Dale, On 19 April 2018 at 17:36, Dale Tronrud <de...@daletronrud.com> wrote:
> The meaning of the term "e/A^3" as used in Coot has nothing to do > with the charge of an electron. The intention of its authors is to > indicate that the value being represented by the map is the density of > electrons. It is the number of electrons per cubic Anstrom at that > point in space. > My intention was certainly not to imply that the unit designated 'e' in the units of electron density used by Coot should be taken to mean charge (in fact quite the opposite!), even though 'e' would normally be interpreted here as the atomic unit of charge (see https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication811e2008.pdf, Table 8). I, generally, find it useless to say that a number is a density > (units of "per volume") without saying density of what. This is not a > density of charge, not a density of mass, not a density of rabbits. It > is a density of electrons. > First, note that (SI Guide cited above, section 7), quote: "The value of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit". So by stating that the electron density is X electrons/A^3, you are claiming that the number in the above recipe is X and that the unit is 'electrons/A^3', correct? I just want to be sure we are talking about the same thing when referring to a 'unit'. There are numerous reasons why this terminology is inconsistent, semantically illogical, unconventional or impermissible under the universally accepted SI system of units (Système international d'unités), as well as possibly unlawful in the US! For starters, the object 'electron' to which we are referring here is already very clear from the definition, so it's quite unnecessary to repeat that object as a component of its units. Here is the definition of 'electron density': "Electron density is the volume number density of electrons, or the number of electrons per unit volume". For the definition of 'number density' see: https://en.wikipedia.org/wiki/Number_density, quote: "Volume number density is the number of specified objects per unit volume <https://en.wikipedia.org/wiki/Volume>: [image: n={\frac {N}{V}},] where *N* is the total number of objects in a volume *V*. In SI <https://en.wikipedia.org/wiki/International_System_of_Units> units, number density is measured in m−3, although cm−3 is often used." There is of course also line number density and area number density, such as population density, i.e. the number of people per unit area, or area number density of people. See https://en.wikipedia.org/wiki/Population_density, quote: "Therefore, the worldwide human population density is around 7,500,000,000 ÷ 510,000,000 = 14.7 per km2 (38 per sq. mi). If only the Earth's land area of 150,000,000 km2 (58,000,000 sq. mi.) is taken into account, then human population density increases to 50 per km2 (129 per sq. mile)." (note that the unit 'sq. mi' is not permitted under SI !). Seems pretty clear from this what the units of area number density are, and by analogy the units of volume number density. There doesn't seem to be any ambiguity here that what is being described is the number density of people, and for anyone who doesn't think Wikipedia has got it right, for an 'official' example of usage of number density, please refer to section 6.2.6 in the SI Guide cited above, quote: "6.2.6 ... Example: the number density of Pb atoms is 5 ×106 /m3 but not: the number density of Pb atoms is 5 M/m3." Furthermore, are you saying that, for example, the statement "the mass of an electron is 9.10938356(11)×10^−31 kg" is not sufficiently clear and that it should be "9.10938356(11)×10^−31 kg/electron" instead? That would follow logically from your claim above. The problem with this is that this usage is clearly not permitted under the SI system, to which all modern scientific institutions and journals are signed up to using, without exception. Please see the above-mentioned SI Guide document, under 'Check List for Reviewing Manuscripts', item 6: "Information is not mixed with unit symbols (or names). For example, the form 'the water content is 20 mL/kg' is used and not '20 mL H2O/kg' or '20 mL of water/kg.' (See Sec. 7.5.)." So mixing the object 'electron' with the other units (A^-3 or kg) is absolutely not permissible in SI. Note that an 'electron' is like 'H2O' in the sense that both are instances of the class 'Matter', and so neither can possibly be units (i.e. instances of the class 'Units'). This makes perfect sense for several reasons: First, the identical unit applied to different objects is 'commensurable', i.e. it can be compared, subtracted or added. So if the number density of electrons is 5 A^-3 and the number density of protons is 10 A^-3 we can just add number densities and get the total number density of particles or matter objects (electrons or protons, or yes even rabbit volume density can be added to electron density!) = 15 A^-3. Different units are 'incommensurable', i.e. can't be compared or equated. Even if there were such units as 'electron', 'proton' or 'rabbit' and if the units of number density really were 'electrons/A^3' or 'protons/A^3' or rabbits/m^3 then these are clearly different units so are incommensurable. It would therefore not be possible to calculate the total number density of matter objects - you may as well ask what is the sum of 5 kg and 10 meters! Second, if you insist that every object has its own collection of units associated with it, a vast plethora of potential units would spring into existence! Surely the whole point of units is to make things simple and keep the number of units to a minimum, not to make things vastly more complicated: the same unit can be applied to many different objects. Or are you saying that this exception to the rule only applies to number densities, in which case what's the justification for making it a special case? Third, we can talk about, 'length', 'mass' etc. as abstract objects divorced from any particular hunks of matter. As you are located in the US, take note also that SI has the force of law, quote from SI Guide: "In January 1991, the Department of Commerce issued an addition to the Code of Federal Regulations entitled 'Metric Conversion Policy for Federal Agencies,' 15 CFR 1170, which removes the voluntary aspect of the conversion to the SI for Federal agencies and gives in detail the policy for that conversion. Executive Order 12770, issued in July 1991, reinforces that policy by providing Presidential authority and direction for the use of the metric system of measurement by Federal agencies and departments". The important phrase here is 'removes the voluntary aspect', which I would say means that you can't make up your own units such as 'electron'. So for Federal agencies and departments it is actually unlawful to use units such as 'electron' that are not permitted under the SI system! Actually it's worse than that since 'electron' is not even a unit, SI or otherwise! Comparing or equating different units, or worse, using something which isn't even a unit where a unit is expected, is called a 'category mistake' (https://en.wikipedia.org/wiki/Category_mistake <https://en.wikipedia.org/wiki/Category_mistake)>), or in common parlance 'comparing apples with oranges'. The concept of 'category mistake' will be familiar to programmers of classed-based object-oriented languages, such as C++ and Java. Here 'category' = 'class', and it's a semantic error (trapped by the compiler) to compare or equate an object instantiated from one class with an object from a different non-inherited class, simply because it's not semantically meaningful to do so. It's also a category mistake to equate the property of an object (or any part of an object) to the object itself, and to attempt to apply a method of one class to objects instantiated from another non-inherited class. The SI system (see the Guide, sections 4 & 5) consists of several tiers: 1. The 7 SI base units: metre, kilogram, second and 4 others. 2. SI derived units: combinations of the base units, such as metre/second. 3. SI coherent derived units: units such as radian with special names and symbols. 4. Decimal multiples and sub-multiples of all the above, such as nanometre. Then outside the SI system, but still acceptable for use in scientific publications, there are: 5. Non-SI units accepted for use with the SI: non-SI units that are acceptable owing to common usage, such as hour, degree, litre. 6. Non-SI units accepted by the International Committee for Weights and Measures, and therefore accepted into SI, such as electronvolt and dalton. 7. Non-SI units that are part of other international standards, such as the electronic charge 'e', being the atomic unit of charge, and the electron rest mass m_e. 8. Other non-SI units also accepted because of common usage in specialised fields, such as Angstrom, rad, barn. Finally we have units that are not permissible in any scientific publication: 9. CGS units, such as erg, dyne. 10. Other obsolete units, such as torr, micron, inch, foot, yard, chain, furlong, mile, pound, poundal, rod, perch, peck, bushel, etc. etc. Notice that the supposedly 'electron' ( or 'proton' or 'rabbit' or anything composed of matter) units, don't even make it into the list of impermissible units - simply because they are not even units! The closest units to 'electron' in the list are electronic charge 'e' and electronvolt 'eV' (unit of energy), but as discussed above, they have nothing to do with electron density. A requirement of SI is that if you use a non-SI unit for some quantity you are _required_ to state the value of that quantity in SI units, or the conversion factor. See above-cited Check List for Reviewing Manuscripts, item 2, quote: "Only SI units and those units recognized for use with the SI are used to express the values of quantities. Equivalent values in other units are given in parentheses following values in acceptable units only when deemed necessary for the intended audience. (See Chapter 2.)". Therefore if you believe that 'electron' is a unit, please state the value of the 'electron' unit in the equivalent SI units. If you have trouble with that, try stating the value of 1 rabbit or 1 person (for the supposed population density of rabbits or people that incorporates 'rabbit' or 'person' in the unit itself) in their equivalent SI units, or in any units for that matter! Of course this is all semantic nonsense because it's plainly a category mistake to attempt to apply the method unitConversion in class Units to an object instantiated from the non-inherited class Matter (e.g. electron, proton, person, rabbit etc.). Let's make it simpler and drop the units of volume, and talk instead about form factors (or structure factors), so the unit of electron density is whatever is the unit of the form factor divided by volume (by the usual equation for electron density). Now a form factor is defined as the _ratio_ of the amplitude of scattering of the specified atom at some specified scattering angle divided by the scattering of a single electron (for which the angle is irrelevant). Being a ratio of two quantities with identical dimensions, it can only be a pure dimensionless and unitless number (just like the unitless refractive index, being the ratio of two speeds). It follows logically from your claim above that the units of electron density are 'electrons/A^3', that the units of form factors would have to be 'electrons', but this is impossible if form factors are unitless pure numbers (and anyway electrons are not units!). A 'reductio ad absurdum' (reduce to absurdity) argument can be used here: the form factor at zero scattering angle is of course the number of electrons in the atom or ion, so do you say 'the number of electrons in a carbon atom is 6 electrons', or 'the number of electrons is 6'? The former follows logically if the units of electron density are 'electrons/A^3', because then the units of a form factor would be 'electrons'. However that is a clear category mistake: it equates a property of the 'electrons' object ('number of electrons'), which is obviously an instance of the 'Number' class, with the object itself ('6 electrons') a blatant commission of a category mistake. The second statement above is the only semantically valid one: we must equate 'number of electrons' (a Number class variable) with '6' (a Number class constant). Doing anything else is 'comparing apples with oranges'! Now the calculations are pretty simple. When the refinement program > produces a model one of its parameters is a scale factor which relates > the relative values of the observed intensity to those calculated from > the model. This scale factor, which is usually not written out in an > obvious place since its value is not very interesting, is just a simple > number, not a function of resolution or anything like that. It's value > is also sensitive to a number of assumptions (like you are modeling > everything in your unit cell) so it is not particularly reliable. This > means you shouldn't take the density values of your map terribly > seriously. The differences in density from place to place is much more > reliable. > > This has led to the practice of converting the density values to > normalized values. Here you calculate the rms value of the points in > the map and divide all the density points by that number. "rms" is > simply what is says "Root of the Mean of the Squares". Again this is > just a number, nothing fancy. > > The only trick is deciding what region of space to sum. The crystal > is composed of regions which are occupied by ordered molecules (One > hopes this includs the molecule you are interested in!) and regions of > bulk solvent. You could reasonably conclude that the rms's of these > regions should be considered different properties of the map. I'm not > aware of any software that actually tries to calculate an rms for just > the region of the map that contains ordered structure. > > What was done in the past was to simply calculate the sum over the > region of the map presented to the program, and this was usually a > rectangular box inscribed over the molecule. The corners of that box > would cover some amount of bulk solvent (and symmetry related molecules) > which depends on a lot of factors which shouldn't be affecting your > choice of contour levels. This method is inconsistent and causes > confusion. > > The conventional method today is to calculate the rms by summing > over the entire asymmetric unit. This, at least, creates consistency, > and can be calculated from the Fourier coefficients making it an easy > number to come up with. Many programs now use this method, including > Coot (As I understand). It does have the drawback that a crystal with a > large amount of solvent will have a lower rms than one that is very dry, > even if the variation within the ordered structure is the same. You > need to be aware of this when interpreting maps whose contours are based > on the rms. Showing the contour level both as rms and electrons/A^3 is > an attempt to provide a fuller description. > > Ian is certainly right -- the rms of a function is not a "sigma". > This confusion is a problem that is endemic in our field! "sigma" is > shorthand for standard deviation which is a measure of the uncertainty > in our knowledge about a value. The rms is not a measure in any way of > uncertainty -- It is simply a description of how variable the values of > the function are. James Holton has written a very nice paper on this > topic, but I don't have the reference on hand. > > The excuse for this confabulation is that people believe, in a Fo-Fc > style map, most of the values should represent factors other than errors > in the structural model and therefore one can estimate the uncertainty > of the map by calculating the rms over the map. This assumption is > highly questionable and unreliable. The major problem then arises when > the same logic is extrapolated to a 2Fo-Fc style map. In these maps the > variability in the ordered region of the crystal is all "signal" so > calculating an rms really has nothing to do with uncertainty. > > Describing your contours or peaks in an Fo-Fc style map by rms can > sort-of, kind-of, be justified, but it makes no sense for a 2Fo-Fc style > map. If I want to be really serious about deciding if a peak in a map > may be missing atoms, I will leave some known atoms out of the model and > see how the heights of their difference peaks compare to the heights of > the mysterious peaks. This method is fairly insensitive to the > systematic problems that affect both rms and electrons/A^2. > On all your remaining points above we are thankfully in complete agreement! Cheers -- Ian
