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* Page found: Spezifische Wärme von Festkörpern (eq math.2577.9)

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Spezifische Wärme von Festkörpern Eq: math.2577.9

Hash: 6d114a6fb0ff594a1304a41165d960ac

TeX (original user input):

\begin{align}

& \sum\limits_{{\bar{q}}}^{{}}{{}}->\frac{V}{{{h}^{3}}}\int_{{}}^{{}}{{}}{{d}^{3}}\left( \hbar \bar{q} \right)=\frac{4\pi V}{{{\left( 2\pi  \right)}^{3}}}\int_{0}^{{{q}_{D}}}{{}}dq{{q}^{2}}=\frac{4\pi V}{{{\left( 2\pi  \right)}^{3}}}\left( \frac{1}{{{v}_{L}}^{3}}+\frac{2}{{{v}_{T}}^{3}} \right)\int_{0}^{{{\omega }_{D}}}{{}}d\omega {{\omega }^{2}} \\

& \left( \frac{1}{{{v}_{L}}^{3}}+\frac{2}{{{v}_{T}}^{3}} \right)\tilde{\ }\frac{3}{{{{\bar{v}}}^{3}}} \\

& \Rightarrow 3N=!=\frac{4\pi V}{{{\left( 2\pi  \right)}^{3}}}\left( \frac{1}{{{v}_{L}}^{3}}+\frac{2}{{{v}_{T}}^{3}} \right)\int_{0}^{{{\omega }_{D}}}{{}}d\omega {{\omega }^{2}}=\frac{4\pi V}{{{\left( 2\pi  \right)}^{3}}}\frac{3}{{{{\bar{v}}}^{3}}}\int_{0}^{{{\omega }_{D}}}{{}}d\omega {{\omega }^{2}}=\frac{4\pi V}{{{\left( 2\pi  \right)}^{3}}}\frac{{{\omega }_{D}}^{3}}{{{{\bar{v}}}^{3}}} \\

\end{align}

TeX (checked):

{\begin{aligned}&\sum \limits _{\bar {q}}^{}{}->{\frac {V}{{h}^{3}}}\int _{}^{}{}{{d}^{3}}\left(\hbar {\bar {q}}\right)={\frac {4\pi V}{{\left(2\pi \right)}^{3}}}\int _{0}^{{q}_{D}}{}dq{{q}^{2}}={\frac {4\pi V}{{\left(2\pi \right)}^{3}}}\left({\frac {1}{{{v}_{L}}^{3}}}+{\frac {2}{{{v}_{T}}^{3}}}\right)\int _{0}^{{\omega }_{D}}{}d\omega {{\omega }^{2}}\\&\left({\frac {1}{{{v}_{L}}^{3}}}+{\frac {2}{{{v}_{T}}^{3}}}\right){\tilde {\ }}{\frac {3}{{\bar {v}}^{3}}}\\&\Rightarrow 3N=!={\frac {4\pi V}{{\left(2\pi \right)}^{3}}}\left({\frac {1}{{{v}_{L}}^{3}}}+{\frac {2}{{{v}_{T}}^{3}}}\right)\int _{0}^{{\omega }_{D}}{}d\omega {{\omega }^{2}}={\frac {4\pi V}{{\left(2\pi \right)}^{3}}}{\frac {3}{{\bar {v}}^{3}}}\int _{0}^{{\omega }_{D}}{}d\omega {{\omega }^{2}}={\frac {4\pi V}{{\left(2\pi \right)}^{3}}}{\frac {{{\omega }_{D}}^{3}}{{\bar {v}}^{3}}}\\\end{aligned}}

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\begin{aligned} \sum_{\bar{q}}^{} - > \frac{V}{h^{3}} \int_{}^{} d^{3} (ℏ \bar{q}) = \frac{4 π V}{{(2 π)}^{3}} \int_{0}^{q_{D}} d q q^{2} = \frac{4 π V}{{(2 π)}^{3}} (\frac{1}{{v_{L}}^{3}} + \frac{2}{{v_{T}}^{3}}) \int_{0}^{ω_{D}} d ω ω^{2} \\ (\frac{1}{{v_{L}}^{3}} + \frac{2}{{v_{T}}^{3}}) \tilde{} \frac{3}{{\bar{v}}^{3}} \\ \Rightarrow 3 N =! = \frac{4 π V}{{(2 π)}^{3}} (\frac{1}{{v_{L}}^{3}} + \frac{2}{{v_{T}}^{3}}) \int_{0}^{ω_{D}} d ω ω^{2} = \frac{4 π V}{{(2 π)}^{3}} \frac{3}{{\bar{v}}^{3}} \int_{0}^{ω_{D}} d ω ω^{2} = \frac{4 π V}{{(2 π)}^{3}} \frac{{ω_{D}}^{3}}{{\bar{v}}^{3}} \end{aligned}

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Identifiers

$\bar{q}$

$V$

$h$

$\bar{q}$

$π$

$V$

$π$

$q_{D}$

$q$

$q$

$π$

$V$

$π$

$v_{L}$

$v_{T}$

$ω_{D}$

$ω$

$ω$

$v_{L}$

$v_{T}$

$\tilde{}$

$\bar{v}$

$N$

$π$

$V$

$π$

$v_{L}$

$v_{T}$

$ω_{D}$

$ω$

$ω$

$π$

$V$

$π$

$\bar{v}$

$ω_{D}$

$ω$

$ω$

$π$

$V$

$π$

$ω_{D}$

$\bar{v}$

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