The nature of a Bragg reflection can be deduced from the shape of its peak base'

1
Introduction

• The nature of a Bragg reflection can be deduced
  from the shape of its peak base.
• An ideal Bragg peak projected in two-dimensions
  (2D) is circular.
• A circle can be described as a geometric
  structure for which the perimeter to area ratio
  is at a minimum.
• For a 2D lattice array of points the area of a
  circular feature can be estimated by the sum of
  the points (Na) enclosed by a set of perimeter
  points (P).
• For a given Na the minimum P/Na ratio can be
  estimated by a circle with radius
  (Na1/2-1)/2, area ?(Na1/2-1)/22 and perimeter
  ?(Na1/2-1) so that
• min. P/Na ?(Na1/2-1)/Na

2
Continued

• The experimental parameters of any given Bragg
  peak projected on to a fixed lattice (2D pixel)
  array, can be determined by Graph Theory.
• Perimeter points, in a 2D array, are described as
  nodes in a structure that have degree of seven or
  less.
• The sum of the perimeter points affords Np. The
  experimental Np/Na ratio is a number that is less
  than 1 and greater than ?(Na1/2-1)/Na
• 1 gt Np/Na gt ?(Na1/2-1)/Na
• A figure of Merit (that is independent of
  statistical determinations) can thus be
  calculated for each Bragg peak (in 2D)
• FOM ?(Na1/2-1)/Np

3
Minimum Perimeter/Area Ratios

• An experimental perimeter to Bragg peak base
  area (Np/Na ratio) will be less than one and
  greater than the min(P/Na).
• The more circular the Bragg Peak Base the
  closer the Np/Na ratio will be to the min(P/Na)
  value.

4
Procedure

• Determine Beak Base and calculate the number of
  points (Na)a
• Calculate Np by counting node degrees.a
• Calculate Np/Na
• From Na determine the min(P/N)
• Assign a FOM to the Bragg peak from Np and Na.
  ??Na1/2-1)/Np
• Employ FOM to downweight suspicious diffraction
  events.

a) Enumeration and Classification of
Anomaly/Peak Bases in Two-Dimensional Intensity
Histograms. Application of Graph Theory to
Crystallographic Data Imaging Reibenspies, J.
J. Appl. Cryst. (1996). 29, 241-245.
5
Test Events

• Event 1
• event
• 000000000 Idealized circular
  structure on 2D grid
• 001111100
• 011111110 Na 38
• 011111110 Np 18
• 011111110 Np/Nq 0.47
• 011111110 minP/Nq 0.43
• 001111100 FOM 0.91
• 000000000 note High FOM, the event is
  circular
• Event 2
• event
• 000000000000000000000 Idealized rectangular
  structure
• 011111111111111111110 Na 38 Np 38
• 011111111111111111110 Np/Na 1.0 minP/Na
  0.43
• 000000000000000000000 FOM 0.43
• note low FOM, the event is not circular

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Example 1
Na 34 Np 17 Np/Na 0.50 min P/Na
0.45 FOM 0.90
Based on FOM Peak base is circular
Bragg reflection not suspicious
7
Example 2
Na 79 Np 31 Np/Na 0.39 min P/Na
0.31 FOM 0.80
Based on low FOM peak base is not circular.
Bragg Reflection is suspicious
8
Conclusion

• Perimeter to area ratios can be employed as a
  measure of peak base circularity.
• A figure of merit can be calculated and compared
  to a standard to determine the nature of the
  Bragg peak shape.
• Natural elongation of the Bragg peak base with
  increasing two-theta can be included in the
  Figure of Merit.