Affiliations: 12773 Seabreeze Farms Dr. # 33, San Diego, CA 92130, USA. E-mail: [email protected]
Abstract: The possibility of condensation of electron pairs has been established by Yang and Coleman. Recent observations of Bose–Einstein condensate (BEC) of alkali atoms, containing odd number of electrons, require some explanation, which also must be based on a correct permutational symmetry of a system, containing different types of fermions. In the framework of the same reduced density matrix formalism, it is proved that aggregates (atoms), containing even number 2f of fermions, can be condensed only in a mixed quantum state in contrast to elementary bosons because a natural occupation number λ is proved to be strictly smaller than their total sum. If fermions are absent in aggregates the reduction of sum of λ to one term becomes possible. The upper bound λmax for a macroscopic ensemble of n aggregates, built of m different sorts of fermions is shown to be m1[((2f−1)!!)3A/(2f)!!B]1/2 where A is the number of all possible aggregates in a system, B is the number of ways to form a given composition of an aggregate from 2f-fermions of m sorts; n is a macroscopic greatest common devisor of fermion numbers of all sorts. The bound λmax increases as nf, while sum of λ grows as n2f. The evenness of the total number of fermions 2f in an aggregate is a necessary and sufficient condition for BEC formation. In particular, the number N of neutrons in neutral atom must be even because of the integrity of f=Z+N/2. The extreme type wave function is built, for which λ is arbitrary close to nf that proves the sufficiency of this criterion. The occupation maximum is achieved when pairs of fermions (identical or different) are condensed independently. The possibility of condensation of bound aggregates only (atoms) is shown arising from nonoverlapping of wave functions of electron relative coordinates of different atoms. This conclusion is based on separation of the centers’ of mass motion, which are proved behaving like a bosonic gas. The remaining product of identical atomic wave functions of relative coordinates, which determine main properties of condensate ingredients, is shown not requiring further antisymmetrization that makes condensation possible.
Keywords: Bose-Einstein condensate, reduced density matrix, upper bound of natural occupation numbers, statistics of different type fermion aggregates
