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原子核单轨道壳模型组态的维数与角动量藕合系数的求和规则 ( dimension of the single-j shell configurations and

原子核单轨道壳模型组态的维数与角动量藕合系数的求和规则 ( dimension of the single-j shell configurations and sum rules of angular momentum recoupling coefficients). 赵玉民 ( Y.M.Zhao). 上海交通大学 物理系 ( Dept. Phys., Shanghai Jiao Tong University) 兰州重离子国家实验室 ( Lanzhou) 理论核物理中心 中国高科技中心 ( World Laboratory,Beijing)

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原子核单轨道壳模型组态的维数与角动量藕合系数的求和规则 ( dimension of the single-j shell configurations and

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  1. 原子核单轨道壳模型组态的维数与角动量藕合系数的求和规则原子核单轨道壳模型组态的维数与角动量藕合系数的求和规则 (dimension of the single-j shell configurations and sum rules of angular momentum recoupling coefficients) 赵玉民(Y.M.Zhao) 上海交通大学 物理系(Dept. Phys., Shanghai Jiao Tong University) 兰州重离子国家实验室(Lanzhou) 理论核物理中心 中国高科技中心(World Laboratory,Beijing) 日本 理化学研究所 (RIKEN Cyclotron Center)

  2. Outline • Simple comments on regularities of many-body systems under random interactions • Number of spin I states for single-j configuration • J-pairing interaction • Sum rules of angular momentum recoupling coefficients • Number of states with given spin and isospin • Nucleon approximation of the shell model • Prospect and summary

  3. Two-body random ensemble Part I A brief introduction to nuclei under random interactions

  4. In 1998, Johnson, Bertsch, and Dean found spin zero ground state dominance can be obtained by random two-body interactions (Phys. Rev. Lett. 80, 2749)． Ref. C. W. Johnson et al., PRL80, 2749 (1998); R.Bijker et al., PRL84, 420 (2000); L. Kaplan et al., PRB65, 235120 (2002).

  5. Why this statistics is interesting？

  6. Intrinsic collectivity based on the sd IBM 从PRC62,014303(2000)摘录. 原作者:R. Bijker and A. Frank

  7. Energy centroids of spin I states

  8. A short summary • Spin 0 ground state dominance for even-even nuclei, regularities for energy centroids with given quantum numbers, collectivity, etc. • Open questions: spin distribution in the ground states；energy centroids；requirement for nuclear collectivity ；etc. For a review, See YMZ, AA, NY, Physics Reports, Volume 400, Page 1 (2004).

  9. Themes and Challenges of Modern Science • Complex systems arising out of basic constituents How the world, with all its apparent complexity and diversity, can be constructed out of a few elementary building blocks and their interactions • Simplicity out of complexity How the world of complex systems can display such remarkable regularity and, often, simplicity • Understanding the nature of the physical universe • Manipulating matter for the benefit of mankind Challenges of modern science reflecting the twin themes of complexity and simplicity in many-body systems Taken from [Brad Sherrill and Rick F. Casten, Frontiers of Nuclear structure: Exotic nuclei. Nuclear Physics News, Vol. 15, No.2, Pp. 13 (2005).]

  10. Part II Number of states for identical particles in a single-j shell • Why we study this number (Ginocchio)?

  11. A simple method in the text-book

  12. Empirical formulas • YMZ and AA, PRC68, 044310 (2003). empirical formulas for n=3,4. For example, For n=4, results are more complicated (omitted).

  13. A new method

  14. Conjugates of

  15. An Example

  16. An example: n=4 • Here we should study bosons with SU(5)symmetry, i.e., d bosons. The number of states for d bosons has been studied in the interacting boson model. • By using results of the IBM, we were able to obtain dimension for d bosons. Then we quickly get the number of states for n=4 of fermions and bosons. • What about odd number of particles? YMZ and AA, PRC71, 047304 (2005)

  17. Other works • Dimension for n>3: J. N. Ginocchio and W. C. Haxton, “Symmetries in Science VI”, Edited by B. Gruber and M. Ramek, (Plenum, New York, 1993). • Number of states for Zamick et al. Physical Review C71, 054308 (2005).

  18. For n=3, we should study SU(4) reduction rule. • However, Talmi proved our results for n=3(Physical Review C72,037302(2005)) . He obtained some recursion formulas and proved the formulas by reduction method. • This method can be used to prove any formulas (in principle) but it can not be used to find new formulas.

  19. The guideline of Talmi’s efforts • First, he assumes the formula is correct for j-1 shell, then it suffices to show that it is also correct to j shell. • Next he enumerates the effect by changing m1= j-1 to j. • Summing this effect, he obtains the dimension of j shell.

  20. Part III J-pairing interaction • Fermions in a single-j shell:

  21. J –pair truncation of the shell model • Empirically we find that J-pair truncation is good for J-pairing interaction.

  22. PartIV Sum rules of angular momentum recoupling coefficients • For n=3, J-pair truncation gives exact solution.

  23. An example

  24. Similar things can be done for n=4 Ref.: YMZ & AA (Physical Review C72, 054307 (2005) • Here we obtain sum rules of 9-j symbols. • The difference is that here situation is more complicated. Generally speaking, the number of nonzero eigenvalues is not always one for J pairing interaction and each of eigenvalue is unknown. • However, the trace of eigenvalues for each I states with only J-pairing interactions is always a constant with respect to orthogonal transformation:

  25. Proof

  26. Sum rules Summing over J, we obtain sum rules of 9-j symbols

  27. Sum rules with odd J and odd K

  28. Sum rules with both even and odd J,K

  29. Two examples

  30. Part V Number of states for nucleons in a single-j shell[An application of these sum rules] • References: L.Zamick et al., Physical Review C72, 044317 (2005); Y.M.Z. and A.A., Physical Review C72, 064333 (2005).

  31. JT-pairing interaction • Nucleons in a single-j shell:

  32. Similarly,

  33. The case of T=0

  34. Part VI Our next effort on pair approximation • IBM SD collective pairs diagonalization of the shell model Hamiltonian in nucleon pair subspace. • How far one can go? • How reliable is “collective pair” approximation? • What can one calculate by using pair appro. ?

  35. We have done following work • Validity of SD pair truncation for special cases • Application to A=130 even-even nuclei (rather successful calculation) • We proposed an efficient algorithm to describe even systems and odd-A (also doubly odd) systems on the same footing.

  36. The nuclei we shall try in the near future • Mass number A around 140, neutron rich side (both even and odd A, both positive and negative parity) • Validity of SD truncation, realistic cases. [How good or not good is the pair truncation?] • Extension of pairs [S1, S2, D1, D2, F pairs, G pairs, etc. ]

  37. Part VII Summary & prospect • Nuclear structure under random interactions • Number of states with given spin (&isospin), sum rule of 6j and 9j symbols • Our next project: Nucleon pair approximation of the shell model: odd and doubly odd nuclei. Applications to realistic systems

  38. Arigado gozaimasu! 谢谢! Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinaga (Saitama) Kengo Ogawa (Chiba) Nobuaki Yoshida (Kansai)

  39. 欢迎参加welcome to participate in International Conference on Nuclear Structure Physics, Shanghai, June 12-17th, 2006. email:ymzhao@sjtu.edu.cn lwchen@sjtu.edu.cn lisheng@sjtu.edu.cn 电话:021-5474-1971 传真:021-3420-2659 www.physics.sjtu.edu.cn/nucl-conf

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