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.
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References
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).
I. A. Kvasnikov, Thermodynamics and Statistical Physics: Theory of Equilibrium Systems [in Russian], Vol. 2, URSS, Moscow (2002).
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).
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).
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.
Yu. E. Pennionzhkevich, “Light nuclei and bounds of neutron stability,” Preprint, Joint Inst. Nucl. Res., Dubna (2016).
R. Gilmore, “Uncertainty relations of statistical mechanics,” Phys. Rev. A, 31, 3237–3239 (1985).
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).
A. D. Bruno, “Self-similar solutions and power geometry,” Russian Math. Surveys, 55, 1–42 (2000).
A. Weinstein, “The Maslov Gerbe,” Lett. Math. Phys., 69, 3–9 (2004).
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.
Y. Mishin, “Thermodynamic theory of equilibrium fluctuations,” Ann. Phys., 363, 48–97 (2015); arXiv: 1507.05662v1 [cond-mat.stat-mech] (2015).
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).
V. P. Maslov, “On mathematical investigations related to the Chernobyl disaster,” Russ. J. Math. Phys., 25, 309–318 (2018).
V. P. Maslov, “Statistics corresponding to classical thermodynamics construction of isotherms,” Russ. J. Math. Phys., 22, 53–67 (2015).
V. P. Maslov, “Locally ideal liquid,” Russ. J. Math. Phys., 22, 361–373 (2015).
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.
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This research was supported by the Federal Target Program (Government Registration Number AAAA-A17-117021310377-1).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 201, No. 1, pp. 65–83, October, 2019.
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Maslov, V.P. Description of Stable Chemical Elements by an aF Diagram and Mean Square Fluctuations. Theor Math Phys 201, 1468–1483 (2019). https://doi.org/10.1134/S0040577919100052
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DOI: https://doi.org/10.1134/S0040577919100052