Question

\(\mathrm{X}\) rays of wavelength \(0.154 \mathrm{nm}\) are diffracted from a crystal at an angle of \(14.17^{\circ} .\) Assuming that \(n=1\), calculate the distance (in pm) between layers in the crystal.

Short answer

The distance between layers in the crystal is approximately 314.29 pm.

Step-by-step solution

Understand the diffraction formula

For X-ray diffraction, the relevant formula is Bragg's Law: \( n\lambda = 2d \sin \theta \). Here, \( n \) is the order of diffraction, \( \lambda \) is the wavelength, \( d \) is the distance between crystal layers, and \( \theta \) is the angle of diffraction.

Convert units

First, convert the wavelength \( \lambda = 0.154 \text{ nm} \) into meters: \( 0.154 \text{ nm} = 0.154 \times 10^{-9} \text{ m} \). To find \( d \) in picometers (pm), remember that 1 nm = 1000 pm, so \( 0.154 \text{ nm} = 154 \text{ pm} \).

Plug values into Bragg's Law

We know \( n = 1 \), \( \lambda = 0.154 \text{ nm} = 154 \text{ pm} \), and \( \theta = 14.17^{\circ} \). Convert \( \theta \) to radians if necessary, then use the equation: \( d = \frac{n\lambda}{2 \sin \theta} \).

Calculate \( \sin \theta \)

Calculate \( \sin 14.17^{\circ} \): \( \sin 14.17^{\circ} \approx 0.245 \).

Solve for \( d \)

Substitute the values into the Bragg's Law rearranged formula to find \( d \):\[ d = \frac{1 \times 154}{2 \times 0.245} = \frac{154}{0.49} \approx 314.29 \text{ pm}.\]

Final Answer

The distance between layers in the crystal is approximately \( 314.29 \text{ pm} \).

Key concepts

Bragg's Law

X-ray diffraction is described by a neat and elegant equation known as Bragg's Law. This formula is crucial for understanding how X-rays bounce off crystal layers. It is stated as:

• \( n\lambda = 2d \sin \theta \)

Let's break it down:

• \( n \) represents the order of diffraction, usually an integer starting with 1.
• \( \lambda \) is the wavelength of the incoming X-ray beam.
• \( d \) stands for the distance between layers in a crystal.
• \( \theta \) is the angle at which the X-rays are diffracted.

Bragg's Law tells us that for certain angles (\( \theta \)), a constructive interference occurs, making certain X-rays detectable. This interference pattern is dependent on the arrangement of atoms in the crystal.

Crystal Layer Spacing

The distance between crystal layers, noted as \( d \), is crucial to understanding how materials interact with X-rays.
Materials have unique, fixed arrangements of atoms in their structure, forming layers.
This layer distance influences the diffraction pattern when an X-ray hits the material.Given that X-ray diffraction aids in determining crystal structures, knowing how to calculate \( d \) helps identify materials. To find \( d \), rearrange Bragg's Law to:

• \( d = \frac{n\lambda}{2 \sin \theta} \)

This equation shows that by measuring the angle of diffraction and knowing the wavelength, you can calculate the layer spacing, crucial for material analysis.

Wavelength Conversion

Understanding the wavelengths in X-ray diffraction is pivotal since they are usually measured in nanometers (nm), and calculations often require conversion.
In our example, the wavelength \( \lambda \) was given as 0.154 nm.To perform calculations, convert to other units:

• From nanometers to meters: Multiply by \( 10^{-9} \).
• From nanometers to picometers: Multiply by 1000, since 1 nm = 1000 pm.

In this step-by-step solution, 0.154 nm was converted to 154 pm. This helps keep calculations simple and accurate, adhering to the units typically used in diffraction calculations.

Trigonometric Calculations

In diffraction problems, trigonometry plays an integral role because it involves angles and their sine values.
Here, we need \( \sin \theta \) for \( \theta = 14.17° \).The sine function is a ratio that helps determine the height of the opposite side of an angle in a right triangle over its hypotenuse:

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• For 14.17°, \( \sin 14.17° \approx 0.245 \).

Thus, the sine value is crucial for placing it into our rearranged Bragg’s Law equation:

• \( d = \frac{n\lambda}{2 \sin \theta} \)

This step ensures that we effectively calculate the spacing \( d \) using precise trigonometric functions.