The X-ray diffraction angles for the first-order diffraction...

The X-ray diffraction angles for the first-order diffraction spot from the 111 planes of a cubic crystal with $a=380.5 \mathrm{pm}$ are observed to be $\alpha=18.79^{\circ}, \beta=0^{\circ}$, and $\gamma=0^{\circ} .$ How is the crystal oriented? Take the wavelength of the X-radiation to be $\lambda=154.433 \mathrm{pm}$.

Key Concepts

Bragg's Law

Bragg's Law is the fundamental principle behind X?ray diffraction. It relates the wavelength of the incident X-rays, the distance between the crystallographic planes (d-spacing), and the angle at which X-rays are diffracted. The law is generally written as n? = 2d sin?, where n is the order of diffraction, ? is the wavelength, d is the interplanar spacing, and ? is the angle of incidence. This law is essential in determining the orientations and spacings of crystal planes.

Miller Indices

Miller indices are a notation system in crystallography used to denote the orientation of atomic planes in a crystal lattice. They are represented by three integers (h, k, l) that describe the intercepts of the plane with the crystallographic axes, and are crucial for identifying the specific set of planes responsible for diffraction in an X-ray experiment. Understanding Miller indices allows one to connect features in diffraction patterns to specific crystallographic directions and planes.

Crystal Orientation

Crystal orientation refers to the alignment of the crystallographic axes relative to a fixed reference frame, such as the incident beam of X-rays. In diffraction experiments, the observed angles can be used to deduce the orientation of the crystal. This includes knowledge about which crystal plane is aligned with respect to the beam, and deviations from central alignment can indicate rotations about certain axes.

Cubic Crystal System

The cubic crystal system is characterized by three equal axes at right angles to each other. This symmetry simplifies the calculation of interplanar spacings and the interpretation of diffraction patterns, since the geometry is highly regular. In a cubic system, formulas for d-spacings and the relation between lattice parameters and Miller indices are straightforward, which is particularly useful when applying Bragg’s Law.

Diffraction Geometry

Diffraction geometry involves the arrangement and angles of the incident X-rays and the crystal planes. It encompasses the orientation of the sample relative to the incident beam, the detector position, and the resulting pattern of diffraction spots. Analyzing the angles from diffraction (such as the given rotation angles about specific axes) is key to determining the spatial orientation of the crystal with respect to the X-ray source.

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