China-Japan collaborative
workshop
Nuclear mass and life for unravelling mysteries of r-process

June 26-28, 2017
Takashi Nakatsukasa (University of Tsukuba)• Bilateral Program• City of Tsukuba• Isospin-symmetry-breaking effect on
nuclear mass

China-Japan Bilateral Program• Microscopic information for r-process– Nuclear mass
– Nuclear life timeJapanese members• 筑波大学（University of Tsukuba）– 中務 孝（Nakatsukasa, Takashi）– 日野原 伸生（Hinohara, Nobuo）– 倪 放（Ni, Fang）– 王 之恒 （Wang, Zhiheng）– 小沢 顕（Ozawa, Akira）– 鈴木 伸司 （Suzuki, Shinji）– 森口 哲朗（Moriguchi, Tetsuro）– 天野 将道（Amano, Masamichi)
Blue: Theory
Red: ExperimentJapanese members• 埼玉大学（Saitama University）– 山口 貴之（Yamaguchi, Takayuki）– 大甕 舜一朗（Omika, Shunichiro）– 若山 清志（Wakayama, Kiyoshi）• 京都大学（Kyoto University）– 吉田 賢市（Yoshida, Kenichi）• 九州大学（Kyushu University）– 小林 良彦（Kobayashi, Yoshihiko）

Japanese members• 日本原子力研究機構（JAEA）– 小浦 寛之（Koura, Hiroyuki）• 高エネルギー加速器研究機構（KEK）– 宮武 宇也（Miyatake, Hiroari）• 理化学研究所（RIKEN）– 梁 豪兆（Liang, Haozhao）– 西村 俊二（Nishimura, Shunji）– 阿部 康志（Abe, Yasushi）

Chinese members• 陈瑞九 [陳瑞九] (CHEN, Ruijiu) 近代物理研究所• 郭璐 (GUO, Lu) 中国科学院大学• 刘敏 [劉敏] (LIU, Min) 広西師範大学• 龙文辉 [龍文輝] (LONG, WenHui) 蘭州大学
• 牛中明 (NIU, ZhongMing)安徽大学• [ 牛一斐(NIU, YiFei) ELI-NP, Romania ]• 欧立 (OU, Li) 広西師範大学• 孙保华 [孫保華] (SUN, BaoHua)北京航空航天大学• 孙向向 [孫向向] (SUN, XiangXiang)理論物理研究所

Chinese members• 王琨 (WANG, Kun) 理論物理研究所• 王猛 (WANG, Meng) 近代物理研究所• 王宁 [王寧] (WANG, Ning) 広西師範大学• 夏铖君 [夏鋮君] (XIA, ChengJun) 浙江大学
• 张鹏 [張鵬] (ZHANG, Peng) 近代物理研究所• 张玉虎 [張玉虎] (ZHANG, YuHu) 近代物理研究所• 周善贵 [周善貴] (ZHOU, ShanGui) 理論物理研究所

Mt.Tsukuba （筑波山）梅園筑波山（紫峰）

Mt. Tsukuba

Isospin symmetry preserving
energy density functionalandits breaking effect on nuclear mass
Sato et al. PRC 88, 061301(R) (2013)
Sheikh et al. PRC 89, 054317 (2014)
Baczyk et al. arXiv: 1701.04628.

Isoscalar and isovector densities
Isospin

11 ! 1 $n = τ = , τ3 ==#&2
2 " 0 %
11 ! 0 $p = τ = , τ3 = − = #&2
2 " 1 %

Nucleon in general! φ $1&
N =## φ2 &"%

One-body isoscalar and isovector density operators"%" 1 0 % cn '++$
ρ̂i ≡ ( cn , c p ) τ i, τ0 = $', τ = (τ 1, τ 2 , τ 3 ) :Pauli matrix$ c '# 0 1 &# p &ρ̂0 , ρˆ++++++++= ( cn cn + c p c p , cn c p + c p cn , i(cn c p − c p cn ), cn cn − c p c p )(Energy density functionals of ( ρn , ρ p ) are not invariant
under the rotation in the isospace.

)Rotational invariance in isospace
Energy density functional (EDF) of ( ρn , ρ p ), or ( ρ0 , ρ3 )202232

ρ = ( ρn + ρ p )ρ = ( ρn − ρ p )The Coulomb energy is anisotropic in theisospin space, but the rests are (almost)isotropic.

EDF of ( ρ0 , ρ )202

ρ = ( ρn + ρ p )2222
ρ = ρ1 + ρ2 + ρ32

2= ( ρnp + ρ pn ) + (−iρ pn + iρnp ) + ( ρn − ρ p )2

General EDF from: Perlinska et al, PRC 69, 014316(2004)pn-mixed orbitals
p-n mixed s.p. wave functions(p,n) è (1,2)

Standard unmixed neutron and proton w. f.

φ1 = φn , φ2 = φ p
Hartree-Fock representation of density(Kohn-Sham)ρ = ∑ φi φii:occ

Diagonal n and p
densities
Off-diagonal p & ndensities

p-n mixing is required in the single-particle orbitals

Isocranking calculation• Standard HF calculation– (N,Z) are specified.

• pn-mixed HF calculation– Only A=N+Z is specified.
– Additional constraint is necessaryTzisospin• Direction of isospin

θ– Cranking terms!
T!

λTy

ϕ– Eigenvalues: Routhians
TxTilted cranking for IAS• Standard HF calculation
(n)
( p)
ε,
ε
– Proton & neutron Fermi levels F F• pn-mixed HF calculation– Nucleon Fermi level only ε F(n)( p)ε≈ε– Cranking term −λzTz should lead to F
F = εF– Change the direction (tilted-axis) to obtain IASwith different Tzλx

ε F(n) − ε F( p)λzWe have developed a code for pnHF by extending an HF(B) solver

HFODD(1997-)
http://www.fuw.edu.pl/~dobaczew/hfodd/hfodd.html
J. Dobaczewski, J. Dudek, Comp. Phys. Comm 102 (1997) 166.
J. Dobaczewski, J. Dudek, Comp. Phys. Comm. 102 (1997) 183.
J. Dobaczewski, J. Dudek, Comp. Phys. Comm. 131 (2000) 164.
J. Dobaczewski, P. Olbratowski, Comp. Phys. Comm. 158 (2004) 158.
J. Dobaczewski, P. Olbratowski, Comp. Phys. Comm. 167 (2005) 214.
J. Dobaczewski, et al., Comp. Phys. Comm. 180 (2009) 2391.
J. Dobaczewski, et al., Comp. Phys. Comm. 183 (2012) 166.

•••••

Skyrme energy density functional
Hartree-Fock or Hartree-Fock-Bogoliubov
No spatial & time-reversal symmetry restriction
Harmonic-oscillator basis
Multi-function (constrained HFB, cranking, angular mom.
projection, isospin projection, finite temperature….)

A=48 isobars
w/o Coulomb

48Ni

48Ca−λ xTx

Tzθ

48Fe−λzTz
48TiTxNo p-n mixing at |Tz|=T(T, Tz) are controlled
by the “isocranking”.
Energies are independent of Tz

In addition to the groundstate, the IAS aredescribed by HF states.

48Cr

T=4

A=48 isobars
with Coulomb
Energies now depend on Tz
because of the Coulomb int.
(almost linear dependence)θ ʹ = 90!

No p-n mixing at |Tz|=T

T=4Single-particle routhians
are roughly independent
on the direction of T.

48Ni

48Ca

A=40 isobars for T=8 IAS’s
pure nTotal energy
T=8Coulomb energy

40Mg

40Ni

pure pEnergy difference mostly comes from the Coulomb
energy difference.

Fig. 3. Single-particle Routhians of the T ≃ 8 states in A = 40 isobars calculated with the Coulombinteraction included. The arrows at the upper left and upper right indicate the positions of the Fermi
energies at θ′ = 0◦ and 180◦ , respectively.

IAS’s in A=40 isobars
Deformation

Radii for T=8 states

0.34
72.2

3.9

w/ Coulomb
w/o Coulomb
T=8A=40, T~840

40Ni

Mg

40Ca

0.28
40Cr

72.1
0.26

40S

0.24
72.05
0.22

T=472
0.18
-8-8-6-6-4-4-2-2

00

<Tz>

224 46 6

8 8

<Tz>

2

3.6

proton (w/ Coulomb)
neutron(w/ Coulomb)
total (w/ Coulomb)

Calc.

-455

Exp.
-460

3.5
3.4-465

3.3

A=40, T~8T~40.2

3.7

Energy (MeV)

β22><T72.15
0.3

3.8rms radii (fm)

0.32

total (w/o Coulomb)

Fig. 4. ⟨T̂ ⟩ calculated for T ≃ 8 IASs withand without the Coulomb interaction.

3.2 -470-8-6-4-1-20
<Tz>
-0.5

204

68

0.5

Fig. 5. Root-mean-square radii<Tz>of the T = 8IASs with A = 54 as functions of ⟨T̂z ⟩.

1Fig. 7. Energies of T ≃ 1 isobaric analog
Fig. 6.
Quadrupoledeformation
β2 calculated
Deformation
stays
almost constantamong IAS’s.
states in A = 54 isobars in comparison with thefor the T ≃ 4 and T ≃ 8 IASs in A = 40 isobars
experimental data [10]. The results of isocranking
withtheCoulombinteraction◦constant.
Protonandbut thematter
is everyroughlyθ ′ , and,unlikeinneutron
the caseradiiof included.
Achanges,= 48 [4], there
is nolargeradiusshell forgap
above30thesurface.
calculationsofFermi
θ′ between
θ ′ = 0◦and 180
are plotted.
Nevertheless, with our choice of ⃗λ, the level crossingsare ◦avoided.
While the s.p. states are◦◦pure proton or neutron states at θ = 0 and 180 , which means that the |Tz | = T states
are nothing but the standard HF states without the p-n mixing, at all other tilting angles,

Comparison with experiments• Energies of T=1 triplets (with SkM*)
A=54 isobars

A=14 isobars

54Ni(g.s)

14O(g.s)54Co(g.s)

14N(excited 0 + )14C(g.s)54Fe(g.s)(The origin of calc. BE is shifted by 3.2 MeV tocorrect the deficiency of SkM* functional in thepanel for A=14)

MDE & TDE• Mirror Energy
Displacement
𝑀𝐷𝐸 ≡ 𝐸 𝑇, 𝑇' = −𝑇−𝐸(𝑇, 𝑇' = +𝑇)• Triple Energy
Displacement

T =1

V̂

MDE/2TT = 12sideIIItIII0V̂
T⇤

Th

MDET =1

𝑇𝐷𝐸 ≡ 𝐸 𝑇 = 1, 𝑇' = −1+𝐸 𝑇 = 1, 𝑇' = +1−2𝐸 𝑇 = 1, 𝑇' = 0T
⇤

A

wh
resp
Eqs
plin
disr
CtheExtension of EDF

2Standard EDF (SLy4 as an example)

sider the following zero-range interactions of class II and• III
Isospin-symmetry-breaking
interaction
with two new low-energy coupling
constants tII0 andtIIIClass II (CIB) and III (CSB)
0– [24]:hi
1
ˆ(i) ~⌧ˆ(j) (3)V̂ II (i, j) = tII(rr)
3ˆ
⌧
(i)ˆ
⌧(j)~⌧,ij
330
21 IIIIIIV̂ (i, j) = t0 (ri rj ) [ˆ⌧3 (i) + ⌧ˆ3 (j)] .
(4)
2
The corresponding contributions to EDF read:– Two parameters, 𝑡233 and 𝑡2333 , are determined
1 II 2
2by fitting
MDEandTDE.
HII =t0 (⇢+⇢2⇢n ⇢p 2⇢np ⇢pnnp2s2ns2p + 2sn · sp + 2snp · spn ),1 III 2
HIII = t0 ⇢n⇢2ps2n + s2p ,(5)(6)

MDE3• Mirror energy
displacementSVSkM*
SLy4T = 12III
upling constants tII
0 and t0 and their uncertainn this work for the Skyrme EDFs SVT , SkM*,

25 ± 8 be well
23 ± 7– Can
5.6 ± 1.4
5.6 ± 1.1
reproduced byre able to show results in extended scale,iled comparison
experimental
𝑡2333with= −7.4
MeV data
fm<isFig. 3, we show results obtained for TDEs,for SV
plementary results
obtainedfor the SkyrmeT
Ly4 EDFs are collected in the Supplemental.
fying to see that the calculated values ofy follow the experimental A-dependence, seeworth noting that a single coupling conproduces both T = 12 and T = 1 MDEs,
ms conclusions of Ref. [9]. In addition, for
MDEs, the SVT results nicely reproduce (i)MDE

17 ± 57.4 ± 1.9

T =1

MDE/2T

fm3 )
fm3 )

TTA

d lack of staggering of the T = 1

Having at hand a model with ISB strong interactions
with fitted parameters we can calculate MDEs for more
massive multiplets and make predictions on binding energies of neutron-deficient (Tz = T ) nuclei. In particular, in Table II we present predictions of mass excesses of 52 Co, 56 Cu, and 73 Rb, whose masses were inAME12 [22] derived from systematics, and 44 V, whose

tifying to see in Fig. 3 that our, with the class-II coupling consolute values as well as staggerwhereas results obtained withoutmall values and show no staggerobtained for the MDEs and TDEsd magnitude of the ISB terms are.
to look at ten outliers which wereCanarebe
wellng procedure.–TheyshownbyT
2 and 3. (i) There are five outliers
reproduced bys of 52 Co, 56 Cu, and 73 Rb, which
he calculated trends for MDEs andwere not directly
𝑡233 =measured
17 MeV butfm<tics [22]. (ii) There are two outforwhose
SVTground-state
he mass of 44 V,
ontaminated by an unresolved isoe di↵erences between experimental
T =1
are found in MDE for A = 16, 67
hese data in the fitting procedure
Acrease the uncertainty of adjusted
he former two, (i) and (ii), call for
FIG. 3. (Color online) Same as in Fig. 2 but for the T = 1al values, whereas the last one (iii)
TDEs with no linear trend subtracted.
uctural e↵ects not included in ourTDE• Triple energy
displacement

Relation to nuclear force• Phase shift analysis– Charge symmetry breaking
Δ𝑎?@A ≡ 𝑎BB − 𝑎CC = 1.5 ± 0.3 fm– Charge independence breaking
1Δ𝑎?3A ≡ 𝑎BB + 𝑎CC − 𝑎CB = 5.7 ± 0.3 fm
2• The present work
– By fitting MDE & TDE
–HIJJ
HIJJJ

=−K LMNJO< LMNPOthe T = 2 MDEs is less pronounced [21].
that all three functionals correctly describe
ence and lack of staggering of the T = 1

model.
Having at hand a model with ISB strong interactions
with fitted parameters we can calculate MDEs for more
massive multiplets and make predictions on binding energies of neutron-deficient (Tz = T ) nuclei. In particular, in Table II we present predictions of mass excesses of 52 Co, 56 Cu, and 73 Rb, whose masses were inAME12 [22] derived from systematics, and 44 V, whoseError in experimental mass?

more gratifying to see in Fig. 3 that ourculations, with the class-II coupling concribe absolute values as well as staggervery well, whereas results obtained without
e too small values and show no staggerreement obtained for the MDEs and TDEs
e role and magnitude of the ISB terms aretablished.
5256
73
structive to look at ten outliers which wereaCSB = app ann and VCIB / aCIB = 12 (app +ann )m TABLEthe fittingprocedure.
They52 are
56 shown
73 by
44TII. Mass excesses
1III
44 of Co, Cu, Rb, and V oba,
which,
inourcase,isequivalenttot/np
02 aCSinFigs.
and3. (i)are with
five thoseoutliers
tainedin 2thisworkandThere
comparedof AME12 [22].
1IIand t0 / 3 aCIB . Assuming further that the propo
Our
predictions
calculated
averages of valon
massesof 52wereCo, 56Cu, andas73weighted
Rb, whichtionality constant is the same, and taking for the eobtainedfrom MDEstrends
and TDEs
for all threeeues
fromthe calculatedfor MDEs
andused Skyrme
perimental values aCSB = 1.5 ± 0.3 fm and aCIB
parametrizations. The AME12 values derived from systemateicsmasses
werenot
directly
measuredbutMass #.
prediction
5.7 ± 0.3 fm [36], one gets:are labelled with symbol
systematics [22]. (ii) There are two outMass excess (keV)
end on theNucleus
mass of 44This
V, whoseground-statetII
2 aCIBwork
AME12 [22]0=
= 2.5 ± 0.5.
(
may be contaminatedby an unresolved isoIII52
3at
CSB
Co
34450(50)33990(200)#0iii) Large di↵erences
between experimental
56
Cu
38720(50)38240(200)#
T =1
d values are73foundinMDEforA=16,67
From the values of coupling constants given in Table
Rb46100(80)46080(100)#44III
sion of these V
data in23770(50)the fitting procedure
24120(180)
we obtain their ratios as tII
4.5 ± 10 / t0 = 2.3 ± 0.9,
A SVT , SkM*, and SLy4 EDFs, r
antly increase the uncertainty of adjusted
and 4.1 ± 1.5 for thetants. The52former
two, (i)MeV
and (ii),forPRL 117, 182503
spectively.
Fig. 4 summarizes these values in comparis
Co: −34361[Xu callet al.,(2016)]
ground-state
mass whereas
measurement
is uncertain.
FIG. 3. (ColorSame asbutweforcanthesee,T =the1 valuwith online)
the estimate
in inEq.Fig.
(7).2 As
perimentalvalues,
the lastone (iii) Recently,
52
theofmass
excess ofCo notwas included
measured inas our34361(8)[34]with determinedTDEs
no linear trend
subtracted.
by ouranalysis of masses of N ' Z nuc
ult
structurale↵ects• Removing fromfitting process––Co,
VCu,

RbSummary• Energy density functional including theproton-neutron mixing in single-particle
orbitals• Natural description of isobaric analogue
states in terms of “Slater determinants”• Extension of the EDF: Isospin symmetry
breaking terms– Good agreement with MDE and TDE– Discrepancy for specific nuclei: 52Co, 56Cu,73Rb, 44V