Class 11 > Unit # 13: Physical Optics > X-Ray Diffraction (Bragg's Law)

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Diffraction of X-Rays through Atomic Crystals & Bragg's Law

Video Lecture: Diffraction of X-Rays

Watch the complete step-by-step video lecture explaining X-ray diffraction through atomic crystals and the derivation of Bragg's Law:

Introduction and Need for Crystal Gratings

“X-rays have extremely short wavelengths (typically less than visible and ultraviolet light), making it impossible to observe their diffraction using conventional ruled gratings because X-rays pass right through the slits.”

To overcome this limitation, crystals such as rock salt are used as natural three-dimensional diffraction gratings. In atomic crystals, atoms are arranged in uniformly spaced parallel planes separated by distances on the order of $2\text{--}5\text{ \AA}$ ($2\text{--}5 \times 10^{-10}\text{ m}$), which matches the short wavelength of X-rays.

Fig: Reflection of X-ray beams from parallel atomic planes separated by interplanar spacing $d$.

Derivation of Path Difference

Consider a parallel beam of X-rays incident at a glancing angle $\theta$ on parallel lattice planes of a crystal separated by an interplanar spacing $d$.

When X-rays strike the atomic planes, reflections occur from successive planes. As seen from the geometry, the second ray penetrates deeper and travels a greater distance than the first ray. The total path difference between two adjacent reflected rays is given by:

$\text{Path Difference} = BC + BD \quad \text{--- (i)}$

From the right-angled triangle $ABC$ formed by the normal and the path:

$\sin\theta = \frac{BC}{d} \implies BC = d \sin\theta$

Similarly, from triangle $ABD$ associated with the lower ray segment:

$\sin\theta = \frac{BD}{d} \implies BD = d \sin\theta$

Substituting these values back into equation (i), the total path difference becomes:

$\text{Path Difference} = d\sin\theta + d\sin\theta = 2d \sin\theta \quad \text{--- (ii)}$

Bragg’s Law for X-Ray Diffraction

For constructive interference to occur, the path difference between waves reflected from adjacent atomic planes must be an integral multiple of the X-ray wavelength $\lambda$:

$\text{Path Difference} = m\lambda \quad \text{--- (iii)} \quad (m = 1, 2, 3, \dots)$

Comparing equation (ii) and equation (iii), we obtain:

$2d \sin\theta = m\lambda$

This fundamental relation is known as Bragg’s Law, named after physicists W.H. Bragg and W.L. Bragg.

Applications: If the interplanar distance $d$ of a crystal is known, and the diffraction angle $\theta$ and order $m$ are measured experimentally, the wavelength $\lambda$ of the X-rays can be accurately calculated. Conversely, if a monochromatic X-ray beam of known wavelength is used, crystal structures and atomic spacings can be precisely mapped.

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