  Motivated by frustrated magnets and quasi-one-dimensional magnetic materials,
we study the magnetic properties of 1D Ising chains with nearest-neighbour (NN)
and weaker next-to-nearest neighbour (NNN) interactions in the presence of
vacancy defects. The effect of a vacancy on the magnetic susceptibility of a
spin chain is two-fold: it reduces the length of the chain by an effective
``vacancy size'' and may also act as a free spin, a ``quasispin'', with a
Curie-type $\chi_\text{quasi}=\langle S^2\rangle/T$ contribution to the
susceptibility. In chains with antiferromagnetic short-range order, the
susceptibility of vacancy-free chains is exponentially suppressed at low
temperatures, and quasispins dominate the effect of impurities on the chains'
magnetic properties. For chains with antiferromagnetic NN interactions, the
quasispin matches the value $\langle S^2\rangle=1$ of the Ising spins in the
chain for ferromagnetic NNN interactions and vanishes for antiferromagnetic NNN
interactions. For chains with ferromagnetic short-range order, quasispin
effects are insignificant due to exponentially large low-temperature
susceptibilities, and the dominant effect of a vacancy is effectively changing
the length of the chain.
