Question

A topaz crystal has an interplanar spacing \((d)\) of \(1.36 Å\) \(\left(1Å =1 \times 10^{-10} \mathrm{m}\right) .\) Calculate the wavelength of the X ray that should be used if \(\theta=15.0^{\circ}\) (assume \(n=1\) ).

Short answer

The wavelength of the X-ray that should be used is approximately 3.99 Å.

Step-by-step solution

Convert the given information into the appropriate units

We are given d as 1.36 Å. We need to convert this into meters. 1 Å = 1 × 10⁻¹⁰ m d = 1.36 Å × 10⁻¹⁰ m/Å = 1.36 × 10⁻¹⁰ m

Convert the angle θ from degrees to radians

To use the sine function in our calculations, we must first convert the angle θ from degrees to radians using the following conversion factor: 1 rad = 180°/π θ = 15.0° × π/180° = \(15.0\pi/180 = \pi/12\) radians

Use Bragg's Law to find the wavelength of the X-ray

Bragg's Law: \(n\lambda = 2d\sin\theta\) We have n = 1, d = 1.36 × 10⁻¹⁰ m, and θ = \(\pi/12\) radians. Plug these values into Bragg's Law and solve for λ: \(1\cdot\lambda = 2(1.36\times 10^{-10}\mathrm{m})\sin(\pi/12)\) Divide both sides by 1 (which doesn't affect the value): \(\lambda = 2(1.36\times 10^{-10}\mathrm{m})\sin(\pi/12)\) Now, calculate the result: \(\lambda = 2(1.36\times 10^{-10}\mathrm{m})\sin(\pi/12) \approx 3.99 \times 10^{-10} \mathrm{m}\)

Convert the wavelength back to Å

Our final step is to convert the result back to Å for easier readability: \(\lambda = 3.99 \times 10^{-10} \mathrm{m} \times \frac{1Å}{1\times 10^{-10}\mathrm{m}} \approx 3.99 Å\) The wavelength of the X-ray that should be used is approximately 3.99 Å.

Key concepts

Interplanar Spacing

Interplanar spacing, often denoted by the symbol \(d\), is a key concept in crystallography. It refers to the distance between adjacent parallel planes of atoms within a crystal lattice. This spacing can significantly influence the properties of a crystal, such as its diffraction pattern when X-rays are applied. Understanding the interplanar spacing is crucial as it helps determine how a crystal will interact with incoming waves, like X-rays. For example, in a topaz crystal, the interplanar spacing is given as 1.36 Å. This precise measurement plays a vital role when using Bragg's law to determine other properties, like wavelength, as seen in the exercise question. The measurement unit for interplanar spacing is often in Ångströms (Å), where 1 Å is equivalent to \(1 \times 10^{-10}\) meters. This smaller unit helps in dealing with the extremely tiny scales associated with atomic structures.

X-ray Wavelength Calculation

X-ray wavelength calculation often involves the application of Bragg's Law, a fundamental principle in the study of crystallography. Bragg's Law is mathematically represented as \[ n\lambda = 2d\sin\theta \]where:

• \( n \) is the order of reflection, usually an integer.
• \( \lambda \) is the wavelength of the X-rays.
• \( d \) is the interplanar spacing within the crystal.
• \( \theta \) is the angle of incidence of the X-rays.

In the given exercise, we need to calculate the X-ray wavelength for a topaz crystal, with an interplanar spacing \(d\) of 1.36 Å and an angle \(\theta\) of 15.0°. The calculation involves converting this angle to radians, as it is necessary for calculating the sine value accurately. The formula becomes: \[ \lambda = 2(1.36 \times 10^{-10} \text{ m})\sin(\pi/12) \]Following through these calculations gives us a wavelength \(\lambda\) of approximately 3.99 Å. Converting this back from meters to Ångströms makes the understanding of results easier since Ångströms are a more natural unit for atomic scale observations.

Topaz Crystal

Topaz is a silicate mineral that often captures interest due to its varied colors and high clarity, but scientifically, it is valued for its well-defined crystal structure. Like many crystals, topaz can be analyzed using X-ray diffraction methods to study its atomic arrangement. These methods rely on understanding key concepts such as interplanar spacing to determine how X-rays will diffract when they pass through it. Due to its relatively large and transparent crystals, topaz provides an excellent study case for Bragg's Law applications. By measuring the diffraction pattern produced by X-rays when passing through a topaz crystal, researchers can learn about the arrangement of atoms and thus deduce important properties of the crystal. These insights are not just of academic interest; they have practical applications. For example, knowing how a crystal structure affects light and other electromagnetic waves can impact the development and refinement of materials for optics, technology, and even jewelry.