# Laue equations

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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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'''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. | ||

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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 | ||

+ | <center> | ||

('''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> | |

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]]. | 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]]. | ||

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- | + | == See also == | |

[[reciprocal lattice]]<br> | [[reciprocal lattice]]<br> | ||

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---- | ---- | ||

- | [[Category: | + | [[Category:X-rays]]<br> |

## Revision as of 08:58, 27 February 2006

## Contents |

# Laue equations

### Other languages

Equations de Laue (*Fr*). 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.

_{o}## 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, 50 Years of X-ray Diffraction, Section 4, page 52.

## See also

reciprocal lattice

The Reciprocal Lattice (Teaching Pamphlet of the *International Union of Crystallography*)