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A systematic comparison is given of the quantum reduction in the ordered spin of quasi one-dimensional Heisenberg antifemmagneis as predicted by spin-wave themy and as m u r e d in ionic systems with known spin-wave panmeters. Simple linear theory yields a leading term [l/(?~r)]ln(CIJ/J'l) -1 where further terms can be neglected for practical purposes if IJ'/Jl < IO-*; snch expressions are listed for ferromagnetic and antifmrmgnetic square and triangular lattices of antiferromagnetic chains with zero anisotmpy. For half-integerspin systems with 6 < S < a. excellent agreement with experiment is found over the range when a known conection for 'kinematical interactions' is applied. It should thus be possible lo identify covalency reductions in excess of 10%. For the S = 1 system CsNiC13, however, the measured spin reduction may indicate exm quantum fluctuations related to the Haldane effect. From the susceptibili behaviour, the quantum ground state for progressively smaller l J ' / J [ is argued to increasingly resemble a singlet separated from other states by an energy gap. > IJ'/JI > * The sw model approximates an actual orthorhombic smchxe. M6sssbauer hyperfine field variation [U] used in extrapalating 4.2 K magnetic moment [24] to T = 0. 'Obtained from TN = IO K and J / k = -9.45 K via the Oguchi relation kTN =Z.IS(S + l)(lJlIJ'l)'''. 8 Includes the effect of easy-axis anisohopy. D/I JI = 2.8 x i Includes the effect of easy-plane anisotropy, D/lJI = -1.6 x I O -' . in compliance [15] with a small spin-flop field of % 20 kG, derived [26] from a spin-flop field of 37 kG. Error as given for the 4.5 K data used [27] in extrapolating to T = 0.The spin-wave energies for the SLF system KCuF3 reveal [ I l l an anisotropy that is negligible compared to J', whereas for TLAP CsMnBr3 [28] an appreciable easy-plane term is found, which has been included in computing AS. For K*FeFS, an easy-axis anisotropy D FX J' derived from the measured spin-flop field [26] has been incorporated in the computation. t The full integral (3) has been used where lJ'/Jl 2 IO-'. For IJ'/Jl = 1.7 x lo-' in TLAF CsNiU, (table 2).for instance, the approximation (5) underestimates ASh by about 0.013, and ASm. (6) by about 0.007.