Question

\(\mathrm{X}\) rays from a copper \(\mathrm{X}\) -ray tube \((\lambda=154 \mathrm{pm})\) were diffracted at an angle of \(14.22\) degrees by a crystal of silicon. Assuming first-order diffraction \((n=1\) in the Bragg equation), what is the interplanar spacing in silicon?

Short answer

The interplanar spacing in the silicon crystal can be found using Bragg's Law: \( n\lambda = 2d\sin\theta \). Plugging in the given values of \(n = 1\), \(\lambda = 154\text{ pm}\), and \(\theta = 14.22^{\circ}\), we can solve for the interplanar spacing \(d\). Upon calculation, we find that the interplanar spacing in the silicon crystal is approximately 315.45 pm.

Step-by-step solution

Write down Bragg's Law

Bragg's Law states that for constructive interference to occur, the path difference between waves scattered from different planes of atoms must be an integral multiple of the wavelength. Mathematically, it is written as: \( n\lambda = 2d\sin\theta \) where, - \(n\) is the order of diffraction (given as 1 in this problem) - \(\lambda\) is the wavelength of the incident wave (154 pm) - \(d\) is the interplanar spacing in the crystal that we need to calculate - \(\theta\) is the angle of diffraction (14.22 degrees)

Input the given values into Bragg's Law

Now, substitute the given values for the wavelength, angle of diffraction, and diffraction order into Bragg's Law: \( 1 \times 154\text{ pm} = 2d \sin 14.22^{\circ} \)

Solve for the interplanar spacing \(d\)

Next, we will solve the equation for the interplanar spacing \(d\): \( d = \frac{1 \times 154\text{ pm}}{2 \sin 14.22^{\circ}} \) Calculate the value of \(d\): \( d \approx 315.45 \text{ pm} \)

State the answer

The interplanar spacing in the silicon crystal is approximately 315.45 pm.

Key concepts

X-ray Diffraction

X-ray diffraction (XRD) is a powerful technique used in material science to uncover the molecular structure of crystalline materials. When X-rays are aimed at a crystal, they may be deflected by the atomic planes in the crystal structure. The pattern of these deflections provides valuable information about the distances between the planes, known as interplanar spacing, and the angles at which planes intersect.

XRD works because X-rays have wavelengths that are on the same order of magnitude as the distance between crystal planes, allowing for the constructive and destructive interference needed to create a diffraction pattern. The conditions for this to occur are succinctly described by Bragg's Law, which serves as the fundamental equation in XRD analysis. By examining the angles and intensities of the diffracted beams, researchers can infer the crystal's structure and properties.

Interplanar Spacing

Interplanar spacing is the physical distance between parallel planes of atoms in a crystal. It's a key concept in crystallography because it helps scientists and engineers determine how atoms are arranged in a solid, which in turn influences the material's properties.

In the context of Bragg's Law, interplanar spacing (\(d\)) is a critical variable used to calculate at which angles X-ray beams will be diffracted by a crystal. Accurately determining interplanar spacing allows for the identification of crystal orientations and can even uncover defects or impurities within the crystal structure. Knowing the interplanar spacing also aids in determining bond lengths and angles within a molecule, thus offering a window into the chemical characteristics of the material.

Crystallography

Crystallography is the scientific study of crystals and their structures. It involves analyzing the arrangement of atoms within a solid and how this arrangement affects the physical properties of the material. Crystallography encompasses various techniques, including X-ray diffraction, to provide a detailed map of the crystal's atomic structure.

The insights gained from crystallography are essential in numerous fields such as chemistry, physics, biology, and materials science. For example, it's crucial in drug design, allowing scientists to determine the precise structure of molecules and how they may interact with biological targets. Moreover, crystallography can be used to predict and engineer the properties of new materials for technological applications, making it an indispensable tool in discovery and innovation.