Dear X-ray Community,
I'm sorry to have dragged you all along on this journey. There is
something to Ian and Marc's arguments but I have been unable to understand
their point. That is my failure and I don't wish to subject the rest
of you to continued argument. Ian, Marc, and I should get together
in a bar somewhere and hash this out.
Dale Tronrud
Gerard Bricogne wrote:
> Dear all,
>
> Two slight confusions seem to have popped up intermittently in this
> thread, in messages other than those included here. The first one was
> related to the charge of the electron - even the colour code according to
> which its electron density should be displayed - and the other one to its
> mass, i.e. the assumption that the word "density" was to be taken literally
> as being in some way connected to a mass by unit volume.
>
> Regarding the first, the sign of the charge plays no role, as shown by
> the fact that the Thomson scattering formula involves the square of the
> electron charge. This could be seen as an instance of TCP invariance: a free
> positron would scatter X-rays to exactly the same degree as a free electron.
> If there is something that would deserve to be called a unit in the original
> context of this question, i.e. a unit for structure factors, it would be (as
> was pointed out by many contributors) the X-ray scattering power of a free
> electron (or positron). A pedantic name for such a unit could be "electron
> qua X-ray scatterer", which the general aversion to Latin would immediately
> shorten to "electron", thereby explaining the current practice. At least the
> pedantic name would have the merit of making it clear that we are not
> referring to the electron in relation to its charge, but in relation to its
> scattering behaviour towards X-rays.
>
> Regarding the second, the word "density" has long been freed from its
> original connection with mass. For instance one speaks about a "probability
> density", which is stripped of any physical association and is related to
> the notion of a measure in the theory of integration. One even encounters
> the expression "number density" independently of any notion of probability,
> to designate a concentration of things that can be counted (as opposed to
> measured). The meaning of density, as has again been explained by several
> contributors, is that the density of 'whatever' is the amount of 'whatever'
> per unit volume; so that integrating it over a specified region delivers a
> result in the unit in which 'whatever' is measured. Replacing 'whatever' by
> X-ray scattering power measured in units of "electron qua X-ray scatterer" -
> or just "electron" - would seem to make everything that has been said
> consistent.
>
> To get back to the original question: if the reference X-ray scatterer
> was taken to be the proton (or antiproton), then the numerical values of f0,
> f' and f" giving the strength of the anomalous scattering of *electrons*
> would obviously change, so there is indeed an underlying dimensionality to
> these numbers via Thomson's formula; but since crystallographers live off
> X-ray scattering from electrons, such a change of units would seem a rather
> daft idea, the possibility of which I would not expect to have occurred to
> any journal editor or reviewer while staring at Table I ... - so if we have
> such a natural unit for what we are looking at, f0, f', f" are, for all
> intents and purposes, pure numbers.
>
> I hope this long message is only minimally eristic, and rather more
> dialectical :-)) ... .
>
>
> With best wishes,
>
> Gerard.
>
> --
> On Sun, Feb 28, 2010 at 07:19:48AM -0800, James Holton wrote:
>> I suppose I could point out that the letter "e" has many more "universal"
>> meanings that just denoting electric charge, such the base of the natural
>> logarithm, the identity element in set theory, or even a musical note.
>> But, today I found that "e" can also stand for "eristic", a new word I
>> learned while reading the following web page:
>>
>> http://en.wikipedia.org/wiki/Flame_war
>>
>> -James Holton
>> MAD Scientist
>>
>>
>> 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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