Overview#

• X rays are electromagnetic waves with wavelengths \(\sim 1\) Å (\(10^{-10}\) m)—comparable to atomic spacing. Optical gratings cannot resolve them.

• A crystal acts as a natural three-dimensional diffraction grating: its regular array of atoms scatters x rays and produces constructive interference at specific angles.

• Bragg’s law \(2d\sin\theta = m\lambda\) gives the angles for intensity maxima. Here \(d\) is the interplanar spacing between parallel crystal planes, and \(\theta\) is measured from the plane surface.

• X-ray diffraction is used to determine both x-ray spectra and crystal structure.

X rays and the need for crystals#

X rays have wavelengths of order 1 Å (\(\sim 0.1\) nm), compared to visible light at \(\sim 550\) nm. For an optical grating with \(d \sim 3000\) nm and \(\lambda \sim 0.1\) nm, the first-order maximum occurs at \(\sin\theta = \lambda/d \approx 3\times 10^{-5}\)—too close to the central maximum to be useful. A grating with \(d \sim \lambda\) would be ideal, but x-ray wavelengths match atomic diameters; such gratings cannot be made mechanically.

von Laue’s insight (1912): A crystalline solid—a regular array of atoms—forms a natural three-dimensional “diffraction grating” for x rays. In crystals such as NaCl, a basic repeating unit called the unit cell (e.g., a cube of side \(a_0\)) repeats throughout the structure.

Bragg scattering and reflecting planes#

When an x-ray beam enters a crystal, x rays are scattered (redirected) in all directions by the atoms. In some directions the scattered waves interfere destructively (minima); in others constructively (maxima). This is a form of diffraction.

Reflecting planes. The maxima occur in directions as if the x rays were reflected by a family of parallel reflecting planes (or crystal planes) that extend through the atoms. (The x rays are not actually reflected; this is a simplifying model.) The spacing between adjacent planes is the interplanar spacing \(d\).

The angle \(\theta\) is defined relative to the plane surface (not the normal), unlike in optics. At each “reflection,” the angle of incidence equals the angle of reflection, both \(\theta\).

Bragg’s law#

Consider two rays that scatter from adjacent planes (Fig. 36-28c). The path length difference between them is \(2d\sin\theta\). For constructive interference (intensity maximum), this must equal an integer number of wavelengths:

This is Bragg’s law. The angle \(\theta\) is called a Bragg angle.

For a given crystal orientation, many families of planes exist—each with its own \(d\) and Bragg angle. As the incident angle changes, a different family of planes may satisfy Bragg’s law.

Unit cell and interplanar spacing#

The unit cell is the smallest repeating unit that defines the crystal structure. For NaCl, it is a cube of side \(a_0\). Different families of planes have different interplanar spacings \(d\), which can be related to \(a_0\) using geometry.

Example: For a family of planes that cut the cube diagonally, the Pythagorean theorem gives \(d = a_0/\sqrt{h^2 + k^2 + \ell^2}\) for certain integer indices \((h,k,\ell)\). Once \(d\) is measured via x-ray diffraction, the unit cell dimensions can be determined.

Applications#

Summary#

• X rays (\(\lambda \sim 1\) Å) require crystals as natural gratings; optical gratings cannot resolve them.

• Bragg’s law \(2d\sin\theta = m\lambda\): intensity maxima when path length difference \(2d\sin\theta\) equals \(m\lambda\).

• Reflecting planes and interplanar spacing \(d\) are defined by the crystal structure; \(\theta\) is measured from the plane.

• X-ray diffraction determines either wavelengths (known \(d\)) or crystal structure (known \(\lambda\)).