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Chapter 11: Problem 9

In Debye's model of a solid, the maximum frequency \(v_{m}\) corresponds to a minimum wavelength. Because of the discrete nature of a solid this minimum wavelength corresponds to a vibration in which adjacent atoms move \(180^{\circ}\) out of phase with one another; that is, the interatomic spacing is half a wavelength. Is this plausible? Explain.

Short Answer

Expert verified
Yes, this is plausible. In Debye's model of a solid, the maximum frequency, which corresponds to the minimum wavelength, is associated with adjacent atoms oscillating 180 degrees out of phase. This means that the distance between the atoms is half of the wavelength, as when one atom is at the maximum displacement, the other is at its minimum.

Step by step solution

01

Understanding Debye's Model and Wave Properties

In Debye's model, solids are represented as a collection of atoms that can vibrate as waves. Waves are characterized by several properties, including their frequency and wavelength. It is important to understand that wavelength and frequency are inversely related; high frequency corresponds to short wavelength and vice versa.
02

Atomic vibrations within the Solid

In a solid, atoms vibrate about their equilibrium positions. These vibrations can be seen as waves, with each atom serving as a point on the wave. The distance between the atoms corresponds to half the wavelength of the wave; this is due to the nature of atomic vibrations in solids
03

Understanding Phase Difference

Phase difference refers to the difference in the positions of two waves at a given moment. Here, a phase difference of \(180^{\circ}\) between adjacent atoms means that when one atom is at its maximum displacement, the adjacent atom is at its minimum. This yields a resultant wave where the wavelength is twice the interatomic distance, corresponding with the notion of maximum frequency and minimum wavelength in Debye's model

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atomic Vibrations
In the context of Debye's model, atomic vibrations are fundamental to understanding how solids behave. Imagine the atoms in a solid as tiny balls connected by springs. These tiny balls are constantly moving back and forth, even in a seemingly stable solid.
Each atom moves around its equilibrium position, creating waves of vibrations throughout the solid. These waves can vary in how fast they move or how far the atoms are displaced.
The fascinating part is that these vibrations are not random. They follow a specific pattern that forms waves. This is crucial in predicting the behavior of solids, especially at different temperatures.
  • Atoms in a solid vibrate around their fixed positions.
  • These vibrations are similar to waves.
  • Atomic vibrations are vital for understanding thermal properties of solids.
Phase Difference
Phase difference is a key concept when exploring wave interactions in a solid. It describes how one wave is shifted with respect to another. In Debye's model, this is particularly important as it explains how adjacent atoms interact.
A phase difference of \(180^{\circ}\) means the two vibrations are exactly out of sync. When one atom moves up, the next atom moves down. Their movements are completely opposite, resembling a seesaw motion.
This specific phase difference leads to notable outcomes in wave behavior. Each wave seems to cancel out its neighbor partially, affecting properties like energy transmission in the solid.
  • Phase difference describes wave shifts between two atoms.
  • \(180^{\circ}\) phase difference means opposite vibrations.
  • This interaction is crucial for understanding energy distribution.
Wavelength and Frequency
Wavelength and frequency are central to understanding waves in solids. They are inversely related, meaning as one increases, the other decreases. This relationship helps explain fundamental properties of solids in Debye's model.
Wavelength refers to the distance between two consecutive points in phase on a wave, like crest to crest. Frequency measures how many waves pass a point per second. In this model, the maximum frequency corresponds to the smallest possible wavelength.
When waves have high frequency, they have a small wavelength. This situation occurs when adjacent atoms vibrate out of phase by \(180^{\circ}\), resulting in a tight, energetic wave pattern.
  • Wavelength is the distance between repeating parts of a wave.
  • Frequency is how often these waves pass a point in time.
  • High frequency means a short wavelength, reflecting intense vibrations.

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Most popular questions from this chapter

Consider the relation \(n_{1} / n_{2}=e^{\left(\mathcal{E}_{2}-\mathscr{E}_{1}\right) / k T}\), the Boltzmann factor for nondegenerate states for systems in equilibrium, where \(\mathscr{E}_{2}>\mathscr{E}_{1}\). (a) Show that \(n_{2}=0\) at \(T=0\). (b) Show that \(n_{1}=n_{2}\) at \(T=\infty\) or \(T=-\infty\). (c) Show that \(n_{2}>n_{1}\) at finite negative temperature \(T\). (d) Show that \(n_{1} \rightarrow 0\) as \(T \rightarrow-0\). (e) Hence, explain the statements, "Negative absolute temperatures are not colder than absolute zero but hotter than infinite temperature," and "One approaches negative temperatures through infinity, not through zero." (f) Can you suggest a change in temperature scale that would avoid temperatures that are negative in this sense?

What examples of a Fermi gas are there other than an electron gas and a gas of \(\mathrm{He}^{3}\) atoms?

Is there a connection between \(V_{0}\), the depth of the potential well for conduction electrons in a metal, and electron diffraction experiments of the Davisson-Germer type? Can we determine \(V_{0}\) from such experiments?

How can the ordered state of the He II explain its lack of resistance to heat conduction?

(a) Determine the order of magnitude of the fraction of hydrogen atoms in a state with principle quantum number \(n=2\) to those in state \(n=1\) in a gas at \(300^{\circ} \mathrm{K}\). (b) Take into account the degeneracy of the states corresponding to quantum numbers \(n=1\) and 2 of atomic hydrogen and determine at what temperature approximately one atom in a hundred is in a state with \(n=2\)
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