What is the mean inner potential in an ideal crystal of finite size?

[Philip Koeck]

In a hypothetical, electrically neutral, ideal crystal, where all unit cells are identical, even the ones at the surface:
What would the average value of the electrostatic potential be compared to that of the vacuum outside the crystal?
Would it be the same or more positive?
As a simple example one could consider a cubic lattice where each unit cell contains a localized positive and a smeared out negative charge.

 

[SpinFlop]

What would the average value of the electrostatic potential be compared to that of the vacuum outside the crystal?

What you are describing is Crystal Field Theory. There is extensive literature on this and the associated energy splitting of electron orbital degeneracy within these crystalline electric fields, ie: CEF levels. In general, the number of levels is determined by the local point group symmetry of the lattice. The exact location of these energy levels is much more difficult to get right.

The simplest approach is to use a point charge model that places the charges at specific locations within the lattice. This can be ramped up to take into account overlap, for instance with Ligand Field Theory.

As a simple example one could consider a cubic lattice where each unit cell contains a localized positive and a smeared out negative charge.

For a cubic lattice you still need to take into account the local coordination of the crystal field point charges, it could be a six-fold coordination due to an octahedral placement of the charges, or four-fold from a tetrahedral placement, or eight-fold if you place a charge at every corner of the cubic cell.

I highly recommend you read the long review "Point-Charge Calculations of Energy Levels of Magnetic Ions in Crystalline Electric Fields" by M. T. Hutchings. This is a fantastic paper that actually works through in detail all three of the above coordinations in the cubic environment; done in both a Cartesian and spherical coordinate system. A full description of this potential is provided in the first half of the paper (within a point charge model). The second half then works through the CEF level scheme by placing magnetic ions within this potential.

The Steven's operator approach for deriving the matrix elements of this crystalline perturbing Hamiltonian is then described and then again the exact elements are all worked out for your cubic system. It is a great read even for those familiar with Crystal Field Theory since it is a rare instance where you can find all of the maths carefully described from start to finish and then applied to a specific system. I cannot recommend this paper enough, regardless of your level of expertise.

 

[Henryk]

What would the average value of the electrostatic potential be compared to that of the vacuum outside the crystal?

The answer is: Work function divided by the electron charge.
If the work function is, say 4 eV, the difference in potential is 4 Volts.

 

[Philip Koeck]

The answer is: Work function divided by the electron charge.
If the work function is, say 4 eV, the difference in potential is 4 Volts.

Yes, that's obvious, but what would the work function of the described crystal be? Would it actually be different from zero?

 

[Philip Koeck]

What you are describing is Crystal Field Theory. There is extensive literature on this and the associated energy splitting of electron orbital degeneracy within these crystalline electric fields, ie: CEF levels. In general, the number of levels is determined by the local point group symmetry of the lattice. The exact location of these energy levels is much more difficult to get right.

I was thinking of a much simpler model without energy levels. Just define a charge distribution and then calculate the potential using the Poisson equation. I tried a cubic lattice with positive point charges (actually one voxel) and completely smeared out negative charge numerically and I get a positive value for the potential inside compared to outside.
I'm uncertain about this result though. I wonder if it shouldn't really be zero.

After reading your post I'm also wondering if my simple electrostatic model could be completely misleading.

According to CFT, would different types of ideal crystals have different values for the potential difference?
Are there some configurations that actually have zero?

 

[Henryk]

My apology, I gave you a simple, easy to understand, wrong answer.
The work function is the minimum energy required to knock an electron out of a crystal and that means the electron of the highest energy, (for simplicity, let's consider metals only and that would be Fermi energy)
Just to give you an idea, work function for simple metals (alkali) is just over 2 eV.
However, there are two components to electron energy: potential and kinetic. Kinetic energy, again for simple metals, is around 4 eV, that would have to be added to the work function. Therefore, the potential energy of an electron at the bottom of the conduction band (zero kinetic energy) would be - 6 eV with respect to vacuum outside. But there is one more snag. The conduction band electrons are prohibited to get to close to the nucleus by the Pauli exclusion principle. Therefore, their potential energy is somewhat higher than the average electrostatic potential within the crystal. Eyeballing this, you can come with a number around 10 eV.
That would mean the average potential within a crystal would be around +10 V (that gives a negative energy for a negatively charged electron).
If you want to be exact, you would need to do calculations based on actual electron density distribution.
of maximum energy (Fermi energy) into the surrounding vacuum. The from the crystal

 

[SpinFlop]

I was thinking of a much simpler model without energy levels.

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.

I tried a cubic lattice with positive point charges (actually one voxel) and completely smeared out negative charge

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.

... I get a positive value for the potential inside compared to outside. I'm uncertain about this result though. I wonder if it shouldn't really be zero.

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?

 

[Philip Koeck]

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?