原子核壳模型及随机矩阵的本征值研究
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摘要
随着世界上大型科学装置(例如美国放射性束流装置（RIA）、日本理化研究所的放射性同位素核束装置)的建立和运行，原子核物理再度成为了科学界的焦点之一，虽然原子核壳模型为我们提供了一个原子核结构理论的坚实基础，但是他的组态空间是非常巨大的以至于当今世界上最强大的超级计算机也无法处理中重核的问题。因此我们对原子核壳模型采取各种组态截断是必须的。其中有大量的工作试图将巨大的原子核壳模型空间截断为核子对子空间。另一方面，慢中子散射实验得到的非常密集的巨共振能谱是壳模型无法解释的，魏格纳等人引入了随机矩阵理论并成功的研究了原子核能谱的统计性质。这两种物理图像完整地描述了原子核结构的图像。
在第一章我们简要的介绍了原子核壳模型和随机矩阵的一些基本概念。在原子核壳模型部分我们简要介绍了单粒子模型以及单粒子能级加上剩余两体相互作用壳模型哈密顿量；在随机矩阵理论部分我们简要的介绍了高斯正交系综和随机两体系综以及他们的应用（包括最近邻能级间隔分布、能级刚度以及随机相互作用下原子核的性质）。
在第二章我们研究了随机矩阵（包括随机两体系综和高斯正交系综）的最小本征值问题。我们给出了以一个使用随机矩阵本征值中心值以及能谱宽度的经验公式。此经验公式同时适用于费米子系统和波色子系统。然后我们通过引入高阶矩（三阶矩）改善了经验公式预言基态能级的精度。最后我们将此经验公式推广到了随机相互作用下原子核低激发态能级上。除了研究随机矩阵系综的本征值问题以外，我们还研究了随机矩阵本征值与对角元的相关性，我们发现壳模型以及随机两体系综的哈密顿量的本征能量与对角元曾在一个非常好的线性关系。通过这个线性关系我们可以不对角化哈密顿量直接预言所有本征能量。我们还发现对于高斯正交系综其本征值与对角元存在着双曲正切的函数关系。
在第三章里我们研究了原子核壳模型哈密顿量矩阵元分布的几何结构。通过将壳模型的基矢按照哈密顿量对角元排序我们可以按照基矢的能量对壳模型空间做组态截断。我们提出了截断空间下哈密顿量本征能量与位数的自然对数的线性外推关系。我们发现线性外推方法可以很好的适用于sd壳和pf壳。在此基础上我们将微扰方法和外推方法结合在一起得到了二次外推方法。我们通过这一结合将我们预言低激发态能谱的精度控制在了40–60keV。最后我们通过将壳模型哈密顿量表示为截断空间下的有效哈密顿量形式的得到了各阶微扰的一般形式。我们举例说明了高阶微扰可以大为改善截断空间下能级和波函数预言的精度。
在附录中给出我们一些进行中的工作：原子核壳模型的随机矩阵模拟（A）；组态近似方法在随机矩阵上的应用（B）；原子核壳模型的配对近似方法在208铅区域中原子核的应用（C）。
As the availability of large-scale facilities (such as RIA in USA and RIKEN-RIBF in Japan)in the last decade, nuclear physics has become one of the focuses in science. Although the nuclearshell model provide us with a firm foundation of nuclear structure theory, its configuration spaceis too gigantic to handle even for the best computers. Therefore, various truncation schemes arenecessary. Many efforts have been devote to truncate the huge shell model space to nucleon-pairsubspaces. In another context, the slow neutron scattering experiments showed numerous andnarrow resonances, due to which Wigner introduce random matrix theory. These two pictures giveus a complementary description of nuclear structure.
In chapter one, we present a short introduction of the nuclear shell model and the randommatrix theory. We introduce the single particle model and the one-body plus two-body shell modelHamiltonian. We also introduce the Gaussian orthogonal ensemble and the two-body randomensemble as well as their applications in nuclear structure studies.
In chapter two, we study the lowest eigenvalues of random Hamiltonian including two-bodyrandom ensemble and Gaussian orthogonal ensemble in both fermion and boson systems. We showthat an empirical formula, which can evaluate the lowest eigenvalues of random Hamiltonian, interms of energy centroid and widths of eigenvalues. It is applicable to many different systems. Weimprove the accuracy of the formula by considering the third central moment. We show that theseformulas are applicable not only to the evaluation of the lowest energy but also to the evaluationof excited energies of systems under random two-body interactions. Moreover, for the two-bodyrandom ensemble, we find a strong linear correlation between eigenvalues and diagonal matrixelements if both of them are sorted from the smaller values to larger ones. By using this linearcorrelation, we are able to reasonably predict all eigenvalues of a given Hamiltonian matrix with-out complicated iterations. For Gaussian orthogonal ensemble matrices, the hyperbolic tangentfunction improves the accuracy of predicted eigenvalues near the minimum and maximum.
In chapter three, we investigate regular patterns of matrix elements of the nuclear shell modelHamiltonian, by sorting the diagonal matrix elements from the smaller to the larger values. Byusing simple plots of nonzero matrix elements and lowest eigenvalues of artificially constructed sub-matrices of Hamiltonian, we propose a new and simple truncation method, which predicts thelowest eigenvalue with remarkable precision. Based on this truncation method, we study extrap-olation approaches to evaluate energies of low-lying states for nuclei in the sd and pf shells, bysorting the diagonal matrix elements of the nuclear shell-model Hamiltonian. We introduce anextrapolation method with perturbation and apply our new method to predict both energies of low-lying states and E2transition rates between these states. Our predicted results arrive at an accuracyof the root-mean-squared deviations40–60keV for low-lying states. We also develop a newperturbation method of obtaining the lowest eigenvalues of the nuclear shell model Hamiltonian.Moreover, we obtain the effective Hamiltonian in the truncated space. We exemplify it by using afew realistic nuclei. Overlaps between the wave functions of our approach and those of the exactshell model calculation are presented. Some of the electromagnetic quantities are also discussed.
In the appendix, we explain a few works in progress: Simulation of realistic shell modelHamiltonian by using random matrices (A); truncation approaches of random matrices (B); paritruncation schemes of some nuclei around208Pb (C).
在第一章我们简要的介绍了原子核壳模型和随机矩阵的一些基本概念。在原子核壳模型部分我们简要介绍了单粒子模型以及单粒子能级加上剩余两体相互作用壳模型哈密顿量；在随机矩阵理论部分我们简要的介绍了高斯正交系综和随机两体系综以及他们的应用（包括最近邻能级间隔分布、能级刚度以及随机相互作用下原子核的性质）。
在第二章我们研究了随机矩阵（包括随机两体系综和高斯正交系综）的最小本征值问题。我们给出了以一个使用随机矩阵本征值中心值以及能谱宽度的经验公式。此经验公式同时适用于费米子系统和波色子系统。然后我们通过引入高阶矩（三阶矩）改善了经验公式预言基态能级的精度。最后我们将此经验公式推广到了随机相互作用下原子核低激发态能级上。除了研究随机矩阵系综的本征值问题以外，我们还研究了随机矩阵本征值与对角元的相关性，我们发现壳模型以及随机两体系综的哈密顿量的本征能量与对角元曾在一个非常好的线性关系。通过这个线性关系我们可以不对角化哈密顿量直接预言所有本征能量。我们还发现对于高斯正交系综其本征值与对角元存在着双曲正切的函数关系。
在第三章里我们研究了原子核壳模型哈密顿量矩阵元分布的几何结构。通过将壳模型的基矢按照哈密顿量对角元排序我们可以按照基矢的能量对壳模型空间做组态截断。我们提出了截断空间下哈密顿量本征能量与位数的自然对数的线性外推关系。我们发现线性外推方法可以很好的适用于sd壳和pf壳。在此基础上我们将微扰方法和外推方法结合在一起得到了二次外推方法。我们通过这一结合将我们预言低激发态能谱的精度控制在了40–60keV。最后我们通过将壳模型哈密顿量表示为截断空间下的有效哈密顿量形式的得到了各阶微扰的一般形式。我们举例说明了高阶微扰可以大为改善截断空间下能级和波函数预言的精度。
在附录中给出我们一些进行中的工作：原子核壳模型的随机矩阵模拟（A）；组态近似方法在随机矩阵上的应用（B）；原子核壳模型的配对近似方法在208铅区域中原子核的应用（C）。
As the availability of large-scale facilities (such as RIA in USA and RIKEN-RIBF in Japan)in the last decade, nuclear physics has become one of the focuses in science. Although the nuclearshell model provide us with a firm foundation of nuclear structure theory, its configuration spaceis too gigantic to handle even for the best computers. Therefore, various truncation schemes arenecessary. Many efforts have been devote to truncate the huge shell model space to nucleon-pairsubspaces. In another context, the slow neutron scattering experiments showed numerous andnarrow resonances, due to which Wigner introduce random matrix theory. These two pictures giveus a complementary description of nuclear structure.
In chapter one, we present a short introduction of the nuclear shell model and the randommatrix theory. We introduce the single particle model and the one-body plus two-body shell modelHamiltonian. We also introduce the Gaussian orthogonal ensemble and the two-body randomensemble as well as their applications in nuclear structure studies.
In chapter two, we study the lowest eigenvalues of random Hamiltonian including two-bodyrandom ensemble and Gaussian orthogonal ensemble in both fermion and boson systems. We showthat an empirical formula, which can evaluate the lowest eigenvalues of random Hamiltonian, interms of energy centroid and widths of eigenvalues. It is applicable to many different systems. Weimprove the accuracy of the formula by considering the third central moment. We show that theseformulas are applicable not only to the evaluation of the lowest energy but also to the evaluationof excited energies of systems under random two-body interactions. Moreover, for the two-bodyrandom ensemble, we find a strong linear correlation between eigenvalues and diagonal matrixelements if both of them are sorted from the smaller values to larger ones. By using this linearcorrelation, we are able to reasonably predict all eigenvalues of a given Hamiltonian matrix with-out complicated iterations. For Gaussian orthogonal ensemble matrices, the hyperbolic tangentfunction improves the accuracy of predicted eigenvalues near the minimum and maximum.
In chapter three, we investigate regular patterns of matrix elements of the nuclear shell modelHamiltonian, by sorting the diagonal matrix elements from the smaller to the larger values. Byusing simple plots of nonzero matrix elements and lowest eigenvalues of artificially constructed sub-matrices of Hamiltonian, we propose a new and simple truncation method, which predicts thelowest eigenvalue with remarkable precision. Based on this truncation method, we study extrap-olation approaches to evaluate energies of low-lying states for nuclei in the sd and pf shells, bysorting the diagonal matrix elements of the nuclear shell-model Hamiltonian. We introduce anextrapolation method with perturbation and apply our new method to predict both energies of low-lying states and E2transition rates between these states. Our predicted results arrive at an accuracyof the root-mean-squared deviations40–60keV for low-lying states. We also develop a newperturbation method of obtaining the lowest eigenvalues of the nuclear shell model Hamiltonian.Moreover, we obtain the effective Hamiltonian in the truncated space. We exemplify it by using afew realistic nuclei. Overlaps between the wave functions of our approach and those of the exactshell model calculation are presented. Some of the electromagnetic quantities are also discussed.
In the appendix, we explain a few works in progress: Simulation of realistic shell modelHamiltonian by using random matrices (A); truncation approaches of random matrices (B); paritruncation schemes of some nuclei around208Pb (C).
引文
[1] MAYER M G. Nuclear Configurations in the Spin-Orbit Coupling Model. I. EmpiricalEvidence[J]. Phys. Rev.,1950,78:16–21. http://link.aps.org/doi/10.1103/PhysRev.78.16.
[2] MAYER M G. On Closed Shells in Nuclei[J]. Phys. Rev.,1948,74:235–239. http://link.aps.org/doi/10.1103/PhysRev.74.235.
[3] MAYER M G. On Closed Shells in Nuclei. II[J]. Phys. Rev.,1949,75:1969–1970. http://link.aps.org/doi/10.1103/PhysRev.75.1969.
[4] MAYER M, JENSEN J. Elementary theory of nuclear shell structure[M]. Structureof matter series.[S.l.]: Wiley,1960. http://books.google.co.jp/books?id=z24sAAAAYAAJ.
[5] AJZENBERG-SELOVE F. Energy levels of light nuclei[J]. Nuclear Physics A,1988,490(1):1–225. http://www.sciencedirect.com/science/article/pii/0375947488901248.
[6] WOODS R D, SAXON D S. Diffuse Surface Optical Model for Nucleon-Nuclei Scatter-ing[J]. Phys. Rev.,1954,95:577–578. http://link.aps.org/doi/10.1103/PhysRev.95.577.
[7] ROSS A A, MARK H, LAWSON R D. Nucleon Energy Levels in a Diffuse Poten-tial[J]. Phys. Rev.,1956,102:1613–1620. http://link.aps.org/doi/10.1103/PhysRev.102.1613.
[8] WALET N. Nuclear and Particle Physics[J]. Physics OER. http://oer.physics.manchester.ac.uk/NP/.
[9] THOMAS L H. The Motion of the Spinning Electron[J]. Nature,1926,117:514. http://adsabs.harvard.edu/abs/1926Natur.117..514T.
[10] RACAH G. Theory of Complex Spectra. III[J]. Phys. Rev.,1943,63:367–382. http://link.aps.org/doi/10.1103/PhysRev.63.367.
[11] M. H. The Nuclear SU3Model[J]. Advances in Nuclear Physics,1968,1:67.
[12] CHEN J Q. The Wick theorem for coupled fermion clusters[J]. Nuclear Physic-s A,1993,562(2):218–240. http://www.sciencedirect.com/science/article/pii/0375947493901976.
[13] JIN-QUAN C, BING-QING C, KLEIN A. Factorization of commutators: The Wick theo-rem for coupled operators[J]. Nuclear Physics A,1993,554(1):61–76. http://www.sciencedirect.com/science/article/pii/0375947493903574.
[14] CHEN J Q. Nucleon-pair shell model: Formalism and special cases[J]. Nuclear Physic-s A,1997,626(3):686–714. http://www.sciencedirect.com/science/article/pii/S0375947497005022.
[15] CHEN J Q, LUO Y A. Nucleon-pair shell model: The effects of the SD pair structure oncollectivity of low-lying states[J]. Nuclear Physics A,1998,639:615–634. http://www.sciencedirect.com/science/article/pii/S0375947498004229.
[16] ZHAO Y M, YOSHINAGA N, YAMAJI S, et al. Nucleon-pair approximation of the shellmodel: Unified formalism for both odd and even systems[J]. Phys. Rev. C,2000,62:014304.http://link.aps.org/doi/10.1103/PhysRevC.62.014304.
[17] ZHAO Y M, YAMAJI S, YOSHINAGA N, et al. Nucleon pair approximation of the nuclearcollective motion[J]. Phys. Rev. C,2000,62:014315. http://link.aps.org/doi/10.1103/PhysRevC.62.014315.
[18] JIA L Y, ZHANG H, ZHAO Y M. Systematic calculations of low-lying states of even-evennuclei within the nucleon pair approximation[J]. Phys. Rev. C,2007,75:034307. http://link.aps.org/doi/10.1103/PhysRevC.75.034307.
[19] JIA L Y, ZHANG H, ZHAO Y M. Systematic calculations of low-lying states in odd-Anuclei within the nucleon pair approximation[J]. Phys. Rev. C,2007,76:054305. http://link.aps.org/doi/10.1103/PhysRevC.76.054305.
[20] XU Z Y, LEI Y, ZHAO Y M, et al. Low-lying states of heavy nuclei within the nucleon pairapproximation[J]. Phys. Rev. C,2009,79:054315. http://link.aps.org/doi/10.1103/PhysRevC.79.054315.
[21] JIANG H, SHEN J J, ZHAO Y M, et al. Low-lying states of valence-hole nuclei in the208Pb region[J]. Journal of Physics G: Nuclear and Particle Physics,2011,38(4):045103.http://stacks.iop.org/0954-3899/38/i=4/a=045103.
[22] YAN-AN L, CHEN J Q. Shell model calculation in the S D subspace[J]. Phys. Rev. C,1998,58:589–592. http://link.aps.org/doi/10.1103/PhysRevC.58.589.
[23] LUO Y A, CHEN J Q, DRAAYER J. Nucleon-pair shell model calculations of the even-even Xe and Ba nuclei[J]. Nuclear Physics A,2000,669:101–118. http://www.sciencedirect.com/science/article/pii/S0375947499008180.
[24] YOSHINAGA N, HIGASHIYAMA K. Systematic studies of nuclei around mass130in thepair-truncated shell model[J]. Phys. Rev. C,2004,69:054309. http://link.aps.org/doi/10.1103/PhysRevC.69.054309.
[25] HIGASHIYAMA K, YOSHINAGA N, TANABE K. New band mechanism of doubly-oddnuclei around mass130[J]. Phys. Rev. C,2005,72:024315. http://link.aps.org/doi/10.1103/PhysRevC.72.024315.
[26] ZHAO Y M, YOSHINAGA N, YAMAJI S, et al. Relationship between the fermion dynam-ical symmetric model Hamiltonian and nuclear collective motion[J]. Phys. Rev. C,2000,62:024322. http://link.aps.org/doi/10.1103/PhysRevC.62.024322.
[27] LEI Y, XU Z Y, ZHAO Y M, et al. Validity of pair truncation of the nuclear shell model in46Ca[J]. Phys. Rev. C,2009,80:064316. http://link.aps.org/doi/10.1103/PhysRevC.80.064316.
[28] LEI Y, XU Z Y, ZHAO Y M, et al. Validity of pair truncations with effective interactionin Ca isotopes[J]. Phys. Rev. C,2010,82:034303. http://link.aps.org/doi/10.1103/PhysRevC.82.034303.
[29] WU C L, FENG D H, CHEN X G, et al. Fermion dynamical symmetry model of nuclei:Basis, Hamiltonian, and symmetries[J]. Phys. Rev. C,1987,36:1157–1180. http://link.aps.org/doi/10.1103/PhysRevC.36.1157.
[30] K. H, Y. S. Projected shell model and high spin spectroscopy[J]. InternationalJournal of Modern Physics E,1995,4:637–785. http://dx.doi.org/10.1142/S0218301395000250.
[31] ARIMA A, IACHELLO F. Collective Nuclear States as Representations of a SU(6) Group[J].Phys. Rev. Lett.,1975,35:1069–1072. http://link.aps.org/doi/10.1103/PhysRevLett.35.1069.
[32] IACHELLO B H P, F., DIEPERINK A E L. Collective aspects of the shell model[J]. Inter-national Conference on Nuclear Structure and Spectroscopy,1974,2:163–181.
[33] JANSSEN D, JOLOS R, DONAU F. An algebraic treatment of the nuclear quadrupoledegree of freedom[J]. Nuclear Physics A,1974,224(1):93–115. http://www.sciencedirect.com/science/article/pii/0375947474901651.
[34] ARIMA A, OHTSUKA T, IACHELLO F, et al. Collective nuclear states as sym-metric couplings of proton and neutron excitations[J]. Physics Letters B,1977,66(3):205–208. http://www.sciencedirect.com/science/article/pii/0370269377908607.
[35] OTSUKA T, ARIMA A, IACHELLO F, et al. Shell model description of interacting boson-s[J]. Physics Letters B,1978,76(2):139–143. http://www.sciencedirect.com/science/article/pii/0370269378902605.
[36] ARIMA A, IACHELLO F. Interacting boson model of collective nuclear states II.The rotational limit[J]. Annals of Physics,1978,111(1):201–238. http://www.sciencedirect.com/science/article/pii/0003491678902282.
[37] ARIMA A, IACHELLO F. Interacting boson model of collective nuclear states IV. The O(6)limit[J]. Annals of Physics,1979,123(2):468–492. http://www.sciencedirect.com/science/article/pii/0003491679903476.
[38] ARIMA A, IACHELLO F. Interacting boson model of collective states I. The vibrationallimit[J]. Annals of Physics,1976,99(2):253–317. http://www.sciencedirect.com/science/article/pii/000349167690097X.
[39] BEHRENS H, GENZ H, CONZE M, et al. Allowed β-transitions, weak magnetism andnuclear structure in light nuclei[J]. Annals of Physics,1978,115(2):276–324. http://www.sciencedirect.com/science/article/pii/0003491678901586.
[40] BOHR A, MOTTELSON B R. Nuclear Structure: Nuclear deformations[M]. NuclearStructure.[S.l.]: W. A. Benjamin,1969. http://books.google.co.jp/books?id=PpofAQAAMAAJ.
[41] D. S B, D. W J. The Nuclear SU3Model[J]. Advances in Nuclear Physics,1986,16:1.
[42] REINHARD P G. The relativistic mean-field description of nuclei and nuclear dynamic-s[J]. Reports on Progress in Physics,1989,52(4):439. http://stacks.iop.org/0034-4885/52/i=4/a=002.
[43] RING P. Relativistic mean field theory in finite nuclei[J]. Progress in Particle and Nucle-ar Physics,1996,37(0):193–263. http://www.sciencedirect.com/science/article/pii/0146641096000543.
[44] MENG J, RING P. Relativistic Hartree-Bogoliubov Description of the Neutron Halo in11Li[J]. Phys. Rev. Lett.,1996,77:3963–3966. http://link.aps.org/doi/10.1103/PhysRevLett.77.3963.
[45] MENG J, RING P. Giant Halo at the Neutron Drip Line[J]. Phys. Rev. Lett.,1998,80:460–463. http://link.aps.org/doi/10.1103/PhysRevLett.80.460.
[46] REN Z, FAESSLER A, BOBYK A. Relativistic mean-field description of a proton haloin the first excited (1/2)+state of17F[J]. Phys. Rev. C,1998,57:2752–2755. http://link.aps.org/doi/10.1103/PhysRevC.57.2752.
[47] ZHANG W, MENG J, ZHANG S, et al. Magic numbers for superheavy nucle-i in relativistic continuum Hartree鈥揃ogoliubov theory[J]. Nuclear Physics A,2005,753:106–135. http://www.sciencedirect.com/science/article/pii/S0375947405002423.
[48] LV H, MENG J, ZHANG S, et al. Neutron halos in hypernuclei[J]. The European PhysicalJournal A-Hadrons and Nuclei,2003,17:19–24. http://dx.doi.org/10.1140/epja/i2002-10136-3.10.1140/epja/i2002-10136-3.
[49] MENG J, SUGAWARA-TANABE K, YAMAJI S, et al. Pseudospin symmetry in Zr and Snisotopes from the proton drip line to the neutron drip line[J]. Phys. Rev. C,1999,59:154–163. http://link.aps.org/doi/10.1103/PhysRevC.59.154.
[50] MIZUSAKI T. Shell model calculation–from basics to the latest methods[J].4th CNS International Summer School. http://www.cns.s.u-tokyo.ac.jp/summerschool/ciss05/lecturenotes/.
[51] BOHR A, MOTTELSON B. Nuclear Structure: Single-particle motion[M]. Nuclear Struc-ture.[S.l.]: W. A. Benjamin,1969. http://books.google.co.jp/books?id=ApoRKQEACAAJ.
[52] WIGNER E P. Random Matrices in Physics[J]. SIAM Rev.,1966,9:1–23. http://dx.doi.org/10.1137/1009001.
[53] SHEN J J, ZHAO Y M, ARIMA A, et al. Lowest eigenvalues of random Hamiltoni-ans[J]. Phys. Rev. C,2008,77:054312. http://link.aps.org/doi/10.1103/PhysRevC.77.054312.
[54] SHEN J J, ARIMA A, ZHAO Y M, et al. Strong correlation between eigenvalues and di-agonal matrix elements[J]. Phys. Rev. C,2008,78:044305. http://link.aps.org/doi/10.1103/PhysRevC.78.044305.
[55] ZHANG L H, SHEN J J, LEI Y, et al. Robustness of Strong Correlation between Eigenval-ues and Diagonal Matrix Elements[J]. International Journal of Modern Physics E,2008,17,Supplement:342–351. http://dx.doi.org/10.1142/S0218301308011975.
[56] YOSHINAGA N, ARIMA A, SHEN J J, et al. Correlation between eigenvalues and sorteddiagonal elements of a large dimensional matrix[J]. Phys. Rev. C,2009,79:017301. http://link.aps.org/doi/10.1103/PhysRevC.79.017301.
[57] SHEN J, ZHAO Y. How random are matrix elements of the nuclear shell model Hamilto-nian?[J]. Science in China Series G: Physics Mechanics and Astronomy,2009,52:1477–1481. http://dx.doi.org/10.1007/s11433-009-0198-7.10.1007/s11433-009-0198-7.
[58] SHEN J J, ZHAO Y M, ARIMA A. Lowest eigenvalue of the nuclear shell model Hamil-tonian[J]. Phys. Rev. C,2010,82:014309. http://link.aps.org/doi/10.1103/PhysRevC.82.014309.
[59] SHEN J J, ZHAO Y M, ARIMA A, et al. New extrapolation method for low-lying states ofnuclei in the sd and the pf shells[J]. Phys. Rev. C,2011,83:044322. http://link.aps.org/doi/10.1103/PhysRevC.83.044322.
[60] SHEN J J, ZHAO Y M, ARIMA A. New perturbation method of diagonalizing the nuclearshell model Hamiltonian[J]. Phys. Rev. C,2012,85:064325. http://link.aps.org/doi/10.1103/PhysRevC.85.064325.
[61] JOHNSON C W, BERTSCH G F, DEAN D J. Orderly Spectra from Random Interaction-s[J]. Phys. Rev. Lett.,1998,80:2749–2753. http://link.aps.org/doi/10.1103/PhysRevLett.80.2749.
[62] ZHAO Y, ARIMA A, YOSHINAGA N. Regularities of many-body systems interacting bya two-body random ensemble[J]. Physics Reports,2004,400(1):1–66. http://www.sciencedirect.com/science/article/pii/S0370157304002972.
[63] ZELEVINSKY V, VOLYA A. Nuclear structure, random interactions and mesoscopic physic-s[J]. Physics Reports,2004,391:311–352. http://www.sciencedirect.com/science/article/pii/S0370157303004319. ce:title From atoms to nuclei toquarks and gluons: the omnipresent manybody theory/ce:title.
[64] PAPENBROCK T, WEIDENM U¨LLER H A. Colloquium: Random matrices and chaos innuclear spectra[J]. Rev. Mod. Phys.,2007,79:997–1013. http://link.aps.org/doi/10.1103/RevModPhys.79.997.
[65] PAPENBROCK T, WEIDENM U¨LLER H A. Distribution of Spectral Widths and Preponder-ance of Spin-0Ground States in Nuclei[J]. Phys. Rev. Lett.,2004,93:132503. http://link.aps.org/doi/10.1103/PhysRevLett.93.132503.
[66] PAPENBROCK T, WEIDENM U¨LLER H A. Two-body random ensemble in nuclei[J]. Phys.Rev. C,2006,73:014311. http://link.aps.org/doi/10.1103/PhysRevC.73.014311.
[67] YOSHINAGA N, ARIMA A, ZHAO Y M. Lowest bound of energies for random interac-tions and the origin of spin-zero ground state dominance in even-even nuclei[J]. Phys.Rev. C,2006,73:017303. http://link.aps.org/doi/10.1103/PhysRevC.73.017303.
[68] ABRAMOWITZ M, STEGUN I. Handbook of Mathematical Functions: With Formulas,Graphs, and Mathematical Tables[M]. Applied mathematics series.[S.l.]: Dover Publi-cations,1964. http://books.google.co.jp/books?id=MtU8uP7XMvoC.
[69] RATCLIFF K F. Applications of Spectral Distributions in Nuclear Spectroscopy[J]. Phys.Rev. C,1971,3:117–143. http://link.aps.org/doi/10.1103/PhysRevC.3.117.
[70] WONG S. Nuclear statistical spectroscopy[M]. Oxford studies in nuclear physic-s.[S.l.]: Oxford University Press,1986. http://books.google.co.jp/books?id=-jZRAAAAMAAJ.
[71] MARGETAN F J, KLAR A, VARY J P. Nuclear binding energies from moment methods:Harmonic oscillator Hamiltonian[J]. Phys. Rev. C,1983,27:852–861. http://link.aps.org/doi/10.1103/PhysRevC.27.852.
[72] VEL A′ZQUEZ V, ZUKER A P. Spectroscopy with Random and Displaced Random Ensem-bles[J]. Phys. Rev. Lett.,2002,88:072502. http://link.aps.org/doi/10.1103/PhysRevLett.88.072502.
[73] IACHELLO F, ARIMA A. The Interacting Boson Model[M]. Cambridge Monographson Mathematical Physics.[S.l.]: Cambridge University Press,1987. http://books.google.co.jp/books?id=fB7Qe73VmsgC.
[74] ZHAO Y M, ARIMA A, YOSHINAGA N. Many-body systems interacting via a two-body random ensemble. I. Angular momentum distribution in the ground states[J]. Phys.Rev. C,2002,66:064322. http://link.aps.org/doi/10.1103/PhysRevC.66.064322.
[75] MON K, FRENCH J. Statistical properties of many-particle spectra[J]. Annals of Physics,1975,95(1):90–111. http://www.sciencedirect.com/science/article/pii/0003491675900457.
[76] PRESS W, TEUKOLSKY S, VETTERLING W, et al. Numerical recipes in C++: the artof scientific computing[M].[S.l.]: Cambridge University Press,2005. http://books.google.co.jp/books?id=DevQtgAACAAJ.
[77] WEIDENM U¨LLER H A, MITCHELL G E. Random matrices and chaos in nuclear physics:Nuclear structure[J]. Rev. Mod. Phys.,2009,81:539–589. http://link.aps.org/doi/10.1103/RevModPhys.81.539.
[78] WILDENTHAL B. Empirical strengths of spin operators in nuclei[J]. Progress in Parti-cle and Nuclear Physics,1984,11(0):5–51. http://www.sciencedirect.com/science/article/pii/0146641084900115.
[79] BROWN B A, WILDENTHAL B H. Status of the Nuclear Shell Model[J]. Annu. Rev. Nucl.Part. Sci.,1988,38:29–66. http://dx.doi.org/10.1146/annurev.nucl.38.1.29.
[80] FRENCH J, WONG S. Validity of random matrix theories for many-particle systems[J].Physics Letters B,1970,33(7):449–452. http://www.sciencedirect.com/science/article/pii/0370269370902133.
[81] BRODY T A, FLORES J, FRENCH J B, et al. Random-matrix physics: spectrum and strengthfluctuations[J]. Rev. Mod. Phys.,1981,53:385–479. http://link.aps.org/doi/10.1103/RevModPhys.53.385.
[82] BOHIGAS O, FLORES J. Two-body random hamiltonian and level density[J]. PhysicsLetters B,1971,34(4):261–263. http://www.sciencedirect.com/science/article/pii/0370269371905983.
[83] ZELEVINSKY V, BROWN B, FRAZIER N, et al. The nuclear shell model as a testing groundfor many-body quantum chaos[J]. Physics Reports,1996,276:85–176. http://www.sciencedirect.com/science/article/pii/S0370157396000075.
[84] ZELEVINSKY V, BROWN B, FRAZIER N, et al. The nuclear shell model as a testing groundfor many-body quantum chaos[J]. Physics Reports,1996,276:85–176. http://www.sciencedirect.com/science/article/pii/S0370157396000075.
[85] ARIMA A, IACHELLO F. Interacting boson model of collective states I. The vibrationallimit[J]. Annals of Physics,1976,99(2):253–317. http://www.sciencedirect.com/science/article/pii/000349167690097X.
[86] ARIMA A, IACHELLO F. Interacting boson model of collective nuclear states II.The rotational limit[J]. Annals of Physics,1978,111(1):201–238. http://www.sciencedirect.com/science/article/pii/0003491678902282.
[87] KUO T, BROWN G. Reaction matrix elements for the0f-1p shell nuclei[J]. NuclearPhysics A,1968,114(2):241–279. http://www.sciencedirect.com/science/article/pii/0375947468903539.
[88] TAKADA K. ftp://ftp.kutl.kyushu-u.ac.jp/pub/takada/jjSMQ/.
[89] TAKADA K, SATO M, YASUMOTO S. A New Formulation of a Many-Level Shell Mod-el[J]. Progress of Theoretical Physics,2000,104(1):173–184. http://ptp.ipap.jp/link?PTP/104/173/.
[90] YASUMOTO S, SHIMIZU Y R, TAKADA K. A Comment on the New Formulation of aMany-Level Shell Model[J]. Progress of Theoretical Physics,2003,110(5):1037–1041.http://ptp.ipap.jp/link?PTP/110/1037/.
[91] TAKADA K, TAZAKI S, YASUMOTO S. Dyson Boson Mapping and Shell-Model Calcula-tionsfor Even-Even Nuclei[J]. Progress of Theoretical Physics,2006,116(1):107–126. http://ptp.ipap.jp/link?PTP/116/107/.
[92] WILDENTHAL B. Empirical strengths of spin operators in nuclei[J]. Progress in Parti-cle and Nuclear Physics,1984,11(0):5–51. http://www.sciencedirect.com/science/article/pii/0146641084900115.
[93] BROWN B A, RICHTER W A. New “USD” Hamiltonians for the sd shell[J]. Phys.Rev. C,2006,74:034315. http://link.aps.org/doi/10.1103/PhysRevC.74.034315.
[94] INOUE T, SEBE T, HAGIWARA H, et al. The structure of the sd-shell nuclei (I). C18,F18, O19, F19and Ne20[J]. Nuclear Physics,1964,59(1):1–32. http://www.sciencedirect.com/science/article/pii/0029558264901105.
[95] AKIYAMA Y, ARIMA A, SEBE T. The structure of the sd shell nuclei:(IV).20Ne,21Ne,22Ne,22Na and24Mg[J]. Nuclear Physics A,1969,138(2):273–304. http://www.sciencedirect.com/science/article/pii/0375947469903364.
[96] HONMA M, OTSUKA T, BROWN B A, et al. New effective interaction for pf-shell nucleiand its implications for the stability of the N=Z=28closed core[J]. Phys. Rev. C,2004,69:034335. http://link.aps.org/doi/10.1103/PhysRevC.69.034335.
[97] KOONIN S, DEAN D, LANGANKE K. Shell model Monte Carlo methods[J]. Physic-s Reports,1997,278(1):1–77. http://www.sciencedirect.com/science/article/pii/S0370157396000178.
[98] OTSUKA T, HONMA M, MIZUSAKI T, et al. Monte Carlo shell model for atomic nucle-i[J]. Progress in Particle and Nuclear Physics,2001,47(1):319–400. http://www.sciencedirect.com/science/article/pii/S0146641001001570.
[99] HOROI M, VOLYA A, ZELEVINSKY V. Chaotic Wave Functions and Exponential Con-vergence of Low-Lying Energy Eigenvalues[J]. Phys. Rev. Lett.,1999,82:2064–2067.http://link.aps.org/doi/10.1103/PhysRevLett.82.2064.
[100] HOROI M, ALEX BROWN B, ZELEVINSKY V. Applying the exponential convergencemethod: Shell-model binding energies of0f7/2nuclei relative to40Ca[J]. Phys. Rev.C,2002,65:027303. http://link.aps.org/doi/10.1103/PhysRevC.65.027303.
[101] HOROI M, BROWN B A, ZELEVINSKY V. Exponential convergence method: Nonyraststates, occupation numbers, and a shell-model description of the superdeformed band in56Ni[J]. Phys. Rev. C,2003,67:034303. http://link.aps.org/doi/10.1103/PhysRevC.67.034303.
[102] HOROI M, KAISER J, ZELEVINSKY V. Spin-and parity-dependent nuclear level densitiesand the exponential convergence method[J]. Phys. Rev. C,2003,67:054309. http://link.aps.org/doi/10.1103/PhysRevC.67.054309.
[103] PAPENBROCK T, JUODAGALVIS A, DEAN D J. Solution of large scale nuclear structureproblems by wave function factorization[J]. Phys. Rev. C,2004,69:024312. http://link.aps.org/doi/10.1103/PhysRevC.69.024312.
[104] WANG W G. Approach to energy eigenvalues and eigenfunctions from nonperturbativeregions of eigenfunctions[J]. Phys. Rev. E,2001,63:036215. http://link.aps.org/doi/10.1103/PhysRevE.63.036215.
[105] YOSHINAGA N, ARIMA A. Extrapolation methods for obtaining low-lying eigenvalues ofa large-dimensional shell model Hamiltonian matrix[J]. Phys. Rev. C,2010,81:044316.http://link.aps.org/doi/10.1103/PhysRevC.81.044316.
[106] LIPKIN H, MESHKOV N, GLICK A. Validity of many-body approximation methods fora solvable model:(I). Exact solutions and perturbation theory[J]. Nuclear Physics,1965,62(2):188–198. http://www.sciencedirect.com/science/article/pii/002955826590862X.
[107] EMERY G T. Perturbation of Nuclear Decay Rates[J]. Annu. Rev. Nucl. Part. Sci.,1972,22:165–202. http://dx.doi.org/10.1146/annurev.ns.22.120172.001121.
[108] PITTEL S, VINCENT C M, VERGADOS J D. Perturbative approximations to the effec-tive interaction: Comparisons with exact results for large matrices[J]. Phys. Rev. C,1976,13:412–429. http://link.aps.org/doi/10.1103/PhysRevC.13.412.
[109] SASAKAWA T, SAWADA T.3H bound state for the Reid soft-core potential: Exact cal-culation by a perturbational approach[J]. Phys. Rev. C,1979,19:2035–2049. http://link.aps.org/doi/10.1103/PhysRevC.19.2035.
[110] GOLDHAMMER P. Perturbation procedure for strong short ranged interactions[J]. Phys.Rev. C,1980,22:287–291. http://link.aps.org/doi/10.1103/PhysRevC.22.287.
[111] STEVENSON P, STRAYER M R, RIKOVSKA STONE J. Many-body perturbation calculationof spherical nuclei with a separable monopole interaction[J]. Phys. Rev. C,2001,63:054309.http://link.aps.org/doi/10.1103/PhysRevC.63.054309.
[112] ROTH R, PAPAKONSTANTINOU P, PAAR N, et al. Hartree-Fock and many body pertur-bation theory with correlated realistic NN interactions[J]. Phys. Rev. C,2006,73:044312.http://link.aps.org/doi/10.1103/PhysRevC.73.044312.
[113] BEANE S R, KAPLAN D B, VUORINEN A. Perturbative nuclear physics[J]. Phys.Rev. C,2009,80:011001. http://link.aps.org/doi/10.1103/PhysRevC.80.011001.
[114] OTSUKA T. Rotational states and interacting bosons[J]. Nuclear Physics A,1981,368(2):244–284. http://www.sciencedirect.com/science/article/pii/0375947481906850.
[115] OTSUKA T. Microscopic Basis of the Proton-Neutron Interacting-Boson Model[J].Phys. Rev. Lett.,1981,46:710–713. http://link.aps.org/doi/10.1103/PhysRevLett.46.710.
[116] WAPSTRA A, AUDI G. The1983atomic mass evaluation:(I). Atomic mass table[J]. Nucle-ar Physics A,1985,432(1):1–54. http://www.sciencedirect.com/science/article/pii/0375947485902830.
[2] MAYER M G. On Closed Shells in Nuclei[J]. Phys. Rev.,1948,74:235–239. http://link.aps.org/doi/10.1103/PhysRev.74.235.
[3] MAYER M G. On Closed Shells in Nuclei. II[J]. Phys. Rev.,1949,75:1969–1970. http://link.aps.org/doi/10.1103/PhysRev.75.1969.
[4] MAYER M, JENSEN J. Elementary theory of nuclear shell structure[M]. Structureof matter series.[S.l.]: Wiley,1960. http://books.google.co.jp/books?id=z24sAAAAYAAJ.
[5] AJZENBERG-SELOVE F. Energy levels of light nuclei[J]. Nuclear Physics A,1988,490(1):1–225. http://www.sciencedirect.com/science/article/pii/0375947488901248.
[6] WOODS R D, SAXON D S. Diffuse Surface Optical Model for Nucleon-Nuclei Scatter-ing[J]. Phys. Rev.,1954,95:577–578. http://link.aps.org/doi/10.1103/PhysRev.95.577.
[7] ROSS A A, MARK H, LAWSON R D. Nucleon Energy Levels in a Diffuse Poten-tial[J]. Phys. Rev.,1956,102:1613–1620. http://link.aps.org/doi/10.1103/PhysRev.102.1613.
[8] WALET N. Nuclear and Particle Physics[J]. Physics OER. http://oer.physics.manchester.ac.uk/NP/.
[9] THOMAS L H. The Motion of the Spinning Electron[J]. Nature,1926,117:514. http://adsabs.harvard.edu/abs/1926Natur.117..514T.
[10] RACAH G. Theory of Complex Spectra. III[J]. Phys. Rev.,1943,63:367–382. http://link.aps.org/doi/10.1103/PhysRev.63.367.
[11] M. H. The Nuclear SU3Model[J]. Advances in Nuclear Physics,1968,1:67.
[12] CHEN J Q. The Wick theorem for coupled fermion clusters[J]. Nuclear Physic-s A,1993,562(2):218–240. http://www.sciencedirect.com/science/article/pii/0375947493901976.
[13] JIN-QUAN C, BING-QING C, KLEIN A. Factorization of commutators: The Wick theo-rem for coupled operators[J]. Nuclear Physics A,1993,554(1):61–76. http://www.sciencedirect.com/science/article/pii/0375947493903574.
[14] CHEN J Q. Nucleon-pair shell model: Formalism and special cases[J]. Nuclear Physic-s A,1997,626(3):686–714. http://www.sciencedirect.com/science/article/pii/S0375947497005022.
[15] CHEN J Q, LUO Y A. Nucleon-pair shell model: The effects of the SD pair structure oncollectivity of low-lying states[J]. Nuclear Physics A,1998,639:615–634. http://www.sciencedirect.com/science/article/pii/S0375947498004229.
[16] ZHAO Y M, YOSHINAGA N, YAMAJI S, et al. Nucleon-pair approximation of the shellmodel: Unified formalism for both odd and even systems[J]. Phys. Rev. C,2000,62:014304.http://link.aps.org/doi/10.1103/PhysRevC.62.014304.
[17] ZHAO Y M, YAMAJI S, YOSHINAGA N, et al. Nucleon pair approximation of the nuclearcollective motion[J]. Phys. Rev. C,2000,62:014315. http://link.aps.org/doi/10.1103/PhysRevC.62.014315.
[18] JIA L Y, ZHANG H, ZHAO Y M. Systematic calculations of low-lying states of even-evennuclei within the nucleon pair approximation[J]. Phys. Rev. C,2007,75:034307. http://link.aps.org/doi/10.1103/PhysRevC.75.034307.
[19] JIA L Y, ZHANG H, ZHAO Y M. Systematic calculations of low-lying states in odd-Anuclei within the nucleon pair approximation[J]. Phys. Rev. C,2007,76:054305. http://link.aps.org/doi/10.1103/PhysRevC.76.054305.
[20] XU Z Y, LEI Y, ZHAO Y M, et al. Low-lying states of heavy nuclei within the nucleon pairapproximation[J]. Phys. Rev. C,2009,79:054315. http://link.aps.org/doi/10.1103/PhysRevC.79.054315.
[21] JIANG H, SHEN J J, ZHAO Y M, et al. Low-lying states of valence-hole nuclei in the208Pb region[J]. Journal of Physics G: Nuclear and Particle Physics,2011,38(4):045103.http://stacks.iop.org/0954-3899/38/i=4/a=045103.
[22] YAN-AN L, CHEN J Q. Shell model calculation in the S D subspace[J]. Phys. Rev. C,1998,58:589–592. http://link.aps.org/doi/10.1103/PhysRevC.58.589.
[23] LUO Y A, CHEN J Q, DRAAYER J. Nucleon-pair shell model calculations of the even-even Xe and Ba nuclei[J]. Nuclear Physics A,2000,669:101–118. http://www.sciencedirect.com/science/article/pii/S0375947499008180.
[24] YOSHINAGA N, HIGASHIYAMA K. Systematic studies of nuclei around mass130in thepair-truncated shell model[J]. Phys. Rev. C,2004,69:054309. http://link.aps.org/doi/10.1103/PhysRevC.69.054309.
[25] HIGASHIYAMA K, YOSHINAGA N, TANABE K. New band mechanism of doubly-oddnuclei around mass130[J]. Phys. Rev. C,2005,72:024315. http://link.aps.org/doi/10.1103/PhysRevC.72.024315.
[26] ZHAO Y M, YOSHINAGA N, YAMAJI S, et al. Relationship between the fermion dynam-ical symmetric model Hamiltonian and nuclear collective motion[J]. Phys. Rev. C,2000,62:024322. http://link.aps.org/doi/10.1103/PhysRevC.62.024322.
[27] LEI Y, XU Z Y, ZHAO Y M, et al. Validity of pair truncation of the nuclear shell model in46Ca[J]. Phys. Rev. C,2009,80:064316. http://link.aps.org/doi/10.1103/PhysRevC.80.064316.
[28] LEI Y, XU Z Y, ZHAO Y M, et al. Validity of pair truncations with effective interactionin Ca isotopes[J]. Phys. Rev. C,2010,82:034303. http://link.aps.org/doi/10.1103/PhysRevC.82.034303.
[29] WU C L, FENG D H, CHEN X G, et al. Fermion dynamical symmetry model of nuclei:Basis, Hamiltonian, and symmetries[J]. Phys. Rev. C,1987,36:1157–1180. http://link.aps.org/doi/10.1103/PhysRevC.36.1157.
[30] K. H, Y. S. Projected shell model and high spin spectroscopy[J]. InternationalJournal of Modern Physics E,1995,4:637–785. http://dx.doi.org/10.1142/S0218301395000250.
[31] ARIMA A, IACHELLO F. Collective Nuclear States as Representations of a SU(6) Group[J].Phys. Rev. Lett.,1975,35:1069–1072. http://link.aps.org/doi/10.1103/PhysRevLett.35.1069.
[32] IACHELLO B H P, F., DIEPERINK A E L. Collective aspects of the shell model[J]. Inter-national Conference on Nuclear Structure and Spectroscopy,1974,2:163–181.
[33] JANSSEN D, JOLOS R, DONAU F. An algebraic treatment of the nuclear quadrupoledegree of freedom[J]. Nuclear Physics A,1974,224(1):93–115. http://www.sciencedirect.com/science/article/pii/0375947474901651.
[34] ARIMA A, OHTSUKA T, IACHELLO F, et al. Collective nuclear states as sym-metric couplings of proton and neutron excitations[J]. Physics Letters B,1977,66(3):205–208. http://www.sciencedirect.com/science/article/pii/0370269377908607.
[35] OTSUKA T, ARIMA A, IACHELLO F, et al. Shell model description of interacting boson-s[J]. Physics Letters B,1978,76(2):139–143. http://www.sciencedirect.com/science/article/pii/0370269378902605.
[36] ARIMA A, IACHELLO F. Interacting boson model of collective nuclear states II.The rotational limit[J]. Annals of Physics,1978,111(1):201–238. http://www.sciencedirect.com/science/article/pii/0003491678902282.
[37] ARIMA A, IACHELLO F. Interacting boson model of collective nuclear states IV. The O(6)limit[J]. Annals of Physics,1979,123(2):468–492. http://www.sciencedirect.com/science/article/pii/0003491679903476.
[38] ARIMA A, IACHELLO F. Interacting boson model of collective states I. The vibrationallimit[J]. Annals of Physics,1976,99(2):253–317. http://www.sciencedirect.com/science/article/pii/000349167690097X.
[39] BEHRENS H, GENZ H, CONZE M, et al. Allowed β-transitions, weak magnetism andnuclear structure in light nuclei[J]. Annals of Physics,1978,115(2):276–324. http://www.sciencedirect.com/science/article/pii/0003491678901586.
[40] BOHR A, MOTTELSON B R. Nuclear Structure: Nuclear deformations[M]. NuclearStructure.[S.l.]: W. A. Benjamin,1969. http://books.google.co.jp/books?id=PpofAQAAMAAJ.
[41] D. S B, D. W J. The Nuclear SU3Model[J]. Advances in Nuclear Physics,1986,16:1.
[42] REINHARD P G. The relativistic mean-field description of nuclei and nuclear dynamic-s[J]. Reports on Progress in Physics,1989,52(4):439. http://stacks.iop.org/0034-4885/52/i=4/a=002.
[43] RING P. Relativistic mean field theory in finite nuclei[J]. Progress in Particle and Nucle-ar Physics,1996,37(0):193–263. http://www.sciencedirect.com/science/article/pii/0146641096000543.
[44] MENG J, RING P. Relativistic Hartree-Bogoliubov Description of the Neutron Halo in11Li[J]. Phys. Rev. Lett.,1996,77:3963–3966. http://link.aps.org/doi/10.1103/PhysRevLett.77.3963.
[45] MENG J, RING P. Giant Halo at the Neutron Drip Line[J]. Phys. Rev. Lett.,1998,80:460–463. http://link.aps.org/doi/10.1103/PhysRevLett.80.460.
[46] REN Z, FAESSLER A, BOBYK A. Relativistic mean-field description of a proton haloin the first excited (1/2)+state of17F[J]. Phys. Rev. C,1998,57:2752–2755. http://link.aps.org/doi/10.1103/PhysRevC.57.2752.
[47] ZHANG W, MENG J, ZHANG S, et al. Magic numbers for superheavy nucle-i in relativistic continuum Hartree鈥揃ogoliubov theory[J]. Nuclear Physics A,2005,753:106–135. http://www.sciencedirect.com/science/article/pii/S0375947405002423.
[48] LV H, MENG J, ZHANG S, et al. Neutron halos in hypernuclei[J]. The European PhysicalJournal A-Hadrons and Nuclei,2003,17:19–24. http://dx.doi.org/10.1140/epja/i2002-10136-3.10.1140/epja/i2002-10136-3.
[49] MENG J, SUGAWARA-TANABE K, YAMAJI S, et al. Pseudospin symmetry in Zr and Snisotopes from the proton drip line to the neutron drip line[J]. Phys. Rev. C,1999,59:154–163. http://link.aps.org/doi/10.1103/PhysRevC.59.154.
[50] MIZUSAKI T. Shell model calculation–from basics to the latest methods[J].4th CNS International Summer School. http://www.cns.s.u-tokyo.ac.jp/summerschool/ciss05/lecturenotes/.
[51] BOHR A, MOTTELSON B. Nuclear Structure: Single-particle motion[M]. Nuclear Struc-ture.[S.l.]: W. A. Benjamin,1969. http://books.google.co.jp/books?id=ApoRKQEACAAJ.
[52] WIGNER E P. Random Matrices in Physics[J]. SIAM Rev.,1966,9:1–23. http://dx.doi.org/10.1137/1009001.
[53] SHEN J J, ZHAO Y M, ARIMA A, et al. Lowest eigenvalues of random Hamiltoni-ans[J]. Phys. Rev. C,2008,77:054312. http://link.aps.org/doi/10.1103/PhysRevC.77.054312.
[54] SHEN J J, ARIMA A, ZHAO Y M, et al. Strong correlation between eigenvalues and di-agonal matrix elements[J]. Phys. Rev. C,2008,78:044305. http://link.aps.org/doi/10.1103/PhysRevC.78.044305.
[55] ZHANG L H, SHEN J J, LEI Y, et al. Robustness of Strong Correlation between Eigenval-ues and Diagonal Matrix Elements[J]. International Journal of Modern Physics E,2008,17,Supplement:342–351. http://dx.doi.org/10.1142/S0218301308011975.
[56] YOSHINAGA N, ARIMA A, SHEN J J, et al. Correlation between eigenvalues and sorteddiagonal elements of a large dimensional matrix[J]. Phys. Rev. C,2009,79:017301. http://link.aps.org/doi/10.1103/PhysRevC.79.017301.
[57] SHEN J, ZHAO Y. How random are matrix elements of the nuclear shell model Hamilto-nian?[J]. Science in China Series G: Physics Mechanics and Astronomy,2009,52:1477–1481. http://dx.doi.org/10.1007/s11433-009-0198-7.10.1007/s11433-009-0198-7.
[58] SHEN J J, ZHAO Y M, ARIMA A. Lowest eigenvalue of the nuclear shell model Hamil-tonian[J]. Phys. Rev. C,2010,82:014309. http://link.aps.org/doi/10.1103/PhysRevC.82.014309.
[59] SHEN J J, ZHAO Y M, ARIMA A, et al. New extrapolation method for low-lying states ofnuclei in the sd and the pf shells[J]. Phys. Rev. C,2011,83:044322. http://link.aps.org/doi/10.1103/PhysRevC.83.044322.
[60] SHEN J J, ZHAO Y M, ARIMA A. New perturbation method of diagonalizing the nuclearshell model Hamiltonian[J]. Phys. Rev. C,2012,85:064325. http://link.aps.org/doi/10.1103/PhysRevC.85.064325.
[61] JOHNSON C W, BERTSCH G F, DEAN D J. Orderly Spectra from Random Interaction-s[J]. Phys. Rev. Lett.,1998,80:2749–2753. http://link.aps.org/doi/10.1103/PhysRevLett.80.2749.
[62] ZHAO Y, ARIMA A, YOSHINAGA N. Regularities of many-body systems interacting bya two-body random ensemble[J]. Physics Reports,2004,400(1):1–66. http://www.sciencedirect.com/science/article/pii/S0370157304002972.
[63] ZELEVINSKY V, VOLYA A. Nuclear structure, random interactions and mesoscopic physic-s[J]. Physics Reports,2004,391:311–352. http://www.sciencedirect.com/science/article/pii/S0370157303004319. ce:title From atoms to nuclei toquarks and gluons: the omnipresent manybody theory/ce:title.
[64] PAPENBROCK T, WEIDENM U¨LLER H A. Colloquium: Random matrices and chaos innuclear spectra[J]. Rev. Mod. Phys.,2007,79:997–1013. http://link.aps.org/doi/10.1103/RevModPhys.79.997.
[65] PAPENBROCK T, WEIDENM U¨LLER H A. Distribution of Spectral Widths and Preponder-ance of Spin-0Ground States in Nuclei[J]. Phys. Rev. Lett.,2004,93:132503. http://link.aps.org/doi/10.1103/PhysRevLett.93.132503.
[66] PAPENBROCK T, WEIDENM U¨LLER H A. Two-body random ensemble in nuclei[J]. Phys.Rev. C,2006,73:014311. http://link.aps.org/doi/10.1103/PhysRevC.73.014311.
[67] YOSHINAGA N, ARIMA A, ZHAO Y M. Lowest bound of energies for random interac-tions and the origin of spin-zero ground state dominance in even-even nuclei[J]. Phys.Rev. C,2006,73:017303. http://link.aps.org/doi/10.1103/PhysRevC.73.017303.
[68] ABRAMOWITZ M, STEGUN I. Handbook of Mathematical Functions: With Formulas,Graphs, and Mathematical Tables[M]. Applied mathematics series.[S.l.]: Dover Publi-cations,1964. http://books.google.co.jp/books?id=MtU8uP7XMvoC.
[69] RATCLIFF K F. Applications of Spectral Distributions in Nuclear Spectroscopy[J]. Phys.Rev. C,1971,3:117–143. http://link.aps.org/doi/10.1103/PhysRevC.3.117.
[70] WONG S. Nuclear statistical spectroscopy[M]. Oxford studies in nuclear physic-s.[S.l.]: Oxford University Press,1986. http://books.google.co.jp/books?id=-jZRAAAAMAAJ.
[71] MARGETAN F J, KLAR A, VARY J P. Nuclear binding energies from moment methods:Harmonic oscillator Hamiltonian[J]. Phys. Rev. C,1983,27:852–861. http://link.aps.org/doi/10.1103/PhysRevC.27.852.
[72] VEL A′ZQUEZ V, ZUKER A P. Spectroscopy with Random and Displaced Random Ensem-bles[J]. Phys. Rev. Lett.,2002,88:072502. http://link.aps.org/doi/10.1103/PhysRevLett.88.072502.
[73] IACHELLO F, ARIMA A. The Interacting Boson Model[M]. Cambridge Monographson Mathematical Physics.[S.l.]: Cambridge University Press,1987. http://books.google.co.jp/books?id=fB7Qe73VmsgC.
[74] ZHAO Y M, ARIMA A, YOSHINAGA N. Many-body systems interacting via a two-body random ensemble. I. Angular momentum distribution in the ground states[J]. Phys.Rev. C,2002,66:064322. http://link.aps.org/doi/10.1103/PhysRevC.66.064322.
[75] MON K, FRENCH J. Statistical properties of many-particle spectra[J]. Annals of Physics,1975,95(1):90–111. http://www.sciencedirect.com/science/article/pii/0003491675900457.
[76] PRESS W, TEUKOLSKY S, VETTERLING W, et al. Numerical recipes in C++: the artof scientific computing[M].[S.l.]: Cambridge University Press,2005. http://books.google.co.jp/books?id=DevQtgAACAAJ.
[77] WEIDENM U¨LLER H A, MITCHELL G E. Random matrices and chaos in nuclear physics:Nuclear structure[J]. Rev. Mod. Phys.,2009,81:539–589. http://link.aps.org/doi/10.1103/RevModPhys.81.539.
[78] WILDENTHAL B. Empirical strengths of spin operators in nuclei[J]. Progress in Parti-cle and Nuclear Physics,1984,11(0):5–51. http://www.sciencedirect.com/science/article/pii/0146641084900115.
[79] BROWN B A, WILDENTHAL B H. Status of the Nuclear Shell Model[J]. Annu. Rev. Nucl.Part. Sci.,1988,38:29–66. http://dx.doi.org/10.1146/annurev.nucl.38.1.29.
[80] FRENCH J, WONG S. Validity of random matrix theories for many-particle systems[J].Physics Letters B,1970,33(7):449–452. http://www.sciencedirect.com/science/article/pii/0370269370902133.
[81] BRODY T A, FLORES J, FRENCH J B, et al. Random-matrix physics: spectrum and strengthfluctuations[J]. Rev. Mod. Phys.,1981,53:385–479. http://link.aps.org/doi/10.1103/RevModPhys.53.385.
[82] BOHIGAS O, FLORES J. Two-body random hamiltonian and level density[J]. PhysicsLetters B,1971,34(4):261–263. http://www.sciencedirect.com/science/article/pii/0370269371905983.
[83] ZELEVINSKY V, BROWN B, FRAZIER N, et al. The nuclear shell model as a testing groundfor many-body quantum chaos[J]. Physics Reports,1996,276:85–176. http://www.sciencedirect.com/science/article/pii/S0370157396000075.
[84] ZELEVINSKY V, BROWN B, FRAZIER N, et al. The nuclear shell model as a testing groundfor many-body quantum chaos[J]. Physics Reports,1996,276:85–176. http://www.sciencedirect.com/science/article/pii/S0370157396000075.
[85] ARIMA A, IACHELLO F. Interacting boson model of collective states I. The vibrationallimit[J]. Annals of Physics,1976,99(2):253–317. http://www.sciencedirect.com/science/article/pii/000349167690097X.
[86] ARIMA A, IACHELLO F. Interacting boson model of collective nuclear states II.The rotational limit[J]. Annals of Physics,1978,111(1):201–238. http://www.sciencedirect.com/science/article/pii/0003491678902282.
[87] KUO T, BROWN G. Reaction matrix elements for the0f-1p shell nuclei[J]. NuclearPhysics A,1968,114(2):241–279. http://www.sciencedirect.com/science/article/pii/0375947468903539.
[88] TAKADA K. ftp://ftp.kutl.kyushu-u.ac.jp/pub/takada/jjSMQ/.
[89] TAKADA K, SATO M, YASUMOTO S. A New Formulation of a Many-Level Shell Mod-el[J]. Progress of Theoretical Physics,2000,104(1):173–184. http://ptp.ipap.jp/link?PTP/104/173/.
[90] YASUMOTO S, SHIMIZU Y R, TAKADA K. A Comment on the New Formulation of aMany-Level Shell Model[J]. Progress of Theoretical Physics,2003,110(5):1037–1041.http://ptp.ipap.jp/link?PTP/110/1037/.
[91] TAKADA K, TAZAKI S, YASUMOTO S. Dyson Boson Mapping and Shell-Model Calcula-tionsfor Even-Even Nuclei[J]. Progress of Theoretical Physics,2006,116(1):107–126. http://ptp.ipap.jp/link?PTP/116/107/.
[92] WILDENTHAL B. Empirical strengths of spin operators in nuclei[J]. Progress in Parti-cle and Nuclear Physics,1984,11(0):5–51. http://www.sciencedirect.com/science/article/pii/0146641084900115.
[93] BROWN B A, RICHTER W A. New “USD” Hamiltonians for the sd shell[J]. Phys.Rev. C,2006,74:034315. http://link.aps.org/doi/10.1103/PhysRevC.74.034315.
[94] INOUE T, SEBE T, HAGIWARA H, et al. The structure of the sd-shell nuclei (I). C18,F18, O19, F19and Ne20[J]. Nuclear Physics,1964,59(1):1–32. http://www.sciencedirect.com/science/article/pii/0029558264901105.
[95] AKIYAMA Y, ARIMA A, SEBE T. The structure of the sd shell nuclei:(IV).20Ne,21Ne,22Ne,22Na and24Mg[J]. Nuclear Physics A,1969,138(2):273–304. http://www.sciencedirect.com/science/article/pii/0375947469903364.
[96] HONMA M, OTSUKA T, BROWN B A, et al. New effective interaction for pf-shell nucleiand its implications for the stability of the N=Z=28closed core[J]. Phys. Rev. C,2004,69:034335. http://link.aps.org/doi/10.1103/PhysRevC.69.034335.
[97] KOONIN S, DEAN D, LANGANKE K. Shell model Monte Carlo methods[J]. Physic-s Reports,1997,278(1):1–77. http://www.sciencedirect.com/science/article/pii/S0370157396000178.
[98] OTSUKA T, HONMA M, MIZUSAKI T, et al. Monte Carlo shell model for atomic nucle-i[J]. Progress in Particle and Nuclear Physics,2001,47(1):319–400. http://www.sciencedirect.com/science/article/pii/S0146641001001570.
[99] HOROI M, VOLYA A, ZELEVINSKY V. Chaotic Wave Functions and Exponential Con-vergence of Low-Lying Energy Eigenvalues[J]. Phys. Rev. Lett.,1999,82:2064–2067.http://link.aps.org/doi/10.1103/PhysRevLett.82.2064.
[100] HOROI M, ALEX BROWN B, ZELEVINSKY V. Applying the exponential convergencemethod: Shell-model binding energies of0f7/2nuclei relative to40Ca[J]. Phys. Rev.C,2002,65:027303. http://link.aps.org/doi/10.1103/PhysRevC.65.027303.
[101] HOROI M, BROWN B A, ZELEVINSKY V. Exponential convergence method: Nonyraststates, occupation numbers, and a shell-model description of the superdeformed band in56Ni[J]. Phys. Rev. C,2003,67:034303. http://link.aps.org/doi/10.1103/PhysRevC.67.034303.
[102] HOROI M, KAISER J, ZELEVINSKY V. Spin-and parity-dependent nuclear level densitiesand the exponential convergence method[J]. Phys. Rev. C,2003,67:054309. http://link.aps.org/doi/10.1103/PhysRevC.67.054309.
[103] PAPENBROCK T, JUODAGALVIS A, DEAN D J. Solution of large scale nuclear structureproblems by wave function factorization[J]. Phys. Rev. C,2004,69:024312. http://link.aps.org/doi/10.1103/PhysRevC.69.024312.
[104] WANG W G. Approach to energy eigenvalues and eigenfunctions from nonperturbativeregions of eigenfunctions[J]. Phys. Rev. E,2001,63:036215. http://link.aps.org/doi/10.1103/PhysRevE.63.036215.
[105] YOSHINAGA N, ARIMA A. Extrapolation methods for obtaining low-lying eigenvalues ofa large-dimensional shell model Hamiltonian matrix[J]. Phys. Rev. C,2010,81:044316.http://link.aps.org/doi/10.1103/PhysRevC.81.044316.
[106] LIPKIN H, MESHKOV N, GLICK A. Validity of many-body approximation methods fora solvable model:(I). Exact solutions and perturbation theory[J]. Nuclear Physics,1965,62(2):188–198. http://www.sciencedirect.com/science/article/pii/002955826590862X.
[107] EMERY G T. Perturbation of Nuclear Decay Rates[J]. Annu. Rev. Nucl. Part. Sci.,1972,22:165–202. http://dx.doi.org/10.1146/annurev.ns.22.120172.001121.
[108] PITTEL S, VINCENT C M, VERGADOS J D. Perturbative approximations to the effec-tive interaction: Comparisons with exact results for large matrices[J]. Phys. Rev. C,1976,13:412–429. http://link.aps.org/doi/10.1103/PhysRevC.13.412.
[109] SASAKAWA T, SAWADA T.3H bound state for the Reid soft-core potential: Exact cal-culation by a perturbational approach[J]. Phys. Rev. C,1979,19:2035–2049. http://link.aps.org/doi/10.1103/PhysRevC.19.2035.
[110] GOLDHAMMER P. Perturbation procedure for strong short ranged interactions[J]. Phys.Rev. C,1980,22:287–291. http://link.aps.org/doi/10.1103/PhysRevC.22.287.
[111] STEVENSON P, STRAYER M R, RIKOVSKA STONE J. Many-body perturbation calculationof spherical nuclei with a separable monopole interaction[J]. Phys. Rev. C,2001,63:054309.http://link.aps.org/doi/10.1103/PhysRevC.63.054309.
[112] ROTH R, PAPAKONSTANTINOU P, PAAR N, et al. Hartree-Fock and many body pertur-bation theory with correlated realistic NN interactions[J]. Phys. Rev. C,2006,73:044312.http://link.aps.org/doi/10.1103/PhysRevC.73.044312.
[113] BEANE S R, KAPLAN D B, VUORINEN A. Perturbative nuclear physics[J]. Phys.Rev. C,2009,80:011001. http://link.aps.org/doi/10.1103/PhysRevC.80.011001.
[114] OTSUKA T. Rotational states and interacting bosons[J]. Nuclear Physics A,1981,368(2):244–284. http://www.sciencedirect.com/science/article/pii/0375947481906850.
[115] OTSUKA T. Microscopic Basis of the Proton-Neutron Interacting-Boson Model[J].Phys. Rev. Lett.,1981,46:710–713. http://link.aps.org/doi/10.1103/PhysRevLett.46.710.
[116] WAPSTRA A, AUDI G. The1983atomic mass evaluation:(I). Atomic mass table[J]. Nucle-ar Physics A,1985,432(1):1–54. http://www.sciencedirect.com/science/article/pii/0375947485902830.