Question

In their study of X-ray diffraction, William and Lawrence Bragg determined that the relationship among the wavelength of the radiation \((\lambda),\) the angle at which the raveation is diffracted \((\theta),\) and the distance between planes of atoms in the crystal that cause the diffraction \((d)\) is given by \(n \lambda=2 d \sin \theta . X\) rays from a copper \(X\) -ray tube that have a wavelength of 1.54\(\hat{\mathrm{A}}\) 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).

Short answer

The distance between the planes of atoms responsible for diffraction in the silicon crystal is approximately \(3.15 \times 10^{-10} m\) or \(3.15 \mathring{A}\).

Step-by-step solution

Write down the given values

We are given the following values: - Wavelength of the radiation \(\lambda = 1.54 \mathring{A}\) (1 angstrom \(\mathring{A}\) = \(1 \times 10^{-10} m\)) - Angle of diffraction \(\theta = 14.22\) degrees - Order of diffraction \(n = 1\)

Substitute the given values into the Bragg equation

Now, we will substitute these values into the Bragg equation: \(n\lambda = 2d \sin \theta\) \(1 \times (1.54 \times 10^{-10}) = 2d \sin(14.22)\)

Solve for the distance between the planes of atoms, \(d\)

We want to solve for \(d\). To do this, we can divide both sides of the equation by \(2\sin(14.22)\): \(d = \frac{1.54 \times 10^{-10}}{2 \sin(14.22)}\) Now, we can plug in the values and find \(d\): \(d \approx \frac{1.54 \times 10^{-10}}{2 \times 0.2444} \approx 3.15 \times 10^{-10} m \) So, the distance between the planes of atoms responsible for diffraction in the silicon crystal is approximately \(3.15 \times 10^{-10} m\) or \(3.15 \mathring{A}\).

Key concepts

X-ray diffraction

X-ray diffraction is a fascinating phenomenon that occurs when X-rays interact with a crystal. In this process, X-rays are scattered by the atoms within a crystal lattice, creating a pattern of interference that can be detected and analyzed. This pattern of scattered X-rays is instrumental in understanding the structure of a crystal.

When X-rays strike a crystal, they are deflected from their original path by the atoms in the crystal. This deflection is what we refer to as diffraction. The way these X-rays diffract depends on the arrangement of atoms in the crystal, and by studying the pattern, scientists can determine the distance between the planes of atoms, which provides vital insights into the crystal structure.

The technique of X-ray diffraction is widely used in various fields, such as chemistry and material science, to identify unknown materials and to elucidate molecular and atomic models.

Crystal lattice

A crystal lattice is an ordered, repeating three-dimensional arrangement of atoms, ions, or molecules within a crystal. This regular pattern, like a grid, extends in all directions and provides the framework that supports the crystal structure. The repeating arrangement of particles within the lattice is what gives a crystal its unique properties and physical characteristics.

Different crystal lattices have varying distances between the planes of atoms known as interplanar spacing. This spacing is one of the key measurements obtained from X-ray diffraction studies.

Understanding the concept of a crystal lattice is essential because it directly influences the diffraction pattern created when X-rays pass through the crystal. Analyzing these patterns helps in identifying and characterizing various materials based on their lattice structures.

Wavelength of X-rays

The wavelength of X-rays is a critical parameter in diffraction studies, as it determines the position and pattern of the diffraction. X-rays are electromagnetic waves with wavelengths in the range of 0.01 to 10 nanometers, which are similar in scale to atomic distances. This makes them particularly suitable for probing the internal structures of crystals.

In X-ray diffraction, the wavelength of the X-rays must be known precisely, as it directly feeds into Bragg's Law, enabling the calculation of distances within the crystal lattice. For instance, in the exercise provided, the wavelength of X-rays used was 1.54 Å (angstroms). This exact wavelength allows us to apply the Bragg equation accurately to determine interplanar distances.

Angle of diffraction

The angle of diffraction (\(\theta\)) is one of the main variables in Bragg's Law and plays a significant role in determining the distance between planes in a crystal. It is the angle at which X-rays are reflected off the planes of atoms within the crystal, leading to constructive interference and the appearance of diffraction spots.

For exact analysis, the diffraction angle must be measured accurately. Any error in this measurement can lead to incorrect calculation of the interplanar spacing. Once the angle is known, it is utilized in the Bragg equation alongside wavelength and order of diffraction to find the distance between atomic layers.

In the example provided, an angle of 14.22 degrees was utilized, illustrating how precise measurements are essential in diffraction studies.