# Laue equations

### From Online Dictionary of Crystallography

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- | = Laue | + | <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''). |

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- | === | + | |

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== Definition == | == Definition == | ||

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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<sub>o</sub>''' and '''s<sub>h</sub>''' 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 '''s<sub>o</sub>''' and '''s<sub>h</sub>''' 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 | ||

+ | <center> | ||

'''a''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''h'' λ | '''a''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''h'' λ | ||

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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> | ||

- | 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 | + | 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 |

- | ('''s | + | <center> |

- | + | ('''s<sub>h</sub>'''/λ - '''s<sub>o</sub>'''/λ) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', | |

- | where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector '''OH''' = '''s | + | </center> |

+ | 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. |

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+ | == 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: | + | [[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 **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)