32
DEi = (exp 2DEi ) + (the 2DEi ) +ror comprising three components:•University	  of	  Tsukuba	  	  |Center	  for	  Computational	  Sciences2

DEmod , is the er-1
T=2

exp
DEi is the experimental error evaluated fromerrors of the binding and excitation energies asgiven in Refs. [11, 12],

Computational	  Nuclear	  Physics
averaging over orientations as described above,

MDE•

theDEi is the theoretical uncertainty related to theHeavy	  element	  synthesis	  in	  giant	  stars•MDE/2T

DEmod is the theoretical model uncertainty,
which
is an unknown
adjustable
1
Thereare about
100 elementsin the parameter.
universe, but how have they been synthesized? Alpha reaction is oneT of=theimportantnuclearreactionstosynthesize
newelementsbyfusionofalphaparticles.
Relevant
nuclearreactions2
The idea of the fit is to minimize with the additional
involvequantummechanicalmotionof
many
protons
andneutrons.
For
microscopic
study,
wehavedevelopeda2
constraint of per degree of freedom being equal to one.
method which selects the relevant reaction path from many degrees of freedom (Fig. 1). We re-­‐quantize theThe latter condition can be obtained by adjusting theKAIWEN AND onTAKASHI
NAKATSUKASAPHYSICALREVIEWtunneling
C 96, 014610(2017)
Hamiltonian
the reaction
path, and succeed to calculate the fusion cross sectionof quantum(Fig.2).

x [fm]44TABLE
III. Values of the model2
2 uncertainties, DEmod , and00RMSD
of MDEs and TDEs obtained
for three EDFs consid-2 in this work. All values are-2 in keV.
ered-4-4SVT

SkM*

SLy4-8 -6 -4 -2MDE0 2 4 6 8190 -8 -6
150-4 -2 01202 4 6 8
modz [fm]RMSD(MDE)

200

150 z [fm]120

TDE
modFIG. 1. Density
distribution
on92the x-z110
plane at 110
four points onFig.
1:Density
plots
along
thepath1620reaction
RMSD(TDE)
95
110
110
the ASCC fusion reaction path of O + α → Ne: (a) R = 7.6 fm,determined(b) R = 5.2
fm, (c) R = self-­‐consistently,
4.2 fm, and (d) R =which
3.8 fmconnects
corresponding
16O+𝜶, to the single
20twoseparatednuclei(a),to the ground state of Ne.
neon nucleus (d).

FIG. 7. (Color online) Theoretical and experimental values ofMDEs-20 2 for the T =161/2 (a) and T = 1 (b) mirror nuclei, shown
O+αwith respect to the average linear trend defined in Fig. 1 of the⇤-40paper.
Calculations were performed using SkM functional
1with-60 the ISB terms added. Shaded bands show theoreticaland0.5 experimental uncertainties. Full (open) symbols denote
4 0.54.5
5 1in5.5
6
6.5 2the7 fitting
7.52.5 procedure.
8data pointsincluded(excludedfrom)
1.5
3R [fm]E [  MeV  ]262FIG.
ωon the10 2. Square of the moving RPA eigenfrequency
with reduced (q)mass16 Finland 20and University of Jyväskylä
thecollective
AcademyofCalculated
O + α → fusion
Ne, shown
as a function
ASCCFig. path
2: ofsection
withcrossASCC
mass of2016
Ne(R=The3.8fm), thisrelative
distanceR.
At
thegroundstateofwithin
theFIDIPRO
program,
byInterdisciplinary
25(astrophysicalS-­‐factor)
forO+𝜶.
dotted10π−=0octupolemode
excitation.
modeComputationalcorresponds
totheK
line showsScience
results ofa conventional
approach
Program
inofCCS,
University ofS factor [MeV b]

(d)

27]
ω[MeVS factor (10 MeV b)(c)

2x [fm]model uncertainty DEmod , which is responsible for the ADIABATIC SELF-CONSISTENT COLLECTIVE PATH IN . . .
spread of the theory-experiment di↵erences. Values of(a)(b)
80⇤model uncertainties obtained for three di↵erent Skyrme
44
5
forcesusedinthiswork
arecollectedinTableIII.
Asit6022with reduced massturns
is the main source of the0 out, the model uncertainty
0with ASCC mass
40 4
total-2 uncertainty, which can-2be checked by comparing itA
20-4 the root-mean-square deviations-4
with(RMSD) between
3
theory-8 andexperiment,
see
Table
III.
-6 -4 -2 0 2 4 6 8-8 -6 -4 -2 0 2 4 6 80

without
dynamicaleffect.
Tsukuba,andbyImPACT
ProgramofCouncilfor
Sci24
10Technology
and Innovation(CabinetOffice, Gov⟨ω|Q̂
|0⟩
=0.
At
largerR,
theoctupole
deformation⇤ and ence,
without
thespin-orbit
interaction.
Toexpress
theorbitalwave
20III. COMPLEMENTARY RESULTS FOR SKM
increases,
thenthe parityWe
is noTheQ30 ernment
231616longer conserved.
functions, the grid representation
isemployed,discretizingofJapan).
acknowledge
theCIŚ Świerk
AND SLY4
10O+ Otransition
strength⟨ω|Q̂|0⟩becomesnonzero,thengradually
the rectangular box into the three-dimensional (3D) Cartesian20
ComputingCenter,
Poland,andtheCSC-ITCenterfor
16
3changes22its character into the relative motion between O andmesh.
Themodelspaceissettobe12×12×18fmfor theQuark	  masses	  v
isible	  in	  heavy	  atomic	  nuclei
ScienceLtd.,
Finland,fortheallocationof
computational
1620
310of the potential energy can be negative,
As stated
the Ne,Letter,calculations
MDEs
O + inα→12 × 12× 24 fm forofthe
systemand α. Since the curvature
reaction
2468
10
216
16
32
leading to imaginary ω(q).
O + withO → theS, andthe meshSkyrme
size is setforces
to be 1.1
fm.
TDEs
extended
involving
the the value of ω (q) can be negative,
EKc. =[MeV]
Sincethegenerators
keep
the0character
alltheway,m.
strong-forceclass
II andtoIIIdistinguishcontact terms
were white
performed
Imagine you
wantedbetweenandthe states |ψ(q)⟩ on the collective path are axially symmetric.
1620
⇤A.
O+α
→
Neforblack
threeseeds
di↵erent
SkyrmeEDFs SVT , without
SkM , andSLy4. Thereinsidea watermelon
actually
appearfive astrophysicalNG modes; namely,modes,FIG.
11. TheS factortwo
for rotational
the subbarrier
fusion of
16
20
1. Collectivepath: Owithin+up
α→groundstate
Ne inside an andAsbreaking
shown
inthedistinguish
Letter,themodeluncertainties,
16 the three translational 16
16 In an actual calculation,
it. Toanddownquarks⇤
modes.
O+α(upperpanel)and
O+O(lowerpanel)asafunctionofIIIIItheresultingcoupling
constants
tandt
in- these
atomicnucleus
seems
tobe
even
more
outlandish
task.
NGenergy
modesEhave
finite energy due to the finite mesh
As a trivial solution of the ASCC0 equations,thefairly
well0 areincident
c.m. . The solid line indicates the results obtained
16
numerical
calculation.
At the thegroundstate,weare
obtainseparatedboththisatBelowthegroundstates 7–10,
canlooking
be forthe initial
Here we Oshow
canbebyat sakethe sizedependent
ofandtheαthatEDF.
in doneFigs.
thewithinthe
ASCC inertial
mass M(R),dashedlines
calculated
ωwith
= 1.9
MeV
for reduced
the rotationalmodes,
ω=3.5
MeV
for
thestate|ψ(q =0)⟩ towestarttheof
iterative
procedure
inSec.
IIEDFsB.
simplest
characteristics
heavy
nuclei−theirmasses.
ofcompleteness,presentresults
obtainedforthe.
theconstantmassµred20
translational
modes
along
thexandydirections,andω=Alternatively,
thegroundstateofNe
canalsobetheinitial⇤
We knowthat theup quark
is about twice
SkMand SLy4,
whereasthe analogous
SVT lighter
resultsthan
were
1.3 MeV for the translational mode along the z direction.
state for the iteration. Although it is not trivial, we find thatthe downquark,
which
leads
to interactions
betweenshowninFigs.
2and3oftheLetter.
Aswecansee,forThe next lowest K = 0 mode of excitation at the groundthe same trajectory is produced starting from these two initialpairsofneutrons
being
slightlystrongerthanthoseallthreeparametrizations,
theagreement
betweenstates. The ASCC collective path smoothly connects the two ex- state of 20 Ne has the positive parity π = + and a transition16
20 is similar.
barrier. ofTheS factor
way thebetween pairsofprotons.
is ground“Canwesee a strength
perimental
andtheoreticalvaluesThis
nicelywell-separated
nuclei,O and Theα,toquestion
Ne at thestate.
operatorofmay
Q̂20 ,reveal⟨ω|Q̂20in|0⟩a more
= 5.3 transparent
fm2 . The RPA
20
influenceωofofthestructure
andwhich
dynamics.
prominent
signatureofaproposed
this quadrupole
asymmetryin isheavier
Ne haslarge
Thegroundstateof method
thisnuclear
state is about
10 MeV,is much higher
confirms
that
thein this deformation.
work
indeed frequency
11 shows
S modesfactor for
scatteringThedensity
profileisindependentshown
in Fig.of1(d).
At thegroundstate,
the octupole
modetheandcalculated
many otherwiththeK ̸=0.
nuclei?”
Wecalculatemassesheavy
nucleiandthus, thanFigure
almostperfectlyontheisoscalarchargeT=11616π161616π−+thephysicalRPA
stateis found tomasses,
be thisthe K= 0 it is Ifofwe O
mode
as the starting
adopt+ α this
andK O=+0 O,
respectively.
For generators,
O + O, thebylowest
comparing
resultswithmeasured
tosensequantify
independent
partof thefunctional.
Inoctupole
excitation,which
has a sizable transition
strengthofWe wecannotconstruct
the collective
pathonconnecting
the groundvaluesoftheS
factor
areplottedalog
scale.
The
dashedstrengths
ofthesymmetry-­‐breaking
effects(Fig.
3).
athequiterobustmethodtoassessthestrong-force
π isospin−
operatorQ̂
definedinEq.
(11).
Choosing
thisK=0state
andtwoseparatednuclei.
Generally
speaking,thehigher30
line
iscalculatedwiththesamepotential
V(R)
but
with theshowedsomeofformerexperimental
data
were
far
A
symmetry-breakinge↵ects in Q̂(q)N ⇡ andZ nuclei.
octupole mode as the generatorsP̂ (q), a series of
theRPA eigenfrequencyis,M(R)the moredifficultit is tovalue
find ofa µ
reducedmass,
replacingbytheconstant
redawayfromthecalculation(opensymbolsinFig.
3).
Fig.
3:
Thenuclear
mass
difference
among
theisospin
tripletsThis
work
wassupportedinpartby
thePolstates can be obtained by iteration, forming a collective fusion
solutionof
themovingmean-field
equation(6).
in Eq.
(30). The called
effect of
the inertial
mass is energy
significant
in the16nuclear masses
20
Thesearerevisited
experimentallyinthe(triangles),tripledisplacement(TDE).
The1
FIG. 38.shows(Colorasofinthe
Fig.
7 but
for the T =Figuretheonline)
potentialSame
energyASCC
collective
path ofO+α ↔
Ne. In Center
the asymptotic[Fig. 1(a)],ishNationalScience
underregion
ContractNos. deep
subbarrier-energy
region,especiallyfor
thewhich
reactionofconventional
theorypredicts
thesolidline,
isveryworld,
andthenewdata
supportourpredictions.
withlinear trend
Eq.16 (9),as noa functionof R. subtracted.
The dashed line shows thethe generators smoothly change
into those representing the by path,
2014/15/N/ST2/03454
and2015/17/N/ST2/04025,
16 TDEs
O + O at E
< 4 MeV. Because of a schematic nature of16

relative motion between O and α. Figure 1 shows density
distributions in the x-z plane (y = 0) at four different points
on the collective path. Figure 1(a) shows the well-separated
1620
O + α, Fig. 1(d) shows Ne at the ground state, and twointermediate
states are shown in Figs. 1(b) and 1(c).
contact	  address:	  nakatsukasa@nucl.ph.tsukuba.ac.jp
Figure 2 shows the square of moving RPA eigenfrequency

Basemethodnuclearnuclear
32S. In
reaction
calculaThe
state ofIt is fouthat ofor octup
16O→161O+
inertial
howevesuch asof 32 S.
In thof the sprincip
superdeno long
to chan
The imsuggest
is possi
Fromand qua
dynamisuggestsignific
energie
Betw
32S, weSince wterms, t
the tria
may bework. I
adoptedmethodand incc.m.experimental
different
fromdata
asymptotic
Coulomb
energy on topof(squares).
the summed groundthe BKN
density
wegroundshouldstate
regardstateenergiesof α functional,
and 16 O. Theof 20this
Ne resultis at as asuggestsbarrier
the significant
Rqualitative= 3.8 fm,one.
and Nevertheless,the top of the itCoulomb
is located effectinertialmassand roughly
basicwithfeatures
atofRthe= 7.7
fm. Tocomparethe ASCCreproducescollective path16
16of experimental
factor for CHF
the calculations,
O- O scattering.
This
http://www.ccs.tsukuba.ac.jp/
thoseobtained with Sconventional
we show This
theoctupole moment
as a functionof Rrequantizationin Fig. 4 for thesedemonstrates
the usefulnessof theapproachNo. 252