http://www.th.phys.titech.ac.jp/~muto/lectures/NPI05/NPI05_chap03.ppt

原子核が存在するとは
 寿命がある程度長くないと，存在が確認できない
ある程度とはどのくらいか？
 光が原子核を通過するのに要する時間程度：
原子核の大きさを L = 10 fm = 10‐14 m として 強い相互作用による直接反応（direct reaction）に要する時間
 速いγ遷移をする励起状態の寿命
 原子番号が 113 の原子核（理研で確認）の寿命

1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

Image of the no‐core shell model• Conventional (cored) shell model vs. No‐core shell model (NCSM)
Conventional Shell model

No‐core shell model

Shell gap
sd‐shell
Shell gap
p‐shells‐shell• Conventional (cored) shell model: inert core + valence shell• No‐core shell model: All nucleons are active

No‐core shell model•

A large sparse matrix eigenvalue problem

Diagonalize••••

Adopt realistic NN (& NNN) interaction(s) renormalization as needed –retain included many‐body interactions: Chiral EFT & JISP16 interaction
Adopt the 3‐D HO for the single‐particle basis states
Evaluate the nuclear Hamiltonian in basis space of HO SDs
Diagonalize this sparse many‐body H in its “m‐scheme” basis
n = 1,2, …, 1010 or more…

•

Evaluate obs & compare w/ exp.

Some comments•••••

Straightforward but computationally demanding‐> new algorithms/computers
Require convergence assessments & extrapolation tools
Achievable for nuclei up to A = 16 (40) today w/ largest computers available
No‐core shell model (NCSM):
effective interaction in the Nmax truncation
No‐core full configuration (NCFC):
bare interaction in the Nmax truncation w/ the model‐space extrp. (Nmax ‐>infinite)

Petascale ~ 200Tbytes formatrix + index arrays

ab inito NCSM
• Effective Hamiltonian for A‐particles(Lee‐Suzuki‐Okamoto method + Cluster decomposition)
P. Navratil, J.P. Vary and B.R. Barrett,Phys. Rev. Lett. 84, 5728(2000); Phys. Rev. C62, 054311(2000)
C. Viazminsky and J.P. Vary, J. Math. Phys. 42, 2055 (2001);
K. Suzuki and S.Y. Lee, Progr. Theor. Phys. 64, 2091(1980);
K. Suzuki, ibid, 68, 246(1982);
K. Suzuki and R. Okamoto, ibid, 70, 439(1983)

Preserves the symmetries of the full Hamiltonian:
Rotational, translational, parity, etc., invariance A( pi  p j )2
 Vij ]  VNNNHA  Trel  V  [2mAi jSelect a finite oscillator basis space (P-space) and evaluate an- body cluster effective Hamiltonian:

H eff  P Trel  V a (N max ,)P

Guaranteed to provide exact answers as a  A or as P  1 .

Effective Hamiltonian in the NCSM
Lee-Suzuki-Okamoto renormalization schemePNmaxQPQHeff00H
QXHX-1Q

H : E1, E 2 , E 3 , E d P , E 
H:E,E,E,
E
eff
123dP
1QXHX P  0H eff  PXHX 1Pmodel spacedimensionunitary X  exp[  arctan h(    )]•••n‐body cluster approximation, 2≤n≤AH(n)eff n‐body operator
Two ways of convergence:– For P → 1 H(n)eff → H– For n → A and fixed P: H(n)eff → Heff

Some NCSM Results

Current Status of No‐Core Shell Model (NCSM)

P. Navratil, Enrico Fermi Lecture (2007)

12Current Status of No‐Core Shell Model (NCSM)•Nmax‐truncation (NCSM, NCFC)

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

P. Navratil, Enrico Fermi Lecture (2007)13

Current Status of No‐Core Shell Model (NCSM)

P. Navratil, Enrico Fermi Lecture (2007)14

Current Status of No‐Core Shell Model (NCSM)

P. Navratil, Enrico Fermi Lecture (2007)15

Slide from J.P. Vary

END