随机相互作用下原子核结构
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随机相互作用下原子核结构 研究的新进展. 赵玉民. 上海交通大学物理系. 提纲. 随机相互作用原子核低激发态主要结果 最近其他研究组几个工作 我们最近的工作 展望. Part I. 随机相互作用下原子核的 规则结构的主要结果. Two-body Random ensemble (TBRE). Wigner introduced Gaussian orthogonal ensemble of random

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随机相互作用下原子核结构

研究的新进展

赵玉民

上海交通大学物理系


提纲

  • 随机相互作用原子核低激发态主要结果

  • 最近其他研究组几个工作

  • 我们最近的工作

  • 展望


Part i
Part I

随机相互作用下原子核的

规则结构的主要结果


Two-body Random ensemble (TBRE)

  • Wigner introduced Gaussian orthogonal ensemble of random

  • matrices (GOE) in understanding the spacings of energy levels

  • observed in resonances of slow neutron scattering on heavy nuclei.

  • Ref: Ann. Math. 67, 325 (1958)

  • 1970’s French, Wong, Bohigas, Flores introduced two-body random

  • ensemble (TBRE)

  • Ref: Rev. Mod. Phys. 53, 385 (1981);

  • Phys. Rep. 299, (1998);

  • Phys. Rep. 347, 223 (2001).

  • Original References:

  • J. B. French and S.S.M.Wong, Phys. Lett. B33, 449(1970);

  • O. Bohigas and J. Flores, Phys. Lett. B34, 261 (1970).

  • Other applications: complicated systems (e.g., quantum chaos)


  • What does 0 g.s. dominance mean ?

  • In 1998, Johnson, Bertsch, and Dean discovered that spin parity =0+ ground state dominance can be obtained by using random two-body interactions.

  • This result is called the 0 g.s. dominance.

  • Similar phenomenon was found in other systems, say, sd-boson systems.

  • C. W. Johnson et al., PRL80, 2749 (1998);

  • R. Bijker et al., PRL84, 420 (2000);

  • L. Kaplan et al., PRB65, 235120 (2002).


Two-body random ensemble(TBRE)

One usually choose Gaussian distribution for two-body random interactions

There are some people who use other distributions, for example,

A uniform distribution between -1 and 1. For our study, it is found that these different distribution present similar statistics.




Available results
Available Results

Empircal method Zhao & Arima & Yoshinaga (2002)

Mean-field method Bijker-Frank (2003)

Geometrid method Chau et al. (2003)

------------------------------------------------------------

Time reversal invariance (TRI) Zuker et al. (2002);

Time reversal invariance? Bijker&Frank&Pittel (1999);

Width ? Bijker&Frank (2000);

off-diagonal matrix elements for I=0 states Drozdz et al. (2001)

Highest symmetry &Time Reveral Otsuka&Shimizu(2004-2007)

Spectral Radius Papenbrock & Weidenmueller (2004-2007)

Semi-empirical formula Yoshinaga, Arima and Zhao(2006-2007)


References after johnson bertsch and dean
References after Johnson, Bertsch and Dean

R. Bijker, A. Frank, and S. Pittel, Phys. Rev. C60, 021302(1999); D. Mulhall, A. Volya, and V. Zelevinsky, Phys. Rev. Lett.85, 4016(2000); Nucl. Phys. A682, 229c(2001); V. Zelevinsky, D. Mulhall, and A. Volya, Yad. Fiz. 64, 579(2001); D. Kusnezov, Phys. Rev. Lett. 85, 3773(2000); ibid. 87, 029202 (2001); L. Kaplan and T. Papenbrock, Phys. Rev. Lett. 84, 4553(2000); R.Bijker and A.Frank, Phys. Rev. Lett.87, 029201(2001); S. Drozdz and M. Wojcik, Physica A301, 291(2001); L. Kaplan, T. Papenbrock, and C. W. Johnson, Phys. Rev. C63, 014307(2001); R. Bijker and A. Frank, Phys. Rev. C64, (R)061303(2001); R. Bijker and A. Frank, Phys. Rev. C65, 044316(2002); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, 061301 (2002); L. Kaplan, T.Papenbrock, and G.F. Bertsch, Phys. Rev. B65, 235120(2002); L. F. Santos, D. Kusnezov, and P. Jacquod, Phys. Lett. B537, 62(2002); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2004); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. C 73 014311 (2006); Y.M. Zhao and A. Arima, Phys. Rev.C64, (R)041301(2001); A. Arima, N. Yoshinaga, and Y.M. Zhao, Eur.J.Phys. A13, 105(2002); N. Yoshinaga, A. Arima, and Y.M. Zhao, J. Phys. A35, 8575(2002); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev.C66, 034302(2002); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev. C66, 064322(2002); Y.M.Zhao, A. Arima, N. Yoshinaga, Phys.Rev.C66, 064323 (2002); Y. M. Zhao, S. Pittel, R. Bijker, A. Frank, and A. Arima, Phys. Rev. C66, R41301 (2002); Y. M. Zhao, A. Arima, G. J. Ginocchio, and N. Yoshinaga, Phys. Rev. C66,034320(2003); Y. M. Zhao, A. Arima, N. Yoshinga, Phys. Rev. C68, 14322 (2003); Y. M. Zhao, A. Arima, N. Shimizu, K. Ogawa, N. Yoshinaga, O. Scholten, Phys. Rev. C70, 054322 (2004); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C71, 017304 (2005); Y. M. Zhao, A. Arima, N. Yoshida, K. Ogawa, N. Yoshinaga, and V.K.B.Kota , Phys. Rev. C72, 064314 (2005); N. Yoshinaga, A. Arima, and Y. M. Zhao, Phys. Rev. C73, 017303 (2006); Y. M. Zhao, J. L. Ping, A. Arima, Phys. Rev. C76, 054318 (2007); J. J. Shen, Y. M. Zhao, A. Arima, N. Yoshinaga, Physic. Rev. C77, 054312 (2008); J. J. Shen, A. Arima, Y. M. Zhao, N. Yoshinagan, Phys. Rev. C78, in press (2008); etc.

Review paper: 

Y.M. Zhao , A. Arima, and N. Yoshinaga,Physics Reports 400, 1 (2004).


  • Phenomenological method by our group (Zhao, Arima and Yoshinaga): reasonably applicable to all systems

  • Mean field method by Bijker and Frank group: sd, sp boson systems (Kusnezov also considered sp bosons in a similar way)

  • Geometric method suggested by Chau, Frank, Smirnova, and Isacker goes along the same line of our method (provided a foundation of our method for simple systems in which eigenvalues are in linear combinations of two-body interactions).




Summary of understandingof the 0 g.s. dominance

  • By using our phenomenological method, one can trace back what interactions, not only monopole pairing interaction but also some other terms with specific features, are responsible for 0 g.s. dominance. We understand that the Imax g.s. probability comes from the Jmax pairing interaction for single-j shell (also for bosons). The phenomenology also predicts spin I g.s. probabilities well. On the other hand, the reason of success of this method is not fully understood at a deep level, i.e., starting from a fundamental symmetry.

  • Bijker-Frank mean field applies very well to sp bosons and reasonably well to sd bosons.

  • Geometry method Chau, Frank, Sminova and Isacker is applicable to simple systems.


Other works

Time reversal invariance Zuker et al. (2002);

Time reversal invariance? Bijker&Frank&Pittel (1999);

Width ? Bijker&Frank (2000);

off-diagonal matrix elements for I=0 states Drozdz et al. (2001),

Highest symmetry hypothesis Otsuka&Shimizu(~2004),

Spectral Radius by Papenbrock & Weidenmueller (2004-2006)

Semi-empirical formula by Yoshinaga, Arima and Zhao(2006).


2. Energy centroids of spin I states

under random interactions


Other works on energy centroids
Other works on energy centroids

  • Mulhall, Volya, and Zelevinsky, PRL(2000)

  • Kota, PRC(2005)

  • YMZ, AA, Yoshida, Ogawa, Yoshinaga, and Kota, PRC(2005)

  • YMZ, AA, and Ogawa PRC(2005)



Collectivity in the ibm under random interactions
Collectivity in the IBM under random interactions

Taken from PRC62,014303(2000), by R. Bijker and A. Frank



Other works

Shell model:

Horoi, Zelevinsky, Volya, PRC, PRL;

Velazquez, Zuker, Frank, PRC;

Dean et al., PRC;

IBM:

Kusnezov, Casten, et al., PRL;

Geometric model:

Zhang, Casten, PRC;


Part II.

Recent efforts on nuclei under

random interactions


Recent efforts on 0 g s dominance
Recent efforts on 0 g.s. dominance

Highest symmetry &Time Reveral

Otsuka & Shimizu(2004-2007)

Spectral Radius

Papenbrock & Weidenmueller (2004-2007)

Semi-empirical formula

Yoshinaga, Arima and Zhao(2006-2007)


集体运动模式

YMZ, Pittel, Bijker, Frank, and AA, PRC66, 041301 (2002).

(By using usual SD pairs)

YMZ, J. L. Ping, and AA, PRC76, 054318 (2007).

(By using symmetry dictated pairs--FDSM)

Calvin W. Johnson, Hai Ah Nam, PRC75, 047305 (2007). Shell model calculations


随机相互作用下宇称分布规律

  • (A) Both protons and neutrons are in the shell which corresponds to nuclei with both proton number Z and neutron number N ~40;

  • (B) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~40 and N~50;

  • (C) Both protons and neutrons are in the shell which correspond to nuclei with Z and N~82;

  • (D) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~50 and N~82.




Part III.

我们最近的工作


我们最近的工作(1):矩阵的本征值问题(最低本征值和所有本征值)

  • “Lowest Eigenvalues of Random Hamiltonians”(2008).

  • J. J. Shen, Y. M. Zhao, A. Arima, and N. Yoshinaga, Physical Review C77, 054312.

  • “Strong Linear Correlation Between Eigenvalues and Diagonal Matrix elements”,

  • J. J. Shen, A. Arima, Y. M. Zhao, and N. Yoshinaga, Physical Review C(2008).

  • N. Yoshinaga, A. Arima, J. J. Shen, and Y. M. Zhao,

  • “Functional Dependence of eignevalues and diagonal matrix elements”,

  • submitted to PRC.

  • J. J. Shen and Y. M. Zhao, in preparation.

  • A. Arima, Inter. J. Mod. Phys. E, in press.

  • 这些工作属于无心插柳的性质。当时(2006年)沈佳杰大学三年级时要做科研, 当时量子力学 还没有学过, 所以只能用计算机玩玩。2006年吉永教授(N. Yoshinaga)、有马教授 (Akito Arima)和我得到了一个最低本征值的、非常简单的半经验公式(平均能量、 分布宽度和维数),我希望能够更加精确一些,比如能否引入高级距修正。但是没有特别的结果。沈佳杰通过有趣的尝试和大量的努力,终于得到 很多结果。


我们这个发现的重要意义

  • 对角化大矩阵是很困难的

  • 我们意外发现本征值与对角元之间存在简单函数关系,壳模型情形呈线性关系。

  • 参考沈佳杰的报告


2 fdsm
我们最近的工作(2):FDSM 内的集体运动

  • FDSM 与 IBM 的相似:

    类似的群结构/ SU(3), SU(5) group chains

    费米子/玻色子自由度;对力+四极力;SD配对

  • FDSM 与 IBM 的不同:

    SO(8) 没有转动极限


SO(8) 极限

SP(6)

极限


总结

  • 随机相互作用原子核的主要结果

  • 最近的主要进展

  • 我们的两个工作

  • 展望


Acknowledgements:

Akito Arima (Tokyo)

Naotaka Yoshinagana (Saitama)

贾力源(上海交大本科生, went to MSU last summer)

张丽华(上海交通大學物理系硕博联读,from April 06)

沈佳杰(上海交通大學物理系直博生,from Sep.07)

雷 扬(上海交通大學物理系直博生,from Sep.07)

徐正宇(上海交通大學物理系直硕生,from Sep.07)

姜 慧(上海交通大學物理系博士生,from Sep.08)

李晨光(上海交大本科生, 预计09年直硕)



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