X-ray source apparatus, being adapted for generating an X-ray beam, comprising: - a laser-plasma accelerator, which includes a first laser source device and a gas channel having a first gas supply conduit and an acceleration section being adapted for flowing a working gas along a flow direction, wherein the laser-plasma accelerator is configured for irradiating a first pulsed laser beam from the first laser source device into the working gas, so that a plasma with an accelerating plasma field can be created in the working gas and an electron beam can be generated by electrons accelerated in the plasma field, and - a Thomson scattering part, which includes a second laser source device, wherein the Thomson scattering part is configured such that the X-ray beam can be generated by Thomson scattering of the electron beam with a second pulsed laser beam from the second laser source device, characterized in that - the gas channel further includes a beam forming section being arranged downstream from the acceleration section and being configured for creating a pressure gradient so that the working gas flowing along the flow direction is subjected to a gas density reduction, and - the beam forming section comprises a portion of the gas channel between a second gas supply conduit and an exit end of the gas channel, wherein - the pressure gradient is formed such that, resulting from the density reduction, the transverse emittance of the electron beam is preserved and, due to an adiabatic variation of transverse focusing forces in the plasma channel, the divergence of the electron beam is reduced in the beam forming section relative to the acceleration section, and the second gas supply conduit is arranged for adjusting the pressure gradient in the beam forming section. - a laser-plasma accelerator, which includes a first laser source device and a gas channel having a first gas supply conduit and an acceleration section being adapted for flowing a working gas along a flow direction, wherein the laser-plasma accelerator is configured for irradiating a first pulsed laser beam from the first laser source device into the working gas, so that a plasma with an accelerating plasma field can be created in the working gas and an electron beam can be generated by electrons accelerated in the plasma field, and - a Thomson scattering part, which includes a second laser source device, wherein the Thomson scattering part is configured such that the X-ray beam can be generated by Thomson scattering of the electron beam with a second pulsed laser beam from the second laser source device, characterized in that - the gas channel further includes a beam forming section being arranged downstream from the acceleration section and being configured for creating a pressure gradient so that the working gas flowing along the flow direction is subjected to a gas density reduction, and - the beam forming section comprises a portion of the gas channel between a second gas supply conduit and an exit end of the gas channel, wherein - the pressure gradient is formed such that, resulting from the density reduction, the transverse emittance of the electron beam is preserved and, due to an adiabatic variation of transverse focusing forces in the plasma channel, the divergence of the electron beam is reduced in the beam forming section relative to the acceleration section, and the second gas supply conduit is arranged for adjusting the pressure gradient in the beam forming section.