Laue equations

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Equations de Laue (Fr). Laue Gleichungen . Ecuaciones de Laue (Sp). Equazioni di Laue (It)

Definition

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 = a, OB = 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 A, O 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/n, k/n, l/n. h, k, l 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 (u, v, w integers) of the direct lattice. This is the case if

(sh/λ - so/λ) = h a* + k b* + l c*,

where h, k, l are integers, namely if the diffraction vector OH = sh,/λ - so/λ is a vector of the reciprocal lattice.

History

The three Laue conditions for diffraction were first given in Laue, M. (1912). Eine quantitative Prüfung der Theorie für die Interferenz-Erscheinungen bei Röntgenstrahlen. Sitzungsberichte 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 Röntgenstrahlen, Sitzungsberichte 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.

See also

reciprocal lattice
The Reciprocal Lattice (Teaching Pamphlet of the International Union of Crystallography)