  A measurement of hafnium foil using a modified ultra-low-background high
purity detector with optimized sample-to-detector geometry was performed at
Laboratori Nazionale del Gran Sasso. Radiopurity of the stock Hf foil was
studied in detail, in addition to an analysis of data collected over 310 days
to search for rare processes that can occur in natural Hf isotopes. Firstly,
limits on alpha decays of all natural Hf isotopes to the first excited state of
the daughter nuclides were established in the range of $10^{16}$-$10^{18}$a
(90% C.L.). Secondly, a search for modes of double electron capture and
electron capture with positron emission in $^{174}$Hf was performed, yielding
half-life limits $10^{16}$-$10^{18}$a (90% C.L.). Lastly, novel dark
matter-induced nuclear excitations in hafnium isotopes were investigated. For
dark matter with 1 TeV/$c^2$ mass, leading limits on the inelastic dark
matter--nucleon cross section are set for mass splittings in the range 428 keV
$< \delta <$ 473 keV.
