Measurements of the Neutron Longitudinal Spin Asymmetry $A_1$ and Structure Function $g_1$ in the Valence Quark Region

The current data for the nucleon-virtual photon longitudinal spin asymmetry $A_1$ on the proton and neutron have shown that the ratio of the polarized-to-unpolarized down-quark parton distribution functions, $\Delta d/d$, tends towards -1/2 at large $x$, in disagreement with the perturbative QCD prediction that $\Delta d/d$ approaches 1. As a part of experiment E06-014 in Hall A of Jefferson Lab, double-spin asymmetries were measured in the scattering of a longitudinally polarized electron beam of energies 4.73 and 5.89 GeV from a longitudinally and transversely polarized $^3$He target in the deep inelastic scattering region, allowing for the extraction of the neutron asymmetry $A_1^n$ and the longitudinal spin structure function $g_1^n$. We will discuss our analysis of the data and present results for $A_1$ and $g_1$ on both $^{3}$He and the neutron in the kinematic range of $0.2 < x < 0.65$ and $2 < Q^2 < 5$ GeV$^2$ for the scattered electrons. Our measurements of $A_1$ and $g_1$ are compared to the world data and a new value of the second moment of $g_1$, namely $a_2$, is evaluated and compared to a lattice QCD calculation. The quantity $a_2$ is needed to extract the twist-4 matrix element known as $f_2$.