3.18: 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 = a , OB = b , OC = c of the crystal and let \(\vec{s}_o\) and \(\vec{s}_h\) 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
\[\vec{a} \cdot (\ce{s}_h - \vec{s}_o) = h λ \nonumber \]
\[\vec{b} \cdot (\vec{s}_h - \vec{s}_o) = k λ \nonumber \]
\[\vec{c} \cdot (\vec{s}_h - \vec{s}_o) = l λ \nonumber \]
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
\[\vec{r} \cdot (\vec{s}_h/λ - \vec{s}_o/λ) \nonumber \]
is an integer for any vector
\[\vec{r} = u \vec{a} + v \vec{b} + w \vec{c} \nonumber \]
where (u, v, w integers) of the direct lattice. This is the case if
( s h /λ - s o /λ) = h a* + k b* + l c* ,
where h , k , l are integers, namely if the diffraction vector OH = s h , /λ - s o /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 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.