Precise determination of the spin structure function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>g</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>of the proton, deuteron, and neutron

Precise measurements of the spin structure functions of the proton ${g}_{1}^{p}(x,{Q}^{2})$ and deuteron ${g}_{1}^{d}(x,{Q}^{2})$ are presented over the kinematic range $0.0041\ensuremath{\le}x\ensuremath{\le}0.9$ and $0.18\text{ }\text{ }{\mathrm{GeV}}^{2}\ensuremath{\le}{Q}^{2}\ensuremath{\le}20\text{ }\text{ }{\mathrm{GeV}}^{2}$. The data were collected at the HERMES experiment at DESY, in deep-inelastic scattering of 27.6 GeV longitudinally polarized positrons off longitudinally polarized hydrogen and deuterium gas targets internal to the HERA storage ring. The neutron spin structure function ${g}_{1}^{n}$ is extracted by combining proton and deuteron data. The integrals of ${g}_{1}^{p,d}$ at ${Q}^{2}=5\text{ }\text{ }{\mathrm{GeV}}^{2}$ are evaluated over the measured $x$ range. Neglecting any possible contribution to the ${g}_{1}^{d}$ integral from the region $x\ensuremath{\le}0.021$, a value of $0.330\ifmmode\pm\else\textpm\fi{}0.011(\mathrm{theo})\ifmmode\pm\else\textpm\fi{}0.025(\mathrm{exp})\ifmmode\pm\else\textpm\fi{}0.028(\mathrm{evol})$ is obtained for the flavor-singlet axial charge ${a}_{0}$ in a leading-twist next-to-next-to-leading-order analysis.