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Spherical Dirac equation on the lattice and the problem of the spurious states

Zhao Bin

Spherical Dirac equation on the lattice and the problem of the spurious states

Zhao Bin
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  • With the development of radioactive ion beam facilities, the study of exotic nuclei with unusual N/Z ratio has attracted much attention. Compared with the stable nuclei, the exotic nuclei have many novel features, such as the halo phenomenon. In order to describe the halo phenomenon with the diffused density distribution, the correct asymptotic behaviors of wave functions should be treated properly. The relativistic continuum Hartree-Bogoliubov (RCHB) theory which provides a unified and self-consistent description of mean field, pair correlation and continuum has achieved great success in describing the spherical exotic nuclei. In order to study the halo phenomenon in deformed nuclei, it is necessary to extend RCHB theory to the deformed case. However, solving the relativistic Hartree-Bogoliubov equation in space is extremely difficult and time consuming. Imaginary time step method is an efficient method to solve differential equations in coordinate space. It has been used extensively in the nonrelativistic case. For Dirac equation, it is very challenging to use the imaginary time step method due to the Dirac sea. This problem can be solved by the inverse Hamiltonian method. However, the problem of spurious states comes out. In this paper, we solve the radial Dirac equation by the imaginary time step method in coordinate space and study the problem of spurious states. It can be proved that for any potential, when using the three-point differential formula to discretize the first-order derivative operator, the energies of the single-particle states respectively with quantum numbers and - are identical. One of them is a physical state and the other is a spurious state. Although they have the same energies, their wave functions have different behaviors. The wave function of physical state is smooth in space while that of spurious state fluctuates dramatically. Following the method in lattice quantum chromodynamics calculation, the spurious state in radial Dirac equation can be removed by introducing the Wilson term. Taking Woods-Saxon potential for example, the imaginary time step method with the Wilson term is implanted successfully and provides the same results as those from the shooting method, which demonstrates its future application to solving the Dirac equation in coordinate space.
      Corresponding author: Zhao Bin, bzhao@buaa.edu.cn
    • Funds: Project supported by the National Basic Research Program of China (Grant No. 2013CB834400), the National Natural Science Foundation of China (Grants Nos. 11175002, 11335002, 11375015), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110001110087).
    [1]

    Tanihata I 1995 Prog. Part. Nucl. Phys. 35 505

    [2]

    Ozawa A, Kobayashi T, Suzuki T, Yoshida K, Tanihata I 2000 Phys. Rev. Lett. 84 5493

    [3]

    Zilges A, Babilon M, Hartmann T, Savran D, Volz S 2005 Prog. Part. Nucl. Phys. 55 408

    [4]

    Meng J, Toki H, Zhou S G, Zhang S Q, Long W H, Geng L S 2006 Prog. Part. Nucl. Phys. 57 470

    [5]

    Meng J, Ring P 1996 Phys. Rev. Lett. 77 3963

    [6]

    Meng J, Ring P 1998 Phys. Rev. Lett. 80 460

    [7]

    Meng J, Toki H, Zeng J Y, Zhang S Q, Zhou S G 2002 Phys. Rev. C 65 041302

    [8]

    Meng J, Tanihata I, Yamaji S 1998 Phys. Lett. B 419 1

    [9]

    Meng J, Zhou S G, Tanihata I 2002 Phys. Lett. B 532 209

    [10]

    Meng J, Sugawara-Tanabe K, Yamaji S, Ring P, Arima A 1998 Phys. Rev. C 58 R628

    [11]

    Meng J, Sugawara-Tanabe K, Yamaji S, Arima A 1999 Phys. Rev. C 59 154

    [12]

    Ginocchio J N 1997 Phys. Rev. Lett. 78 436

    [13]

    Ginocchio J N, Leviatan A, Meng J, Zhou S G 1997 Phys. Rev. C 69 034303

    [14]

    Guo J Y 2012 Phys. Rev. C 85 021302

    [15]

    Lu B N, Zhao E G, Zhou S G 2012 Phys. Rev. Lett. 109 072501

    [16]

    Liang H Z, Shen S H, Zhao P W, Meng J 2013 Phys. Rev. C 87 014334

    [17]

    Shen S H, Liang H Z, Zhao P W, Zhang S Q, Meng J 2013 Phys. Rev. C 88 024311

    [18]

    Guo J Y, Chen S W, Niu Z M, Li D P, Liu Q 2014 Phys. Rev. Lett. 112 062502

    [19]

    Liang H Z, Meng J, Zhou S G 2015 Phys. Rep. 570 1

    [20]

    Zhang M C 2009 Acta Phys. Sin. 58 61 (in Chinese) [张民仓 2009 物理学报 58 61]

    [21]

    Lu H F, Meng J 2002 Chin. Phys. Lett. 19 1775

    [22]

    Lu H F, Meng J, Zhang S Q, Zhou S G 2003 Eur. Phys. J. A 17 19

    [23]

    Zhang W, Meng J, Zhang S Q, Geng L S, Toki H 2005 Nucl. Phys. A 753 106

    [24]

    Qu X Y, Chen Y, Zhang S Q, Zhao P W, Shin I J, Lim Y, Kim Y, Meng J 2013 Sci. China. Phys. Mech. 56 2031

    [25]

    Sun B H, Meng J 2008 Chin. Phys. Lett. 25 2429

    [26]

    Li Z, Niu Z M, Sun B H, Wang N, Meng J 2012 Acta Phys. Sin. 61 072601 (in Chinese) [李竹, 牛中明, 孙保华, 王宁, 孟杰2012 物理学报61 072601]

    [27]

    Price C E, Walker G E 1987 Phys. Rev. C 36 354

    [28]

    Meng J, Lu H F, Zhang S Q, Zhou S G 2003 Nucl. Phys. A 722 C366

    [29]

    Zhou S G, Meng J, Ring P 2003 Phys. Rev. C 68 034323

    [30]

    Zhou S G, Meng J, Ring P, Zhao E G 2010 Phys. Rev. C 82 3481

    [31]

    Davies K T R, Flocard H, Krieger S, Weiss M S 1980 Nucl. Phys. A 342 111

    [32]

    Bonche P, Flocard H, Heenen P H 2005 Comput. Phys. Commun. 171 49

    [33]

    Zhang Y, Liang H Z, Meng J 2010 Int. J. Mod. Phys. E 19 55

    [34]

    Hagino K, Tanimura Y 2010 Phys. Rev. C 82 057301

    [35]

    Grant I P 1982 Phys. Rev. A 25 1230

    [36]

    Salomonson S, ster P 1989 Phys. Rev. A 40 5548

    [37]

    Tanimura Y, Hagino K, Liang H Z 2015 Prog. Theor. Exp. Phys. 2015 073D01

    [38]

    Zhao S 2007 Comput. Method. Appl. M. 196 5031

    [39]

    Shabaev V M, Tupitsyn I I, Yerokhin V A, Plunien G, Soff G 2004 Phys. Rev. Lett. 93 130405

    [40]

    Pestka G 2003 Phys. Scripta. 68 254

    [41]

    Mller C, Grn N, Scheid W 1998 Phys. Lett. A 242 245

    [42]

    Wilson K G 1977 Proceedings of the First Half of the 1975 International School of Subnuclear Physics Erice, Sicily, July 11-August 1, 1975 p69

    [43]

    Serot B D, Walecka J D 1986 Adv. Nucl. Phys. 16

    [44]

    Reinhard P G 1989 Rep. Prog. Phys. 52 439

    [45]

    Meng J 1998 Nucl. Phys. A 635 3

    [46]

    Abramowitz M, Stegun I A 1964 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (New York: Dover Publications) p914

    [47]

    Koepf W, Ring P 1991 Z. Phys. A: Hadrons Nucl. 339 81

  • [1]

    Tanihata I 1995 Prog. Part. Nucl. Phys. 35 505

    [2]

    Ozawa A, Kobayashi T, Suzuki T, Yoshida K, Tanihata I 2000 Phys. Rev. Lett. 84 5493

    [3]

    Zilges A, Babilon M, Hartmann T, Savran D, Volz S 2005 Prog. Part. Nucl. Phys. 55 408

    [4]

    Meng J, Toki H, Zhou S G, Zhang S Q, Long W H, Geng L S 2006 Prog. Part. Nucl. Phys. 57 470

    [5]

    Meng J, Ring P 1996 Phys. Rev. Lett. 77 3963

    [6]

    Meng J, Ring P 1998 Phys. Rev. Lett. 80 460

    [7]

    Meng J, Toki H, Zeng J Y, Zhang S Q, Zhou S G 2002 Phys. Rev. C 65 041302

    [8]

    Meng J, Tanihata I, Yamaji S 1998 Phys. Lett. B 419 1

    [9]

    Meng J, Zhou S G, Tanihata I 2002 Phys. Lett. B 532 209

    [10]

    Meng J, Sugawara-Tanabe K, Yamaji S, Ring P, Arima A 1998 Phys. Rev. C 58 R628

    [11]

    Meng J, Sugawara-Tanabe K, Yamaji S, Arima A 1999 Phys. Rev. C 59 154

    [12]

    Ginocchio J N 1997 Phys. Rev. Lett. 78 436

    [13]

    Ginocchio J N, Leviatan A, Meng J, Zhou S G 1997 Phys. Rev. C 69 034303

    [14]

    Guo J Y 2012 Phys. Rev. C 85 021302

    [15]

    Lu B N, Zhao E G, Zhou S G 2012 Phys. Rev. Lett. 109 072501

    [16]

    Liang H Z, Shen S H, Zhao P W, Meng J 2013 Phys. Rev. C 87 014334

    [17]

    Shen S H, Liang H Z, Zhao P W, Zhang S Q, Meng J 2013 Phys. Rev. C 88 024311

    [18]

    Guo J Y, Chen S W, Niu Z M, Li D P, Liu Q 2014 Phys. Rev. Lett. 112 062502

    [19]

    Liang H Z, Meng J, Zhou S G 2015 Phys. Rep. 570 1

    [20]

    Zhang M C 2009 Acta Phys. Sin. 58 61 (in Chinese) [张民仓 2009 物理学报 58 61]

    [21]

    Lu H F, Meng J 2002 Chin. Phys. Lett. 19 1775

    [22]

    Lu H F, Meng J, Zhang S Q, Zhou S G 2003 Eur. Phys. J. A 17 19

    [23]

    Zhang W, Meng J, Zhang S Q, Geng L S, Toki H 2005 Nucl. Phys. A 753 106

    [24]

    Qu X Y, Chen Y, Zhang S Q, Zhao P W, Shin I J, Lim Y, Kim Y, Meng J 2013 Sci. China. Phys. Mech. 56 2031

    [25]

    Sun B H, Meng J 2008 Chin. Phys. Lett. 25 2429

    [26]

    Li Z, Niu Z M, Sun B H, Wang N, Meng J 2012 Acta Phys. Sin. 61 072601 (in Chinese) [李竹, 牛中明, 孙保华, 王宁, 孟杰2012 物理学报61 072601]

    [27]

    Price C E, Walker G E 1987 Phys. Rev. C 36 354

    [28]

    Meng J, Lu H F, Zhang S Q, Zhou S G 2003 Nucl. Phys. A 722 C366

    [29]

    Zhou S G, Meng J, Ring P 2003 Phys. Rev. C 68 034323

    [30]

    Zhou S G, Meng J, Ring P, Zhao E G 2010 Phys. Rev. C 82 3481

    [31]

    Davies K T R, Flocard H, Krieger S, Weiss M S 1980 Nucl. Phys. A 342 111

    [32]

    Bonche P, Flocard H, Heenen P H 2005 Comput. Phys. Commun. 171 49

    [33]

    Zhang Y, Liang H Z, Meng J 2010 Int. J. Mod. Phys. E 19 55

    [34]

    Hagino K, Tanimura Y 2010 Phys. Rev. C 82 057301

    [35]

    Grant I P 1982 Phys. Rev. A 25 1230

    [36]

    Salomonson S, ster P 1989 Phys. Rev. A 40 5548

    [37]

    Tanimura Y, Hagino K, Liang H Z 2015 Prog. Theor. Exp. Phys. 2015 073D01

    [38]

    Zhao S 2007 Comput. Method. Appl. M. 196 5031

    [39]

    Shabaev V M, Tupitsyn I I, Yerokhin V A, Plunien G, Soff G 2004 Phys. Rev. Lett. 93 130405

    [40]

    Pestka G 2003 Phys. Scripta. 68 254

    [41]

    Mller C, Grn N, Scheid W 1998 Phys. Lett. A 242 245

    [42]

    Wilson K G 1977 Proceedings of the First Half of the 1975 International School of Subnuclear Physics Erice, Sicily, July 11-August 1, 1975 p69

    [43]

    Serot B D, Walecka J D 1986 Adv. Nucl. Phys. 16

    [44]

    Reinhard P G 1989 Rep. Prog. Phys. 52 439

    [45]

    Meng J 1998 Nucl. Phys. A 635 3

    [46]

    Abramowitz M, Stegun I A 1964 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (New York: Dover Publications) p914

    [47]

    Koepf W, Ring P 1991 Z. Phys. A: Hadrons Nucl. 339 81

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  • Received Date:  15 October 2015
  • Accepted Date:  30 November 2015
  • Published Online:  05 March 2016

Spherical Dirac equation on the lattice and the problem of the spurious states

    Corresponding author: Zhao Bin, bzhao@buaa.edu.cn
  • 1. School of Physics and Nuclear Energy Engineering, International Research Center for Nuclei and Particles in the Cosmos, Beihang University, Beijing 100191, China
Fund Project:  Project supported by the National Basic Research Program of China (Grant No. 2013CB834400), the National Natural Science Foundation of China (Grants Nos. 11175002, 11335002, 11375015), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110001110087).

Abstract: With the development of radioactive ion beam facilities, the study of exotic nuclei with unusual N/Z ratio has attracted much attention. Compared with the stable nuclei, the exotic nuclei have many novel features, such as the halo phenomenon. In order to describe the halo phenomenon with the diffused density distribution, the correct asymptotic behaviors of wave functions should be treated properly. The relativistic continuum Hartree-Bogoliubov (RCHB) theory which provides a unified and self-consistent description of mean field, pair correlation and continuum has achieved great success in describing the spherical exotic nuclei. In order to study the halo phenomenon in deformed nuclei, it is necessary to extend RCHB theory to the deformed case. However, solving the relativistic Hartree-Bogoliubov equation in space is extremely difficult and time consuming. Imaginary time step method is an efficient method to solve differential equations in coordinate space. It has been used extensively in the nonrelativistic case. For Dirac equation, it is very challenging to use the imaginary time step method due to the Dirac sea. This problem can be solved by the inverse Hamiltonian method. However, the problem of spurious states comes out. In this paper, we solve the radial Dirac equation by the imaginary time step method in coordinate space and study the problem of spurious states. It can be proved that for any potential, when using the three-point differential formula to discretize the first-order derivative operator, the energies of the single-particle states respectively with quantum numbers and - are identical. One of them is a physical state and the other is a spurious state. Although they have the same energies, their wave functions have different behaviors. The wave function of physical state is smooth in space while that of spurious state fluctuates dramatically. Following the method in lattice quantum chromodynamics calculation, the spurious state in radial Dirac equation can be removed by introducing the Wilson term. Taking Woods-Saxon potential for example, the imaginary time step method with the Wilson term is implanted successfully and provides the same results as those from the shooting method, which demonstrates its future application to solving the Dirac equation in coordinate space.

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