  Electronic states in the gap of a superconductor inherit intriguing many-body
properties from the superconductor. Here, we study in-gap states created by
manipulating Cr atomic chains on the $\beta$-Bi$_2$Pd superconductor. We find
that the electronic properties of the in-gap states can greatly vary depending
on the crafted spin chain. These systems make an ideal platform for the hosting
of Majorana bound states because of the large atom-superconductor interactions
and the existence of a large Rashba coupling at the Bi-terminated surface. As
the Cr chains become more compact, the atomic manipulation process becomes more
involved, making precise atomic-scale manipulation of long chains very
difficult. We study two spin chains, one with atoms two-lattice-parameter apart
and one with square-root-of-two lattice parameters. We show that only the
second one can undergo a topological transition, despite its more compact
structure and more complex assembly.
