  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. The model is now being used to calibrate pixel hit
reconstruction algorithms for CMS.
