素核宇宙融合 レクチャーシリーズ

第4回「原子核殻模型の基礎と応用」
阿部 喬（東大CNS）
京大基研
2012年1月11,12日
1目次• 核子の一粒子運動と原子核での殻構造• 閉殻を仮定する（芯のある）殻模型計算の基礎• 閉殻を仮定しない（芯のない）殻模型による第一原理計算の概要• モンテカルロ殻模型

2

stSummary of 1 lecture• Nucleon: Single‐particle motion• Independent Particle ModelHO (central) potential + LS splitting‐> Mayer‐Jensen’s magic #3

Bohr ‐Mottelson

Single‐particle orbits

WS (HO) central potential + spin‐orbit interaction

4

Schematic picture of the single‐particle potential
WS (HO) central potential + spin‐orbit interactionShell gap

Shell gap

Shell gap

0f7/2
0d3/2
1s1/20d5/220
sd‐shellSpin‐orbit splitting

8Magic number

0p1/2
0p3/2p‐shell

0s1/2s‐shell2

Closed shell(Closed core)‐50 MeV
NLJ

Single‐particle orbit
5

Conventional shell model• (Conventional) Shell model: Interacting shell model w/ a core‐> Independent particle model + Configuration‐mixing

Single‐particle orbits

Shell gap
Shell gap

Model space(Valence space)
Closed core
(Inert core)6

Shell‐Model Hamiltonian: Single‐particle energy: Two‐body matrix element (TBME)• w/ some assumptions/approximations

7

Image of the shell‐model Hamiltonian• Hamiltonian:• Shell‐model Hamiltonian:

One‐body (mean) field (一体場)Residual interaction (残留相互作用；摂動項)
8Shell‐model Hamiltonian• Shell‐model Hamiltonian(previous slide)9

Shell‐Model Hamiltonian: Single‐particle energy: Two‐body matrix element (TBME)• w/ some assumptions/approximations

10

Procedure
1.
2.
3.
4.

Set the model space (valence space)
Get the single‐particle energies (SPEs) from the exp. values
Compute the two‐body matrix elements (TBMEs)
Diagonalize the Hamiltonian matrix (to get the energyeigenvalues & eigenfunctions) ‐> Configuration mixing
5. Calculate the other obs. (w/ the w.f. obtained above)11

Model space (single‐particle states)

12

Bohr ‐Mottelson

Single‐particle orbits
NLJ

N = 2n + l (+ 1)= 1, 2, 3, …
(= 0, 1, 2, …)

J=l+s

Orbital Angular Momentum

Symbol

l=0s

l=1

p

l=2

d

l=3

f

l=4

g

l=5h

l=6i

13

Model space for conventional shell model
50

28
2050

2820

Model Space(Valence Space)

80g9/2
1p1/2
0f5/2
1p3/2
0f7/2
0d3/2
1s1/2
0d5/2

2

sd‐shell(1 major shell)

8
0p1/2

Closed Core(Inert Core)

pf‐shell

20p3/20s1/2

p‐shells‐shell14

Separation energy• Separation energy: minimum energy to take a neutron/proton out

Shell gap

Separation energy
sd‐shell

Shell gap

Shell gap Separation energy
Shell gap

p‐shell
Shell gaps‐shell

Shell gap

• Decrease suddenly after crossing the shell gap
15

Configuration mixing

16

Configuration mixing• Configuration: pattern of occupation

20

20

Model Space(Valence Space)

8Configuration Mixing(thru. the two‐body int.) 20
Model Space(Valence Space)

8++.
..

0d3/2
1s1/2
0d5/2

sd‐shell(1 major shell)

0d3/2
1s1/2
0d5/2

sd‐shell(1 major shell)

820

817

Many‐body wave function• Single‐particle Hamiltonian:• Eigenvalue Equation:
Single‐particle w.f.

Single‐particle energy• Many‐body wave function:• Correlations are included by the configuration mixing.
18

Basis state• Basis state (function):

Combination:

一粒子波動関数の積であるSlater 行
列式はJをよい量子数としてもたない（異なるJの成分が混ざる）

Conservation of M:
Possible configuration:の中でMを満たすもの:

19

Eigenvalue Equation• Many‐body wave function:: #(basis)• Eigenvalue equation:

20

Simultaneous Equation• Nb‐dimentional linear simultaneous equations:• Diagonalization

21

Hamiltonian matrix• Hamiltonian matrix is decomposed into sub matrices
belonging to each value of M
M=‐1‐1

M = ‐1

=0+1

0+10

M=00

M = +1Conservation of M (= Jz) :あ22

What about J?
0f7/2• An exercise: 2 neutrons in 0f7/2 orbitJ+: Angular momentum raising operator
m1m2

m1m2

m1m2

7/2
5/2
3/2
1/2

‐ 7/2
‐ 5/2
‐ 3/2
‐ 1/2

7/2
5/2
3/2

‐ 5/2
‐ 3/2
‐ 1/2

7/2
5/2
3/2

‐ 3/2
‐ 1/2
1/2

M=0

M=1
J = 0 two‐body state is lost

M=2
J = 1 can be eliminated,
but is not contained.23

Bohr ‐Mottelson

Single‐particle orbits
NLJ

N = 2n + l (+ 1)= 1, 2, 3, …
(= 0, 1, 2, …)

J=l+s

Orbital Angular Momentum

Symbol

l=0s

l=1

pl=2

dl=3fl=4gl=5hl=6i#(sps) = 2J + 1 = 2x7/2 + 1 = 824
m = ‐7/2, ‐5/2, …, 0, … 7/2

J components in M
2 neutrons in 0f7/2 orbit
M=0
M=1
M=2
M=3
M=4
M=5
M=6Dimension
43322
1
10f7/2
J components
J = 0, 2, 4, 6
J = 2, 4, 6
J = 2, 4, 6
J = 4, 6
J = 4, 6
J=6
J=6

M’s are conserved w/ the SDs, but J’s are mixed for M fixed.
25

Diagonalization of the Hamiltonian matrix
2 neutrons in 0f7/2 orbit0f7/2• By diagonalizing the Hamiltonian matrix, we can obtain wave
functions of good J values by superposing Slater determinants(SDs).

M=0

=*
*
*
*

*
*
*
*

*
*
*
*

*
*
*
*

E(J=0) 0 0 00 E(J=2) 0 00 0 E(J=4) 00 0 0 E(J=6)

This property is general,and valid for the cases with more than 2 particles.
26

Numerical methods of diagonalization• Algorithm:• Householder method for smaller matrix‐size• Lanczos method: for larger matrix‐size27

Traditional diagonalization method• Lanczos method:– Obtain a few eigenvalues from the lowest energy state– Start from the initial vector v1
Hv1  1v1  1v2
Orthogonalization of Hvi & vi , vi-1 -> vi+1
Hv2  1v1   2 v2   2 v3
 2 v2   3v3   3v4
Hv3 
– Diagonalization of the ith tridiagonal matrix
Convergence of the eigenvalues, even @ i << n1 1  22
 1Ji  2 i 1i 1  i 1  i 28

29

Example (18O = 16O core + 2 n)30

Shell‐model caluculation• Example: 18O (Z = 8, N = 10)• Assume 16O (Z = 8, N = 8) as the doubly‐closed core#(Valence Nucleons):• Take the sd‐shell model space
Model Space#(Single‐particle state):
0d3/2
1s1/2
0d5/2
0p1/2

Closed Core

0p3/2
0s1/2

31

Single‐particle energy• Single‐particle energies: taken from the 17O
energy spectra

Experimentallymeasured values32

Two‐body Matrix Elements• TBME: Phenomenological Effective Interaction

Effective Interaction(determination is explained later)

33

Basis state• #(Slater determinant):• Conservation of M:• For M = 0,
• Possible M = 0 2‐particle states(j1, m1)(j2, m2)(j1, m1)(j2, m2)

1(5/2, +5/2)(5/2, ‐5/2)

8(5/2, ‐1/2)(3/2, +1/2)

2(5/2, +3/2)(5/2, ‐3/2)

9(5/2, ‐3/2)(3/2, +3/2)

3(5/2, +1/2)(5/2, ‐1/2)

10 (1/2, +1/2) (1/2, ‐1/2)

4(5/2, +1/2)(1/2, ‐1/2)

11 (1/2, +1/2) (3/2, ‐1/2)

5(5/2, ‐1/2)(1/2, +1/2) 12 (1/2, ‐1/2)

6(5/2, +3/2)(3/2, ‐3/2)

13 (3/2, +3/2) (3/2, ‐3/2)

7(5/2, +1/2)(3/2, ‐1/2)

14 (3/2, +1/2) (3/2, ‐1/2)(3/2, +1/2)

34

18O energy spectra• Diagonalization of 14x14 Hamiltonian matrix

35Some remarks
1. TBMEs
2. Determination of TBMEs
3. Scheme
36

TBMEs37

Two‐body matrix elements (TBMEs)• A two‐body state is written as• Two‐body matrix elements (TBMEs)

TBMEs can be non‐zero, only if J = J’ & M = M’,
because V is rotationally invariant (a scalar w.r.t. the rotation).
38

Two‐body matrix elements (TBMEs)• Two‐body matrix elements (TBMEs)
are independent of M values, also because V is rotationally invariant.

• Two‐body matrix elements are assigned byandBecause of the complexity of nuclear force, one cannot
express all TBMEs by few empirical parameters.
39

Example of the TBMEs• An example of TBMEs: USD interaction (Wildenthal & Brown)• Model space: sd‐shell (d5/2, d3/2 and s1/2)• 3 single‐particle energies (SPEs):,• 63 TBMEs:(Isospin is also a good quantum #)20

20

Model Space(Valence Space)880d3/21s1/2
0d5/2

andsd‐shell(1 major shell)
40

USD interaction

i,j,k,l1: 0d3/2
2: 0d5/2
3: 1s1/2..
.

41

Dimensions• Dimension: Maximum # of the basis• sd‐shell: 8 < (Z, N) < 20 ((sd)n configuration)• pf‐shell: 20 < (Z, N) < 40 ((pf)n configuration)
M‐scheme
J‐scheme
JT‐scheme42

Progress in shell‐model calculations & computers

Dimension

Dimension of Hamiltonian matrix(publication year of “pioneer” papers)

MCSM
Conventional

Birth of shell model(Mayer & Jensen)

Year

Floating point operations per second

x 105 / 30 years

K computer
Blue Gene
Earth Simulator

Year

43

Determination of the TBMEs

44

Shell‐Model Hamiltonian: Single‐particle energy: Two‐body matrix element (TBME)

45

Determination of the TBMEs•

Early time: Closed shell + Experimental levels of 2 valence particles

TBME•

Example: 0+, 2+, 4+, 6+ in 42Ca: well isolated 0f7/2 orbit

0f7/2
VJ =
are determined directly by the experimental observations.

Experimental energy of state J
E(J) = 2 ε(0f7/2) + VJ
Experimental single‐particle energy of 0f7/2
46

Bohr ‐Mottelson

Single‐particle orbits
NLJ

N = 2n + l (+ 1)= 1, 2, 3, …
(= 0, 1, 2, …)

J=l+s

Orbital Angular Momentum

Symbol

l=0s

l=1

p

l=2d

l=3f

l=4g

l=5h

l=6i#(sps) = 2J + 1 = 2x7/2 + 1 = 847
m = ‐7/2, ‐5/2, …, 0, … 7/2

Empirical determination• The isolation of 0f7/2 is special. In the other cases, several orbits
must be taken into account.
• In general, chi^2 fit is made as follows,(1) TBMEs are assumed.
(2) chi^2 fit is calculated btw theor. & exp. energy levels.
(3) TBMEs are modified.
(4) Go to (1), & iterate (1)‐(3) until chi^2 becomes small enough48

• Example: 0+, 2+, 4+ in 18O: 0d5/2 & 1s1/2..
.
Arima, Cohen, Lawson & McFarlane (1968)49

• At the beginning, it was a perfect chi^2 fit.
• As heavier nuclei are studied,(1) the number of TBMEs increases,(2) shell‐model calculations become huge

• Complete fit becomes more difficult and finally impossible

Hybrid version
50Hybrid version• Microscopically calculated TBMEseg.) G‐matrix (Kuo‐Brown, H.‐Jensen, …)
G‐matrix based TBMEs are not perfecxt
direct use for shell model calculation is only disaster• Use G‐matrix‐based TBMEs as a starting point of the chi^2 fit,and do the chi^2 fit to the experiments.
(Consider some linear combinations of TBMEs and fit them)51cont’d•

The chi^2 fit method produces, as a result of minimization,a set of linear equations of TBMEs•

Some linear combinations of TBMEs are sensitive
to available experimental data (ground & low‐lying states).
The others are insensitive. Those are assumed to be given by G‐matrix‐
based calculation (i.e. no fit).

•

Firstly done for sd‐shell: Wildenthal & Brown’s USD interaction
47 linear combinations (1970)
Recent version of USD: G‐matrix‐based TBMEs have been improved‐> 30 linear combinations fitted52

Example• An example from pf‐shell (f7/2, f5/2, p3/2, p1/2)
G‐matrix + polarization correction + empirical refinement
Microscopic
Empirical (Phenomenological)• GXPF1 interaction: M. Honma et al., PRC65 (2002) 061301(R)• start from a realistic microscopic interaction
 M. Hjorth‐Jensen, et al., Phys. Rept. 261 (1995) 125‐ Bonn‐C potential‐ 3rd order Q‐box + folded diagram 195 TBMEs & 4 SPEs are calculated ‐> not completely good(theory imperfect) Vary 70 linear combinations of 195 TBME2 & 4 SPEs
 Fit to 699 exp. energy data of 87 nuclei

53

Bohr ‐Mottelson

Single‐particle orbits
NLJ

N = 2n + l (+ 1)= 1, 2, 3, …
(= 0, 1, 2, …)

J=l+s

Orbital Angular Momentum

Symbol

l=0s

l=1

p

l=2

d

l=3

f

pf‐shell
l=4#(sps) = 20 l = 5h

l=6ig

54

G‐matrix vs. GXPF1A
7= f7/2, 3= p3/2, 5= f5/2, 1= p1/2

Output• TBMEs: <ab; JT | V | cd ; JT >• T=0 … attractive• T=1 … repulsive• Relatively large modifications inV(abab ; J0 ) with large J
V(aabb ; J1 ) pairing

Input55

Scheme56

Prescriptions of Shell‐model calc.
• M‐scheme– basis function: Slater determinant– SD does not hold J as a good quantum #, but eigen function after thediagonalization holds J as a good quantum #.

• J‐scheme– basis function:• JT‐scheme• M‐scheme + J‐projection• …

57

J‐scheme• M=0
J=0
J=1Configuration

J=0(d5/2)2

J=1

J=2

J=4

(d5/2)(d3/2)

(s1/2)(d3/2)

(d3/2)2

J=3



(d5/2)(s1/2)(s1/2)2

J=2







J=3J=458

Shell‐model caluculation• Example: 18O (Z = 8, N = 10)• Assume 16O (Z = 8, N = 8) as the doubly‐closed core#(Valence Nucleons):• Take the sd‐shell model space
Model Space#(Single‐particle state):
0d3/2
1s1/2
0d5/2
0p1/2

Closed Core

0p3/2
0s1/2

59

Basis state• #(Slater determinant):• Conservation of M:• For M = 0,• Possible M = 0 2‐particle states(j1, m1)(j2, m2)(j1, m1)(j2, m2)

1(5/2, +5/2)(5/2, ‐5/2)

8(5/2, ‐1/2)(3/2, +1/2)

2(5/2, +3/2)(5/2, ‐3/2)

9(5/2, ‐3/2)(3/2, +3/2)

3(5/2, +1/2)(5/2, ‐1/2)

10 (1/2, +1/2) (1/2, ‐1/2)

4(5/2, +1/2)(1/2, ‐1/2)

11 (1/2, +1/2) (3/2, ‐1/2)

5(5/2, ‐1/2)(1/2, +1/2) 12 (1/2, ‐1/2)

6(5/2, +3/2)(3/2, ‐3/2)

13 (3/2, +3/2) (3/2, ‐3/2)

7(5/2, +1/2)(3/2, ‐1/2)

14 (3/2, +1/2) (3/2, ‐1/2)(3/2, +1/2)

60

Summary• (Conventional) Shell model: Interacting shell model w/ a core‐> Independent particle model + Configuration‐mixing

Single‐particle orbits

Shell gap
Shell gap

Model space(Valence space)
Closed core
(Inert core)61

さらに遠くへ

Slide from M. Honma• ｆ５ｐｇ９‐殻 ＪＵＮ４５相互作用– １３７パラメータ中４５線形結合を４００データにフィット– 平均誤差１８５keV– g9/2軌道（異パリティ）の影響

f5pg9-shellpf-shell• ｓｄ‐ｐｆ 殻– 40Ca閉殻の構造– 重いｓｄ殻中性子過剰核

sd-shell• ｐｆ‐ｓｄｇ 殻– Ｎ＝Ｚ近傍核の陽子過剰核– 変形共存と形状相転移– 重いｐｆ殻中性子過剰核– 100Snの閉殻構造62

End63