Difference between revisions of "Laue equations"
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| − | < | + | <font color="blue">Equations de Laue</font> (''Fr''). <font color="red">Laue-Gleichungen</font> (''Ge''). <font color="black">Equazioni di Laue</font> (''It''). <font color="purple">ラウエ方程式</font> (''Ja''). <font color="brown">уравнения Лауэ</font> (''Ru''). <font color="green">Ecuaciones de Laue</font> (''Sp''). |
== Definition == | == Definition == | ||
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'''b''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''k'' λ | '''b''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''k'' λ | ||
| − | '''c''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''l'' λ | + | '''c''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''l'' λ. |
</center> | </center> | ||
| − | 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<sub>h</sub>'''/λ - '''s<sub>o</sub>'''/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''' . ('''s<sub>h</sub>'''/λ - '''s<sub>o</sub>'''/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 | ||
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('''s<sub>h</sub>'''/λ - '''s<sub>o</sub>'''/λ) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', | ('''s<sub>h</sub>'''/λ - '''s<sub>o</sub>'''/λ) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', | ||
</center> | </center> | ||
| − | where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector '''OH''' = '''s<sub>h</sub> | + | where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector '''OH''' = '''s<sub>h</sub>'''/λ - '''s<sub>o</sub>'''/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 | + | 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]] | ||
| + | *[[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) | ||
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[[Category:X-rays]]<br> | [[Category:X-rays]]<br> | ||
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.
See also
- Bragg's law
- Reciprocal lattice
- The Reciprocal Lattice (Teaching Pamphlet No. 4 of the International Union of Crystallography)