Self-Assembly of Synthetic and Biological Polymeric Systems of ...

Figure 2.27. The Ewald sphere and Bragg condition in reciprocal space.
On the other hand, the relationship between the Bragg condition and the reciprocal lattice can
be explained visually by the Ewald sphere, also referred as reflection sphere. Ewald came up
with a geometrical construction to help the visualization of which Bragg planes are in the
correct orientation to diffract. In Figure 2.27, the diffracting crystal is located in the center of
the Ewald sphere, C. The radius of the Ewald sphere is defined as
. The incident beam can
be visualized as the vector from I to C, and the diffracted beam is the vector from C to P. Both
the incident and diffracted beams form an angle θ from a set of crystal planes (hkl). The d-
spacing of the crystal planes is dhkl In the Ewald sphere; both incident beam vector
diffracted beam vector
and
start at the poin C and end at the point O and P, respectively. The
vector from O to P is the reciprocal lattice vector and is perpendicular to the crystal
planes. The three vectors have the following relationship:
and their magnitude can be expressed based on the Bragg law as:
The point O is the origin of the reciprocal lattice, and the point P is the reciprocal point (hkl).
The Bragg condition is satisfied only when the reciprocal lattice point falls on the Ewald sphere.
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