Question

\(X\) rays from a copper 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 is approximately \(191.5 \, \text{pm}\).

Step-by-step solution

Write down Bragg's Law

Bragg's Law relates the X-ray wavelength, diffraction angle, order of diffraction, and interplanar spacing. It is given by: \(2 d \sin{\theta}=n\lambda\) where, - \(d\) is the interplanar spacing - \(\theta\) is the diffraction angle - \(n\) is the order of diffraction - \(\lambda\) is the X-ray wavelength.

Write down the given values

We are given the following values: - X-ray wavelength, \(\lambda = 154 \, \text{pm}\) - Diffraction angle, \(\theta = 14.22^{\circ}\) - Order of diffraction, \(n = 1\)

Convert the angle to radians

In order to use Bragg's Law, we need to convert the given angle in degrees to radians. The conversion formula is: \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\) Therefore, we can calculate the angle in radians as follows: \(\theta_{\text{radians}} = 14.22^{\circ} \times \frac{\pi}{180} \approx 0.248 \, \text{radians}\)

Use Bragg's Law to find the interplanar spacing

Now, we can use Bragg's Law to find the interplanar spacing, using the given values: \(2d \sin{\theta_{\text{radians}}} = n\lambda\) Solving for \(d\): \(d = \frac{n\lambda}{2\sin{\theta_{\text{radians}}}}\) Plug in the values: \(d = \frac{1 \times 154 \, \text{pm}}{2\sin{0.248 \, \text{radians}}}\) Calculate \(d\): \(d \approx 191.5 \, \text{pm}\)

Write the final answer

The interplanar spacing in the silicon crystal is approximately \(191.5 \, \text{pm}\).

Key concepts

Interplanar Spacing

Interplanar spacing refers to the distance between parallel planes of atoms or ions in a crystalline solid. It is a key concept in crystallography, as it helps determine how X-rays are diffracted by a crystal lattice. This spacing is intrinsic to the crystal structure and varies with different minerals and compounds.

Finding the interplanar spacing can be vital in characterizing the material properties, like how the material reacts under stress or its thermal expansion properties. The significance of this spacing lies in its ability to provide insights into the arrangement and bonding potential within the crystal structure.

X-ray Diffraction

X-ray diffraction is a technique used to explore the atomic and molecular structure of a crystal by directing X-rays at it. The process begins with X-rays, which have wavelengths comparable to the distances between lattice planes in the crystal. When X-rays strike a crystal, they are scattered in various directions.

• Upon hitting the crystal, X-rays can be either absorbed, transmitted, or scattered.
• The diffraction pattern produced is characteristic of the crystal structure.

By analyzing the angles and intensities of the scattered beams, we can determine the distances between atoms, referred to as interplanar spacing. X-ray diffraction is instrumental in identifying unknown substances, assessing purity, and elucidating structural details of materials.

First-order Diffraction

In the realm of X-ray crystallography, the term "first-order diffraction" is crucial. This relates to Bragg's Law, which describes the condition for diffracted rays to constructively interfere. First-order diffraction means the path difference between scattered waves is equivalent to one full wavelength, expressed as \( n = 1 \).

It's the simplest case of diffraction, making analysis straightforward. Higher orders of diffraction, where \( n \) equals 2, 3, etc., are possible but tend to be less intense due to their secondary nature. Utilizing first-order diffraction makes calculations more manageable as we deal with a single wave cycle interfering constructively, offering a clear depiction of crystal structures.

Silicon Crystal

A silicon crystal is a solid material with silicon atoms arranged in a repeating pattern. In electronics, silicon crystals are foundational due to their semiconducting properties, enabling the manufacturing of microchips and solar cells. The atomic structure of silicon is symmetrical, forming a diamond cubic lattice.

This lattice structure is significant in X-ray diffraction studies as it provides a clear pattern of how X-rays interact with the crystal. With its tetrahedral bonding, each silicon atom is covalently bonded to four neighboring silicon atoms, facilitating the precise measurement of interplanar spacing. Silicon's capacity to form high-quality crystals is crucial for many technological applications, from integrated circuits to photonics.

Diffraction Angle

The diffraction angle, denoted as \( \theta \), is an essential parameter in Bragg's Law for X-ray diffraction. This angle determines the direction in which the diffracted rays are observed and is directly involved in calculating interplanar distances. Calculating the actual angle involves converting degrees into radians, as radians are the standard unit in mathematical equations.

• Diffraction angles help identify the crystal's orientation and symmetry.
• These angles vary with different crystal planes, offering a detailed map of atomic arrangement.

Understanding the diffraction angle is pivotal in interpreting X-ray diffraction patterns and can highlight any imperfections or variations in crystal structures.