Detailed measurements of $t$-channel single top-quark production are presented. They use 20.2 fb$^{-1}$ of data collected by the ATLAS experiment in proton-proton collisions at a centre-of-mass energy of 8 TeV at the LHC. Total, fiducial and differential cross-sections are measured for both top-quark and top-antiquark production. The fiducial cross-section is measured with a precision of 5.8 % (top quark) and 7.8 % (top antiquark), respectively. The total cross-sections are measured to be $\sigma_{\mathrm{tot}}(tq) = 56.7^{+4.3}_{-3.8}\;$pb for top-quark production and $\sigma_{\mathrm{tot}}(\bar{t}q) = 32.9^{+3.0}_{-2.7}\;$pb for top-antiquark production, in agreement with the Standard Model prediction. In addition, the ratio of top-quark to top-antiquark production cross-sections is determined to be $R_t=1.72 \pm 0.09$, with an improved relative precision of 4.9 % since several systematic uncertainties cancel in the ratio. The differential cross-sections as a function of the transverse momentum and rapidity of both the top quark and the top antiquark are measured at both the parton and particle levels. The transverse momentum and rapidity differential cross-sections of the accompanying jet from the $t$-channel scattering are measured at particle level. All measurements are compared to various Monte Carlo predictions as well as to fixed-order QCD calculations where available.
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