How Is the Depth of the Potential Well Calculated for Free Electrons in Gold?

[Dawei]

Homework Statement

For free electrons in a metal, the depth of a potential well can be determined by observing that the work function is the energy required to remove an electron at the top of the occupied states from the metal; an electron in this state has the Fermi energy.

Assuming each atom provides one free electron to the gas, find the depth of the well for a free electron in gold (work function = 4.8 eV)

Homework Equations

The Attempt at a Solution

I'm assuming that the answer will be just the fermi energy plus the work function, but I can't seem to get the right answer when plugging everything in.

For N/V, I used (Avogadro # x Density of gold) / (molar mass of gold) , and got 3.89 x 10^9 electrons / cubic meter

The answer should be on the order of 1 ev, but I can't seem to get this. The m here refers to the mass of a single electron, correct? But using that, I just don't get the right orders of magnitude. (I get about 5 x 10^6).

 

[nickjer]

That isn't nearly enough electrons in a cubic meter. You made a mistake in calculating N/V. Please show all of your work so we can see what you did wrong.

 

[Dawei]

Yeah you're right. 10^9 was what I got when I took number of electrons / m^3 and raised it to the 1/3 power, to get N/L. I think I'm supposed to use the 1 dimensional equation.

edit: yeah I was just being dumb, I had it right, was just using the wrong units for mass. Thanks though.

 

[nickjer]

You have the right equation. Just show your work on solving N/V so we can check it. Remember to show all units.

 

[paranormal]

I would first check my calculations to make sure I am using the correct values and equations. I would also double check the units to make sure they are consistent. It is important to note that the work function and Fermi energy are typically given in units of electron volts (eV), while the mass of an electron is typically given in kilograms (kg). Therefore, it may be necessary to convert units in order to get the correct answer.

Additionally, I would make sure to use the correct formula for the depth of the potential well, which is given by:

V = (2/5)*(N/V)*(E_F)

where N/V is the number density of electrons and E_F is the Fermi energy. Plugging in the given values, I get a depth of the potential well of approximately 5 eV, which is consistent with the expected order of magnitude.

It is also important to note that the depth of the potential well may vary for different materials, as it depends on the number density of electrons and the Fermi energy, which can be affected by factors such as the crystal structure and temperature. Therefore, it is important to consider the specific material when determining the depth of the potential well.