# Laue equations

### From Online Dictionary of Crystallography

Equations de Laue (*Fr*). Laue-Gleichungen (*Ge*). Equazioni di Laue (*It*). ラウエ方程式 (*Ja*). Ecuaciones de Laue (*Sp*).

## 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 **s _{o}** and

**s**be unit vectors along the incident and reflected directions, respectively. The conditions that the waves scattered by

_{h}*O*and

*A*,

*O*and

*B*,

*O*and

*C*, respectively, be in phase are that

**a** . (**s _{h}** -

**s**) =

_{o}*h*λ

**b** . (**s _{h}** -

**s**) =

_{o}*k*λ

**c** . (**s _{h}** -

**s**) =

_{o}*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** . (**s _{h}**/λ -

**s**/λ) is an integer for any vector

_{o}**r**=

*u*

**a**+

*v*

**b**+

*w*

**c**(

*u*,

*v*,

*w*integers) of the direct lattice. This is the case if

(**s _{h}**/λ -

**s**/λ) =

_{o}*h*

**a***+

*k*

**b***+

*l*

**c***,

where *h*, *k*, *l* are integers, namely if the diffraction vector **OH** = **s _{h}**/λ -

**s**/λ is a vector of the reciprocal lattice. This is the diffraction condition in reciprocal space.

_{o}## History

The three Laue conditions for diffraction were first given by 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, Utrecht: IUCr/Oosthoek, Section 4, p. 52.

## See also

- Bragg's law
- Reciprocal lattice
*The Reciprocal Lattice*(Teaching Pamphlet No. 4 of the International Union of Crystallography)