Solution to Set 5: Difference between revisions

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<math>(2k-m_1\omega^2)(2k-m_2\omega^2)- k(1)(-k) =0 \;</math>
<math>(2k-m_1\omega^2)(2k-m_2\omega^2)- k(1)(-k) = 0 \;</math>


<math>(4k^2 - 2k(m_1+m_2)\omega^2 + m_1m_2\omega ^2) + k^2 = 0 \;</math>
<math>(m \omega^2 + 2k^2)^2 = k^2 \;</math>
<math>-m \omega^2 + 2k = k \;</math>
<math> \omega = \sqrt{\frac{k}{m}} \;</math>
[[Image:Dispersionrelation.jpg]]
[[Image:Dispersionrelation.jpg]]


Revision as of 17:54, 2 March 2009

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

Diatomic harmonic chain

Problem 1

I found this site somewhat helpful and explanatory:

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


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 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_2\;}
  • connected with elastic springs with constant 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 K\;}
  • moving only in the x-direction

Chainatoms.jpg

Derive the dispersion relation 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 \omega^{\alpha} (k)\;} for this chain

Equations of motion

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 \ddot{u}_n = - k_1 [2u_n - v_{n} - v_{n-1}] \;}

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_2 \ddot{v}_n = - k_2 [2v_n - u_{n} - u_{n+1}] \;}

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 u_n = u \mathrm{exp}(i(kna-\omega t))\;}

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 v_n = v \mathrm{exp}(i(kna-\omega t))\;}

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 -\omega^2 m u \mathrm{exp}(i(kna-\omega t)) = k (v\mathrm{exp}(i(kna-\omega t)) + v\mathrm{exp}(i(k(n-1)a-\omega t)) - 2 u \mathrm{exp}(i(kna-\omega t))) \;}

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 -\omega^2 m_1 u = k (v + v\mathrm{exp}(-ika)) - 2 k u \;}

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 -\omega^2 m_2 v = k (u\mathrm{exp}(ika) + u) - 2 k v \;}

Set determinant to 0

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 \omega^2 = k(\frac{m_1+m_2}{m_1 m_2}) \pm k\sqrt{(\frac{m_1 + m_2}{m_1 m_2})^2 - \frac{4}{m_1 m_2} sin^2(\frac{ka}{2})]} \;}

Step 1:

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 \begin{vmatrix} 2k-m_1\omega^2 & -C[1-\mathrm{exp}(-ika)]\\ -C[1+\mathrm{exp}(-ika)] & 2k-m_2\omega^2 \end{vmatrix} =0 }

Step 2:

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 \begin{vmatrix} 2k-m_1\omega^2 & -1\\ -k & 2k-m_2\omega^2 \end{vmatrix} =0 \;}

Step 3:

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 (2k-m_1\omega^2)(2k-m_2\omega^2)- k(1)(-k) = 0 \;}

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 (4k^2 - 2k(m_1+m_2)\omega^2 + m_1m_2\omega ^2) + k^2 = 0 \;}

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 \omega^2 + 2k^2)^2 = k^2 \;}

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 \omega^2 + 2k = k \;}

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 \omega = \sqrt{\frac{k}{m}} \;} 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?

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 \omega _{\alpha }(k)\approx C_{\alpha } k \;}

where

  • 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 \omega_{\alpha } \;} = frequency
  • 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 C_{\alpha } \;} = speed of sound
  • 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 k \;} = ???

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 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 k \rightarrow 0 } 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 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 C_V (T)} in the limits of high and low temperatures.

Debye Temperature 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 T_D \;}

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

From our class lectures:

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 k_B T_D = \hbar \omega_D = \hbar c k_D \;}

From Wikipedia:

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 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}}}

Specific Heat 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 C_V \;}

Low Temperature Limit

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 \frac{C_V}{Nk} \sim {12\pi^4\over5} \left({T\over T_D}\right)^3}

High Temperature Limit

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 T_{high} >> \frac{\hbar\omega}{k_B} \;}

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 \frac{\hbar\omega}{k_B T} << 1 \;}

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 e^{\frac{\hbar\omega}{k_B T}} \approx 1 + \frac{\hbar\omega}{k_B T} + (\frac{\hbar\omega}{k_B T})^2 \;}

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 \frac{C_V}{Nk} \sim 3\, }

Net Result (Classical Limit)

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 C_V \approx k_B \;}

Problem 5

Consider low temperatures (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 T \ll T_D\;} ) 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}}

Given

  • Bose-Einstein Distribution 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 f(E)=\frac{1}{e^{E / k_B T}-1} \;}
    • 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 f(E) \;} = probability that a particle will have energy E
    • 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 k_B \;} = Boltzmann constant
    • 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 T \;} = Temperature
  • Planck's Radiation Formula 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 E = \frac{h \omega}{e^{h \omega / k_B T} - 1} \;}
    • Density by frequency: 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 \rho (\omega) = \frac{dn_s}{d\omega} = \frac{8 \pi}{c^3} \omega^{2} \;}
    • Density by wavelength: 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 \rho (\lambda) = \frac{dn_s}{d\lambda} = \frac{8 \pi}{\lambda^{4}} \;}
  • Wien's law 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_{\mathrm{max}} = \frac{b}{T} \;}
    • 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 \;} is the peak wavelength in meters,
    • 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 T \;} = temperature of the blackbody in Kelvin
    • 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 b \;} = Wien's displacement constant
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