We present a multi-frequency Electron Spin Resonance (ESR) study in the range of 4 GHz to 420 GHz of the quasi-one-dimensional, non-dimerized, quarter-filled Mott insulators, delta-(EDT-TTF-CONMe_2)_2X (X=AsF_6, Br). In the high temperature orthorhombic phase above T~190 K, the magnitude and the temperature dependence of the high temperature spin susceptibility are described by a S = 1/2 Heisenberg antiferromagnetic chain with J_AsF6=298 K and J_Br=474 K coupling constants for X=AsF_6 and Br respectively. We estimate from the temperature dependence of the line width an exchange anisotropy, J/J of ~2 * 10^{-3}. The frequency dependence of the line width and the g-shift have an unusual quadratic dependence in all crystallographic orientations that we attribute to an antisymmetric exchange (Dzyaloshinskii--Moriya) interaction.
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