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Ch.12 - Solids and Modern Materials
Chapter 12, Problem 120

In their study of X-ray diffraction, William and Lawrence Bragg determined that the relationship among the wavelength of the radiation 1l2, the angle at which the radiation is diffracted 1u2, and the distance between planes of atoms in the crystal that cause the diffraction (d) is given by nl = 2d sin u. X rays from a copper X-ray tube that have a wavelength of 1.54 Å are diffracted at an angle of 14.22 degrees by crystalline silicon. Using the Bragg equation, calculate the distance between the planes of atoms responsible for diffraction in this crystal, assuming n = 1 (first-order diffraction).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bragg's Law

Bragg's Law describes the relationship between the wavelength of X-rays, the angle of diffraction, and the distance between atomic planes in a crystal. It is mathematically expressed as nλ = 2d sin(θ), where n is the order of diffraction, λ is the wavelength, d is the distance between planes, and θ is the angle of diffraction. This law is fundamental in crystallography for determining the structure of crystalline materials.
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Wavelength of X-rays

The wavelength of X-rays is a critical parameter in X-ray diffraction studies, influencing how X-rays interact with the crystal lattice. In this context, a wavelength of 1.54 Å (angstroms) is commonly used, particularly for copper X-ray tubes. The wavelength determines the resolution and the ability to distinguish between different atomic planes in the crystal structure.
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Frequency-Wavelength Relationship

Angle of Diffraction

The angle of diffraction (θ) is the angle at which X-rays are scattered by the crystal planes. It is essential for applying Bragg's Law, as it directly affects the calculation of the interplanar spacing (d). In the given problem, the angle of 14.22 degrees is used to find the distance between the planes of atoms, showcasing how the geometry of the crystal influences diffraction patterns.
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Related Practice
Textbook Question

Silicon carbide, SiC, has the three-dimensional structure shown in the figure.

(b) Would you expect the bonding in SiC to be predominantly ionic, metallic, or covalent?

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Textbook Question

Energy bands are considered continuous due to the large number of closely spaced energy levels. The range of energy levels in a crystal of copper is approximately 1 * 10–19 J. Assuming equal spacing between levels, the spacing between energy levels may be approximated by dividing the range of energies by the number of atoms in the crystal. (b) Determine the average spacing in J between energy levels in the copper metal in part (a).

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Textbook Question

Sodium oxide (Na2O) adopts a cubic structure with Na atoms represented by green spheres and O atoms by red spheres.

(c) The unit cell edge length is 5.550 Å. Determine the density of Na2O.

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Textbook Question

Germanium has the same structure as silicon, but the unit cell size is different because Ge and Si atoms are not the same size. If you were to repeat the experiment described in the previous problem but replace the Si crystal with a Ge crystal, would you expect the X rays to be diffracted at a larger or smaller angle u?

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Textbook Question

(a) The density of diamond is 3.5 g>cm3, and that of graphite is 2.3 g>cm3. Based on the structure of buckminsterfullerene, what would you expect its density to be relative to these other forms of carbon?

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Textbook Question

The karat scale used to describe gold alloys is based on mass percentages. (a) If an alloy is formed that is 50 mol% silver and 50 mol% gold, what is the karat number of the alloy? Use Figure 12.18 to estimate the color of this alloy.

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