abc-αβγ from 2 tilts

Determining lattice parameters from diffraction and/or lattice fringe information taken with the specimen at two or more orientations (specifically from information on the relative direction of three or more non-coplanar lattice periodicities) is not trivial.  However with modern computers analysis can be made increasingly routine. 

Three (relatively large) such lattice spacings span a subset of (and often the full) reciprocal lattice, and any points beyond three facilitate an empirical analysis of lattice parameter uncertainties. The exercise incidentally also returns the oriented basis triplet for the specimen in the stage goniometer, should one wish to go to a selected orientation after the fact.

Related references:

• • P. Fraundorf (1981) "Stereo analysis of single crystal electron diffraction data", Ultramicroscopy 6, 227-236.

  • P. Fraundorf (1981) "Stereo analysis of electron diffraction patterns from knowns", Ultramicroscopy 7, 203-206.

  • P. Fraundorf (1987) "Determining the 3D lattice parameters of nanometer-sized single crystals from images" Ultramicroscopy 22, 225-230.

  • Wentao Qin (2000) Crystal lattice parameters from direct-space images at two tilts, Dissertation in partial fulfillment of the requirements for a Ph.D. in Physics and Astronomy at UM-StL and UM-Rolla. (pdf)

  • Wentao Qin and P. Fraundorf (2003) "Lattice parameters from direct-space images at two tilts", Ultramicroscopy 94:3-4, 246-262 (pdf).