I'm not aware that anyone has suggested the notation "rho e/Å^3".
I think you misunderstood my point, I certainly didn't mean to imply that anyone
had suggested or used that notation, quite the opposite in fact. My point was that
you said that you use the term 'electron density' to define two different things
either at the same time or on different occasions, but that to resolve the
ambiguity you use labels such as 'e/Å^3' or 'sigma/Å^3' attached to the values. My
point was that if I needed to use these quantities in equations then the rules of
algebra require that distinguishable symbols (e.g. rho and rho') be assigned,
otherwise I would be forced into the highly undesirable situation of labelling the
symbols with their units in the equations in the way you describe in order to
distinguish them. Then in my 'Notation' section my definitions of rho & rho'
would need to be different in some way, again in order to distinguish them: I could
not simply call both of them 'electron density' as you appear to be doing.
The question of whether your units of electron density are '1/Å^3' or 'e/Å^3' clearly comes down to
definition, nothing more. If we can't agree on the definition then we are surely not going to agree on
the units! Actually we don't need to agree on the definition: as long as I know what precisely your
set of definitions is, I can make the appropriate adjustments to my units & you can do the same if
you know my definitions; it just makes life so much easier if we can agree to use the same definitions!
Again it comes down to the importance of having a 'Notation' section so everyone knows exactly what
the definitions in use are. My definition of electron density is "number of electrons per unit
volume" which I happen to find convenient and for which the appropriate units are '1/Å^3'. In
order for your choice of units 'e/Å^3' to be appropriate then your definition would have to be
"electric charge per unit volume", then you need to include the conversion factor 'e' (charge
!
on the electron) in order to convert from my "number of electrons" to your "electric charge", otherwise your values
will all be very small (around 10^-19 in SI units). I would prefer to call this quantity "electric charge density" since
"electron density" to me implies "density of electrons" not "density of charge". I just happen to think that
it's easier to avoid conversion factors unless they're essential.
Exactly the same thing of course happens with the scattering factor: I'm using
what I believe is the standard definition (i.e. the one given in International
Tables), namely the ratio of scattered amplitude to that for a free electron
which clearly must be unitless. So I would say 'f = 10' or whatever. I take
it that you would say 'f = 10e'. Assuming that to be the case, then it means
you must be using a different definition consistent with the inclusion of the
conversion factor 'e', namely that the scattering factor is the equivalent
point electric charge, i.e. the point charge that would scatter the same X-ray
amplitude as the atom. I've not seen the scattering factor defined in that way
before: it's somewhat more convoluted than the standard definition but still
usable. The question remains of course - why would you not want to stick to
the standard definitions?
BTW I assume your 'sigma/Å^3' was a slip and you intended to write just 'sigma'
since sigma(rho) must have the same units as rho (being its RMS value), i.e.
1/Å^3, so in your second kind of e.d. map rho/sigma(rho) is dimensionless (and
therefore unitless). However since rho and sigma(rho) have identical units I
don't see how their ratio rho/sigma(rho) can have units of 'sigma', as you seem
to imply if I've understood correctly?
What I'm more concerned about is when you assign a numerical value to
a quantity. Take the equation E=MC^2. The equation is true
regardless
of how you measure your energy, mass, and speed. It is when you say
that M = 42 that it becomes important to unambiguously label 42 with
its units. It is when you are given a mass equal to 42 newtons, the
speed of light in furlongs/fortnight, and asked to calculate
the energy
in calories that you have to track your units carefully and
perform all
the proper conversions to calculate the number of calories.
I can only agree with you there, but I never suggested or implied that a mass
value (or speed or energy) should be given without the appropriate units
specification, or that one should not take great care to track the units
conversions.
Actually many equations in crystallography are not as friendly as
this one since they have conversion factors built into their standard
formulations. With the conversion factor built in you are then
restricted to use the units that were assumed. The example of this
that I usually use is the presence of the factor of 1/V in the Fourier
synthesis equation. It is there only because our convention is to
measure scattering in e/Unit Cell and electron density in e/Å^3. The
factor of 1/V is simply the conversion factor that changes
these units.
I don't go along with you on that: the factor V is there simply because the
definition of electron density we are using requires it, without it you get
something other than the electron density as usually defined. Also without V in
the equation you would not be able to compare electron density values for crystals
with different values of V - a true apples & oranges situation! V is not a
true unitless conversion factor like 'pi', 'radian', 'degree' etc, which are all
constants: V is a variable and moreover it is not unitless so its value will depend
both on the crystal and on the assumed units.
Mathematicians use the same units in reciprocal and real space and do
not have this term in their Fourier synthesis equation.
I can only conclude that the mathematicians you are referring to work with
idealised crystals whose unit cell volumes are always the same.
Since the conventional forms of the equations in our field often
have conversion factors built in (e.g. 1/V or 2 Pi radians/cycle),
we have to worry about the units of the variables in ways that pure
physics people usually don't.
I don't see why we have to worry any less or more about units than the
physicists: crystallography is essentially a branch of physics (the biologists
contributing to this forum may disagree!), so I don't see why the problems of
dealing with units should be any different.
When calculating structure factors from
coordinates we can't just say that "x" is the x coordinate of an atom,
we have to specify that this "x" is measured in fractional
coordinates.
I can only re-iterate the importance of definitions. So in my notation section
I might have:
x : fractional co-ordinate of an atom (unitless),
xo: orthogonal co-ordinate of an atom (Å units).
Then I can use both 'x' (e.g. x = 0.1234) and 'xo' (e.g. xo = 1.234 Å) in the body of my paper without fear of ambiguity.
>in which case
>one needs to be careful to avoid ambiguous definitions.
Which is exactly what I've been advocating. I'm glad we
have reached agreement.
Excellent! I take it then that henceforth you won't be using two incompatible
definitions of electron density?
Cheers
-- Ian
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