In their study of X-ray diffraction, William and Lawrence Bragg determined that the relationship among the wavelength of the radiation 1l2, the angle at which the radiation is diffracted 1u2, and the distance between planes of atoms in the crystal that cause the diffraction (d) is given by nl = 2d sin u. X rays from a copper X-ray tube that have a wavelength of 1.54 Å are diffracted at an angle of 14.22 degrees by crystalline silicon. Using the Bragg equation, calculate the distance between the planes of atoms responsible for diffraction in this crystal, assuming n = 1 (first-order diffraction).

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Identify the given values: wavelength (\( \lambda \)) = 1.54 Å, angle (\( \theta \)) = 14.22 degrees, and order of diffraction (n) = 1.

Convert the angle from degrees to radians if necessary, as trigonometric functions typically use radians. However, for this problem, you can use degrees directly in the sine function.

Use the Bragg equation: \( n\lambda = 2d \sin \theta \).

Rearrange the equation to solve for the distance between planes (d): \( d = \frac{n\lambda}{2 \sin \theta} \).

Substitute the known values into the equation: \( d = \frac{1 \times 1.54 \text{ Å}}{2 \sin(14.22^\circ)} \) and calculate the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bragg's Law

Bragg's Law describes the relationship between the wavelength of X-rays, the angle of diffraction, and the distance between atomic planes in a crystal. It is mathematically expressed as nλ = 2d sin(θ), where n is the order of diffraction, λ is the wavelength, d is the distance between planes, and θ is the angle of diffraction. This law is fundamental in crystallography for determining the structure of crystalline materials.

Wavelength of X-rays

The wavelength of X-rays is a critical parameter in X-ray diffraction studies, influencing how X-rays interact with the crystal lattice. In this context, a wavelength of 1.54 Å (angstroms) is commonly used, particularly for copper X-ray tubes. The wavelength determines the resolution and the ability to distinguish between different atomic planes in the crystal structure.

Angle of Diffraction

The angle of diffraction (θ) is the angle at which X-rays are scattered by the crystal planes. It is essential for applying Bragg's Law, as it directly affects the calculation of the interplanar spacing (d). In the given problem, the angle of 14.22 degrees is used to find the distance between the planes of atoms, showcasing how the geometry of the crystal influences diffraction patterns.