Bragg's law in reciprocal lattice

Bragg's Law in Reciprocal Lattice

Introduction to Bragg's Law

• The discussion begins with Bragg's law, which relates to x-ray diffraction by crystal planes, expressed as 2d sin theta = mlambda.

• A geometrical representation of this relationship is introduced, emphasizing the sine function in relation to the Miller indices (d_hkl).

Reciprocal Lattice and Ewald's Model

• The concept of drawing a reciprocal lattice is explained using Ewald's model, illustrating how diffraction occurs within this framework.

• A circle is drawn with center point C representing the crystal position; parallel planes are formed when x-ray beams pass through these planes.

Diffraction Conditions

• When an x-ray beam reflects off the crystal at angle 2theta, it intersects at point O on the sphere surface, marking the origin in reciprocal space.

• The vector OP connects point O to another reciprocal lattice point P; if this vector intersects the sphere’s surface, diffraction can occur.

Scattering Vectors and Elastic Scattering

• The incident wave vector (K) and reflected wave vector (K') are defined; their relationship is given by K' = K + G, where G represents a reciprocal lattice vector.

• This equation indicates that scattering changes only in direction while maintaining magnitude during elastic scattering conditions.

Deriving Bragg's Law in Vector Form

• By squaring both sides of K' = K + G, we derive an important equation: G^2 + 2K cdot G = 0.

• This equation encapsulates Bragg’s law in vector form within the context of reciprocal lattices.

Geometrical Representation and Conditions for Diffraction

• Earlier discussions highlighted that geometry aligns with Bragg’s relation; specifically, sine theta correlates with plane spacing and wavelength.