How Does Room Temperature Affect Electron States in an Energy Band?

[nadeemo]

Homework Statement

Given the width of an energy band of electrons is typically ~10eV, calculate the number of states per unit volume within a small energy range KT about the centre of the band at room temperature.

K is boltzmanss constant = 8.617*10^-5 eV/K

Homework Equations

volume of a sphere = 4*pi*(p^2)*dp

p^2 = 2mE

g(E) dE = 2*B*(m^3/2)*sqrt E* dE

p is momentum, E is energy, m is mass, B is some constant (not given)

The Attempt at a Solution

so I've let E = 10eV, and m as the mass of an electron, but we still don't know B, and how room temp plays into the equation...my relevant equations may be wrong, also if you solve for E = p^2 / 2m, and plug it into volume and solve for g(E) dE you get, B/(4*pi)

 

[Jhero]

* (m^3/2) * (2m/KT)^3/2, and then you can multiply by the small energy range dE, but I'm not sure if this is correct.

Thank you for your question. To calculate the number of states per unit volume within a small energy range KT about the centre of the band at room temperature, we can use the density of states formula:

g(E) = (2*m^3/2)/sqrt(2*pi*h^3)*(E)^(1/2)

Where m is the mass of an electron and h is Planck's constant. We can then integrate this formula over the energy range KT to get the total number of states within that range.

g(KT) = (2*m^3/2)/sqrt(2*pi*h^3)*∫(E)^(1/2)dE from 0 to KT

Simplifying this integral, we get:

g(KT) = (2*m^3/2)/sqrt(2*pi*h^3)*(2/3)*(KT)^(3/2)

Substituting in the given values for m, h, and K, we get:

g(KT) = (2*(9.109*10^-31)^3/2)/sqrt(2*pi*(6.626*10^-34)^3)*(2/3)*(8.617*10^-5)^(-3/2)*(KT)^(3/2)

g(KT) = 4.258*10^47*(KT)^(3/2)

To get the number of states per unit volume, we divide this value by the volume of a sphere with radius p = √(2mE):

g(KT)/V = 4.258*10^47*(KT)^(3/2)/(4*pi*(√(2mE))^3)

g(KT)/V = 2.129*10^47*(KT)^3/(8*pi*(m^3)*E^(3/2))

Substituting in the given value for E = 10eV and the mass of an electron, we get:

g(KT)/V = 2.129*10^47*(KT)^3/(8*pi*(9.109*10^-31)^3*(10*1.602*10^-19)^(3/2))

g(KT