Question: # of particles within an energy range below Fermi Energy.

[Lil Dansn Man]

Homework Statement

Here's what the question says.

"Consider a cube of gold 1 mm on an edge. calculate the approximate number of conduction electrons in this cube whose energies lie in the range from 4.000 to 4.025 at 300k. Assume Ef(300K) = Ef(0)."

Homework Equations

Well, I know that n(E)dE = g(E)Ffd(E)dE, where g(E) is the density of states, and Ffd is the Fermi-Dirac probability.

Also, N/V = 0-inf∫ n(E)dE

and the Fermi Energy for gold is 5.53 ev

The Attempt at a Solution

Finding the Volume is easy, which is just V = (1e-3)^3

Then, I try to find g(E), which is g(E)dE = DE^(1/2)dE, where D =$\frac{8*sqrt(2)*pi*Me^(3/2)}{h^3}$

Using 4.00eV for E^(1/2) (and consequently converting it to Joules) I get g(E) = 6.79^37 Energy States/m^3.

Here's what I did in wolfram alpha language:

here's D: http://www.wolframalpha.com/input/?i=(8*sqrt(2)*pi*(9.109e-31)^(3/2))/(6.626e-34)^3)

Now here's D*E^(1/2): http://www.wolframalpha.com/input/?i=1.06e56*(4*1.602e-19)




Pretty much my question is "What do I do next?" Due to N/V equaling an integral, I'm a little iffy on what to do.

Thanks :)

 

[Lil Dansn Man]

Hmm sorry, I actually only multiplied by E, not E^1/2...


Here's E^1/2: 1.06e56*((4*1.602e-19)^(1/2))