From 08a1d77053372b310055ddde8621c6b8a206ef50 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Thu, 19 Mar 2009 14:32:23 -0400 Subject: corrections to fermi, TODO check second to last line --- physics/quantum/fermigas | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) (limited to 'physics/quantum') diff --git a/physics/quantum/fermigas b/physics/quantum/fermigas index b91c67c..0114b43 100644 --- a/physics/quantum/fermigas +++ b/physics/quantum/fermigas @@ -39,14 +39,13 @@ correction can be important. There will then be a second correction due to removing the states along the individual axes twice (once for each side-surface), u.s.w. -The surface of this shell is called the Fermi surface -and represents the most excited states in the gas. The radius can be derived -by calculating the total volume enclosed: each block has volume -:m:$\frac{\pi^3}{l_x l_y l_z}=\frac{pi^3}{V}$ and there are N/2 blocks occupied by N -fermions, so: +The surface of this shell is called the Fermi surface and represents the most +excited states in the gas. The radius can be derived by calculating the total +volume enclosed: each block has volume :m:$\frac{\pi^3}{l_x l_y +l_z}=\frac{\pi^3}{V}$ and there are N/2 blocks occupied by N fermions, so: :m:$$\frac{1}{8}(\frac{4\pi}{3} |k_{F}|^{3})&=&\frac{Nq}{2}(\frac{\pi^{3}}{V})\\|k_{F}|&=&\sqrt{\frac{3Nq\pi^2}{V}}^3=\sqrt{3\pi^2\rho}^3$$ :m:$\rho$ is the "free fermion density". The corresponding energy is: -:m:$$E_{F}=\frac{\hbar^{2}}{2m}|k_{F}|^{2}=\frac{\hbar^{2}}{2m}\sqrt{3\rho \pi}^3$$ +:m:$$E_{F}=\frac{\hbar^{2}}{2m}|k_{F}|^{2}=\frac{\hbar^{2}}{2m}(3\rho \pi)^{2/3}$$ -- cgit v1.2.1