In this problem, we will derive the Bragg equation, Equation...

In this problem, we will derive the Bragg equation, Equation 29.12. William and Lawrence Bragg (father and son) assumed that X-rays are scattered by successive planes of atoms within a crystal (see the following figure). Each set of planes reflects the X-rays specularly; that is, the angle of incidence is equal to the angle of reflection, as shown in the figure. The X-radiation reflected from the lower plane in the figure travels a distance $P Q R$ longer than the $\mathrm{X}$-radiation reflected by the upper layer. Show that $P Q R=2 d \sin \theta$, and argue that $2 d \sin \theta$ must be an integral number of wavelengths for constructive interference and hence a diffraction pattern to be observed.

In this problem, we will derive the Bragg equation, Equation 29.12. William and Lawrence Bragg (father and son) assumed that X-rays are scattered by successive planes of atoms within a crystal (see the following figure).
Each set of planes reflects the X-rays specularly; that is, the angle of incidence is equal to the angle of reflection, as shown in the figure. The X-radiation reflected from the lower plane in the figure travels a distance $P Q R$ longer than the $\mathrm{X}$-radiation reflected by the upper layer. Show that $P Q R=2 d \sin \theta$, and argue that $2 d \sin \theta$ must be an integral number of wavelengths for constructive interference and hence a diffraction pattern to be observed.

Key Concepts

Path Difference

Path difference refers to the difference in the travel distances of two waves emanating from the same source before they meet. In X-ray diffraction, the extra distance that one wave travels relative to another, due to reflection from a deeper atomic plane, must be an integer multiple of the wavelength for constructive interference to take place.

Constructive Interference

Constructive interference is the process in which two or more waves combine to produce a wave with greater amplitude. In the context of X-ray diffraction, it occurs when the path difference between waves reflected from successive crystal planes is a whole number multiple of the wavelength, resulting in intense, well-defined diffraction peaks.

X-ray Diffraction

X-ray diffraction is a phenomenon where X-rays interact with the periodic structure of a crystal, causing the waves to scatter in specific, predictable ways. By analyzing the resulting diffraction pattern, scientists can deduce the crystal’s structure, as the pattern is directly related to the arrangement of atoms within the material.

Bragg's Law

Bragg's Law establishes the condition for constructive interference of X-rays scattered by parallel planes of atoms in a crystal. It states that when the extra path length traveled by an X-ray, quantified as 2d sin(?), is an integer multiple of the wavelength, constructive interference occurs, leading to a detectable diffraction peak.

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