An X-ray beam of unknown wavelength is diffracted from a NaCl surface. If the interplanar distance in the crystal is 286 pm, and the angle of maximum reflection is found to be 7.23°, what is the wavelength of the X-ray beam? (Assume n = 1.)

Verified step by step guidance

Identify the relevant formula for X-ray diffraction, which is Bragg's Law: \( n\lambda = 2d\sin\theta \).

Substitute the given values into Bragg's Law: \( n = 1 \), \( d = 286 \text{ pm} \), and \( \theta = 7.23^\circ \).

Convert the angle from degrees to radians if necessary, or use the sine of the angle directly in degrees.

Rearrange the formula to solve for the wavelength \( \lambda \): \( \lambda = \frac{2d\sin\theta}{n} \).

Calculate the sine of the angle and substitute all known values into the rearranged formula to find the wavelength \( \lambda \).

Key Concepts

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

Bragg's Law

Bragg's Law relates the wavelength of X-rays to the angle of diffraction and the interplanar spacing in a crystal. It is expressed as nλ = 2d sin(θ), where n is the order of reflection, λ is the wavelength, d is the interplanar distance, and θ is the angle of diffraction. This principle is fundamental in determining the structure of crystalline materials using X-ray diffraction.

Interplanar Distance

The interplanar distance (d) is the distance between parallel planes of atoms in a crystal lattice. It is a critical parameter in crystallography, influencing how X-rays interact with the crystal. In the context of Bragg's Law, knowing the interplanar distance allows for the calculation of the wavelength of X-rays based on the diffraction angle.

Diffraction Angle

The diffraction angle (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 constructive interference of the diffracted waves. The angle of maximum reflection indicates the specific conditions under which the X-rays are most effectively diffracted, allowing for the determination of the wavelength.