Realizing square and diamond lattice S = 1 / 2 Heisenberg antiferromagnet models in the α and β phases of the coordination framework, K Ti ( C 2 O 4 ) 2 · x H 2 O

We report the crystal structures and magnetic properties of two pseudopolymorphs of the $S=1/2 {\\mathrm{Ti}}^{3+}$ coordination framework, $\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}x{\\mathrm{H}}_{2}\\mathrm{O}$. Single-crystal x-ray and powder neutron diffraction measurements on $\\ensuremath{\\alpha}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}x{\\mathrm{H}}_{2}\\mathrm{O}$ confirm its structure in the tetragonal $I4/mcm$ space group with a square planar arrangement of ${\\mathrm{Ti}}^{3+}$ ions. Magnetometry and specific heat measurements reveal weak antiferromagnetic interactions, with ${J}_{1}\\ensuremath{\\approx}7$ K and ${J}_{2}/{J}_{1}=0.11$ indicating a slight frustration of nearest- and next-nearest-neighbor interactions. Below 1.8 K, $\\ensuremath{\\alpha}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}x{\\mathrm{H}}_{2}\\mathrm{O}$ undergoes a transition to G-type antiferromagnetic order with magnetic moments aligned along the $c$ axis of the tetragonal structure. The estimated ordered moment of ${\\mathrm{Ti}}^{3+}$ in $\\ensuremath{\\alpha}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}x{\\mathrm{H}}_{2}\\mathrm{O}$ is suppressed from its spin-only value to $0.62(3)\\phantom{\\rule{0.28em}{0ex}}{\\ensuremath{\\mu}}_{B}$, thus verifying the two-dimensional nature of the magnetic interactions within the system. $\\ensuremath{\\beta}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}2{\\mathrm{H}}_{2}\\mathrm{O}$, on the other hand, realizes a three-dimensional diamondlike magnetic network of ${\\mathrm{Ti}}^{3+}$ moments within a hexagonal $P{6}_{2}22$ structure. An antiferromagnetic exchange coupling of $J\\ensuremath{\\approx}54$ K---an order of magnitude larger than in $\\ensuremath{\\alpha}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}x{\\mathrm{H}}_{2}\\mathrm{O}$---is extracted from magnetometry and specific heat data. $\\ensuremath{\\beta}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}2{\\mathrm{H}}_{2}\\mathrm{O}$ undergoes N\\\u0027eel ordering at ${T}_{N}=28$ K, with the magnetic moments aligned within the $ab$ plane and a slightly reduced ordered moment of $0.79\\phantom{\\rule{0.28em}{0ex}}{\\ensuremath{\\mu}}_{B}$ per ${\\mathrm{Ti}}^{3+}$. Through density-functional theory calculations, we address the origin of the large difference in the exchange parameters between the $\\ensuremath{\\alpha}$ and $\\ensuremath{\\beta}$ pseudopolymorphs. Given their observed magnetic behaviors, we propose $\\ensuremath{\\alpha}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}x{\\mathrm{H}}_{2}\\mathrm{O}$ and $\\ensuremath{\\beta}\\ensuremath{-}\\mathrm{K}\\mathrm{Ti}{({\\mathrm{C}}_{2}{\\mathrm{O}}_{4})}_{2}\\ifmmode\\cdot\\else\\textperiodcentered\\fi{}2{\\mathrm{H}}_{2}\\mathrm{O}$ as close to ideal model $S=1/2$ Heisenberg square and diamond lattice antiferromagnets, respectively.