are summarized in Table III and are compared
neutron thresholds increases. As a result the neutron to data and to previous theoretical estimates in Fig.
emission probability is strongly reduced in these nuclei. 9. Compared to experiment, the half-life of 131In is
9

100 1000
FRDM+QRPA FRDM+QRPA
DF3+QRPA DF3+QRPA
SM HFB+QRPA
Exp. SM
)
% s)100 Exp.
P (n50 (m2
1/
T
10
0
24 26 28 30 32
42 44 46 48
Charge Number Z Charge number Z
FIG. 7. (Color online) Neutron emission probability for N= FIG.9. (Coloronline)Comparisonofhalf-livesoftheN =82
50 isotones from FRDM+QRPA [14], DF3+QRPA [20, 56], isotones as calculated in the FRDM, HFB, DF3+QRPA [20]
thepresent shell model and experiment [53]. and thepresent shell model approaches with data.

18 TABLE III. Comparison of the present shell model half-lives
and the ones of reference [19] with experiment [51, 59]. All
FRDM half-lives are in ms.
HFB
15 SM Nucleus Half-Life (ms)
) Exp. Expt. present shell model (ref [19])
V
e 131In 280±30 247.53 260
M
( 12 130Cd 162±7 164.29 162
Q 129Ag 46+5 69.81 70
−9
128Pd 47.25 46
9 127Rh 27.98 27.65
126Ru 20.33 19.76
125Tc 9.52 9.44
124Mo 6.21 6.13
42 44 46 48
Charge Number Z
FIG.8. (Coloronline)ComparisonofQ valuesoftheN =82
β
isotones between theoretical results from the FRDM [14], prediction of an 18% contribution stemming from first-
HFB[16],thepresentshellmodelapproachesandexperimen- forbidden transitions to the decay of the 131In ground
tal data [53]. state is in agreement with the experimental limit of
20%. We further add that we calculate a Gamow-
≤ −
Teller contribution to the half-life of the 1/2 isomer
well reproduced, while the one for 129Ag is somewhat in 131In which is less than 1%, confirming our assump-
too long. This shortcoming had already been observed tion to fix the quenching of the first-forbiddentransition
in the previous shell model calculations. In fact, the to this decay. Fig. 11 shows the partial decay rates
present shell model results, including contributions from to different final states for the nuclei 124Mo, 126Ru and
first-forbidden transitions, agree very well with the shell 128Pd. WenotethatGamow-Tellertransitionsarelarger
model results of Ref. [19]. This, however, does not thanfirst-forbiddentransitions,which, however,proceed
mean that first-forbidden transitions are negligible. As tolevelsatlowerexcitationenergieswhichenhancesthem
is shown in Fig. 10 first-forbidden transitions contribute by phase space.
about 13% to the half-life. However, this value is nearly The β-delayed neutron emission probabilities, i.e. the
the same for all N = 82 waiting point nuclei explaining probabilitiesthat the decayleads to states inthe daugh-
the similarity between the present shell model results to ter nucleus above the neutron separation threshold and
those of Ref. [19]. Only for 131In, first-forbidden tran- hence is followed by the emission of a neutron, is obvi-
sitions contribute somewhat more, resulting in a slightly ously sensitiveto a gooddescriptionof both the neutron
smaller half-life than in the shell model study based separation energies and the β strength functions in the
solely on Gamow-Teller transitions. We note that our Q window. As has been stressed in Refs. [17, 19] the
β
10

100 TABLE IV. Comparison of the present shell model half-lives
and theone of reference [60]. All half-lives are in ms.
FRDM+QRPA
Nucleus Half-Life (ms)
DF3+QRPA
SM present SM (ref [60])
199Ta 286.17 278.88
198Hf 193.28 129.65
50
/F 197Lu 107.85 84.81
F
196Yb 68.98 44.18
195Tm 36.03 29.49
194Er 24.58 18.11
193Ho 13.58 10.94
0 192Dy 10.10 7.75
42 44 46 48
Charge Number Z
FIG. 10. (Color online) Percentage of thecontributionsfrom
approachontop of the density functional DF3, showing,
first-forbidden transitions to the half-lives of the N= 82 iso-
tonesfromFRDM+QRPA[14],DF3+QRPA[20,56]andthe however, a similar dependence with charge number [56].
present shell model data. Adopting a different parameterization, Borzov has also
calculated half-lives for N = 126 isotones with Z < 70
whichareslightlyfasterthanthe shellmodelvalues[62].
improveddescriptionofcorrelationsinshellmodelcalcu- Theshellmodelhalf-livesarenoticeablyfasterthanthose
lations gives a more realistic account of the fragmenta- predictedpreviouslybyglobalmodels,e.g. bytheQRPA
tion of the strength function than is obtained in QRPA calculationontopofthemicroscopic-macroscopicFRDM
studies. Fig. 12 compares the present shell model prob- or ETFSI approaches.
abilities to those obtained in Ref. [19]. We find that the RecentlySuzukietal. [60]havepresentedthefirstshell
inclusionoffirst-forbiddentransitionsleadsonlytominor model half-lives for N = 126 r-process nuclei, includ-
changes. Adetailedcomparisonoftheshellmodelresults ingbothGTandfirst-forbiddencontributions. However,
[19] to those obtained in other theoretical approaches is our present model space including the (0g , 1d ,
7/2 5/2,3/2
given in [17, 19]. 0h , 2s ) protonorbits is noticeably largerthan the
11/2 1/2
one used in Ref. [60] (the 1d , 0h , 2s protonor-
3/2 11/2 1/2
bits). Relatedlythetwoshellmodelcalculationsdifferin
VI. HALF-LIVES OF THE N =126 WAITING the residual interaction and additionally in the adopted
POINT NUCLEI quenching scheme for first-forbidden transitions. Never-
theless, as is shown in Table IV and Fig. 14, both shell
Fig. 13 compares the calculated Q values of the model calculations predict very similar half-lives for the
β
N=126 isotones with other theoretical models. While N = 126 nuclei. Both studies do not predict the strong
thegeneraltrendoftheQ isquitesimilarthanobtained odd-even staggering in the half-lives as observed in the
β
in the FRDM model, the shell model values are slightly QRPA results on top of the FRDM model.
smallerthan those fromthe FRDM model. As, however, As already noted in Ref. [20] based on the density
no experimental data exist for these very neutron-rich functionalcalculations,first-forbiddentransitionsareex-
N =126nuclei, itis notpossible to decide whichQ are pected to contribute significantly to the half-lives of the
β
morerealistic. Inthefollowingwewillusetheshellmodel N = 126 r-process nuclei. This finding is supported by
values to calculate the half-lives and β-delayed neutron our shell model calculations (Fig. 15). One observes an
emission probabilities for the N =126 r-process waiting increasing contribution from the first-forbidden transi-
point nuclei. tions with increasing proton number. In fact, for nu-
Theshellmodelhalf-livesarelistedinTableIVandare clei with proton number Z 70, contributions from
≥
comparedtoothertheoreticalpredictionsinFig. 14. Al- first-forbidden transitions to the half-life dominate over
though recently researchers at GSI have been successful Gamow-Teller transitions. This behavior can be under-
to measure half-lives of nuclei close to N = 126 with stood by inspecting the partial decay rates arising from
charge numbers below lead [12], supplying important Gamow-Teller and first-forbidden transitions which are
constraintsaboutthehalf-lifetrendtowardsther-process shown in Fig. 16 for selected nuclei. We note that
nuclei, experimental data for the N = 126 r-process nu- Gamow-Teller transitions are related to the change of
clei do yetnot exist. Hence our results can only be com- a neutron in the 0h orbit to a proton in the 0h
9/2 11/2
pared to other theoretical predictions. We note that the orbit which, however, is fragmented over several states
presenthalf-livesforZ >70arefaster,byaboutafactor in the daughter nucleus due to correlations. Neverthe-
of two, than those obtained by Borzov within an QRPA less, for the nuclei studied here these final proton states