Sayre equation

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<font color="blue">Équation de Sayre</font> (''Fr''); <font color="black">Equazione di Sayre</font> (''It'').
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<font color="blue">Équation de Sayre</font> (''Fr''). <font color="red">Sayre-Gleichung</font> (''Ge''). <font color="black">Equazione di Sayre</font> (''It''). <font color="green">Ecuación de Sayre</font> (''Sp'').
In [[direct methods]], the '''Sayre equation''' is used to calculate probable values for the phases of some reflections. Its formulation is the following:
In [[direct methods]], the '''Sayre equation''' is used to calculate probable values for the phases of some reflections. Its formulation is the following:
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<math>F_{hkl} = \sum_{h'k'l'} F_{h'k'l'}F_{h-h',k-k',l-l'} </math>
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<math>F_{hkl} = \sum_{h'k'l'} F_{h'k'l'}F_{h-h',k-k',l-l'}</math>
 +
 
and states that the structure factor of a reflection ''hkl'' can be calculated as a function of structure factors whose [[Laue indices]] sum to the desired values of ''hkl''.
and states that the structure factor of a reflection ''hkl'' can be calculated as a function of structure factors whose [[Laue indices]] sum to the desired values of ''hkl''.
-
In particular, in a centrosymmetric structure, the phases of three reflections satisfying the above relation of Laue indices can only be 0 or π and the Sayre equation reduces to a relation between signs of structure factors:
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In particular, in a centrosymmetric structure, the phases of three reflections satisfying the above relation of Laue indices can only be 0 or <math>\pi</math> and the Sayre equation reduces to a relation between signs of structure factors:
<math>S_{h} \approx S_{h'} S_{h-h'} </math>
<math>S_{h} \approx S_{h'} S_{h-h'} </math>
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where the signs S are positive if the phase is 0 and negative if it is π and the <math>\approx</math> symbol indicates a certain degree of probability that the relationship is true, which becomes higher the stronger the reflections are.
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where the signs S are positive if the phase is 0 and negative if it is <math>\pi</math> and the <math>\approx</math> symbol indicates a certain degree of probability that the relationship is true, which becomes higher the stronger the reflections are.
[[Category: Structure determination]]
[[Category: Structure determination]]

Latest revision as of 11:06, 15 December 2017

Équation de Sayre (Fr). Sayre-Gleichung (Ge). Equazione di Sayre (It). Ecuación de Sayre (Sp).


In direct methods, the Sayre equation is used to calculate probable values for the phases of some reflections. Its formulation is the following:

Fhkl = Fh'k'l'Fhh',kk',ll'
h'k'l'

and states that the structure factor of a reflection hkl can be calculated as a function of structure factors whose Laue indices sum to the desired values of hkl. In particular, in a centrosymmetric structure, the phases of three reflections satisfying the above relation of Laue indices can only be 0 or π and the Sayre equation reduces to a relation between signs of structure factors:

S_{h} \approx S_{h'} S_{h-h'}

where the signs S are positive if the phase is 0 and negative if it is π and the \approx symbol indicates a certain degree of probability that the relationship is true, which becomes higher the stronger the reflections are.