Strong-force isospin-symmetry breaking in masses of $N \sim Z$ nuclei

0.3 MB·

by  Pawel Baczyk

Strong-force isospin-symmetry breaking in masses of N ∼ Z nuclei
P. Ba¸czyk,1 J. Dobaczewski,1,2,3,4 M. Konieczka,1 W. Satul a,1,4 T. Nakatsukasa,5 and K. Sato6
1Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland
2Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom
3Department of Physics, P.O. Box 35 (YFL), University of Jyva¨skyla¨, FI-40014 Jyva¨skyla¨, Finland
4Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
5Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
6Department of Physics, Osaka City University, Osaka 558-8585, Japan
(Dated: February 2, 2017)
Effectsoftheisospin-symmetrybreakingduetothestronginteractionaresystematically studied
for nuclear masses near the N =Z line, using extendedSkyrmeenergy density functionals (EDFs)
7
withproton-neutron-mixeddensitiesandnewtermsbreakingtheisospinsymmetry. Twoparameters
1
associated with the new terms are determined by fitting mirror and triplet displacement energies
0
2 (MDEs and TDEs) of isospin multiplets. The new EDFs reproduce the MDEs for T = 21 doublets
andT =1triplets,aswellasthestaggeringofTDEforT =1triplets. Therelativestrengthsofthe
b obtainedisospin-symmetry-breakingtermsareconsistent withthedifferencesin theNN scattering
Fe lengths, ann, app, and anp.

1
Similarity between the neutron-neutron (nn), proton- Refs. [3, 4].

] proton (pp), and proton-neutron (pn) nuclear forces, Theisospinformalismoffersaconvenientclassification
h
t commonlyknownastheirchargeindependence,hasbeen of different components of the nuclear force by dividing
-
wellestablishedexperimentallyalreadyin1930’s,leading themintofourdistinctclasses. ClassIisoscalarforcesare
l
c
totheconceptofisospinsymmetryintroducedbyHeisen- invariantunderanyrotationintheisospinspace. ClassII
u
berg[1]andWigner[2]. Sincethen,theisospinsymmetry isotensorforcesbreakthechargeindependencebutarein-
n
[ has beentested andwidely used intheoreticalmodelling variantunder a rotationby π with respectto the y−axis

of atomic nuclei, with explicit violation by the Coulomb intheisospacepreservingthereforethechargesymmetry.
2
interaction. In addition, the nuclear force also weakly Class III isovectorforces break both the chargeindepen-
v
8 violates the isospin symmetry. There exists firm experi- denceandthechargesymmetry,andaresymmetricunder
2 mental evidence in the nucleon-nucleon(NN) scattering interchangeoftwointeractingparticles. Finally,forcesof
6 data that it also contains small charge-dependent (CD) class IV break both symmetries and are anti-symmetric
4
components. The differences in the NN phase shifts in- undertheinterchangeoftwoparticles. Thisclassification
0
dicatethatthenn interaction,V ,isabout1%stronger was originally proposed by Henley and Miller [4, 5] and
. nn
1 than the pp interaction, V , and that the np interac- subsequently used in the framework of potential models
pp
0
tion,V ,isabout2.5%strongerthantheaverageofV basedonboson-exchangeformalism,likeCD-Bonn[3]or
7 np nn
and V [3]. These are called charge-symmetry breaking AV18[6]. TheCSBandCIBwerealsostudiedintermsof
1 pp
: (CSB) and charge-independence breaking (CIB), respec- the chiraleffective field theory [7, 8]. So far, the Henley-
v
tively. In this paper, we show that the manifestation of Millerclassificationhasbeenratherrarelyutilizedwithin
i
X the CSB and CIB in nuclear masses can systematically the nuclear DFT [9, 10], which is usually based on the
r be accounted for in extended nuclear density functional charge-independent strong forces.
a
theory (DFT). The mostprominentmanifestationofthe isospinsym-
metrybreaking(ISB)isinthe mirrordisplacementener-
Thechargedependenceofthenuclearforcefundamen-
gies (MDEs) defined as the differences between binding
tallyoriginatesfrommassandchargedifferencesbetween
energies of mirror nuclei:
u and d quarks. The strong and electromagnetic inter-
actions among these quarks give rise to the mass split- MDE=BE(T,T =−T)−BE(T,T =+T). (1)
z z
ting among the baryonic and mesonic multiplets. The
A systematic study by Nolen and Schiffer [11] showed
neutron is slightly heavier than the proton. The pions,
that the MDEs cannot be reproduced by using models
which are the Goldstone bosons associated with the chi-
involving Coulomb interaction as the only source of the
ral symmetry breaking and are the primary carriers of
ISB, see also Refs. [9, 12, 13]. Another source of infor-
the nuclear force at low energy, also have the mass split-
mation on the ISB is the so-called triplet displacement
ting. The CSB mostly originates from the difference in
energy (TDE):
massesofprotonsandneutrons,leadingtothedifference
in the kinetic energies and influencing the one- and two-
TDE=BE(T =1,T =−1)+BE(T =1,T =+1)
boson exchange. On the other hand, the major cause of z z
−2BE(T =1,T =0), (2)
the CIB is the pion mass splitting. For more details, see z
2
sider the following zero-rangeinteractions of classII and
12 III with two new low-energy coupling constants tII and
0
(a) tIII [26]:
0
10 T = 1 T = 1
2
eV) and VˆII(i,j)=tI0Iδ(ri−rj)h3τˆ3(i)τˆ3(j)−~τˆ(i)◦~τˆ(j)i,(3)
M 8
( VˆIII(i,j)=tIIIδ(r −r )[τˆ (i)+τˆ (j)]. (4)
T 0 i j 3 3
2
E/ 6 The corresponding contributions to EDF read:
D T
M SV∗
1
4 SkM H = tII(ρ2 +ρ2−2ρ ρ −2ρ ρ
SLy4 II 2 0 n p n p np pn
experiMmDenEt −s2 −s2+2s ·s +2s ·s ), (5)
2 n p n p np pn
1
T = 1 H = tIII ρ2 −ρ2−s2 +s2 , (6)
V) 0.6 III 2 0 (cid:0) n p n p(cid:1)
Me where ρ and s are scalar and spin (vector) densities,
E( 0.4 SVT∗ respectively. Inclusion of the spin exchange terms in
D
T 0.2 SkM Eqs. (3) and (4) leads to trivial rescaling of the cou-
0.0 (b) experiSmLeyn4t pling constants tI0I and tI0II, see [26]. Hence, it can be
disregarded.
10 20 30 40A 50 60 70 Contributions of class III force to EDF (6) depend on
the standard nn and pp densities and, therefore, can be
taken into account within the conventional pn-separable
FIG.1. (Color online) Calculated (noISBterms) and exper-
imental values of MDEs (a) and TDEs (b). The values of DFT approach [9]. In contrast, contributions of class II
MDEs for triplets are divided by two to fit in the plot. Thin force (5) depend explicitly on the mixed densities, ρ
np
dashed line shows the average linear trend of experimental and s , and require the use of pn-mixed DFT [27, 28],
np
MDEs in doublets, defined as MDE = 0.137A + 1.63 (in augmentedbythe isospincrankingto controlthe magni-
MeV). Measured values of binding energies were taken from
tude and direction of the isospin (T,T ).
Ref. [24] and the excitation energies of the T = 1, Tz = 0 WeimplementedthenewtermsofthzeEDFinthecode
states from Ref. [25]. Open squaresdenote datathat depend
HFODD [19, 20], where the isospin degree of freedom is
on masses derived from systematics [24].
controlled within the isocranking method [27, 29, 30] –
ananalogueofthecrankingtechniquethatiswidelyused
in high-spin physics. The isocranking method allows us
which is a measure of the binding-energy curvature
to calculate the entire isospin multiplet, T, by starting
within the isospin triplet. The TDE also cannot be re-
from an isospin-aligned state |T,T = Ti and isocrank-
producedbymeansofconventionalapproachesdisregard- z
ing it around the y-axis in the isospace. The method
ing nuclear CIB forces, see [14]. In these definitions the
can be regarded as an approximate isospin projection.
binding energies are negative (BE < 0) and the proton
Arigoroustreatmentofthe isospinsymmetrywithinthe
(neutron) has isospin projection of t =−1(+1).
z 2 2 pn-mixedDFTformalismrequiresfull,three-dimensional
In Fig. 1 we show MDEs and TDEs calculated fully isospinprojection,whichiscurrentlyunderdevelopment.
self-consistently using three different standard Skyrme Physicallyrelevantvalues of tII andtIII turnoutto be
∗ 0 0
EDFs; SVT [15, 16], SkM [17], and SLy4 [18]. Details fairly small [26], and thus the new terms do not impair
of the calculations, performed using code HFODD [19, the overall agreement of self-consistent results with the
20], are presented in the Supplemental Material [21]. In standardexperimentaldata. Moreover,calculatedMDEs
Fig.1(a),weclearlyseethatthevaluesofobtainedMDEs and TDEs depend on tII and tIII almost linearly, and, in
0 0
are systematically lower by about 10% than the experi- addition, MDEs (TDEs) depend very weaklyontII (tIII)
0 0
mentalones. Evenmorespectaculardiscrepancyappears [26]. This allowsus to use the standardlinear regression
in Fig. 1(b) for TDEs – their values are underestimated method, see, e.g. Refs. [31, 32], to independently adjust
by about a factor of three and the characteristic stag- tII and tIII to experimental values of TDEs and MDEs,
0 0
gering pattern seen in experiment is entirely absent. It respectively. SeeSupplementalMaterial[21]fordetailed
is also very clear that the calculated MDEs and TDEs, description of the procedure. Coupling constants tII
0
which are specific differences of binding energies, are in- and tIII resulting from such an adjustment are collected
0
dependentofthechoiceofSkyrmeEDFparametrization, in Table I.
that is, of the isospin-invariantpart of the EDF. In Fig. 2, we show values of MDEs calculated within
We aim at comprehensive study of MDEs and TDEs our extended DFT formalism for the Skyrme SV EDF.
T
based on extended Skyrme pn-mixed DFT [16, 19, 20] By subtracting an overall linear trend (as defined in
that includes zero-range class II and III forces. We con- Fig. 1) we are able to show results in extended scale,
3
TABLEI.CouplingconstantstI0IandtI0IIandtheiruncertain- 0.8 (a) T = 21
ties obtained in this work for the SkyrmeEDFs SVT, SkM*,
and SLy4. 0.4
SVT SkM* SLy4
0.0
tI0I (MeV fm3) 17±5 25±8 23±7 )
tI0II (MeV fm3) −7.4±1.9 −5.6±1.4 −5.6±1.1 MeV -0.4
(
E
D -0.8
M
where a detailed comparison with experimental data is
− T = 1
possible. In Fig. 3, we show results obtained for TDEs, 2T (b)
whereas complementary results obtained for the Skyrme /
E 0.4
SkM* and SLy4 EDFs are collected in the Supplemental D
M
Material [21]. d
0.0 e
It is gratifying to see that the calculated values of d (cid:28)tt SB
MDEscloselyfollowtheexperimentalA-dependence,see -0.4 tte ot oI
(cid:28) n n
Fig. 2. It is worth noting that a single coupling con-
T
stant tIII reproduces both T = 1 and T = 1 MDEs, -0.8 SV
0 2 experiment
which confirms conclusions of Ref. [9]. In addition, for
the T = 21 MDEs, the SVT results nicely reproduce (i) 10 20 30 40A 50 60 70
changes in experimental trend that occur at A=15 and
39, (ii) staggering pattern between A = 15 and 39, and
FIG.2. (Coloronline)Calculated andexperimentalvaluesof
(iii) disappearance of staggering between A=41 and 49 MDEs for the T = 1 (a) and T =1 (b) mirror nuclei, shown
2
(the f7/2 nuclei). Wenotethatthesefeaturesarealready with respect to the average linear trend defined in Fig. 1.
present in the SVT results without the ISB terms, and CalculationswereperformedforfunctionalSVT withtheISB
that adding this terms increases amplitude of the stag- terms added. Shaded bands show theoretical uncertainties.
gering. However,fortheSkM*andSLy4functionals,the Experimentalerrorbarsareshown onlywhentheyarelarger
staggering of the T = 1 MDEs is less pronounced [21]. than thecorresponding symbols. Full (open) symbols denote
2 datapointsincludedin(excludedfrom)thefittingprocedure.
We alsonote thatallthree functionals correctlydescribe
the A-dependence and lack of staggering of the T = 1
MDEs.
Having at hand a model with ISB strong interactions
It is even more gratifying to see in Fig. 3 that our
with fitted parameters we can calculate MDEs for more
pn-mixed calculations, with the class-II coupling con-
massive multiplets and make predictions on binding en-
stant, tII, describe absolute values as well as stagger-
0 ergies of neutron-deficient (T = −T) nuclei. In par-
ing of TDEs very well, whereas results obtained without z
ticular, in Table II we present predictions of mass ex-
ISB terms give too small values and show no stagger-
cesses of 52Co, 56Cu, and 73Rb, whose masses were in
ing. Good agreement obtained for the MDEs and TDEs
AME12 [24] derived from systematics, and 44V, whose
showsthat the role andmagnitude ofthe ISB terms are
now firmly established.
Itisveryinstructivetolookattenoutlierswhichwere
excluded from the fitting procedure. They are shown by 1.2
open symbolsin Figs. 2 and3. (i) There are five outliers (cid:28)tted not(cid:28)tted noISB
T
that depend on masses of 52Co, 56Cu, and 73Rb, which 1.0 SV
experiment
clearly deviate from the calculatedtrends for MDEs and 0.8
)
TDEs. These masses were not directly measured but V
e
derived from systematics [24]. (ii) There are two out- M 0.6
(
liersthatdependonthemassof44V,whoseground-state E
D 0.4
measurementmaybecontaminatedbyanunresolvediso- T
mer[33–35]. (iii)Largedifferencesbetweenexperimental 0.2
and calculated values are found in MDE for A = 16, 67 T = 1
0.0
and 69. Inclusion of these data in the fitting procedure
would significantly increase the uncertainty of adjusted -0.2
coupling constants. The former two, (i) and (ii), call for 10 20 30 A 40 50
improvingexperimentalvalues, whereasthe lastone (iii)
may be a result of structural effects not included in our FIG. 3. (Color online) Same as in Fig. 2 but for the T = 1
model. TDEs with no linear trend subtracted.
4
which, in our case, is equivalent to tIII ∝−1∆a and
TABLE II. Mass excesses of 52Co, 56Cu, 73Rb, and 44V ob- tII ∝ 1∆a . Assuming further th0at the p2ropCorStBional-
tained in thiswork andcompared with those of AME12 [24]. 0 3 CIB
ityconstantisthe same,andtakingforthe experimental
Our predictions were calculated as weighted averages of val-
values∆a =1.5±0.3 fmand∆a =5.7±0.3 fm[?
uesobtainedfromMDEsandTDEsforallthreeusedSkyrme CSB CIB
parametrizations. TheAME12valuesderivedfromsystemat- ], one gets:
ics are labelled with symbol #.
tII 2∆a
Mass excess (keV) 0 =− CIB =−2.5±0.5. (7)
tIII 3∆a
Nucleus This work AME12 [24] 0 CSB
52Co −34450(50) −33990(200)# From the values of coupling constants given in Table I,
56Cu −38720(50) −38240(200)# weobtaintheirratiosastII/tIII =−2.3±0.9,−4.5±1.8,
73Rb −46100(80) −46080(100)# 0 0
44V −23770(50) −24120(180) and −4.1±1.5 for the SVT, SkM*, and SLy4 EDFs, re-
spectively. Fig.4summarizesthese valuesincomparison
with the estimate in Eq. (7). As we can see, the values
ground-state mass measurement is uncertain. Recently, determined by our analysis of masses of N ≃ Z nuclei
the mass excess of 52Co was measuredas −34361(8)[36] with 10 ≤ A ≤ 75 agree very well with estimates based
on properties of the NN forces deduced from the NN
or−34331.6(66)keV[37]. These values areinfair agree-
ment with our prediction (1.8 or 2.4σ difference with scattering experiments.
In summary, we showed that the pn-mixed DFT with
respect to our estimated theoretical uncertainty), even
added two new terms related to the ISB interactions of
thoughthedifferencebetweenthemisstillfarbeyondthe
class II and III is able to systematically reproduce ob-
estimated (much smaller) experimental uncertainties.
served MDEs and TDEs of T = 1 and T =1 multiplets.
2
Adjusting only two coupling constants tII and tIII, we
0 0
34 reproduced not only the magnitudes of the MDE and
3m) (a) TDE but also their characteristic staggering patterns.
eVf 26 TheobtainedvaluesoftI0I andtI0II turnoutto agreewith
M the NN ISB interactions(NN scatteringlengths)in the
IIt0( 18 1S0 channel. We predicted mass excesses of 52Co, 56Cu,
73Rb, and 44V, and for 52Co we obtained fair agreement
10
withtherecentlymeasuredvalues[36,37]. Tobetter pin
3m) -4 (b) downtheISBeffects,accuratemassmeasurementsofthe
f
V -6 other three nuclei are very much called for.
e
M This work was supported in part by the Pol-
( -8
IIIt0-10 average i2s0h14/N1a5t/ioNn/aSlT2S/c0ie3n4c5e4 aCnednt2e0r15u/n1d7/erN/CSTon2t/r0a4c0t25N, obsy.
the Academy of Finland and University of Jyva¨skyla¨
0
(
) froms
atteringlengths within the FIDIPRO program, by Interdisciplinary
IIIt0 -2 Computational Science Program in CCS, University of
/ Tsukuba, and by ImPACT Program of Council for Sci-
IIt0 -4 ence, Technology and Innovation (Cabinet Office, Gov-
ernment of Japan). We acknowledge the CIS´ S´wierk
-6
∗ Computing Center, Poland, and the CSC-IT Center for
T
SV SkM SLy4 ScienceLtd.,Finland,fortheallocationofcomputational
resources.
FIG.4. (Coloronline)CouplingconstantstI0I (a)andtI0II (b),
together with their average values and uncertainties. Panel
(c) shows ratios of coupling constants tI0I/tI0II compared with
thevalue of Eq. (7).
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Supplemental material for: Strong-force isospin-symmetry breaking in masses of
N ∼ Z nuclei
P. Ba¸czyk,1 J. Dobaczewski,1,2,3,4 M. Konieczka,1 W. Satul a,1,4 T. Nakatsukasa,5 and K. Sato6
1Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland
2Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom
3Department of Physics, P.O. Box 35 (YFL), University of Jyva¨skyla¨, FI-40014 Jyva¨skyla¨, Finland
4Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
5Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
6Department of Physics, Osaka City University, Osaka 558-8585, Japan
(Dated: February 2, 2017)
7
1
0 This SupplementalMaterialexplainstechnicalaspects A. Selecting HF solutions used for fitting
2 of the method presented in the Letter and provides nu-
merical results that complement the results published
b Calculations of MDEs and TDEs involve ground-state
e therein. We start by providing the reader with details energiesineven-even,odd-A,andodd-oddnuclei. Inthe
F concerning the choice of the basis size. Next, we dis-
even-evenA=4n+2triplets,theHFsolutionsrepresent-
1 cuss the subtleties relatedto the fitting procedure of the ing the 0+ ground states are unambiguously defined by
two new coupling constants. Finally, we present results
filling the Kramers-degenerated levels from the bottom
] obtained for the MDEs and TDEs with the SkM∗ [1] ofthepotentialwelluptotheFermienergy. However,in
h and SLy4 [2] Skyrme functionals that mirror those for theodd-AT = 1 doubletsandA=4nT =1triplets,the
-t SVT [3, 4], which were presented in the Letter. HFsolutionsrep2resentinggroundstatesarenotuniquely
l
c defined. The ambiguity is related to the orientation of
u
a current generated by unpaired nucleon(s) with respect
n
to the nuclear shape, which gives rise, in triaxial nuclei,
[
I. CHOICE OF THE BASIS SIZE to three different solutions that have angular momenta

2 orientedalongtheintermediate(X), short(Y), andlong
v
(Z) axes of the nuclear shape [8]. No tilted-axis solu-
8 The code HFODD [5, 6] used in this work solves the
tions were found. In A = 4n |T = 1,Tz = ±1i states,
2 Hartree-Fock (HF) equations in the Cartesian harmonic
one should consider, additionally, the aligned |ν⊗πi (or
6 oscillator (HO) basis. In this work, we use the spher-
4 |ν¯⊗π¯i)andanti-aligned|ν⊗π¯i(or|ν¯⊗πi)arrangements
ical HO basis whose size is determined by the num-
0 of the valence neutron (ν) and proton (π). Let us men-
ber of shells, N , taken into account. The numeri-
. 0 tionthatboththeorientationandalignmentambiguities
1 cal stability with respect to N was tested in details
0 in Ref. [7]. It turned out that0the optimum choice of arerelatedtothetime-oddpartsofthefunctional,which
7 are very poorly constrained.
number of HO shells (in terms of the trade-off between
1 To test dependence of MDEs or TDEs on the orienta-
precision and computing time) is N = 10 for light nu-
v: clei (10 ≤ A ≤ 30), N = 12 for m0edium-mass nuclei tion and alignment of the HF solutions, for SVT we per-
0
Xi (31≤A≤ 56) and N0 =14 for heavier nuclei (A ≥57). ftohremoreedticthael McoDmEpsleitne bseotthofTca=lcu1laatniodnsT. W=e1fomuunldtiptlheatst
IntheLetter,wefollowedtheseguidelineswithexception 2
r weakly depend on both the orientation and alignment.
a oftheA=58triplet,whichwascalculatedwithN0 =12
Thecorrespondinguncertaintyturnedouttobeoftheor-
to maintain consistency with other triplets.
derof10keV,andthusinthetotaluncertaintybudgetof
MDE, it could be neglected. Accordingly, we performed
calculations of MDEs using only the lowest-energy HF
solutions obtained for any given Skyrme functional.
II. ADJUSTMENT OF THE MODEL
At variance with the results obtained for MDEs, for
COUPLING CONSTANTS
TDEs the influence of the orientations of the HF solu-
tions turned out to be significant. Hence, in this case
Results of fit of the model coupling constants tII and we computed TDEs for each of the three orientations
0
tIII, together with essentialelements ofthe fitting proce- separately, and then we averaged out the obtained re-
0
dure,aregivenintheLetter. Inthefollowingsubsections sults. The corresponding standard deviations associated
we provide further details concerning the methodology with the averagingprocedure was carried over to the to-
used. In particular, we discuss arguments justifying the tal uncertainty budget. For SV , the fitting procedure
T
splitting of the two-dimensional (2D) fit into two one- wasdoneseparatelyforthealignedandanti-alignedcon-
dimensional (1D) fits. We also discuss and quantify the figurations. These two sets of solutions led to similar
sources of potential errors and uncertainties. valuesoftII couplingconstant,andtosimilarpredictions
0
2
for the TDEs. Based on this observation, we decided to
perform the calculations for other parametrizations only 10
withalignedsolutionswhicharemucheasierto converge
at the HF level.
Unfortunately, for certain values of A, we were unable 8
to find solutions for all orientations. The missing solu-
tions were: for SV , A = 12 and 16; for SkM∗, A = 12, )
T V
16, 20, 24, 28, 32, 40, 44, and 52; for SLy4, A = 12, 16, Me 6
28, 32, 40, and 52. Furthermore, for SkM∗ and SLy4, (
E
solutions for A = 30 and 36 triplets turned out to be D
M
4
spherically symmetric, which led to unexpectedly large
values of TDE. Therefore, those points were excluded
from fitting.
2
A=11 A=31 A=51
A=21 A=41
B. The fitting strategy
0
-8 -6 -4 -2 0
Ourmodelhastwofreecouplingconstants,tII andtIII tI0II (MeVfm3)
0 0
(see Eqs. (5) and (6) of the paper), which are adjusted
FIG.5. (Coloronline)MDEsofT = 1 doubletscalculatedfor
to reproduce all available data on MDEs and TDEs in 2
isospin doublets and triplets. The most general strat- differentvaluesoftI0II(tI0I=0)andforSVT. Calculatedpoints
are connected with line segments that are almost perfectly
egy would be to perform the 2D fit to the data, that is,
parallel to one another.
to fit both coupling constants simultaneously. However,
the 2D procedure is very costly computationally, as it
requires both the doublets and triplets to be calculated
0.8 A=12
in the pn-mixing formalism. Fortunately, coupling con-
stantstII andtIII aretoalargeextentindependentofone A=22
0 0 0.7 A=32
another. Indeed,testcalculationswithoutCoulomb,per-
A=42
formedinRef.[9],clearlyindicatethatcouplingconstant
tII does not affect MDEs. The same calculations showed 0.6 A=52
t0hat the influence of coupling constant tIII on TDEs is V)
0 e
also very weak, and comes mostly from a subtle modifi- M 0.5
(
cation of the HF solution due to the self-consistency of E
D
solutions. Thispropertyallowsustoreplacetherigorous T 0.4
2D procedure by two constitutive 1D fits.
Toreduceapossiblediscrepancybetweenthefull2Dfit 0.3
and our simplified fitting procedure, we first performed
the adjustment of tIII coupling constant to MDEs. In 0.2
0
the next step, for the physical value of tIII obtained pre-
0
viously, we performed the adjustment of tI0I to TDEs. 0.1
The adopted strategy allowed us to compute the dou- 0 10 tI0I 20 3 30 40
blets within the standard pn-separable formalism, thus (MeVfm )
reducing the computational effort considerably. More-
FIG. 6. (Color online) TDEs of T = 1 triplets calculated
over, the calculations showed almost perfect linearity of for different values of tI0I (tI0II =−7.4MeV fm3) and for SVT.
MDEs (TDEs) as functions of tI0II (tI0I). For typical cal- Calculated points are connected with line segments.
culations, this fact is illustrated in Figs. 5 and 6. Such
linearity, in turn, allowed us to compute derivatives of
MDEs and TDEs over the coupling constants by using function χ2 in the following way:
only two values of the coupling constants. Actually, to
verify the linearity and to double-check the obtained re- χ2(t )= Nd (DEi(t0)−DEeixp)2, (8)
sbuylttsh,ewaesususemdetdhrlieneeoarriftoyutruvranleudeso.uTthteoebreronresgilnigtriboldeu.ced 0 Xi=1 ∆DE2i
where t represents the actual coupling constants, tII or
0 0
tI0II, used in the 1D fitting, Nd is the number of experi-
C. Penalty function and minimization procedure mental data points, DEi(t0) represents the displacement
energies, MDEs or TDEs, calculated for a given value
The 1D fitting was based on the χ2 minimization pro- of coupling constant t , and DEexp are the experimental
0 i
cedure described in Ref. [10]. We defined the penalty values of the displacement energies. The denominator,
3
∆DE2 = (∆DEexp)2+(∆DEthe)2+∆DE2 , is the er-
i i i mod 0.8 T = 1
ror comprising three components: (a) 2
• ∆DEexp is the experimental error evaluated from 0.4
i
errors of the binding and excitation energies as
given in Refs. [11, 12], 0.0
)
V
• ∆DEtiheisthetheoreticaluncertaintyrelatedtothe Me -0.4
averagingover orientations as described above, (
E
D -0.8
• ∆DE is the theoretical model uncertainty, M
mod
which is an unknown adjustable parameter. − T = 1
2T (b)
Theideaofthefitistominimizeχ2withtheadditional E/ 0.4
D
constraintofχ2perdegreeoffreedombeingequaltoone. M
d
The latter condition can be obtained by adjusting the 0.0 e
msporedaedl uonfcethrteaitnhteyo∆ryD-eExmpeordi,mwehnitchdiiffserreesnpcoens.sibVlealfuoresthoef -0.4 tted ot(cid:28)tt oISB
(cid:28) n n
model uncertainties obtained for three different Skyrme ∗
SkM
forces used in this work are collected in Table III. As it -0.8
experiment
turnsout,themodeluncertaintyisthemainsourceofthe
total uncertainty, which can be checked by comparing it 10 20 30 40A 50 60 70
with the root-mean-square deviations (RMSD) between
theory and experiment, see Table III.
FIG.7. (Coloronline)Theoreticalandexperimentalvaluesof
MDEsfortheT =1/2(a)andT =1(b)mirrornuclei,shown
TABLEIII.Valuesofthemodeluncertainties,∆DEmod,and withrespecttotheaveragelineartrenddefinedinFig.1ofthe
∗
RMSD of MDEs and TDEs obtained for three EDFs consid- paper. Calculations were performed using SkM functional
ered in this work. All valuesare in keV. with the ISB terms added. Shaded bands show theoretical
uncertainties. Experimental error bars are shown only when
SVT SkM* SLy4
they are larger than the corresponding symbols. Full (open)
∆MDEmod 190 150 120 symbols denote data points included in (excluded from) the
RMSD(MDE) 200 150 120
fittingprocedure.
∆TDEmod 92 110 110
RMSD(TDE) 95 110 110
the Academy of Finland and University of Jyva¨skyla¨
within the FIDIPRO program, by Interdisciplinary
Computational Science Program in CCS, University of
III. COMPLEMENTARY RESULTS FOR SKM∗
Tsukuba, and by ImPACT Program of Council for Sci-
AND SLY4
ence, Technology and Innovation (Cabinet Office, Gov-
ernment of Japan). We acknowledge the CIS´ S´wierk
As stated in the Letter, calculations of MDEs and
TDEs with the extended Skyrme forces involving the
strong-forceclassIIandIIIcontacttermswereperformed
∗
for three different Skyrme EDFs SV , SkM , and SLy4.
T 1.2
As shown in the Letter, within the model uncertainties, ∗ (cid:28)tted not(cid:28)tted noISB
the resulting coupling constants tII and tIII are fairly in- 1.0 SkM
0 0
dependent of the EDF. Below in Figs. 7–10, for the sake experiment
0.8
ofcompleteness,wepresentresultsobtainedfortheEDFs )
V
SkM∗ andSLy4,whereastheanalogousSVT resultswere Me 0.6
shown in Figs. 2 and 3 of the Letter. As we can see, for (
E
all three parametrizations, the agreement between ex- D 0.4
T
perimental and theoretical values is similar. This nicely
0.2
confirmsthatthemethodproposedinthisworkisindeed
almost perfectly independent on the isoscalar charge- T = 1
0.0
independent part of the functional. In this sense it is
a quite robust method to assess the strong-forceisospin- -0.2
symmetry-breaking effects in N ≈Z nuclei. 10 20 30 A 40 50
This work was supported in part by the Pol-
ish National Science Center under Contract Nos. FIG. 8. (Color online) Same as in Fig. 7 but for the T = 1
2014/15/N/ST2/03454 and 2015/17/N/ST2/04025, by TDEs with no linear trend subtracted.
4
Computing Center, Poland, and the CSC-IT Center for
0.8 T = 1
(a) 2 ScienceLtd.,Finland,fortheallocationofcomputational
resources.
0.4
0.0
)
V
Me -0.4
(
E 1.2
D -0.8
M (cid:28)tted not(cid:28)tted noISB
− T = 1 1.0 SLy4
E2/T 0.4 (b) V) 0.8 experiment
D e
M M 0.6
d (
0.0 e E
-0.4 tted ot(cid:28)tt oISB TD 0.4
(cid:28) n n 0.2
SLy4 T = 1
-0.8 0.0
experiment
-0.2
10 20 30 40A 50 60 70 10 20 30 A 40 50
FIG. 9. (Color online) Same as in Fig. 7 but for the SLy4 FIG. 10. (Color online) Same as in Fig. 9 but for the T = 1
EDF TDEs with no linear trend subtracted.
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