Using a diffractometer and a radiation of wavelength \(1.54 \AA\) only one reflection from an fce crystal is observed when \(2 \theta\) is \(121^{\circ}\). What are the indices of reflection? What is the interplanar spacing?

Understanding Bragg's Law is essential for interpreting X-ray diffraction in crystalline materials. Bragg's Law provides the condition for constructive interference of the X-rays reflected from the lattice planes in a crystal.
It is given by the formula:
\[ n\lambda = 2d \sin(\theta) \]
where:

• \lambda is the wavelength of the incident X-ray.
• n is the order of reflection (usually taken as 1).
• d is the spacing between the planes in the atomic lattice.
• \theta is the angle between the incident X-rays and the crystallographic plane.

In our given exercise, we use a radiation of wavelength 1.54 Å and observe the angle 2\theta = 121°. Bragg's Law helps us calculate the interplanar spacing by rearranging it to solve for d.

Interplanar spacing, denoted as d, is the distance between two adjacent planes in a crystal lattice. It’s an important factor in determining the crystal structure, as it affects how X-rays are diffracted.
Using the modified Bragg's Law, we can calculate interplanar spacing with:
\[ d = \frac{n\lambda}{2\sin(\theta)} \]
Here, we have all the required values:

• n = 1
• \lambda = 1.54 \AA
• \theta = 60.5\textdegree

First, we find the sine of the angle:
\[ \sin(60.5\textdegree) \approx 0.8746 \]
Substituting back into the equation gives:
\[ d \approx \frac{1.54\backslash\backslash mathrm{\AA}}{2\times0.8746} \ \backslash approx\backslash 0.88\backslash \mathrm{\AA} \]
Therefore, the interplanar spacing for this crystal is approximately 0.88 Å.

Indices of reflection, also known as Miller indices, are a notation to denote the orientation of a crystal plane. They are represented as (hkl).
These indices are derived from the intersection of a plane with the crystallographic axes.

• h, k, and l are the intercepts of the plane with the x, y, and z axes respectively.
• In the exercise, the indices of reflection found were (332).

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For an FCC crystal structure, these indices help identify the exact plane being reflected in the diffraction experiment.

An FCC (Face-Centered Cubic) crystal structure is one of the most common types of crystal arrangements found in metals.
The properties are:

• Atoms are located at each of the corners and the centers of all the faces of the cube.
• Each face-centered atom is shared between two adjacent unit cells.

For FCC crystals, the Miller indices relate to the specific planes of atoms that the X-ray diffraction measures. By using Bragg's Law and the observed diffraction angles, we determine these indices, which in turn tell us about the atomic arrangement, such as the (332) reflection in the exercise.
This aids in understanding material properties and behaviors based on their atomic structure.