We can derive the Bragg equation from the von Laue equations...

We can derive the Bragg equation from the von Laue equations derived in the previous problem. First show that $\mathbf{S}=\mathbf{s}-\mathbf{s}_{0}$ bisects the angle between $\mathbf{s}_{0}$ and $\mathbf{s}$ and is normal to the plane from which the X-radiation would be specularly reflected (the angle of incidence equals the angle of reflection). Now show that the distance from the origin of the $\mathbf{a}, \mathbf{b}$, and c axes to the $h k l$ plane is given by $$ d=\frac{\mathbf{a}}{h} \cdot \frac{\mathbf{S}}{|\mathbf{S}|}=\frac{\mathbf{b}}{k} \cdot \frac{\mathbf{S}}{|\mathbf{S}|}=\frac{\mathbf{c}}{l} \cdot \frac{\mathbf{S}}{|\mathbf{S}|}=\frac{\lambda}{|\mathbf{S}|} $$ Last, show that $|\mathbf{S}|=\left[\left(\mathbf{s}-\mathbf{s}_{0}\right) \cdot\left(\mathbf{s}-\mathbf{s}_{0}\right)\right]^{1 / 2}=[2-2 \cos 2 \theta]^{1 / 2}=2 \sin \theta$, which leads to the Bragg equation $d=\lambda / 2 \sin \theta$

We can derive the Bragg equation from the von Laue equations derived in the previous problem. First show that $\mathbf{S}=\mathbf{s}-\mathbf{s}_{0}$ bisects the angle between $\mathbf{s}_{0}$ and $\mathbf{s}$ and is normal to the plane from which the X-radiation would be specularly reflected (the angle of incidence equals the angle of reflection). Now show that the distance from the origin of the $\mathbf{a}, \mathbf{b}$, and c axes to the $h k l$ plane is given by
$$
d=\frac{\mathbf{a}}{h} \cdot \frac{\mathbf{S}}{|\mathbf{S}|}=\frac{\mathbf{b}}{k} \cdot \frac{\mathbf{S}}{|\mathbf{S}|}=\frac{\mathbf{c}}{l} \cdot \frac{\mathbf{S}}{|\mathbf{S}|}=\frac{\lambda}{|\mathbf{S}|}
$$
Last, show that $|\mathbf{S}|=\left[\left(\mathbf{s}-\mathbf{s}_{0}\right) \cdot\left(\mathbf{s}-\mathbf{s}_{0}\right)\right]^{1 / 2}=[2-2 \cos 2 \theta]^{1 / 2}=2 \sin \theta$, which leads to the Bragg equation $d=\lambda / 2 \sin \theta$

Key Concepts

Bragg's Law

Bragg's Law relates the wavelength of the incident radiation, the interplanar spacing (d), and the angle of incidence/reflection (?) to establish the condition for constructive interference of X-rays reflected from crystal planes. It is fundamental in analyzing and interpreting X-ray diffraction patterns in crystallography.

Crystallographic Planes (Miller Indices and Interplanar Spacing)

Crystallographic planes are described by Miller indices (h, k, l), which specify the orientation of the planes in the crystal lattice. The spacing between these planes (d-spacing) is determined by the lattice parameters and is critical in predicting the diffraction angles through Bragg's Law.

Scattering Vector

The scattering vector, defined as the difference between the outgoing and incoming unit wavevectors (S = s - s?), is central to diffraction theory. It represents the momentum transfer during the scattering process and determines the conditions under which constructive interference occurs in a periodic structure.

Specular Reflection and Geometric Conditions

Specular reflection in diffraction refers to the condition where the angle of incidence equals the angle of reflection. In this context, the scattering vector bisects the angle between the incident and reflected waves, providing a geometric framework that simplifies the derivation of the Bragg equation from the diffraction conditions.

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