Slow strain rate testing: Difference between revisions

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In [[crystallography]], the '''Sayre equation''', named after [[David Sayre]] who introduced it in 1952, is a mathematical relationship that allows to calculate probable values for the [[Phase problem|phases]] of some diffracted beams. It is used when employing [[direct methods (crystallography)|direct methods]] to solve a structure and its formulation is the following:
 
<math> F_{hkl} = \sum_{h'k'l'} F_{h'k'l'}F_{h-h',k-k',l-l'} </math>
 
which states how the [[structure factor]] for a beam can be calculated as the sum of the products of pairs of structure factors whose indices sum to the desired values of <math>h,k,l</math>. Since weak diffracted beams will contribute a little to the sum, this method can be a powerful way of finding the phase of related beams, if some of the initial phases are already known by other methods.
 
In particular, for three such related beams in a [[centrosymmetric]] structure, the phases can only be 0 or <math>\pi</math> and the Sayre equation reduces to the triplet relationship:
 
<math>S_{h} \approx S_{h'} S_{h-h'} </math>
 
where the <math>S</math> indicates the sign of the structure factor (positive if the phase is 0 and negative if it is <math>\pi</math>) and the <math>\approx</math> sign indicates that there is a certain degree of [[probability]] that the relationship is true, which becomes higher the stronger the beams are.
 
== References ==
*{{cite doi|10.1107/S0365110X52000137}}
*{{cite book |title=Crystal Structure Determination |last= Werner |first=Massa |year= 2004|publisher=Springer|isbn=3540206442 |page= 102}}
 
[[Category:Crystallography]]

Latest revision as of 21:24, 7 March 2013

In crystallography, the Sayre equation, named after David Sayre who introduced it in 1952, is a mathematical relationship that allows to calculate probable values for the phases of some diffracted beams. It is used when employing direct methods to solve a structure and its formulation is the following:

Fhkl=∑h′k′l′Fh′k′l′Fh−h′,k−k′,l−l′

which states how the structure factor for a beam can be calculated as the sum of the products of pairs of structure factors whose indices sum to the desired values of h,k,l. Since weak diffracted beams will contribute a little to the sum, this method can be a powerful way of finding the phase of related beams, if some of the initial phases are already known by other methods.

In particular, for three such related beams in a centrosymmetric structure, the phases can only be 0 or π and the Sayre equation reduces to the triplet relationship:

Sh≈Sh′Sh−h′

where the S indicates the sign of the structure factor (positive if the phase is 0 and negative if it is π) and the ≈ sign indicates that there is a certain degree of probability that the relationship is true, which becomes higher the stronger the beams are.

References

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