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Shells, orbit bifurcations, and symmetry restorations in Fermi systems

  • Nuclei
  • Theory
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Abstract

The periodic-orbit theory based on the improved stationary-phase method within the phase-space path integral approach is presented for the semiclassical description of the nuclear shell structure, concerning themain topics of the fruitful activity ofV.G. Soloviev. We apply this theory to study bifurcations and symmetry breaking phenomena in a radial power-law potential which is close to the realistic Woods–Saxon one up to about the Fermi energy. Using the realistic parametrization of nuclear shapes we explain the origin of the double-humped fission barrier and the asymmetry in the fission isomer shapes by the bifurcations of periodic orbits. The semiclassical origin of the oblate–prolate shape asymmetry and tetrahedral shapes is also suggested within the improved periodic-orbit approach. The enhancement of shell structures at some surface diffuseness and deformation parameters of such shapes are explained by existence of the simple local bifurcations and new non-local bridge-orbit bifurcations in integrable and partially integrable Fermi-systems. We obtained good agreement between the semiclassical and quantum shell-structure components of the level density and energy for several surface diffuseness and deformation parameters of the potentials, including their symmetry breaking and bifurcation values.

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References

  1. M. Gutzwiller, J. Math. Phys. 12, 343 (1971); Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).

    Article  ADS  Google Scholar 

  2. R. B. Balian and C. Bloch, Ann. Phys. (N.Y.) 69, 76 (1972).

    Article  ADS  Google Scholar 

  3. V. M. Strutinsky, Nukleonika 20, 679 (1975)

    Google Scholar 

  4. V. M. Strutinsky and A. G. Magner, Sov. J. Part. Nucl. 7, 138 (1976).

    Google Scholar 

  5. M. V. Berry and M. Tabor, Proc. R. Soc. London Ser. A 349, 101 (1976); 356, 375 (1977).

    Article  ADS  Google Scholar 

  6. V. M. Strutinsky, A. G. Magner, S. R. Ofengenden, and T. Døssing, Z. Phys. A 283, 269 (1977).

    Article  ADS  Google Scholar 

  7. S. C. Creagh, J. M. Robbins, and R. G. Littlejohn, Phys. Rev. A 42, 1907 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  8. S. C. Creagh and R. G. Littlejohn, Phys. Rev. A 44, 836 (1991); J. Phys. A 25, 1643 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  9. M. Brack and R. K. Bhaduri, Semiclassical Physics, rev. ed. (Westview Press, Boulder, 2003).

    MATH  Google Scholar 

  10. A. G. Magner, I. S. Yatsyshyn, K. Arita, and M. Brack, Phys. Atom. Nucl. 74, 1445 (2011).

    Article  ADS  Google Scholar 

  11. V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967); 122, 1 (1968).

    Article  ADS  Google Scholar 

  12. M. Brack, J. Damgaard, A. S. Jensen, et al., Rev. Mod. Phys. 44, 320 (1972).

    Article  ADS  Google Scholar 

  13. W. D. Myers and W. J. Swiatecki, Ann. Phys. (N.Y.) 55, 395 (1969); 84, 186 (1974).

    Article  ADS  Google Scholar 

  14. M. Brack, C. Guet, and H.-B. Håkansson, Phys. Rep. 123, 275 (1985).

    Article  ADS  Google Scholar 

  15. A. B. Migdal, The Finite Fermi–System Theory and Properties of Atomic Nuclei (Intersci., New York, 1967; Nauka, Moscow, 1983).

    Google Scholar 

  16. V. A. Khodel and E. E. Saperstein, Phys. Rep. 92, 183 (1982).

    Article  ADS  Google Scholar 

  17. L. D. Landau, Sov. Phys. JETP 3, 920 (1956); 8, 70 (1959).

    Google Scholar 

  18. A. A. Abrikosov and I. M. Khalatnikov, Rep. Prog. Phys. 22, 329 (1959).

    Article  ADS  Google Scholar 

  19. A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, New York, 1975), Vol. 2.

    MATH  Google Scholar 

  20. P. Ring and P. Schuck, The Nuclear Many–Body Problem (Springer, New York, Heisenberg, Berlin, 1980).

    Book  Google Scholar 

  21. H. Hofmann, The Physics of Warm Nuclei with Analogies to Mesoscopic Systems (Oxford, Univ. Press, Oxford, 2008).

    Book  Google Scholar 

  22. V. V. Pashkevich and S. Frauendorf, Yad. Fiz. 20, 1122 (1974) [Sov. J. Nucl. Phys. 20, 588 (1974)].

    Google Scholar 

  23. I. N. Mikhailov, K. Neergard, V. V. Pashkevich, and S. Frauendorf, Sov. J. Part. Nucl. 8, 550 (1977).

    Google Scholar 

  24. S. Frauendorf, arXiv: 1209.5816 [nucl-th].

  25. A. G. Magner, V. M. Kolomietz, and V. M. Strutinskiĭ, Sov. J. Nucl. Phys. 28, 764 (1978).

    Google Scholar 

  26. V. M. Kolomiets, A. G. Magner, and M. Strutinskiĭ, Sov. J. Nucl. Phys. 29, 758 (1979).

    Google Scholar 

  27. A. G. Magner, V. M. Kolomiets, and V. M. Strutinsky, Bull. Acad. Sci. USSR, Phys. Ser. 43, 142 (1979).

    Google Scholar 

  28. K. Richter, D. Ulmo, and R. A. Jalabert, Phys. Rep. 276, 1 (1996).

    Article  ADS  Google Scholar 

  29. S. Frauendorf, V. M. Kolomietz, A. G. Magner, and A. I. Sanzhur, Phys. Rev. B 58, 5622 (1998).

    Article  ADS  Google Scholar 

  30. M. A. Deleplanque, S. Frauendorf, V. V. Pashkevich, et al., Phys. Rev. C 69, 044309 (2004).

    Article  ADS  Google Scholar 

  31. A. G. Magner, A. S. Sitdikov, A. A. Khamzin, et al., Nucl. Phys. At. Energy 10, 239 (2009); Int. J. Mod. Phys. E 19, 735 (2010); Phys. Atom. Nucl. 73, 1398 (2010).

    Google Scholar 

  32. A. G. Magner, A. S. Sitdikov, A. A. Khamzin, and J. Bartel, Phys. Rev. C 81, 064302 (2010).

    Article  ADS  Google Scholar 

  33. A. G. Magner, D. V. Gorpinchenko, and J. Bartel, Phys. Atom. Nucl. 77, 1229 (2014).

    Article  Google Scholar 

  34. D. V. Gorpinchenko, A. G. Magner, J. Bartel, and J. P. Blocki, Phys. Scr. 90, 114008 (2015).

    Article  ADS  Google Scholar 

  35. D. V. Gorpinchenko, A. G. Magner, J. Bartel, and J. P. Blocki, Phys. Rev. C 93, 024304 (2016).

    Article  ADS  Google Scholar 

  36. S. T. Belyaev and V. G. Zelevinsky, Sov. Phys. Usp. 28, 854 (1985).

    Article  ADS  Google Scholar 

  37. V. G. Solovjov, Fiz. Elem. Chastits At. Yadra 9, 580 (1978).

    Google Scholar 

  38. L. A. Molov and V. G. Soloviev, Fiz. Elem. Chastits At. Yadra 11, 301 (1980).

    Google Scholar 

  39. V. G. Soloviev, Theory of Atomic Nucleus. Nuclear Models (Energoizdat, Moscow, 1981) [in Russian].

    Google Scholar 

  40. A. I. Vdovin and V. G. Soloviev, Fiz. Elem. Chastits At. Yadra 14, 237 (1983).

    Google Scholar 

  41. A. I. Vdovin and V. G. Soloviev, Fiz. Elem. Chastits At. Yadra 14, 1380 (1983).

    Google Scholar 

  42. A. I. Vdovin, V. V. Voronov, V. G. Soloviev, and Ch. Stojanov, Fiz. Elem. Chastits. At. Yadra 16, 245 (1985).

    Google Scholar 

  43. V. G. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons (Institute of Physics, Bristol; Philadelphia, 1992).

    Google Scholar 

  44. V. G. Soloviev, Structure of Even Deformed Nuclei (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  45. V. G. Solovjov, Theory of Complex Nuclei (Pergamon Press, New York, 1976).

    Google Scholar 

  46. V. G. Soloviev, Nucl. Phys. A 9, 655 (1958).

    Article  Google Scholar 

  47. S. T. Belyaev, Sov. Phys. JETP 13, 470 (1961).

    Google Scholar 

  48. V. G. Solovjov, Effect of Superconducting Pairing Correlations on Nuclear Properties; in Selected Topics in Nuclear Theory (IAEA, Vienna, 1963).

    Google Scholar 

  49. S. T. Belyaev and V. G. Zelevinskiĭ, Sov. J. Nucl. Phys. 11, 416 (1970); 16, 657 (1973); 17, 269 (1973).

    Google Scholar 

  50. D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626 (1972).

    Article  ADS  Google Scholar 

  51. M. Brack and P. Quentin, Nucl. Phys. A 361, 35 (1981).

    Article  ADS  Google Scholar 

  52. V. I. Abrosimov, D. M. Brink, A. Delafiore, and F. Matera, Nucl. Phys. A 864, 38 (2011); and the works cited therein.

    Article  ADS  Google Scholar 

  53. H. Hofmann, F. A. Ivanyuk, and S. Yamaji, Nucl. Phys. A 598 187 (1996).

    Article  ADS  Google Scholar 

  54. F. A. Ivanyuk, H. Hofmann, V. V. Pashkevich, and S. Yamaji, Phys. Rev. C 55, 1730 (1997).

    Article  ADS  Google Scholar 

  55. A. G. Magner, S. M. Vydrug-Vlasenko, and H. Hofmann, Nucl. Phys. A 524, 31 (1991); Izv. Akad. Nauk SSSR, Ser. Fiz. 54, 148 (1990).

    Article  ADS  Google Scholar 

  56. A. M. Gzhebinsky, A. G. Magner, and S. N. Fedotkin, Phys. Rev. C 76, 064315 (2007).

    Article  ADS  Google Scholar 

  57. A. G. Magner, A. M. Gzhebinsky, and S. N. Fedotkin, Yad. Fiz. 70, 677, 1910 (2007) [Phys. Atom. Nucl. 70, 647, 1859 (2007)].

    Google Scholar 

  58. J. P. Blocki, A. G. Magner, and I. S. Yatsyshyn, Int. J. Mod. Phys. E 21, 1250034 (2012).

    Article  ADS  Google Scholar 

  59. J. P. Blocki and A. G. Magner, Phys. Scr. T 154, 014006 (2013).

    Article  ADS  Google Scholar 

  60. G. Lazzari, H. Nishioka, E. Vigezzi, and R. A. Broglia, Phys. Rev. B 53, 1064 (1996).

    Article  ADS  Google Scholar 

  61. A. G. Magner, S. N. Fedotkin, F. A. Ivanyuk et al., Ann. Phys. (Berlin) 509, 555 (1997).

    Article  ADS  Google Scholar 

  62. M. Brack, S. M. Reimann, and M. Sieber, Phys. Rev. Lett. 79, 1817 (1997).

    Article  ADS  Google Scholar 

  63. A. G. Magner, S. N. Fedotkin, K. Arita, et al., Phys. Rev. E 63, 065201(R) (2001).

    Article  ADS  Google Scholar 

  64. A. G. Magner, K. Arita, S. N. Fedotkin, and K. Matsuyanagi, Prog. Theor. Phys. 108, 853 (2002).

    Article  ADS  Google Scholar 

  65. A. G. Magner, Nucl. Phys. At. Energy 11, 227 (2010).

    ADS  Google Scholar 

  66. A. G. Magner, Yad. Fiz. 28, 1477 (1978) [Sov. J. Nucl. Phys. 28, 759 (1978)].

    Google Scholar 

  67. M. V. Fedoryuk, Zh. Vychisl. Mat. Mat. Fiz. 2(1), 145 (1962); 4(4), 671 (1964) [Sov. J. Comput. Math. Math. Phys. 2(1), 152 (1963); 4(4), 66 (1964)].

    Google Scholar 

  68. M. P. Maslov, Theor. Math. Phys. 2, 21 (1970).

    Article  Google Scholar 

  69. M. V. Fedoryuk, The Saddle-Point Method (Nauka, Moscow, 1977) [in Russian].

    MATH  Google Scholar 

  70. C. Chester, B. Friedmann, and F. Ursell, Proc. Cambridge Philos. Soc. 53, 599 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  71. A. G. Magner, K. Arita, and S. N. Fedotkin, Prog. Theor. Phys. 115, 523 (2006).

    Article  ADS  Google Scholar 

  72. A. G. Magner, S. N. Fedotkin, K. Arita, et al., Prog. Theor. Phys. 102, 551 (1999).

    Article  ADS  Google Scholar 

  73. M. V. Koliesnik, Ya. D. Krivenko-Emetov, A. G. Magner, K. Arita, and M. Brack, Phys. Scr. 90, 114011 (2015).

    Article  ADS  Google Scholar 

  74. A. M. Ozorio de Almeida, Hamiltonian Systems: Chaos and Quantization (Cambridge Univ. Press, Cambridge, 1988).

    MATH  Google Scholar 

  75. D. Ullmo, M. Grinberg, and S. Tomsovic, Phys. Rev. E 54, 136 (1996).

    Article  ADS  Google Scholar 

  76. S. C. Creagh, Ann. Phys. (N.Y.) 248, 60 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  77. M. Sieber, J. Phys. A 30, 4563 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  78. M. Brack, J. Blaschke, S. C. Creagh, et al., Z. Phys. D 40, 276 (1997).

    Article  ADS  Google Scholar 

  79. H. Schomerus and M. Sieber, J. Phys. A 30, 4537 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  80. M. Sieber and H. Schomerus, J. Phys. A 31, 165 (1998).

    Article  ADS  Google Scholar 

  81. H. Schomerus, J. Phys. A 31, 4167 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  82. M. Brack, P. Meier, and K. Tanaka, J. Phys. A 32, 331 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  83. M. Brack, Found. Phys. 31, 209 (2001); nlin.CD/0006034.

    Article  MathSciNet  Google Scholar 

  84. M. Brack, M. Mehta, and K. Tanaka, J. Phys. A 34, 8199 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  85. J. Kaidel and M. Brack, Phys. Rev. E 70, 016206 (2004).

    Article  ADS  Google Scholar 

  86. K. Arita and M. Brack, J. Phys. A 41, 385207 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  87. K. Arita, Phys. Rev. C 86, 034317 (2012).

    Article  ADS  Google Scholar 

  88. S. N. Fedotkin, A. G. Magner, and M. Brack, Phys. Rev. E 77, 066219 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  89. K. Arita, Int. J. Mod. Phys. E 13, 191 (2004).

    Article  ADS  Google Scholar 

  90. A. G. Magner, A. A. Vlasenko, and K. Arita, Phys. Rev. E 87, 062916 (2013).

    Article  ADS  Google Scholar 

  91. M. Brack and K. Tanaka, Phys. Rev. E 77, 046205 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  92. M. Brack, R. K. Bhaduri, J. Law, and M. V. N. Murthy, Phys. Rev. Lett. 70, 568 (1993).

    Article  ADS  Google Scholar 

  93. J. P. Blocki, A. G. Magner, and I. S. Yatsyshyn, Nucl. Phys. At. Energy 11, 239 (2010).

    Google Scholar 

  94. J. P. Blocki, A. G. Magner, and I. S. Yatsyshyn, Int. J. Mod. Phys. E 20, 292 (2011).

    Article  ADS  Google Scholar 

  95. J. P. Blocki and A. G. Magner, Phys. Rev. C 85, 064311 (2012).

    Article  ADS  Google Scholar 

  96. M. Brack, J. Kaidel, P. Winkler, and S. N. Fedotkin, Few-Body Syst. 38, 147 (2006).

    Article  ADS  Google Scholar 

  97. C. Gustafsson, P. Möller, and S. G. Nilsson, Phys. Lett. B 34, 349 (1971).

    Article  ADS  Google Scholar 

  98. S. Cohen and W. J. Swiatecki, Ann. Phys. (N.Y.) 22, 406 (1963).

    Article  ADS  Google Scholar 

  99. P. J. Richens, J. Phys. A. 15, 2101 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  100. H. Frisk, Nucl. Phys. A 511, 309 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  101. I. Hamamoto and B. R. Mottelson, Phys. Rev. C 79, 034317 (2009).

    Article  ADS  Google Scholar 

  102. N. Tajima and N. Suzuki, Phys. Rev. C 64, 037301 (2001)

    Article  ADS  Google Scholar 

  103. N. Tajima, Y. R. Shimizu, and N. Suzuki, Prog. Theor. Phys. Suppl. 146, 628 (2002).

    Article  ADS  Google Scholar 

  104. S. Takahara, N. Onishi, Y. R. Shimizu, and N. Tajima, Phys. Lett. B 702, 429 (2011)

    Article  ADS  Google Scholar 

  105. S. Takahara, N. Tajima, and Y. R. Shimizu, Phys. Rev. C 86, 064323 (2012).

    Article  ADS  Google Scholar 

  106. K. Arita, to appear in Phys. Scr. (2016).

    Google Scholar 

  107. P. A. Butler and W. Nazarewicz, Rev. Mod. Phys. 68, 349 (1996).

    Article  ADS  Google Scholar 

  108. J. Blocki et al., Ann. Phys. (N.Y.) 113, 330 (1978).

    Article  ADS  Google Scholar 

  109. J. Blocki, J. Skalski, and W. J. Swiatecki, Nucl. Phys. A 594, 137 (1995)

    Article  ADS  Google Scholar 

  110. J. Blocki, J.-J. Shi, W. J. Swiatecki, Nucl. Phys. A 554, 387 (1993)

    Article  ADS  Google Scholar 

  111. C. Jarzynski and W. J. Swiatecki, Nucl. Phys. A 552, 1 (1993)

    Article  ADS  Google Scholar 

  112. P. Magierski, J. Skalski, and J. Blocki, Phys. Rev. C 56, 1011 (1997).

    Article  ADS  Google Scholar 

  113. J. P. Blocki, A. G. Magner, and I. S. Yatsyshyn, Nucl. Phys. At. Energy 11, 239 (2010); Int. J. Mod. Phys. E 20, 292 (2011); 21, 1250034 (2012).

    Google Scholar 

  114. I. Hamamoto, B. R. Mottelson, H. Xie, and X. Z. Zhang, Z. Phys. D 21, 163 (1991).

    Article  ADS  Google Scholar 

  115. S. M. Reimann, M. Koskinen, H. Häkkinen, et al., Phys. Rev. B 56, 12147 (1997).

    Article  ADS  Google Scholar 

  116. S. Takami, K. Yabana, and M. Matsuo, Phys. Lett. B 431, 242 (1998).

    Article  ADS  Google Scholar 

  117. J. Dudek, A. Goźdź, N. Schunck, and M. Miśkiewicz, Phys. Rev. Lett. 88, 252502 (2002).

    Article  ADS  Google Scholar 

  118. K. Arita and Y. Mukumoto, Phys. Rev. C 89, 054308 (2014).

    Article  ADS  Google Scholar 

  119. M. Brack, M. Ögren, Y. Yu, and S. M. Reimann, J. Phys. A 38, 9941 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  120. H. Frisk and T. Guhr, Ann. Phys. (N.Y.) 221, 229 (1993).

    Article  ADS  Google Scholar 

  121. Ch. Amann and M. Brack, J. Phys. A 35, 6009 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  122. M. Brack, Ch. Amann, M. Pletyukhov, and O. Zaitsev, Int. J. Mod. Phys. E 13, 19 (2004).

    Article  ADS  Google Scholar 

  123. K. Bencheikh, P. Quentin, and J. Bartel, Nucl. Phys. A 571, 518 (1994).

    Article  ADS  Google Scholar 

  124. V. M. Strutinskii and A. S. Tyapin, Sov. Phys. JETP 18, 664 (1964).

    Google Scholar 

  125. V. M. Strutinsky, A. G. Magner, and M. Brack, Z. Phys. A 319, 205 (1984).

    Article  ADS  Google Scholar 

  126. V. M. Strutinsky, A. G. Magner, and V. Yu. Denisov, Z. Phys. A 322, 149 (1985); Sov. J. Nucl. Phys. 42, 690 (1985).

    Article  ADS  Google Scholar 

  127. A. G. Magner, A. I. Sanzhur, and A. M. Gzhebinsky, Int. J. Mod. Phys. E 18, 885 (2009).

    Article  ADS  Google Scholar 

  128. J. P. Blocki, A. G. Magner, P. Ring, and A. A. Vlasenko, Phys. Rev. C 87, 044304 (2013).

    Article  ADS  Google Scholar 

  129. J. P. Blocki, A. G. Magner, and P. Ring, Phys. Scr. 89, 054019 (2014).

    Article  ADS  Google Scholar 

  130. J. P. Blocki, A. G. Magner, and P. Ring, Phys. Scr. 90, 114009 (2015).

    Article  ADS  Google Scholar 

  131. J. P. Blocki, A. G. Magner, and P. Ring, Phys. Rev. C 92, 064311 (2015).

    Article  ADS  Google Scholar 

  132. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed (Sprinder-Verlag, Berlin, Heidelberg, 1971).

    Book  MATH  Google Scholar 

  133. S. C. Milne, in Developments in Mathematics (Kluwer Academic Publ., Dordrecht, 2002); Ramanujan J. 6, 7 (2002).

    Google Scholar 

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  1. Institute for Nuclear Research, NASU, Kiev, Ukraine

    A. G. Magner & M. V. Koliesnik

  2. Department of Physics, Nagoya Institute of Technology, Nagoya, Japan

    K. Arita

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  1. A. G. Magner
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Magner, A.G., Koliesnik, M.V. & Arita, K. Shells, orbit bifurcations, and symmetry restorations in Fermi systems. Phys. Atom. Nuclei 79, 1067–1123 (2016). https://doi.org/10.1134/S1063778816060181

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