The theory of diffusive shock acceleration provides for several limitations to the maximum energy particles can reach in suprnova-remnant shocks. Two limits result from the acceleration time being limited by the remnant age or by the timescale for synchrotron losses, while particles with gyroradii longer than wavelengths of any MHD waves present will not scatter and will escape freely. For remnants younger than about 5000 yr, the acceleration-time limits imply maximum electronic energies of order 10^12-10^14 eV, high enough to produce keV synchrotron radiation. I describe a modeling program to calculate X-ray images and spectra from supernova remnants. Spectra result from a superposition of emmision from regions with different maximum energies, and curve gently through the X-ray bandpass, approximating power-laws. Most young remnants show obviously thermal emmision which is the upper limit on a possible nonthermal component. Hard components sometimes seen in SNR X-ray spectra could be due to the synchrotron process. X-ray emitting electrons will have long diffusion lengths ahead of the shock and should produce a synchrotron "halo" outside the main shock, which may be detectable. Detections or limits on these evidences of nonthermal X-rays provide surprisingly strong constraints on the microphysical properties of electron scattering and diffusion, and on the external magnetic-field strength.
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