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The spin-structure functions \(g_1\) and \(g_2\), and the spin-dependent partial cross-section \(\sigma_\mathrm{TT}\) have been extracted from the polarized cross-sections differences, \(\Delta \sigma_{\parallel}\hspace{-0.06cm}\left(\nu,Q^{2}\right)\) and \(\Delta \sigma_{\perp}\hspace{-0.06cm}\left(\nu,Q^{2}\right)\) measured for the \(\vec{^\textrm{3}\textrm{He}}(\vec{\textrm{e}},\textrm{e}')\textrm{X}\) reaction, in the E97-110 experiment at Jefferson Lab. Polarized electrons with energies from 1.147 to 4.404 GeV were scattered at angles of 6\(^{\circ}\) and 9\(^{\circ}\) from a longitudinally or transversely polarized \(^{3}\)He target. The data cover the kinematic regions of the quasi-elastic, resonance production and beyond. From the extracted spin-structure functions, the first moments \(\overline{\Gamma_1}\hspace{-0.06cm}\left(Q^{2}\right)\), \(\Gamma_2\hspace{-0.06cm}\left(Q^{2}\right)\) and \(I_{\mathrm{TT}}\hspace{-0.06cm}\left(Q^{2}\right)\) are evaluated with high precision for the neutron in the \(Q^2\) range from 0.035 to 0.24~GeV\(^{2}\). The comparison of the data and the chiral effective field theory predictions reveals the importance of proper treatment of the \(\Delta\) degree of freedom for spin observables.
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