Question

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 \((n=1 \text { in the Bragg equation), what is }\) the interplanar spacing in silicon?

Short answer

The interplanar spacing in the silicon crystal is approximately \(314 \times 10^{-12} m\) or \(314 \mathrm{pm}\).

Step-by-step solution

Understand Bragg's Law

Bragg's Law relates the wavelength of the X-rays (\(\lambda\)), the diffraction angle (\(\theta\)), the interplanar spacing in the crystal (d), and the diffraction order (n). It is given by the formula: \[n\lambda = 2d \sin\theta\] In this exercise, we are given the wavelength (\(\lambda = 154 pm\)), the diffraction angle (\(\theta = 14.22^\circ\)), and are told to assume first-order diffraction (n = 1). Our task is to find the interplanar spacing (d).

Plug in Known Values

Now, we can plug in the known values into Bragg's Law equation: \[1(154 \times 10^{-12} m) = 2d \sin(14.22^\circ)\]

Solve for the Interplanar Spacing (d)

Next, we want to isolate the variable d in the equation. To do this, let's first calculate the value of \(\sin(14.22^\circ)\): \[\sin(14.22^\circ) \approx 0.2458\] Now we can substitute this value back into the equation: \[154 \times 10^{-12} m = 2d (0.2458)\] Since we want to find the value of d, we can divide both sides of the equation by \(2(0.2458)\): \[d = \frac{154 \times 10^{-12} m}{2(0.2458)}\] Finally, we can calculate the value of d: \[d \approx 314 \times 10^{-12} m\]

Report the Answer

The interplanar spacing in the silicon crystal is approximately \(314 \times 10^{-12} m\) or \(314 \mathrm{pm}\).

Key concepts

X-ray diffraction

X-ray diffraction is a fascinating phenomenon that occurs when X-rays pass through a crystal lattice. This process helps us to study the arrangement of atoms within a crystal.
X-rays have very short wavelengths, which are similar in size to the distance between atoms in a crystal.
When X-rays strike a crystal, they are scattered in various directions. However, in certain specific directions, the scattered waves interfere constructively, creating a stronger intensity. This is what we refer to as diffraction.

• X-ray diffraction is crucial for understanding the structure of materials.
• Studying diffraction can reveal information about the size and shape of the unit cell in a crystal lattice.

The pattern and angles of diffraction are key to analyzing the structure of solid materials.

Interplanar spacing

Interplanar spacing, often denoted as "d" in equations, is the distance between parallel planes of atoms in a crystal. Understanding this spacing is essential when using X-ray diffraction to determine the crystal structure.
In a crystal, atoms are arranged in a highly ordered, repeating pattern. These repetitive layers are what we refer to as planes.
The distance between these planes, or interplanar spacing, influences how and where X-rays are diffracted.
When applying Bragg's Law, which relates the angle of diffraction with the interplanar spacing, it's possible to use the observed diffraction pattern to determine the values of "d".

• Smaller interplanar spacing will lead to diffraction at larger angles.
• The magnitude of spacing affects the angles where constructive interference occurs in the diffraction pattern.

In this context, calculating interplanar spacing helps us accurately determine the internal arrangement of atoms in crystals, like silicon.

Crystal lattice

A crystal lattice is the organized, repeating structure of atoms within a crystal. This structure is foundational to the physical properties and behaviors of the material. In a crystal lattice, each atom is arrayed in a three-dimensional repeated pattern.
There are different types of lattice structures, such as face-centered cubic, body-centered cubic, or hexagonal close-packed. Each type has unique symmetry and arrangement properties.

• Lattice structure dictates many properties, including density and melting point.
• Observation of the lattice is possible through techniques like X-ray diffraction.

Overall, the crystal lattice provides a snapshot into the material's characteristic properties by revealing how the atoms are spatially arranged.

First-order diffraction

First-order diffraction refers to the initial set of visible diffraction peaks observed when X-rays interact with a crystal. It is fundamental in the application of Bragg’s Law.
In the context of Bragg's Law, first-order diffraction occurs when the diffraction order "n" is set to 1. This means the path difference between waves scattered from successive planes differs by exactly one wavelength.
First-order peaks are typically used because they are the most straightforward to measure and interpret.

• This is the simplest way to assess the crystal structure using diffraction data.
• Using first-order diffraction minimizes complexity in calculations associated with Bragg's equation.

By analyzing these peaks, scientists can determine important structural information about the crystal, such as the interplanar spacing.