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Bose gas specific heat above condensation temperature

Posted by peeterjoot on May 9, 2013

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Question: Bose gas specific heat above condensation temperature ([1] section 7.1.37)

Equation 7.1.33 provides a relation for specific heat

\begin{aligned}\frac{C_{\mathrm{V}}}{N k_{\mathrm{B}}} = \left(\frac{\partial {}}{\partial {T}}\left( \frac{3}{2} T \frac{ g_{5/2}(z) } { g_{3/2}(z) } \right)\right)_v.\end{aligned} \hspace{\stretch{1}}(1.0.1)

Fill in the details showing how this can be used to find

\begin{aligned}\frac{C_{\mathrm{V}}}{N k_{\mathrm{B}}} = \frac{15}{4} \frac{ g_{5/2}(z) }{ g_{3/2}(z) }-\frac{9}{4} \frac{ g_{3/2}(z) }{ g_{1/2}(z) }.\end{aligned} \hspace{\stretch{1}}(1.0.2)

With

\begin{aligned}g_{{3/2}}(z) = \frac{\lambda^3}{v} = \frac{h^3}{\left( 2 \pi m k_{\mathrm{B}} T \right)^{3/2}}\end{aligned} \hspace{\stretch{1}}(1.0.3)

we have for constant $v$

\begin{aligned}\left({\partial {g_{3/2}}}/{\partial {T}}\right)_{{v}}= -\frac{3}{2}\frac{h^3}{\left( 2 \pi m k_{\mathrm{B}} \right)^{3/2} T^{5/2}}= -\frac{3}{2 T} g_{{3/2}}(z).\end{aligned} \hspace{\stretch{1}}(1.0.3)

From the series expansion

\begin{aligned}g_{{\nu}}(z) = \sum_{k = 1}^\infty \frac{z^k}{k^\nu},\end{aligned} \hspace{\stretch{1}}(1.0.5)

we have

\begin{aligned}z \frac{\partial {}}{\partial {z}} g_{{\nu}}(z) = z\sum_{k = 1}^\infty k \frac{z^{k-1}}{k^\nu}=\sum_{k = 1}^\infty \frac{z^{k}}{k^{\nu-1}}= g_{{\nu-1}}(z).\end{aligned} \hspace{\stretch{1}}(1.0.5)

Taken together we have

\begin{aligned}-\frac{3}{2 T} g_{{3/2}}(z) &=\left({\partial {g_{3/2}}}/{\partial {T}}\right)_{{v}} \\ &=\left({\partial {z}}/{\partial {T}}\right)_{{v}}\frac{\partial {}}{\partial {z}} g_{{3/2}}(z) \\ &=\frac{1}{{z}} \left({\partial {z}}/{\partial {T}}\right)_{{v}}z \frac{\partial {}}{\partial {z}} g_{{3/2}}(z) \\ &=\frac{1}{{z}} \left({\partial {z}}/{\partial {T}}\right)_{{v}}g_{{1/2}}(z),\end{aligned} \hspace{\stretch{1}}(1.0.5)

or

\begin{aligned}\frac{1}{{z}} \left({\partial {z}}/{\partial {T}}\right)_{{v}} = -\frac{3}{2 T} \frac{g_{{3/2}}(z)}{g_{{1/2}}(z)}.\end{aligned} \hspace{\stretch{1}}(1.0.5)

We are now ready to evaluate the derivative and find the specific heat

\begin{aligned}\frac{C_{\mathrm{V}}}{N k_{\mathrm{B}}} &= \left(\frac{\partial {}}{\partial {T}}\left( \frac{3}{2} T \frac{ g_{5/2}(z) } { g_{3/2}(z) } \right)\right)_v \\ &=\frac{3}{2} \frac{ g_{5/2}(z) }{ g_{3/2}(z) }+\frac{3 T}{2} \left({\partial {z}}/{\partial {T}}\right)_{{v}}\frac{\partial {}}{\partial {z}}\left( \frac{ g_{5/2}(z) } { g_{3/2}(z) } \right) \\ &=\frac{3}{2} \frac{ g_{5/2}(z) }{ g_{3/2}(z) }-\frac{9 T}{4} \frac{g_{{3/2}}(z)}{g_{{1/2}}(z)}z\frac{\partial {}}{\partial {z}}\left( \frac{ g_{5/2}(z) } { g_{3/2}(z) } \right) \\ &=\frac{3}{2} \frac{ g_{5/2}(z) }{ g_{3/2}(z) }-\frac{9 }{4} \frac{g_{{3/2}}(z)}{g_{{1/2}}(z)}\not{{\frac{ g_{3/2}(z) }{ g_{3/2}(z) }}}+\frac{9 }{4} \frac{\not{{g_{{3/2}}(z)}}}{\not{{g_{{1/2}}(z)}}}\frac{ g_{5/2}(z) \not{{g_{1/2}(z)}}}{ \left( g_{3/2}(z) \right)^{\not{{2}}} } \\ &=\frac{3}{2} \frac{ g_{5/2}(z) }{ g_{3/2}(z) }-\frac{9 }{4} \frac{g_{{3/2}}(z)}{g_{{1/2}}(z)}+\frac{9 }{4} \frac{ g_{5/2}(z) }{ g_{3/2}(z) } \\ &=\frac{15}{4} \frac{ g_{5/2}(z) }{ g_{3/2}(z) }-\frac{9 }{4} \frac{g_{{3/2}}(z)}{g_{{1/2}}(z)}.\end{aligned} \hspace{\stretch{1}}(1.0.5)

This is the desired result.

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.