We report on simulation results relevant both for EuPRAXIA@SPARC LAB [1] and EuPRAXIA [2] showing how to boost a high brightness electron bunch, delivered by a conventional accelerator at 500 MeV, up to 5 GeV and more by external injection in a laser driven plasma wave. Beam slice quality is preserved along acceleration and the bunch is shown to be able to drive a Free Electron Laser at 1 Å wavelength. We also show two methods to evaluate global bunch parameters, one for energy spread, the other for emittance, allowing to exclude particles in halos and tails in a consistent way. Both methods are easily implemented in computational codes, are statistically robust and failure free.
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