# Fermi Dirac- missing something from Ashcroft derivation

• Benindelft

#### Benindelft

[SOLVED] Fermi Dirac- missing something from Ashcroft derivation

## Homework Statement

Deriving Fermi Dirac function
following ashcroft all good up to equation 2.43 but then it does the folowing at 2.44
and I can't see how you reach 2.44.

## Homework Equations

as

(2.43) f_{i}^{N}= 1- sum(P_{N}(E_{alpha}^{N+1}-E_{i}) which is 'summation over all (N+1) electron states alpha in which there is an electron in the one electron level i)
Then because
P_{N} (E)= exp(-(E-F_{N})/kT)

We may write
(2.44) P_{N}(E_{alpha}^{N+1}- E_{i})=exp((E_{i}-u)/kt)P_{N+1}(E_{alpha}^{N+1})

as u=F_{N+1}-F_{N}

## The Attempt at a Solution

I tried just subbing in but I am missing some important point and end up with rubbish...

they substitute in Eq. 2.40 and then add 0=F_{N+1}-F_{N+1} in the exponent
$$P_N(E_\alpha^{N+1}-\epsilon_i)=e^{-(E_\alpha^{N+1}-\epsilon_i-F_N)/T} =e^{-(E_\alpha^{N+1}-\epsilon_i-F_N+F_{N+1}-F_{N+1})/T}$$
then factor out part
$$=e^{-(E_\alpha^{N+1}-F_{N+1})/T}e^{(\epsilon_i-[F_{N+1}-F_{N}])/T}$$
and use the defintion of P (Eq.2.40) again
$$=P_{N+1}(E_{\alpha}^{N+1})e^{(\epsilon_i-[F_{N+1}-F_{N}])/T}$$
and the defintion of \mu
$$=P_{N+1}(E_{\alpha}^{N+1})e^{(\epsilon_i-[\mu])/T}$$

Thanks! that looks pretty clear to me now.

BTW how do you get the equations to looks so nice? Is that info some where on this website?

Benindelft said:
Thanks! that looks pretty clear to me now.

BTW how do you get the equations to looks so nice? Is that info some where on this website?

I'm using LaTeX commands. LaTeX is a typesetting program that's really good for math. just google "LaTeX tutorial" or something and you will find a lot of information. To use LaTeX on physics forums you have to enclose the commands in between tags... put your mouse over the following equation and then
click on it to see the code (enclosed in tex and /tex tags...in square brackets) which created it
$$\frac{1}{c}\frac{\partial \vec E}{\partial t}$$
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