Hi Silvia, I don't know about sophisticated, but it's certainly interminable!
Whether you need to modify your slides (actually only your 3rd slide seems to be relevant to the discussion) obviously depends on which definition you choose to along with. If you use what I can call the 'dimensionless' definition you need do nothing, since you are already using that definition, i.e. f is a ratio of amplitudes (or sqrt of ratio of intensities), therefore any units must cancel out and the result must be dimensionless and unitless, just as in the definition of refractive index (as Marc pointed out). If you accept the alternative where f is not dimensionless (it has dimensions of an amplitude, i.e. length), you need to re-define f as the amplitude (or sqrt of the intensity) of scattering by the atom, i.e. the sqrt of the numerator in your equation for 'f^2'. Then you need to define a new unit of f (call it [f]) as the amplitude of scattering by a free electron, i.e. the sqrt of the denominator of your equation: this is what pretty well everyone is calling an 'electron' - though definitely not to be confused with a real electron! Finally f is expressed as a multiple of its unit in the usual way: f = n[f] where 'n' is the numerical value of f (for example: "f = 10 electrons" if n=10). So to summarise, using this definition f has dimensions of 'length' and units of 'electrons'. Further for the 2nd definition, if you're tempted to abbreviate 'electrons' to 'e' as is often seen, you also have to remember that 'e' here has nothing to do with the conventional physicists' definition, namely the charge on the electron (though it seems I'm the only one here who thinks this is very confusing notation). If you do decide to go down this path I would recommend following the notation in Bernhard Rupp's book: he uses the correct abbreviation for the electron, namely 'e-' avoiding the confusion with 'e', but again don't confuse this with the real electron! As Gerard pointed out you could equally well use the terms 'positron' and 'e+' since the sign is irrelevant for scattering. Note that both definitions are perfectly internally self-consistent - no dimensionality issues - so the choice is purely a matter of definition, certainly neither can be said to be right or wrong, and therefore the definitions are essentially totally arbitrary. It really comes down to whether you wish to change the definition you're accustomed to, and whether you can live with the ambiguities in the definition of 'electron' (and 'e' if you use it). It seems to me that it's this ambiguity that actually instigated this whole thread! One point highlighted by your slides: you say that at zero scattering angle f0 = Z (atomic number). This is perfectly correct, however note that Z being a pure number is never expressed as say 'Z = 10 electrons', always as just 'Z = 10' so if you express f in 'e-' units you need to say 'f0 = Ze-', otherwise the equation is dimensionally inconsistent. Anyway I beginning to sound like Microsoft explaining how to install their competitors' browsers in as fair a way as possible in order to placate the European Commission, so I'll stop there! Cheers -- Ian > On Mon, Mar 1, 2010 at 7:06 PM, Silvia Onesti > <silvia.one...@elettra.trieste.it> wrote: > I feel uneasy at entering this sophisticated discussion, but since it looks > like the very interesting, learned but subtle and complex statements by Ian, > Marc, Gerard & co seem unable to shake people assumptions that the atomic > scattering factor is expressed in electrons, can I provide a couple of very > basic slides that I have been using for teaching an undergraduate course? > > As far as I know that is the DEFINITION of the atomic scattering factor. The > scattering-equivalent of one electron is just a convenient unit. Analogous > to saying that the charge of a proton is +1, rather than 1.6x10-19 Coulomb. > > Now I hope that the experts are not going to find other mistakes in my > slides! > > Silvia > > ------------------------------------------------------------------------------------------------ > Silvia Onesti > > Sincrotrone Trieste S.C.p.A. > SS 14 - km 163,5 - AREA Science Park, 34149 Basovizza, Trieste ITALY > > Email: silvia.one...@elettra.trieste.it > Tel. +39 040 3758451 > Mob +39 366 6878001 > > http://www.elettra.trieste.it/PEOPLE/index.php?n=SilviaOnesti.HomePage > http://www.sissa.it/sbp/web_2008/research_structuralbio.html > ------------------------------------------------------------------------------------------------ > > On Mon, 1 Mar 2010 09:10:44 -0800 > Ronald E Stenkamp <stenk...@u.washington.edu> wrote: > Hi. > I'm a little reluctant to get into this discussion, but I'm greatly > confused > by > it all, and I think much of my confusion comes from trying to understand > one > of Ian's assumptions. > Why are the scattering factors viewed as dimensionless quantities? In > the International Tables (for example, Table 6.1.1.1 in the blue books), > the > scattering factors are given "in electrons". In the text for that > section, > the scattering factors are obtained from an integral (over space) of the > electron density. So there's some consistency there between scattering > factors > in units of electrons and electron density in electrons/(Angstrom**3). > What's > gained at this point by dropping the word "electron" from all of these > dimensions? > Ron > On Sat, 27 Feb 2010, Ian Tickle wrote: > >>> >>> 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 >> >> >> Disclaimer >> This communication is confidential and may contain privileged information >> intended solely for the named addressee(s). It may not be used or >> disclosed >> except for the purpose for which it has been sent. If you are not the >> intended recipient you must not review, use, disclose, copy, distribute or >> take any action in reliance upon it. 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