Calculate the density of states g(E)g(E) for the free-electron model of a metal if E=7.0E = 7.0 eV and V=1.0V = 1.0 cm3. Express your answer in units of states per electron volt.

Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 42: Molecules and Condensed Matter

Problem 21

Chapter 42, Problem 21

Calculate the density of states $g (E)$ for the free-electron model of a metal if $E equals 7.0$ eV and $V equals 1.0$ cm³. Express your answer in units of states per electron volt.

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Step 1: Recall the formula for the density of states in the free-electron model: $ g(E) = \frac{V}{2\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{E} $, where $ V $ is the volume, $ m $ is the electron mass, $ \hbar $ is the reduced Planck's constant, and $ E $ is the energy.

Step 2: Convert the given volume $ V = 1.0 \, \text{cm}^3 $ into cubic meters (SI units). Use the conversion factor $ 1 \, \text{cm}^3 = 10^{-6} \, \text{m}^3 $.

Step 3: Substitute the known constants into the formula: $ m = 9.11 \times 10^{-31} \, \text{kg} $, $ \hbar = 1.055 \times 10^{-34} \, \text{J·s} $, and $ E = 7.0 \, \text{eV} $. Convert $ E $ from electron volts to joules using $ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} $.

Step 4: Calculate the term $ \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{E} $. This involves squaring $ \hbar $, dividing $ 2m $ by $ \hbar^2 $, taking the cube root, and multiplying by $ \sqrt{E} $.

Step 5: Multiply the result from Step 4 by $ \frac{V}{2\pi^2} $ to find $ g(E) $. Ensure the final units are in states per electron volt by verifying the dimensional consistency of the formula.

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Key Concepts

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

Density of States

The density of states (g(E)) is a crucial concept in solid-state physics that quantifies the number of electronic states available at a given energy level per unit volume. It helps in understanding how many electrons can occupy specific energy levels in a material, which is essential for calculating various properties of metals and semiconductors.

Free-Electron Model

The free-electron model is a simplified representation of electrons in a metal, treating them as non-interacting particles in a uniform potential. This model assumes that electrons can move freely throughout the metal, allowing for the derivation of important properties such as conductivity and the density of states, which are foundational for understanding metallic behavior.

Energy and Volume Relationship

In the context of the density of states, the relationship between energy (E) and volume (V) is significant. The density of states is often expressed in terms of energy levels per unit volume, and knowing the volume of the system allows for the calculation of how many states are available at a specific energy, which is critical for determining electronic properties in materials.

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