Ian Tickle wrote:
>>    For maps it is important for people to know if their map is
>> in e/Å^3 or sigma/Å^3.  Both maps are commonly encountered in
>> this field and both are called electron density maps.  I could
>> put a note on my home page stating that whenever I talk about
>> a map I give numbers in e/Å^3 but it is more convenient for the
>> reader if I just put the convention next to the number.
> 
> The problem obviously arises here because of the ambiguity of using the 
 >same name ('electron density') to define two different quantities
 >simultaneously.  You'll recall I pointed out that "Problems with units
 >often stem from an ambiguous definition"!  Let's suppose you were writing
 >equations involving both of these quantities, and let's say 'electron
 >density' = 'rho'.  Then you might write in a computer program the perfectly
 >valid statement:
> 
> 	rho = rho/sigma
> 
> where the 'rho' on each side means different things.  However this is 
 >certainly not valid as an algebraic statement as it stands (where sigma may
 >take any value).  Note that it is not the usual practice to carry the units
 >with the variables in the equations in the way you suggest in order to
 >allow you to distinguish them (it would make the equations pretty

    I'm not aware that anyone has suggested the notation "rho e/Å^3".
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.

    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.
Mathematicians use the same units in reciprocal and real space and do
not have this term in their Fourier synthesis equation.

    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.  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.
The way we write the equation forces us to use this particular
coordinate system in a way that E=MC^2 does not.

 >unreadable!), so the units are actually irrelevant are far as equations are
 >concerned.  So you would have to write something like:
> 
> 	rho' = rho/sigma
> 
> then the ambiguity is resolved.  Moreover you would now need to define 
 >rho and rho' in a way that the reader would be able to distinguish them,
 >for example you might say "rho = electron density = number of
 >electrons/Angstroem^3" and "rho' = electron density Z-score = rho/sigma(rho)".
 >So now you have not only been forced to distinguish the variable names,
 >you also have had to distinguish them in their definitions, hence it
 >should no longer be necessary to distinguish them by labelling their units.
> 
> Now of course labelling things in a computer output in order to distinguish 
 >things that might be confused is quite a different issue from distinguishing
 >things in equations, for one thing you're free to write anything you want in
 >your own program, but equations have to obey the rules of algebra.  I don't
 >know what was the context that gave rise to the original question but I think
 >it's quite likely to have been the equation f = f0 + f' + if", 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.

Dale Tronrud

P.S. to respond out-of-band to Dr. Schiltz: On the US flag I see 7 "red stripes",
6 "white stripes", and 50 "stars".  If I state "I see 7" I have conveyed no
useful information.


> 
> Cheers
> 
> -- Ian
> 
> 
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