Shortrange and tensor correlations in light nuclei studied
- Slides: 39
Short-range and tensor correlations in light nuclei studied with antisymmetrized molecular dynamics (TOAMD) Takayuki MYO Mengjiao LYU Masahiro ISAKA Hiroshi TOKI Hisashi HORIUCHI Kiyomi IKEDA Tadahiro SUHARA Taiichi YAMADA QNP 2018. 11, Tsukuba (RCNP) (Hosei) (RCNP) (RIKEN) (Matsue) (Kanto Gakuin)
Outline • F
Deuteron properties & tensor force Energy S D -2. 24 Me. V S 11. 31 D 8. 57 Kinetic 19. 88 Central -4. 46 SD -18. 93 Tensor -16. 64 DD 2. 29 r [fm] LS -1. 02 P(L=2) S D Radius 5. 77% 1. 96 fm S 2. 00 D 1. 22 d-wave is “spatially compact” (high-momentum)
Prog. Theor. Exp. Phys. 2015, 073 D 02 (38 pages) DOI: 10. 1093/ptep/ptv 087 Tensor-optimized antisymmetrized molecular dynamics in nuclear physics (TOAMD) Takayuki Myo, Hiroshi Toki, Kiyomi Ikeda, Hisashi Horiuchi, and Tadahiro Suhara 4
General formulation of TOAMD tensor • short-range tensor short-range Power series expansion, but, all F are independent nucleon w. f. Gaussian wave packet spin-isospin dependent Eigenvalue problem
Matrix elements of correlated operator Correlated Hamiltonian Correlated Norm Classify the connections of F, H into many-body operators using cluster expansion method F† F 3 -body (2 -body)2 4 -body bra V ket i j k l (2 -body)3 relative single particle
Results PLB 769 (2017) 213 PRC 95 (2017) 044314 PRC 96 (2017) 034309 PTEP (2017) 073 D 01 PTEP (2017) 111 D 01 • • Single F AMD Double F F F 2 p-2 h 3 p-3 h F F F 4 p-4 h
AV 8’ F F S D SS SD DD DS F are independent AMD 3 H Correlation functions
AV 8’ F F S D SS SD DS DD PLB 769 (2017) 213 3 H Kinetic/2 F are independent • Reproduce the Hamiltonian components of 3 H Central Tensor Correlation functions
AV 8’ F F S D SS SD DD DS F are independent AMD 4 He Correlation functions
Amplitude FS 3 E intermediate FS 1 E • r 2×FD 3 E long
3 H Energy 3 -body 4 He 3 -body 4 -body Energy 2 -body • Many-body terms play a decisive role for energy saturation • NO energy saturation within 2 -body term 12 PTEP (2017) 073 D 01
High-Momentum Antisymmetrized Molecular Dynamics (HM-AMD) TM et al. Lyu et al. TM Lyu et al. Zhao et al. PTEP (2017) 111 D 01 Tensor correlation vs. TOSM PTEP (2018) 011 D 01 HM-AMD + TOAMD with AV 8’ PTEP (2018) 031 D 01 Short-range correlation with AV 4’ ar. Xiv: 1807. 11213 Submitted to RPC ar. Xiv: 1811. 00271 Submitted to RPC 13
TOAMD for p-shell nuclei • 14
High-momentum AMD (HM-AMD) • cf. TOAMD PTEP (2017) 111 D 01, (2018) 011 D 01
Lyu’s work: HM-AMD + TOAMD (HM-TOAMD) (RCNP, Osaka) • F
Energy, radius and Hamiltonian components • 4 He with AV 8’ bare interaction • Superpose the basis states with different D’s for pairs. 4 He Add bases successively Lyu et al. PTEP 2018 (2018) 011 D 01, PRC submitted
Energy, radius in 4 He with HM-TOAMD • 4 He with AV 8’ bare interaction • Superpose the basis states with different D’s for pairs. HM-TOAMD-F 2 GFMC Energy -24. 74 Kinetic 95. 17 97. 06 102. 3 Central -52. 33 -53. 12 -55. 05 Tensor -63. 80 -64. 84 -68. 05 LS -3. 77 -3. 83 -4. 75 Radius 1. 51 1. 50 -25. 93 [Me. V] 1. 49 [fm] Lyu et al. PTEP 2018 (2018) 011 D 01, PRC • p-shell nuclei with HM-TOAMD is in progress.
Summary • Tensor-Optimized AMD (TOAMD) – Successive variational method for nuclei to treat VNN directly. – Correlation functions : FD (tensor) , FS (short-range). – F are independently optimized, better than Jastrow method. PLB 769 (2017) 213 PRC 96 (2017) 034309 • High-momentum AMD (HM-AMD) PTEP (2017) 111 D 01 – High-momentum pairs using imaginary centroids in Gaussian wave packets – Comparison with shell model : “One high-momentum pair” = 2 p-2 h correlation – Hybrid : HM-TOAMD using VNN PTEP (2018) 011 D 01
Backup 20
AV 8’ F F S 4 He D SS SD Kinetic/2 DD DS F are independent Central Tensor Correlation functions
TOAMD & Jastrow method • R. Jastrow, Phys. Rev. 98, 1479 (1955) f f
TOAMD & VMC with Jastrow VNN : AV 6 for central & tensor forces Energy (Me. V) 3 H Few-body omit LS, L 2, (LS)2 from AV 14 TOAMD (power series) gives better energy than VMC (Jastrow) from variational point of view Independent optimization of all F 4 He PRC 96 (2017) 034309 Correlation functions JPS magazine (2017) Dec. 867
AV 6 S D SS SD DS DD same Few-body F Kinetic/2 F Central 3 H Tensor Correlation functions Free F
Diagrams of cluster expansion - VNN bra V 2 -body F F V V ket 3 -body F 5 -body 6 -body 4 -body
Diagrams of cluster expansion, Kinetic energy 1 -body 2 -body bra T ket F F T T F uncorrelated kinetic energy 5 -body 3 -body 4 -body
Convergence of HM-AMD • VNN : Central : Volkov No. 2 with M=0. 6 LS & Tensor : G 3 RS bare force 4 He HM-AMD with 1 pair (0 s)4 Add bases successively Tensor-optimized shell model (TOSM), complete 2 p-2 h. “Criterion”
Energy surface with the shift Dz • VNN : Central Volkov No. 2 with M=0. 6, Coulomb, LS & Tensor : G 3 RS bare force (0 s)4 TM, K. Kato, K. Ikeda 2 bases : (0 s)4 + pn pair with Dz 4 He Superpose by increasing Dz
Hamiltonian Component in HM-AMD 4 He 29
HM-AMD & TOSM for 4 He • Tensor-Optimized Shell Model (Me. V) HM-AMD TOSM Energy − 64. 7 − 65. 2 Kinetic 83. 6 84. 8 Central − 83. 6 − 84. 1 Tensor − 66. 3 − 67. 7 LS 0. 8 0. 9 Coulomb 0. 9 Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 NO truncation of particle states in TOSM using Gaussian expansion with high-L more than 10ℏ HM-AMD = TOSM (1 pair) (full 2 p-2 h) Next, bare VNN (Lyu, Isaka)
Tensor-optimized Shell Model for 4 He NO truncation of particle states using Gaussian expansion Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 • 9 Gaussian bases for each orbit with L • “Criterion” to examine 2 p-2 h effect (0 s)4 (single particle orbit)
AV 8’ F F Single SS SD DS 3 H Single +SS +SD+DS +DD DD F are independent • Getting close to the GFMC energy. • n-dependence is small due to the flexibility of F.
Kinetic 3 H F Energy 4 He LS Central Tensor • 33
uncorrelated kinetic Kinetic Full T+ 2 -body 3 -body same trend in central & tensor 2 -body > 3 -body not negligible Tensor Central 34
Many-body Hamiltonian terms in 4 He PTEP (2017) 073 D 01 Kinetic Full 1+2 body 3 -body 4 -body Central Higher-body term tends to give smaller scale, but, not ignored. Tensor 35
AV 8’ S D SS SD DS DD PTEP (2017) 073 D 01 cf. Jastrow ansatz 3 H DE=1. 4 Me. V Correlation functions F are independent as a full variation
AV 8’ S D SS SD DS DD same Kinetic/2 3 H F F Few-body Correlation functions Central Tensor Free F
AV 8’ S D SS SD DS DD PTEP (2017) 073 D 01 cf. Jastrow ansatz 4 He DE=2. 3 Me. V Correlation functions F are independent as a full variation
AV 8’ S D SS SD 4 He DS DD Kinetic/2 same F F Central Tensor Correlation functions Free F Independent optimization of all F
- Associations and correlations in data mining
- Associations and correlations in data mining
- Mining frequent patterns associations and correlations
- Korelasi 6 pair
- Quantum correlations with no causal order
- Spurious correlations
- Thinking critically
- Light light light chapter 23
- Into the light chapter 22
- Chapter 22
- Data collection instruments in qualitative research
- Qualitative vs quantitative
- Traction stress tensor
- Put out the light othello
- Membrane-bound organelles
- It is the bouncing off of light
- . . . . they do their homework last night
- Sociologist who studied dating patterns
- The most studied group is
- Been ving
- I studied hard because i knew that the test would
- Natural history of disease is best studied by
- Richard dugdale theory
- Dugdale criminology
- We've recently studied the
- The motion of a projectile is often studied in terms of
- The scientist mathias schleiden studied _______ in ______.
- We've recently studied the
- Tpcastt attitude
- Diagram studied in requirement analysis
- Interpersonal attraction psychology
- As darwin studied fossils what new questions arose
- Sprawl geography definition
- Multiple nuclei model
- Multiple nuclei model year
- Homer hoyt sector model
- Concentric zone mode
- Urban realms model
- Nmr active and inactive nuclei
- How unstable atoms gain stability