  Harmonic lasing of low-gain free electron laser oscillators has been
experimentally demonstrated in the terahertz and infrared regions. Recently,
the low-gain oscillator has been reconsidered as a promising candidate for hard
x-ray free electron lasers, through the use of high reflectivity, high
resolution x-ray crystals. In this letter, it is proposed to utilize a
crystal-based cavity resonant at a higher harmonic of the undulator radiation,
together with phase shifting, to enable harmonic lasing of the x-ray free
electron laser oscillator, and hence allow the generation of hard x-ray
radiation at a reduced electron beam energy. Results show that fully coherent
free electron laser radiation with megawatt peak power, in the spectral region
of 10-25keV, can be generated with a 3.5GeV electron beam.
