Neutron scattering sum rules, symmetric exchanges, and helicoidal magnetism in MnSb$_2$O$_6$

MnSb$_{2}$O$_{6}$ is based on the noncentrosymmetric $P321$ space group with magnetic Mn$^{2+}$ ($S={5/2}$, $L\approx 0$) spins ordering below $T_{\mathrm{N}}=12$ K in a helicoidal structure. The ground state magnetic structure, expected to be built and originate from 7 Heisenberg exchange constants, has been shown to be coupled to the underlying crystallographic chirality with polar domain switching being reported. We apply neutron spectroscopy to extract these symmetric exchange constants. Given the high complexity of the magnetic exchange network, crystallographic structure and complications fitting linear spin-wave models, we take advantage of multiplexed neutron instrumentation to use the first moment sum rule of neutron scattering to estimate the 7 exchange constants. We then use these parameters to calculate the low-energy spin-waves in the N\'eel state to reproduce the neutron response without strong antisymmetric coupling. Using Green's response functions, the stability of long-wavelength excitations in the context of proposed magnetic structures is then discussed. The results show the presence of strong exchange constants for the chiral exchange pathways and illustrate an underlying coupling between crystallographic and magnetic ``chirality" through predominantely symmetric exchange.