1V��$��T��v4�[�������

�� �

<@������e^$H9<z2�Ff2017.12.6-8 @<@

•Basic properties of nuclei

•TDDFT (TDMF) for nuclear collective motion

•Re-quantization of collective submanifold

•Application to alpha reaction in giant stars

Saturation properties of nuclear matter

• Symmetric nuclear matter w/o Coulomb

–

• Constant binding energy per nucleon

– Constant separation energy

• Saturation density

– Fermi energy

MeV 16)( ≈≈ pnSAB

13 fm 35.1fm 16.0 −− ≈⇒≈ Fkρ

2AZN ==

MeV 402

22

≈=mkT F

F!

 �)�\�����(AME2016)

Neutron number N

S 2n (M

eV)

Fig. 7. Two-neutron separation energies N = 122 to 145

120 125 130 135 140 145

120 125 130 135 140 145

6

8

10

12

14

16

18

20

6

8

10

12

14

16

18

20

208Pt

200Pt

210Au

201Au

216Hg

202Hg

218Tl

203Tl

220Pb

204Pb

224Bi

205Bi

227Po

206Po

229At

207At

231Rn

208Rn

232Fr

209Fr

233Ra

210Ra

234Ac

211Ac

235Th

212Th

236Pa

213Pa

237U

217U

238Np

221Np

239Pu

229Pu

240Am

231Am

241Cm

233Cm

242Bk

235Bk

243Cf

239Cf

244Es

241Es

245Fm

243Fm

Magic number: N=126

Neutron number N

S 2n (M

eV)

Fig. 4. Two-neutron separation energies N = 62 to 85

60 65 70 75 80 85

60 65 70 75 80 85

5

10

15

20

25

5

10

15

20

25

103Rb

99Rb

107Sr

100Sr

109Y

101Y

112Zr

102Zr

115Nb

103Nb

118Mo

104Mo

121Tc

105Tc

124Ru

106Ru

127Rh

107Rh

129Pd

108Pd

132Ag

109Ag

133Cd

110Cd

134In

111In

135Sn

112Sn

136Sb

113Sb

137Te

114Te

138I

115I

139Xe

116Xe

140Cs

117Cs

141Ba

118Ba

142La

119La

143Ce

121Ce

144Pr

123Pr 126Nd

146Pm

128Pm130Sm

148Eu

132Eu 135Gd

150Tb

137Tb 140Dy

152Ho

142Ho144Er

154Tm

146Tm

155Yb

150Yb

156Lu

152Lu

157Hf

155Hf

 �)�\�����(AME2016)

Magic number: N=82

Failure of the mean-field models

• In order to explain the nuclear saturation

within the mean-field picture, we need an

extremely small value of the effective

mass.

– Inconsistent with the experimental data.

• A solution

– Energy density functional

(Rearrangement terms)

[ ] [ ]

[ ]δρδ

ρ

φεφρρ

Eh

hE iii

≡

=⇒

4.0125

23

1*

≈⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−

FTAB

mm

Nakatsukasa et al., RMP 88, 045004 (2016)

Nuclear energy density functional

• Spin & isospin degrees of freedom

– Spin-current density is indispensable.

• Nuclear superfluidity à Kohn-Sham-

Bogoliubov eq.

– Pair density (tensor) is necessary for heavy

nuclei. [ ]qqqq JE κτρ ;,,!

spin-current

kineticpair density

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−−Δ−

Δ−

)()(

)()(

)()()()(

** tVtU

EtVtU

thttth

µ

µµ

µ

µ

λ

λ

Nuclear deformation

S. Ebata, T. Nakatsukasa

2016 71st JPS meeting @ Touhoku Gakuin Univ.

3D HF+BCS Cal. w/ SkM* From N=Z to N=2Z, Z=6-92 even-even (Total # 1005)

ResultsEbata, Nakatsukasa, Phys. Scr. 92 (2017) 064005

Deformation landscape Quadrupole deformation

S. Ebata, T. Nakatsukasa

2016 71st JPS meeting @ Touhoku Gakuin Univ.

3D HF+BCS Cal. w/ SkM* From N=Z to N=2Z, Z=6-92 even-even (Total # 1005)

Results

Nuclear deformation predicted by DFT

IntrinsicQmoment

Deformation landscape

N = 82

Z = 50

Time-dependentdensityfunctionaltheory(TDDFT)fornuclei

• Time-odddensities(currentdensity,spindensity,etc.)

• TDKohn-Sham-Bogoliubov-de-Gennes eq.

[ ])();(),(),(),(),(),( ttTtstjtJttE qqqqqqq κτρ!!!"

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−−Δ−

Δ−=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂)()(

)()()()(

)()(

** tVtU

thttth

tVtU

ti

µ

µ

µ

µ

λ

λ

spin-current

current

spin

spin-kinetickinetic

pair density

Deformation effects for photoabsorption cross section

SkM*functional

IntrinsicQmoment

Yoshida and Nakatsukasa,

Phys. Rev. C 83, 021404 (2011)

Real-timesimulation

K. Sekizawa and K. Yabana, Phys. Rev. C 88, 014614 (2013)

● 衝突径数の大きいとき (3 fm < b)

● 衝突径数の小さいとき (b < 3 fm)

K. Sekizawa

58Ni +208Pb at Elab=328.4 MeV28 8230 126TDHF計算の結果:

核子の移行確率核子の移行確率

: 粒子数射影法

208Pb

58Ni

+符号

―符号

荷電平衡

くびれ構造の形成と断裂

Wed., 18 February, 2015MNT reactions and QF processes in TDHF theory 11/26

“Partial”-spaceparticle-numberprojection

9

10-310-210-1100101102103

120 135 150

(+4p; Ce)

σ (m

b)

10-310-210-1100101102103

120 135 150 120 135 150

(+3p; La)

Expt. Y.X. Watanabe et al.

120 135 150 120 135 150

(+2p; Ba)

TDHFTDHF+GEMINI

120 135 150 120 135 150

(+1p; Cs)

136Xe+198Pt (Ec.m.≈ 644.98 MeV)

GRAZING w/ evap.

120 135 150

120 135 150 120 135 150

(0p; Xe)

120 135 150

(-1p; I)

120 135 150 120 135 150

(-2p; Te)

MASS NUMBER 120 135 150 120 135 150

(-3p; Sb)

120 135 15010-310-210-1100101102103

120 135 150

(-4p; Sn)

σ (m

b)

10-310-210-1100101102103

120 135 150

FIG. 6. Same as Figs. 1–5, but for the 136Xe+198Pt reaction at Ec.m. ≃ 644.98 MeV. The horizontal axis is the mass numberof the fragments. Cross sections for secondary products evaluated by TDHF+GEMINI (with E∗ and J) are shown by bluesolid lines. The experimental data were reported in Ref. [66].

excited compared to those in proton-stripping channels.Note that GRAZING also provides similar magnitude ofevaporation effects, although the absolute value of thecross sections are substantially underestimated.

IV. SUMMARY

In this paper, a method, called TDHF+GEMINI, hasbeen proposed, which enables us to evaluate productioncross sections for secondary products in low-energy heavyion reactions. In the method, the reaction dynamics, onthe timescale of 10−21–10−20 sec, is described microscopi-cally based on the time-dependent Hartree-Fock (TDHF)theory. Production probabilities, total angular momenta,and excitation energies of primary reaction products areextracted from the TDHF wavefunction after collision,using the particle-number projection method. Based onthose quantities derived from TDHF, secondary deexci-tation processes of primary reaction products, both par-ticle evaporation and fission, are described employing theGEMINI++ compound-nucleus deexcitation model.The method was applied to 40,48Ca+124Sn,

40Ca+208Pb, 58Ni+208Pb, 64Ni+238U, and 136Xe+198Ptreactions for which precise experimental cross sectionsare available. The inclusion of deexcitation effects,which are dominated by neutron evaporation, changesthe cross sections toward the direction consistent withthe experimental data. However, there remain discrep-ancies between the measured cross sections and theTDHF+GEMINI results, especially for multi-proton

stripping processes. It may indicate the importance ofdescription going beyond the standard self-consistentmean-field theory to correctly describe multinucleontransfer processes in low-energy heavy ion reactions.Finally, it is to be reminded that, in the proposed

method, there is no room to adjust the model param-eters specific to each reaction: energy density functionalis determined so as to reproduce known properties of fi-nite nuclei and nuclear matter [54]; GEMINI++ [46] andits ongoing developments [59, 60] allow a systematic re-production of a large body of data. Therefore, it will be apromising tool that can predict, in a non-empirical way,optimal reaction mechanisms to produce new neutron-rich isotopes that have not yet been produced to date.

ACKNOWLEDGMENTS

The author is grateful to Lorenzo Corradi (INFN-LNL) and Yutaka Watanabe (KEK-Japan) for providingthe experimental data. Lu Guo (the University of Chi-nese Academy of Sciences) is thanked for useful informa-tion on GEMINI++. The author acknowledges supportof Polish National Science Centre (NCN) Grant, deci-sion No. DEC-2013/08/A/ST3/00708. This work is in-debted for continuous usage of computational resourcesof the HPCI system (HITACHI SR16000/M1) providedby Information Initiative Center (IIC), Hokkaido Uni-versity, through the HPCI System Research Projects(Project IDs: hp120204, hp140010, 150081, hp160062,and hp170007).

Sekizawa, Phys. Rev. C 96, 014615 (2017)

ReactionabovetheCoulombbarrier

Simenel, C., 2010, Phys. Rev. Lett. 105, 192701.

136Xe + 198Pt

Large amplitude collective motion

• Decay modes

– Spontaneous fission

– Alpha decay

• Low-energy reaction

– Sub-barrier fusion reaction

– Alpha capture reaction (element synthesis in

the stars)

Missing quantum fluctuation, tunneling, etc.

LG#0¡������

�����(

• 12C��4j=*o��vd

– 12C + & → 16O+ *

– 16O + & → 20Ne+ *– 20Ne + & → 24Mg + *– …

• ����Z�j�kmx�zudSg�Ey!pt{Mn�ymmx�

/_BdS-factor

• ����Z�

– 1 2 = 454676

8

– ���� 3

–OQ;

• /_B

9(;) = exp(−2@A)

A =BCBDE

D

ℏG

H ; =1;9 ; ×J(;) Astrophysical S-factor

�45Pz 3�

�kx�������)

Classical Hamiltonas form

The TDDFT can be described by the classical form.

The canonical variables

Number of variables = Number of ph degrees of freedom

ξ ph =∂H∂π ph

π ph = −∂H∂ξ ph

Blaizot, Ripka, bQuantum Theory of Finite Systemsc (1986)

Yamamura, Kuriyama, Prog. Theor. Phys. Suppl. 93 (1987)

K L, @ = ; N(L, @)

NOOP = L + Q@ L + Q@ ROOP

LOS,@OS

NSSP = 1 − L + Q@ R L + Q@SSP

NOS = L + Q@ 1 − L + Q@ R L + Q@OS

�.�. COLLECTIVE SUBSPACE 19

multi-dimensional TDDFT phase space. In this category of theories for the large amplitudecollective motion, the principal issue is to identify a subspace, which is approximatelydecoupled from the other (non-collective) degrees of freedom, and canonical variables whichspan the subspace. Once we have done this task, the requantization is much more feasible,although it is not unique and we need to choose the way of quantization. For instance,if we obtain a decoupled subspace for the quadrupole-shape degrees of freedom, we mayconstruct the collective Hamiltonian of the Bohr model with the deformation parametersof ↵2µ (µ = �2, · · · , 2). The Bohr model usually adopts the Pauli’s prescription for thequantization.

According to the fundamental theorems of the TDDFT [2, 3], the TDDFT should be ableto probe all the excited states. In this sense, the linear response calculation (Chapter 1)should not be an approximation, but the exact linear response. However, in practice,currently available density functionals used in the TDDFT cannot reproduce low-lying excited states associated with strong anharmonicity and quantum fluctuations.Thus, in nuclear physics, theoretical tools capable of describing such descrete quantumlevels, corresponding to the stationary collective eigenstates, are also desirable. Theextraction of the collective subspace and its requantization may serve this purpose.

2.1.1 Classical Hamilton’s form of TDDFTIn order to understand the concept of the decoupled collective subspace, it may be useful torewrite the time-dependent Kohn-Sham equations in a form of the classical Hamilton’s form.In this section, let us show that this can be done, with the classical equations of motion withcanonical variables whose number is twice of the particle-hole degrees of freedom. Thereare many di�erent ways to define the classical canonical variables, among which we showhere the simplest one, the instantaneous (local) canonical variables. For other definition, seeRef. [4], for instance.

At each instantaneous time t, we define (occupied/hole) natural orbitals h(t) as thesolution of the time-dependent Kohn-Sham (TDKS) equation (eq-no). The unoccupied(particle) orbitals p(t) are arbitrary as far as they are orthogonal to all the occupied orbitals.In this instantaneous basis { k (t)}, the density ⇢(t) is diagonal,

⇢kl =8><>:�kl k N0 k > N

, (2.1)

where N = Nh is the particle number that equals to the number of hole orbitals. Calculatingthe matrix elements of The TDKS equation for the density matrix

i@

@t⇢(t) =

⇥h[⇢(t)], ⇢(t)

⇤, (2.2)

among unoccupied (particle) and occupied (hole) states, we can easily obtain

i ⇢hh0 = i ⇢pp0 = 0, i ⇢ph = hph, i ⇢hp = �hhp. (2.3)

20 CHAPTER �. LARGE AMPLITUDE COLLECTIVE MOTION

Here, the Kohn-Sham signle-particle Hamiltonian is given by the derivative of the total energywith respect to the density,

hkl ⌘@E[⇢]@⇢lk

. (2.4)

The latter two equations in Eq. (2.3) are identical to

i@

@t⇢ph =

@E[⇢]@⇢⇤ph

, i@

@t⇢⇤ph = �

@E[⇢]@⇢ph

, (2.5)

Therefore, we can identify the real and imaginary parts of the ph matrix elements of thedensity with the coordinates ⇠ and momenta ⇡, respectively; ⇢ph = (⇠ph + i⇡ph)/

p2. This

leads to the Hamilton’s equation of motion for the canonical variables (⇠, ⇡), with the classicalHamiltonianH (⇠, ⇡) = E[⇢].

ddt⇠↵ (t) =

@H (⇠, ⇡)@⇡↵

,ddt⇡↵ (t) = �@H (⇠, ⇡)

@⇠↵, (2.6)

where the index ↵ here denotes a pair of particle and hole ph. Thus, the TDKS equation isidentical to the classical Hamilton’s equation of motion with the canonical variables (⇠↵, ⇡↵).Note that the dimension of the phase space in this classical dynamics is very large, 2M = 2Nph,where Nph = Nh Np is the number of possible particle-hole excitations.

In case for the presence of the pairing correlations, usins a similar argument, we ends upwith the analogous equations for the generalized density R(t).

i@

@tRµ⌫ =

@E[R]@R⇤µ⌫

, i@

@tR⇤µ⌫ = �

@E[R]@Rµ⌫

. (2.7)

Here, Rµ⌫ are defined as Rµ⌫ ⌘ �†µR�⌫ where �µ and �⌫ are “unoccupied” and “occupied”quasiparticle states. At time t, the generalized density R(t) is expressed by the time-dependentquasiparticle orbits as R(t) =

P⌫ �⌫ (t)�

†⌫ (t) = 1 �Pµ�µ(t)�†µ(t). Identifying the real and

imaginary parts of R⌫⌫0 with the coordinates ⇠µ⌫ and the momenta ⇡µ⌫, respectively, the time-dependent Kohn-Sham-Bogoliubov-de-Gennes equations are expressed by the Hamilton’sequation (2.6), with replacement of indices, ph ! µ⌫. In this case, the index ↵ of thecanonical variables (⇠↵, ⇡↵) corresponds to the two-quasiparticle index µ⌫. The dimensionof the phase space (⇠, ⇡) is, in this case, 2 ⇥ Nm(Nm � 1) where Nm is the dimension of thesingle-particle model space.

2.1.2 Basic concept of a decoupled subspaceWe use the fact that the TDDFT equations of motion can be written in the classical form of Eq.(2.6), with a large number of classical variables (⇠↵, ⇡↵). This may facilitate understanding ofthe concept of the collective submanifold, in terms of the suitable coordinate transformationof a classical system. Let us start with 2M-dimensional phase space with canonical variables(⇠↵, ⇡↵), ↵ = 1, · · · .M and the equations of motion (2.6). Then, we consider a pointtransformation to the new coordinates qµ, µ = 1, · · · ,M , and its inverse transformation.

qµ = f µ(⇠), ⇠↵ = g↵ (q). (2.8)

Strategy

• Purpose

– Recover quantum fluctuation effect

associated with “slow” collective motion

• Difficulty

– Non-trivial collective variables

• Procedure

1. Identify the collective subspace of such slow

motion, with canonical variables (T, U)

2. Quantize on the subspace T, U = Qℏ

Expansion for “slow” motion

• Hamiltonian

K = K L, @ ≈12XYZ L @Y@Z + 1(L)

expanded up to 2nd order in @ [α, \ = (Uℎ)]

• Point transformation LY, @Y → T^, U^

U^ =_`a

_bc@Y , @Y =

_bc

_`aU^

• Hamiltonian

Kd = Kd T, U ≈12Xe^f T U^Uf + 1(T)

Decoupled submanifold

• Collective canonical variables (T, U)– LY, @Y → T, U; Th , Uh; i = 2,⋯ ,kOS– Decoupled collective subspace Σ– Σ defined by (Th, Uh) = (0,0)

• Decoupled eq. of motion

Uh = − _n

_bo− C

D

_pe

_boUD = 0

Th = XehCU = 0

[ Kd = Kd T, U ≈ C

DXe^f T U^Uf + 1(T) ]

P(q) = i d/dq

q1q2

P2(q) = i d/dq2

P1(q) = i d/dq1

(a)

(b)

Symplectic formulation

• Collective canonical variables (T, U)– LY, @Y → T, U; Th , Uh; i = 2,⋯ ,kOS

• Equations for a decoupled submanifold

_n

_`a− _n

_b

_b

_`a= 0 → LYonΣ

XZs ts_n

_`a_b

_`u= vD _b

_`a→ _b

_`a

ts_n_`a

≡_6n

_`x_`a− ΓYs

Z _n_`u

Covariant derivative

with ΓYsZ =

CDzZ{ zY{,s + zs{,Y − zYs,{

using the metric zYZ ≡ ∑ _bc

_`a_bc

_`u^

Riemannian formulation

• Rewriting a curvature term _b

_`a_`uin ΓZs

Yby

using the decoupled condition

• Equations for a decoupled submanifold

_n

_`a− _n

_b

_b

_`a= 0 Moving mean-field eq.

XZs ts_n

_`a_b

_`u= vD _b

_`aMoving RPA eq.

ts_n_`a

≡_6n

_`x_`a− ΓYs

Z _n_`u

Covariant derivative

with ΓYsZ =

CDzZ{ zY{,s + zs{,Y − zYs,{

using the metric zYZ ≡ XYZ , XYsXsZ = δY

Z

Numerical procedure

LT,Y =~T~LY

_n

_`a− _n

_b

_b

_`a= 0 Moving mean-field eq.

XZs ts_n

_`a_b

_`u= vD _b

_`aMoving RPA eq.

Move to the next point

LY + �LY = LY + ÄL,bY

Moving MF eq. to

determine the point: LY

L,bY =

~LY

~T

Tangent vectors (Generators)

Canonical variables and quantization

• Solution

– 1-dimensional state: ξ T

– Tangent vectors: _b

_`aand

_`a

_b

– Fix the scale of Tby making the inertial mass

Xe = _b

_`aXYZ _b

_`a= 1

• Collective Hamiltonian

– KdÇÉÑÑ T, U = C

DUD + 1e(T), 1e T = 1(ξ T )

– Quantization T, U = Qℏ

3D real space representation

X [ fm ]

y [

fm

]

Wen, T.N., 96, 014610 (2017)

Wen, T.N., PRC 94, 054618 (2016).

Wen, Washiyama, Ni, T.N., Acta Phys. Pol.

B Proc. Suppl. 8, 637 (2015)

• 3D space discretized in lattice

• BKN functional

• Moving mean-field eq.: Imaginary-time method

• Moving RPA eq.¡ Finite amplitude method (PRC

76, 024318 (2007) )

At a moment, no pairing

1-dimensional reaction path

extracted from the Hilbert space of

dimension of 104 ~105.

• Reaction path

• After touching

– No bound state, but

– a resonance state in 8Be

Simple case: α + α scattering

α particle�4He� α particle�4He�

8Be: Tangent vectors (generators)

7

3.95

4

4.05

4.1

4.15

4.2

0.6 0.8 1 1.2 1.4

M/m

Mesh Size [fm]

α

FIG. 1. (Color online) Calculated translational mass of the ↵particle in units of nucleon’s mass m, as functions of adoptdmesh size h.

with various mesh sizes h = 0.5 ⇠ 1.4 fm. Note thatthe ground state of the system is a trivial solution of theASCC equation (6). We can clearly identify the threetranslational modes for x, y, and z directions, degener-ated in energy at !com 1 MeV. Using smaller meshsize, the eigenfrequency of the translational motion ap-proaches to zero. There are no low-lying excited states inthe ↵ particle because of its compact and doubly-closedcharacters. The calculated energy of the lowest excitedstate is larger than 20 MeV.

Using Eqs. (19) and (22) with R as the center of mass,we calculate the inertial mass of the translational motionof the ↵ particle. Figure 1 shows the results calculatedwith di↵erent mesh size h of the 3D grid. Since this isthe trivial center-of-mass motion of the total system, thisshould equal the total mass,M = Am with A = 4. As themesh size decreases, the total mass certainly converges tothe value of 4m. In the follwoing, we adopt the mesh sizeh = 0.8 fm.

2. Relative motion of two ↵ particles in 8Be

Figure 2 shows the calculated eigenfrequencies for theground state of 8Be and the two well separated ↵’s atdistance R = 7.2 fm. Since the ground state of 8Beis deformed, there appear the rotational modes of exci-tation as the zero modes, in addition to the three in-dependent modes of the translational motion. Becauseof the axial symmetry of the ground state, the rota-tion about the symmetry axis (z axis) does not ap-pear. In Fig. 2 the calculation produces two rotationalmodes of excitation around 2.8 MeV with large transi-tion matrix element of the K = 1 quadrupole operator,Q2±1 ⌘ R

r2Y2±1(r) †(~r) (~r)d~r. The finite energy ofthese rotational modes comes from the finite mesh sizediscretizing the space. Besides these five zero modes,the lowest mode of excitation turns out to have a sizable

0

5

10

15

R = 7.20 fmR = 3.54 fm

ω [

MeV

]

α + α 8Be

Rel - Q20

Rot - Q21

Rot - Q2-1

Trans - Y

Trans - X

Trans - Z

FIG. 2. (Color online) Calculated eigenfrequencies for theground state of 8Be (left column) and the two well-separated↵’s at distance R = 7.2 fm (right column). The three modesof translational motion and two modes of rotational motionare shown by thin lines, while the thick line indicates theK = 0 quadrupole oscillation.

δρ at G.S.

-6-3 0 3 6

y [fm]

-6 -4 -2 0 2 4 6

z [fm]

-0.021

-0.014

-0.007

0

0.007

0.014

0.021

Den

sity

[fm

-3]

ρ at G.S.

-6-3 0 3 6

y [fm]

-6 -4 -2 0 2 4 6

z [fm]

0

0.04

0.08

0.12

0.16

Den

sity

[fm

-3]

δρ at R = 7.2 fm

-6-3 0 3 6

y [fm]

-6 -4 -2 0 2 4 6

z [fm]

-0.06

-0.03

0

0.03

0.06

Den

sity

[fm

-3]

ρ at R = 7.2 fm

-6-3 0 3 6

y [fm]

-6 -4 -2 0 2 4 6

z [fm]

0

0.04

0.08

0.12

0.16

Den

sity

[fm

-3]

FIG. 3. (Color online) The density distribution ⇢(~r) for 8Bein the upper panels, and the transition density �⇢(~r) of thelowest mode of excitation in the lower panels. The left panelsshow those at the ground state and the right at R = 7.2 fm.Those on the y � z plane are plotted.

transition strength of the K = 0 quadrupole operatorQ20 ⌘ R

r2Y20(r) †(~r) (~r)d~r. This mode correspondsto the elongation of 8Be. The transition density is givenby

�⇢(~r) ⌘ h!| (~r) †(~r)|0i = h0|h

⌦, (~r) †(~r)i

|0i

=

r

2

!

X

i

Pi

(~r)'i

(~r). (41)

The left panels of Fig. 3 show the density profile of 8Beand the transition density �⇢(r) corresponding to the low-est RPA normal mode. We can see an elongated struc-ture along the z direction in the ground state. The lowestmode of excitation corresponds to the change of its elon-gation (�-vibration).

N(2)

�N(2)

Tangent vectors (Generators)

8Be: Collective potential

8

-40

-38

-36

-34

-32

-30

3 3.5 4 4.5 5 5.5 6 6.5 7

V(R

) [M

eV]

R [fm]

ASCC

4e2/R + 2E

α

FIG. 4. (Color online) Potential energy as a function of therelative distance R. The solid (blue) line corresponds to V (R)on the ASCC collective path, while the dashed (red) lineshows 4e2/R+ 2E↵ for reference.

We also perform the same calculation for the state inwhich two ↵ particles are located far away, at the rel-ative distance R = 7.2 fm. In the right panel of Fig.3, we clearly see that the two ↵ particles are well sepa-rated, and the quadrupole mode in fact corresponds tothe translational motion of the ↵ particles in the oppositedirections, namely, the relative motion of two ↵’s. Theexcitation energy almost vanishes for this normal mode(Fig. 2).

B. Results of the ASCC method

In Sec. III A 2, we show that the the lowest quadrupolemode of excitation at the ground state of 8Be may changeits character and lead to the relative motion of two ↵’sat the asymptotic region. We adopt this mode as thegenerators (Q(q), P (q)) of the collective variables (q, p),then, construct the collective path.

1. Collective path, potential, and inertial mass

We successfully derive the collective path {| (q)i; q =0, �q, 2�q, · · · } connecting the ground state of 8Be into thewell-separated two ↵ particles. The inertial mass M(q)is taken as unity and the collective potential is calcu-lated according to Eq. (9). Then, according to Sec. II B,the collective coordinate q is mapped onto the relativedistance R ⌘ h (q)|R| (q)i with Eq. (18). Figure 4shows the obtained potential energy along the ASCCcollective path. As a reference, we also show the pureCoulomb potential between two ↵ particles at distanceR, 4e2/R+2E

↵

, where E↵

is the calculated ground stateenergy of the isolated ↵ particle. Apparently, it asymp-totically approaches the pure Coulomb potential. As two↵’s get closer, the potential starts to deviate from theCoulomb potential at R < 6 fm and finally reaches the

-100

0

100

200

300

400

3.5 4 4.5 5 5.5 6 6.5

ω2 [

MeV

2]

R [fm]

∂2V/∂q

2

ω2

FIG. 5. (Color online) !2 in Eq. (13) and @2V/@q2 of theASCC calculation as a function of relative distance R.

ground state of 8Be. The ground state is at R = 3.54fm, and the top of the Coulumb barrier is at R = 6.6fm. Note that the path is determined self-consistentlywithout any a priori assumption.With this calculated potential, we may check the self-

consistency of the ASCC potential and the eigenfre-quency. If the collective path perfectly follows the di-rection defined by the local generators (Q(p), P (q)) ateach point of q, the second derivative of the potentiald2V/dq2 should coincide with the eigenfrequency !2 ofthe moving RPA equation. The almost perfect agree-ment between these is shown in Fig. 5.For the region of R < 3.5 fm, there exists some discrep-

ancy between d2V/dq2 and !2. In this region, the 8Benucleus has even more compact shapes than the groundstate, then, the coordinate q and R become almost or-thogonal to each other, losing the one-to-one correspon-dence between them. In other words, the states | (q)ichange as q gets smaller, but keep R = h (q)|R| (q)i al-most constant. In addition, the moving RPA frequency !becomes larger than the particle threshold energy, enter-ing in the continuum. Thus, in this region of R < 3.5 fm,the results somewhat depend on the adopted box size.Figure 6 shows the obtained inertial mass M(R) as a

function of R for the scattering between two ↵’s As thetwo ↵’s are far away, the ASCC inertial mass asymp-totically produces the exact reduced mass of 2m. Thismeans that the collective coordinate q becomes parallelto the relative distance R, even though we do not assumeso. At R < 3.54 fm, the value of inertial mass M(R) in-creases. This is due to the decrease of the factor dR/dqin Eq. (19). Making the sytem even more compact thanthe ground state, M(R) rises up drastically, which meansthat the coordinates q and R become almost orthogonal.

2. Phase shift for ↵� ↵ scattering

The ASCC calculation provides us the collectiveHamiltonian along the optimal reaction path. Using this,

Represented by the relative distance RTransformation: T → Ü

1 Ü = 1(T Ü )

R

8Be: Collective inertial mass

Ground(resonance)state

9

2

2.5

3

3.5

4

3 3.5 4 4.5 5 5.5 6 6.5 7

M(R

)/m

R [fm]

FIG. 6. (Color online) Inertial mass in units of the nucleon’smass m for the collective path of ↵ + ↵ $8Be, as a functionof the relative distance R.

-150

-100

-50

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

δ [

deg

]

E [MeV]

L = 0

L = 2

L = 4L = 6

FIG. 7. (Color online) Nuclear phase shift for the scatteringbetween two ↵ particles, as a function of incident energy E.The solid lines indicate the results obtained with the ASCCinertial massM(R), while the dashed lines are calculated withthe constant reduced mass 2m.

we demonstrate the calculation of nuclear phase shift. Weshould take this result in a qualitative sense, because ofa schematic nature of the BKN interaction.

Using the collective potential V (R) and the inertialmass M(R) obtained in the ASCC calculation, the nu-clear phase shift for the angular momentum L at incidentenergy E is calculated in the WKB approximation as [41]

�L

(E) =

Z 1

R0

k(R)dR�Z 1

Rc

kc

(R)dR, (42)

with

k2(R) = 2M(R)

(

E � V (R)��

L+ 12

�2

4mR2

)

,

k2c

(R) = 4m

(

E � 4e2

R�

�

L+ 12

�2

4mR2

)

, (43)

where k(R) and kc

(R) are the wave numbers in theradial motion with and without the nuclear potential.

R0 and Rc are the outer turning points for the po-tentials V (R) and 4e2/R, respectively, i.e. k(R0) =kc

(Rc

) = 0. The centrifugal potential is approximatedas (L + 1/2)2/(2µR2) with the reduced mass µ = 2mand the semiclassical approximation for L(L+ 1).Figure 7 shows the calculated nuclear phase shifts for

the scattering between two ↵’s. The dashed line is calcu-lated with the same potential V (R) but with the constantreduced mass, M(R) ! µ = 2m. We can see the promi-nent increase of the nuclear phase shift caused by thecoordinate-dependent ASCC inertial mass M(R). Weshould remark that the energy of the resonance in 8Be isnot reproduced with the BKN interaction. In fact, thepresent calculation leads to the stable ground state for8Be; E8Be < 2E

↵

. Thus, we should regard this resultas a quatlitative one. Nevertheless, the basic features ofphase shifts for the ↵�↵ scattering are reproduced. Thisdemonstrates the usefulness of the requantization usingthe ASCC calculation.

C. Comparison with other approaches

We compare the present ASCC results with those ob-tained with other approaches: (i) CHF + cranking in-ertia, (ii) CHF + local RPA, and (iii) ATDHF. Weadopt the same model space as the ASCC calculationsfor these calculations. For the constraint operators ofCHF calculation in (i) and (ii), we adopt the K = 0mass quadrupole operator Q20 and the relative distanceR.

1. CHF + cranking inertia

Since 8Be is the simplest system and has a promi-nent ↵ + ↵ structure even at the ground state, thecollective path can be approximated by more conven-tional CHF calculations with a constraint operator aseither Q20 or R. The potential is defined as VCHF(R) =h CHF(R)|H| CHF(R)i. For the inertial mass, the In-glis’s cranking formula is widely used. There are twokinds of cranking formulae: The original formula is de-rived by the adiabatic perturbation, which is given forthe 1D collective motion as

MNPcr (R) = 2

X

m,i

|h'm

(R)|@/@R|'i

(R)i|2em

(R)� ei

(R), (44)

where the single-particle states and energies are definedwith respect to hCHF(R) = hHF[⇢]� �(R)O as

hCHF(R)|'µ

(R)i = eµ

(R))|'µ

(R)i, µ = i,m. (45)

Note that, depending on choice of the constraint oper-ator, O = (Q20, R), we obtain slightly di↵erent |'

i

(R)ieven at the same R.Another formula, which is more frequently used in

many applications and also called the cranking inertial

Reducedmass

Xe(Ü) =~Ü~T

Xe~Ü~T

=~Ü~T

D

ád(Ü) =1

Xe(Ü)

Transformation: T → Ü

ád(Ü) → 2à

α + α scattering (phase shift)

9

2

2.5

3

3.5

4

3 3.5 4 4.5 5 5.5 6 6.5 7

M(R

)/m

R [fm]

FIG. 6. (Color online) Inertial mass in units of the nucleon’smass m for the collective path of ↵ + ↵ $8Be, as a functionof the relative distance R.

-150

-100

-50

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

δ [

deg

]

E [MeV]

L = 0

L = 2

L = 4L = 6

FIG. 7. (Color online) Nuclear phase shift for the scatteringbetween two ↵ particles, as a function of incident energy E.The solid lines indicate the results obtained with the ASCCinertial massM(R), while the dashed lines are calculated withthe constant reduced mass 2m.

we demonstrate the calculation of nuclear phase shift. Weshould take this result in a qualitative sense, because ofa schematic nature of the BKN interaction.

Using the collective potential V (R) and the inertialmass M(R) obtained in the ASCC calculation, the nu-clear phase shift for the angular momentum L at incidentenergy E is calculated in the WKB approximation as [41]

�L

(E) =

Z 1

R0

k(R)dR�Z 1

Rc

kc

(R)dR, (42)

with

k2(R) = 2M(R)

(

E � V (R)��

L+ 12

�2

4mR2

)

,

k2c

(R) = 4m

(

E � 4e2

R�

�

L+ 12

�2

4mR2

)

, (43)

where k(R) and kc

(R) are the wave numbers in theradial motion with and without the nuclear potential.

R0 and Rc are the outer turning points for the po-tentials V (R) and 4e2/R, respectively, i.e. k(R0) =kc

(Rc

) = 0. The centrifugal potential is approximatedas (L + 1/2)2/(2µR2) with the reduced mass µ = 2mand the semiclassical approximation for L(L+ 1).Figure 7 shows the calculated nuclear phase shifts for

the scattering between two ↵’s. The dashed line is calcu-lated with the same potential V (R) but with the constantreduced mass, M(R) ! µ = 2m. We can see the promi-nent increase of the nuclear phase shift caused by thecoordinate-dependent ASCC inertial mass M(R). Weshould remark that the energy of the resonance in 8Be isnot reproduced with the BKN interaction. In fact, thepresent calculation leads to the stable ground state for8Be; E8Be < 2E

↵

. Thus, we should regard this resultas a quatlitative one. Nevertheless, the basic features ofphase shifts for the ↵�↵ scattering are reproduced. Thisdemonstrates the usefulness of the requantization usingthe ASCC calculation.

C. Comparison with other approaches

We compare the present ASCC results with those ob-tained with other approaches: (i) CHF + cranking in-ertia, (ii) CHF + local RPA, and (iii) ATDHF. Weadopt the same model space as the ASCC calculationsfor these calculations. For the constraint operators ofCHF calculation in (i) and (ii), we adopt the K = 0mass quadrupole operator Q20 and the relative distanceR.

1. CHF + cranking inertia

Since 8Be is the simplest system and has a promi-nent ↵ + ↵ structure even at the ground state, thecollective path can be approximated by more conven-tional CHF calculations with a constraint operator aseither Q20 or R. The potential is defined as VCHF(R) =h CHF(R)|H| CHF(R)i. For the inertial mass, the In-glis’s cranking formula is widely used. There are twokinds of cranking formulae: The original formula is de-rived by the adiabatic perturbation, which is given forthe 1D collective motion as

MNPcr (R) = 2

X

m,i

|h'm

(R)|@/@R|'i

(R)i|2em

(R)� ei

(R), (44)

where the single-particle states and energies are definedwith respect to hCHF(R) = hHF[⇢]� �(R)O as

hCHF(R)|'µ

(R)i = eµ

(R))|'µ

(R)i, µ = i,m. (45)

Note that, depending on choice of the constraint oper-ator, O = (Q20, R), we obtain slightly di↵erent |'

i

(R)ieven at the same R.Another formula, which is more frequently used in

many applications and also called the cranking inertial

Nuclearphaseshift

EffectofdynamicalchangeoftheinertialmassDashedline:Constantreducedmass(á Ü → 2à)

16O + α scattering

• Important reaction to synthesize heavy

elements in giant stars

– Alpha reaction

16O4He

20Ne

16O + α to/from 20Ne

Reaction path

20Ne

20Ne: Inertial mass

Reducedmassá Ü → 3.2à

Strong increase in the mass

near the ground state of 20Ne

T

Ü

á(Ü)

á T = 1

Ü

20Ne: Collective potential

-146-144-142-140-138-136-134-132-130

4 4.5 5 5.5 6 6.5 7 7.5 8

V(R

) [M

eV]

R [fm]

16O + α

Quantum tunneling

Alpha reaction¡16O + α

Synthesis of 20Ne

9

pecially near the SD state. Our result shows a peculiarincrease in the inertial mass near the SD local minimum(R = 4.9 fm). On the contrary, the ATDHF result ofRef. [32] even shows a decrease near the ending point atR ≈ 5 fm. In our previous study on α + α →8Be, wehave also found that the ATDHF potential is relativelysimilar to that of the ASCC, while the inertial masses aredifferent.

C. Sub-barrier fusion cross section

The ASCC calculation provides us the collectiveHamiltonian on the optimal reaction path. Using this,we demonstrate the calculation of sub-barrier fusion crosssection for 16O+α→ 20Ne and 16O+16O→32S. We followthe procedure in Ref. [32].Using the collective potential V (R) and the inertial

mass M(R) obtained in the ASCC calculation, the sub-barrier fusion cross section is evaluated with the WKBapproximation. The transmission coefficient for the par-tial wave L at incident energy Ec.m. is given by

TL(Ec.m.) = [1 + exp(2IL)]−1, (29)

with

IL(Ec.m.) =

! b

adR

"2M(R)

×#V (R) +

L(L+ 1)

2µredR2− Ec.m.

$%1/2, (30)

where a and b are the classical turning points on the innerand outer sides of the barrier respectively. The centrifu-gal potential is approximated as L(L+1)/(2µredR2). Thefusion cross section is given by

σ(Ec.m.) =π

2µredEc.m.

&

L

(2L+ 1)TL(Ec.m.). (31)

For identical incident nuclei, Eq. (31) must be modifiedaccording to the proper symmetrization. Only the partialwave with even L contribute to the cross section as

σ(Ec.m.) =π

2µredEc.m.

&

L

[1 + (−)L](2L+ 1)TL(Ec.m.).

(32)

Instead of σ(Ec.m.), one usually refers to the astrophys-ical S factor defined by

S(Ec.m.) = Ec.m.σ(Ec.m.) exp[2πZ1Z2e2/!v], (33)

where v is the relative velocity at R → ∞. The as-trophysical S factor is preferred for sub-barrier fusionbecause it removes the change by tens of orders of mag-nitude present in the cross section due to the trivial pen-etration through the Coulomb barrier. The S factor mayreveal in a more transparent way the influence of the nu-clear structure and dynamics.

0.5

1

2

3

4

5

0.5 1 1.5 2 2.5 3

S fa

ctor

(107 M

eV b

)

16O + α

with reduced masswith ASCC mass

1022

1023

1024

1025

1026

2 4 6 8 10

S fa

ctor

[MeV

b]

Ec. m. [MeV]

16O + 16O

with reduced masswith ASCC mass

FIG. 11. (Color online) The astrophysical S factor for the sub-barrier fusion of 16O+α (upper panel) and 16O+16O (lowerpanel), as a function of incident energy Ec.m.. The solid lineindicates the results obtained with the ASCC inertial massM(R), the dashed lines are calculated with the constant re-duced mass µred.

Figure 11 shows the calculated S factor for the scatter-ing of 16O+α and 16O+16O, respectively. For 16O+16O,the values of the S factor are plotted in log scale. Thedashed line is calculated with the same potential V (R)but with the reduced mass, replacing M(R) by the con-stant value of µred in Eq. (30). Effect of the inertialmass is significant in the deep sub-barrier energy region,especially for the reaction of 16O+16O at Ec.m. < 4 MeV.Because of a schematic nature of the BKN density func-tional, we should regard this result as a qualitative one.Nevertheless, it suggests the significant effect of the iner-tial mass and roughly reproduces basic features of exper-imental S factor for the 16O-16O scattering. This demon-strates the usefulness of the requantization approach us-ing the ASCC collective Hamiltonian.

IV. SUMMARY

Based on the ASCC method we developed a numericalmethod to determine the collective path for the largeamplitude nuclear collective motion. We applied thismethod to the nuclear fusion reactions; 16O+α →20Neand 16O+16O→32S. In the grid representation of the 3Dcoordinate space, the reaction paths, collective poten-tials, and the inertial masses are calculated.

The ASCC collective path smoothly connects the ini-

E [ MeV ]

H ; =1;9 ; ×J(;)

Fusion reaction:

Astrophysical S-factor

EffectofdynamicalchangeoftheinertialmassDashedline:Constantreducedmass(á Ü → 3.2à)

Withá ÜWithã

16O+16O → 32S: Reaction path

Starting from two 16O configuration

-250

-240

-230

-220

4 5 6 7 8 9

E [M

eV]

R [fm]

16O + 16O

16O+16O → 32S: Collective potential

Superdeformed state

Groundstate

16O+16O → 32S: Collective mass

Reducedmass

áå8çé(Ü)

áå8é (Ü)

á(Ü)-250

-240

-230

-220

0 200 400 600 800 1000 1200 1400

E [M

eV]

⟨ r2Y20 ⟩

16O + 16O

ADIABATIC SELF-CONSISTENT COLLECTIVE PATH IN . . . PHYSICAL REVIEW C 96, 014610 (2017)

0.5

1

2

3

4

5

0.5 1 1.5 2 2.5 3S

fact

or (1

07 MeV

b)

16O + α

with reduced masswith ASCC mass

1022

1023

1024

1025

1026

2 4 6 8 10

S fa

ctor

[MeV

b]

Ec. m. [MeV]

16O + 16O

with reduced masswith ASCC mass

FIG. 11. The astrophysical S factor for the subbarrier fusion of16O + α (upper panel) and 16O + 16O (lower panel) as a function ofincident energy Ec.m.. The solid line indicates the results obtainedwith the ASCC inertial mass M(R), the dashed lines are calculatedwith the constant reduced mass µred.

barrier. The S factor may reveal in a more transparent way theinfluence of the nuclear structure and dynamics.

Figure 11 shows the calculated S factor for the scatteringof 16O + α and 16O + 16O, respectively. For 16O + 16O, thevalues of the S factor are plotted on a log scale. The dashedline is calculated with the same potential V (R) but with thereduced mass, replacing M(R) by the constant value of µredin Eq. (30). The effect of the inertial mass is significant in thedeep subbarrier-energy region, especially for the reaction of16O + 16O at Ec.m. < 4 MeV. Because of a schematic nature ofthe BKN density functional, we should regard this result as aqualitative one. Nevertheless, it suggests the significant effectof the inertial mass and roughly reproduces basic featuresof experimental S factor for the 16O-16O scattering. Thisdemonstrates the usefulness of the requantization approachusing the ASCC collective Hamiltonian.

IV. SUMMARY

Based on the ASCC method we developed a numericalmethod to determine the collective path for the large-amplitudenuclear collective motion. We applied this method to thenuclear fusion reactions; 16O + α → 20Ne and 16O + 16O →32S. In the grid representation of the 3D coordinate space, thereaction paths, collective potentials, and the inertial masses arecalculated.

The ASCC collective path smoothly connects the initialstate of 16O + α to the ground state of the fused nucleus 20Ne.It is found the self-consistent collective path is different fromthat of the conventional CHF calculation with the quadrupoleor octupole moment as the constraint. For the reaction of 16O +16O → 32S, we succeed to obtain the 1D reaction path between16O + 16O and a superdeformed state in 32S. The calculatedinertial mass asymptotically coincides with the reduced mass;however, it shows a peculiar increase near equilibrium states,such as the ground state of 20Ne and the superdeformed stateof 32S.

In the present work, we continue to choose the generatorsof the same symmetry type to construct the collective path. Inprinciple we may lift this restriction. For instance, inside thesuperdeformed state of 32S, the Kπ = 0+ quadrupole mode isno longer favored in energy, which may suggest the necessityto change the generator Q of quadrupole type to octupole type.The importance of the octupole shape in this region was alsosuggested in Ref. [31]. The bifurcation of the collective pathis possible in the ASCC and will be a future issue.

From the ASCC results, it is straightforward to constructand quantize the collective Hamiltonian to study the collectivedynamics microscopically. The calculated fusion cross sectionsuggests that the behavior of the inertial mass may have asignificant impact on the fusion probability at deep subbarrierenergies.

Between the superdeformed and triaxial ground states in32S, we cannot find a 1D collective path to connect them.Since we made an approximation neglecting the curvatureterms, the mixture of the rotational NG modes takes place inthe triaxial states. The multidimensional collective subspacemay be necessary, which is beyond the scope of the presentwork. In the present study, the schematic EDF of the BKN isadopted. To make more quantitative discussion and apply themethod to heavier nuclei, it is necessary to use realistic EDFsand include the pairing correlation. These are our future tasks.

ACKNOWLEDGMENTS

This work is supported in part by JSPS KAKENHI GrantNo. 25287065 and by Interdisciplinary Computational ScienceProgram in CCS, University of Tsukuba.

[1] N. Schunck and L. M. Robledo, Rep. Prog. Phys. 79, 116301(2016).

[2] M. Brack, J. Damgaard, A. S. Jensen, H. C. PauIi,V. M. Strutinsky, and C. Y. Wong, Rev. Mod. Phys. 44, 320(1972).

[3] D. M. Brink, M. J. Giannoni, and M. Veneroni, Nucl. Phys. A258, 237 (1976).

[4] F. Villars, Nucl. Phys. A 285, 269 (1977).[5] M. Baranger and M. Vénéroni, Ann. Phys. (NY) 114, 123

(1978).

014610-9

Fusion reaction¡16O + 16O

Effectofdynamicalchangeoftheinertialmasshindersthefusioncrosssectionby2ordersofmagnitude.

Withá Ü

Withã

Summary

• Missing correlations in nuclear density

functional

– Correlations associated with low-energy

collective motion

• Re-quantize a specific mode of collective

motion

– Derive the slow collective motion

– Quantize the collective Hamiltonian

– Applicable to nuclear structure and reaction

|v}

• T�,�lzJNy�`

–45P��������

–4�(��������

–��4�(d�)�0����5P

• ���rT�,�lz 3�wh~st�'q�i

– %7W.z-�

–[�RCVzT��