ALGEBRA OF COUPLED ELECTRONS IN THE CO-EXISTENCE OF
Superconductivity is a phenomenon in which the d.c electrical resistance of a material vanishes completely and instantly rather than gradually when it is cooled below a certain temperature called the superconducting transition or critical temperature, Both experimental and theoretical studies have been carried out in the last few years on materials that exhibit co-existence of superconductivity and magnetism. The compounds that exhibited such properties included and among others.. In the conventional superconductivity theories, such a co-existence was ruled out, Since superconductivity depends on the nature of electron-electron coupling, weak coupling leading to BCS theory, and strong coupling leading to high- superconductivity, it is necessary to understand the nature of electron-electron coupling that can lead to the coexistence of superconductivity and ferromagnetism. In BCS theory no attempt was made to study the commutation laws that the operator, , that constitutes Cooper pair, should obey. It was also not pointed out as to the kind of statistics that the Cooper pairs will obey. These inconsistencies were pointed out latter. It was, therefore, felt necessary to look into the algebra of coupled electrons that lead to superconductivity and to see simultaneously if such an algebra can lead to the understanding of superconductivity and ferromagnetism. Isolated electrons obey anti-commutation laws, whereas Cooper pairs ( ) will behave as bosons that obey commutation laws for Bose particles. The algebra developed correlates the operators associated with the electrons (Fermions) to the operators associated with the bi-linear electron operators that correspond to a pair of electrons. Effect of spin-fluctuation and electron-phonon coupling on the transition temperature has also been studied, and it has been established that is finite and it increases as the values of and increase showing thereby that superconductivity and ferromagnetism can co-exist.
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