On 01/03/2010, at 20.44, Dale Tronrud wrote:

> Morten Kjeldgaard wrote:
>>
>> On 01/03/2010, at 19.01, James Holton wrote:
>>
>>> personal discourse.  If I review a paper that lists electron density
>>> in "1/A^3", I will tell the authors to fix it.  If a reviewer  
>>> tells me
>>> to change my "electron/A^3" to "A^-3", then I will simply tell the
>>> editor that the reviewer is mistaken.  Nothing I read on the BB is
>>> going to convince me otherwise, and
>>
>> Both are correct. "Electrons" is a nominal unit, you can omit it if  
>> you
>> wish. Mathematically, "electrons" never enter the electron density
>> equation, because the atomic scattering factor is dimensionless, so  
>> the
>> dimension of rho is given by the 1/V term.
>>
>   This is seriously close to a circular argument.  You can leave off
> "electrons" because they weren't there in the scattering factor?  We
> "unit lovers" are not proposing to put it in one place and not the  
> other.

The ambiguity probably arises because some authors choose to tabulate  
the atomic form factor in units of  "electrons". If you use that unit  
in the electron density equation, rho gets a unit of "electrons per  
cubic Ångstrøm". Mathematically, the atomic form factor is an function  
that -- for a specific atom type -- specifies the scattering  
efficiency, and it has no unit. Whether you specify the unit  
"electrons" or not doesn't matter, since there is no other unit that  
makes sense for the problem at hand. What *is* important is to specify  
what basic unit of length you are using to compute the unit cell  
volume. Thus, electron density could in principle be given in meter  
**-3 which would be correct according the the SI standard (but  
admittedely weird to a crystallographer.)

The same is true when talking about population density... you can  
specify it as 10 per square mile or 10 humans per square mile or 10  
persons per square mile. It's all the same, "persons" and "humans" are  
nominal units for the number and you can optionally omit it.  
Specifically, concerning dimensionless quantities, read section 1.3 of  
"The International System of Units" [1].

>  I'm puzzled by how you define "nominal" units.  Certainly it cannot
> be related to whether the variable is declared int or float?  I don't
> see a logical connection and I don't see an operational difference
> between the units you consider nominal and those you do not.  This
> distinction is at the heart of the discussion here: which units can
> be dropped at will and which must be kept?

It doesn't necessarily need to be a counting variable (integer). The  
atom form factor is an example of a variable that has the nominal unit  
of "electrons". An angle can be measured in degrees, but if specified  
in radians, that unit is nominal and can be omitted (I have personally  
heard you say "the phase angle is pi" but never "the phase angle is pi  
radians".)

>   If "countability" is the principal difference between the two
> classes of units what do you think of the unit "mole" (and I don't
> mean the animal that burrows underground ;-) ).  This is just a
> count of molecules and yet it is used pretty consistently.  I
> don't see people talking of the energy of a reaction in "Kcal" with
> out the "per mole" always being there.  Are you in favor of reporting
> reaction energies in "Kcal"?

This whole thing has to do with what units have been defined and  
standardized. One mole is a defined unit consisting of NA molecules,  
it's a basic unit in the SI system, and so when we work with  
properties of molecules, we need to use it. If we didn't have a  
definition for a mole, I suppose we would specify reaction energies  
"per molecule" in which case the unit would be implicitly understood  
and could be omitted. When we work with lengths, there are several  
standardized measures we can use, and we need to choose one, and we  
need to specify which one we are using. When working with volumes or  
areas, we need to derive the unit from the basic unit of length. This  
is specified and defined by whatever unit standard we choose to work  
with (SI, CGS, ...). So to answer your question:  being an SI man, I'm  
in favour of reporting Gibbs free energy in kJ/mol ;-)

Cheers,
Morten

[1] http://physics.nist.gov/Pubs/SP330/sp330.pdf