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Laue equations

The three Laue equations give the conditions to be satisfied by an incident wave to be diffracted by a crystal. Consider the three basis vectors, OA = aOB = b , OC = c of the crystal and let so and sh be unit vectors along the incident and reflected directions, respectively. The conditions that the waves scattered by O and AO and B,O and C, respectively, be in phase are that

a . (sh - so) = h λ

b . (sh - so) = k λ

c . (sh - so) = l λ

If these three conditions are simultaneously satisfied, the incoming wave is reflected on the set of lattice planes of Miller indices h/nk/nl/nhkl are the indices of the reflection.

The three Laue equations can be generalized by saying that the diffraction condition is satisfied if the scalar product r . (sh/λ - so/λ) is an integer for any vector r = u a + v b + w c (uvw integers) of the direct lattice. This is the case if

(sh/λ - so/λ) = h a* + k b* + l c*,

where hkl are integers, namely if the diffraction vector OH = sh,/λ - so/λ is a vector of the reciprocal lattice. This is the diffraction condition in reciprocal space.

History

The three Laue conditions for diffraction were first given in Laue, M. (1912). Eine quantitative Prüfung der Theorie für die Interferenz-Erscheinungen bei RöntgenstrahlenSitzungsberichte der Kgl. Bayer. Akad. der Wiss 363--373, reprinted in Ann. Phys. (1913), 41, 989-1002 where he interpreted and indexed the first diffraction  diagram (Friedrich, W., Knipping, P., and Laue, M. (1912). Interferenz-Erscheinungen bei RöntgenstrahlenSitzungsberichte der Kgl. Bayer. Akad. der Wiss, 303--322, reprinted in Ann. Phys., (1913), 41, 971-988, taken with zinc-blende, ZnS. For details, see P. P. Ewald, 1962, IUCr, 50 Years of X-ray Diffraction, Section 4, page 52.