# Laue equations

(Difference between revisions)
 Revision as of 15:55, 6 February 2012 (view source)m← Older edit Latest revision as of 15:09, 8 May 2018 (view source)m (Lang (Ru)) (4 intermediate revisions not shown) Line 1: Line 1: - Equations de Laue (''Fr''). Laue Gleichungen . Ecuaciones de Laue (''Sp''). Equazioni di Laue (''It'') + Equations de Laue (''Fr''). Laue-Gleichungen (''Ge''). Equazioni di Laue (''It''). ラウエ方程式 (''Ja''). уравнения Лауэ (''Ru''). Ecuaciones de Laue (''Sp''). == Definition == == Definition == Line 11: Line 11: '''b''' . ('''sh''' - '''so''') = ''k'' λ '''b''' . ('''sh''' - '''so''') = ''k'' λ - '''c''' . ('''sh''' - '''so''') = ''l'' 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. + 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 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 Line 21: Line 21: ('''sh'''/λ - '''so'''/λ) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', ('''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]]. This is the [[Reciprocal lattice#Diffraction condition in reciprocal space|diffraction condition in reciprocal space]]. + where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector '''OH''' =  '''sh'''/λ - '''so'''/λ is a vector of the [[reciprocal lattice]]. This is the [[Reciprocal lattice#Diffraction condition in reciprocal space|diffraction condition in reciprocal space]]. == History == == 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, [http://www.iucr.org/iucr-top/publ/50YearsOfXrayDiffraction/ 50 Years of X-ray Diffraction], Section 4, page 52. + 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, [http://www.iucr.org/iucr-top/publ/50YearsOfXrayDiffraction/ 50 Years of X-ray Diffraction], Utrecht: IUCr/Oosthoek, Section 4, p. 52. == See also == == See also == - + *[[Bragg's law]] - [[Bragg's law]]
+ *[[Reciprocal lattice]] - [[reciprocal lattice]]
+ *[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ ''The Reciprocal Lattice''] (Teaching Pamphlet No. 4 of the International Union of Crystallography) - [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice] (Teaching Pamphlet of the ''International Union of Crystallography'') + [[Category:X-rays]]
[[Category:X-rays]]

## Latest revision as of 15:09, 8 May 2018

Equations de Laue (Fr). Laue-Gleichungen (Ge). Equazioni di Laue (It). ラウエ方程式 (Ja). уравнения Лауэ (Ru). 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 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. This is the diffraction condition in reciprocal space.

## 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.