# Laue equations

(Difference between revisions)
 Revision as of 12:08, 25 January 2006 (view source)← Older edit Latest revision as of 15:09, 8 May 2018 (view source)m (Lang (Ru)) (12 intermediate revisions not shown) Line 1: Line 1: - = Laue equations = + 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''). - + - === Other languages === + - + - Equations de Laue (''Fr''). Ecuaciones de Laue (''Sp''). + - + == Definition == == Definition == Line 10: Line 5: 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 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 waves scattered by ''O'' and ''A'', ''O'' and ''B'', ''O'' and ''C'', respectively, be in phase are that +
'''a''' . ('''sh''' - '''so''') = ''h'' λ '''a''' . ('''sh''' - '''so''') = ''h'' λ Line 15: 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''' . ('''s,,h,,'''/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 - ('''s,,h,,'''/λ - '''s,,o,,'''/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''' =  '''s,,h,,'''/λ - '''so'''/λ is a vector of the [[reciprocal lattice]]. +
+ 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. von (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 where he interpreted and indexed the first diffraction diagram (Friedrich, W., Knipping, P., and Laue, M. von (1912). ''Interferenz-Erscheinungen bei Röntgenstrahlen'', ''Sitzungsberichte der Kgl. Bayer. Akad. der Wiss'', 303--322, reprinted in ''Ann. Phys.'', (1913), '''41''', 971, 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 === + - + - [[reciprocal lattice]]
+ - [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'') + + == See also == + *[[Bragg's law]] + *[[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) - ---- - [[Category:Fundamental crystallography]]
+ [[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.