  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.
