 Magic angle (EELS)

This article is about the magic angle as defined in the field of electron energyloss spectroscopy. For the magic angle as defined in the field of nuclear magnetic resonance spectroscopy, see magic angle.
The magic angle is a particular value of the collectionangle of an electron microscope at which the measured energyloss spectrum "magically" becomes independent of the tiltangle of the sample with respect to the beam direction. The magic angle is not not uniquely defined for isotropic samples, but the definition is unique in the (typical) case of small angle scattering on materials with a "caxis" such as graphite.
The "magic" angle depends on both the incoming electron energy (which is typically fixed) and on the energyloss suffered by the electron. The ratio of the magicangle θ_{M} to the characteristicangle θ_{E} is roughly independent of the energyloss and more interestingly is roughly independent of the particular type of sample considered.
Mathematical definition
For the case of a relativistic incident electron, the "magic" angle is defined by the equality of two different functions (denoted below by A and C) of the collectionangle α:
and
where β is the speed of the incoming electron divided by the speed of light (N.B., the symbol β is also often used in the older literature to denote the collectionangle instead of α).
Of course, the above integrals may easily be evaluated in terms of elementary functions, but they are presented as above because in the aboveform it is easier to see that the former integral is due to momentumtransfers which are perpendicular to the beamdirection whereas the latter is due to momentumtransfers parallel to the beamdirection.
Using the above definition it is then found that
References
 Daniels H, et al. (2003). "Experimental and theoretical evidence for the magic angle in transmission electron energy loss spectroscopy". Ultramicroscopy 96 (34): 523–534. doi:10.1016/S03043991(03)00113X. PMID 12871813.
 Schattschneider P, et al. (2005). "Anisotropic relativistic crosssections for inelastic electron scattering, and the magic angle". Phys. Rev. B 72 (4): 045142. doi:10.1103/PhysRevB.72.045142.
 Jouffrey B, et al. (2004). "The Magic Angle: a solved mystery". Ultramiscoscopy 102 (1): 61–66. doi:10.1016/j.ultramic.2004.08.006. PMID 15556701.
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