# The average potential inside an electrically neutral solid

If a solid (a few nm diameter) was placed in vacuum inside a grounded hollow sphere, but without touching the sphere (zero gravity), qualitatively what would the potential inside this solid be on average? In other words I don't want to look so closely that I see the potential wells of the nuclei. Would an electron wave propagating through this solid experience a phase shift?

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hutchphd
If a solid (a few nm diameter) was placed in vacuum inside a grounded hollow sphere, but without touching the sphere (zero gravity), qualitatively what would the potential inside this solid be on average? In other words I don't want to look so closely that I see the potential wells of the nuclei. Would an electron wave propagating through this solid experience a phase shift?
Is the solid charged? A conductor? Can you be a little more transparent as to where you are going here? Need more to go on.

The solid is not charged. As an example one could think of a piece of amorphous carbon.
The reason for my question is that I'm wondering about the usual explanation how an electron optical Zernike phase plate works. I can't quite make sense of it.

hutchphd
The solid is not charged. As an example one could think of a piece of amorphous carbon.
The reason for my question is that I'm wondering about the usual explanation how an electron optical Zernike phase plate works. I can't quite make sense of it.
what would the potential inside this solid be on average
What do you mean by "average" and "inside" ? Even inside a Faraday cage if I look at a simple electric dipole the near potential will vary but average to zero far away.
Materials develop surface potentials that look like ~dipole layers for various reasons. If a material looks like a negative slab of stuff with thin positive surface (or vice-versa) it will phase shift the electron passing through. It can even produce resonances if the geometry matches. Does that help?

Philip Koeck
Materials develop surface potentials that look like ~dipole layers for various reasons. If a material looks like a negative slab of stuff with thin positive surface (or vice-versa) it will phase shift the electron passing through. It can even produce resonances if the geometry matches. Does that help?
Yes, I think that might be the explanation. The idea is that carbon has a "mean inner potential" of about 10 V compared to vacuum, so that a 200 keV electron wave going through 20 nm of carbon is phase shifted by pi/2 compared to one going through vacuum. That's how a phase plate for an electron microscope is supposed to work. If I think of a solid as a smeared out negative charge with positive charges distributed evenly in it I can't get that to work. The average charge density (averaged over sufficiently large volumes) would be about zero everywhere and I would expect a mean inner potential equal to that of vacuum (based on the Poisson equation). If instead I assume a negative surface charge balanced by a positive charge right below it one could get a positive inner potential (compared to vacuum).
Do you agree with my reasoning or am I way off?

hutchphd