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Theoretical and Mathematical Physics

, Volume 201, Issue 1, pp 1468–1483 | Cite as

Description of Stable Chemical Elements by an aF Diagram and Mean Square Fluctuations

  • V. P. MaslovEmail author
Article
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Abstract

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes a boson at the instant of separating from a fermionic nucleus. We study the further transformations of boson and fermion states of separation in a small neighborhood of zero pressure and obtain new important parastatistical relations between the temperature and the chemical potential when a nucleon separates from an atomic nucleus. The obtained relations allow constructing a new diagram (an aF diagram) or isotherms of very high temperatures corresponding to nuclear matter. We mathematically prove that the transition of particles from the domain governed by Fermi-Dirac statistics to the domain governed by Bose-Einstein statistics near the zero pressure P occurs in the neutron uncertainty domain or halo domain. We obtain equations for the chemical potential that allow determining the width of the uncertainty domain. Based on the calculated values of the minimum intensivity for Bose particles, the chemical potential, the compressibility factor, and the minimum mean square fluctuation of the chemical potential, we construct a table of stable nuclei of chemical elements, demonstrating a monotonic relation between the nucleus mass number and the other parameters.

Keywords

energy of neutron separation from the atomic nucleus nucleus binding energy neutron domain of uncertainty parastatistics (Gentile statistics) mean square fluctuations of energy and chemical potential nuclear matter 

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Notes

Acknowledgments

The author is deeply grateful to E. I. Nikulin, who recently listened to a course of lectures by I. A. Kvasnikov, for the help in representing some of his ideas underlying this paper.

Conflicts of interest. The author declares no conflicts of interest.

References

  1. 1.
    W.-S. Dai and M. Xie, “Gentile statistics with a large maximum occupation number,” Ann. Phys., 309, 295–305 (2004); arXiv:cond-mat/0310066v3 (2003).ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    I. A. Kvasnikov, Thermodynamics and Statistical Physics: Theory of Equilibrium Systems [in Russian], Vol. 2, URSS, Moscow (2002).Google Scholar
  3. 3.
    V. P. Maslov, “Extremal values of activity for nuclear matter when a nucleon separates from the atomic nucleus,” Russian J. Math. Phys., 26, 50–54 (2019).ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    L. D. Faddeev and O. A. Yakubovskii, Lectures on Quantum Mechanics for Mathematical Students [in Russian], Leningrad Univ. Press, Leningrad (1980); English transl. (Student Math. Libr., Vol. 47), Amer. Math. Soc., Providence, R. I. (2009).zbMATHGoogle Scholar
  5. 5.
    G. L. Litvinov, “The Maslov dequantization, idempotent and tropical mathematics: A very brief introduction,” in: Idempotent Mathematics and Mathematical Physics (Contemp. Math., Vol. 377, G. L. Litvinov and V. P. Maslov, eds.), Amer. Math. Soc., Providence, R. I. (2005), pp. 1–18.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Yu. E. Pennionzhkevich, “Light nuclei and bounds of neutron stability,” Preprint, Joint Inst. Nucl. Res., Dubna (2016).Google Scholar
  7. 7.
    R. Gilmore, “Uncertainty relations of statistical mechanics,” Phys. Rev. A, 31, 3237–3239 (1985).ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    V. P. Maslov, “Case of less than two degrees of freedom, negative pressure, and the Fermi–Dirac distribution for a hard liquid,” Math. Notes, 98, 138–157 (2015).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. D. Bruno, “Self-similar solutions and power geometry,” Russian Math. Surveys, 55, 1–42 (2000).ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Weinstein, “The Maslov Gerbe,” Lett. Math. Phys., 69, 3–9 (2004).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    N. J. Davidson, H. G. Miller, R. M. Quick, B. J. Cole, R. H. Lemmer, and R. Tegen, “Specific heat of strongly interacting matter,” in: Phase Structure of Strongly Interacting Matter (J. Cleymans, ed.), Springer, Berlin (1990), pp. 216–250.Google Scholar
  12. 12.
    Y. Mishin, “Thermodynamic theory of equilibrium fluctuations,” Ann. Phys., 363, 48–97 (2015); arXiv: 1507.05662v1 [cond-mat.stat-mech] (2015).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    V. P. Maslov, V. P. Myasnikov, and V. G. Danilov, Mathematical Modeling of the Damaged Block of the Chernobyl Atomic Electric Power Station [in Russian], Nauka, Moscow (1987).zbMATHGoogle Scholar
  14. 14.
    V. P. Maslov, “On mathematical investigations related to the Chernobyl disaster,” Russ. J. Math. Phys., 25, 309–318 (2018).MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. P. Maslov, “Statistics corresponding to classical thermodynamics construction of isotherms,” Russ. J. Math. Phys., 22, 53–67 (2015).MathSciNetCrossRefGoogle Scholar
  16. 16.
    V. P. Maslov, “Locally ideal liquid,” Russ. J. Math. Phys., 22, 361–373 (2015).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Tikhonov Moscow Institute of Electronics and MathematicsNational Research University “Higher School of Mathematics”MoscowRussia

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