On some generalized stopping power sum rules

The Lindhard-Winther (LW) equipartition sum rule shows that within the linear response theory, the stopping power of an energetic point-charge projectile in a degenerate electron gas medium, receives equal contributions from single-particle and collective excitations in the medium. In this paper we show that the LW sum rule does not necessarily hold for an extended projectile ion and for ion-clusters moving in a fully degenerate electron gas. We have derived a generalized equipartition sum rule and some related sum rules for this type of projectiles. We also present numerical plots for He$^{+}$ ion and He$^{+}$ ion-clusters.