This is a god example for what happens if mathematics get translated into
language. It is not a lossless transform.
1) "Rupp says a structure fator is a vector"
I hope I did not make that statement ( because it is wrong). If I explicitly
did (or that impression emerged) , it would be good to know where so this
can be addressed.
In sidebar 6-2 there is the explicit statement
'Note that complex numbers are not really vectors' and an explanation about
the differences between vectors and complex numbers follows.
It just happens that when complex numbers are represented as vectors, the
addition rules are the same. In
crystallography we use the Argand diagrams in the complex plane to visualize
that, and sometimes colloquially and
in principle misleading call these 'vector diagrams'
Also the preceding figure captions in BMC are quite clear about the
'interpretation' part:
6-5: We can represent any plane wave ...as a vector in in the complex
plane.
6-6 Superposition of waves becomes convenient using the vector
representation....
I must credit Ian Tickle with pointing out to me that this misconception
'complex number = vector' will happen, prompting the warning statements in
sidebar 6-2. Maybe bold font would be in order.
2) How is that a wave? r and S are constant vectors.
Because of the small i in front of it. The phase itself is a number, but
when the entire complex term ends up in the exponent, you get the
representation (there we go again) of a periodic process.
(this is simple math, Euler). This is entirely independent of what that
periodic process is, it says nothing about its physical meaning.
3) " temporal cosine wave at the surface of the detector"
Does not happen. Your brain will explode if you take the partial wave
picture verbatim. Scattering is a single photon process, and there have
been heated threads on this subject before.
All these multi-wave and water wave with slit explanations of diffraction
will invariably lead to confusion. Diffraction is a probabilistic process,
where you can
calculate the PDF a.k.a. the diffraction pattern by (here we go again)
interpreting or representing the process as a superposition of partial
waves.
Simply speaking, a structure factor is simply a measure for the probability
that a photon will be scattered in a given direction.
It is not the mathematical (or structure) models that are to blame, but
almost always their physical interpretation that causes confusion.
Lost in translation.
Best, BR
-----Original Message-----
From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of Edward
A. Berry
Sent: Mittwoch, 2. April 2014 04:40
To: [log in to unmask]
Subject: [ccp4bb] Structure factor equation
Encouraged by recent help from the BB in filling in gaps in my
understanding, maybe I can get help with another question:
At the top of page 121 in Blundell and Johnson, it is written:
"The total wave scattered by a small unit of volume dv at a position r
relative to the wave scattered from the origin will therefore have an
amplitude proportional to Rho(r)dv and phase 2Pi i(r.S)dv" (OK so far) "i.e.
wave scattered = Rho(r)exp(2Pi i r.S)dv"
How is that a wave? r and S are constant vectors.
My best explanation so far is to say this is a complex coefficient that will
adjust the weight and phase of the wave scattered by this point.
Say the wave scattered from one electron at the origin will result in a
temporal cosine wave at the surface of the detector:
E = exp(2Pi i wt) = cos(2Pi wt)
(not sure if 2Pi is needed when w is radians/sec)
Then the wave at the same point, scattered by dv at r, would be the
same multiplied by the quantity in question:
E = rho(r)exp(2Pi i r.s)dv * exp(2pi i wt)
= rho(r)exp(2pi i (wt - r.S))
i.e. phase-shifted by 2Pi (r.S), and multiplied by Rho(r)dv
Is that more or less it?
(since these quantities add up to the Structure Factor F(s), I guess I'm
really asking what a structure factor is.
Rupp says a structure fator is a vector "representing"
the diffracted X-rays", which i take to be consistent with this if vector is
in the complex plane)
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