Calculating the Relation between Electrons at Vakuum Level and Fermilevel

[magnusse]

I posted this in the math forum by mistake, so posting it here now instead

Homework Statement

The cutoff frequency for the photoelectric effect for silver is 1.089*10^15 Hz and its Fermienergy at T = 0 degrees Celsius is 5.5 eV.

Calculate the relation between the number of electrons in a very small energyinterval at vakuumlevel (that is, the energylevel at which electrons are "freed" from the metal) and a small interval att the Fermilevel.
Assume T = 0 degrees Celsius.

Homework Equations

I THINK theses are the relevant equations.

Fermi-dirac distribution
f(E) = 1 / (e^(E-E_F)/(kT)+1)

(density of states, free-electron model)
g(E) =((2m)^3/2 * V) / (2*Pi²*(h/2*pi)³) * E^1/2

V = L^3 (L = length of cubic "box")


The Attempt at a Solution

I know that because the intervals are small, there should be some way to use the derivate to find what I'm seeking. And I think I need to combine the two equations i wrote earlier.

But I'm stuck and now time is running out. I would really appreciate some help.



Sorry for my english, it's not my native language.

 

[mark726]

Thank you for reaching out for help with your question. I can understand how confusing and frustrating it can be to get stuck on a problem, especially when time is running out. I will do my best to assist you in finding a solution.

To start off, I would recommend reviewing the Fermi-Dirac distribution and the density of states equations that you have mentioned. These are indeed the relevant equations for this problem. The Fermi-Dirac distribution describes the probability of an electron occupying a certain energy level, while the density of states equation gives the number of available energy states at a given energy level.

To calculate the relation between the number of electrons in a small energy interval at vacuum level and at the Fermi level, we can use the fact that the total number of electrons in the system is conserved. This means that the number of electrons in the small energy interval at vacuum level must be equal to the number of electrons in the small energy interval at the Fermi level.

To find the number of electrons in a small energy interval, we can use the density of states equation. Since the energy intervals are small, we can approximate the density of states as a constant value within the interval. This means that we can simply multiply the density of states by the size of the energy interval to find the number of available energy states in that interval.

Using this approach, we can set up an equation where the number of electrons in the small energy interval at vacuum level is equal to the number of electrons in the small energy interval at the Fermi level. We can then solve for the relation between these two intervals.

I hope this explanation helps you in finding the solution to your problem. If you need any further clarification or assistance, please do not hesitate to ask. Good luck!