The magic angle is a particular value of the collection angle of an electron microscope at which the measured energy-loss spectrum "magically" becomes independent of the tilt angle of the sample with respect to the beam direction. The magic angle is not uniquely defined for isotropic samples, but the definition is unique in the (typical) case of small angle scattering on materials with a "c-axis", such as graphite.
The "magic" angle depends on both the incoming electron energy (which is typically fixed) and the energy loss suffered by the electron. The ratio of the magic angle  to the characteristic angle  is roughly independent of the energy loss and roughly independent of the particular type of sample considered.