Electric field above surface

[aksonik]

I intend to calculate electric field in certain point above surface of crystal, which is infinite in directions of surface and below. Crystal is composed of unit cells containing equal amount of positive and negative ions (+1e and -1e). Distances between ions and between point and surface (jons at the top) are in the same order of magnitude.

 

Should I simply calculate electric field from ions containing in some radii of the point - like from half of a spheres, and extrapolate results to infinity? Or mayby consider some thin discs or whatever else making use of Gauss's Law?

 

Best regards,

Aksonik

[elfmotat]

Well, if there' no charge other than the ions then they're all going to cancel when you're calculating flux through any symmetrical Gaussian surface (I would use a cylinder).

[aksonik]

Thank you for replay.

 

If I consider uniformly charged infinite disc, lines of electric field are perpendicular to it. Then, I can use any symmetrical Gaussian surface and find out that electric field doesn't depend on distance. Now, If I treated my crystal as a set of parallel, infinite and charged uniformly, positively and nagatively discs arranged alternatelyelectric field outside would be zero - that is okay.

 

The thing is, a point very close to the surface doesn't "see" nearest part of the crystal as uniformly charged. I expect differences in electric field depending on positions in the surface plane - in the same distance but above positive or negative ion.

 

I wonder, is a reason why I should differently consider ions in the surface plane then in deep (cylindrically), not uniformly in all directions (spherically)?

 

Best regards,

Aksonik

[Enthalpy]

If I understand your quest, you want to evaluate the periodic field very near to the surface and to individual ions. Then, I wouldn't first group ions into planes, hemisphere or cylinders, but just sum up the contributions or all individual ions.

 

This may raise difficulties when integrating potentials in 1/R, but you might first compute the field in 1/R² and deduce later the potential if needed. Better: you could first couple the ions in pairs, and then the far field will decrease in 1/R³, helping the sum converge.

 

In a real case you would need the precise positions of the ions and their charge (which isn't an integer number of q), and then it would get seriously dificult. Even for a simple cubic crystal like table salt, you could seek help from a computation software like Maple.

[aksonik]

Hi Enthalpy - thank you for raplay.

 

I know the positions and charges of all ions in the crystal. If I want to sum up the contributions of individual ions I have to decide which take into consideration and which don't - amount of them is infinite, so ions in a long distance have to be neglected.

 

Single unit cell contain equal number of positive and negative ions. That is why I intend to consider only integer amount of them. I suppose, this is what you meant writing "couple in pairs".

 

Best regards,

Aksonik

[Enthalpy]

Or a complete neutral cell, yes: better than a pair or ions, for being more general.

 

You could compute an equivalent of a cell in terms of dipole and tetrapole moments, which would tell you from which distance all the cells can be neglected, since these multipolar moments give a finite sum when integrating to infinite distance. Taking higher-order multipolear mometns would also reduce the number of cells needed to achieve a given accuracy.