Testing the predictive power of nuclear mass models

Nuclear Physics A 812 (2008) 28–43

www.elsevier.com/locate/nuclphysa

Testing the predictive power of nuclear mass models

J. Mendoza-Temis a, I. Morales a, J. Barea b, A. Frank a, J.G. Hirsch a,∗,J.C. López Vieyra a, P. Van Isacker c, V. Velázquez d

a Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,Apdo. Postal 70-543, México 04510 D.F., Mexico

b Center for Theoretical Physics, Sloane Physics Laboratory, Yale University,PO Box 208120, New Haven, CT 06520-8120, USA

c Grand Accélérateur National d’Ions Lourds, CEA/DSM–CNRS/IN2P3, BP 55027, F-14076 Caen cedex 5, Franced Facultad de Ciencias, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, México 04510 D.F., Mexico

Received 23 May 2008; received in revised form 12 August 2008; accepted 12 August 2008

Available online 23 August 2008

Abstract

A number of tests are introduced which probe the ability of nuclear mass models to extrapolate. Threemodels are analyzed in detail: the liquid drop model, the liquid drop model plus empirical shell correctionsand the Duflo–Zuker mass formula. If predicted nuclei are close to the fitted ones, average errors in predictedand fitted masses are similar. However, the challenge of predicting nuclear masses in a region stabilizedby shell effects (e.g., the lead region) is far more difficult. The Duflo–Zuker mass formula emerges as apowerful predictive tool.© 2008 Elsevier B.V. All rights reserved.

PACS: 21.10.Dr; 21.60.Cs; 21.60.Fw

Keywords: Nuclear mass models; Binding energies; Extrapolation; Tests

1. Introduction

A large number of processes in nuclear physics depend crucially on an accurate knowledge ofnuclear masses, in particular processes occurring in astrophysical phenomena [1]. Though much

* Corresponding author.E-mail address: [email protected] (J.G. Hirsch).

0375-9474/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2008.08.008

J. Mendoza-Temis et al. / Nuclear Physics A 812 (2008) 28–43 29

progress has been made in measuring the masses of exotic nuclei, theoretical models are stillnecessary to predict them in regions far from stability [2]. Advances in the calculation of atomicmasses have been hampered by the absence of an exact theory of the nuclear interaction andby the difficulties inherent to quantum many-body calculations. There has been much work indeveloping mass formulas with both microscopic and macroscopic input, on one side, and on thederivation of masses in a fully microscopic framework, on the other.

The simplest approach is that of the liquid-drop model (LDM) [3]. It incorporates the essentialmacroscopic terms, which means that the nucleus is pictured as a very dense, charged liquiddrop. Inclusion of the discrete character of the nucleons and their basic interactions requiresmore sophisticated treatments. The finite-range droplet model (FRDM) [4], which combines themacroscopic effects with microscopic shell and pairing corrections, has become the de factostandard for mass formulas. Also, a microscopically inspired model was introduced by Duflo andZuker (DZ) [5–7] with positive results. Finally, among the mean-field methods it is also worthmentioning the Skyrme–Hartree–Fock approach [8,9]. All these mass formulas can calculateand predict the masses (and often other properties) of as many as 8979 nuclides [2] but it is ingeneral difficult to match theory and experiment (for all known nuclei) to an average precisionbetter than about 0.5 MeV [2]. This minute quantity, corresponding to less than a part in 105

of the mass of a typical nucleus, still represents a significant fraction of the energy released innuclear decays, strongly affecting the extrapolations of proton and neutron separation energiesrequired in astrophysical processes [1,2].

In the present article a number of tests are introduced which probe the ability of nuclear massmodels to extrapolate. In all cases the full set consists of the measured nuclear masses given inAME03, the 2003 Atomic Mass Evaluation [10]. This set is then partitioned in two, one used tofit the model parameters and the remainder to compare measured masses with those predictedby extrapolation. Three models are analyzed in detail: the liquid drop model, the liquid dropplus empirical shell corrections and the Duflo–Zuker mass formula. Seven different probes (i.e.,partitions of the AME03 set) are introduced which test both short- and long-range predictions ofthe three mass models.

2. The tests

Throughout this paper nuclear masses M(N,Z) will be used, being N the total number ofneutrons and Z the atomic number. Given that in Ref. [10] atomic masses Matomic(N,Z) arelisted, we relates them through

Matomic(N,Z) = M(N,Z) + Zme − Bel(Z), (1)

where me is the electron mass, and Bel(Z) is the total electronic binding energy which can beparametrized as

Bel(Z) = 1.44381 × 10−5Z2.39 + 1.55468 × 10−12Z5.35 MeV, (2)

with an estimated error of 150 eV [2]. The nuclear binding energies BE(N,Z) can be obtainedfrom the atomic binding energies BEatomic(N,Z) through the relation

BE(N,Z) = BEatomic(N,Z) + ZBel(Z = 1) − Bel(Z). (3)

The figure of merit to evaluate the quality of a given set of fitted or predicted binding energiesis its root-mean-square (RMS) deviation, defined as

30 J. Mendoza-Temis et al. / Nuclear Physics A 812 (2008) 28–43

Table 1Number of fitted and predicted nuclei in each probe

Test N � 8, Z � 8 N � 28, Z � 28

Fit Prediction Fit Prediction

AME95-03 1760 389 1454 371Border 1790 359 1524 301A � 160 1339 810 1015 810A � 170 1449 700 1125 700A � 180 1564 585 1240 585A � 190 1672 477 1348 477A � 200 1773 376 1449 376

RMS ={∑[BEexp(N,Z) − BEth(N,Z)]2

Nnucl

}1/2

, (4)

where BEexp(N,Z) and BEth(N,Z) are the experimental and theoretical binding energies, re-spectively, and Nnucl is the number of nuclei included in the set.

To gauge the quality of the predictions obtained from different nuclear mass models, we haveconstructed a set of probes, all based on nuclear masses taken from Ref. [10]. The tests areperformed for the 2149 nuclei with N � 8,Z � 8 or the 1825 nuclei with N � 28, Z � 28. Theyare:

• AME95-03: The subset of nuclei with measured masses in the AME95 compilation [11] isfitted. This test was used in Ref. [2] to compare predictions of different models. In this workthe actual masses used in the fit are taken from AME03, only the set of nuclei to be fitted isbased on AME95.

• Border region: Nuclei which are furthest removed from stability are excluded from the fitand subsequently predicted by extrapolation.

• Lead region: Nuclei with mass number A(= N + Z) � 160, 170, 180, 190 or 200 are fittedand the remaining ones, which always include the region around 208Pb, are predicted byextrapolation.

The number of nuclei included in the fitted and predicted sets, for each probe and for N � 8,Z � 8 and N � 28, Z � 28, is listed in Table 1. The ratio of predicted to fitted nuclei varies from1/5 for the border test and N � 8, Z � 8, to 4/5 for the A � 160 probe with N � 28, Z � 28.The effects on the predictions of these different ratios are discussed below.

The first two tests probe the short-range predictions, since the predicted nuclei are relativelyclose to the fitted ones. In the AME95-03 test the predicted nuclei are concentrated in the regionclose to the proton drip line and confined to medium- to heavy-mass nuclei, as can be seen inupper line of Fig. 1. In the border test there is a uniform distribution of predicted nuclei lying allaround the fitted ones, as shown in second inset of Fig. 1. The five probes A � 160, 170, 180,190 and 200 test the long-range predictive power of the mass models, including their ability topredict a region which is stabilized through shell effects. The fit and prediction regions on theprobes A � 160 and 200 are shown in the lower line of Fig. 1. This is clearly relevant in thecontext of the long-standing search for the super-heavy island(s) of stability.


Fig. 1. Fit (red) and prediction (yellow) regions, in the 95-03, border, A � 160 and A � 200 probes. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this paper.)

3. The models

Three models were tested with the probes described above. They are:

• LDM: The Bethe–Weizsäcker liquid drop mass (LDM) formula, modified to include volumeand surface contributions in the symmetry energy, and a Wigner correction [12,13]:

BE(N,Z) = avA − asA2/3 − ac

Z(Z − 1)

A1/3+ ap

δ(N,Z)√A

− Sv

1 + ysA−1/3

4T (T + r)

A, (5)

with T = |N − Z|/2 and δ(N,Z) = 1 for N , Z even, −1 for N , Z odd and 0 for A odd.• LDMM: The LDM modified with empirical shell corrections, linear and quadratic in the

number of valence protons nπ and neutrons nν , defined as the number of particles or holesrelative to closest shell closure,

BE(N,Z) = avA − asA2/3 − ac

Z(Z − 1)

A1/3+ ap

δ(N,Z)√A

− Sv−1/3

4T (T + r) − afF + affF2 (6)

1 + ysA A


with

F = nν + nπ

2−

⟨nν + nπ

2

⟩

and

FF =(

nν + nπ

2

)2

−⟨(

nν + nπ

2

)2⟩.

For details see [13,14]:• DZ-33: The Duflo–Zuker (DZ) model [7] proposes a functional of the occupation numbers

and includes explicitly deformation effects. It is inspired by the multi-particle shell model byincorporating its monopole contributions [15,16], and considers pairing and Wigner terms aswell as the isospin dependence of the nuclear radius [5,6]. The version employed here has33 parameters.

4. Root-mean-square deviations

For the three models independent fits were performed for the full set of AME03 data, and foreach of the seven probes, in all cases for N � 8, Z � 8 as well as N � 28, Z � 28, and theirpredictions were confronted with the experimental binding energies.

The task of finding the parameters which minimize the RMS deviation between measuredand predicted values for each model and each reference set is performed employing the CERNminimization code MINUIT [17]. “Minuit is conceived as a tool to find the minimum value ofa multi-parameter function and analyze the shape of the function around the minimum. It isespecially suited to compute the best-fit parameter values and uncertainties, including correla-tions between the parameters, to handle difficult problems, including those which may requireguidance in order to find the correct solution” [17].

The RMS deviations of the fits and the predictions are listed below, and also plotted using acolor code to display the differences between measured and calculated individual binding ener-gies.

In Tables 2, 3 and 4 the first line, denoted “Full Set”, refers to a fit of the entire data set, whilethe columns labeled “fit” and “prediction” contain the RMS of the fitting and predicted subsets.Note that, for each model, the RMS deviation of this fit is very similar to the one obtained ineach of the seven different probes. The LDM can describe all nuclear masses with an averageerror of 2.4 MeV, the LDMM gives 1.2–1.3 MeV and the DZ model leads to an RMS deviationof 0.30–0.36 MeV. When only nuclei with N � 28, Z � 28 are included in the fit, the LDMMand DZ models exhibit a 10–15% reduction in the RMS error,

In the AME95-03 probe all models perform pretty well, with an RMS error in the predictionsof the same size as the one in the fit, or even smaller in the case of LDM. The border probe is

Table 2RMS deviation, in MeV, of the fits and predictions for LDM

Test N � 8, Z � 8 N � 28, Z � 28


Full set 2.401 2.421AME95-03 2.498 2.070 2.587 1.931Border 2.313 2.882 2.389 2.776


Table 3RMS deviation, in MeV, of the fits and predictions for LDMM

Test N � 8, Z � 8 N � 28, Z � 28


Full set 1.332 1.209AME95-03 1.368 1.318 1.318 0.996Border 1.269 2.157 1.236 1.980A � 160 1.174 3.271 1.131 3.394A � 170 1.183 1.719 1.120 2.607A � 180 1.198 1.642 1.141 2.451A � 190 1.209 1.710 1.155 2.536A � 200 1.170 2.992 1.123 3.618

Table 4RMS deviation, in MeV, of the fits and predictions for the DZ model

Test N � 8, Z � 8 N � 28, Z � 28


Full set 0.352 0.308AME95-03 0.345 0.414 0.296 0.337Border 0.323 0.504 0.286 0.404A � 160 0.372 0.756 0.297 0.823A � 170 0.366 0.770 0.295 0.951A � 180 0.366 0.489 0.292 0.717A � 190 0.361 0.912 0.298 1.173A � 200 0.359 1.357 0.302 1.429

Table 5RMS deviation, in MeV, of the fits and the predictions of various models in the AME95-03 probe

Model Fit Prediction

HFBCS-1 [8] 0.718 1.115HFB-1 [18] 0.740 1.123HFB-2′ [2,19] 0.651 0.857FRDM [4] 0.678 0.655TF-FRDM [20] 0.662 0.655Duflo–Zuker [7] 0.346 0.479

harder, and for all models the RMS error in the predictions is between 30% and 70% larger thanthe one in the fit. While both tests predict masses of nuclei which are very close to the fitted ones,the AME95-03 probe has a majority of predicted nuclei concentrated in a reduced and concavearea (see Fig. 1). This case is in fact more an interpolation than an extrapolation, and this factperhaps explains the relative ease with which the nuclear models deal with the AME95-03 probe.In both short-range probes, AME95-03 and border, the RMS error in the predictions is, again,consistently smaller if only nuclei with N � 28, Z � 28 are included in the fit.

For comparison, the AME95-03 RMS errors reported in Table I, Appendix D of Ref. [2] arepresented in Table 5. While in this case the masses in the fit were taken from the AME95 data


set [11], making them not directly comparable with the AME95-03 probe reported here, they arein any case instructive. Note that in all HFB calculations and in the DZ model the RMS error inthe prediction is larger than in the fit, while for the two versions of the FRDM the RMS erroris of the same size. In the DZ model the RMS error in the fit is nearly the same as in Table 4for the AME95-03 probe and N � 8, Z � 8, but the average error in the prediction is larger inthe Lunney compilation, Table 5. The fact that the DZ predictions have smaller errors when themasses reported in the AME03 data set are employed, than when the AME95 are used, seems toreflect that the masses modified from AME95 to AME03 have a better consistency with the newones.

The present analysis illustrates the predictive power of the DZ model. It also allows to set aconservative lower bound on the average precision of its long range predictions, which is at least1 MeV. This is in sharp contrast with the short-range mass predictions, which have an averageerror of 0.4 MeV.

Predicting the region around 208Pb is, by far, the hardest challenge for any model. Results forLDM are not presented because it has no microscopic information and hence fails to predict theextra binding energy associated with shell closures. The LDMM and DZ mass predictions haveaverage errors which are 1.5 to 4 times larger than the errors in the fits. There is no identifiabletrend in the size of the RMS error in the fit as the cut moves from A = 160 to A = 200. In fact,for both models the largest error in the predictions is found for the cut A = 200, when only 376nuclei are predicted, a figure very close to the number of nuclei predicted in the AME95-03 andthe border probes. While it would be expected that increasing the ratio between the numbers offitted and predicted nuclei would diminish the average error in the prediction, this is not the casefor the cuts from A = 160 to A = 200.

The probes involving the prediction of nuclei around the lead region have RMS errors witha paradoxical behavior: if only nuclei with N � 28,Z � 28 are included in the fit, its RMSdiminishes as compared with the fit to the full set N � 8, Z � 8. However, the RMS error in thepredictions exhibits the opposite behavior. It seems that, in the long-range probes, the eliminationof the lighter nuclei makes it easier to fit their masses but harder predict the masses of those nucleithat are far from the fitted region.

5. Graphical analysis

Fig. 2 exhibits the differences between experimental and calculated binding energies for LDM,LDMM and the DZ model. For LDM the extra binding energy around shell closures is clearlyseen in red, with its characteristic rhomboidal form. Note the energy scale, ranging from −8 to8 MeV. Shell closures are schematically included in LDMM and as a result they are less visible.There are, however, remaining correlations which can be recognized in the figure, like somemissing binding energy around N = 50 and N = 82, for 90 < N < 126 and some extra bindingenergy for 84 < N < 90 and 126 < N < 136. The energy scale for LDMM ranges from −2 to2 MeV. In the DZ model the differences are pretty small, most of them in blue or light blue(−0.5 to 0.5 MeV). The differences exhibit very little structure and are consistent with whitenoise [21].

Figs. 3 and 4 exhibit the differences between experimental and calculated binding energiesfor LDM, LDMM and the DZ model, in the fitted (left-hand side) and predicted (right-hand side)regions in the AME95-03 and border probes, respectively. They help to visualize the fitted andpredicted regions of nuclei and the color code emphasizes the fact that for both tests errors are ofthe same size in both regions.


Fig. 2. Differences between experimental and fitted binding energies, in MeV in the color code, for the LDM, LDMMand DZ models. (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this paper.)

Fig. 5 displays the differences between experimental binding energies and fitted or predictedones, depending on which side of the cut (indicated by the line) they are, for LDMM for theA � 160, 170, 180, 190 and 200 probes. Comparing with the differences between experimentaland fitted binding energies in Fig. 2, in the LDMM case it can be seen that in most cases theerrors in the predictions, Fig. 5 are similar to those appearing in the fit. In the predicted regionmost errors are large, represented by red and green areas.

Fig. 6 exhibits similar differences for the DZ model. The color code in all cases reflects thenoticeable increase in the errors in the predicted region, as compared with the fitted one and withthe differences shown in Fig. 2. The green and red areas are those with bigger differences and


Fig. 3. Differences between experimental binding energies and their fit (left) and prediction (right), for the LDM, for theLDMM and for DZ, in the 95-03 probe.

they are larger if only nuclei with N � 28, Z � 28 are included in the fit, which is consistentwith the RMS errors reported in Table 4.

6. Global analysis

For each mass model, sixteen sets of parameters have been found, eight for fits includingnuclei with N � 8, Z � 8 and eight for fits including nuclei with N � 28, Z � 28. In each caseone set is associated with the fit of the full AME03 data set and the other seven sets arise forthe seven probes described above. For each model, a detailed comparison of the values obtainedfor each parameter provides valuable information to set limits on the stability of this parameter,on its range of variation and on the estimated changes if data obtained from new measurements


Fig. 4. Differences between experimental binding energies and their fit (left) and prediction (right), for the LDM, for theLDMM and for DZ, in the border probe.

are included. Such analysis proceeds along the lines proposed recently by Kirson concerning theparameters of the LDM [22].

The following tables display a statistical analysis of the variation of the parameters of the threemodels. The first line, “full set”, displays the values of the parameters which provide the best fitemploying the full AME03 data set, starting with N,Z � 8. The second line, “mean”, show theaverage value of each parameter for the 16 different fits, the third, “sigma”, their dispersion, andthe fourth the ratio between the dispersion and the mean, given as a percentage.

In Table 6 it can be seen that the first three parameters, associated with the volume, surface andCoulomb terms in the liquid drop model, vary less than a fraction of a percent along the sixteenfits. the following three, associated with the volume and surface contribution to the symmetryterm and with pairing have a dispersion slightly smaller than 5%, and the Wigner term changes


in average 10% along all the fits. As mentioned above, these findings are consistent with theextensive study reported in Ref. [22].

It is worth noting that by adding to the LDM two microscopic terms, the LDMM can fit allmasses with half the RMS error, but the dispersion in the parameters has clearly increased. Whilethe two new parameters only change by 3–7%, fluctuations in the others can be as large as 22%,as can be seen in Table 7.

Table 8 exhibits the dispersion in the parameters of the DZ model. From the 33 parameters,only three have fluctuations smaller than 1%, 12 between 1% and 10%, and 14 between 10%and 50%. The other four parameters have larger fluctuations: ss by 51%, qq+ by 120%, d3 by492% and D3 by 2064%, although in this last case the average value is very close to zero andthe relative fluctuation could provide an exaggerated representation. These large fluctuations in

Fig. 5. Differences between experimental and fitted or predicted binding energies for N � 8, Z � 8 (left) and N � 28,Z � 28 (right), for the LDMM, for the A � 160, 170, 180, 190, 200 probes.


Fig. 5. (continued)

the parameters could be behind the errors of around 1 MeV for the predicted masses in the leadprobe in the DZ model. More research in this direction should be necessary to clarify this point.

It is also possible to perform some global tests on a full set of binding energies predicted in anymodel between the proton and neutron drip lines. The Garvey–Kelson (GK) mass relations [23]provide empirical constraints on the mass formulas. The GK relations are very accurately fulfilledthroughout most of the periodic table, with errors typically of the order of 100 to 200 keV [21,24], much less than those of mass formulas obtained either from macroscopic/microscopic orpurely microscopic considerations. While the masses predicted employing formulas of the typeconsidered here satisfy the GK relations with even smaller errors, of the order of 1 keV for LDMand 40 keV for the DZ model, most other mass formulas, like the FRDM [4] or Skyrme–Hartee–Fock [8,9] do not typically satisfy them, in particular for nuclei not included in their fits [24].

The fact that the current mass formulae derived either from macroscopic/microscopic orpurely microscopic considerations do not typically satisfy the GK relations when extrapolatedto unmeasured masses, suggests that it might be of interest to modify these formulations toexplicitly include the GK constraints. In the context of the present article, it can be seen as a con-sistency check: all mass predictions between the drip lines should fulfill the GK relations withan average error of the order of 100 keV.

7. Conclusions

A number of tests have been presented which probe the ability of nuclear mass models toextrapolate. These probes can be applied to any mass model able to perform global predictions,


provided the task of obtaining a parameter set from a fit to a large set of nuclei is feasible ina reasonable amount of time. We have analyzed in detail three models with this property: theliquid drop model (LDM), the liquid drop model plus empirical shell corrections (LDMM) andthe Duflo–Zuker (DZ) mass formula.

Two of the probes, the AME95-03 and border tests, predict masses of nuclei close to thosebeing fitted. The AME95-03 probe is the easiest one because it involves interpolations rather thanextrapolations. In these two short-range probes the average error in the predictions is similar tothe error in the fits: 2.4 MeV for LDM, 1.2 MeV for LDMM and 0.3 MeV for the DZ model.The other five probes involve the prediction of masses of nuclei in the lead region. The fittedand predicted sets are separated by lines along A = 160, 170, 180, 190 and 200. Extrapolationof masses to the lead region represents a very hard challenge for any model, with average errors

Fig. 6. Differences between experimental and fitted or predicted binding energies for N � 8, Z � 8 (left) and N � 28,Z � 28 (right), for the DZ model, for the A � 160, 170, 180, 190, 200 probes.


Fig. 6. (continued)

Table 6Reference and average values of the parameters, their dispersion, and the ratio between the dispersion and the mean, forthe 7 parameters in the LDM

LDM av as ac Sv ys ap r

Full set 15.454 17.053 0.6891 44.507 6.4437 12.444 2.2437Mean 15.439 16.985 0.688 48.430 7.674 12.046 2.554Sigma 0.0108 0.0501 0.0018 2.2444 0.3240 0.5778 0.2628

100 × sigmamean 0.07 0.29 0.26 4.63 4.22 4.80 10.3

in the predicted region up to four times larger than those in the fitted one. It is found that theDZ model is least unstable and is the best suited for predicting nuclear masses beyond the leadregion. While excluding the lightest nuclei improves in all cases the quality of the fit, for thelong-range predictions around the lead region, the RMS deviation in the predicted region in factincreases when the restricted set of nuclei is used in the fit. Fluctuations in the model parametersalong the different fits were reported. They can be large in some cases, limiting the predictivepower of the different mass models.

Acknowledgements

This work was supported in part by Conacyt, México, and DGAPA, UNAM.


Table 7The same as in Table 6, for the 9 parameters in the LDMM

LDMM av as ac Sv ys

Full set 15.454 17.053 0.6891 44.507 6.4437Mean 15.369 16.739 0.684 42.684 6.0880Sigma 0.0645 0.2447 0.0042 4.8981 1.3301

100 × sigmamean 0.42 1.46 0.61 11.5 21.8

COEFF ap r af aff

Full set 12.444 2.2437 1.3349 0.0469Mean 12.653 2.346 1.402 0.051Sigma 0.6848 0.2652 0.0432 0.0033

100 × sigmamean 5.41 11.3 3.08 6.47

Table 8The same as in Table 6, for the 33 parameters in the DZ model

DZ FM+ f m+ FS+ f s+ FS− f s− FC+Full set 8.9771 6.0829 4.9275 21.92 1.7951 8.0271 −5.6213Mean 8.9573 5.9962 5.0259 21.9637 1.9509 8.7341 −5.9258Sigma 0.0879 0.3721 0.1703 0.6178 0.1537 0.7452 1.6659

100 × sigma|mean| 0.98 6.21 3.39 2.81 7.88 8.53 28.1

COEFF f c+ PM+ pm+ PS+ ps+ PS− ps−Full set −38.042 −0.2780 1.1051 −0.8023 −4.0274 −0.1249 −0.6355Mean −39.510 −0.2631 1.1787 −0.7618 −3.7906 −0.1266 −0.6288Sigma 6.8486 0.0724 0.3057 0.0609 0.3357 0.0292 0.1392

100 × sigma|mean| 17.3 27.5 25.9 7.99 8.86 23.1 22.1

COEFF S3 s3 SQ− sq− D3 d3 QQ+Full set 0.4346 1.9750 0.3415 1.3921 −2.3015 −4.8308 7.6667Mean 0.4078 1.9722 0.3765 1.5498 −0.2566 5.5880 7.7214Sigma 0.1027 0.0882 0.0452 0.2096 5.2965 27.5231 2.2880

100 × sigma|mean| 25.2 4.47 12.0 13.5 2064 492 29.6

COEFF qq+ D0 d0 QQ− qq− T T tt

Full set 12.422 −33.471 −158.61 −5.0282 −35.613 −37.329 −52.041Mean 9.6921 −34.233 −160.58 −5.5738 −37.253 −37.234 −51.790Sigma 11.652 1.6092 6.8870 1.7068 9.5367 0.1288 0.6089

100 × sigma|mean| 120 4.70 4.29 30.6 25.6 0.35 1.17

COEFF SS ss COU P 0 P 1

Full set 1.2432 4.1378 0.6995 6.1660 17.909Mean 0.9951 3.3039 0.6976 6.2208 14.479Sigma 0.3899 1.6921 0.0012 0.0850 4.2439

100 × sigma|mean| 39.2 51.2 0.17 1.37 29.3

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Testing the predictive power of nuclear mass models