SpinFlop said:
As I stated above, the paper I recommended works out the potential for you cubic system in the first half. The second half then works out the energy level scheme due to splitting the orbital degeneracy of electrons placed within this crystal field. So there is no simpler model, just disregard the second half of the paper if you are not interested in the CEF level splitting due to these fields.In fact, what you describe is a more complicated model since you want to include a screening effect in the form of a negative charge distribution that intermixes with the positive point charges at cubic vertices.
For simplicity let's assume your negative charge distribution is a uniform spherical volume. Thus, inside the sphere the potential due to the negative distribution scales as ##r^2##. However, all of your positive point charges scale as ##-r^{-1}_{i}##. Given that the full potential can be taken as the direct superposition of these two contributions, how could you possibly hope to produce a net zero field throughout the entire volume?
Thanks! Now I believe I've straightened out my thinking. I'll try to get hold of the paper too and see what they do.
Using just the Poisson equation I see things this way:
A model atom consisting of a spherical, negative charge distribution with a localized positive charge in the center has to have average potential above the vacuum value (zero, let's say).
Outside the atom the potential is constant (zero, let's say). In the negative sphere it curves upwards according to Poissons equation. In the localized positive charge it curves sharply downwards. the total result is a peak which is above zero everywhere inside the atom.
Assembling atoms like this in a cubic lattice without overlap doesn't change this result.
The only thing I need to do now to get to the situation I calculated numerically is fill in the gaps between the spheres of negative charge and reduce the negative charge density slightly to maintain a neutral solid, but that can't make a qualitative difference.
Interestingly I get roughly 10 V mean inner potential in my simulations if I use about 1 to 2 elementary charges of each sign per unit cell, so I'm close to the values Henryk gives in post 6 (based on proper solid state physics, I believe).
The reason I'm thinking about this is that I want to understand Zernike phase plates in electron microscopy.
The understanding is that carbon film has an inner potential of about 10 V, which leads to a phase shift in the electron wave of pi/2 if the thickness is chosen correctly.
I was wondering how much of this inner potential comes from the bulk of the carbon and how much is added by surface dipole layers.
It looks like a large part of the potential could be explained by the distribution of charges inside the carbon and surface effects might not account for very much.
Any thoughts on this?