We then have \[\begin{split} n&=\!\int\limits_{-\infty}^\infty\!\!\!d\ve\>g\ns_\uar(\ve)\,f(\ve-\mu) \> + \int\limits_{-\infty}^\infty\!\!\!d\ve\>g\ns_\dar(\ve)\,f(\ve-\mu) \\ &=\half\!\int\limits_{-\...We then have \[\begin{split} n&=\!\int\limits_{-\infty}^\infty\!\!\!d\ve\>g\ns_\uar(\ve)\,f(\ve-\mu) \> + \int\limits_{-\infty}^\infty\!\!\!d\ve\>g\ns_\dar(\ve)\,f(\ve-\mu) \\ &=\half\!\int\limits_{-\infty}^\infty\!\!\!d\ve\>\Big\{ g(\ve-\mutB H) + g(\ve+\mutB H)\Big\}\,f(\ve-\mu)\\ &=\!\int\limits_{-\infty}^\infty\!\!\!d\ve\>\Big\{g(\ve) + (\mutB H)^2\,g''(\ve) + \ldots\Big\}\, f(\ve-\mu)\ . \end{split}\] We now invoke the Sommerfeld expension to find the temperature dependence: \[\begin{split…
