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<p>Hi Philip,</p>
<p><br>
</p>
<p>Let me then be more precise. You're right that the formula I
mentioned is valid far from the dipole and that we should consider
layers instead of single dipoles. The analytical solution for two
dipole layers, valid far from the layers shows that the potential
is proportional to the distance between the layers (see little
calculations at <a class="moz-txt-link-freetext" href="https://datashare.biochem.mpg.de/s/4Xc18s6G9ydbAiK">https://datashare.biochem.mpg.de/s/4Xc18s6G9ydbAiK</a>
), so no change there.<br>
</p>
<p><br>
</p>
<p>However strange it sounds, an electron approaching the dipoles
will feel the dipole potential and its linear dependence on the
distance between the layers. I don't expect this potential to be
strong because it falls off pretty fast with the distance. <br>
</p>
<p><br>
</p>
<p>I also provide there the solution for the potential generated by
a disk-shaped double layer of dipoles, which is valid for any
distance to the layers, distance between layers, distance between
charges within a dipole and the radius of the dipole layers. One
can use this to calculate the potential due to dipole layers
between the layers. The way I see it, it will not be constant
between the layers, the highest value should be in the middle, but
I didn't calculate it exactly.<br>
</p>
<p><br>
</p>
<p>However, my understanding is that all this doesn't matter (much)
for the phase change of an electron passing through a material
having dipole layers on the surfaces, because the potential
between the layers strongly depends on the material between the
dipoles (eg some type of C). This potential can be approximated by
the mean inner potential of the material, which was the original
reason for starting this thread. Reading the literature, it seems
that dipoles formed on the surface have two effects: they add
their potential, but also induce a rearrangement of the electrons
within the material, thus modifying the mean inner potential. <br>
</p>
<p> <br>
</p>
<p>My main point in this discussion was the sentence I quoted below
that (erroneously or not) mentions dipoles perpendicular to the
electron beam, which to me appears similar to an explanation for
the Volta phase plate that was recently published (mentioned
somewhere below).<br>
</p>
<p><br>
</p>
<p>Best,</p>
<p>Vladan<br>
</p>
<p><br>
</p>
<div class="moz-cite-prefix">On 6/28/19 2:08 PM, Philip Köck wrote:<br>
</div>
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cite="mid:HE1PR0802MB2586E82E2749531F4E8B5C4783FC0@HE1PR0802MB2586.eurprd08.prod.outlook.com">
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<p style="margin-top:0; margin-bottom:0">Hi Vladan.</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">I think I've spotted
the problems in your argument.</p>
<p style="margin-top:0; margin-bottom:0">One is that the
formula for the potential from a dipole is only valid at a
large distance from the dipole, r much bigger than the
distance between + and - in the dipole.<br>
The inside of a slab of matter is not necessarily very far
from the surface. Think of a 10 nm carbon film coated with
water molecules. A lot of it is quite close to the surface
(compared to the charge separation in a water molecule).</p>
<p style="margin-top:0; margin-bottom:0">The second problem is
that you would have to integrate over the surface if you
want to apply your reasoning to an extended double layer of
dipoles.</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">I would suggest a
very simple model in stead: Just think of two infinitely
large charged planes on each face of the slab, both with the
negative charge closer to the inside. If you come from the
vacuum the potential decreases within the first bilayer and
then remains constant within the slab and then increases
back to the vacuum value in the opposite bilayer. The value
inside the slab will be constant and will not depend on the
thickness of the slab, but only on what happens in the
surface bilayers.</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">There's an easy way
to calculate potentials numerically with an image processing
package (MATLAB maybe). You just create a charge
distribution and then apply the Poisson equation by a simple
manipulation in Fourier space and you get the potential.</p>
<p style="margin-top:0; margin-bottom:0">I put the formulas in
my recent Ultramicroscopy paper, but you can find them in
many other places (papers by Spence, e.g.).</p>
<p style="margin-top:0; margin-bottom:0">If you want to try
this I'd like to here what you come up with.</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">There's one more
thing I might have to point out since I'm not sure about
your last statement quoted below:<br>
The phase shift is proportional to the projected potential.
This means that it's proportional to the thickness if the
potential is constant inside the slab.</p>
<p style="margin-top:0; margin-bottom:0">If the potential was
proportional to the thickness the phase shift would be
proportional to the square of the thickness.</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">All the best,</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">Philip</p>
</div>
<hr tabindex="-1" style="display:inline-block; width:98%">
<div id="divRplyFwdMsg" dir="ltr"><font style="font-size:11pt"
face="Calibri, sans-serif" color="#000000"><b>From:</b> 3dem
<a class="moz-txt-link-rfc2396E" href="mailto:3dem-bounces@ncmir.ucsd.edu"><3dem-bounces@ncmir.ucsd.edu></a> on behalf of Vladan
Lucic <a class="moz-txt-link-rfc2396E" href="mailto:vladan@biochem.mpg.de"><vladan@biochem.mpg.de></a><br>
<b>Sent:</b> Thursday, 27 June 2019 19:08:51<br>
<b>To:</b> <a class="moz-txt-link-abbreviated" href="mailto:3dem@ncmir.ucsd.edu">3dem@ncmir.ucsd.edu</a><br>
<b>Subject:</b> Re: [3dem] (mean Inner potential) Re: 3dem
Digest, Vol 142, Issue 38</font> <br>
</div>
<div style="background-color:#FFFFFF">
<blockquote type="cite">
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<p style="margin-top:0; margin-bottom:0">I don't see how
the potential can be proportional to the distance
between the dipole layers. If we think of a slab of
neutral matter (as a thought experiment), which is
covered first by a layer of positive charges and then an
equal amount of negative charges on top of that, the
following should happen: The negative charge curves the
potential upwards and the potential increases, then the
positive charge curves the potential down again by an
equal amount. The total effect is that the potential is
constant inside the slab and independent of the slab's
thickness and therefore the phase shift is proportional
to the thickness.<br>
</p>
</div>
</blockquote>
Let me try to expand on your model. Adsorbed H2O forms dipoles
on the surface, with positive charge towards the vacuum and
the negative towards the material. A potential arising from
one dipole (having electric dipole moment p, vector) that an
electron at a distance (vector) r sees is proportional to:<br>
<br>
scalar_product(r, p) / magnitude(r)^3 <br>
<br>
So you're right that it is proportional to the distance
between the positive and the negative charge (contained in p),
but that's not all. H2O dipoles form on both sides of the
material and they're oriented in the opposite direction. The
contribution to the potential from the "other side dipol" has
the same form, but r needs to be replaced by r+d, where
(vector) d is the thickness of the material. Adding the two
terms (with opposite signs because of the opposite
orientation), when (magnitudes) r>>d leaves in the first
approximation a term proportional to d. That's the reason for
my proportionality statement.
<br>
<br>
In any case, we agree that it is clear why the phase should be
proportional to the thickness, at least for flat surfaces.<br>
<br>
Best,<br>
Vladan<br>
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<pre class="moz-signature" cols="72">--
Vladan Lucic, PhD
Max Planck Institute of Biochemistry
Department of Molecular Structural Biology
Am Klopferspitz 18
82152 Martinsried, Germany
Phone +49 89 8578-2647
Fax: +49 89 8578-2643
Web: <a class="moz-txt-link-freetext" href="http://www.biochem.mpg.de/277447/15_ContentSynCompl">http://www.biochem.mpg.de/277447/15_ContentSynCompl</a></pre>
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