  The crystalline undulator is a single crystal with periodically bent
crystallographic planes. If ultrarelativistic charged particles channel through
such a crystal, they emit hard radiation of undulator type. A crystalline
undulator with a bending amplitude smaller than the distance between the bent
planes and a bending period shorter than the period of channeling oscillations
is proposed. Heretofore, it was believed that such a range of bending
parameters was unsuitable for a crystalline undulator. This point of view is
refuted. In fact, the undulator with a small amplitude and a short period is
far superior to what was proposed previously. It requires much lower beam
energy for production of photons of the same frequency. Such an undulator
allows for a larger effective number of undulator periods. It is predicted to
emit intense undulator radiation in the forward direction with a narrow
spectral distribution and a lower and softer background. The undulator effect
is seen for both positron and electron beams. Using positrons is, however,
preferable because they enable one to obtain higher intensity of the undulator
radiation with lower background.
