The Anderson-Rickayzen equations of motion for a superconductor derived within the random-phase approximation (RPA) are used to investigate the collective excitations of superconductors. A spherical harmonic expansion is made of the two-body interaction potential $V(\mathrm{k}, {\mathrm{k}}^{\prime })$ and a spectrum of excitations whose energies lie within the energy gap $2\Delta$ is obtained. These excitations may be characterized by the quantum numbers $L$ and $M$ involved in the potential expansion. For an $L$-state exciton to exist, the $L$-wave part of the potential must be attractive at the Fermi surface. Odd-$L$ excitons have unit spin and may be considered as spin waves. For $s$-state pairing in the superconducting ground state, the plasmon mode corresponds to the $L=0\mathrm{}$ exciton whose energy is strongly modified by the long-range Coulomb interaction. For a general potential several bound states may exist for given $L$ and $M$. If the $L$-wave potential is stronger than the $s$-wave part of the potential, the system is unstable with respect to formation of $L$-state excitons. In this case, the ground state is formed with $L$-state pairing, special cases of which are the $p$-state pairing considered by Fisher and the $d$-state pairing proposed recently by several authors for the ground state of ${\mathrm{He}}^3$ and nuclear matter. Corrections to the Anderson-Rickayzen equations are discussed which lead to a new set of exciton states if the $L$-wave potential is repulsive. These excitons are interpreted as bound electron-hole pairs, as opposed to the particle-particle excitons present with an attractive $L$-wave potential.

• Received 13 October 1960

DOI:https://doi.org/10.1103/PhysRev.121.1050