Dear Ian,
Perhaps I should have made a more explicit connection to your message
in what I wrote yesterday. I do not think there is any paradox, or apples
vs. oranges problem, in this situation.
The structure factor is a count of "electrons as X-ray scatterers", so
that the Fourier synthesis computed from them is a number density for these
unit scatterers. The density can get clothed with a charge a-posteriori,
because we know what the charge of an electron is, but it is not that charge
as such that is sensed by the diffraction experiment: it is the complicated
combination of charge and mass and various physical constants that ends up
determining an electron's ability to scatter X-rays.
I think that if one bears this in mind at all times, paradoxes never
appear.
With best wishes,
Gerard.
--
On Sun, Feb 28, 2010 at 02:40:15PM -0000, Ian Tickle wrote:
>
> > Yes, I think this is exactly the point. 'Electrons' gives the whole
> > thing a consistent meaning.
>
> The big problem with statements like 'f = 10e' or 'rho = 1.5e/Å^3' is of course that they are dimensionally invalid, and I'm surprised that people are not doing such simple checks! For example I think we've all agreed that 'f' is defined as the ratio of two amplitudes and is therefore dimensionless, whereas 'e' is universally defined as the electronic charge, which in SI units has the value 1.602176487×10^−19 coulombs, but obviously has the dimensions of electric charge (time*electric current in terms of the base SI dimensions). So we have a real apples & oranges situation!
>
> You could of course get around this by redefining 'f' as I suggested previously, as the free point equivalent charge, but to avoid confusion we should call it something else, so let's say:
>
> Notation
> ========
> f: "atomic scattering factor", defined as the ratio of scattered amplitude for an atom to that for a free electron (dimensionless).
> g: "atomic scattering free point equivalent charge", defined as the free point charge which scatters with the same amplitude as the atom (dimensions of electric charge).
>
> Now we can validly write 'g = 10e' since we have dimensions of charge on both sides.
>
> This again highlights the importance of 1) rigorously defining all quantities in use, and 2) that the definition and the dimensions are linked: you cannot arbitrarily change the dimensions of some quantity without also changing its definition, or vice versa; and in particular you can't mix the definition of 'f' with the units of 'g', which is what seems to be happening here!
>
> This logical inconsistency can only be resolved by recognising that 'f' is a pure number so removing the 'e' unit. The same argument obviously applies to anything derived from 'f' such as the structure factor and the electron density.
>
> Cheers
>
> -- Ian
>
>
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