Dept. Phys., Shanghai Jiao Tong Univ., China

1. Nucleon pair approximation of the shell model and its applications to heavy nuclei Y. M. Zhao Dept. Phys., Shanghai Jiao Tong Univ., China

2. Outline • Background • Previous results • Our recent results • Future

3. Part IBackground

4. The configuration space of the shell model for heavy nuclei is usually to huge to handle even for the most advanced computers. One has to truncate the shell model space. • Pair approximation is an old idea. However, one faces the problem of computing matrix elements of the shell model hamiltonian in the pair subspace. Here one can not apply the technique of coefficients of fractional parentage which is well known and an extremely useful in the quantum mechanics. It is this difficulty that we should overcome in order to apply the pair approximation. • The simplest pair approximation is S pair (spin zero) approximation. For the simplest single-j shell, Racah invented the seniority scheme which was generalized to many-j shells by Talmi in the early 70’s. • Broken pair approximation was developed by Allart, Boeker, Bonsignori, Gambhir and collaborators. In the broken pair approximation, one can calculate the cases in which there are one or two at most to pairs are not S pairs.

5. SD pair approximation Through the great success of the interacting boson model, developed by Arima and Iachello, it was realized that S and D nucleon pairs play the most important role in low-lying states of atomic nuclei. In the interacting boson model, SD pairs are approximated as sd bosons, for simplicity.

6. why SD pair approximation? The most important part of the residual interactions is the monopole pairing. If we had this interaction only, the ground states of even-even nuclei would be consisted of spin zero pairs. The quadrupole correlation is found to be also important, the quadrupole correlation leads to quadrupole excitations (and deformation). Then spin D pairs is also important in the low-lying states of atomic nuclei. For simplicity, let us neglect other correlation.

7. In the 80’s, many groups studied the microscopic foundation of the IBM. The first step along this line is to compute some matrix elements of shell model Hamiltonian in pair subspace. At that time, diagonalization of the shell model Hamiltonian in SD pair subspace was performed mostly in schematic cases. In 1992-1995, J. Q. Chen developed the Wick theorem for coupled clusters. In principle we can apply it to calculate all matrix elements of the shell model Hamiltonian. In 1998-2000, Chen and I, and other collaborators further developed the technique and established the model, called Nucleon pair approximation of the shell model, or Nucleon pair shell model (pair truncated shell model in References).

8. Framework

9. Formulation Pair subspace is constructed by operators acting on vacuum:

10. Hamiltonian： Separable form of the shell model Hamiltonian

11. J. Q. Chen, Nucl.Phys. A626, 686(1997)Y. M. Zhao et al., Phys.Rev.C62, 014304 (2000) Formulation One can apply the Nucleon pair approximation to even-even, odd-A, odd-odd nuclei on the same footing, with CPU decreased drastically.

12. Part IIPrevious results (what was done + what was open?)

13. Old works • Generalized seniority; • IBM microscopic foundation (v=4) • FDSM (symmetry dictated pairs) • Other schematic pairs

14. Higashiyama-Yoshinaga

15. Luo and collaborators

16. Problems How to determine y(abr) ? How do we believe that SD pairs are good or not good ? How to explain special cases by using valence pairs beyond SD ? We should compare the Shell model results and pair Approximation explicitly but this can not be done for heavy nuclei.

17. Preliminary calculations (only SD)

19. Prediction fit the experimental data accidentally ?

20. I thought over this question for some time without answer. • In October 2004, I came back from Japan to China, and I could find two excellent undergraduate students (Mr. Jia and Mr. Zhang, they are from another department but joined my lecture) who worked with me. Their purpose was to get basic training of scientific research, my purpose was to realize the pair truncation for odd and doubly odd nuclei (suggested by me in 2000). I can not work out the code by myself because I am very weak at writing numerical programs. There are not many calculations for odd-A and doubly odd nuclei by using pairs. We finished two codes (now we have six independent codes) and wrote two papers in Physical Review C.

21. SD pairs for even-even nuclei

22. SD配对，这里80个奇A核结构 磁矩计算(至今160核素)至今没有大误差！

23. What is new here? Nothing new in pair subspace, we find an efficient way to fix our parameters of the Hamiltonian in a large region. We calculated structure of many odd-A nuclei.

24. NPSMI 是我们(第一作者和第二作者 这是我带的本科生)计算结果。

25. 2008年韩国原子能研究所Kim等人发表在Nuclear Physics A 上论文

26. Why am I convinced by pair approximation ? Not by the citations but by our very recent work below: • We go beyond SD pair approximation. • We use low-energy pairs (SD are low energy pairs). In some states we should introduce other pairs. • We study pair structure coefficients in various truncated subspace (carefully). Now we are able to diagonalize full SD pair subspace (any SD pairs are included); We are able to add other pairs (subspace dimension around 10^4) to see the difference.

27. Validity of pair approximation Y. Lei, Z. Y. Xu, Y. M. Zhao, D. H. Lu, and A. Arima, Validity of pair approximation of the shell model (I): SD pairs approximation, in preparation; Z. Y. Xu, Y. Lei, Y. M. Zhao, and A. Arima, Validity of pair approximation of the shell model (II): effects of non-SD pairs, in preparation. As well as the validity of the truncation, one should pay attention to “systematics”. Pair approximation provides us with an opportunity to calculate low-lying states systematically.

28. Why pair approximation is applicable to transitional nuclei ? • Indeed low spin pairs are important; • Systematic calculations might be the key; • Non-SD pairs can be readily considered. • Pair structure are better known.

29. Next (hard) problem: the Shell Model Hamiltonian. What is the correct shell model Hamiltonian ? We know roughly about it.

30. Part III Our recent results

32. 取自雷杨和徐正宇的计算(配对J=0,2,4,6,8)：

33. 2000 年合作者: 吉永尚孝、山路修平(S.Yamaji)、 陈金全、有马朗人/ 4 papers in PRC62, 2000. 2003年合作者：Joe Ginocchio, 1 paper in PRC68, 2003. 2004年合作者：Arima, 1 paper in PRC70, 2004. 2006年-2007年(2007年初奇A系统程序首次完成)： 合作者：贾力源、张赫 (本科生、这个程序对我们很重要) 2 papers in PRC, 2007. 2007年-至今 (2007年底独立程序完成)： 合作者：雷杨、徐正宇(研究生,他们已经熟悉); 1 paper in PRC, 2009 + 2 preprints (2009). 2009年-至今 (2009年5月初独立程序完成)： 合作者：沈佳杰、姜慧(研究生，刚开始还不熟练).

34. 举例

35. 举例

36. (2008,unpublished)

38. Questions raised by Y. H. Zhang from Lanzhou : (work in progress) 1) d5/2h11/2, I=3-,4-,5-,6-,7-,8- . 2) g7/2h11/2, I=2-,3-,4-,5-,6-,7-,8-,9-. 3) h11/2h11/2, I=0+-----11+ . 4) d5/2h11/2与g7/2h11/2的mixing.

39. Part IV future works • Pair structure (hard question) • A~210 (odd and doubly odd, in preparation) ------------------------------------------------------------- • A~80-110, with S. Pittel • A~120, doubly odd, 1+ cluster (与Fujita) • Odd-odd nuclei

40. What can we do ? ？ Xu future Jia odd-odd Are you interested in low-lying states of these nuclei ?

41. Merit and demerit of NPA Calculated results are consistent with experimental data, predictions look good. We did not go to details of odd A yet. So far no prediction of new features of collective motion.

42. Another work by us Eigenvalue (of large matrices) J. J. Shen, Y. M. Zhao, A. Arima, and N. Yoshinaga, PRC77, 054312 (2008); PRC78, 044305 (2008); submitted to PRC; N. Yoshinaga, A. Arima, J. J. Shen, and Y. M. Zhao, PRC79, 017301 (2009); J. J. Shen and Y. M. Zhao, Science in China G.

43. Summary (pair approximation) • Background • Previous results • Our recent results • Future

44. 谢 谢 大 家 !

45. sd-IBM (也称作玻色子近似) 壳模型 配对近似 (直接用配对的价核子对 角化壳模型哈密顿量)

46. 与IBM的关系 • IBM中，SD价核子配对简化为sd 玻色子。 哈密顿量和组态空间全部由sd 玻色子构造。 极大简化计算 物理的美 (动力学对称性) 普遍应用和发展 SO(6) 极限、剪刀模式、F-旋、 超对称、新对称性

47. 已有数值计算 • S 配对近似: Talmi 等人 • 破对近似：Allaart, Gamhbir等 • SD 配对近似： 赵玉民及其合作者、罗延安及其合作者、 吴成礼(FDSM)等、日本Yoshinaga及其合作者、 罗马尼亚 Kwasniewicz、 法国Piepenbring、俄罗斯 Protosov 等 IBM 微观基础(主要是八十年代)： 国内杨立铭以及合作者、徐躬耦以及合作者等； 国际上在那个年代很多, 这里不列出。