YONUPA “Young Nuclear and Particle Physicist Group of Japan”
Summer School 2021: August 6-10, 2021 (via Zoom)
8/7 (Sat) 13:00-14:30 Sekizawa - Lecture 2

時間依存密度汎関数法で探る原子核ダイナミクス:
原子核反応から超流動現象，中性子星まで

Kazuyuki Sekizawa

Department of Physics, School of Science
Tokyo Institute of Technology

Lecture Plan

Theory (8/7 10:00-11:30)#1. An introduction to microscopic mean-field approaches and (TD)DFT

Nuclear reactions (8/7 13:00-14:30)#2. Recent advances in microscopic approaches for heavy-ion reactions
Superheavy element synthesis & deep-inelastic collisions

Neutron stars (8/8 10:00-11:30)#3. Neutron-star “glitch” and neutron superfluid
Dynamics of quantized vortices in the inner crust of neutron stars

Lecture Plan

Theory (8/7 10:00-11:30)#1. An introduction to microscopic mean-field approaches and (TD)DFT

Nuclear reactions (8/7 13:00-14:30)#2. Recent advances in microscopic approaches for heavy-ion reactions
Superheavy element synthesis & deep-inelastic collisions

Neutron stars (8/8 10:00-11:30)#3. Neutron-star “glitch” and neutron superfluid
Dynamics of quantized vortices in the inner crust of neutron stars

Voyage towards the limit of existence

The continent of stability has been well explored..

The Great Wall of China by Stacy Funderburke @shutterstockNow we are sailing towards the edge of the nuclear landscape..
Stable nuclei: 288
Experiment: ~3300
Theory: ~7000-10000 drip lines shell structure deformation
 skin, halo

 nuclear matter properties
 nucleosynthesis ...
“Flat Earth” by iStock; “Fanciful view of ship sailing over edge of Earth” by Georgia Studies Images

How can we create yet-unknown neutron-rich nuclei?
Z=114

Proton number

Theory: ~7,000
Expt.: ~3,300

New magic numbers?

Impact on r-process?
Heaviest element?
Island of stability?

Neutron number
M. Thoennessen, Rep. Prog. Phys. 76, 056301 (2013)

Multinucleon transfer (MNT) and quasi-fission (QF) may be useful
Production cross section for N=126 isotones in 136Xe+208Pb

101

N=126

σ (mb)

100
MNT: 136Xe+208Pb(Langevin calc.)

10-1

10-2
10-3

Fragmentation (Expt.):
208Pb (1GeV/A) +Be

10-4
10-5
76 77 78 79 80 81 82
Proton number

Langevin: V.I. Zagrebaev and W. Greiner, PRC83(2011)044618
Fragmentation: T. Kurtukian-Nieto et al., PRC89(2014)024616

Multinucleon transfer (MNT) and quasi-fission (QF) may be useful
Production cross section for N=126 isotones in 136Xe+198Pt
Y.X. Watanabe et al., PRL115(2015)172503

σ (mb)

N=126

Fragmentation (Expt.):
208Pb (1GeV/A) +Be

MNT: 136Xe+198Pt(8 MeV/A)

Proton number
Fragmentation: T. Kurtukian-Nieto et al., PRC89(2014)024616

Multinucleon transfer (MNT) and quasi-fission (QF) may be useful
Production cross section for primary products in 238U+248Cm
238U 248Cm

103

Transcurium
Z >96

σ (mb)

102
101

U

238

248

Cm

Pb

208

278

SHE

100
10-1

208Pb

10-2
210 220 230 240 250 260 270 280 290

Mass number
Langevin calculation by: V.I. Zagrebaev and W. Greiner, PRC87(2013)034608

Let’s see how TDDFT works in practice“TDHF” - past and present

For reviews, see: J.W. Negele, Rev. Mod. Phys. 54, 913 (1982); C. Simenel, Eur. Phys. J. A 48, 152 (2012); T.
Nakatsukasa et al., Rev. Mod. Phys. 88, 045004 (2016); C. Simenel and A.S. Umar, Prog. Part. Nucl. Phys. 103, 19(2018); P.D. Stevenson and M.C. Barton, Prog. Part. Nucl. Phys. 104, 142 (2019); K.S., Front. Phys. 7, 20 (2019)✓ Various symmetry restrictions✓ Simplified interactions✓ Full 3D✓ Modern interactions (or EDF)

✓ Limited computational power✓ Accurate numerical methods
and use of supercomputers

84Kr+209Bi, E = 850 MeV
lab

238U+238U, E = 900 MeV
lab

K.T.R. Davis, K.R.S. Devi, and M.R. Strayer, PRL44(1980)23

C. Golabek and C. Simenel, PRL103(2009)042701

Remarks on TDDFT (or TDHF, TDEDF)✓ There is no adjustable parameter on reaction dynamics
Parameters in an EDF are determined to reproduce known nuclear properties
Effective interaction

Proton number: TDKS eq.

Energy Density Functional (EDF)

N, Z

Neutron number
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

How to describe heavy-ion reactions
Once an EDF is given, there are no empirical parameters
1. Prepare ground states of projectile and target nuclei
KS equations

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

How to describe heavy-ion reactions
Once an EDF is given, there are no empirical parameters
1. Prepare ground states of projectile and target nuclei
KS equations

2. Put them into a larger box
3. Give relative momentum, and let them collide
TDKS equations

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

by Physics Fun @PhysicsVideo_ on Twitter

Complex
Quantum Mechanical
Processes

TDDFT
Distribution ofreaction outcomes1. Multinucleon transfer reactions

58Ni+208Pb at E =328.4 MeV
lab

K.S. and K. Yabana, PRC88(2013)014614

TDDFT predicts two distinct transfer mechanisms
b=1.6 fm

b=4 fmNeck formation/breaking

Fast charge equilibration

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

Neutrons

ΔN

Protonsb (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

ΔN

中性子
陽子

b (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

Charge equilibration process

中性子
陽子

Transfer towards opposite directions

Ni

ΔN

Pb

n
p

b (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

ΔN

中性子
陽子

b (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

Transfer towards the same direction

ΔN

中性子
陽子

Neck formation/breakingb (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

Transfer towards the same direction

ΔN

中性子
陽子

Neck formation/breakingnpb (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Average number of transferred nucleons vs. impact parameter
There appear two transfer mechanisms

Neck formation/breaking
Transfer towards the same direction

Charge equilibration process
Transfer towards opposite directions

Ni

Pb

n
p

n
p

Small
K. Sekizawa

Impact parameter
Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

LargeSat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Particle-number projection method
V: Particle-number projection operator

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)

K.S. and K. Yabana, PRC88(2013)014614

Particle-number projection method
V: Particle-number projection operator

Protons

Pn

Neutronsb (fm)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactionsb (fm)
Sat., Aug. 7, 2021

TDDFT results:

58Ni+208Pb(Elab=328.4 MeV)(TDDFT+GEMINI: K.S., PRC96(2017)014615)
K.S. and K. Yabana, PRC88(2013)014614

Transfer cross sections: TDDFT vs. Expt.
Comparable to measurements, disagreements for many-nucleon transfer

Expt.
TDDFT pre-evap.
TDDFT post-evap.
(-5p; V)(-4p; Cr)(-3p; Mn)(-2p; Fe)(-1p; Co)

(0p; Ni)σ (mb)(-6p; Ti)

Neutron number
Expt.: L. Corradi et al., PRC66(2002)024606
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

2. Quasifission processes

Towards synthesis of superheavy nuclei

Quasifission process
A fast (~10-21-10-20 sec) fission process before compound nucleus formation (fusion)

CaptureFusionQuasifission

CN

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Towards synthesis of superheavy nuclei

Quasifission process
A fast (~10-21-10-20 sec) fission process before compound nucleus formation (fusion)

Capture✓ QF is the predominant
process that hinders
SHE synthesis

FusionQuasifission

CN

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

64Ni+238U at E =390 MeV
lab

K.S. and K. Yabana, PRC93(2016)054616

Quasifission dynamics in TDDFT
Tip collision

Side collision

Shell effects of 208Pb

More mass-symmetric

24 Neutrons
12 Protons

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

32 Neutrons
18 Protons

Sat., Aug. 7, 2021

64Ni+238U at E =390 MeV
lab

K.S. and K. Yabana, PRC93(2016)054616

TDDFT provides quantitative description of quasifission dynamics

TKE-A distribution: Comparison with experimental data
Experiment

TKE (MeV)

TDDFT

A
Expt.: E.M. Kozulin et al., PLB686(2010)227

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

However, TDDFT can not properly describe
the distribution of observables

Stochastic extensions

by Physics Fun @PhysicsVideo_ on Twitter

Complex
Quantum Mechanical
Processes

TDDFT
Distribution ofreaction outcomes

SMF

Stochastic Mean-Field (SMF) approach

S. Ayik, Phys. Lett. B658, 174 (2008)
D. Lacroix and S. Ayik, Eur. Phys. J. A 50, 95 (2014)

Observables are evaluatedby an ensemble average

Sakir Ayik
(Tennessee Tech.)

TDDFT✓ In the small fluctuation limit, SMF approachanalytically reproduces the TDRPA formula: Stochasticity in the density matrix
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

SMF approach: Quantal diffusion description

K.S. and S. Ayik, PRC102(2020)014620

The widths develop in time by nucleon exchanges through the window*Drift and diffusion coefficients (ν & D) are determined by TDHF

64Ni+208Pb (Ec.m.=268 MeV)

K.S. and S. Ayik, PRC102(2020)014620

The SMF approach predicts a much wider distribution*PN,Z(b): transfer probability
Particle-number projection for TDHF
Correlated Gaussian for SMF

Primary production cross sections for lighter fragments

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

K.S. and S. Ayik, PRC102(2020)014620

Secondary production cross sections for heavier fragments
Expt.
SMF
TDDFT

Expt.: W. Królas et al., Nucl. Phys. A 724, 289 (2010)

At the frontiers in nuclear physics:

3. Synthesis of superheavy elements→ also relevant to chemistry

Nucleosynthesis at accelerator facilities

Collide!

Fusion!

The world-leading factoryof unstable nuclei!
RI Beam Factory (RIBF) @RIKENSuperconducting Ring Cyclotron (SRC)

The periodic table is growing!
Synthesis of “superheavy element” is the most challenging subject in nuclear physics

The Periodic Table

November 2016
Four new elements were added,
completing the seventh row!!
Nh

Mc

Ts

Og

113: Nh (Nihonium)
115: Ms (Moscovium)
117: Ts (Tennessine)
118: Og (Oganesson)

Nihonium is the only one
discovered by Asian country:
The element 113

Nov. 30, 2016: Name confirmed
June 8, 2016: Name announced
Dec. 31, 2015: Naming rights given

2019 was the 150th anniversary of
the Periodic Table!

Dec. 5, 2019@Tokyo Prince HotelBismuth

Zinc

Collide!

Fusion!

Nihonium

A long way to synthesize
Nihonium
Atomic No. 83

Sep. 2003
expt. started~100 days irradiation~100 days irradiation

July 23, 2004

~ 2.4×1012
nuclei per second

Atomic No. 30~360 days irradiation

April 2, 2005
The 2nd event!

The 1st event!

Atomic No. 113

Aug. 12, 2012
The 3rd event!

Not confident enough as
Bh → Db α-decay wasmeasured only once inthe past

fission

1/3 → fission
2/3 → α-decay

fissionBoth Db and Lr decayed
via an alpha emission→ consistent with known fact

The north-east part of the nuclear map*108 sec ~ 3 years

Proton number

Half-lives

Neutron number
Yuri Oganessian, Pure Appl. Chem. 78, 889 (2006)

The north-east part of the nuclear map
We do need theoretical predictions!
Z = 119, 120, ...
Cold fusion

Light-ion induced

Neutron capture

Hot fusion

The next is: 119 or 120
TDDFT simulation for SHE synthesis
on a supercomputer
51V+248Cm → 119?

Collaborative works are in progress with Kyoto U., Kindai U., and Kyushu U.
cf. K.S. and K. Hagino, Phys. Rev. C 99, 051602(R) (2019)

Towards synthesis of superheavy nuclei

Quasifission process
A fast (~10-21-10-20 sec) fission process before compound nucleus formation (fusion)

CaptureFusionQuasifission

CN

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Towards synthesis of superheavy nuclei

Quasifission process
A fast (~10-21-10-20 sec) fission process before compound nucleus formation (fusion)

Capture✓ The rare process of CN formation
can not be described in TDDFT

Fusion✓ QF is the predominant
process that hinders
SHE synthesis

Quasifission

CN

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

K.S. and K. Hagino, Phys. Rev. C 99, 051602(R) (2019)
Rapid Comm. with Editors’ Suggestion

TDDFT + Langevin approach: Idea

Kouichi Hagino(Kyoto U.)

Input: Distance of closest approach➢ Analytical formula can be derived:

We use fusion-by-diffusion model to describe
the “up-hill diffusion” over the inner barrier
Fusion-by-diffusion model:
W.J. Świątecki, K. Siwek-Wilczyńska, and J. Wilczyński,Acta Phys. Pol. B 34(2003)2049; PRC71(2005)014602
K. Hagino, PRC98(2018)014607

Formation probability of Z=120
K.S. and K. Hagino, Phys. Rev. C 99, 051602(R) (2019)

Magicity of 48Ca affects the survival probability via the lower excitation energy
TDDFT FBD

Very similar
for all systems

Thus, PCN isalso similar

Note:
Survival probability also depends
on the fission barrier height!

(higher E*, larger PCN)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

How much does pairing affect the reaction dynamics?

4. Solitonic excitations in heavy-ion reactions

TDSLDA (Time-Dependent Superfluid Local Density Approximation)
TDSLDA: TDDFT with local treatment of pairing
Kohn-Sham scheme is extended for non-interacting quasiparticles➢ TDSLDA equations (formally equivalent to TDHFB or TD-BdG equations): number density: s.p. Hamiltonian: anomalous density: pairing field: current

A large number (104-106) of 3D coupled non-linear PDEs have to be solved!!
# of qp orbitals ~ # of grid points
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

What we have investigated
3D, TDSLDA calculations (Fayans w/o LS) predicts
novel phenomena associated with solitonic excitations!!

Q. What happens when two superfluid nuclei with different phases collide??

?
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDSLDA results: 240Pu+240Pu head-on collisions (E/VBass=1.1)
The phase difference changes kinetic energy of the fragments

Why??

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDSLDA results: 240Pu+240Pu head-on collisions (E/VBass=1.1)
The phase difference changes kinetic energy of the fragments
Note:
It cannot be explained by Josephson effect!

Why??

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Solitonic excitations
Additional energy is required to attach two superfluids with different phases➢ The additional energy (derived from Ginzburg-Landau theory)*It does not depend on the absolute value of the pairing!
e.g.) S=πR2, L~R=6 fm, ns=0.08 fm-3 → E~30 MeV

S: Attaching area
L: Length scale over which the phase varies
ns: Superfluid density

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Solitonic excitations
Additional energy is required to attach two superfluids with different phases➢ The additional energy (derived from Ginzburg-Landau theory)
240Pu+240Pu (E/V
Bass=1.1),*It does not depend on the absolute value of the pairing!
e.g.) S=πR2, L~R=6 fm, ns=0.08 fm-3 → E~30 MeV

S: Attaching area
L: Length
scalefieldover which
the phase varies
Pairing
is vanishing!
ns: Superfluid density(due to phase discontinuity)

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDSLDA results: 90Zr+90Zr head-on collisions
Fusion reaction is suppressed by the phase difference

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDSLDA results: 90Zr+90Zr head-on collisions
Fusion reaction is suppressed by the phase difference

Non-Fusion

FusionP. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

TDSLDA results: 90Zr+90Zr head-on collisions
Fusion reaction is suppressed by the phase difference

Efusion (MeV)*Efusion: the lowest energy at which fusion reaction is observed

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Summary of the findings

When two superfluid nuclei with different phases collide
solitonic excitations might be induced

solitonic
excitations

P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Summary of the findings

When two superfluid nuclei with different phases collide
solitonic excitations might be induced

It may affect:

solitonic
excitations

V(R) TKE (~10-30 MeV) Fusion dynamics
 Neck formation
 Contact timeR
K. Sekizawa

 Scattering angle
P. Magierski, K.S., G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017)
Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

SummarySummary of Lecture 2
Takeaway message✓ TDDFT can properly describe the main reaction outcomes in low-energy
heavy-ion reactions, and may be further improved with extended techniques

Main reaction outcomes

TDDFT
Average number of nucleons,
TKE, scattering angles,
contact times, ...

Fluctuations/correlations

Further extensions

SMF (or TDRPA)

e.g. TDDFT+Langevin

Width of mass, charge distributions,correlation between n/p transfers,...

Providing input information forother theoretical approaches

K. Sekizawa

Lecture 2: Recent advances in microscopic approaches for heavy-ion reactions

Sat., Aug. 7, 2021

Lecture Plan

Theory (8/7 10:00-11:30)#1. An introduction to microscopic mean-field approaches and (TD)DFT

Nuclear reactions (8/7 13:00-14:30)#2. Recent advances in microscopic approaches for heavy-ion reactions
Superheavy element synthesis & deep-inelastic collisions

Neutron stars (8/8 10:00-11:30)#3. Neutron-star “glitch” and neutron superfluid
Dynamics of quantized vortices in the inner crust of neutron stars

Kazuyuki Sekizawa
Associate Professor
Department of Physics, School of Science
Tokyo Institute of Technology
2-12-1 O-Okayama, Meguro, Tokyo 152-8551, Japansekizawa @ phys.titech.ac.jp
http://sekizawa.fizyka.pw.edu.pl/english/