  Parametric X-ray radiation (PXR) from relativistic electrons moving in a
crystal along the crystal-vacuum interface is considered. In this geometry the
emission of photons is happening in the regime of extremely asymmetric
diffraction (EAD). In the EAD case the whole crystal length contributes to the
formation of X-ray radiation opposed to Laue and Bragg geometries, where the
emission intensity is defined by the X-ray absorption length. We demonstrate
that this phenomenon should be described within the dynamical theory of
diffraction and predict a radical increase of the PXR intensity. In particular,
under realistic electron-beam parameters, an increase of two orders of
magnitude in PXR-EAD intensity can be obtained in comparison with conventional
experimental geometries of PXR. In addition we discuss in details the
experimental feasibility of the detection of PXR-EAD.
