  We present results from monitoring observations of the gravitationally lensed
quasar RX J1131-1231 performed with the Chandra X-ray Observatory. The X-ray
observations were planned with relatively long exposures that allowed a search
for energy-dependent microlensing in the soft (0.2-2 keV) and hard (2-10 keV)
light curves of the images of RX J1131-1231. We detect significant microlensing
in the X-ray light-curves of images A and D, and energy-dependent microlensing
of image D. The magnification of the soft band appears to be larger than that
in the hard band by a factor of ~ 1.3 when image D becomes more magnified. This
can be explained by the difference between a compact, softer-spectrum corona
that is producing a more extended, harder spectrum reflection component off the
disk. This is supported by the evolution of the fluorescent iron line in image
D over three consecutive time-averaged phases of the light curve. In the first
period, a Fe line at E = 6.36(-0.16,+0.13) keV is detected (at > 99%
confidence). In the second period, two Fe lines are detected, one at E =
5.47(-0.08,+0.06) keV (detected at > 99% confidence) and another at E =
6.02(-0.07,+0.09) keV (marginally detected at > 90% confidence), and in the
third period, a broadened Fe line at 6.42(-0.15,+0.19) keV is detected (at >
99% confidence). This evolution of the Fe line profile during the microlensing
event is consistent with the line distortion expected when a caustic passes
over the inner disk where the shape of the fluorescent Fe line is distorted by
General Relativistic and Doppler effects.
