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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