Question

The second-order Bragg reflection \((n=2)\) from a copper crystal for X-rays with a wavelength of \(166 \mathrm{pm}\) is \(27.35^{\circ}\). Calculate the spacing between the planes of copper atoms.

Short answer

The spacing between copper crystal planes is approximately 360.87 pm.

Step-by-step solution

Understand Bragg's Law

Bragg's Law relates the wavelength of electromagnetic waves such as X-rays to the angle of diffraction and the spacing between planes in a crystal lattice. The formula is given by: \( n \cdot \lambda = 2 \cdot d \cdot \sin(\theta) \), where \(n\) is the order of reflection, \(\lambda\) is the wavelength, \(d\) is the spacing between the planes, and \(\theta\) is the angle of reflection.

Identify Given Values

In the problem, the given values are: the wavelength \(\lambda = 166 \, \text{pm}\), the Bragg angle \(\theta = 27.35^{\circ}\), and the order of reflection \(n = 2\).

Rearrange Bragg's Law to Solve for \(d\)

Rearrange Bragg's Law to solve for \(d\): \( d = \frac{n \cdot \lambda}{2 \cdot \sin(\theta)} \). This formula allows us to calculate the spacing between the crystal planes.

Calculate \(\sin(\theta)\)

Convert the angle \(\theta\) from degrees to radians, if necessary. However, most calculators will handle the conversion internally when computing sine. Calculate \(\sin(27.35^{\circ})\).

Plug in the Values and Calculate \(d\)

Plug the given values into the rearranged Bragg's equation: \( d = \frac{2 \times 166 \, \text{pm}}{2 \cdot \sin(27.35^{\circ})} \). Simplify and compute the result to find \(d\).

Final Calculation

Perform the calculation: \( \sin(27.35^{\circ}) \approx 0.460\). So, \( d = \frac{2 \times 166}{2 \times 0.460} = \frac{332}{0.92} \approx 360.87 \, \text{pm} \).

Key concepts

X-ray diffraction

X-ray diffraction is a powerful technique used to study the structure of crystalline materials. When X-rays interact with a crystal, they can scatter off its atomic planes. This scattering produces a specific pattern known as a diffraction pattern, which can be analyzed to provide information about the atomic structure of the crystal.
The key to understanding X-ray diffraction is recognizing how X-rays, with their very short wavelengths, interact with the regular and periodic arrangement of atoms in a crystal lattice. These interactions create constructive and destructive interference.

• Constructive interference: Occurs when scattered X-rays coincide in phase, resulting in detectable peaks on a detector.
• Destructive interference: Occurs when scattered X-rays are out of phase, cancelling each other out.

The diffraction pattern thus contains valuable information about the crystal's internal structure, including the spacing between atomic planes.

crystal lattice

A crystal lattice is a highly ordered structure consisting of atoms, ions, or molecules arranged in a repeating pattern. This orderly arrangement is what gives crystals their unique properties. Each point in a crystal lattice, called a lattice point, represents an identical environment in the crystal.
This regular arrangement of lattice points can be thought of as a 3D grid extending throughout the entire crystal. The simplest repeating unit in a lattice is known as the unit cell, which contains the arrangement of lattice points that describes the entire crystal structure when repeated in space.

• Crystal lattices are classified into seven crystal systems: cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.
• Each crystal system has distinctive lattice parameters, which define the unit cell's dimensions and angles.

Understanding crystal lattices is crucial for interpreting diffraction patterns because the arrangement dictates the planes that X-rays interact with. The regularity and symmetry in crystal lattices directly influence how X-rays are diffracted and can be studied using Bragg's Law.

atomic plane spacing

Atomic plane spacing, often denoted as "d" in equations, refers to the distance between parallel planes of atoms within a crystal lattice. This spacing is a fundamental characteristic of the crystal structure and can be determined using techniques like X-ray diffraction.
Bragg's Law provides a direct relationship between the atomic plane spacing, the wavelength of the incoming X-rays, and the angle at which they are diffracted:\[ n \cdot \lambda = 2 \cdot d \cdot \sin(\theta)\]Where:

• \(n\) is the order of reflection (e.g., 1, 2, 3, ...).
• \(\lambda\) is the wavelength of the X-rays.
• \(d\) is the spacing between atomic planes.
• \(\theta\) is the Bragg angle, or the angle at which diffraction occurs.

In an analysis like the Bragg reflection problem, the atomic plane spacing can be calculated by rearranging Bragg's Law. Given that X-ray diffraction studies hinge on precisely knowing these spacings, accurate calculations help scientists understand material properties at the atomic level, which is crucial for various applications such as materials science, chemistry, and crystallography.