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Display information for equation id:math.2555.63 on revision:2555

* Page found: Das ideale Fermigas (eq math.2555.63)

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Das ideale Fermigas Eq: math.2555.63

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\begin{align}

& \bar{N}=\frac{2}{3}\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2m\mu  \right)}^{\frac{3}{2}}}\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right]=\frac{2}{3}\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2m{{E}_{F}} \right)}^{\frac{3}{2}}} \\

& \Rightarrow {{\left( \mu  \right)}^{\frac{3}{2}}}\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right]\approx {{\left( {{E}_{F}} \right)}^{\frac{3}{2}}} \\

& \Rightarrow \mu \approx {{E}_{F}}{{\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right]}^{-\frac{2}{3}}} \\

\end{align}

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\begin{aligned} \bar{N} = \frac{2}{3} \frac{(2 s + 1)}{2} (\frac{V}{h^{3}}) 4 π {(2 m μ)}^{\frac{3}{2}} [1 + \frac{π^{2}}{8} {(\frac{k T}{μ})}^{2}] = \frac{2}{3} \frac{(2 s + 1)}{2} (\frac{V}{h^{3}}) 4 π {(2 m E_{F})}^{\frac{3}{2}} \\ \Rightarrow {(μ)}^{\frac{3}{2}} [1 + \frac{π^{2}}{8} {(\frac{k T}{μ})}^{2}] \approx {(E_{F})}^{\frac{3}{2}} \\ \Rightarrow μ \approx E_{F} {[1 + \frac{π^{2}}{8} {(\frac{k T}{μ})}^{2}]}^{- \frac{2}{3}} \end{aligned}

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