Solution to Set 5: Difference between revisions

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===Debye Temperature <math>T_D \;</math>===
===Debye Temperature <math>T_D \;</math>===
The Debye temperature, aka the effective sonic velocity, is a measure of the hardness of the crystal
The Debye temperature, aka the effective sonic velocity, is a measure of the hardness of the crystal
From our class lectures:
<math>k_B T_D = \hbar \omega_D = \hbar c k_D \;</math>
From Wikipedia:


<math>T_D\ \stackrel{\mathrm{def}}{=}\  {hc_sR\over2Lk} = {hc_s\over2Lk}\sqrt[3]{6N\over\pi} = {hc_s\over2k}\sqrt[3]{{6\over\pi}{N\over V}}</math>
<math>T_D\ \stackrel{\mathrm{def}}{=}\  {hc_sR\over2Lk} = {hc_s\over2Lk}\sqrt[3]{6N\over\pi} = {hc_s\over2k}\sqrt[3]{{6\over\pi}{N\over V}}</math>

Revision as of 07:00, 2 March 2009

I have no idea what I'm doing - KimberlyWynne 03:11, 2 March 2009 (EST)

I found this site somewhat helpful and explanatory:

http://newton.ex.ac.uk/teaching/resources/rjh/phy2009/PHY2009handout13.pdf

Diatomic harmonic chain

Problem 1

Given:

  • a chain of atoms
  • with alternating masses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1\;} and
  • connected with elastic springs with constant
  • moving only in the x-direction

Chainatoms.jpg

Derive the dispersion relation for this chain

Index for acoustic branch

Potential Energy

Eigenvectors of Modes A and B (defined arbitrarily)

Band Matrix

Running waves through a solid

where

= distance on some coordinate system

Derive and get:

Index for optical branch

Dispersionrelation.jpg

Problem 2

Determine the speed of sound for this chain. What is the lowest frequency of long-wavelength sound corresponding to the optical branch?

where

  • = frequency
  • = speed of sound
  • = ???

Problem 3

Sketch the motion of the atoms corresponding to the edge of the Brillouin zone, both for the optical and the acoustic branch.

Dispersionbranches.jpg

  • Acoustic Branch: lower branch
  • Optical Branch: upper branch, as on this branch the vibrations of the 2 types of atom are in antiphase and the resulting charge oscillation in an ionic craystal give a strong coupling to electromagnetic waves at the frequency of point A.

Problem 4

Determine the Debye temperature for this system, and determine the form of the specific heat in the limits of high and low temperatures.

Debye Temperature

The Debye temperature, aka the effective sonic velocity, is a measure of the hardness of the crystal

From our class lectures:

From Wikipedia:

Specific Heat

Low Temperature Limit

High Temperature Limit

Net Result (Classical Limit)

Problem 5

Consider low temperatures () and determine the wavelength of the most abundant phonons Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_{max}} (Hint: note the analogy with Wien's Law!)