Isbell conjugacy: Difference between revisions

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{{underlinked|date=November 2012}}
}}
 
'''Electron-longitudinal acoustic phonon interaction''' is an equation concerning [[atom]]s.
 
==Displacement operator of the longitudinal acoustic phonon==
 
The equation of motions of the atoms of mass M which locates in the periodic lattice is
 
: <math>M \frac {d^{2}} {dt^{2}} u_{n} = -k_{0} ( u_{n-1} + u_{n+1} -2u_{n} )</math>,
 
where <math>u_{n}</math> is the displacement of the nth atom from their equilibrium positions.
 
If we define the displacement <math>u_{l}</math> of the nth atom by <math>u_{l}= x_{l} - la</math>, where <math>x_{l}</math> is the coordinates of the lth atom and a is the lattice size,
 
the displacement is given by <math>u_{n}= A e^{i q l a - \omega t}</math>
 
Using Fourier transform, we can define
 
: <math>Q_{q} = \frac {1} {\sqrt {N}} \sum_{l} u_{l} e^{- i q a l } </math>
 
and
 
: <math>u_{l} = \frac {1} {\sqrt {N}} \sum_{q} Q_{q} e^{ i q a l }</math>.
 
Since <math>u_{l}</math> is a Hermite operator,
 
: <math>u_{l} = \frac {1} {2 \sqrt{N}} \sum_{q} (Q_{q} e^{iqal} + Q^{\dagger}_{q} e^{-iqal} )</math>
 
From the definition of the creation and annihilation operator    <math>a^{\dagger}_{q} = \frac {q} {\sqrt{2M\hbar\omega_{q}}}(M\omega_{q}Q_{-q}-iP_{q}), \; a_{q} = \frac {q} {\sqrt{2M\hbar\omega_{q}}}(M\omega_{q}Q_{-q}+iP_{q})</math>
 
: <math>Q_{q}</math> is written as
 
: <math>Q_{q} = \sqrt { \frac {\hbar} {2M\omega_{q}}}(a^{\dagger}_{-q}+a_{q})</math>
 
Then <math>u_{l}</math> expressed as
 
: <math>u_{l} = \sum_{q} \sqrt {\frac {\hbar} {2MN\omega_{q}}} (a_{q} e^{iqal} + a^{\dagger}_{q} e^{-iqal})</math>
 
Hence, when we use continuum model, the displacement for the 3-dimensional case is
 
: <math>u(r) = \sum_{q} \sqrt{ \frac {\hbar}{2M N \omega_{q} } } e_{q} [ a_{q} e^{ i q \cdot r} + a^{\dagger}_{q} e^{-i q \cdot r}  ] </math>,
 
where <math>e_{q}</math> is the unit vector along the displacement direction.
 
==Interaction Hamiltonian==
 
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as '''<math>H_{el}</math>'''
 
: <math>H_{el} = D_{ac} \frac{\delta V}{V} = D_{ac} \, div \, u(r)</math>,
 
where <math>D_{ac} </math> is the deformation potential for electron scattering by acoustic phonons.<ref>Hamaguchi 2001, p. 208.</ref>
 
Inserting the displacement vector to the Hamiltonian results to
 
: <math>H_{el} = D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } ( i e_{q} \cdot q ) [ a_{q} e^{i q \cdot r} - a^{\dagger}_{q} e^{-i q \cdot r} ]</math>
 
==Scattering probability==
 
The scattering probability for electrons from <math>|k \rangle </math> to <math>|k' \rangle</math> states is
 
: <math>P(k,k') = \frac {2 \pi} {\hbar} \mid \langle k' , q' | H_{el}| \ k , q \rangle \mid ^ {2} \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ]</math>
 
: <math>= \frac {2 \pi} {\hbar} \left| D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } ( i e_{q} \cdot q ) \sqrt { n_{q} + \frac {1} {2} \mp \frac {1} {2} } \, \frac {1} {L^{3}} \int d^{3} r \, u^{\ast}_{k'} (r) u_{k} (r) e^{i ( k - k' \pm q ) \cdot r } \right|^2 \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ] </math>
 
Replace the integral over the whole space with a summation of unit cell integrations
 
: <math>P(k,k') = \frac {2 \pi} {\hbar} \left( D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } | q | \sqrt { n_{q} + \frac {1} {2} \mp \frac {1} {2} } \, I(k,k') \delta_{k' , k \pm q } \right)^2 \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ],</math>
 
where <math>I(k,k') = \Omega \int_{\Omega} d^{3}r \, u^{\ast}_{k'} (r) u_{k} (r) </math>,  '''<math> \Omega </math>''' is the volume of a unit cell.
 
: <math>P(k,k') = \begin{cases}
\frac {2 \pi} {\hbar} D_{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 n_{q} & (k' = k + q ; \text{absorption}), \\
\frac {2 \pi} {\hbar} D_{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 ( n_{q} + 1 ) & (k' = k - q ; \text{emission}).
\end{cases}
</math>
 
== Notes ==
{{Reflist|2}}
 
==References==
{{More footnotes|date=June 2009}}
*{{cite book | author=C. Hamaguchi | title=Basic Semiconductor Physics | publisher=Springer | year=2001 | pages= 183&ndash;239}}
 
*{{cite book | author=Yu, Peter Y. and Cardona, Manuel | title=Fundamentals of Semiconductors | edition = 3rd | publisher=Springer | year=2005}}
 
{{DEFAULTSORT:Electron-Longitudinal Acoustic Phonon Interaction}}
[[Category:Atomic physics]]

Latest revision as of 22:58, 17 November 2014

They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. This one is one of the most beneficial features of Word - Press as this feature allows users to define the user roles. This CMS has great flexibility to adapt various extensions and add-ons. After confirming the account, login with your username and password at Ad - Mob. If you have any type of inquiries regarding where and how you can make use of wordpress backup plugin, you could call us at the web site. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme.

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But before choosing any one of these, let's compare between the two. s cutthroat competition prevailing in the online space won. We can active Akismet from wp-admin > Plugins > Installed Plugins. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business.

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