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Possible N\'eel orderings of antiferromagnetically coupled spins on a kagom\'e lattice are studied using linear-spin-wave theory and high-temperature expansions. Spin-wave analysis, applied to q=0 (three spins per magnetic unit cell) and to 3 3 (nine spins per cell) N\'eel orderings yield identical excitation spectra with twofold-degenerate linear modes and a dispersionless zero-energy mode. This dispersionless mode is equivalent to an excitation localized to an arbitrary hexagon of nearest-neighbor spins. Second- (J₂) and third- (J₃) neighbor interactions are shown to stabilize the q=0 state for J₂>J₃ and the 3 3 state for J₂J₃. A high-temperature expansion of the spin-spin susceptibility _ (q) is performed to order 1/T^8, for n-component, classical spins with nearest-neighbor interactions only. To order 1/T^7 the largest eigenvalue of the susceptibility matrix is found to be independent of wave vector with an eigenvector that corresponds to the dispersionless mode of the ordered phase. This degeneracy is removed at order 1/T^8. For n=0, the q=0 mode is favored; for n=1, the band is flat; and, for n>1, the maximum susceptibility is found for a 3 3 excitation. Similar results are found for the three-dimensional pyrochlore lattice. The high-temperature expansion is used to interpret experimental data for the uniform susceptibility and powder-neutron-diffraction spectrum for the kagom\'e-lattice system SrCr₈-ₗGa₄+ₗO₁₉.
Harris et al. (Sat,) studied this question.