Using a mixed-ligand synthetic scheme, we create a family of quasi-two-dimensional antiferromagnets, namely, [Cu(HF$_2$)(pyz)$_2$]ClO$_4$ [pyz = pyrazine], [Cu$L_2$(pyz)$_2$](ClO$_4$)$_2$ [$L$ = pyO = pyridine-N-oxide and 4-phpyO = 4-phenylpyridine-N-oxide. These materials are shown to possess equivalent two-dimensional [Cu(pyz)$_2$]$^{2+}$ nearly square layers, but exhibit interlayer spacings that vary from 6.5713~AA ~to 16.777~AA, as dictated by the axial ligands. We present the structural and magnetic properties of this family as determined via x-ray diffraction, electron-spin resonance, pulsed- and quasistatic-field magnetometry and muon-spin rotation, and compare them to those of the prototypical two-dimensional magnetic polymer Cu(pyz)$_2$(ClO$_4$)$_2$. We find that, within the limits of the experimental error, the two-dimensional, {it intralayer} exchange coupling in our family of materials remains largely unaffected by the axial ligand substitution, while the observed magnetic ordering temperature decreases slowly with increasing layer separation. Despite the structural motifs common to this family and Cu(pyz)$_2$(ClO$_4$)$_2$, the latter has significantly stronger two-dimensional exchange interactions and hence a higher ordering temperature. We discuss these results, as well as the mechanisms that might drive the long-range order in these materials, in terms of departures from the ideal $S=1/2$ two-dimensional square-lattice Heisenberg antiferromagnet. In particular, we find that both spin exchange anisotropy in the intralayer interaction and interlayer couplings (exchange, dipolar, or both) are needed to account for the observed ordering temperatures, with the intralayer anisotropy becoming more important as the layers are pulled further apart.
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