In the late nineteenth century, Joseph John Thomson moved away from Maxwell's specific theoretical models of matter and energy, even though he continued to rely on the general framework of Maxwell's electromagnetic theory. In his 1893 book, he accomplished the conceptual drift towards a discrete model for matter, electricity, and fields. In Thomson's model, energy was linked to tubes of force, in particular to the aether contained in them and surrounding them: the energy was the kinetic energy of aether, of both a rotational and translational kind. Starting from Maxwell's electromagnetic fields, namely stresses propagating through a continuous solid medium, Thomson arrived at a representation of fields as a sea of discrete units carrying energy and momentum. He tried to transform Maxwell's theory into a unified picture in which atomic models of matter stood beside atomic models of fields. In 1904 his interpretation of X-rays was based on the integration between two complementary features of electromagnetic radiation, the continuity and the discreteness, and on some kind of fibrous aether. In recent secondary literature, the problematic conceptual link between J. J. Thomson's theory and contemporary theories on electromagnetic radiation has been underestimated. On the contrary, in the first half of the twentieth century, some physicists inquired into that link, and a widespread debate emerged, misunderstandings included.