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In the following we would like to observe that in a hypothetical atom with Z the atomic number being very very large, due to a very large number of protons, the outer electrons would behave like an almost free bosonic assembly. Naively, we would expect that the energy levels are given by,
[absolute value of [E.sub.n]] [varies] [(Z[[alpha]]).sup.2]/2[n.sup.2], n = 1, 2, ... (Z [much greater than] 1)
But let us examine this situation more closely.
We consider the interesting case of a " Super Atom", one in which there are a very large number of protons in the nucleus. We use the fact that when a particle is far from a cloud of charged particles, and moves slowly, the potential is approximately spherically symmetric . So if there is an atom with atomic number Z [much greater than] 1, so that the outer most electrons are far away, such an atom is like a Hydrogen atom . In this case, if [rho] is the (reduced) distance from the nucleus then we have (Cf. ref. )
<[rho]> = [[3[N.sup.2]--l ([l + 1])]/2Z] [less than or equal to] [N.sup.2] (1)
where N represents the number of states and remembering that Z itself is large. We also have that the energy of the Nth level is given instead by
[[epsilon].sub.N] = -[Z.sup.2]/2[N.sup.2] [less than or equal to] -1/[N.sup.2] (2)
From (1) it follows that the reduced volume V is given by
V [less than or equal to] [N.sup.6] (3)
It then follows using (3) that the Fermi energy is given by 
[[epsilon].sub.F] [varies] [(N/V).sup.2/3] [less than or equal to] 1/[N.sup.3.3] (4)
We can see from (2) and (4) that the outer electron cloud for large N behaves as if it is a degenerate electron gas below the Fermi energy (or temperature). The electrons in such a "Super Atom" are like a Fermi gas except for their negative energy which is approximately zero in any case. …