  The inverse Compton scattering of the cosmic microwave background (CMB)
radiation with electrons in the intracluster medium which has a temperature
gradient, was examined by the third-order perturbation theory of the Compton
scattering. A new type of the spectrum distortion of the CMB was found and
named as gradient T Sunyaev-Zel'dovich effect (gradT SZE). The spectrum has an
universal shape. The spectrum crosses over zero at 326GHz. The sign of the
spectrum depends on the relative direction of the line-of-sight to the
direction of the temperature gradient. This unique spectrum shape can be used
to detect the gradT SZE signal by broad-band or multi-frequency observations of
the SZE. The amplitude of the spectrum distortion does not depend on the
electron density and is proportional to the heat conductivity. Therefore, the
gradT SZE provides an unique opportunity to measure thermally nonequilibrium
electron momentum distribution function when the ICM has a temperature gradient
and the heat conductivity in the ICM. However, the expected amplitude of the
signal is very small. The modifications to the thermal SZE spectrum due to
variety of known effects, such as relativistic correction etc., can become
problematic when using multi-frequency separation techniques to detect the
gradT SZE signal.
