An X-ray beam with λ = 154 pm incident on the surface of a crystal produced a maximum reflection at an angle of θ = 28.3°. Assuming n = 1, calculate the separation between layers of atoms in the crystal.

Verified step by step guidance

1. Identify the relevant formula for X-ray diffraction, which is Bragg's Law: \( n\lambda = 2d\sin\theta \).
2. Substitute the given values into Bragg's Law: \( n = 1 \), \( \lambda = 154 \text{ pm} \), and \( \theta = 28.3^\circ \).
3. Rearrange the formula to solve for \( d \), the separation between layers of atoms: \( d = \frac{n\lambda}{2\sin\theta} \).
4. Convert the angle \( \theta \) from degrees to radians if necessary, or use the sine of the angle directly in degrees.
5. Calculate \( d \) using the rearranged formula and the substituted values.

Key Concepts

Bragg's Law

Bragg's Law relates the wavelength of X-rays to the angle of diffraction and the distance between atomic layers in a crystal. It is expressed as nλ = 2d sin(u), where n is the order of reflection, λ is the wavelength, d is the interplanar spacing, and u is the angle of incidence. This law is fundamental in crystallography for determining the arrangement of atoms within a crystal.

Wavelength

Wavelength is the distance between successive peaks of a wave, commonly denoted by the symbol λ. In the context of X-rays, it is crucial for understanding how X-rays interact with matter, particularly in diffraction experiments. The wavelength influences the resolution and the ability to distinguish between different atomic layers in a crystal.

Interplanar Spacing

Interplanar spacing (d) refers to the distance between parallel planes of atoms in a crystal lattice. This spacing is a key parameter in determining how X-rays are diffracted by the crystal. By applying Bragg's Law, one can calculate the interplanar spacing using the wavelength of the incident X-ray and the angle of maximum reflection.