Magic angle (EELS)

Magic angle (EELS)

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 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 on the energy-loss suffered by the electron. The ratio of the magic-angle θM to the characteristic-angle θE is roughly independent of the energy-loss 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 collection-angle α:


A(\alpha)=\frac{1}{2}\int_0^{\alpha^2}dx\frac{x}{(x+\theta_E^2{(1-\beta^2))}^2}

and


C(\alpha)=\theta_E^2{(1-\beta^2)}^2\int_0^{\alpha^2}dx\frac{1}{{(x+\theta_E^2(1-\beta^2))}^2}

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 collection-angle 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 above-form it is easier to see that the former integral is due to momentum-transfers which are perpendicular to the beam-direction whereas the latter is due to momentum-transfers parallel to the beam-direction.

Using the above definition it is then found that 
\theta_M\approx 2\theta_E

References

  • Daniels H, et al. (2003). "Experimental and theoretical evidence for the magic angle in transmission electron energy loss spectroscopy". Ultramicroscopy 96 (3-4): 523–534. doi:10.1016/S0304-3991(03)00113-X. PMID 12871813. 
  • Schattschneider P, et al. (2005). "Anisotropic relativistic cross-sections 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. 

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Magic angle — This article is about the magic angle as defined in the field of nuclear magnetic resonance spectroscopy. For the magic angle as defined in the field of electron energy loss spectroscopy, see magic angle (EELS). The magic angle is a precisely… …   Wikipedia

  • Bo Diddley — à Prague en 2005. Nom Ellas Otha Bates McDaniel Naissance …   Wikipédia en Français

  • Ellas McDaniel — Bo Diddley Bo Diddley Bo Diddley à Prague en 2005. Nom Ellas Otha Bates McDaniel Naissance …   Wikipédia en Français

  • Ellas Otha Bates McDaniel — Bo Diddley Bo Diddley Bo Diddley à Prague en 2005. Nom Ellas Otha Bates McDaniel Naissance …   Wikipédia en Français

  • TRIP - Remix Your Experience — TRIP – Remix Your Experience is a German Gesamtkunstwerk, a synthesis of the arts, authored and produced by Frank Otto and Bernt Köhler Adams in 2005. As early as December 2004, it was premiered for the very first time as a multimedia… …   Wikipedia

  • List of alternative rock artists — This is a list of alternative rock artists. This includes artists who have either been very important to the genre or have had a considerable amount of exposure (such as in the case of one that has been on a major label, but not limited to such) …   Wikipedia

  • Список исполнителей альтернативного рока —   Это служебный список статей …   Википедия

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”