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Hi Ed,

your explanation E = rho(r)exp(2pi i (wt - r.S)) is nearly correct, as
Blundell and Johnson are nearly correct. In Physics any function
f(x,y,z,t) depending on three coordinates and time that can be
expressed such that f(x,y,z,t) = f(\vec k.\vec x - \omega t) is a
wave. In books on electrodynamics I often find the time dependency
\omega t dropped, which you had to put back in. The reason for its
dropping is probably that the time variation of the wave is only a
mathematical concept in order to explain interference, but not a
physical one: the intensity is the mean over time, i.e. an idealised
light source does not flicker.

So in your above equation you must replace S with k to get a spherical
wave. Once you take the Laue equations into account, i.e. you realise
your single scatterer is member of a crystal, you do not need to
measure the scattered wave in all directions around the crystal but
only in the directions governed by the Laue equations. Then you can
replace k with S and find spots on your detector.

Regards,
Tim

P.S.: complex numbers together with the operation '+' defined in the
canonical way fulfill the axioms of a vector space, hence complex
number are vectors. If you also take the operation '*' into account,
defined in the canonical way, (C, +, *) fulfills the axioms of a
field, which is an algebra, but not a vector space. So it depends on
your point of you, just like with photons and their interpretation as
waves or as particles: pick what suits you best depending on what you
want to describe.

On 04/02/2014 04:39 AM, Edward A. Berry wrote:
> 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)
>

- --
- --
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen

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