The ATLAS Collaboration Inclusive and differential cross-sections for the production of a top-quark pair in association with a photon are measured with proton-proton collision data corresponding to an integrated luminosity of 36.1 fb −1 , collected by the ATLAS detector at the LHC in 2015 and 2016 at a centre-of-mass energy of 13 TeV. The measurements are performed in single-lepton and dilepton final states in a fiducial volume. Events with exactly one photon, one or two leptons, a channel-dependent minimum number of jets, and at least one b-jet are selected. Neural network algorithms are used to separate the signal from the backgrounds. The fiducial cross-sections are measured to be 521 ± 9(stat.) ± 41(sys.) fb and 69 ± 3(stat.) ± 4(sys.) fb for the single-lepton and dilepton channels, respectively. The differential cross-sections are measured as a function of photon transverse momentum, photon absolute pseudorapidity, and angular distance between the photon and its closest lepton in both channels, as well as azimuthal opening angle and absolute pseudorapidity difference between the two leptons in the dilepton channel. All measurements are in agreement with the theoretical predictions.The ATLAS detector [11] consists of three main components. The innermost component is the Inner Detector (ID), which is used for tracking charged particles. It surrounds the beam pipe and is located inside a superconducting solenoid, operating with a magnetic field of 2 T. An additional silicon pixel layer, the insertable B-layer, was added between 3 and 4 cm from the beam line to improve b-hadron tagging [12, 13] for Run 2. The calorimeter outside the ID is divided into two subsystems. The inner subsystem is the electromagnetic calorimeter (ECAL) and the second is the hadronic calorimeter (HCAL). The outermost layer is the third main component of the ATLAS detector: the muon spectrometer (MS), which is within a magnetic field provided by air-core toroid magnets with a bending integral of about 2.5 Tm in the barrel and up to 6 Tm in the end-caps. The ID provides tracking information from silicon pixel and silicon microstrip detectors in the pseudorapidity range |η| < 2.5 and from a transition radiation tracker (TRT) covering |η| < 2.0. The magnetic field of the superconducting solenoid bends charged particles for the momentum measurement. The ECAL uses lead absorbers and liquid argon (LAr) as active medium and is divided into barrel (|η| < 1.475) and end-cap (1.375 < |η| < 3.2) regions. The HCAL is composed of a steel/scintillating-tile calorimeter, segmented into three barrel structures within |η| < 1.7, and two copper/LAr hadronic endcap calorimeters, that cover the region 1.5 < |η| < 3.2. The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules, optimised for electromagnetic and hadronic measurements respectively, and covering the region 3.1 < |η| < 4.9. The MS measures the deflection of muon tracks within |η| < 2.7 using multiple layers of high-precision tracking chambers in toroidal...

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