- #1
Teymur
- 16
- 3
- Homework Statement
- Show that:
$$<\:K.E.>\:=E_c+3/2\:k_B\:T$$
- Relevant Equations
- $$<\:K.E.>\:=\frac{\left(total\:K.E.\right)}{\left(no.of\:electrons\right)}$$
$$<\:K.E.>\:=\:\frac{\int \:\left(E-E_c\right)g\left(E\right)f\left(E\right)dE}{\int \:g\left(E\right)f\left(E\right)dE}$$
Hello,
I've seen in a few books on solid state physics that one can deduce an expression for average K.E.:
$$<\:K.E.>\:=E_c+3/2\:k_B\:T$$
from the following:
$$<\:K.E.>\:=\:\frac{\int \:\left(E-E_c\right)g\left(E\right)f\left(E\right)dE}{\int \:g\left(E\right)f\left(E\right)dE}$$
I can't, however, find any work through of how to do so. I've had a go at the bottom part:
where ##n=\int g\left(E\right)f\left(E\right)dE## and ##\int \:x^{\frac{1}{2}}exp\left(-x\right)dx=\frac{\pi \:^{\frac{1}{2}}}{2}##
and
##g\left(E\right)=\frac{\left(2m_e\right)^{\frac{3}{2}}\left(E-E_c\right)^{\frac{1}{2}}}{2\pi ^2ℏ^3}## and ##f\left(E\right)\approx exp\left(\frac{\mu -E}{k_B\:T}\right)##
to get:
$$n=2\left(\frac{m_ek_B\:T}{2\pi ℏ^2}\right)^{\frac{3}{2}}\:exp\left(\frac{\mu -E_c}{k_B\:T}\right)$$
But how does one integrate the numerator with the ##\left(E\:-E_c\right)## term and simplify to the desired result?
I've seen in a few books on solid state physics that one can deduce an expression for average K.E.:
$$<\:K.E.>\:=E_c+3/2\:k_B\:T$$
from the following:
$$<\:K.E.>\:=\:\frac{\int \:\left(E-E_c\right)g\left(E\right)f\left(E\right)dE}{\int \:g\left(E\right)f\left(E\right)dE}$$
I can't, however, find any work through of how to do so. I've had a go at the bottom part:
where ##n=\int g\left(E\right)f\left(E\right)dE## and ##\int \:x^{\frac{1}{2}}exp\left(-x\right)dx=\frac{\pi \:^{\frac{1}{2}}}{2}##
and
##g\left(E\right)=\frac{\left(2m_e\right)^{\frac{3}{2}}\left(E-E_c\right)^{\frac{1}{2}}}{2\pi ^2ℏ^3}## and ##f\left(E\right)\approx exp\left(\frac{\mu -E}{k_B\:T}\right)##
to get:
$$n=2\left(\frac{m_ek_B\:T}{2\pi ℏ^2}\right)^{\frac{3}{2}}\:exp\left(\frac{\mu -E_c}{k_B\:T}\right)$$
But how does one integrate the numerator with the ##\left(E\:-E_c\right)## term and simplify to the desired result?