On "box" models of shock acceleration and electron synchrotron spectra

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The recent detection of high energy gamma-rays coming from supernova remnants and active galactic nuclei has revived interest in the diffusive shock acceleration of electrons. In the present paper we examine the basis of the so-called ``box'' model for particle acceleration and present a more physical version of it. Using this we determine simple criteria for the conditions under which ``pile-ups'' can occur in shock accelerated electron spectra subject to synchrotron or inverse Compton losses (the latter in the Thompson limit). An extension to include nonlinear effects is proposed.