Ewald Construction

Evolved Construction in X-ray Diffraction

Geometrical Representation of the Brazilian Second Construction

• The discussion begins with an overview of evolved construction, focusing on geometrical representation and its application to X-ray diffraction.

• Bragg's law is introduced as a fundamental principle, expressed as 2d sin theta = nlambda, where d is the interplanar spacing, theta is the Bragg angle, n is the order of diffraction, and lambda is the wavelength of X-rays.

• A modified equation for wavelength is presented: lambda = 2(d/n)sin theta, simplifying further to relate sine terms geometrically using Miller indices (hkl).

Constructing Evolved Sphere

• The geometric representation involves drawing a right triangle within a circle where the diameter represents 2/lambda.

• The perpendicular component related to plane hkl is defined as 1/d, leading to a geometric interpretation that satisfies Bragg's law.

• The evolved sphere concept emerges from this geometry, visualizing crystal planes at the center and marking points where X-ray beams enter.

Conditions for Diffraction

• When X-rays hit crystal planes, they diffract along specific directions; angles between incident and diffracted beams are crucial for understanding diffraction conditions.

• An evolved sphere represents loci where diffracted beams intersect reciprocal lattice points. This intersection indicates potential diffraction events.

• If certain conditions are met (e.g., magnitude relationships involving wavelengths), diffraction may or may not occur based on crystal orientation relative to incident beams.

Animation of Evolved Model

• An animation illustrates how the evolved model demonstrates possible diffraction scenarios within reciprocal lattice space.

• It shows lattice planes rotating about a reciprocal lattice vector while an incident beam creates an imaginary sphere aiding visualization of diffraction events.