We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping of charge carriers can be tuned to produce a good description of the measured charge collection profiles in the fluence range from 0.5x10^{14} Neq/cm^2 to 5.9x10^{14} Neq/cm^2. The model correctly predicts the variation in the profiles as the temperature is changed from -10C to -25C. The measured charge collection profiles are inconsistent with the linearly-varying electric fields predicted by the usual description based upon a uniform effective doping density. This observation calls into question the practice of using effective doping densities to characterize irradiated silicon.
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