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Comment: 8 pages, 3 figures, 2 tables
We present for the first time the concept of a seeded {\gamma} quantum Free-Electron-Laser (QFEL) at 478 keV, which has very different properties compared to a classical. The basic concept is to produce a highly brilliant {\gamma} beam via SASE. To produce highly intense and coherent {\gamma} beam, we intend to use a seeded FEL scheme. Important for the production of such a {\gamma} beam are novel refractive {\gamma} -lenses for focusing and an efficient monochromator, allowing to generate a very intense and coherent seed beam. The energy of the {\gamma} beam is 478 keV, corresponding to a wavelength in the sub-{\AA}ngstr{\o}m regime (1/38 {\AA}). To realize a coherent {\gamma} beam at 478 keV, it is necessary to use a quantum FEL design. At such high radiation energies a classical description of the {\gamma}-FEL becomes wrong.
When a rather cold electron bunch is transported during laser bubble acceleration in a strongly focusing plasma channel with typical forces of 100 GeV/m, it will form partially ordered long electron cylinders due to the relativistically longitudinal reduced repulsion between electrons, resulting in a long-range pair correlation function, when reaching energies in the laboratory above 0.5 GeV. During Compton back-scattering with a second laser, injected opposite to the electron bunch, the electron bunch will be further modulated with micro bunches and due to its ordered structure will reflect coherently, M\"ossbauer-like, resulting in a \gamma free electron laser (\gamma-FEL). Increasing the relativistic \gamma factor, the quantal regime becomes more dominant. We discuss the scaling laws with \gamma.
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