Apparatus for the controllable downhole production of ionizing radiation, wherein the apparatus includes: - a fluid-tight, cylindrical housing arranged to be electrically grounded; - an electrically insulated vacuum container suspended within the housing; - at least a thermionic emitter which is arranged in a first end portion of the electrically insulated vacuum container, the thermionic emitter being configured for generating leptons, and - a lepton target which is arranged in a second end portion of the electrically insulated vacuum container; the thermionic emitter being connected to a series of serially connected negative electrical-potential-increasing elements (14 1, 14 2, 14 3, 14 4), each of said electrical-potential-increasing elements (14 1, 14 2, 14 3, 14 4) being arranged to increase an applied direct-current potential (δV 0, δV 1, δV 1+2,..., δV 1+2+3) by transforming an applied, driving voltage (V AC), and to transmit the increased, negative direct-current potential (δV 1, δV 1+2,..., δV 1+2+3+4) and also the driving voltage (V AC) to the next unit in the series of serially connected elements, and the ionizing radiation exceeding 200 keV with a predominant portion of the spectral distribution within the Compton range, wherein each negative electrical-potential-increasing element (14 1 -14 4) is centrally arranged and supported within the electrically grounded housing by a rotationally symmetrical support structure made of a material or composite of materials with high dielectric resistivity and good thermal conductivity, wherein the rotationally symmetrical support structure is configured in such a way that the distance that electrical energy will have to cover along the surface of the support structure from the negative electrical-potential-increasing elements (14 1 -14 4) to the grounded surrounding housing is much larger than the physical radial distance between the negative electrical-potential-increasing elements (14 1 -14 4) and the housing, so that electrical flashover or sparking between conductors with large differences in voltage is inhibited, wherein a cylindrical field controller is arranged on the outside of each electrical-potential-increasing element (14 1 -14 4) to ensure that the radial potential between each of the electrical-potential-increasing elements (14 1 -14 4) and the enveloping housing remains constant across the entire axial extent of the electrical-potential-increasing element (14 1 -14 4), thereby forming a homogeneous field towards ground regardless of the electrical potential (δV 1, δV 1+2, δV 1+2+3, δV 1+2+3+4) of the specific electrical-potential-increasing element (14 1 -14 4). - a fluid-tight, cylindrical housing arranged to be electrically grounded; - an electrically insulated vacuum container suspended within the housing; - at least a thermionic emitter which is arranged in a first end portion of the electrically insulated vacuum container, the thermionic emitter being configured for generating leptons, and - a lepton target which is arranged in a second end portion of the electrically insulated vacuum container; the thermionic emitter being connected to a series of serially connected negative electrical-potential-increasing elements (14 1, 14 2, 14 3, 14 4), each of said electrical-potential-increasing elements (14 1, 14 2, 14 3, 14 4) being arranged to increase an applied direct-current potential (δV 0, δV 1, δV 1+2,..., δV 1+2+3) by transforming an applied, driving voltage (V AC), and to transmit the increased, negative direct-current potential (δV 1, δV 1+2,..., δV 1+2+3+4) and also the driving voltage (V AC) to the next unit in the series of serially connected elements, and the ionizing radiation exceeding 200 keV with a predominant portion of the spectral distribution within the Compton range, wherein each negative electrical-potential-increasing element (14 1 -14 4) is centrally arranged and supported within the electrically grounded housing by a rotationally symmetrical support structure made of a material or composite of materials with high dielectric resistivity and good thermal conductivity, wherein the rotationally symmetrical support structure is configured in such a way that the distance that electrical energy will have to cover along the surface of the support structure from the negative electrical-potential-increasing elements (14 1 -14 4) to the grounded surrounding housing is much larger than the physical radial distance between the negative electrical-potential-increasing elements (14 1 -14 4) and the housing, so that electrical flashover or sparking between conductors with large differences in voltage is inhibited, wherein a cylindrical field controller is arranged on the outside of each electrical-potential-increasing element (14 1 -14 4) to ensure that the radial potential between each of the electrical-potential-increasing elements (14 1 -14 4) and the enveloping housing remains constant across the entire axial extent of the electrical-potential-increasing element (14 1 -14 4), thereby forming a homogeneous field towards ground regardless of the electrical potential (δV 1, δV 1+2, δV 1+2+3, δV 1+2+3+4) of the specific electrical-potential-increasing element (14 1 -14 4).