Finite-size dimensional analysis of the superconducting vibronic interaction

Even in a finite-size system, the vibronic interaction acts as the attractive force to bind a pair of electrons. For small-size systems, the electron repulsion overwhelms the vibronic attraction. As the size of the system becomes large, the electronic repulsion diminishes to zero in proportion to the volume of the system, whereas the vibronic attraction (1) grows to infinity for a one-dimensional system, (2) converges to a finite value for a two-dimensional system, and (3) diminishes to zero for a three-dimensional system. Even for a three-dimensional system, the vibronic attraction diminishes much slower than does the electronic repulsion. This brings about a concept of the critical size for an any-dimensional system, over which size the vibronic attraction overwhelms the electronic repulsion, thereby creating purely attractive interaction for a pair of electrons, which may lead to superconductivity.