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
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<center>
'''a''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''h'' &#955;
'''a''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''h'' &#955;
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'''c''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''l'' &#955;
'''c''' . ('''s<sub>h</sub>''' - '''s<sub>o</sub>''') = ''l'' &#955;
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</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>'''/&#955; - '''s<sub>o</sub>'''/&#955;) 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>'''/&#955; - '''s<sub>o</sub>'''/&#955;) 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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<center>
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''',
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''',
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</center>
where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector '''OH''' =  '''s<sub>h</sub>,'''/&#955; - '''s<sub>o</sub>'''/&#955; is a vector of the [[reciprocal lattice]].  
where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector '''OH''' =  '''s<sub>h</sub>,'''/&#955; - '''s<sub>o</sub>'''/&#955; is a vector of the [[reciprocal lattice]].  
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=== See also ===
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== See also ==
[[reciprocal lattice]]<br>
[[reciprocal lattice]]<br>
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[[Category:Fundamental crystallography]]<br>
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[[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 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.

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)