Origin of the anomalous long lifetime of 14C

0.12 MB·English

by  P. Maris

Origin of the anomalous long lifetime of 14C
P. Maris,1 J.P. Vary,1 P. Navra´til,2,3 W. E. Ormand,3,4 H. Nam,5 and D. J. Dean5
1Department of Physics and Astronomy, Iowa State University, Ames, Ia 50011-3160, USA
2TRIUMF, 4004 Wesbrook Mall, Vancouver BC, V6T 2A3, Canada
3Lawrence Livermore National Laboratory, L-414, P.O. Box 808, Livermore, CA 94551, USA
4Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
5Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, USA
Wereportthemicroscopicoriginsoftheanomalouslysuppressedbetadecayof14Cto14Nusingthe
abinitiono-coreshellmodel(NCSM)withtheHamiltonian fromchiraleffectivefieldtheory(EFT)
includingthree-nucleonforce(3NF)terms. The3NFinducesunexpectedlylargecancellationswithin
the p-shell between contributions to beta decay, which reduce the traditionally large contributions
from theNN interactions byan order of magnitude, leading to thelong lifetime of 14C.
1
1
PACSnumbers: 21.30.Fe,21.60.Cs,23.40.-s
0
2
n Themeasuredlifetimeof14C,5730±30years,isavalu- the GT transitions to excited A=14 states [8].
a ablechronometerformanypracticalapplicationsranging ChPT provides a theoretical framework for inter-
J
from archeology to physiology. It is anomalously long nucleoninteractionsbasedontheunderlyingsymmetries
6 comparedtolifetimesofotherlightnucleiundergoingthe of QCD. Beginning with pionic or the nucleon-pion sys-
2
same decay process, allowed Gamow-Teller (GT) beta- tem [9] one works consistently with systems of increas-

] decay,anditposesamajorchallengetotheorysincetra- ing nucleon number [10–12]. One makes use of the ex-
h ditionalrealisticnucleon-nucleon(NN)interactionsalone plicit and spontaneous breaking of chiral symmetry to
t
- appear insufficient to produce the effect[1]. Since the expandthe stronginteractionin termsofa genericsmall
l
c transition operator, in leading approximation, depends momentum. The ChPT expansion conveniently divides
u onthe nucleonspin andisospinbut notthe spatialcoor- the interactions into perturbative and non-perturbative
n
dinate, this decayprovidesa precisiontoolto inspect se- elements. The latter are represented by a finite set of
[
lectedfeaturesofthe initialandfinalstates. To convinc- constants at each order of perturbation theory that are

1 ingly explain this strongly inhibited transition, we need not presently obtainable from QCD, but can be fixed by
v a microscopic description that introduces all physically- experiment. These constants should be of order unity
4
relevant14-nucleonconfigurationsinthe initial andfinal (“naturalness”). Once determined, the resulting Hamil-
2
1 states and a realistic Hamiltonian that governs the con- tonian predicts all other nuclear properties. We have
5 figuration mixing. demonstratedthat this reductiveprogramworkswellfor
1. We report the first no-core solutions of 14C and 14N light nuclei when the 3NF is included. [7].
0 usingaHamiltonianwithfirmtiestotheunderlyingthe- We adopt NN and 3NF potentials of ChPT [13, 14]
1 ory of the strong interaction, Quantum Chromodynam- and the no-core shell model (NCSM) to solve the many-
1
ics (QCD), which allows us to isolate the key canceling body Schr¨odinger equation for A=14 nuclei while pre-
:
v contributions involved in this beta decay. We find that servingallHamiltoniansymmetries [5,15,16]. The non-
i the three-nucleonforce(3NF) ofchiralperturbationthe- perturbativecoupling constants ofthe 3NF, notfixedby
X
ory (ChPT) plays a major role in producing a transition π−N or NN data, are c for the N −π−NN contact
r D
a rate that is near zero, needed for the anomalous long term and cE for the NNN contact term. We previously
lifetime. A chiral 3NF with coupling constants consis- fitc andc toA=3bindingenergiesandselecteds-shell
D E
tentwithotherworksandwithintheirnaturalrangecan and p-shell nuclear properties and showed the resulting
provide the precise lifetime. This indicates that correc- interactions worked well for A=10-13 nuclei [7]. Here,
tions to the lifetime that arise from increasing the basis we present results for (c ,c ) = (−0.2,−0.205) and for
D E
space,from including additionalmany-body interactions (−2.0,−0.501). Both sets, labeled by their c value be-
D
and from corrections to the GT operator in ChPT [2] low, are allowed by the naturalness criterium and fit the
may be absorbed into an allowed choice of the 3NF. A=3 binding energies. The former also produces a pre-
Our work features two major advances over recent al- cisefittothetritonhalflife[17]. Thelatterproducesthe
ternativeexplanations[3,4]: (1)wetreatallnucleonson tritonhalf life within 20%ofexperiment but is preferred
the same dynamicalfooting with the no-coreshellmodel by the 14C half life as we show here.
(NCSM) [5], and (2) we include the 3NF of ChPT [6] as We select a basis space of harmonic oscillator (HO)
a full 3-nucleon interaction. This follows previous work single-particle states specified by the HO energy ¯hΩ
detailing the structure and electroweak properties of se- and by N , the limit on the number of HO quanta,
max
lected A=10-13 nuclei [7] with the same chiral NN + (2n +l ), above the minimum for that nucleus. n is
Pi i i i
3NF. We also establish a foundation for future work on the principal quantum number of the orbital of the ith
2
nucleon and l is its orbital angular momentum. Thus interaction alone (“N3LO” columns) and the chiral NN
i
N =0isthesmallestbasisspaceforeachnucleuscon- + 3NF (“N3LO+3NF” columns).
max
sistentwith the Pauliprinciple. ForeachchoiceofN ThetrendsinexcitationenergieswithincreasingN
max max
andh¯Ωwecarryoutawell-establishedfinite-basisrenor- suggest reasonable convergence. We note that the level
malization[18]oftheHamiltoniantoarriveatanHermi- orderings are the same with and without the 3NF in the
tianeffective3-bodyHamiltonian[5,16]. Allsymmetries largestbasisspaces(N =8). Overall,the spectraare
max
oftheunderlyingHamiltonianarepreservedthroughout. more compressed with NN alone than with NN + 3NF,
We adopt ¯hΩ=14 MeV, which corresponds to the min- a pattern seen before [7, 16]. However, the 3NF appears
imum in the binding energy obtained for N =8. We tohaveovercompensatedforthe compressioninthe NN
max
have checked that results for 13≤ ¯hΩ≤ 18 MeV do not spectra when compared with experiment. On the other
qualitatively alter our conclusions. handthe3NFimprovesthebindingenergies. Theexper-
Basis space dimensions grow rapidly with N and imental (14C, 14N) binding energies are (105.28, 104.66)
max
provideamajortechnicalchallenge[19]. Themany-body MeV. Without the 3NF, we obtain (97.20, 96.36) MeV.
matrix dimension in the M-scheme basis (total magnetic Whenweincludethe3NFthebindingenergiesincreaseto
projection quantum number is fixed) for 14C (14N) is (108.43,108.47)MeVforcD =−0.2and(108.30,108.41)
872,999,912 (1,090,393,922) at Nmax = 8 and the asso- MeV for cD =−2.0
ciated number of non-vanishing 3NF matrix elements on We note that the binding energies for both 14C and
and below the diagonal ≃2.9×1013 (≃3.9×1013). 14N are about 8 MeV or 8% larger at Nmax = 6 than
We obtained our results on the Jaguarsupercomputer they are at Nmax = 8. Similar calculations for lighter
at Oak Ridge National Lab [22] using up to 35,778 hex- nucleishowthattheconvergenceofthebindingenergyis
core processors (214,668 cores) and up to 6 hours of not monotonic, hampering an extrapolation to the com-
elapsed time for each set of low-lying eigenvalues and plete (infinite-dimensional) basis. On the other hand,
eigenvectors. The number of non-vanishing matrix ele- the excitation spectra appear rather stable upon reach-
ments exceeded the total memory available andrequired ing Nmax = 8. We take results in the largest available
matrix element recomputation“on-the-fly”for the itera- basisspaceasestimatesoftheconvergedresults. Wealso
tive diagonalization process (Lanzos algorithm). show in Fig. 1 that the spectra are relatively insensitive
to a significant range of values for (c ,c ).
D E
Using Fermi’s Golden rule for the transition rate, the
20 ((31++,,01)) half life T1/2 for 14C is given by
(2+,1)
MeV)15 ((20++,,01)) T1/2 = f(Z1,E0)2mπ5e3c¯h47Gln2V2gA2|M1GT|2 , (1)
y ( (1+,0)
g where M is the reduced GT matrix element; f(Z,E )
er GT 0
n en10 is the Fermi phase-space integral; E0 =156 keV is the β
atio endpoint; GV = 1.136 10−11 MeV−2 is the weak vector
xcit coupling constant; gA =1.27 is the axial vector coupling
e
5 constant;andme istheelectronmass. MGT forthetran-
sitionfromthe initial 14C (Jπ,T)=(0+,1)groundstate
(Ψ ) to the 14N (1+,0) ground state (Ψ ) is defined by
i f
0 2 4 6 8 expt 8 6 4 2 thespin-isopinoperatorσ(k)τ+(k)actingonallnucleons,
N3LO + 3NF N3LO k:
FIG. 1: (Color online) The excitation spectrum of 14N ob- MGT = XhΨf||σ(k)τ+(k)||Ψii. (2)
tainedintheNCSMusingchiralinteractionsasafunctionof
k
basisspacecutoffNmax,thenumberthatlabelsthecolumns.
(Jπ,T) values are in the legend. The center column displays Since the initial state has total spin zero, MGT is equal
theexperimentalspectrum for14N.Columns totheright are to the M-dependent GT matrix element MM ,
GT
obtained with the chiral NN interaction alone while those
ocDn =th−e0le.2ft(saorleidolbintaeisn)eadndwictDh=th−e2c.h0ir(adlasNhNed+lin3esN)F. using MGMT = Xhα|στ+|βiραβ, (3)
α,β
where hα|στ |βi is the one-body matrix element be-
+
Fig. 1 compares our spectra for 14N with experiment. tween HO single-particle states α and β, which is non-
The T = 1 isobaric analog states serve as good proxies vanishing only when both single-particle states are in
forthe14Cspectra. Wepresentthespectraasafunction the same shell, and the one-body density matrix ρ ≡
αβ
oftheN value(columnlabels)forboththechiralNN Ψ |a†a |Ψ .
max (cid:10) f α β i(cid:11)
3
In order to reproduce the measured half life of T ≃ supports a recurring theme of 3NF’s in p-shell nuclei -
1/2
5730years,theGTmatrixelementmustbeanomalously they play a significant role in the spin-sensitive proper-
small, |MM |≃2×10−3,incontrastwithaconventional ties of spin-orbit pairs. We note that our observedshifts
GT
strong GT transition in a light nucleus with |M |≃1. within the p-shell due to the 3NF is similar to what Ref.
GT
[3] accomplished with a density-dependent effective NN
0.2924 interaction obtained by modeling leading contributions
fromthechiral3NF.However,ournetcontributionsfrom
0.03 other shells, which are absent in Ref. [3], overwhelm the
no 3NF forces
net p-shell contribution as seen in Fig. 2. That is, while
with 3NF forces (c = -0.2)
nt 0.02 with 3NF forces (cD= -2.0) the s-shell and sd-shell contributions nearly cancel, all
me 0.01 D the shells above the sd-shell contribute about a factor 2
e
el greater than the now-suppressed p-shell contribution.
x 0
matri-0.01 amTinoefuthrtehceornutnridbeurtsitoannsdtothMe rGoTleinoftthheeL3SN-sFc,hwemeecawnheexre-
GT the single-particlequantumnumbers now involvethe or-
-0.02
bital angular momentum projection m and spin projec-
l
-0.03 tionmsreplacingthetotalangularmomentumprojection
m . This is a convenient representation for this transi-
0.3 j
tionsincem andm mustbethesameforincomingand
l s
0.2
outgoing single-particle states. For the p-shell contribu-
0.1 tions, the resulting decomposition to LS-scheme yields
0 results shown in Table I. Note that there is nearly per-
-0.1 s p sd pf sdg pfh sdgi pfhj sdgikpfhjl fectcancellationbetweenthe ml =0andml =±1terms
once the 3NF is included.
Given the overall effects on M by inclusion of the
shell GT
3NF as seenthrough Fig. 2 and in Table I, one may fine
FIG. 2: (Color online) Contributions to the 14C beta decay tune cD to reproduce the value 2×10−3 consistent with
matrixelementasafunctionoftheHOshellinwhichtheyare the 14C lifetime. This is analogous to the fine tuning of
evaluatedwhenthenuclearstructureisdescribedbythechiral c in Ref. [17] to fit the 3H lifetime. Our estimate that
D
interaction. Top panel displays the contributions with (two the desired suppression of M occurs with c ≃ −2.0
GT D
right bars of each triplet) and without (leftmost bar of each illustrates this point. Note that the spectra of 14C and
triplet) the3NFat Nmax =8. Allcontributionsare summed 14N are rather insensitive to this range of c values. We
within the shell to yield a total for that shell. The bottom D
show the resulting M in the final row of Table I.
panel displaystherunningsum of theGT contributionsover GT
the shells included in the sum. Note the order-of-magnitude
suppressionofthep-shellcontributionsarisingfromthe3NF. (ml,ms) NNonly NN + 3NF NN+ 3NF
cD =−0.2 cD =−2.0
(1,+1) 0.015 0.009 0.009
2
(1,−1) −0.176 −0.296 −0.280
In Fig. 2, we present our main results for M . We 2
GT (0,+1) 0.307 0.277 0.283
decompose MGT at Nmax = 8 into the contributions (0,−12) 0.307 0.277 0.283
arising from each HO shell for two cases with the 3NF (−1,+21) −0.176 −0.296 −0.280
2
(cD = −0.2,−2.0) and one without. The largest effect (−1,−1) 0.015 0.009 0.009
2
occurs in the p-shell where, for both values of cD, the Subtotal 0.292 −0.019 0.024
3NFreduces the contributionsby anorder-of-magnitude Total Sum 0.275 −0.063 −0.013
from the result with the NN interaction alone. The con-
tributionsofeachofthe9additionalshellsisenhancedby TABLE I: Decomposition of p-shell contributions to MGT in
the LS-scheme for the beta decay of 14C without and with
uptoafactorof2bythe 3NF.Thecumulativecontribu-
3NF. The 3NF is included at two values of cD where cD ≃
tions to the GT matrix element is displayedin the lower −0.2ispreferredbythe3HlifetimeandcD ≃−2.0ispreferred
panelofFig. 2whereoneseesclearlythenetsuppression by the 14C lifetime. The calculations are performed in the
to MGT due to the 3NF. Nmax =8 basis space with h¯Ω=14 MeV.
Looking into the detailed changes within the p-shell,
one finds that the 3NF introduces a systematic shift of
strength away from 1-body density matrix terms involv- Next, we studied the sensitivity at (c ,c ) =
D E
ing transitions between the 0p and the 0p orbits (−2.0,−0.501) and at N = 6 by successively setting
3/2 1/2 max
to 1-body density matrix elements involving transitions each to zero while keeping the other fixed. The larger
within the 0p orbits. The shift is about 30% of the effect on M appears with c = 0. However, the re-
1/2 GT D
magnitudes of these 1-body density matrix terms and sultingshiftsinthemagnitudeofM areapproximately
GT
4
proportional to the magnitude of the changes in c and suppression of M consistent with the anomalous long
D GT
c which implies M has about the same sensitivity to lifetime of 14C.
E GT
each. This sensitivity differs from that of Ref. [3] where
the c term was found to play a leading role in M . This work was supported in part by US DOE Grants
E GT
Since both appear natural, we conclude that the dif- DE-FC02-09ER41582 (UNEDF SciDAC Collaboration),
ferent values of c from the beta decays of 3H and 14C DE-FG02-87ER40371, and by US DOE Contract No.
D
accommodate the sum of higher-order effects: meson- DE-AC52-07NA27344 and No. DE-AC05-00OR22725 .
exchange currents, additional ChPT interaction terms, Computational resources were provided by Livermore
and contributions from larger basis spaces. Additional Computing at LLNL and by a “Petascale Early Science
3NFterms,e.g. thosearisingattheN3LOlevelofChPT Award”andanINCITEAwardontheJaguarsupercom-
that do not impact the 3H lifetime [20], and/or four nu- puter at the Oak Ridge Leadership Computing Facility
cleon interaction terms may impact 14C and 14N. at ORNL [22] which is supported by the DOE Office of
Inordertofurthertesttheab initiowavefunctionsand Science under Contract DE-AC05-00OR22725.
the effects of the 3NF, we present in Table II the con-
tributions to selected electromagnetic properties of 14N.
We see that the 3NF has substantial influence on the
magnetic-dipole transition in 14N from the ground state
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