  We investigate the isotope effect on the London penetration depth of a
superconductor which measures $n_S/m^*$, the ratio of superfluid density to
effective mass. We use a simplified model of electrons weakly coupled to a
single phonon frequency $\omega_E$, but assume that the energy gap $\Delta$
does not have any isotope effect. Nevertheless we find an isotope effect for
$n_S/m^*$ which is significant if $\Delta$ is sufficiently large that it
becomes comparable to $\omega_E$, a regime of interest to high $T_c$ cuprate
superconductors and possibly other families of unconventional superconductors
with relatively high $T_c$. Our model is too simple to describe the cuprates
and it gives the wrong sign of the isotope effect when compared with
experiment, but it is a proof of principle that the isotope effect exists for
$n_S/m^*$ in materials where the pairing gap and $T_c$ is not of phonon origin
and has no isotope effect.
