Re: [ccp4bb] units of the B factor
This is absolutely correct. Radian is in fact just another symbol for 1. Thus : 1 rad = 1 From the official SI documentation (http://www.bipm.org/en/si/si_brochure)(section 2.2 - table 3) : The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities. Marc Quoting Ian Tickle i.tic...@astex-therapeutics.com: Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] units of the B factor
James Holton wrote: No No No! This is not what I meant at all! I am not suggesting the creation of a new unit, but rather that we name a unit that is already in widespread use. This unit is A^2/(8*pi^2) which has dimensions of length^2 and it IS the unit of B factor. That is, every PDB file lists the B factor as a multiple of THIS fundamental quantity, not A^2. If the unit were simply A^2, then the PDB file would be listing much smaller numbers (U, not B). Hi James I must confess that I do not understand your point. If you read a value from the last column of a PDB file, say 27.34, then this really means : B = 27.34 Å^2 for this atom. And, since B=8*pi^2*U, it also means that this atom's mean square atomic displacement is U = 0.346 Å^2. It does NOT mean : B = 27.34 Born = 27.34 A^2/(8*pi^2) = 27.34/(8*pi^2) A^2 = 0.346 Å^2 as you seem to suggest. If it was like this, the mean square atomic displacement of this atom would be U = 0.00438 Å^2 (which would enable one to do ultra-high resolution studies). (Okay, there are a few PDBs that do that by mistake, but not many.) As Marc pointed out, a unit and a dimension are not the same thing: millimeters and centimeters are different units, but they have the same dimension: length. And, yes, dimensionless scale factors like milli and centi are useful. The B factor has dimension length^2, but the unit of B factor is not A^2. For example, if we change some atomic B factor by 1, then we are actually describing a change of 0.013 A^2, because this is equal to 1.0 A^2/(8*pi^2). What I am suggesting is that it would be easier to say that the B factor changed by 1.0, and if you must quote the units, the units are B, otherwise we have to say: the B factor changed by 1.0 A^2/(8*pi^2). Saying that a B factor changed by 1 A^2 when the actual change in A^2 is 0.013 is (formally) incorrect. The unfortunate situation however is that B factors have often been reported with units of A^2, and this is equivalent to describing the area of 80 football fields as 80 and then giving the dimension (m^2) as the units! It is better to say that the area is 80 football fields, but this is invoking a unit: the football field. The unit of B factor, however does not have a name. We could say 1.0 B-factor units, but that is not the same as 1.0 A^2 which is ~80 B-factor units. Admittedly, using A^2 to describe a B factor by itself is not confusing because we all know what a B factor is. It is that last column in the PDB file. The potential for confusion arises in derived units. How does one express a rate-of-change in B factor? A^2/s? What about rate-of-change in U? A^2/s? I realized that this could become a problem while comparing Kmetko et. al. Acta D (2006) and Borek et. al. JSR (2007). Both very good and influential papers: the former describes damage rates in A^2/MGy (converting B to U first so that A^2 is the unit), and the latter relates damage to the B factor directly, and points out that the increase in B factor from radiation damage of most protein crystals is almost exactly 1.0 B/MGy. This would be a great rule of thumb if one were allowed to use B as a unit. Why not? The authors of both papers make it perfectly clear what their quantities are, so there is no risk of confusion. Borek et. al. (2007) systematically use change of B per dose, reported in units of A^2/MGy. Kmetko et. al. (2006), use change of U per dose (in their table 2), also repoprted in units of A^2/MGy. There is nothing wrong with this. If two teams of scientists investigate how the size of a margherita pizza changes when irradiated with microwaves, and one of them reports the change of radius per dose, in m/Gy, whereas the other one reports the change in circumference per dose, also in m/Gy, would you get confused because their values are not the same, but their units are ? Would you think that, because the results systematically differ by a factor 2*pi, there is a problem with the units ? Interesting that the IUCr committee report that Ian pointed out stated we recommend that the use of B be discouraged. Hmm... Good luck with that! I agree that I should have used U instead of u^2 in my original post. Actually, the u should have a subscript x to denote that it is along the direction perpendicular to the Bragg plane. Movement within the plane does not change the spot intensity, and it also does not matter if the x displacements are instantaneous, dynamic or static, as there is no way to tell the difference with x-ray diffraction. It just matters how far the atoms are from their ideal lattice points (James 1962, Ch 1). I am not sure how to do a symbol with both superscripts and subscripts AND inside brackets that is legible in all email clients. Here is a try: B = 8*pi*usubx/sub^2. Did that work? I did find it interesting that the 8*pi^2 arises from the fact that diffraction occurs in angle space, and so factors of 4*pi
Re: [ccp4bb] units of the B factor
On Sun, 2009-11-22 at 23:33 -0800, Dale Tronrud wrote: I could be describing my angle as 1.5 radians, 1.5 degrees, or 1.5 cycles (or 1.5 of the mysterious grad on my calculator). I thought that use of degrees is based on dividing a circle into 360 parts - roughly one per day (then in geography they somehow divide a degree day into 60 minutes - not 24 hours, go figure). Such agreement, while workable, leads to some ugly results (just like geocentric system is actually workable but cumbersome). For instance, if angles are measured in degrees and x1 sin x ~ pi * x / 180 Not a big deal, really, but this is one reason to use radians instead, since then you get sin x ~ x On the other hand, Christoffa Corombo would add 12 minutes to Santa Maria's longitude when traveling 12 extra nautical miles westward on his voyage to what he thought was India. I wonder how enraged he would be should mathematician appear on the deck and start arguing that he should add 0.00349 radians instead. There are many other examples of such agreements. For instance, the singular choice of axes permutation in P21212 is to make sure that two-fold is along c. We could agree instead to always have abc - it's workable, but we would have to keep two more space groups around (hope I didn't make too many factual mistakes here). I witnessed once a physics professor having a psychotic break when someone mentioned SI units during discussion of Maxwell's equations. Well, electrical engineer will probably try to electrocute you if forced to measure current in franklins per second (i.e. statamperes) because a theoretician wants to get rid of Coulomb's force constant. PS. By the way, did you notice that pi^2 ~ g ? I remember reading long time ago that this is due to the early choice of the meter length as that of a pendulum with two seconds period of motion. Talk about anthropic principle... --
Re: [ccp4bb] units of the B factor
Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability
Re: [ccp4bb] units of the B factor
will be something like B in A^2 radian^2 and u_x^2 in A^2 cycle^2. It would be much clearer it someone figured out exactly what those units are and we started properly stating the units of each. I'm sorry that I don't have the time myself for this project. Dale Tronrud P.S. As for your distinction between the convenience units used to measure angles and the absolutely required units of length and mass: all units are part of the coordinate systems that we humans impose on the universe. Length and mass are no more fundamental than angles. Feet and meters are units chosen for our convenience and one converts between them using an arbitrary scaling constant. In fact the whole distinction between length and mass is simply a matter of convenience. In the classic text on general relativity Gravitation by Miser, Thorne and Wheeler they have a table in the back of Some Useful Numbers in Conventional and Geometrized Units where it lists the mass of the Sun as 147600 cm and and the distance between the Earth and Sun as 499 sec. Those people in general relativity are great at manipulating coordinate systems! -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company
Re: [ccp4bb] units of the B factor
Zitat von marc.schi...@epfl.ch: Dale Tronrud wrote: While it is true that angles are defined by ratios which result in their values being independent of the units those lengths were measured, common sense says that a number is an insufficient description of an angle. If I tell you I measured an angle and its value is 1.5 you cannot perform any useful calculation with that knowledge. I disagree: you can, for instance, put this number x = 1.5 (without units) into the series expansion for sin X : x - x^3/(3!) + x^5/(5!) - x^7/(7!) + ... and compute the value of sin(1.5) to any desired degree of accuracy (four terms will be enough to get an accuracy of 0.0001). Note that the x in the series expansion is just a real number (no dimension, no unit). ... However you get this Taylor expansion under the assumption that sin'(0)=1 sin''(0)=0, sin'''(0)=-1, ... this only holds true under the assumption that the sin function has a period of 2pi and this 'angle' is treated as unitless. Taking e. g. the sine function with a 'degree' argument treated properly as 'unit' will result in a Taylor expansion showing terms with this unit sticking to them. Clemens
Re: [ccp4bb] units of the B factor
Marc SCHILTZ wrote: Hi James I must confess that I do not understand your point. If you read a value from the last column of a PDB file, say 27.34, then this really means : B = 27.34 Å^2 for this atom. And, since B=8*pi^2*U, it also means that this atom's mean square atomic displacement is U = 0.346 Å^2. It does NOT mean : B = 27.34 Born = 27.34 A^2/(8*pi^2) = 27.34/(8*pi^2) A^2 = 0.346 Å^2 as you seem to suggest. Marc, Allow me to re-phrase your argument in a slightly different way: If we replace the definition B=8*pi^2*U, with the easier-to-write C = 100*M, then your above statement becomes: It does NOT mean : C = 27.34 millimeters^2 = 27.34 centimeter^2/100 = 27.34/100 centimeter^2 = 0.2734 centimeter^2 Why is this not true? If it was like this, the mean square atomic displacement of this atom would be U = 0.00438 Å^2 (which would enable one to do ultra-high resolution studies). I feel I should also point out that B = 0 is not all that different from B = 2 (U = 0.03 A^2) if you are trying to do ultra-high resolution studies. This is because the form factor of carbon and other light atoms are essentially Gaussians with full-width at half-max (FWHM) ~0.8 A (you can plot the form factors listed in ITC Vol C to verify this), and blurring atoms with a B factor of 2 Borns increases this width to only ~0.9 A. This is because the real-space blurring kernel of a B factor is a Gaussian function with FWHM = sqrt(B*log(2))/pi Angstrom. The root-mean-square RMS width of this real-space blurring function is sqrt(B/8*pi^2) Angstrom, or sqrt(U) Angstrom. This is the real-space size of a B factor Gaussian, and I, for one, find this a much more intuitive way to think about B factors. I note, however, that the real-space manifestation of the B factor is an object that can be measured in units of Angstrom with no funny scale factors. It is only in reciprocal space (which is really angle space) that we see all these factors of pi popping up. More on that when I find my copy of James... -James Holton MAD Scientist
Re: [ccp4bb] units of the B factor
Ed, For instance, if angles are measured in degrees and x1 sin x ~ pi * x / 180 sin x ~ x Your equations cannot simultaneously be true in fact the 1st one is obviously wrong, the 2nd is right. In the 1st case I think you meant (substituting 'x*deg' for 'x' in your correct 2nd equation): sin(x*deg) ~ (x*deg) for (x*deg) 1 where 'deg' = pi/180. Therefore we have: sin(x*deg) ~ pi*x/180 There are many other examples of such agreements. For instance, the singular choice of axes permutation in P21212 is to make sure that two-fold is along c. We could agree instead to always have abc - it's workable, but we would have to keep two more space groups around (hope I didn't make too many factual mistakes here). Just one: it's a=b=c. In any case, this comment is analogous to Henry Ford's famous sales pitch for the Model T: You can have a car in any colour so long as it's black. Tell me, which would you say makes more sense: a) 1 person spends 10 secs adding 10 lines to the syminfo file once and for all, or b) many people post queries to CCP4BB about re-indexing their MTZ files because the processing mis-identified 2-fold screw axes from the systematic absences? PS. By the way, did you notice that pi^2 ~ g ? I ... and did you notice that e^(i*pi) + 1 = 0 connects the 5 fundamental mathematical constants? - that also has nothing whatsoever to do with this thread ;-). 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] units of the B factor
of 2 Pi radian/cycle or 1/(2 Pi) cycle/radian when switching between. I agree with Ian that the 8 Pi^2 factor in the conversion of u_x^2 to B looks suspiciously like 2 (2 Pi)^2 and it is likely a conversion of cycle^2 to radian^2. I can even imagine that the derivation of effect of distortions of the lattice points that lead to these parameters would start with a description of these distortions in cycles, but I also have enough experience with this sort of problem to know that you can only be certain of these units after going back to the root definition and tracking the algebra forward. In my opinion the Mad Scientist is right. B and u_x^2 represent the same quantity reported with different units (or conventions if you will) and the answer will be something like B in A^2 radian^2 and u_x^2 in A^2 cycle^2. It would be much clearer it someone figured out exactly what those units are and we started properly stating the units of each. I'm sorry that I don't have the time myself for this project. Dale Tronrud P.S. As for your distinction between the convenience units used to measure angles and the absolutely required units of length and mass: all units are part of the coordinate systems that we humans impose on the universe. Length and mass are no more fundamental than angles. Feet and meters are units chosen for our convenience and one converts between them using an arbitrary scaling constant. In fact the whole distinction between length and mass is simply a matter of convenience. In the classic text on general relativity Gravitation by Miser, Thorne and Wheeler they have a table in the back of Some Useful Numbers in Conventional and Geometrized Units where it lists the mass of the Sun as 147600 cm and and the distance between the Earth and Sun as 499 sec. Those people in general relativity are great at manipulating coordinate systems! -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex
Re: [ccp4bb] units of the B factor
James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit
Re: [ccp4bb] units of the B factor
Argument from authority, from the omniscient Wikipedia: http://en.wikipedia.org/wiki/Radian Although the radian is a unit of measure, it is a dimensionless quantity. The radian is a unit of plane angle, equal to 180/pi (or 360/(2 pi)) degrees, or about 57.2958 degrees, It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level. … the radian is now considered an SI derived unit. On Nov 23, 2009, at 1:31 PM, Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has
Re: [ccp4bb] units of the B factor
So... how do you measure or report a solid angle without invoking the steradian? sterdegrees? Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B
Re: [ccp4bb] units of the B factor
No, just like this: 'solid angle = 1.234' (or whatever its value is). Since the conversion unit 'steradian' = 1 (i.e. the dimensionless pure number 1) identically, 'a solid angle of 1.234 steradians' is identical to 'a solid angle of 1.234': the unit 'steradian' is redundant. Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 19:07 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor So... how do you measure or report a solid angle without invoking the steradian? sterdegrees? Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think
Re: [ccp4bb] units of the B factor
Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674 -- Marc SCHILTZ http://lcr.epfl.ch
Re: [ccp4bb] units of the B factor
That's still only by convention. Which was the point of this thread to begin with: let's settle on a convention. I'm surprised this is contentious. phx. Ian Tickle wrote: No, just like this: 'solid angle = 1.234' (or whatever its value is). Since the conversion unit 'steradian' = 1 (i.e. the dimensionless pure number 1) identically, 'a solid angle of 1.234 steradians' is identical to 'a solid angle of 1.234': the unit 'steradian' is redundant. Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 19:07 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor So... how do you measure or report a solid angle without invoking the steradian? sterdegrees? Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself
Re: [ccp4bb] units of the B factor
Ian, On Mon, 2009-11-23 at 17:34 +, Ian Tickle wrote: Ed, For instance, if angles are measured in degrees and x1 sin x ~ pi * x / 180 sin x ~ x Your equations cannot simultaneously be true in fact the 1st one is obviously wrong, the 2nd is right. In the 1st case I think you meant (substituting 'x*deg' for 'x' in your correct 2nd equation): Hmm... It's not the same x in these two equations - one is measured in degrees, the other in radians. Just one: it's a=b=c. In any case, this comment is analogous to Henry Ford's famous sales pitch for the Model T: You can have a car in any colour so long as it's black. Tell me, which would you say makes more sense: a) 1 person spends 10 secs adding 10 lines to the syminfo file once and for all, or b) many people post queries to CCP4BB about re-indexing their MTZ files because the processing mis-identified 2-fold screw axes from the systematic absences? Tough call. On one hand, refusing P22121's right to exist is discrimination, on the other - these are the subtleties that help understanding so this has some educational value. Then there is Ockham's razor (which I personally believe people sometimes take too far). I think you pose the question in the way which pushes towards certain answer, let me try it differently: Which one makes more sense: 1) people learning more about space groups and reading the manuals of the software they are using to process data or 2) adding more space groups and using more paper to print the International Tables for Crystallography (gently hugs an imaginary tree)? Seriously though, I think it makes sense to keep just P21212, because you don't get a different crystal form by axes permutation. PS. By the way, did you notice that pi^2 ~ g ? I ... and did you notice that e^(i*pi) + 1 = 0 connects the 5 fundamental mathematical constants? - that also has nothing whatsoever to do with this thread ;-). Oh yeah - e^(i*pi)=-1 is my favorite meditation object :-) Nicely connects arithmetics, geometry, calculus and complex analysis. Cheers, Ed. --
Re: [ccp4bb] units of the B factor
Nice Scott On Mon, Nov 23, 2009 at 1:07 PM, Ed Pozharski epozh...@umaryland.eduwrote: Ian, On Mon, 2009-11-23 at 17:34 +, Ian Tickle wrote: Ed, For instance, if angles are measured in degrees and x1 sin x ~ pi * x / 180 sin x ~ x Your equations cannot simultaneously be true in fact the 1st one is obviously wrong, the 2nd is right. In the 1st case I think you meant (substituting 'x*deg' for 'x' in your correct 2nd equation): Hmm... It's not the same x in these two equations - one is measured in degrees, the other in radians. Just one: it's a=b=c. In any case, this comment is analogous to Henry Ford's famous sales pitch for the Model T: You can have a car in any colour so long as it's black. Tell me, which would you say makes more sense: a) 1 person spends 10 secs adding 10 lines to the syminfo file once and for all, or b) many people post queries to CCP4BB about re-indexing their MTZ files because the processing mis-identified 2-fold screw axes from the systematic absences? Tough call. On one hand, refusing P22121's right to exist is discrimination, on the other - these are the subtleties that help understanding so this has some educational value. Then there is Ockham's razor (which I personally believe people sometimes take too far). I think you pose the question in the way which pushes towards certain answer, let me try it differently: Which one makes more sense: 1) people learning more about space groups and reading the manuals of the software they are using to process data or 2) adding more space groups and using more paper to print the International Tables for Crystallography (gently hugs an imaginary tree)? Seriously though, I think it makes sense to keep just P21212, because you don't get a different crystal form by axes permutation. PS. By the way, did you notice that pi^2 ~ g ? I ... and did you notice that e^(i*pi) + 1 = 0 connects the 5 fundamental mathematical constants? - that also has nothing whatsoever to do with this thread ;-). Oh yeah - e^(i*pi)=-1 is my favorite meditation object :-) Nicely connects arithmetics, geometry, calculus and complex analysis. Cheers, Ed. -- -- Scott D. Pegan, Ph.D. Assistant Professor Chemistry Biochemistry University of Denver
Re: [ccp4bb] units of the B factor
James, I don't think that you are re-phrasing me correctly. At least I can not understand how your statement relates to mine. You simply have to tell us whether a value of 27.34 read from the last column of a PDB file means : (1) B = 27.34 Å^2 , as I (and hopefully some others) think, or (2) B = 27.34 A^2/(8*pi^2) = 0.346 Å^2 , as you seem to suggest Once you have settled for one of the two options, you can convert your B to U and you will get for either choice : (1) U = 0.346 Å^2 (2) U = 0.00438 Å^2 Even small-molecule crystallographers (who almost always compute and refine U's) rarely see values as low as U = 0.00438 Å^2. Cheers Marc Quoting James Holton jmhol...@lbl.gov: Marc SCHILTZ wrote: Hi James I must confess that I do not understand your point. If you read a value from the last column of a PDB file, say 27.34, then this really means : B = 27.34 Å^2 for this atom. And, since B=8*pi^2*U, it also means that this atom's mean square atomic displacement is U = 0.346 Å^2. It does NOT mean : B = 27.34 Born = 27.34 A^2/(8*pi^2) = 27.34/(8*pi^2) A^2 = 0.346 Å^2 as you seem to suggest. Marc, Allow me to re-phrase your argument in a slightly different way: If we replace the definition B=8*pi^2*U, with the easier-to-write C = 100*M, then your above statement becomes: It does NOT mean : C = 27.34 millimeters^2 = 27.34 centimeter^2/100 = 27.34/100 centimeter^2 = 0.2734 centimeter^2 Why is this not true? If it was like this, the mean square atomic displacement of this atom would be U = 0.00438 Å^2 (which would enable one to do ultra-high resolution studies). I feel I should also point out that B = 0 is not all that different from B = 2 (U = 0.03 A^2) if you are trying to do ultra-high resolution studies. This is because the form factor of carbon and other light atoms are essentially Gaussians with full-width at half-max (FWHM) ~0.8 A (you can plot the form factors listed in ITC Vol C to verify this), and blurring atoms with a B factor of 2 Borns increases this width to only ~0.9 A. This is because the real-space blurring kernel of a B factor is a Gaussian function with FWHM = sqrt(B*log(2))/pi Angstrom. The root-mean-square RMS width of this real-space blurring function is sqrt(B/8*pi^2) Angstrom, or sqrt(U) Angstrom. This is the real-space size of a B factor Gaussian, and I, for one, find this a much more intuitive way to think about B factors. I note, however, that the real-space manifestation of the B factor is an object that can be measured in units of Angstrom with no funny scale factors. It is only in reciprocal space (which is really angle space) that we see all these factors of pi popping up. More on that when I find my copy of James... -James Holton MAD Scientist
Re: [ccp4bb] units of the B factor
Not at all ! If I want to compute the sinus of 15 degrees, using the series expansion, I write X = 15 degrees = 15 * pi/180 = 0.2618 because, 1 degree is just a symbol for the unitless, dimensionless number pi/180. I plug this X into the series expansion and get the right result. Marc Quoting Clemens Grimm clemens.gr...@biozentrum.uni-wuerzburg.de: Zitat von marc.schi...@epfl.ch: Dale Tronrud wrote: While it is true that angles are defined by ratios which result in their values being independent of the units those lengths were measured, common sense says that a number is an insufficient description of an angle. If I tell you I measured an angle and its value is 1.5 you cannot perform any useful calculation with that knowledge. I disagree: you can, for instance, put this number x = 1.5 (without units) into the series expansion for sin X : x - x^3/(3!) + x^5/(5!) - x^7/(7!) + ... and compute the value of sin(1.5) to any desired degree of accuracy (four terms will be enough to get an accuracy of 0.0001). Note that the x in the series expansion is just a real number (no dimension, no unit). ... However you get this Taylor expansion under the assumption that sin'(0)=1 sin''(0)=0, sin'''(0)=-1, ... this only holds true under the assumption that the sin function has a period of 2pi and this 'angle' is treated as unitless. Taking e. g. the sine function with a 'degree' argument treated properly as 'unit' will result in a Taylor expansion showing terms with this unit sticking to them.
Re: [ccp4bb] units of the B factor
I would believe that the official SI documentation has precedence over Wikipedia. In the SI brochure it is made quite clear that Radian is just another symbol for the number one and that it may or may no be used, as is convenient. Therefore, stating alpha = 15 (without anything else) is perfectly valid for an angle. Marc Quoting Douglas Theobald dtheob...@brandeis.edu: Argument from authority, from the omniscient Wikipedia: http://en.wikipedia.org/wiki/Radian Although the radian is a unit of measure, it is a dimensionless quantity. The radian is a unit of plane angle, equal to 180/pi (or 360/(2 pi)) degrees, or about 57.2958 degrees, It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level. … the radian is now considered an SI derived unit. On Nov 23, 2009, at 1:31 PM, Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real
Re: [ccp4bb] units of the B factor - resolved?
I would like to apologize to everyone for creating such a busy thread (an what could perhaps be construed as an occasionally belligerent tone), but I really do want to know the right answer to this! I am trying to model radiation damage from first principles, and in such models you cannot have arbitrary scale factors. And I really do appreciate the effort Dale, Ian, Marc, and many others, put into their posts. Taking bits from many of them, I think I can say that: The unit of B factor is: hemi-(cycle/Angstrom)^-2 and the dimensions of the B factor are length^2 Apparently, the B factor is derived from the square of a spatial frequency, which has fundamental units cycles per meter. However, there is an extra factor of two that makes the B factor incompatible with merely spatial frequency squared (with no scale prefix) as the unit, so I think we have to include the prefix hemi before we can make the 2*pi radians/cycle go away. Marc and Ian I imagine will tell me that cycle = 1 and hemi = 1 and therefore we have Angstrom^2 and they are more than welcome to do that in their papers, but I think it important here to clarify exactly what one B factor unit means. -James Holton MAD Scientist
Re: [ccp4bb] units of the B factor
I agree that the official SI documentation has priority, but as I read it there is no discrepancy between it and Wikipedia. The official SI position (and that of NIST and IUPAC) is that the radian is a dimensionless unit (i.e., a unit of dimension 1). Quoting at length from the SI brochure: 2.2.3 Units for dimensionless quantities, also called quantities of dimension one Certain quantities are defined as the ratio of two quantities of the same kind, and are thus dimensionless, or have a dimension that may be expressed by the number one. The coherent SI unit of all such dimensionless quantities, or quantities of dimension one, is the number one, since the unit must be the ratio of two identical SI units. The values of all such quantities are simply expressed as numbers, and the unit one is not explicitly shown. Examples of such quantities are refractive index, relative permeability, and friction factor. There are also some quantities that are defined as a more complex product of simpler quantities in such a way that the product is dimensionless. Examples include the 'characteristic numbers' like the Reynolds number Re = ρvl/η, where ρ is mass density, η is dynamic viscosity, v is speed, and l is length. For all these cases the unit may be considered as the number one, which is a dimensionless derived unit. Another class of dimensionless quantities are numbers that represent a count, such as a number of molecules, degeneracy (number of energy levels), and partition function in statistical thermodynamics (number of thermally accessible states). All of these counting quantities are also described as being dimensionless, or of dimension one, and are taken to have the SI unit one, although the unit of counting quantities cannot be described as a derived unit expressed in terms of the base units of the SI. For such quantities, the unit one may instead be regarded as a further base unit. In a few cases, however, a special name is given to the unit one, in order to facilitate the identification of the quantity involved. This is the case for the radian and the steradian. The radian and steradian have been identified by the CGPM as special names for the coherent derived unit one, to be used to express values of plane angle and solid angle, respectively, and are therefore included in Table 3. The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities. pp 119-120, The International System of Units (SI). International Bureau of Weights and Measures (BIPM). http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf also see http://physics.nist.gov/cuu/Units/units.html http://www.iupac.org/publications/books/gbook/green_book_2ed.pdf On Nov 23, 2009, at 4:03 PM, marc.schi...@epfl.ch wrote: I would believe that the official SI documentation has precedence over Wikipedia. In the SI brochure it is made quite clear that Radian is just another symbol for the number one and that it may or may no be used, as is convenient. Therefore, stating alpha = 15 (without anything else) is perfectly valid for an angle. Marc Quoting Douglas Theobald dtheob...@brandeis.edu: Argument from authority, from the omniscient Wikipedia: http://en.wikipedia.org/wiki/Radian Although the radian is a unit of measure, it is a dimensionless quantity. The radian is a unit of plane angle, equal to 180/pi (or 360/(2 pi)) degrees, or about 57.2958 degrees, It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level. … the radian is now considered an SI derived unit. On Nov 23, 2009, at 1:31 PM, Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me
Re: [ccp4bb] units of the B factor
Quoting James Holton jmhol...@lbl.gov: Now the coefficients of a Taylor polynomial are themselves values of the derivatives of the function being approximated. Each time you take a derivative of f(x), you divide by the units (and therefore dimensions) of x. So, Pete's coefficients below: 1, -1/6, and 1/120 have dimension of [X]^-1, [X]^-2, [X]^-3, respectively. James, The the factors 1, 1/6, 1/120, etc. in the Taylor series of a funcion f(x) do not come from the derivatives of that function. They simply come from the coefficients 1/(n!) that pre-multiply each term (each derivative) in the series. They are, of course, dimensionless (note that n is just an integer number). Marc
Re: [ccp4bb] units of the B factor - resolved?
James, I could not help typing something! Consider a circle of radius R, its circumstance L is then 2*Pi*R. Both R and L have the same unit, the 2*Pi angle is unitless. SI defines the unit of angle to be Ran just because this unitless number is different because it is obtained by the length of an arc over a fragment of straight line, not like sin/cos which are given by straightline fragments. The unit of L is not Ran*unit(R) but unit(R). OK, L = 2*Pi*R.(1) Now B = 8*Pi*Pi*U*U. (2) (Isotropic) B is defined as above. U is the average displacement from the miller plane. B function is defined to be amplified by U*U by 8*pi*pi. If you do not agree, apply your rule to (1). Using the rule of (1) to (2), B has a unit of A*A, while the unit(U) is A. The 8*pi*pi is a convenient amplifier. From U to B, this is a one single factor to another single factor function. In this case, to describe an amount of physical meaning, both factors (U and B) are logically, equivalent, depending on which one is more convenient. Lijun On Nov 23, 2009, at 1:11 PM, James Holton wrote: I would like to apologize to everyone for creating such a busy thread (an what could perhaps be construed as an occasionally belligerent tone), but I really do want to know the right answer to this! I am trying to model radiation damage from first principles, and in such models you cannot have arbitrary scale factors. And I really do appreciate the effort Dale, Ian, Marc, and many others, put into their posts. Taking bits from many of them, I think I can say that: The unit of B factor is: hemi-(cycle/Angstrom)^-2 and the dimensions of the B factor are length^2 Apparently, the B factor is derived from the square of a spatial frequency, which has fundamental units cycles per meter. However, there is an extra factor of two that makes the B factor incompatible with merely spatial frequency squared (with no scale prefix) as the unit, so I think we have to include the prefix hemi before we can make the 2*pi radians/cycle go away. Marc and Ian I imagine will tell me that cycle = 1 and hemi = 1 and therefore we have Angstrom^2 and they are more than welcome to do that in their papers, but I think it important here to clarify exactly what one B factor unit means. -James Holton MAD Scientist Lijun Liu Cardiovascular Research Institute University of California, San Francisco 1700 4th Street, Box 2532 San Francisco, CA 94158 Phone: (415)514-2836
Re: [ccp4bb] units of the B factor
Sorry I'm not clear exactly what your question is, but it seems to me that my paper will actually need fewer words than yours, since I can leave out all occurrences of 'radian' and 'steradian' with no loss of meaning! This quantity you're talking about presumably has a name (otherwise how are we going to talk about it?), so to avoid me having to guess what its name is, for the sake of argument let's say it's called the 'foobar density'. First, the 'foobar density' has to be defined somewhere (e.g. 'photons per unit area per unit solid angle' or whatever). This definition will obviously be the same regardless of the exact words we use to express the measurements, so we can't save any words there! - and anyway it only needs to be defined once and for all. Then '1234 photons/metre^2 in a solid angle of 1.234' is simply 'a foobar density of 1000 photons/metre^2'. You can make your paper longer by appending '/steradian' if you wish, perhaps to remind yourself and the reader of the definition, but it's not essential since the fact that it's for 1 unit of solid angle is clear from the definition (which as I said need only appear once!). If you want to use a non-SI unit of solid angle then you absolutely must state that, but I would advise sticking to the SI unit. Cheers -- Ian -Original Message- From: James Holton [mailto:jmhol...@lbl.gov] Sent: 23 November 2009 19:53 To: Ian Tickle Subject: Re: [ccp4bb] units of the B factor .. and if you have 1234 photons scattered into a solid angle of 1.234 per incident photon per square meter on the sample? And you are pushing the word limit of your paper? Ian Tickle wrote: No, just like this: 'solid angle = 1.234' (or whatever its value is). Since the conversion unit 'steradian' = 1 (i.e. the dimensionless pure number 1) identically, 'a solid angle of 1.234 steradians' is identical to 'a solid angle of 1.234': the unit 'steradian' is redundant. Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 19:07 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor So... how do you measure or report a solid angle without invoking the steradian? sterdegrees? Ian Tickle wrote: James, I think you misunderstood, no-one is suggesting that we can do without the degree (minute, second, grad, ...), since these conversion units have considerable practical value. Only the radian (and steradian) are technically redundant, and as Marc suggested we would probably be better off without them! Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 23 November 2009 16:35 To: CCP4BB@jiscmail.ac.uk Subject: Re: [ccp4bb] units of the B factor Just because something is dimensionless does not mean it is unit-less. The radian and the degree are very good examples of this. Remember, the word unit means one, and it is the quantity of something that we give the value 1.0. Things can only be measured relative to something else, and so without defining for the relevant unit, be it a long-hand description or a convenient abbreviation, a number by itself is not useful. It may have meaning in the metaphysical sense, but its not going to help me solve my structure. A world without units is all well and good for theoreticians who never have to measure anything, but for those of us who do need to know if the angle is 1 degree or 1 radian, units are absolutely required. -James Holton MAD Scientist Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc
Re: [ccp4bb] units of the B factor
Hi, There's no real conflict at all here, and I am surprised at the amount of time spent on this subject :) I hope that people *do* mention which units they refer to and that they *don't* name new units without reasonable justification. If I encounter a situation where a number that is relevant to my work is mentioned without any units of measure then I am likely going to assume something based on what is customary, or perhaps on how I feel that day. If it's really important I would dig deeper. If things go bad I would of course blame it on the supplier of the ill-defined number. Maybe even write a snooty email or something. Or at least think about writing one, while eating icecream in bed directly from the container at 2AM at night. On the other hand if things go well then I will naturally be sure to mention how I bravely tackled the issue and won. Win-win either way, sorry about the expensive Mars probe. Personally I prefer to measure angles as pizza slices - 1 slice defined as about 36 degrees, but of course also depending on how hungry I am (sometimes one slice may be 180 or even 270 degrees). This convention also works well for temperature by the way. I do not intend to propose, insinuate, proselytize or enforce it in any way, for now... Artem -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Dale Tronrud Sent: Monday, November 23, 2009 1:33 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Artem Evdokimov wrote: The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle While it is true that angles are defined by ratios which result in their values being independent of the units those lengths were measured, common sense says that a number is an insufficient description of an angle. If I tell you I measured an angle and its value is 1.5 you cannot perform any useful calculation with that knowledge. Yes it's true that the confusion does not arise from a mix up of feet and meters. I would have concluded my angle was 1.5 in either case. The confusion arises because there are differing conventions for describing that unitless angle. I could be describing my angle as 1.5 radians, 1.5 degrees, or 1.5 cycles (or 1.5 of the mysterious grad on my calculator). For me to communicate my result to you I would need to also tell you the convention I'm using, and you will have to perform a conversion to transform my value to your favorite convention. If it looks like a unit, and it quacks like a unit, I think I'm free to call it a unit. I think you will agree that if we fail to pass the convention along with it value our space probe will crash on Mars just as hard as if we had confused feet and meters. The result of a Sin or Cos calculation can be treated as unitless only because there is 100% agreement on how these results should be represented. Everyone agrees that the Sin of a right angle is 1. If I went off the deep end I could declare that the Sin of a right angle is 12 and I could construct an entirely self-consistent description of physics using that convention. In that case I would have to be very careful to keep track of when I was working with traditional Sin's and when with crazy Tronrud Sin's. When switching between conventions I would have to careful to use the conversion factor of 12 crazy Tronrud Sin's/traditional Sin and I'd do best if I put a mark next to each value indicating which convention was used for that particular value. Sounds like units to me. Of course no one would create crazy Tronrud Sin's because the pain created by the confusion of multiple conventions is not compensated by any gain. When it comes to angles, however, that ship has sailed. While mathematicians have very good reasons for preferring the radian convention you are never going to convince a physicist to change from Angstrom/cycle to Angstrom/radian when measuring wavelengths. You will also fail to convince a crystallographer to measure fractional coordinates in radians. We are going to have to live in a world that has some angular quantities
Re: [ccp4bb] units of the B factor
Interesting that the IUCr committee report that Ian pointed out stated we recommend that the use of B be discouraged. Hmm... Good luck with that! You seem to be implying, if I understand you correctly, that the IUCr report recommends that the use of the equivalent isotropic B be discouraged, but that's not what it says! The recommendation concerning B comes in section 2.1 Anisotropic displacement parameters, just after eqn. 2.1.27. But in fact it's clear from that equation, where B is a tensor, that it's talking about the *anisotropic* B tensor. In the following main section 2.2 Equivalent isotropic displacement parameters no such recommendation appears. Also in section 4 at the end where the recommendations are summarised it explicitly says (point 7) Avoid using the Gaussian anisotropic parameters that are now usually symbolized as B^ij and are defined in eq. (2.1.26). These quantities are directly proportional to the recommended U^ij , the ratio being 8pi^2. Again, no mention of a recommendation concerning the equivalent isotropic B. Indeed of course, the PDB follows the IUCr recommendations (actually it was more a case that the IUCr accepted the de facto existence of the PDB!), i.e. equivalent isotropic B's and anisotropic U's. B^ij's are indeed used internally by some software for convenience in intermediate calculations, but since the output values are U's there's no problem with that. This raises a point relevant to your original suggestion concerning a new name for the unit of B: in a PDB file the U^ij values are actually 1*U^ij, in order to save space by eliminating non-significant digits, as I pointed out previously. However, does this mean that one should think of the values in the file as being in units of picometres^2 (it took me a few moments to work that out!), or does it mean that the values are to be thought of as 1*U^ij so that the units are still the familiar A^2? So by analogy values of B are to be thought of as 8pi^2*U (that's what the equation B = 8pi^2*U means after all!), but still in units of A^2. I suspect that most people, like me, would think of it in those terms. Cheers -- Ian I agree that I should have used U instead of u^2 in my original post. Actually, the u should have a subscript x to denote that it is along the direction perpendicular to the Bragg plane. Movement within the plane does not change the spot intensity, and it also does not matter if the x displacements are instantaneous, dynamic or static, as there is no way to tell the difference with x-ray diffraction. It just matters how far the atoms are from their ideal lattice points (James 1962, Ch 1). I am not sure how to do a symbol with both superscripts and subscripts AND inside brackets that is legible in all email clients. Here is a try: B = 8*pi*usubx/sub^2. Did that work? 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] units of the B factor
Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] units of the B factor
The angle value and the associated basic trigonometric functions (sin, cos, tan) are derived from a ratio of two lengths* and therefore are dimensionless. It's trivial but important to mention that there is no absolute requirement of units of any kind whatsoever with respect to angles or to the three basic trigonometric functions. All the commonly used units come from (arbitrary) scaling constants that in turn are derived purely from convenience - specific calculations are conveniently carried out using specific units (be they radians, points, seconds, grads, brads, or papaya seeds) however the units themselves are there only for our convenience (unlike the absolutely required units of mass, length, time etc.). Artem * angle - the ratio of the arc length to radius of the arc necessary to bring the two rays forming the angle together; trig functions - the ratio of the appropriate sides of a right triangle -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number
Re: [ccp4bb] units of the B factor
using an arbitrary scaling constant. In fact the whole distinction between length and mass is simply a matter of convenience. In the classic text on general relativity Gravitation by Miser, Thorne and Wheeler they have a table in the back of Some Useful Numbers in Conventional and Geometrized Units where it lists the mass of the Sun as 147600 cm and and the distance between the Earth and Sun as 499 sec. Those people in general relativity are great at manipulating coordinate systems! -Original Message- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle Sent: Sunday, November 22, 2009 10:57 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] units of the B factor Back to the original problem: what are the units of B and u_x^2? I haven't been able to work that out. The first wack is to say the B occurs in the term Exp( -B (Sin(theta)/lambda)^2) and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom and the argument of Exp, like Sin, must be radian. This means that the units of B must be A^2 radian. Since B = 8 Pi^2 u_x^2 the units of 8 Pi^2 u_x^2 must also be A^2 radian, but the units of u_x^2 are determined by the units of 8 Pi^2. I can't figure out the units of that without understanding the defining equation, which is in the OPDXr somewhere. I suspect there are additional, hidden, units in that definition. The basic definition would start with the deviation of scattering points from the Miller planes and those deviations are probably defined in cycle or radian and later converted to Angstrom so there are conversion factors present from the beginning. I'm sure that if the MS sits down with the OPDXr and follows all these units through he will uncover the units of B, 8 Pi^2, and u_x^2 and the mystery will be solved. If he doesn't do it, I'll have to sit down with the book myself, and that will make my head hurt. Hi Dale A nice entertaining read for a Sunday afternoon, but I think you can only get so far with this argument and then it breaks down, as evidenced by the fact that eventually you got stuck! I think the problem arises in your assertion that the argument of 'exp' must be in units of radians. IMO it can also be in units of radians^2 (or radians^n where n is any unitless number, integer or real, including zero for that matter!) - and this seems to be precisely what happens here. Having a function whose argument can apparently have any one of an infinite number of units is somewhat of an embarrassment! - of course that must mean that the argument actually has no units. So in essence I'm saying that quantities in radians have to be treated as unitless, contrary to your earlier assertions. So the 'units' (accepting for the moment that the radian is a valid unit) of B are actually A^2 radian^2, and so the 'units' of 8pi^2 (it comes from 2(2pi)^2) are radian^2 as expected. However since I think I've demonstrated that the radian is not a valid unit, then the units of B are indeed A^2! 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] units of the B factor
To avoid the creation of a cumbersome new unit everyone will need to keep track of, can we just come up with a prefix that means 0.013 of something? Perhaps we could give it the symbol b and then we could say the B-factor is 20 bA^2.* James *Seemed like 76.92 b humor units when I wrote it. On Nov 20, 2009, at 11:22 PM, James Holton wrote: No No No! This is not what I meant at all! I am not suggesting the creation of a new unit, but rather that we name a unit that is already in widespread use. This unit is A^2/ (8*pi^2) which has dimensions of length^2 and it IS the unit of B factor. That is, every PDB file lists the B factor as a multiple of THIS fundamental quantity, not A^2. If the unit were simply A^2, then the PDB file would be listing much smaller numbers (U, not B). (Okay, there are a few PDBs that do that by mistake, but not many.) As Marc pointed out, a unit and a dimension are not the same thing: millimeters and centimeters are different units, but they have the same dimension: length. And, yes, dimensionless scale factors like milli and centi are useful. The B factor has dimension length^2, but the unit of B factor is not A^2. For example, if we change some atomic B factor by 1, then we are actually describing a change of 0.013 A^2, because this is equal to 1.0 A^2/(8*pi^2). What I am suggesting is that it would be easier to say that the B factor changed by 1.0, and if you must quote the units, the units are B, otherwise we have to say: the B factor changed by 1.0 A^2/ (8*pi^2). Saying that a B factor changed by 1 A^2 when the actual change in A^2 is 0.013 is (formally) incorrect. The unfortunate situation however is that B factors have often been reported with units of A^2, and this is equivalent to describing the area of 80 football fields as 80 and then giving the dimension (m^2) as the units! It is better to say that the area is 80 football fields, but this is invoking a unit: the football field. The unit of B factor, however does not have a name. We could say 1.0 B-factor units, but that is not the same as 1.0 A^2 which is ~80 B-factor units. Admittedly, using A^2 to describe a B factor by itself is not confusing because we all know what a B factor is. It is that last column in the PDB file. The potential for confusion arises in derived units. How does one express a rate-of-change in B factor? A^2/s? What about rate-of-change in U? A^2/s? I realized that this could become a problem while comparing Kmetko et. al. Acta D (2006) and Borek et. al. JSR (2007). Both very good and influential papers: the former describes damage rates in A^2/MGy (converting B to U first so that A^2 is the unit), and the latter relates damage to the B factor directly, and points out that the increase in B factor from radiation damage of most protein crystals is almost exactly 1.0 B/MGy. This would be a great rule of thumb if one were allowed to use B as a unit. Why not? Interesting that the IUCr committee report that Ian pointed out stated we recommend that the use of B be discouraged. Hmm... Good luck with that! I agree that I should have used U instead of u^2 in my original post. Actually, the u should have a subscript x to denote that it is along the direction perpendicular to the Bragg plane. Movement within the plane does not change the spot intensity, and it also does not matter if the x displacements are instantaneous, dynamic or static, as there is no way to tell the difference with x- ray diffraction. It just matters how far the atoms are from their ideal lattice points (James 1962, Ch 1). I am not sure how to do a symbol with both superscripts and subscripts AND inside brackets that is legible in all email clients. Here is a try: B = 8*pi*usubx/sub^2. Did that work? I did find it interesting that the 8*pi^2 arises from the fact that diffraction occurs in angle space, and so factors of 4*pi steradians pop up in the Fourier domain (spatial frequencies). In the case of B it is (4*pi)^2/2 because the second coefficient of a Taylor series is 1/2. Along these lines, quoting B in A^2 is almost precisely analogous to quoting an angular frequency in Hz. Yes, the dimensions are the same (s^-1), but how does one interpret the statement: the angular frequency was 1 Hz. Is that cycles per second or radians per second? That's all I'm saying... -James Holton MAD Scientist Marc SCHILTZ wrote: Frank von Delft wrote: Hi Marc Not at all, one uses units that are convenient. By your reasoning we should get rid of Å, atmospheres, AU, light years... They exist not to be obnoxious, but because they're handy for a large number of people in their specific situations. Hi Frank, I think that you misunderstood me. Å and atmospheres are units which really refer to physical quantities of different dimensions. So, of course, there must be
Re: [ccp4bb] units of the B factor
This question by the Mad Scientist (here after the MS) has provoked me to give the topic a lot of thought. I think I can provide some direction towards the solution, but I'm not adept enough with The Optical Principles of the Diffraction of X-rays (Which people on this BB should refer to simply as OPDXr because it is so fundamental to most topics discussed here.) to come up with a final answer to the question of the units of B and u_x^2. My hope is that the MS, who is much better with OPDXr than I will finish the job. I have been a big fan of Dimensional Analysis since high school and have found its rigorous application to be very useful in verifying algebraic derivations. I learned quite early that quite a few quantities that people usually say are unitless can be usefully given meaningful units. I think this is the root of the current issue - There are units present in the definition of these terms that are ignored by traditional dimensional analysis. As a first example, I'll consider Bragg's Law: 2 d Sin(theta) = n lambda. Traditionally, the units are (d - A, theta - unitless, Sin(theta) - unitless, n - unitless, lambda - A). While the units on each side of the equation match (Angstrom) that's a lot of unitless quantities. These unit assignments also create problems. With the wavelength, lambda, is measured in Anstrom: does that mean Anstrom/cycle, Anstrom/radian, Anstrom/degree? Just defining a wave length as a length is not good enough, you have to define a length per something. I've created these additional rules for my personal Dimensional Analysis. 1) Angles have units. Either radians, degrees, cycles, or (a button on my calculator tells me) grd. There are well-known conversion factors between these units that appear, unexplained, in popular equations. For example, there are 2 Pi radians per cycle. We see the term 2 Pi in many equations and usually this should be assigned its units. 2) Trigonometric functions have arguments that must be measured in radians and their results are unitless (yes, I still have unitless quantities). In Bragg's Law, I have the new unit assignments of (theta - radian, n - cycle, and lambda - A/cycle). Tracking these additional units allows for tighter checking of the validity of equations. It is difficult to determine the units of quantities in derived equations: you need to concentrate on the defining equations, like Bragg's Law. Why? If you see Sin(theta)/lambda in some other equation, and it comes up a lot, and you try to assign units you will say that Sin(theta) is unitless and lambda is A/cycle so the units of Sin(theta)/lambda is cycle/A. Wrong! You've forgotten that there was an n in the original equation that was assumed to be 1. It is still there and its unit of cycle persists, invisibly, in Sin(theta)/lambda. The unit of Sin(theta)/lambda is 1/Angstrom. Another interesting term to analyse is 2 Pi I (hx + ky + lz). The traditional approach is to say that fractional coordinates are unitless, Miller indices are unitless, and the 2 Pi is just there, don't ask. I have additional rules: 3) The fractional coordinate x has the unit a cell edge, y is b cell edge and z is c cell edge. A location that has x = 0.5 actually means that the location is 0.5 along the a cell edge. This value can be converted to Angstrom with a conversion factor with units of Angstrom/a cell edge, and we call that conversion factor the A cell constant. 4) The unit of h is cycle/a cell edge. When you think about the definition of Miller indices this makes sense. When h = 5 we mean that there are five cycles of that set of planes along the a cell edge of the unit cell. The application of these rules shows why you never see the term x + y unless the symmetry of the crystal includes an equivalence of the a and b edges. You can't add two numbers unless their units match and they don't, unless the symmetry causes the units a cell edge and b cell edge to be equivalent. This is also true for h + k. Writing the units explicitly for our little term results in 2 Pi I (h (cycle/a cell edge) x (a cell edge) + k (cycle/b cell edge) y (b cell edge) + l (cycle/c cell edge) z (c cell edge)) and all the cell edge stuff cancels to cycle. Wait! Didn't I say that the argument of a Sin or Cos function has to be in radian, and this term is usually such an argument? Yes, the factor of 2 Pi is actually 2 Pi radian/cycle and converts the unit of the term to radian. If you read a lot of math books you will be confused because their Fourier transform kernel don't include the 2 Pi that ours does. Mathematicians are cleaver enough to define their reciprocal space coordinates in radians from the start so they don't need to change units later on. Whenever you see an equation where something is actually calculated from h you will see it present as 2 Pi h because the math wants
Re: [ccp4bb] units of the B factor
On Thu, 19 Nov 2009 23:13:53 -0800, James Holton jmhol...@lbl.gov said: should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This There is already the unit barn (b) for area - about the cross section of an uranium nucleus, it is 1E-8 A^2 (100 fm^2). http://en.wikipedia.org/wiki/Barn_%28unit%29 So a Born would be somewhat more than a Megabarn. -Christoph -- | Dr Christoph Best b...@ebi.ac.uk http://www.ebi.ac.uk/~best | Project Leader Electron Microscopy Data Bank, PDB Europe | European Bioinformatics Institute, Cambridge, UK +44-1223-492649
Re: [ccp4bb] units of the B factor
I second that... are there committees that ratify these things? phx James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? -James Holton MAD Scientist
Re: [ccp4bb] units of the B factor
Hi James, James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. There is nothing wrong with this. In the case of derived units, there is almost never a univocal relation between the unit and the physical quantity that it refers to. As an example: from the unit kg/m^3, you can not tell what the physical quantity is that it refers to: it could be the density of a material, but it could also be the mass concentration of a compound in a solution. Therefore, one always has to specify exactly what physical quantity one is talking about, i.e. B/dose or u^2/dose, but this is not something that should be packed into the unit (otherwise, we will need hundreds of different units) It simply has to be made clear by the author of a paper whether the quantity he is referring to is B or u^2. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. This is like claiming that the radius and the circumference of a circle would need different units because they are related by the scale factor 2*pi. What matters is the dimension. Both radius and circumference have the dimension of a length, and therefore have the same unit. Both B and u^2 have the dimension of the square of a length and therefoire have the same unit. The scalefactor 8*pi^2 is a dimensionless quantity and does not change the unit. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? Good luck in submitting your proposal to the General Conference on Weights and Measures. -- Marc SCHILTZ http://lcr.epfl.ch
[ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
Hi Marc Not at all, one uses units that are convenient. By your reasoning we should get rid of Å, atmospheres, AU, light years... They exist not to be obnoxious, but because they're handy for a large number of people in their specific situations. Sounds familiar... phx Marc SCHILTZ wrote: Hi James, James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. There is nothing wrong with this. In the case of derived units, there is almost never a univocal relation between the unit and the physical quantity that it refers to. As an example: from the unit kg/m^3, you can not tell what the physical quantity is that it refers to: it could be the density of a material, but it could also be the mass concentration of a compound in a solution. Therefore, one always has to specify exactly what physical quantity one is talking about, i.e. B/dose or u^2/dose, but this is not something that should be packed into the unit (otherwise, we will need hundreds of different units) It simply has to be made clear by the author of a paper whether the quantity he is referring to is B or u^2. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. This is like claiming that the radius and the circumference of a circle would need different units because they are related by the scale factor 2*pi. What matters is the dimension. Both radius and circumference have the dimension of a length, and therefore have the same unit. Both B and u^2 have the dimension of the square of a length and therefoire have the same unit. The scalefactor 8*pi^2 is a dimensionless quantity and does not change the unit. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? Good luck in submitting your proposal to the General Conference on Weights and Measures.
Re: [ccp4bb] units of the B factor
I think that you should suggest a new unit of 10^(-11) m, a JHm perhaps. If it is convenient to have B in A^2 then u^2 should be in JHm^2. Adam On Thu, 19 Nov 2009, James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? -James Holton MAD Scientist
Re: [ccp4bb] units of the B factor
Hi James If we're going to sort out the units we need to get the terminology right too. The mean square atomic displacement already has a symbol U = u^2 (or to be precise Ueq as we're talking about isotropic displacements here), and u is conventionally not defined as the RMS displacement as you seem to be implying, but the *instantaneous* displacement (otherwise you then need another symbol for the instantaneous displacement!). See: http://www.iucr.org/resources/commissions/crystallographic-nomenclature/ adp (or Acta Cryst. (1996). A52, 770-781). My theory is that B became popular over U because it needs 1 fewer digit to express it to a given precision, and this was important given the limited space available in the 80-column PDB format. So a B of 20.00 to 4 sig figs requires 5 columns, whereas the equivalent U of 0.2500 to 4 sig figs requires 6 columns (personally I've got nothing against '.2500' but many compiler writers don't see it my way!). Interestingly the IUCr commission in their 1996 report did not address the question of units for B and U. Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 20 November 2009 07:14 To: CCP4BB@jiscmail.ac.uk Subject: units of the B factor Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? -James Holton MAD Scientist 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] units of the B factor
Of course, for SI political correctness we should be using nm^2 anyway. This would add more confusion to a situation that most people don't worry about anyway. Pete On 20 Nov 2009, at 11:05, Ian Tickle wrote: Hi James If we're going to sort out the units we need to get the terminology right too. The mean square atomic displacement already has a symbol U = u^2 (or to be precise Ueq as we're talking about isotropic displacements here), and u is conventionally not defined as the RMS displacement as you seem to be implying, but the *instantaneous* displacement (otherwise you then need another symbol for the instantaneous displacement!). See: http://www.iucr.org/resources/commissions/crystallographic-nomenclature/ adp (or Acta Cryst. (1996). A52, 770-781). My theory is that B became popular over U because it needs 1 fewer digit to express it to a given precision, and this was important given the limited space available in the 80-column PDB format. So a B of 20.00 to 4 sig figs requires 5 columns, whereas the equivalent U of 0.2500 to 4 sig figs requires 6 columns (personally I've got nothing against '.2500' but many compiler writers don't see it my way!). Interestingly the IUCr commission in their 1996 report did not address the question of units for B and U. Cheers -- Ian -Original Message- From: owner-ccp...@jiscmail.ac.uk [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton Sent: 20 November 2009 07:14 To: CCP4BB@jiscmail.ac.uk Subject: units of the B factor Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? -James Holton MAD Scientist 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. If you have received this communication in error, please notify Astex Therapeutics Ltd by emailing i.tic...@astex-therapeutics.com and destroy all copies of the message and any attached documents. Astex Therapeutics Ltd monitors, controls and protects all its messaging traffic in compliance with its corporate email policy. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
Frank von Delft wrote: Hi Marc Not at all, one uses units that are convenient. By your reasoning we should get rid of Å, atmospheres, AU, light years... They exist not to be obnoxious, but because they're handy for a large number of people in their specific situations. Hi Frank, I think that you misunderstood me. Å and atmospheres are units which really refer to physical quantities of different dimensions. So, of course, there must be different units for them (by the way: atmosphere is not an accepted unit in the SI system - not even a tolerated non SI unit, so a conscientious editor of an IUCr journal would not let it go through. On the other hand, the Å is a tolerated non SI unit). But in the case of B and U, the situation is different. These two quantities have the same dimension (square of a length). They are related by the dimensionless factor 8*pi^2. Why would one want to incorporate this factor into the unit ? What advantage would it have ? The physics literature is full of quantities that are related by multiples of pi. The frequency f of an oscillation (e.g. a sound wave) can be expressed in s^-1 (or Hz). The same oscillation can also be charcterized by its angular frequency \omega, which is related to the former by a factor 2*pi. Yet, no one has ever come up to suggest that this quantity should be given a new unit. Planck's constant h can be expressed in J*s. The related (and often more useful) constant h-bar = h/(2*pi) is also expressed in J*s. No one has ever suggested that this should be given a different unit. The SI system (and other systems as well) has been specially crafted to avoid the proliferation of units. So I don't think that we can (should) invent new units whenever it appears convenient. It would bring us back to times anterior to the French revolution. Please note: I am not saying that the SI system is the definite choice for every purpose. The nautical system of units (nautical mile, knot, etc.) is used for navigation on sea and in the air and it works fine for this purpose. However, within a system of units (whichever is adopted), the number of different units should be kept reasonably small. Cheers Marc Sounds familiar... phx Marc SCHILTZ wrote: Hi James, James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. There is nothing wrong with this. In the case of derived units, there is almost never a univocal relation between the unit and the physical quantity that it refers to. As an example: from the unit kg/m^3, you can not tell what the physical quantity is that it refers to: it could be the density of a material, but it could also be the mass concentration of a compound in a solution. Therefore, one always has to specify exactly what physical quantity one is talking about, i.e. B/dose or u^2/dose, but this is not something that should be packed into the unit (otherwise, we will need hundreds of different units) It simply has to be made clear by the author of a paper whether the quantity he is referring to is B or u^2. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. This is like claiming that the radius and the circumference of a circle would need different units because they are related by the scale factor 2*pi. What matters is the dimension. Both radius and circumference have the dimension of a length, and therefore have the same unit. Both B and u^2 have the dimension of the square of a length and therefoire have the same unit. The scalefactor 8*pi^2 is a dimensionless quantity and does not change the unit. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have
[ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
Eh? m and Å are related by the dimensionless quantity 10,000,000,000. Vive la révolution! Marc SCHILTZ wrote: Frank von Delft wrote: Hi Marc Not at all, one uses units that are convenient. By your reasoning we should get rid of Å, atmospheres, AU, light years... They exist not to be obnoxious, but because they're handy for a large number of people in their specific situations. Hi Frank, I think that you misunderstood me. Å and atmospheres are units which really refer to physical quantities of different dimensions. So, of course, there must be different units for them (by the way: atmosphere is not an accepted unit in the SI system - not even a tolerated non SI unit, so a conscientious editor of an IUCr journal would not let it go through. On the other hand, the Å is a tolerated non SI unit). But in the case of B and U, the situation is different. These two quantities have the same dimension (square of a length). They are related by the dimensionless factor 8*pi^2. Why would one want to incorporate this factor into the unit ? What advantage would it have ? The physics literature is full of quantities that are related by multiples of pi. The frequency f of an oscillation (e.g. a sound wave) can be expressed in s^-1 (or Hz). The same oscillation can also be charcterized by its angular frequency \omega, which is related to the former by a factor 2*pi. Yet, no one has ever come up to suggest that this quantity should be given a new unit. Planck's constant h can be expressed in J*s. The related (and often more useful) constant h-bar = h/(2*pi) is also expressed in J*s. No one has ever suggested that this should be given a different unit. The SI system (and other systems as well) has been specially crafted to avoid the proliferation of units. So I don't think that we can (should) invent new units whenever it appears convenient. It would bring us back to times anterior to the French revolution. Please note: I am not saying that the SI system is the definite choice for every purpose. The nautical system of units (nautical mile, knot, etc.) is used for navigation on sea and in the air and it works fine for this purpose. However, within a system of units (whichever is adopted), the number of different units should be kept reasonably small. Cheers Marc Sounds familiar... phx Marc SCHILTZ wrote: Hi James, James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. There is nothing wrong with this. In the case of derived units, there is almost never a univocal relation between the unit and the physical quantity that it refers to. As an example: from the unit kg/m^3, you can not tell what the physical quantity is that it refers to: it could be the density of a material, but it could also be the mass concentration of a compound in a solution. Therefore, one always has to specify exactly what physical quantity one is talking about, i.e. B/dose or u^2/dose, but this is not something that should be packed into the unit (otherwise, we will need hundreds of different units) It simply has to be made clear by the author of a paper whether the quantity he is referring to is B or u^2. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. This is like claiming that the radius and the circumference of a circle would need different units because they are related by the scale factor 2*pi. What matters is the dimension. Both radius and circumference have the dimension of a length, and therefore have the same unit. Both B and u^2 have the dimension of the square of a length and therefoire have the same unit. The scalefactor 8*pi^2 is a dimensionless quantity and does not change the unit. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B
Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
But in this case you are no longer defining distances but some other arbitrary
quantity, like vendors do when they convert a small computer speaker into a
rockband PA by using PMPO instead of music power.
Herman
-Original Message-
From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Frank von
Delft
Sent: Friday, November 20, 2009 1:11 PM
To: CCP4BB@JISCMAIL.AC.UK
Subject: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] {Spam?} Re: {Spam?} Re:
[ccp4bb] units of the B factor
Eh? m and Å are related by the dimensionless quantity 10,000,000,000.
Vive la révolution!
Marc SCHILTZ wrote:
Frank von Delft wrote:
Hi Marc
Not at all, one uses units that are convenient. By your reasoning we
should get rid of Å, atmospheres, AU, light years... They exist not
to be obnoxious, but because they're handy for a large number of
people in their specific situations.
Hi Frank,
I think that you misunderstood me. Å and atmospheres are units which
really refer to physical quantities of different dimensions. So, of
course, there must be different units for them (by the way: atmosphere
is not an accepted unit in the SI system - not even a tolerated non SI
unit, so a conscientious editor of an IUCr journal would not let it go
through. On the other hand, the Å is a tolerated non SI unit).
But in the case of B and U, the situation is different. These two
quantities have the same dimension (square of a length). They are
related by the dimensionless factor 8*pi^2. Why would one want to
incorporate this factor into the unit ? What advantage would it have ?
The physics literature is full of quantities that are related by
multiples of pi. The frequency f of an oscillation (e.g. a sound wave)
can be expressed in s^-1 (or Hz). The same oscillation can also be
charcterized by its angular frequency \omega, which is related to the
former by a factor 2*pi. Yet, no one has ever come up to suggest that
this quantity should be given a new unit. Planck's constant h can be
expressed in J*s. The related (and often more useful) constant h-bar =
h/(2*pi) is also expressed in J*s. No one has ever suggested that this
should be given a different unit.
The SI system (and other systems as well) has been specially crafted
to avoid the proliferation of units. So I don't think that we can
(should) invent new units whenever it appears convenient. It would
bring us back to times anterior to the French revolution.
Please note: I am not saying that the SI system is the definite choice
for every purpose. The nautical system of units (nautical mile, knot,
etc.) is used for navigation on sea and in the air and it works fine
for this purpose. However, within a system of units (whichever is
adopted), the number of different units should be kept reasonably small.
Cheers
Marc
Sounds familiar...
phx
Marc SCHILTZ wrote:
Hi James,
James Holton wrote:
Many textbooks describe the B factor as having units of square
Angstrom (A^2), but then again, so does the mean square atomic
displacement u^2, and B = 8*pi^2*u^2. This can become confusing if
one starts to look at derived units that have started to come out
of the radiation damage field like A^2/MGy, which relates how much
the B factor of a crystal changes after absorbing a given dose. Or
is it the atomic displacement after a given dose? Depends on which
paper you are looking at.
There is nothing wrong with this. In the case of derived units,
there is almost never a univocal relation between the unit and the
physical quantity that it refers to. As an example: from the unit
kg/m^3, you can not tell what the physical quantity is that it
refers to: it could be the density of a material, but it could also
be the mass concentration of a compound in a solution. Therefore,
one always has to specify exactly what physical quantity one is
talking about, i.e. B/dose or u^2/dose, but this is not something
that should be packed into the unit (otherwise, we will need
hundreds of different units)
It simply has to be made clear by the author of a paper whether the
quantity he is referring to is B or u^2.
It seems to me that the units of B and u^2 cannot both be A^2
any more than 1 radian can be equated to 1 degree. You need a
scale factor. Kind of like trying to express something in terms of
1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the
dimensions of cm^2, but you can't just say 1 cm^2 when you really
mean 1 mm^2! That would be silly. However, we often say B = 80
A^2, when we really mean is 1 A^2 of square atomic displacements.
This is like claiming that the radius and the circumference of a
circle would need different units because they are related by the
scale factor 2*pi.
What matters is the dimension. Both radius and circumference have
the dimension of a length, and therefore have the same unit. Both B
and u^2 have the dimension
Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
Yes, but Å is really only just tolerated. It has evaded the Guillotine - for the time being ;-) Frank von Delft wrote: Eh? m and Å are related by the dimensionless quantity 10,000,000,000. Vive la révolution! Marc SCHILTZ wrote: Frank von Delft wrote: Hi Marc Not at all, one uses units that are convenient. By your reasoning we should get rid of Å, atmospheres, AU, light years... They exist not to be obnoxious, but because they're handy for a large number of people in their specific situations. Hi Frank, I think that you misunderstood me. Å and atmospheres are units which really refer to physical quantities of different dimensions. So, of course, there must be different units for them (by the way: atmosphere is not an accepted unit in the SI system - not even a tolerated non SI unit, so a conscientious editor of an IUCr journal would not let it go through. On the other hand, the Å is a tolerated non SI unit). But in the case of B and U, the situation is different. These two quantities have the same dimension (square of a length). They are related by the dimensionless factor 8*pi^2. Why would one want to incorporate this factor into the unit ? What advantage would it have ? The physics literature is full of quantities that are related by multiples of pi. The frequency f of an oscillation (e.g. a sound wave) can be expressed in s^-1 (or Hz). The same oscillation can also be charcterized by its angular frequency \omega, which is related to the former by a factor 2*pi. Yet, no one has ever come up to suggest that this quantity should be given a new unit. Planck's constant h can be expressed in J*s. The related (and often more useful) constant h-bar = h/(2*pi) is also expressed in J*s. No one has ever suggested that this should be given a different unit. The SI system (and other systems as well) has been specially crafted to avoid the proliferation of units. So I don't think that we can (should) invent new units whenever it appears convenient. It would bring us back to times anterior to the French revolution. Please note: I am not saying that the SI system is the definite choice for every purpose. The nautical system of units (nautical mile, knot, etc.) is used for navigation on sea and in the air and it works fine for this purpose. However, within a system of units (whichever is adopted), the number of different units should be kept reasonably small. Cheers Marc Sounds familiar... phx Marc SCHILTZ wrote: Hi James, James Holton wrote: Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. There is nothing wrong with this. In the case of derived units, there is almost never a univocal relation between the unit and the physical quantity that it refers to. As an example: from the unit kg/m^3, you can not tell what the physical quantity is that it refers to: it could be the density of a material, but it could also be the mass concentration of a compound in a solution. Therefore, one always has to specify exactly what physical quantity one is talking about, i.e. B/dose or u^2/dose, but this is not something that should be packed into the unit (otherwise, we will need hundreds of different units) It simply has to be made clear by the author of a paper whether the quantity he is referring to is B or u^2. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. This is like claiming that the radius and the circumference of a circle would need different units because they are related by the scale factor 2*pi. What matters is the dimension. Both radius and circumference have the dimension of a length, and therefore have the same unit. Both B and u^2 have the dimension of the square of a length and therefoire have the same unit. The scalefactor 8*pi^2 is a dimensionless quantity and does not change the unit. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature
Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
What a funny pleasant piece of discussion ! Given any physical quantity Something, having any kind of dimension (even as awful as inches^2*gallons*pounds^-3) Would it exist any room for a discussion about the dimension of 2*Something ? And what about 1*Something ? Philippe Dumas attachment: p_dumas.vcf
Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] {Spam?} Re: {Spam?} Re: [ccp4bb] units of the B factor
What a funny pleasant piece of discussion ! Given any physical quantity Something, having any kind of dimension (even as awful as inches^2*gallons*pounds^-3) Would it exist any room for a discussion about the dimension of 2*Something ? And what about 1*Something ? (1) You can always convert anything into anything else (related to it by a scale factor) using Google, e.g.: http://www.google.com/search?hl=enq=2+fortnights+in+msec http://www.google.com/search?hl=enq=7+furlongs+in+mm http://www.google.com/search?hl=enq=7+square+angstrom+in+cm%5E2 To answer your question: http://www.google.com/search?hl=enq=1+inches%5E2*gallons*pounds%5E-3 So: 1 inches^2*gallons*pounds^-3 = 2.61687719 10^-5 m^5 / kg^3 (assuming US gallons! If you meant imperial gallons, the answer is 3.14273976 10^-5 m^5 / kg^3). (2) With respect to the subject of this thread, can I have my spam, spam, spam, spam and units with eggs, please? (http://www.youtube.com/watch?v=cFrtpT1mKy8) --dvd ** Gerard J. Kleywegt Dept. of Cell Molecular Biology University of Uppsala Biomedical Centre Box 596 SE-751 24 Uppsala SWEDEN http://xray.bmc.uu.se/gerard/ mailto:ger...@xray.bmc.uu.se ** The opinions in this message are fictional. Any similarity to actual opinions, living or dead, is purely coincidental. **
Re: [ccp4bb] units of the B factor
No No No! This is not what I meant at all! I am not suggesting the creation of a new unit, but rather that we name a unit that is already in widespread use. This unit is A^2/(8*pi^2) which has dimensions of length^2 and it IS the unit of B factor. That is, every PDB file lists the B factor as a multiple of THIS fundamental quantity, not A^2. If the unit were simply A^2, then the PDB file would be listing much smaller numbers (U, not B). (Okay, there are a few PDBs that do that by mistake, but not many.) As Marc pointed out, a unit and a dimension are not the same thing: millimeters and centimeters are different units, but they have the same dimension: length. And, yes, dimensionless scale factors like milli and centi are useful. The B factor has dimension length^2, but the unit of B factor is not A^2. For example, if we change some atomic B factor by 1, then we are actually describing a change of 0.013 A^2, because this is equal to 1.0 A^2/(8*pi^2). What I am suggesting is that it would be easier to say that the B factor changed by 1.0, and if you must quote the units, the units are B, otherwise we have to say: the B factor changed by 1.0 A^2/(8*pi^2). Saying that a B factor changed by 1 A^2 when the actual change in A^2 is 0.013 is (formally) incorrect. The unfortunate situation however is that B factors have often been reported with units of A^2, and this is equivalent to describing the area of 80 football fields as 80 and then giving the dimension (m^2) as the units! It is better to say that the area is 80 football fields, but this is invoking a unit: the football field. The unit of B factor, however does not have a name. We could say 1.0 B-factor units, but that is not the same as 1.0 A^2 which is ~80 B-factor units. Admittedly, using A^2 to describe a B factor by itself is not confusing because we all know what a B factor is. It is that last column in the PDB file. The potential for confusion arises in derived units. How does one express a rate-of-change in B factor? A^2/s? What about rate-of-change in U? A^2/s? I realized that this could become a problem while comparing Kmetko et. al. Acta D (2006) and Borek et. al. JSR (2007). Both very good and influential papers: the former describes damage rates in A^2/MGy (converting B to U first so that A^2 is the unit), and the latter relates damage to the B factor directly, and points out that the increase in B factor from radiation damage of most protein crystals is almost exactly 1.0 B/MGy. This would be a great rule of thumb if one were allowed to use B as a unit. Why not? Interesting that the IUCr committee report that Ian pointed out stated we recommend that the use of B be discouraged. Hmm... Good luck with that! I agree that I should have used U instead of u^2 in my original post. Actually, the u should have a subscript x to denote that it is along the direction perpendicular to the Bragg plane. Movement within the plane does not change the spot intensity, and it also does not matter if the x displacements are instantaneous, dynamic or static, as there is no way to tell the difference with x-ray diffraction. It just matters how far the atoms are from their ideal lattice points (James 1962, Ch 1). I am not sure how to do a symbol with both superscripts and subscripts AND inside brackets that is legible in all email clients. Here is a try: B = 8*pi*usubx/sub^2. Did that work? I did find it interesting that the 8*pi^2 arises from the fact that diffraction occurs in angle space, and so factors of 4*pi steradians pop up in the Fourier domain (spatial frequencies). In the case of B it is (4*pi)^2/2 because the second coefficient of a Taylor series is 1/2. Along these lines, quoting B in A^2 is almost precisely analogous to quoting an angular frequency in Hz. Yes, the dimensions are the same (s^-1), but how does one interpret the statement: the angular frequency was 1 Hz. Is that cycles per second or radians per second? That's all I'm saying... -James Holton MAD Scientist Marc SCHILTZ wrote: Frank von Delft wrote: Hi Marc Not at all, one uses units that are convenient. By your reasoning we should get rid of Å, atmospheres, AU, light years... They exist not to be obnoxious, but because they're handy for a large number of people in their specific situations. Hi Frank, I think that you misunderstood me. Å and atmospheres are units which really refer to physical quantities of different dimensions. So, of course, there must be different units for them (by the way: atmosphere is not an accepted unit in the SI system - not even a tolerated non SI unit, so a conscientious editor of an IUCr journal would not let it go through. On the other hand, the Å is a tolerated non SI unit). But in the case of B and U, the situation is different. These two quantities have the same dimension (square of a length). They are related by the
[ccp4bb] units of the B factor
Many textbooks describe the B factor as having units of square Angstrom (A^2), but then again, so does the mean square atomic displacement u^2, and B = 8*pi^2*u^2. This can become confusing if one starts to look at derived units that have started to come out of the radiation damage field like A^2/MGy, which relates how much the B factor of a crystal changes after absorbing a given dose. Or is it the atomic displacement after a given dose? Depends on which paper you are looking at. It seems to me that the units of B and u^2 cannot both be A^2 any more than 1 radian can be equated to 1 degree. You need a scale factor. Kind of like trying to express something in terms of 1/100 cm^2 without the benefit of mm^2. Yes, mm^2 have the dimensions of cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That would be silly. However, we often say B = 80 A^2, when we really mean is 1 A^2 of square atomic displacements. The B units, which are ~1/80th of a A^2, do not have a name. So, I think we have a new unit? It is A^2/(8pi^2) and it is the units of the B factor that we all know and love. What should we call it? I nominate the Born after Max Born who did so much fundamental and far-reaching work on the nature of disorder in crystal lattices. The unit then has the symbol B, which will make it easy to say that the B factor was 80 B. This might be very handy indeed if, say, you had an editor who insists that all reported values have units? Anyone disagree or have a better name? -James Holton MAD Scientist
