Re: I need help.

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Mon, 25 Feb 2002, David Bowman wrote:

> > ...  It is approximated here by a calculation based on the image
> > charge technique.  I have superimposed the potential contributions
> > from five other mirror dipoles that abut this region ....
>
> John, how did you justify truncating the contributions from the
> lattice of images after including only 5 image dipoles?

I didn't.  That's why I used the all purpose hedge word
"approximated."  Nevertheless, the proof may be in the pudding.
Since the resulting potential visibly (almost) satisfies the
required boundary conditions, the uniqueness property
(approximately) guarantees the solution.  How's that for more
hedging?

>  I believe the Ewald sum of potentials over the lattice of
> images converges quite slowly with distance from the original
> unit cell of the actual paper.  In fact the Ewald sum is only
> conditionally convergent.  If all the positive charges are
> summed first and then the negative ones are subtracted from
> them the result is a difference of infinities. If the image
> contributions are summed as charge neutral dipoles the sum is
> still conditionally convergent.  The sum is only significantly
> convergent when the dipoles are summed in nearest neighbor
> pairs of dipoles whose overall dipole moment vanishes for each
> dipole pair.  This requires a sum over a lattice of clusters
> of (at least) 4 point charges whose net charge and dipole
> moment vanish for each cluster.

In my case I used 3 pairs of oppositely directed dipoles
satisfying these conditions.  I came to that solution only after
trying lots of other configurations that clearly did not even
approximately satisfy the desired boundary conditions.  So it's
interesting to me that you are providing (something like) a
theoretical justification for what I observed by trial and error.

> Even in this case I don't think the rate of convergence is
> very impressive.  It doesn't seem to me that adding up 5 image
> dipoles plus the original dipole in the paper's unit cell is
> going far enough out to get a decent approximation for the
> total potential.  Did you maybe approximate the contribution
> from the more distant dipole pairs (actually these pairs have
> a quadrupole moment as their lowest order nonvanishing
> multipole moment) by some sort of continuum approximation
> scheme?

Me?  Not likely!  I *am* going to go back and add another bunch of
dipoles to see if the is noticeably altered.  My current system
looks like this:

    -    -    -

    +    +    +


    +    +    +

    _    _    _

With the simulation region centered on the lower middle positive
charge.  I am going to extend it to look like this:

    -    -    -    -    -    -    -    -    -    -    -

    +    +    +    +    +    +    +    +    +    +    +


    +    +    +    +    +    +    +    +    +    +    +

    -    -    -    -    -    -    -    -    -    -    -

still with the simulation region centered on the lower middle
positive charge.

Back soon ...

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm