Direct detection of dark matter axions with directional sensitivity

We study the directional effect of the expected axion dark matter signal in a resonant cavity of an axion haloscope detector, for cavity geometries not satisfying the condition that the axion de Broglie wavelength lambda(a) is sufficiently larger than the cavity dimensions L for a fully coherent conversion, i.e. lambda(a) greater than or similar to 2 pi L. We focus on long thin cavities immersed in dipole magnets and find, for appropriately chosen cavity lengths, an O(1) modulation of the signal with the cavity orientation with respect the momentum distribution of the relic axion background predicted by the isothermal sphere model for the galactic dark matter halo. This effect can be exploited to design directional axion dark matter detectors, providing an unmistakable signature of the extraterrestrial origin of a possible positive detection. Moreover, the precise shape of the modulation may give information of the galactic halo distribution and, for specific halo models, give extra sensitivity for higher axion masses.