  We performed systematic studies of the combined effects of
annealing/quenching temperature ({\itshape T}$_{A/Q}$) and T = Ni, Rh
substitution ({\itshape x}) on the physical properties of
Ca(Fe$_{1-x}$T$_{x}$)$_{2}$As$_{2}$. We constructed two-dimensional, {\itshape
T}$_{A/Q}$-{\itshape x} phase diagrams for the low-temperature states for both
substitutions to map out the relations between ground states and compared them
with that of Co-substitution. Ni-substitution, which brings one more extra
electron per substituted atom and suppresses the {\itshape c}-lattice parameter
at roughly the same rate as Co-substitution, leads to a similar parameter range
of antiferromagnetic/orthorhombic in the {\itshape T}$_{A/Q}$-{\itshape x}
space as that found for Co-substitution, but has the parameter range for
superconductivity shrunk (roughly by a factor of two). This result is similar
to what is found when Co- and Ni-substituted BaFe$_{2}$As$_{2}$ are compared.
On the other hand, Rh-substitution, which brings the same amount of extra
electrons as does Co-substitution, but suppresses the {\itshape c}-lattice
parameter more rapidly, has a different phase diagram. The collapsed tetragonal
phase exists much more pervasively, to the exclusion of the normal,
paramagnetic, tetragonal phase. The range of antiferromagnetic/orthorhombic
phase space is noticeably reduced, and the superconducting region is
substantially suppressed, essentially truncated by the collapsed tetragonal
phase. In addition, we found that whereas for Co-substitution there was no
difference between phase diagrams for samples annealed for one or seven days,
for Ni- and Rh- substitutions a second, reversible, effect of annealing was
revealed by seven-day anneals.
