Solution to Set 5

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_A\;} 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_B\;}
• 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 Q\;}
• moving only in the x-direction

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

When solving problem 1 one can arrive at the same solution in two different ways. Both methods will be solved for completeness of this solution.

Model 1

Model 1 involves having two separate displacement vectors so that the equations of motion may be solved independently. Let u be the displacement vector for type A atoms and v be the displacement vector for the B atoms. The equations of motion for any of these particles is given by:

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_A \ddot{u}_n = - Q [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_B \ddot{v}_n = - Q [2v_n - u_{n} - u_{n+1}] \;}

Then since this is a chain of harmonic oscillators one can assume a solution for u and v for a given n.

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 = C e^{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 = D e^{i(kna-\omega t)}\;}

These equations add the wavenumber k, the periodicity length a, and the 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 \omega} . Then substituting these solutions into the equations of motion one gets:

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_A C e^{i(kna-\omega t)} = Q (De^{i(kna-\omega t)} + De^{i(k(n-1)a-\omega t)} - 2 C e^{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_B D e^{i(kna-\omega t)} = Q (Ce^{i(kna-\omega t)} + Ce^{i(k(n+1)a-\omega t)} - 2 D e^{i(kna-\omega t)}) \;}

Then dividing both equations by 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^{i(kna-\omega t)}} one gets:

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_A C = Q (D + De^{-ika)} - 2 Q C \;}

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_B D = Q (Ce^{ika} + C - 2 D) \;}

Rearranging these equations one gets:

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} m_A\omega^2-2Q & Q+Qe^{-ika}\\ Q+Qe^{-ika} & m_B\omega^2-2Q \end{vmatrix} \begin{vmatrix} C \\ D \end{vmatrix} =0 }

Since we are concerned with the non-trivial solutions of these equations the determinant must be zero. Therefore one gets 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_A\omega^2-2Q)(m_B\omega^2-2Q) - Q^2\left(1+e^{-ika}\right)(1+e^{ika}) = 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_Am_B\omega^4 - 2Q\left(m_A + m_B\right)\omega^2 + 4Q^2 - 2Q^2 - Q^2e^{-ika} - Q^2e^{ika}}

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_Am_B\omega^4 - 2Q\left(m_A + m_B\right)\omega^2 + 2Q^2 - Q^2e^{-ika} - Q^2e^{ika}}

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_Am_B\omega^4 - 2Q(m_A + m_B)\omega^2 + 2Q^2 - 2Q^2cos^2\left({ka \over 2}\right)}

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_Am_B\omega^4 - 2Q(m_A + m_B)\omega^2 + 2Q^2sin^2\left({ka \over 2}\right)}

Then one needs to solve this quadratic equation for omega squared.

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 = \frac{2Q(m_A+m_B)\pm\sqrt{4Q^2(m_A+m_B)^2-4m_Am_B(4Q^2sin^2(ka/2))} }{2m_1m_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 \omega^2 = Q({1 \over m_B} + {1 \over m_A}) \pm Q\sqrt{( {1 \over m_B}+ {1 \over m_A})^2 - {4 \over m_A m_B} sin^2({ka \over 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 \omega^2 = \frac{k(m_1+m_2) \pm k(m_1-m_2)}{m_1m_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 \omega^2 = \frac{2k}{m_1} or \frac{2k}{m_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 \omega = \sqrt{\frac{2k}{m_1}} or \sqrt{\frac{2k}{m_2}} \;}

Problem 2

Determine the speed of sound for this chain.

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

Therefore the 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 C_{\alpha} = \frac{\omega}{k} = a \sqrt{\frac{k}{2(m_1+m_2)}} \;}

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_{low} = \sqrt{\frac{2k}{m}}}

Problem 3

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

• 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: $\rho (\lambda )={\frac {dn_{s}}{d\lambda }}={\frac {8\pi }{\lambda ^{4}}}\;$

• Wien's law $\lambda _{\mathrm {max} }={\frac {b}{T}}\;$
  • $\lambda _{m}ax\;$ is the peak wavelength in meters,
  • $T\;$ = temperature of the blackbody in Kelvin
  • $b\;$ = Wien's displacement constant

Planck's Radiation Formula

$S_{\lambda }={\frac {8\pi hc}{\lambda ^{3}}}{\frac {1}{e^{\frac {hc}{\lambda kt}}-1}}\;$

Take the derivative and set = 0 to find the maximum

${\frac {\partial S_{\lambda }}{\partial \lambda }}=-{\frac {(3kt\lambda -ch)e^{\frac {ch}{kt\lambda }}-3kt\lambda }{kt\lambda ^{5}e^{\frac {2ch}{kt\lambda }}-2kt\lambda ^{5}e^{\frac {2ch}{kt\lambda }}+kt\lambda ^{5}}}\;$

$0=-{\frac {(3kt\lambda -ch)e^{\frac {ch}{kt\lambda }}-3kt\lambda }{kt\lambda ^{5}e^{\frac {2ch}{kt\lambda }}-2kt\lambda ^{5}e^{\frac {2ch}{kt\lambda }}+kt\lambda ^{5}}}\;$

$0=-(3kt\lambda -ch)e^{\frac {ch}{kt\lambda }}-3kt\lambda \;$

$\lambda _{max}T=2.898\times 10^{-3}m\cdot K\;$