密度汎関数理論による核構造と核反応

Takashi NAKATSUKASA （中務 孝）

Theoretical Nuclear Physics Laboratory RIKEN Nishina Center

•Walter Kohn’s biography •Bohr-Mottelson ↔ Kohn-Sham • 最近の発展 • 時間依存密度汎関数理論による光核反応断面積 2009.8.11-13 ＫＥＫ理論センター研究会「原子核・ハドロン物理」

Walter Kohn

• Nobel Prize in Chemistry 1998 • Harvard (Van Vleck, Schwinger) – “While I did not yet know in what subfield of physics I wanted to do my thesis, I was sure it would not be in solid state physics.” – Green's function variational method for low-energy neutron-deuteron scattering (→ Failed?) – QED, Field theory of nucleons and mesons (→ Feel completely useless) • Polaroid Laboratory – charged particles falling on a photographic plate lead to a photographic image (→ Solid-state physics, Van Vleck) – “Since you are familiar with solid state physics, ..." • Copenhagen – No one , including Niels Bohr, had even heard the expression "Solid State Physics“.

– Very exciting work was going on in Copenhagen, which eventually led to the great "Collective Model of the Nucleus" of A. Bohr and B. Mottelson, both of whom had become close friends. • École Normale Supérieure (Nozières) – I knew that there was a 1-to-1 correspondence between a weak perturbing potential δv(r) and the corresponding small change δn(r) of the density distribution. – It seemed such a remarkable result that I did not trust myself.

Nucleonic single-particle motion in nucleus

Neutron Separation energy Neutron # N Bohr & Mottelson, Nuclear Structure Vol.1

Mayer Jensen’s Shell Model

Harmonic oscillator potential + spin-orbit force

V

(

r

)  1 2

M

 2

r

2 

v ll

 2 

v ls

  

s

 → Correct magic numbers.

Nucleons as “free gas” in the potential.

Nucleus is gas?

 free 

R

0

Liquid-drop Model

Binding energy B/A ≈ 8 MeV Density ρ ≈ 0.14 fm -3 d ≈ 2 fm Bethe-Weizs äcker mass formula

B

(

N

,

Z

) 

a V A



a S A

2 / 3 

a sym

(

N



Z

) 2

A



a C Z

2

A

1 / 3   (

A

) Nuclear fission

x



E C

2

E S

~

Z

2

A

Bohr & Mottelson Nuclear Structure Vol.1

Nucleus as a quantum liquid • Classical vs Quantum

– Strength of interaction vs Zero-point kinetic energy

V

0

vs

 2

2

Mc

2

c

: Length scale of the interactio n

V

0

: Energy scale of the interactio n

Nuclear force vs molecular force

Bohr, Mottelson, Nucl. Str. Vol.1 < T > « <– V > < T > ≈ < –V > Crystallized at low temperature Classical → MD Liquid at low temperature Quantum

Collective (Unified) Model

by Bohr & Mottelson • Nucleons are

independently

moving in a potential that

slowly

changes.

– Collective motion ↔ oscillation/rotation of the potential.

– The fluctuation of the potential changes the nucleonic single-particle motion.

H



H

vib 

H

part 

H

coupl

H

vib      1 2

B

  2   1 2

C

  2    In modern approaches, microscopic construction

H H

part coupl   

i



i

 

p

2 

V i

2

M



r

i

 ; 

V

0   

i

   

V

0  

i

   

i k

(

r i

)    * 

Y

 ( 

i

, 

i

)

Time-dependent density functional theory

Hohenberg-Kohn, Kohn-Sham, Runge-Gross • “Nucleons” are

independently

moving in a potential that depends on nucleon densities – Collective motion ↔ Density oscillation/rotation – The density oscillation induces oscillation of the potential that affects the nucleonic motion.

• ( 見かけ上 ) Identical to the time-dependent mean-field theory

i

 

t



i

(

r

,

t

) 

p

2 2

M



V



r

;  (

r

,

t

)  

i

(

r

,

t

)  (

r

,

t

)  

i



i

(

r

,

t

) 2 密度の表現（ Kohn-Sham 軌道）

全核種を対象に研究できる。“原理的”には exact たり得る。

密度汎関数理論によっ て記述しようとするエネ ルギー範囲

GeV ~ keV

Hohenberg-Kohn Theorem

The theorem tells us that Density ρ(r) determines

v

(

r

) , Hohenberg & Kohn (1964) except for arbitrary choice of zero point.

A system with a one-body potential

v H v



v gs



v E gs



v gs H v

 

w

( 

r i

,

r



j

) 

i

  

j k w

( 

r i

,

r



j

, 

r k

)  

i v

 

i

Existence of one-to-one mapping ：

v

 

v gs

 

v

(

r

 ) Strictly speaking, one-to-one or one-to-none v-representative

Kohn-Sham Scheme: Ground state

Real interacting system

V

(

r

 ) TD state 

V

TD density  Virtual non-interacting system

V s

(

r

 ) TD state 

S

TD density 

Kohn-Sham theory

Since we know the ground state of the non-interacting system (Slater det.), we obtain the exact density as  

i N

  1 

i

2 solving the Kohn-Sham (KS) equation     2 2

m

 2 

v s

[  ](

r

)   

i

(

r

)  

i



i

(

r

)

v s

[  ](

r

)  

E

 (

r

) Assuming non-interacting v-representability

基底状態に関する計算

• • • 系統的計算が汎用的に可能に – Ab-inito 計算の助けを借りて、 Energy functional を最適 化（目標：質量誤差 500 keV 以下） 偶核の計算 – （１００テラ級のＰＣクラスターがあれば）、１，２時間で偶 偶核全種の結果を出せる汎用コード 奇核の計算 – Odd-even mass difference は実験をよく再現 channel の functional には依存 ) (Pairing – Filling approx.

の正当性 – スピン・パリティを正確に予言することは未だ困難な課題

Jaguar Cray XT4 at ORNL No. 2 on Top500 • 11,706 processor nodes • Each compute/service node contains 2.6 GHz dual-core AMD Opteron processor and 4 GB/8 GB of memory • Peak performance of over 119 Teraflops • 250 Teraflops after Dec.'07 upgrade • 600 TB of scratch disk space

励起状態に関する計算

Collective Hamiltonian in 5-dim. quadrupole collective coordinates constructed by the constrained HFB calculation

coll

  2 2

k

3   1

I

ˆ

k

2

J k

  2 2 m, n   0 and 2

D

 1 / 2  

a m D

1 / 2  

mn

 1  

a n



V



a

0 ,

a

2   

V



a

0 ,

a

2  a 0   cos  a 2   sin  J k (a 0 , a 2 ): moment of inertia → « Thouless-Valatin » B mn (a 0 , a 2 ): collective mass (vibration) D(a 0 , a 2 ): metric D ( a 0 , a 2 )  k   3 1 , J k ( a → 0 , V ( a 0 , a 2 )   q  q a « Cranking » 2 ) det( B )

Gogny-HFB

 V ( a 0 , a 2 )  ZPE ( rot .

 vib .) ZPE pot neglected

Delaroche et al, 2009 Pack Forest Meeting

Time-dependent density functional theory

(3D lattice simulation for Skyrme functionals) Mostly the functional is local in density →Appropriate for coordinate-space representation Kinetic energy, current densities, etc. are estimated with the finite difference method

Time-dependen Kohn-Sham Scheme

Real interacting system

V

(

r

 ,

t

) TD state  (

t

)

V

TD density  Virtual non-interacting system

V s

(

r

 ,

t

) TD state  (

t

)

S

TD density 

Time-dependent Kohn-Sham theory

Assuming non-interacting v-representability  

i N

  1 

i

2 Time-dependent Kohn-Sham (TDKS) equation

i

 

t



i

(

r

,

t

)      2 2

m

 2 

v s

[  ](

r

,

t

)   

i

(

r

,

t

)

v s

[  ](

r

,

t

)  

S

[  ]  (

r

,

t

)

S

[  ] 

S

[  ]  

t

0

t

1 

D

[  ](

t

)

i

 

t



T



D

[  ](

t

) Solving the TDKS equation, in principle, we can obtain the exact time evolution of many-body systems.

The functional depends on ρ(

r

,

t

） and the initial state Ψ 0 .

Skyrme TDDFT in real space Time-dependent Kohn-Sham equation

i

 

t



i

(

r

 ,

t

)  

h

KS [  ,  ,

j

,

s

, 

J

](

t

) 

V

ex

t



i

   (

t

)  

i

(

r

 ,

t

) 3D space is discretized in lattice Single-particle orbital: 

i

(

r

,

t

)  { 

i

(

r

k

,

t n

)}

n k

  1 , 1 , 

Mt



Mr

,

i

 1 ,  ,

N N:

Number of particles

Mr

: Number of mesh points

Mt

: Number of time slices Spatial mesh size is about 1 fm.

Time step is about 0.2 fm/c Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301 X [ fm ]

Real-time calculation of response functions

1. Weak instantaneous external perturbation

V

ext (

t

)  

F

ˆ  (

t

)  (

t

) 2. Calculate time evolution of  (

t

)  (

t

) 3. Fourier transform to energy domain

dB

(  ;

d

 )   1  Im   (

t

)

F

ˆ  (

t

)

e i



t dt

 (

t

)

dB

(  ;

d



F

ˆ ) ω [ MeV ]

Neutrons 

n

(

t

)

 

n

(

t

)

   0

n

16 O Time-dep. transition density δρ> 0 δρ< 0 

p

(

t

)  

p

(

t

)    0

p

Protons

16 O 18 O Prolate 10 20 30 E x [ MeV ] 40

Finite Amplitude Method

T.N., Inakura, Yabana, PRC76 (2007) 024318.

A method to avoid the explicit calculation of the residual fields (interactions)  

X i

(  )

Y i

(  )    

h

0  

i



Y i

(  )

X i

(  ) 

h

0  

i

   

i

 

h

(  )  

h

(  ) 

V

ext 

V

ext (  )  

i

(  ) 

Q

ˆ (1) Residual fields can be estimated by the finite difference method: 

h

(  )  1  

h

  ' ,  

i

 

i

 

X i

(  ) ,  

h

0  

i

'  

i

 

Y i

(  ) Starting from initial amplitudes X (0) and Y (0) , one can use an iterative method to solve eq. (1).

Programming of the RPA code becomes very much trivial, because we only need calculation of the single-particle potential, with different bras and kets .

Skyrme FAM in 3D real space Linear response equations    *

X Y i i

( (   ; ;

r



r

 ) )    

h

0

h

0    

i i

 

Y X i i

( (   ; ;

r



r

 ) )   3D space is discretized in lattice F & B amplitudes:  

h

(   

h

 (  ) ) 

V

ext 

V

 ext

X i

(

r

,  )  {

X i

(

r

k

, 

n

)}

n k

  1 1 , ,  

ME Mr

, (  (  )  

i

)  

i

(

r

 ) (

r

 )

i

 1 ,  ,

N

z

N:

Number of particles

Mr

: Number of mesh points

ME

: Number of energy points y Spatial mesh size is about 0.8 fm.

Energy mesh size is about 0.3 MeV Inakura, Nakatsukasa, Yabana, arXive:0906.5239

x

Electric dipole responses

Finite amplitude method to Skyrme-HF+RPA

Inakura, Nakatsukasa, Yabana, arXive:0906.5239

•

SkM*

interaction •

3D mesh

• R box = 15 fm

Low-energy strength

Low-lying strengths

He Be C O Ne Mg Si S Ar Ca Ti Cr Fe

Low-energy strengths quickly rise up beyond N=14, 28

PDR: impact on the r-process

S. Goriely, Phys. Lett.

B436

, 10.

24 Mg Prolate 26 Mg Triaxial 10 20 E x [ MeV ] 30 40 10 20 E x [ MeV ] 30 40

28 Si Oblate 30 Si Oblate 10 20 E x [ MeV ] 30 40 10 20 E x [ MeV ] 30 40

40 Ar Oblate 10 20 30 E x [ MeV ] 40

44 Ca Prolate 40 Ca 48 Ca 10 20 E x 30 [ MeV ] 40 10 20 E x [ MeV ] 30 10 20 E x 30 [ MeV ] 40

Cal. vs. Exp.

Summary 原子核密度汎関数理論の現状 • 核質量（束縛エネルギー）については、実験との誤差を解 消するため、 Kohn-Sham スキームでは組み入れられていな い相関エネルギーの計算法を開発中 • 効率の高い計算コードの開発 • 励起状態・遷移強度の記述については、ＴＤＤＦＴの線形 応答に基づく記述が標準的： 変形（Ｑ）ＲＰＡコード • 有限振幅法によるアプローチ • ＢＣＳ形式による実時間法（ → 江幡さんの講演） • 調和近似を超えた扱い • ＴＤＤＦＴに準拠し、断熱型自己無撞着集団座標法に基 づく大振幅集団運動 （ → 日野原さんの講演）