Free electron model (Sommerfeld model)

[steroidjunkie]

Homework Statement

Using free electron model find the number of electron quantum states per unit volume in $[\varepsilon_F, \varepsilon_F + \Delta \varepsilon]$ energy interval of sodium. Fermi energy of sodium is $\varepsilon_F = 3.22 eV$, and energy band width is $\Delta \varepsilon=0.02 eV$.

Homework Equations

$N=2V \cdot \frac{2 \pi}{h^3}(2m)^{\frac{3}{2}} E^{\frac{1}{2}} dE$ - number of electrons in $[\varepsilon_F, \varepsilon_F + \Delta \varepsilon]$
$E_{TOT}=\int_{\varepsilon_F}^{\varepsilon_F + \Delta \varepsilon} \frac{4 \pi V}{h^3}(2m)^{\frac{3}{2}} E^{\frac{1}{2}} E dE$

The Attempt at a Solution

[/B]
$E_{TOT}=\frac{4 \pi V 2m^{\frac{3}{2}}}{h^3} \cdot \frac{2}{5}( E_{F+\Delta \varepsilon}^{\frac{5}{2}} - E_{F}^{\frac{5}{2}})$
$E_{TOT}= \frac{4 \pi V 2m^{\frac{3}{2}}}{h^3} \cdot \frac{2}{5}(3.24^{\frac{5}{2}} - 3.22^{\frac{5}{2}})$
$E_{TOT}= \frac{4 \pi V 2m^{\frac{3}{2}}}{h^3} \cdot 0.12$

I don't know if this makes any sense so some input would be great.

 

[TSny]

Note that the question asks for the number of electron states that have energies between $\varepsilon_F$ and $\varepsilon_F + \Delta \varepsilon$ for a unit volume of sodium assuming the free electron model. It does not ask for $E_{TOT}$ for this energy range.

$N=2V \cdot \frac{2 \pi}{h^3}(2m)^{\frac{3}{2}} E^{\frac{1}{2}} dE$ - number of electrons in $[\varepsilon_F, \varepsilon_F + \Delta \varepsilon]$

This equation will give you the answer directly if you consider $\Delta \varepsilon$ to be small enough that you can approximate $dE$ as $\Delta \varepsilon$. No integration is needed.