Propagation of uncertainties in the Skyrme energy-density-functional model

0.33 MB·English

by  Y. Gao

Propagation of uncertainties in the Skyrme energy-density-functional model
Y. Gao,1 J. Dobaczewski,1,2 M. Kortelainen,1 J. Toivanen,1 and D. Tarpanov2,3
1Department of Physics, P.O. Box 35 (YFL), FI-40014 University of Jyva¨skyla¨, Finland
2Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Hoz˙a 69, PL-00-681 Warsaw, Poland
3Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria
(Dated: January 28, 2013)
Parameters of nuclear energy-density-functionals (EDFs) are always derived by an optimization
to experimental data. For the minima of appropriately defined penalty functions, a statistical sen-
sitivity analysis provides the uncertainties of the EDF parameters. To quantify theoretical errors
of observables given by the model, we studied the propagation of uncertainties within the unedf0
Skyrme-EDFapproach. We found that typically the standard errors rapidly increase towards neu-
3 tron richnuclei. This can belinkedtolarge uncertaintiesof theisovector coupling constantsof the
1 currently used EDFs.
0
2 PACSnumbers: 21.60.Jz, 21.10.Dr,21.10.Gv,21.10.Ft,21.30.Fe

n
a
J Introduction. The nuclear energy-density-functional gateduncertaintiessimplymeanthatrelativelysmallde-
5 (EDF)approachistheonlymicroscopicmethodthatcan viations of model parameters, within the bounds of the
2 be applied throughout the entire nuclear chart. Owing adjustment, strongly influence the predicted values. On
totheexistingandfutureradioactive-beamfacilities,the theonehand,this maygiveusinvaluableinformationon

] experimentally known region of the nuclear landscape whichparametersofthemodelaregrosslyundetermined,
h
will be pushed towards more and more exotic systems. and on the other hand, this may indicate which observ-
t
- New experimental data on nuclei with a large neutron ables, when measured, would have the strongest impact
l
c or proton excess will put predictions of the current and on decreasing the errors of parameters.
u
future EDF models to test. Therefore,it is important to Thereis,ofcourse,anotherimportantclassoftheoreti-
n
[ assess the uncertainties and predictive power of models caluncertainties,whichisnotdirectlyaccessiblethrough
[1–4]. the statistical analysis of parameters, and which is very
1
The nuclear EDF models can best be formulated in difficult to estimate, namely, the one stemming from the
v
terms of effective theories, tailored to low-energy phe- definition and limitation of the model itself, see discus-
1
nomena with high-energy effects taken into account sioninRefs.[1,3,4,10]. However,thestatisticalanalysis
5
1 through model parameter adjustments. Although a rig- may alsohere give us interesting information. Indeed, in
6 orous applicationof the effective-theory methodology [5] cases when the propagated uncertainties turn out to be
. isnotyetfullyimplemented,theaspectofsystematicad- significantly smaller than the fit residuals, one may sus-
1
0 justmentsofmodelparameterstodataisalwaysacrucial pect that the model lacksimportant physics and/ornew
3 element of the approach. Only very recently, such ad- terms in the Hamiltonian (functional) and extended pa-
1 justments areaccompaniedwiththefullcovarianceerror rameter freedom. This is so, because we are then in the
: analysis [6–9]. situation of model parameters well constrained by some
v
i The aim of the present work is to study how uncer- data, but not describing some other data correctly. This
X
tainties in the model parameters propagate to observ- canbecontrastedwiththesituationofpropagateduncer-
r ables calculated within the model. This appears to be taintieslargerthanthefitresiduals,whenonecannottell
a
the simplestandmostfundamentalmethodtoassessthe whether the model is deficientor whether its parameters
predictive power of the model. First, the model is char- are underdetermined.
acterizednotonly by the fit residuals,that is,deviations Method. A good measure of theoretical uncertainty is
between measured and calculated observables, but also to calculate the statistical standard deviation of an ob-
by propagated uncertainties, which carry the errors of servable, predicted by the model. In the present, work
model parametersoverto calculatedobservables. In this we used the Skyrme EDFs [11] within the Hartree-Fock-
way,eventhedatapointsthathavebeenusedforthepa- Bogoliubov(HFB)nuclearstructurecalculations. Tode-
rameteradjustmentsarecharacterizedbythepropagated termine the propagationof uncertainties from the model
uncertainties. parameters to some observable, predicted by the model,
A properly balanced parameter adjustment should, in oneneedsinformationaboutthestandarddeviationsand
principle, lead to propagated uncertainties, which are correlationsbetween the model parameters. Some of the
comparable in magnitude to fit residuals. Second, ex- recent Skyrme-EDF parameter optimizations have pro-
trapolations of the model to unmeasured data points, videdthisinformation[6,7,9]. Inthiswork,todetermine
which is, in fact, one of the principal goals of formulat- thepropagationoftheoreticaluncertainties,wehavecho-
ingmodels,bearnomeaningiftheyarenotaccompanied sen to use the unedf0 parameterization of the Skyrme
by their propagated uncertainties. Indeed, large propa- EDF [7].
2
The calculated statistical error of an observable is lack deformation effects. A priori, one can expect that
linked with the covariance matrix of the nuclear EDF thestandarderrorsdeterminedinsphericalanddeformed
model parameters. From the Taylor expansion, we can nuclei should be similar, see also Ref. [18], although in
find that the square of statistical error reads, the future this conjecture must be tested in more exten-
sive calculations. The aim of this work is to quantify a
n
∂y ∂y typicalmagnitudeofthestandarderrorrelatedtoanob-
σ2(y)= Cov(x ,x ) , (1)
i j
(cid:20)∂x (cid:21)(cid:20)∂x (cid:21) servable, and thus assess the overall predictive power of
iX,j=1 i j
the model.
where y is the calculated observable, x is the parame- Results. In Fig. 1, we present the calculated stan-
i
ter of the EDF and Cov(x ,x ) is the covariance matrix darderrorsofbindingenergiesofsemi-magicnuclei. The
i j
between parameters x and x . In the present work we upper panel shows the isotonic chains with magic neu-
i j
use the covariance matrix that was obtained from the tron numbers of N = 20, 28, 50, 82, and 126 and the
sensitivity analysis of the unedf0 parameterization, see lowerpanelshowstheCa,Ni,Sn,andPbisotopicchains.
Tables VIII and X of Ref. [7]. Another use of the covari- Withintheexperimentallyknownregionsofnuclei,where
ance matrix consists in calculating covariance ellipsoids also data points were used during the EDF optimiza-
betweenpairsofobservables[3,8]andanalyzingcorrela- tion, standard error usually stays rather constant and is
tions between observables [12, 13]. typically of the order of 1MeV. Moving to the neutron
Calculation of the standard error (1) requires calcula- rich regime, calculated standard error increases rather
tion of derivatives of observable y with respect to model steadily. This is directly related to the fact that the
parameters x . As the relations between parameters isovector parameters of the EDF are not constrained as
i
and observables can be complicated, to calculate the well as the isoscalar ones [3, 7]. With isotonic (isotopic)
derivatives one has to use numerical methods. By def- chains,thenucleiwithsmallZ (largeN)ofFig.1arethe
inition, the derivative of a function f can be approxi- ones with most neutron excess, and therefore standard
mated simply by a finite difference, f′(a) ≈ G (h) = deviation is largest in these nuclei. The values reached
0
(f(a+h)−f(a−h))/2h. To improve the accuracy, we there can be of the order of 10MeV, which means that
used the Richardson extrapolation method [14], forthetotalbindingenergies,whenapproachingtheneu-
trondripline the predictivepowerofthe modelisrather
4mGm−1(h/2)−Gm−1(h) poor.
G (h)= , (2)
m 4m−1
For the Pb and N = 126 nuclei we also show, as
with m=1,2,..., which then satisfies, f′(a)−G (h)= an illustration, the moduli of fit residuals with respect
m
O(h2(m+1)). Generally speaking, a larger m means im- to experimental data [19]. We note immediately, that
the propagated error of the binding energy of 208Pb
proved accuracy on derivatives with an increased cost of
of 0.41MeV is much smaller than the fit residual of
computational time. In this work we extrapolate up to
4.07MeV. This clearly indicates that no refined fits can
m=2 as a balanced point.
probably account for this experimental data point, be-
Large values h may lead to inaccurate derivatives,
cause the fit parameters are already quite rigidly con-
whereas small values may suffer from rounding errors.
strained by other data. This result points to a miss-
In this paper, we checked that when h changes for any
ing physics in the description of the 208Pb mass, which
parameter between 0.001% and 0.1% of the parameter’s
is most probably related to a poor description of the
value, the calculated errorsof observables do not change
ground-statescollectivecorrelationsindoublymagicsys-
much. Thus values of h fixed in this region were used.
tems, see discussion in Refs. [20–22].
The HFB calculations were carried out by using the
computer code hosphe [15, 16], which solves the HFB InFig.2,thestandarderrorsoftwo-neutron(S2n)and
equations in a spherical harmonic-oscillator basis. We two-proton (S2p) separation energies are shown. Sim-
have run it within exactly the same conditions as those ilarly as for the binding energies, within the experi-
defined for the optimization of unedf0. In particular, mentally know region, these standard errors stay rather
calculationsweredoneinthespaceof20majoroscillator small, that is, in heavy nuclei below 0.5MeV for S2p
shellsandthe Lipkin-NogamiprocedureofRef.[17]with and below 0.25MeV for S2n. Then, they start to in-
the pairing cut-off window of E = 60MeV was used. crease rapidly when the calculations are extrapolated to
cut
We determined ground-state properties of all even-even neutron-rich regime, especially after the doubly magic
semi-magic nuclei with N = 20, 28, 50, 82, and 126 and gaps are crossed. Owing to the cancellation of errors,
Z =20,28,50,and82extendingbetweenthetwo-proton even in very exotic nuclei, the standard errors of two-
and two-neutron driplines. particle separation energies are much smaller than those
For unedf0, some of the very neutron rich Ca, Sn, ofthebindingenergies. Thisnicelyillustratestherobust-
and Pb isotopes turn to be deformed, see, e.g., supple- ness ofthe position oftwo-neutrondriplines discussedin
mentary material of Ref. [4]. The same also holds for Ref. [4].
some semimagic isotonic chains at the very neutron-rich InFigs.3and4,weplotthestandarderrorsofneutron
regime. Since the current calculations are done with a and proton rms radii, respectively. One can see that the
spherical HFB solver, the calculated uncertainties may uncertainties related to the proton rms radii are much
3

(a) (a)
15 N=20 3 N=20
N=28 N=28
V) NN==5802 eV) NN==5802
Me10 NR=es1i2d6ue M 2 N=126
E) ( ) (2p
( 5 (S 1
0 0
10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100
Z Z

Ca (b) (b)
15 NSni 1.5 Ca
Pb Ni
) Residue V) Sn
V e Pb
e10 M1.0
M
E) ( ) (2n
( 5 (S0.5
0 0.0
20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200
N N
FIG. 1: (Color online) Standard errors of binding energies FIG.2: (Coloronline)(a)Standarderrorsoftwo-protonsep-
of semi-magic nuclei, calculated for (a) isotonic chains with aration energiesS2p forneutronsemi-magicnuclei. (b)Stan-
magic neutron numbers and (b) isotopic chains with magic darderrorsoftwo-neutronseparationenergiesS2n forproton
proton numbers. For the Pb and N = 126 nuclei, moduli of semi-magic nuclei.
fit residuals with respect to experimental data [19] are also
shown.
errors to be 0.0035fm−3 or less. In the same region, the
incoherent sum of experimental and model errors [23] is
smaller than those corresponding to neutron rms radii. roughly 0.0075fm−3. Even though the density profile
This is directly related to the fact that in the optimiza- of Fig. 5 is for a point-neutron density, folding it with
tionoftheunedf0parameterset,experimentalinputon a finite weak-charge distribution of neutrons, or includ-
the charge radii was used. For example, in 208Pb we ob- ing in the experimentalestimate other smallcorrections,
tainσ(R )=0.062fmandσ(R )=0.006fm. Thestandard should not significantly change our result of propagated
n p
errorsofneutronradiisteadilyincreasetowardsthe neu- theoretical uncertainties being about twice smaller than
tron drip line, again illustrating the large uncertainties the experimental ones.
in the isovector coupling constants of the functional. In Owing to the correlations between the model parame-
heavy nuclei, the same is also true for protons, however, ters, the standard errors calculated by using Eq. (1) can
in light nuclei, the standard errors of proton radii also be strongly affected by the off-diagonal components of
increase towards the proton drip line. This effect can thecovariancematrix. ThisisillustratedinFig.6,where
be related to the weak binding of protons within a low variouscomponentsofthesumcontributingtosquaresof
Coulombbarrier. Uncertaintiesrelatedtotheprotonand standarderrorsofthebindingenergyE (a),two-neutron
neutron rms radii are closely linked to the uncertainties separation energy S (b), neutron rms radius R (c),
2n n
of the neutron skin, see discussion in Ref. [18]. and proton rms radius R (d) in 208Pb are plotted. The
p
In Fig. 5, we present standard errors of the calculated order of the parameters used in Fig. 6 is the same as
mean-field (point-particle) proton and neutron densities that used in the definition of the correlation matrix of
in208Pb. Againweseethatthestandarderrorsofproton unedf0, see Tables VIII and X of Ref. [7]. For the sake
densities are significantly smaller than for neutrons, and of completeness, these parameters are also listed in the
that the neutron ones extend to much higher distances. caption of Fig. 6.
The neutron density of Fig. 5d can be compared with One sees from Fig. 6a that the standard error of the
the experimental weak-charge profile presented in Fig. 1 binding energy E results from a strong cancellation be-
of Ref. [23], obtained from the PREX measurement at tweenlargepositive(>500MeV2)diagonalcontributions
Jefferson Lab [24]. In the region of relative flat neutron coming from aNM and LNM (parameters No. 3 and 4),
sym sym
density profile (that is, R < 4fm) we find our standard andlarge negative(<−500MeV2) contributions coming
4

(a) 0.03 (a)
0.15
N=20 N=20
N=28 N=28
N=50 N=50
m) N=82 m) 0.02 N=82
) (fn0.10 N=126 ) (fp N=126
R R
( 0.05 ( 0.01
0.00 0.00
10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100
Z Z

0.03
0.15 (b) (b)
m) m) 0.02
0.10
) (fn ) (fp
R R
( 0.05 ( 0.01
Ca Ca
Ni Ni
Sn Sn
Pb Pb
0.00 0.00
20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200
N N
FIG. 3: (Color online) Same as in Fig. 1 but for the neutron FIG. 4: (Color online) Same as in Fig. 1 but for the proton
rms radii. rms radii.
from the off-diagonal correlations between these two pa-
R (fm) R (fm)
rameters. The remaining parameters of the functional 0 4 8 12 16 0 4 8 12 16
give much smaller positive and negative contributions, 3)0.0015 (a) (b) 0.003
bringing the finalvalue to σ2(E)=0.17MeV2. It is inter- -m (
esting to note that even in this well-bound nucleus, the )( fp0.0010 0.002nmf ()
largestuncertaintiesofthebindingenergycomefromthe (0.0005 0.001 -3
poorsymmetry-energycharacterizationofthefunctional. )
0.0000 0.000
The situation obtained for the standard error of S
(Fig. 6b) is quite similar. Although the value 2onf 3) 0.065 (c) (d) 0.10
σbtri2inb(dSui2tnnigo)n=es0n.ec0ro2gmyM,ineigVt2rfoeirssmulmtstuhfceohramsbmotvaheleletrcwatonhcapenallraatthmioaentteofrofsrcaotnhnde- -( fmp 000...000556050 000...000789 nmf )(-3
0.045 0.06
C1ρ∆ρ (parametersNo.6),whichistheparametercharac- 0.040 0.05
terizingtheisovectorsurfacepropertiesofthefunctional. 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Again,the uncertaintiesintheisovectorcharacterization R (fm) R (fm)
of the functional dominate the standard error here. FIG. 5: Standard errors of densities in 208Pb for (a) protons
The pattern obtained for the standard error of the and(b)neutronsasfunctionsoftheradialvariableR. Panels
neutron rms radius of σ2(R )=3.8×10−3fm2 (Fig. 6c) is (c) and (d) show the proton and neutron densities, respec-
n
markedlydifferent, withthe singleparameterLNM dom- tively,with thestandard errors represented byerror bars.
sym
inating the uncertainty of this variable, although again
the isovector properties are to be blamed. Only for the
proton rms radius (Fig. 6d), its fairly small standard er- teresting to discuss their standard errors and check how
ror of σ2(Rp)=4.1×10−5fm2 is given by contributions tightly their values are fixed by the adjustment of the
comingfromaNM andisoscalarparametersρ andCρ∆ρ. functional to other observables, see also discussion in
sym c 0
Even though the single-particle (s.p.) energies are Ref. [6]. Since our calculations were performed with
not strictly-speaking observables, they are nevertheless pairingcorrelationsandLipkin-Nogamicorrectionstaken
closely connected to observable levels of neighboring intoaccount,weperformedtheanalysisforthes.p.ener-
even-oddnucleiandtoothernuclearpropertieslike,e.g., gies e defined as the eigenstates of the mean-field part
HF
deformations of open-shell systems. It is, therefore, in- of the HFB Hamiltonian.
5

10 -5(a0)0 10 -0(.b2)0 TABLEI:The208Pbproton(top)andneutron(bottom)s.p.
68 ---0321752505 68 ---0000.0...1100505 (teehn)ee,rePgmiVepsCiersHicδFael(PbvV)aClau(nefd)s,t(ahdne)dir[2rs5ets]a,indrdueasarilddsuoeafrlrsto,hre∆s(P(ceV)H,CFa)-sc=ocorermeHcpFtae−rdeedse.xtppo.
4 122550 4 00..0150 energies ∆(ePVC) = eHF+δePVC−eexp (g). Where applica-
2 375 2 0.15 ble,wealso givetheroot-mean-square(rms)valuesofentries
500 0.20
2 4 6 8 10 (MeV2) 2 4 6 8 10 (MeV2) shown in a given column. Allenergies are in MeV.
orbital eHF σ(eHF) eexp ∆(eHF) δePVC ∆(ePVC)
10 -0(.c2) 10 -0(.d0)15 (a) (b) (c) (d) (e) (f) (g)
0.0 -0.010
8 0.2 8 -0.005 π3p 0.219 0.181 −0.16 0.38 −0.440 −0.06
1/2
0.4 0.000
6 0.6 6 0.005 π3p3/2 −1.045 0.139 −0.68 −0.37 −0.662 −1.03
0.8 0.010
4 1.0 4 0.015 π2f5/2 −1.284 0.229 −0.97 −0.31 −0.480 −0.79
1.2 0.020
2 1.5 2 0.032 π1i13/2 −2.794 0.238 −2.10 −0.69 −0.221 −0.92
2 4 6 8 10 (10-3fm2) 2 4 6 8 10 (10-3fm2) π1h9/2 −3.501 0.282 −3.80 0.30 −0.280 0.02
π2f −3.725 0.115 −2.90 −0.83 −0.284 −1.11
FIG. 6: Contributions of various components of the sum (1) 7/2
π3s −8.036 0.140 −8.01 −0.03 −0.108 −0.13
tosquaresofstandarderrorsofthebindingenergyE(a),two- 1/2
neutronseparationenergyS2n(b),neutronrmsradiusRn(c), π2d3/2 −8.378 0.199 −8.36 −0.02 0.220 0.20
andprotonrmsradiusRp (d)in208Pb. Thenumbersgivenin π1h11/2 −9.153 0.207 −9.36 0.21 −0.141 0.07
the ordinates and abscissas refer to the order of parameters π2d −10.117 0.117 −9.82 −0.30 0.116 −0.18
Cdeρfi∆nρe,dVinn,RVepf,.C[7ρ]∇,Jt,haantdisC, ρρ∇c,J.ENM/A, aNsyMm, LNsyMm, C0ρ∆ρ, π1g75//22 −10.908 0.229 −12.00 1.09 0.131 1.22
1 0 0 0 1 rms n.a. 0.196 n.a. 0.52 0.328 0.70
ν3d −1.856 0.250 −1.40 −0.46 −0.104 −0.56
3/2
ν4s −2.051 0.235 −1.90 −0.15 −0.668 −0.82
1/2
InTableI,welistedthes.p.energiesandtheirstandard
ν2g −2.141 0.258 −1.44 −0.70 −0.280 −0.98
errors calculated in 208Pb. We see that the proton (neu- 7/2
ν1f −2.231 0.167 −2.51 0.28 −0.226 0.05
tron)rmsofstandarderrors,equalto0.196(0.176)MeV, 15/2
is significantly smaller than the rms deviations with re- ν3d5/2 −2.549 0.166 −2.37 −0.18 −0.384 −0.56
spect to empirical data [25] of 0.52 (0.61)MeV. The rms ν1i11/2 −2.680 0.223 −3.16 0.48 −0.271 0.21
deviations obtained in lead are still much smaller than ν2g −4.336 0.071 −3.94 −0.40 −0.183 −0.58
9/2
those obtained when all doubly-magic nuclei are con- ν3p −7.855 0.109 −7.37 −0.49 0.152 −0.33
1/2
sidered simultaneously, which typically equal to about ν3p −8.503 0.065 −8.26 −0.24 −0.119 −0.36
3/2
1.2MeV [26], quite independently of which functional ν2f −8.519 0.143 −7.94 −0.58 0.156 −0.42
5/2
andwhichdataevaluationisused. Anyhow,weseeagain
ν1i −8.603 0.162 −9.24 0.64 −0.130 0.51
thatthes.p.energies,whichinthefitoftheunedf0func- 13/2
ν1h −9.922 0.190 −11.40 1.48 0.120 1.60
tionalwerenottakenintoaccount,arerathertightlycon- 9/2
ν2f −10.407 0.103 −9.81 −0.60 0.179 −0.42
strained by the fit to other observables. It is, therefore, 7/2
unlikelythat,withintheparameterspaceofthestandard rms n.a. 0.176 n.a. 0.61 0.273 0.68
Skyrmefunctional,theiragreementwithdatacanbeim-
proved, which supports conclusions reached in Ref. [26].
Ofcourse,inprinciple, the obtainedinsufficientagree-
ment with data can also signal some important miss- are very much required.
ing physics. For the s.p. energies, an obvious and of- Summary and Conclusions. In the framework of the
ten invoked effect is the well-known particle-vibration- unedf0 Skyrme-EDF model, we studied propagation of
coupling (PVC) [27], which continues to be the subject uncertainties from model parameters to calculated ob-
of recent works [28–30]. Before studying this problem servables. We found that the magnitude of standard
for the Skyrme functionals in full detail [31], in Table I errors increases towards neutron rich nuclei. This is di-
wepresentedthePVCcorrectionstos.p.energies,δe , rectlylinkedtotheisovectorpartofthefunctional,where
PVC
calculated with the standard methodology reviewed re- coupling constants have large uncertainties.
cently in, e.g., Ref. [29]. We see that in the particular For binding energies, the standard errors are in the
case studied here, the inclusion of the PVCs does not range of one to several MeV – towards the neutron drip
improve the agreement with data, but, in fact, the rms line rapidlyincreasingto about10Mev. Fortwo-particle
deviation obtained for protons (neutrons) even increases separation energies, they range from a few hundred keV
to 0.70 (0.68)MeV. An improved agreement with data to a few MeV. Compared to the deviations from experi-
should probably be soughtfor by using richer parameter mentalvalues,thestandarderrorsofbindingenergiesare,
spaces of the functional, such as, e.g., those proposed in inthe experimentallyknownregion,ofthe sameorderof
Refs. [5, 32]. Evidently, further studies of this problem magnitude or somewhat smaller. For two-particle sepa-
6
ration energy, standard errors are typically smaller, but analysesofthe obtainedminima. Withthis information,
stillcomparable. Itisworthstressingthatthetheoretical theoreticaluncertaintiesrelatedtothemodelparameters
uncertainties of binding energies and two-particle sepa- and predictions should be routinely assessed.
ration energies are still much larger than the present ex- Acknowledgments. Interesting comments by Witek
perimentalerrorsinPenning-trapmeasurements[33,34]. Nazarewiczaregratefullyacknowledged. WethankGillis
Forprotonrmsradii,ourstandarderrorsarealsocompa- Carlssonforhishelpwithupgradingthe numericalcode.
rable, but slightly smaller than the rms deviations from ThisworkwassupportedinpartbytheAcademyofFin-
experimentalvalues. Theoreticaluncertaintiesrelatedto land and University of Jyva¨skyla¨ within the FIDIPRO
the neutron skin are actually smaller than the current programme, by the Centre of Excellence Programme
experimental estimates [18]. 2012–2017(Nuclear and Accelerator Based Physics Pro-
In conclusion, determination of theoretical uncertain- grammeatJYFL),andbytheEuropeanUnion’sSeventh
ties in terms of standard statistical errors of observables Framework Programme ENSAR (THEXO) under Grant
provides a way to assess the predictive power of nuclear No. 262010. We acknowledge the CSC - IT Center for
models. In the future EDF parameter optimizations, ScienceLtd,Finland,fortheallocationofcomputational
it would be useful to perform by default the sensitivity resources.
[1] J. Toivanen, J. Dobaczewski, M. Kortelainen, and K. M.MacCormick,X.Xu,andB.Pfeiffer,ChinesePhysics
Mizuyama, Phys. Rev.C 78, 034306 (2008). C36, 1287 (2012); M. Wang, G. Audi, A.H. Wapstra,
[2] J. Dudek, B. Szpak, M.-G. Porquet, H. Molique, K. Ry- F.G. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer,
bak,and B. Fornal, J. Phys.G 37, 064031 (2010). Chinese Physics C36, 1603 (2012).
[3] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 81, [20] M. Bender, G. Bertsch, and P.-H.Heenen,Phys.Rev.C
051303 (2010). 73, 034322 (2006).
[4] J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. [21] J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hi-
Olsen, A. M. Perhac, and M. Stoitsov, Nature 486, 509 laire, S. P´eru, N.Pillet, and G.F. Bertsch, Phys. Rev.C
(2012). 81, 014303 (2010).
[5] J. Dobaczewski, K. Bennaceur, and F. Raimondi, J. [22] B.G. Carlsson, J. Toivanen, and U. von Barth,
Phys.G 39, 125103 (2012). arXiv:1212.2050.
[6] P. Klu¨pfel, P.-G. Reinhard, T.J. Burvenich, and J.A. [23] C.J. Horowitz, Z.Ahmed,C.-M.Jen, A.Rakhman,P.A.
Maruhn,Phys. Rev.C 79, 034310 (2009). Souder, M.M. Dalton, N. Liyanage, K.D. Paschke, K.
[7] M. Kortelainen, T. Lesinski, J. Mor´e, W.Nazarewicz, J. Saenboonruang, R. Silwal, G.B. Franklin, M. Friend, B.
Sarich, N. Schunck, M.V. Stoitsov, and S. Wild, Phys. Quinn, K.S. Kumar, D. McNulty, L. Mercado, S. Rior-
Rev.C 82, 024313 (2010). dan,J.Wexler,R.W.Michaels,andG.M.Urciuoli,Phys.
[8] F.J. Fattoyev and J. Piekarewicz, Phys. Rev. C 84, Rev. C 85, 032501 (2012).
064302 (2011). [24] S.Abrahamyan,Z.Ahmed,H.Albataineh, et. al.,Phys.
[9] M. Kortelainen, J. McDonnell, W. Nazarewicz, P.-G. Rev. Lett.108, 112502 (2012).
Reinhard, J. Sarich, N. Schunck, M.V. Stoitsov, and S. [25] N. Schwierz, I. Wiedenhover, and A. Volya,
Wild, Phys.Rev. C 85, 024304 (2012). arXiv:0709.3525.
[10] J. Piekarewicz, B.K. Agrawal, G. Col`o, W. Nazarewicz, [26] M. Kortelainen, J. Dobaczewski, K. Mizuyama, and J.
N. Paar, P.-G. Reinhard, X. Roca-Maza, and D. Vrete- Toivanen, Phys. Rev.C 77, 064307 (2008).
nar, Phys.Rev.C 85, 041302(R) (2012). [27] A. Bohr and B.R. Mottelson, Nuclear Structure (Ben-
[11] M.Bender,P.-H.Heenen,andP.-G.Reinhard,Rev.Mod. jamin, New York,1975), Vol.II.
Phys.75, 121 (2003). [28] E.Litvinova,P.Ring,andV.Tselyaev,Phys.Rev.C75,
[12] F.J. Fattoyev and J. Piekarewicz, Phys. Rev. C 86, 064308 (2007).
015802 (2012). [29] G. Col`o, H. Sagawa, and P.-F. Bortignon, Phys. Rev. C
[13] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 87, 82, 064307 (2010).
014324 (2013). [30] K.Mizuyama,G.Col`o,andE.Vigezzi,Phys.Rev.C86,
[14] L.F. Richardson and J.A. Gaunt, Phil. Trans. R. Soc. 034318 (2012).
Lond.A 226 299 (1927). [31] D. Tarpanov et al.,unpublished.
[15] B.G. Carlsson, J. Dobaczewski, J. Toivanen, and P. [32] B.G. Carlsson, J. Dobaczewski, and M. Kortelainen,
Vesely´,Comput. Phys. Commun. 181, 1641 (2010). Phys. Rev.C 78, 044326 (2008); 81, 029904(E) (2010).
[16] B.G. Carlsson, J. Toivanen, P. Vesely´, and Y. Gao, un- [33] J. Hakala, J. Dobaczewski, D. Gorelov, T. Eronen, A.
published. Jokinen, A. Kankainen, V.S. Kolhinen, M. Kortelainen,
[17] M.V.Stoitsov,J.Dobaczewski,W.Nazarewicz,S.Pittel, I.D. Moore, H. Penttila¨, S. Rinta-Antila, J. Rissanen,
and D.J. Dean, Phys. Rev.C 68, 054312 (2003). A. Saastamoinen, V. Sonnenschein, and J. A¨yst¨o, Phys.
[18] M. Kortelainen, J. Erler, N. Birge, Y. Gao, W. Rev. Lett.109, 032501 (2012).
Nazarewicz, and E. Olsen, unpublished. [34] A. Kankainen,J. A¨yst¨o, and A. Jokinen, J. Phys. G39,
[19] G. Audi, F.G. Kondev, M. Wang, B. Pfeiffer, X.Sun, J. 093101 (2012).
Blachot,andM.MacCormick,ChinesePhysicsC36,1157
(2012);G.Audi,M.Wang,A.H.Wapstra,F.G.Kondev,