- CERN-LHC. This article presents measurements of the $t$-channel single top-quark $(t)$ and top-antiquark $(\bar t)$ total production cross sections $\sigma(tq)$ and $\sigma(\bar tq)$, their ratio $R_{t}=\sigma(tq)/\sigma(\bar tq)$, and a measurement of the inclusive production cross section $\sigma(tq+\bar tq)$ in proton-proton collisions at $\sqrt{s}~=~7~\mathrm{TeV}$ at the LHC. Differential cross sections for the $tq$ and $\bar tq$ processes are measured as a function of the transverse momentum and the absolute value of the rapidity of $t$ and $\bar t$, respectively. The analyzed data set was recorded with the ATLAS detector and corresponds to an integrated luminosity of $4.59~\mathrm{fb}^{-1}$. Selected events contain one charged lepton, large missing transverse momentum, and two or three jets. The cross sections are measured by performing a binned maximum-likelihood fit to the output distributions of neural networks. The resulting measurements are $\sigma(tq)=46 \pm 1(\mathrm{stat})\pm 6(\mathrm{syst})~\mathrm{pb}$, $\sigma(\bar tq)=23 \pm 1(\mathrm{stat})\pm 3(\mathrm{syst})~\mathrm{pb}$, $R_{t}=2.04 \pm 0.13(\mathrm{stat})\pm 0.12(\mathrm{syst})$, and $\sigma(tq+\bar tq)=68 \pm 2(\mathrm{stat})\pm 8(\mathrm{syst})~\mathrm{pb}$, consistent with the Standard Model expectation. The uncertainty on the measured cross sections is dominated by systematic uncertainties, while the uncertainty on $R_{t}$ is mainly statistical. Using the ratio of $\sigma(tq+\bar tq)$ to its theoretical prediction, and assuming that the top-quark-related CKM matrix elements obey the relation $|V_{tb}|\gg |V_{ts}|$, $|V_{td}|$, we determine $|V_{tb}|=1.02 \pm 0.07$.

Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured $\dfrac{\mathrm{d}\sigma}{\mathrm{d}|y(t)|}$ distribution given in percent for each bin. The list includes only those uncertainties that contribute with more than $1\%$. The following uncertainties contribute to the total uncertainty with less than $1\%$ to each bin content$:$ JES detector, JES statistical, JES physics modeling, JES mixed detector and modeling, JES close-by jets, JES pileup, JES flavor composition, JES flavor response, jet-vertex fraction, $b/\bar{b}$ acceptance, $E_{\mathrm{T}}^{\mathrm{miss}}$ modeling, $W+$ jets shape variation, $t \bar{t}$ generator, $t \bar{t}$ ISR/FSR, and unfolding. In cases when the uncertainty is report to be "$<1\%$" in the table of the paper the uncertainty is approximated by a value of $0.5\%$.