Ch.12 - Solids and Modern Materials

Problem 112b

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Chapter 12, Problem 112b

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, one can approximate the spacing between energy levels 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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Identify the given information: the range of energy levels in a crystal of copper is approximately \(1 \times 10^{-19}\) J.

Assume that the energy levels are equally spaced.

To find the average spacing between energy levels, divide the total range of energy levels by the number of atoms in the crystal.

Express the formula for average spacing as: \( \text{Average Spacing} = \frac{\text{Total Energy Range}}{\text{Number of Atoms}} \).

Substitute the given values into the formula to find the average spacing between energy levels.

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Key Concepts

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

Energy Bands

Energy bands in solids arise from the overlap of atomic orbitals in a crystal lattice, leading to a continuum of energy levels. In metals like copper, these bands allow for the conduction of electricity as electrons can move freely between closely spaced energy levels. Understanding energy bands is crucial for analyzing the electronic properties of materials.

Energy Level Spacing

The spacing between energy levels in a solid can be approximated by dividing the total range of energy levels by the number of available states, which is often related to the number of atoms in the crystal. This concept is essential for determining how energy is quantized in a material and influences its electrical and thermal properties.

Crystal Structure

The arrangement of atoms in a crystal, known as its crystal structure, significantly affects its physical properties, including electrical conductivity. In metals like copper, the face-centered cubic (FCC) structure allows for a high density of atoms, contributing to the closely spaced energy levels that define the material's behavior in electronic applications.

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Pure iron crystallizes in a body-centered cubic structure, shown in the figure. but small amounts of impurities can stabilize a facecentered cubic structure. Which form of iron has a higher density?

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Textbook Question

What type of lattice—primitive cubic, body-centered cubic, or face-centered cubic—does each of the following structure types possess: (e) ZnS?

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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

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

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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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 Additional Exercise 12.117, but replace the Si crystal with a Ge crystal, would you expect the X rays to be diffracted at a larger or smaller angle 𝜃?