Mkn 421 was repeatedly observed with BeppoSAX in 1997-1998. The source showed a very rich phenomenology, with remarkable spectral variability. This is the second of two papers presenting the results of a thorough temporal and spectral analysis of all the data available to us, focusing in particular on the flare of April 1998, which was simultaneously observed also at TeV energies. The spectral analysis and correlations are presented in this paper, while the data reduction and timing analysis are the content of the companion paper. The spectral evolution during the flare has been followed over few ks intervals, allowing us to detect for the first time the peak of the synchrotron component shifting to higher energies during the rising phase, and then receding. This spectral analysis nicely confirms the delay of the flare at the higher energies, which in Paper I we quantified as a hard lag of a few ks. Furthermore, at the highest energies, evidence is found of variations of the inverse Compton component. The spectral and temporal information obtained challenge the simplest models currently adopted for the (synchrotron) emission and most importantly provide clues on the particle acceleration process. A scenario accounting for all the observational constraints is discussed, where electrons are injected at progressively higher energies during the development of the flare, and the achromatic decay is ascribed to the source light crossing time exceeding the particle cooling timescales.
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