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Theory of Diffraction: Bragg Law
Incident X‐rays
• Single plane of infinite lattice points separated by .
• Incident beam “reflects” off of array (why?)
• Condition for constructive interference:
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Theory of Diffraction: Von Laue Conditions
From van Holde, p. 295.
Key difference: waves are not reflecting off of a plane of atoms; instead, atoms are point sources for scattered waves (see blue circles on previous slide)
Conditions for Constructive Interference:
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1‐Dimensional Diffraction
From Principles of Physical Biochemistryvan Holde, et al., Chapt. 6, p. 295
( =2)
( =1)
( =‐1)
( =0)
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2‐Dimensional Diffraction
Von Laue Conditions must be met for both cones: result is a line
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Diffraction in 2D and 3D• Multiple scattering planes are
possible; each has functional form:
cos coscos cos
l cos cos
• For indices h, k, and l (Miller indices) plane spacing of a, b, and c are observed
• Miller indices define the plane of scattering (reciprocal of intercept on a axes; 1 1
“Miller Index,” Wikipedia.
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Diffraction in 2D and 3D
• Define a scattering vector (S) with direction normal to the Bragg plane
• Then, condition for diffraction is given by the von Laueequation:
/
• It can be shown that this is equivalent to Bragg’s law in 1D
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Measuring Reflections
http://www.stolaf.edu/people/hansonr/mo/xray.gif
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Measuring Reflections
Detector(CCD, Image Plate, or Film)
2θ
R (precisely known)θ
SSource
Reflection (intensity only; no phase)
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Measuring Reflections
From Principles of Physical Biochemistryvan Holde, et al., Chapt. 6, p. 298
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Ewald Sphere and Reflections
From Principles of Physical Biochemistryvan Holde, et al., Chapt. 6, p. 303
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Data Collection
From Diffractionhttp://en.wikipedia.org/wiki/Diffraction
Beam Stop
hkl reflections in reciprocal space (each one has an
index!)
Crystal morphology (unit cell dimensions & symmetry) can be determined by assigning indices and observing patterns in reflections –typically done by computer.
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Summary
• Bragg Planes and Von Laue Conditions are two ways of determining diffraction angle
• Each reflection corresponds to a miller index, and a set of planes responsible for diffraction
• Reciprocal space and Ewald sphere are a graphical representation von Laue conditions
• Detectors only measure intensity, not phase of scattered light
• Looking at reflections allows measurement of unit cell shape/symmetry