Solution to Set 5: Difference between revisions

From PhyWiki
Jump to navigation Jump to search
Line 13: Line 13:


'''Derive the dispersion relation <math>\omega^{\alpha} (k)\;</math> for this chain'''
'''Derive the dispersion relation <math>\omega^{\alpha} (k)\;</math> for this chain'''
* Index <math>\alpha = 1\;</math> for acoustic branch
 
* Index<math>\alpha = 2\;</math> for optical branch
* '''Index <math>\alpha = 1\;</math> for acoustic branch'''


Potential Energy <math>U = 1, 2, 3, ... n \;</math>
Potential Energy <math>U = 1, 2, 3, ... n \;</math>
Line 20: Line 20:
<math>U \cong \frac{1}{2} k \sum_{n} (U_n - U_{n-1}) \;</math>
<math>U \cong \frac{1}{2} k \sum_{n} (U_n - U_{n-1}) \;</math>


<math>\Rightarrow m \ddot{U}_n = - k [2U_n - U_{n-1} - U_{n+1}] \;</math>
<math>\Rightarrow m \ddot{U}_n = - k [2U_n - U_{n-1} - U_{n+1}] = -m \omega U(t) \;</math>


Eigenvector Value for modes
Eigenvectors of Modes A and B (defined arbitrarily)


<math>u_m(t)=e^{i\omega t} u_m \;</math>
<math>u_m(t)=e^{i\omega t} u_m \;</math>


<font color="blue">What does the subscript m stand for? Modes? What are modes? - [[User:KimberlyWynne|KimberlyWynne]] 03:36, 2 March 2009 (EST)</font>
<math>\Rightarrow -m \omega ^2 \ddot{\vec{u}}_n = - k \mathbf{M} \vec{u} \;</math>


<math>\Rightarrow -m \omega ^2 \vec{u} = - k \mathbf{M} \vec{u} \;</math>
Band Matrix


Band Matrix
<math>
<math>
\mathbf{M} = \begin{vmatrix}
\mathbf{M} = \begin{vmatrix}
Line 40: Line 39:
</math>
</math>


<math>u_m (t) = e^{ik(na)- \omega t} \;</math> where <math>u_m \rightarrow  u_n e^{ik(na)} \;</math>
<math>u_m (t) = e^{ik(na)- \omega t} = e^{i k R_n} \;</math>  
 
where  
 
<math>u_m \rightarrow  u_n e^{ik(na)} \;</math>
 
<math>R_n \;</math> = distance on some coordinate system


<math>\Rightarrow -m \omega ^2 \vec{u} = - k [2 - e^{ik \alpha} - e^{-ik \alpha}] \;</math>
<math>\Rightarrow -m \omega ^2 \vec{u} = - k [2 - e^{ik \alpha} - e^{-ik \alpha}] \;</math>
Line 47: Line 52:


<math>\Rightarrow \omega (k) = 2 \sqrt{ \frac{k}{m} } |sin(ka)|</math>
<math>\Rightarrow \omega (k) = 2 \sqrt{ \frac{k}{m} } |sin(ka)|</math>
* '''Index<math>\alpha = 2\;</math> for optical branch'''
<math>u(R_n) \equiv e^{i k R_n} = cos (k a) \;<>
<math>u_m (t) = e^{ik(na)- \omega t} = e^{i k R_n} \;</math>
<math>\Rightarrow - m \omega^2 e^{ikna} = -k [2e^{ik(na)} - e^{ik(n+1)a} - e^{ik(n-1)a}] \;</math>
<math>\Rightarrow m \omega^2 = k [2 - (e^{ika}  + e^{-ika}] \;</math>
<math>\Rightarrow \omega^2 = \frac{2k}{m} [1 - cos(ka)] =  \;</math>
[[Image:Dispersionrelation.jpg]]
[[Image:Dispersionrelation.jpg]]


Revision as of 05:43, 2 March 2009

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

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 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

  • Index 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 \alpha = 1\;} for acoustic branch

Potential Energy 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 = 1, 2, 3, ... 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 \cong \frac{1}{2} k \sum_{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 \Rightarrow m \ddot{U}_n = - k [2U_n - U_{n-1} - U_{n+1}] = -m \omega U(t) \;}

Eigenvectors of Modes A and B (defined arbitrarily)

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_m(t)=e^{i\omega t} u_m \;}

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 \Rightarrow -m \omega ^2 \ddot{\vec{u}}_n = - k \mathbf{M} \vec{u} \;}

Band Matrix

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 \mathbf{M} = \begin{vmatrix} 2 & 1 & 0 & 0\\ 1 & 2 & 1 & 0\\ 0 & 1 & 2 & 1\\ 0 & 0 & 1 & 2 \end{vmatrix} }

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_m (t) = e^{ik(na)- \omega t} = e^{i k R_n} \;}

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 u_m \rightarrow u_n e^{ik(na)} \;}

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 R_n \;} = distance on some coordinate system

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 \Rightarrow -m \omega ^2 \vec{u} = - k [2 - e^{ik \alpha} - e^{-ik \alpha}] \;}

Derive and get:

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 \Rightarrow \omega (k) = 2 \sqrt{ \frac{k}{m} } |sin(ka)|}

  • IndexFailed 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 \alpha = 2\;} for 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 u(R_n) \equiv e^{i k R_n} = cos (k a) \;<> <math>u_m (t) = e^{ik(na)- \omega t} = e^{i k R_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 \Rightarrow - m \omega^2 e^{ikna} = -k [2e^{ik(na)} - e^{ik(n+1)a} - e^{ik(n-1)a}] \;}

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 \Rightarrow m \omega^2 = k [2 - (e^{ika} + e^{-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 \Rightarrow \omega^2 = \frac{2k}{m} [1 - cos(ka)] = \;} 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?

From my lecture notes: 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 C_{\alpha } } = 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.

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.

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}} (Hint: note the analogy with Wien's Law!)

Retrieved from "http://wiki.physics.fsu.edu/index.php?title=Solution_to_Set_5&oldid=2568"