Transition from the Bose gas to the Fermi gas through a Nuclear Halo
The first part of the paper deals with the behavior of the Bose--Einstein distribution as the activity $a\to 0$. In particular, the neighborhood of the point $a=0$ is studied in great detail, and the expansion of both the Bose distribution and the Fermi distribution in powers of the parameter $a$ is used. This approach allows to find the value of the parameter $a_0$, for which the Bose distribution (in the statistical sense) becomes zero. In the second part of the paper, the process of separation of a nucleon from the atom's nucleus is studied from the mathematical point of view. At the moment when the nucleon tears away from the fermionic nucleus, the nucleus becomes a boson. We investigate the further transformations of bosonic and fermionic separation states in a small neighborhood of the pressure $P$ equal to zero. We use infinitely small quantities to modify the parastatistical distribution. Our conception is based on interpolation formulas yielding expansions in powers of the density. This method differs from those in other models based on the interaction potential between two or three particles. We obtain new important relations connecting the temperature with the chemical potential during the separation of a nucleon from the atom's nucleus. The obtained relations allow us to construct, on an antipode of sorts of the Hougen--Watson P-Z chart, the very high temperature isotherms corresponding to nuclear matter. It is proved mathematically that the passage of particles satisfying the Fermi--Dirac distribution to the Bose--Einstein distribution in the neighborhood of pressure $P$ equal to zero occurs in a region known as the "halo".