Fission fragment "Energy amplifier" rockets

Most proposed fission fragment rockets use a self sustaining nuclear
reaction to generate the fission fragments and neutrons. However,
this requires very high temperatures of tiny particles. Why not
operate in a non-self sustaining mode where the reaction is started by
an external neutron source. Each external neutron produces a shower
of fission neutrons but it is not self sustaining, each shower dies
out. However, they do produce fission fragments that are the reaction
mass.
The external source of neutrons could be something like Americium that
would be placed closer to the fissionable material when you wanted
more power and further away when you wanted less. This would allow
the fissionable material to be made from a less refractory material as
the fissionable material particles would not get very hot.