[3dem] RE: Williams and Carter

Philip Köck Philip.Koeck at ki.se
Thu May 31 04:34:08 PDT 2012

Thanks for the nice answers, Terje and Qing.

I'll stick to the main problems I have to start with.

Actually the text never mentions that h is a phase factor (a complex exponential).
Only if I know that I can break it up into cos + i sin as Terje points out.

Thinking about this I've just realized that I'm not convinced that h is a phase factor.
The corresponding expression in Fourier space, exp(i X(u)), cannot be inversely FTd analytically as far as I know.
This is only possible (as far as I know) if Cs is zero. Then you get the Fresnel propagator apart from some
constant factors, and that is actually a quadratic phase factor.
If Cs is not zero it's not obvious to me that h is a phase factor.

If I accept (28.23) anyway everything is clear up to (28.25).
Then they say that since in real space only the imaginary part of h is left after taking the absolute
square, the same thing must be true in Fourier space.
This I cannot see, especially since in real space we have a convolution whereas in Fourier space
we have a multiplication.

Terje puts his finger right on the sore spot in his first sentence. What good is a text book if you need
several other books and a lot of good will to follow the derivations?

Now to some "small" things:
What is cos(x,y) in (28.23). cos is only defined for scalar variables. Surely they mean cos(a(x,y)) where
a is a scalar function of x and y. This is only a formality, but it must be very confusing to students.

In (28.22) a "1" is missing that suddenly reappears in (28.23). In (28.3) it should say G(ux,uy) and not
G(ux-uy). Two typos in formulas on 3 pages in the second edition of a text book.

In the text after (28.21) they say "the amplitude of the wave is linearly related to the projected potential".
I thought they were discussing the phase object approximation!?

The problems continue in section 28.6. To get from (28.29) to (28.30) the authors integrate, because "there
is a range of angles". I don't understand why they do that other than that it somehow gives the right result.
I sent an e-mail about this particular question to the authors some years ago and never got an answer, although
the preface encourages contacting them by e-mail with comments.
I also discussed the derivation with an expert on electron optics, but he couldn't make sense of it either
and suggested another derivation.

Here is the e-mail that I sent to Professors Williams and Carter at the time:

(beginning of insert)

Dear professors,

I really have a hard time making sense of your derivation of the lens aberration function (28.34) in
"Transmission Electron Microscopy".

You start by saying that (28.29) gives the radius of the disk that is the image of a point.
In fact if you go back to (6.14) for the part of this expression that relates to spherical aberration you see
(from the derivation of (6.14)) that the angle theta is an arbitrary angle and not necessarily the largest (beta).
Only if the largest theta is chosen does (6.14) give the radius of the disk!

However, the goal of this derivation is to find the phase-difference between a beam travelling at a particular
angle theta and a beam travelling at theta=0. Therefore it should be correct to use an arbitrary angle theta.
This of course makes it very hard to understand why you integrate (or average) to get from (28.29) to

My third and last question: To get from (28.30) to (28.34) you replace the scattering angle by the corresponding
reciprocal lattice vector and then multiply by 2pi/lambda to convert a path difference into a phase difference.
The problem here is that your derivation of (28.30) doesn't indicate that (28.30) actually describes a path difference.
(Of course the formula is correct, so it does describe a path difference, but this is not apparent from your derivation.)
According to your own explanation (28.30) is some sort of averaged disk radius.

I hope you can bring some clarity into these matters.

(end of insert)

I think I'll give it break here.

Thank you for your patience,


From: Terje Dokland [mailto:dokland at uab.edu]
Sent: 30 May 2012 16:55
To: Philip Köck
Cc: 3dem at ncmir.ucsd.edu; microscopy at microscopy.com
Subject: Re: [3dem] Williams and Carter

Which part is it that you are having problems with? I agree it can be a bit hard to follow unless you also read Reimer and/or Spence to fill in the blanks!

I'm no expert, but to me Equation 28.22 is just saying that the image function is the convolution of the point spread function h(x,y) with the 2D projection of the 3D specimen, which is [1- isV(x,y)] according to the WPOA in their lingo.

Since the PSF is a phase shift it can be broken into sin and cos components (28.23).

Then when you take the square of that  (28.24) you  get a lot of second order terms that are ignored because they are too small, yielding 28.25.

The next part is a bit hard to follow, but basically you take the fourier transform of that, which turns the convolution into multiplication, which would give you

FTimage(u,v) = d(0) + 2s FTspecimentprojection(u,v) sin(Chi)

since they already showed previously the relationship between Chi and h (28.6)

and Chi is of course the phase shift due to spherical aberration and defocus.

They then multiply that with the aperture function A and the envelope function E as well.

On May 30, 2012, at 9:16 AM, Philip Köck wrote:

I've been trying to read chapter 28 in the second edition of "Transmission Electron Microscopy" by
Williams and Carter.

Is there anybody out there who can make sense of the derivation in section 28.4?

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