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

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

原子核分野におけるHPCI戦略活動
清水@核理懇、日本物理学会（弘前、2011）
革新的ハイパフォーマンス・コンピューティング・インフラ（HPCI）の構築
次世代スーパーコンピュータ「京」を中核として、多様なユーザーニーズに応える革新的な計算環境を実現するHPCI(革新的ハイパフォーマンス・コンピューティング・インフラ)を構築するとともに、その利用を促進する。 今年度から5年間。

計算基礎科学研究連携拠点（筑波大学CCS、KEK、天文台）
http://www.jicfus.jp/field5/jp/
拠点長 青木慎也（筑波大）

開発課題責任者

課題１ 格子QCDによる物理点でのバリオン間相互作用の決定（藏増嘉伸）
課題２ 大規模量子多体計算による核物性の解明とその応用（大塚孝治）
課題３ 超新星爆発およびブラックホール誕生過程の解明（柴田大）
課題４ ダークマターの密度ゆらぎから生まれる第１世代天体形成 （牧野淳一郎）
計算科学技術推進体制の構築（橋本省二）

清水@核理懇、日本物理学会（弘前、2011）
HPCI戦略分野５課題２「大規模量子多体計算による核物性の解明とその応用」
大塚 孝治 (開発課題責任者)
清水 則孝
阿部 喬
東大CNSに月山 幸志郎Ｈ２３．４．１着任
12Cなどの計算
吉田
HPCI専従江幡 修一郎吉田 亨
軽い核の芯を仮定しない殻模型計算
阿部、大塚、清水角田 直文 (東大理)412
第一原理的、p殻核、 He～ C, sd殻核角田 佑介 (東大理)殻模型計算コード開発
本間 道雄 （会津大） アルゴリズム
大規模計算に必要な有効相互作用
宇都野 穣 （JAEA)
月山、角田
清水、宇都野、阿部、大塚
中務 孝
（理研）
中重核の殻模型計算 清水、宇都野、本間、大塚、角田鈴木 俊夫 (日大)
芯を仮定した有効相互作用
中田 仁
(千葉大)
Cr, Ni, Sn, Xe, Nd, ...
梶野 敏貴 (天文台)r-process, 二重ベータ崩壊, 原子力工学, ...
密度汎関数法江幡、中務Outline of this part••••

Motivation
Monte Carlo Shell Model (MCSM)
Benchmark Results
Summary & Outlook

Current status of ab inito approaches•

Major challenge of the nuclear structure theory‐ Understand the nuclear structures from the first principle ofquantum many‐body theory by ab‐initio calc w/ realistic nuclear forces‐ Standard approaches: GFMC, NCSM (up to A ~ 12‐14), CC (closed shell +/‐ 1,2)
• demand for extensive computational resources ab‐initio(‐like) approaches (which attempt to go) beyond standard methods
No‐core Monte Carlo shell model (MCSM)
 Another approaches beyond standard NCSM:‐ IT‐NCSM, IT‐CI: R. Roth (TU Darmstadt), P. Navratil (TRIUMF)‐ Sp‐NCSM: T. Dytrych, K.D. Sviratcheva, J.P. Draayer, C. Bahri, and J.P. Vary
6
(Louisiana State U, Iowa State U)Current status of some ab initio calc
• GFMC• NCSM• IT‐NCSM• CC

Current Status of Green’s Function Monte Carlo (GFMC)

S.C. Pieper, Enrico Fermi Lecture (2007)8

Current Status of Green’s Function Monte Carlo (GFMC)

S.C. Pieper, Annual UNEDF Collaboration Meetings (2011)
http://unedf.org/content/annual_mtg.php

9Current Status of No‐Core Shell Model (NCSM)

14N

P. Maris, Annual UNEDF Collaboration Meetings (2011)
http://unedf.org/content/annual_mtg.php

10Current Status of Importance‐Truncated No‐Core Shell
Model (IT‐NCSM)•First ab initio NCSM calculations w/ SRG‐evolved chiral NN+3N interaction
throughout p‐shell nuclei– Surpass all previous NCSM calc. incl. 3N int. regarding A & Nmax
Previous study: Nmax = 8 for 14C w/ NN + 3N (NCSM) [P. Maris, et al., PRL 106, 202502 (2011)]
This study: Nmax = 12 for 12C & 16O w/ NN + 3N (IT‐NCSM)

R. Roth, J. Langhammer, A. Calci, S. Binder, and P. Navratil, arXiv:1105.3173 [nucl‐th]R. Roth, J. Langhammer, A. Calci, S. Binder, and P. Navratil, arXiv:1105.3173 [nucl‐th]Ground‐state energy for lower p‐shell nuclei
NN onlyalpha dependent‐> SRG‐induced 3N contrib.
originating initial NN int.

α : flow parameterc.f.) Λ = α-1/4 =2.24 fm-1
α = 0.04 fm42.11 fm-1
0.05 fm4
2 fm-1
0.0625 fm4
1.88 fm-1
0.08 fm4
1.58 fm-1
0.16 fm4

NN+3N‐inducedalpha independent‐> negligible SRG‐induced
higher many‐body contrib.
originating initial NN int.

induced 4N (4He)
ind. 4N, 5N, & 6N (6Li)
originating initial NN int.

NN+3N‐fullalpha independent
reproduce exp data

For lower p‐shell nuclei, induced 3N terms originating from the initial NN interaction are important,but induced 4N (and higher) terms are not important

R. Roth, J. Langhammer, A. Calci, S. Binder, and P. Navratil, arXiv:1105.3173 [nucl‐th]Ground‐state energy for upper p‐shell nuclei
NN onlyalpha dependent‐> SRG‐induced 3N contrib.
originating initial NN int.

NN+3N‐inducedalpha independent‐> negligible SRG‐induced
higher many‐body contrib.
originating initial NN int.

induced 4N‐12N (12C)
ind. 4N‐16N (16O)
originating initial NN int.

NN+3N‐fullalpha dependent‐> SRG‐induced 4N contrib.
originating initial 3N int.
For upper p‐shell nuclei, induced 4N terms originating from the initial 3N interaction are sizable

R. Roth, J. Langhammer, A. Calci, S. Binder, and P. Navratil, arXiv:1105.3173 [nucl‐th]Excitation spectra of carbon‐12First six excited states of positive parity for fixed alpha = 0.08 fm4
Induced 3N terms
Initial (genuine) 3N terms‐> over‐all compression of the spectrum‐> different behavior among the different states
2+ & 4+ states: improved
1+ & 0+2 (Hoyle) states: not well described

Excited states: alpha dependence is much weaker than that in ground states (not shown in Fig.4, though)~ a few 100 keV for E*(0+2) w/ NN+3N‐full ‐> negligible induced 4N contrib.

Current Status of Coupled Cluster (CC) Theory
Saturation of N3LO (NN only) in medium mass nuclei
Benchmarks in light nuclei:
Coupled‐cluster meets few‐body benchmarks for
4He. Recent IT‐NCSM and UMOA calculations of
16O agree with CCM.
R. Roth et al, arXiv:1105.3173 (2011)
Fujii et al, PRL 103, 182501(2009)

G. Hagen, T. Papenbrock, D. J. Dean, M.
Hjorth‐Jensen, Phys. Rev. C 82, 034330 (2010).

G. Hagen, Annual UNEDF Collaboration Meetings (2011)

Current Status of Coupled Cluster (CC) Theory
Going beyond closed-shell nuclei. Low-lying states in
18O and 26F (Preliminary)Two‐particle attached coupled‐cluster works very well for
low‐lying states in open‐shell nuclei like 18O and 26F.
Our results for 26F seem to suggest a more compressed
spectrum as compared to USDA/USDB calculations.
G. Jansen, M. Hjorth‐Jensen, G. Hagen, T. Papenbrock, Phys.
Rev. C 83, 054306 (2011).

G. Hagen, Annual UNEDF Collaboration Meetings (2011)

Current status of some ab initio calc
• GFMC: A = 12 w/ NN + NNN• NCSM: A = 14 w/ NN + NNN @ Nmax = 8• IT‐NCSM: A = 16 w/ NN + NNN @ Nmax = 12
• CC: Closed core +/‐ A = 2 w/ NN (+NNN)M‐scheme dimension
No-core MCSM

DM

16O (0+)12C (0+)10B (3+)Current FCI limit

8Be (0+)7Li(3/2-)
6He (0+), 6Li(1+)
4He (0+).
.
.
Moore’s law:#transistors doubles every two years. (p = 2n/2)
x 5.7 after 5 yrs
x 32 after 10 yrs.
.
.
Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1 18

UNEDF SciDAC Collaboration: http://unedf.org/

DFT
CI
Ab initio

19

UNEDF SciDAC Collaboration: http://unedf.org/

MCSM
No‐core MCSM

20

NCSM, FCI, NCFC, MCSM
Truncation

Interaction

NCSM

Nmax

Bare/Effective

FCI

Nshell

Bare

NCFC

Nmax

Bare

MCSM

NshellBare/Effective

 NCSM,FCI,NCFC does exact diagonalization of large Hamilotonian matrices, while MCSM
utilizes the diagonalization of smaller matrices w/ importance‐truncated bases.
 NCSM,NCFC uses Nmax truncation, while FCI, MCSM does Nshell truncation.
 Nmax is the sum of the HO excitation quanta from the reference state.
 Nshell is the # of major shells included as the model space.
 NCSM usually employs effective interactions for getting faster convergence wrt themodel space.
 FCI,NCFC employs bare interactions & extrapolates into the infinite model space (Nshell,
Nmax ‐> ∞).
 Treatment of spurious CM effect is exact in NCSM,NCFC , while it is approximate in
FCI,MCSM (by using Gloeckner‐Lawson method).
21

Truncations of the Model Space in NCSM & FCI•

Nshell (FCI, MCSM, IT‐CI, …)

 Max. # of HO quanta of many‐body basis
Nmax = 4 (A = 4)

 Max. # of HO quanta of single‐particle basis
Nshell = 5 (A = 4)

.
.
.

.
.
.

hw

.
.
.

N = 4 (2s, 1d, 0g)
N = 3 (1p, 0f)
N = 2 (1s, 0d)
N = 1 (0p)N = 0 (0s)N = ∑i 2ni + li <= Nmax

A=3
Nmax

Nk

A=2

Nmax
Nj

N = Ni + Nj

N = Ni + Nj + Nk

Ni = 2ni + li

Ni

Nj

.
.
.

N=4
N= 3
N= 2
N=1
N = 2n + l = 0

Nshell – 1

N = 2n + l <= Nshell - 1

A=2
Nj = 2nj + lj•

Nmax (NCSM, NCFC, IT‐NCSM, …)

A=3
N
Nshell - 1k
Nj

Ni

22

Ni

Truncations of Model Space: Nshell & Nmax
Nmax :
Nshell:Nmax
Nshell< Nshell < 2 x Nshell (for Nshell:even) E (hω)
Nshell – 1 < Nshell < 2 x Nshell (for Nshell:odd)
Nshell + 1 ~Nmax Nshell Nmax
0 < 1 < 2
2 < 2 < 4
4 ~? 3 < 6
4 < 4 < 8
6 ~? 5 < 10
6 < 6 < 12

N = 2n + l[N = 6] 2d, 1g, 0i (+)[N = 5] 2p, 1f, 0h (‐)[N = 4] 2s, 1d, 0g (+)[N = 3] 1p, 0f (‐)

8 < 8 < 16
[N = 2] 1s, 0d (+)
10 < 10 < 20
[N = 1] 0p (‐)
Nmax
Nshell 1

2
2

Nshell = N + 1

3

4
4

6
5

8

[N = 0] 0s (+)(N+1)(N+2)(56) 168(42) 112(30) 70(20) 40(12) 20
(6) 8
(2) 2

(N+1)(N+2)(N+3)/3

P. Maris, EFES‐Iowa workshop (2010)

Monte Carlo Shell Model (MCSM)

Review: T. Otsuka , M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319 (2001)

Monte Carlo shell model (MCSM)• Importance truncation
Standard shell model

H=
Diagonalization

All Slater determinantsMonte Carlo shell model

H~

Diagonalization

Important bases stochastically selected

dMCSM ~ O(10‐100)26

Hamiltonian & wave function• Second‐quantized Hamiltonian (up to two‐body int.)• Many‐body wave function: superposition of non‐orthogonal SDs• Angular‐momentum & parity projected MCSM basis• Deformed SDs( cα† … HO basis)

Monte Carlo Shell Model• Deformed Slater determinant basis( cα† … HO basis)• MCSM basisc.f.) Imaginary‐time evolution & Hubbard‐Stratonovich transf.

P. Ring, P. Schuck, The Nuclear Many‐Body Problem, Springer

Monte Carlo Shell Model• Symmetry restoration by projection method

Euler angles

Wigner function– Angular momentum projection operator (same as the parity)
Unitary rotational operator– General (GCM) ansatz– Projected energy

Kernels

Hot spot in MCSM:~ 30 x 30 x 30 mesh points

Monte Carlo Shell Model
Hamiltonian
kernel• Basis search(n-1)*(n-1)matrix– Fix the n‐1 basis states already taken
fixed
H(,’)=– Requirement for the new basis:
atopt the basis which makes theenergy (of a many‐body state) as low
as possible by a stochastic samplingn-th(to be optimized) ( )   n e h ( n ) h( n )  hHF     nO
Estarting    abandoned
basis






HF
HF

adoptedRecent developments in MCSM•

Acceleration of the computation of two‐body matrix elements



 1
1
ˆ
 V  '    ki   vijkl  lj     ( ki )   v( ki ),(lj )  (lj ) 
2 i ,k jl j ,l 2 ( ki )Matrix product is performed w/ bundled density matrices by DGEMM
subroutine in BLAS library
800 % performance improvement from the original MCSM code
Y. Utsuno, N. Shimizu, T. Otsuka, and T. Abe, in preparation.

•

Extrapolation method by the energy variance(naively) 8‐fold loops ‐> (effectively) 6‐fold loops by the factorization
N. Shimizu, Y. Utsuno, T.Mizusaki, T. Otsuka, T. Abe, & M. Honma, Phys. Rev. C82, 061305(R) (2010)31

Computation of the TBMEs• hot spot: Computation of the TBMEs(w/o projections, for simplicity)c.f.) Indirect‐index method(list‐vector method)• non‐zero ME: jz(i) + jz(j) = jz(k) + jz(l) ‐> jz(i) ‐ jz(k) = ‐ (jz(j) – jz(l))

Operations: sparse matrix ‐> dense matrix
sparsedenseSchematic illustration of the computation of TBMEs• Matrix‐vector method

Δm =

‐1

‐1

0

0

+1

x

x 0
+1

0

Schematic illustration of the computation of TBMEs• Matrix‐matrix method

Δm =

‐1

‐1

0

0

+1

x 0
+1

BLAS Level 3x0Comparison of the performance
theoretical peak performanceIntel Xeon Harpertown E5440
2.83GHz, 12MB LS cache,
ifort 11.1 + Intel MKL 10.2

Y. Utsuno, N. Shimizu, T. Otsuka, and T. Abe, in preparation.

Energy‐variance extrapolation-80
Carbon‐12 g.s. energy@ Nshell = 4, hw = 30 MeV
w/o Coulomb int & spurious CoM treatment

Energy (MeV)

-82
-84

Converged or not?

-86
-88

?

-90
-92
0

25

50
75
MCSM basis dim.

100

DM ~ 6 x 1011

Energy variance extrapolation• Originally proposed in condensed matter physics
Path Integral Renormalization Group method
M. Imada and T. Kashima, J. Phys. Soc. Jpn 69, 2723 (2000)• Imported to nuclear physics
Lanczos diagonalization with particle‐hole truncation
T. Mizusaki and M. Imada Phys. Rev. C65 064319 (2002)
T. Mizusaki and M. Imada Phys. Rev. C68 041301 (2003)

single deformed Slater determinant
T. Mizusaki, Phys. Rev. C70 044316 (2004)

Apply to the MCSM

What is the energy‐variance extrapolation?
Demonstrated by Mizusaki in the framework of conventional shell model
Energy variance is defined as

H 2  H 2  H

2If the wave function is an exact
eigenstate of the Hamiltonian,energy variance is exactly zero

H 2  0“A series ofapproximation”
H 2  H 2  H

A series of approximated wave functions:

H  E0  a H2

 b H2 2

 ...

Ref. T. Mizusaki and M. Imada, Phys. Rev. C65 064319 (2002)

2

With a sequence of approximate
energies,

extrapolate H  0 so thatH becomes E0 , true energy.
2Why we need to extrapolate the energies•

Definition: (Correlation Energy)
10B (1+, 4shl)
10B (3+, 4shl)
7Li (1/2‐, 4shl)

CI

7Li (3/2‐, 4shl)

6Li (1+, 4shl)
64Ge (pfg9) 6He (0+, 4shl)
56Ni (pf)

12C (0+, 4shl)

4He (0+, 4shl)

8Be (0+, 4shl)

FCI

NCSM wf w/ realistic NN int is more correlated (complicated) than SSM wf w/ effective int

Need energy‐variance extrapolation for No‐Core MCSM calc

39

Numerical effort
8‐folded loop
~O(Nsps^8)

6‐folded loop
~O(Nsps^6)

Extrapolation of 12C Energy
DJ =2,936,582 = 2.9 x 106

E = ‐76.621 MeV (exact FCI)
E = ‐76.621 MeV (quadratic fit)
E = ‐76.740 MeV (linear fit)
E = ‐76.243 MeV (MCSM) [52 dim]
12C
Nshell = 3 (s,p,sd)
hw = 30 MeV
w/o Coulomb force41

DM ~ 6 x 1011

Extrapolation of 12C Energy

DJ = 11,384,214,614 ~ 1.1 x 1010

Exact value is unknown
E = ‐90.030 MeV (MCSM) [81 dim]
Variational upper boundEstimated error ~ 144 keV

Energy (MeV)

12C (0+)-80

E = ‐92.18(14) MeV (quadratic)
E = ‐92.58 MeV (linear)-82

Eeffective lower bound-84

Nshell = 4 (spsdpf)
hw = 30 MeV
w/o Coulomb force

-86
-88

?

-90
-92
0

25

50
75
MCSM basis dim.

100

42

Benchmark results‐ Energy‐ RMS‐ Q‐moment‐ μ‐moment

What we have calculated as Benchmark••

Comparison btw MCSM & FCI (exact diag.) calc
Nuclei (JP): s‐ & p‐shell nuclei:
Our test set up:‐ NN interaction: JISP16‐ 4He(0+)‐ model space: Nshell = 2, 3, 4
‐ 6He(0+)‐ optimal hw selected for states & Nshell’s‐ 6Li(1+)‐ w/o Coulomb‐ 7Li(1/2‐, 3/2‐)‐ w/o Gloeckner‐Lawson prescription‐ 8Be(0+)
MCSM: Abe, Otsuka, Shimizu, Utsuno (Tokyo)‐ 10B(1+, 3+)
T2K (Tokyo, Tsukuba), BX900 (JAEA)
FCI: Maris, Vary (Iowa)‐ 12C(0+)
Jaguar, Franklin (NERSC, DOE)
• Observables:
JISP16:‐ BE
A.M. Shirokov, J.P. Vary, A. I. Mazur, T.A. Weber,Phys. Lett. B644, 33 (2007)‐ Point‐particle RMS radius (matter)
NCFC calc of light nuclei w/ JISP16:‐ Electromagnetic moments (Q, μ)P. Maris, J.P. Vary, A.M. Shirokov,
Phys. Rev. C 79, 014308 (2009)44

Helium‐4 & carbon‐12 gs energies
Nshell = 2
4He(0+;gs)

w/ optimum hw
w/o Coulomb force
w/o spurious CoM treatment

Nshell = 3
Nshell = 4
Nshell = 5

Nshell = 2
12C(0+;gs)

Nshell = 3
Nshell = 4

Exact result is unknown

T. Abe, P. Maris, T. Otsuka, N. Shimizu, Y. Utsuno, J. P. Vary

Energies of Light Nuclei
6He (0+)MCSM
FCI

7Li (1/2‐)

Extrp.

8Be (0+)10B (3+)

6Li (1+)

4He (0+).
.
.

7Li (3/2‐).
.
.

Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1

10B (1+)

12C (0+)Nshell = 2 (sp)Nshell = 3 (spsd)

Performed only by MCSM
Nshell = 4 (spsdpf)

MCSM & FCI results are consistent within the size of lines46

Convergence pattern of the 4He point‐particle RMS
radius w.r.t. MCSM basis dimension• Comparison of MCSM (solid symbols) w/ FCI (dashed lines)@ Nshell = 2 (sp), 3 (spsd), & 4 (spsdpf).
.
.

Good agreement w/ FCI within 0.001 fm up to Nshell = 4
H = Hint + β Hcm, (β = 0).
.
.

Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1

Nshell = 4 (spsdpf)
1.379 fm (MCSM)
1.379 fm (FCI)
4He
hw = 30 MeV
w/o Coulomb force

Nshell = 3 (spsd)
1.355 fm (MCSM)
1.355 fm (FCI)
Nshell = 2 (sp)1.301 fm (MCSM)
1.301 fm (FCI)
47

Point‐particle RMS matter Radius

w/ energy‐variance extrapolationby 1st‐order polynomial

Performed only by MCSM

7Li (1/2‐)

8Be (0+)

6He (0+)6Li (1+)

7Li (3/2‐)

MCSM
FCI

10B (3+)

10B (1+)

Nshell = 4 (spsdpf)
Nshell = 3 (spsd)
Nshell = 2 (sp)
12C (0+)4He (0+)MCSM & FCI results are consistent within the size of symbols

48

Convergence pattern of the 6Li Q‐moment
w.r.t. MCSM basis dimension• Comparison of MCSM (solid symbols) w/ FCI (dashed lines)@ Nshell = 2 (sp), 3 (spsd), & 4 (spsdpf).
.
.

Good agreement w/ FCI within 0.01 efm2 up to Nshell = 4
H = Hint + β Hcm, (β = 0)

w/o Coulomb force.
.
.

Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1

Nshell = 2 (s,p)
0.044 efm2 (MCSM)
0.043 efm2 (FCI)
Nshell = 3 (s,p,sd)‐0.260 efm2 (MCSM)‐0.259 efm2 (FCI)
Nshell = 4 (s,p,sd,pf)‐0.280 efm2 (MCSM)‐0.285 efm2 (FCI)49

Q moment

MCSM
FCI
exp.
Nshell = 4 (spsdpf)
Nshell = 3 (spsd)
Nshell = 2 (sp)

MCSM & FCI results are consistent within the size of symbols

Convergence pattern of the 6Li μ‐moment
w.r.t. MCSM basis dimension• Comparison of MCSM (solid symbols) w/ FCI (dashed lines)@ Nshell = 2 (s,p), 3 (s,p,sd), & 4 (s,p,sd,pf).
.
.

Good agreement w/ FCI within 0.01 μN up to Nshell = 4
H = Hint + β Hcm, (β = 0)

w/o Coulomb force.
.
.

Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1

Nshell = 2 (sp)0.852 μN (MCSM)
0.852 μN (FCI)
Nshell = 3 (spsd)‐0.836 μN (MCSM)‐0.833 μN (FCI)
Nshell = 4 (spsdpf)‐0.835 μN (MCSM)‐0.832 μN (FCI)51

μ moment
67

μ (μN)4
6

2-

Li(3/2 )
B(3+)

10

B(1+)

Li(1+)

0
7

-2

10

Li(1/2-)

-4
5

10
A

MCSM & FCI results are consistent with each other, andμ moments are well‐reproduced even at small Nshell.

MCSM
FCI

Summary• MCSM can be applied to the no‐core calculations & thebenchmarks for the p‐shell nuclei have been performed.
‐ MCSM & FCI results are consistent with each other.

Outlook••••Larger model spaces
Inclusion of the 3‐body force in the MCSM algorithm
Coupling to the continuum states
Search for the cluster states• Tuning of the MCSM code on the K ComputerBridging the nuclear physics scales

QCD

Nuclear
Structure

Applications in astrophysics,
defense, energy, and medicine
Adapted from D. Dean, JUSTIPEN Meeting, 200954

http://www.rarf.riken.go.jp/pub/newcontents/contents/sisetu/RIBF.html

Table of Nuclides (Nuclear Chart)55

http://www.sci.tohoku.ac.jp/mediaoffce_s/_src/sc911/3Dchart.jpg

3D Nuclear Chart

56

UNEDF SciDAC Collaboration: http://unedf.org/

Ex. energy

DFTCI
Ab initio57

END