  In a cluster crystal, each lattice site is occupied by multiple soft-core
particles. As the number density is increased at zero temperature, a `cascade'
of isostructural phase transitions can occur between states whose site
occupancy differs by unity. For low but finite temperature, each of these
transitions terminates in a critical point. Using tailored Monte Carlo
simulation techniques we have studied such demixing cascades in systems of soft
particles interacting via potentials of the generalized exponential form
$u(r)=\epsilon\exp[-(r/\sigma)^n]$. We have estimated the critical parameters
of the first few transitions in the cascade as a function of the softness
parameter $n$. The critical temperature and pressure exhibit non-monotonic
behaviour as $n$ is varied, although the critical chemical potential remains
monotonic. The trends for the pressure and chemical potential are confirmed by
cell model calculations at zero temperature. As $n\to 2^+$, all the transitions
that we have observed are preempted by melting although we cannot rule out that
clustering transitions survive at high density.
