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*<u<sub>x</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 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 units?
>>>>>
>>>>> Anyone disagree or have a better name?
>>>>
>>>> Good luck in submitting your proposal to the General Conference  
>>>> on Weights and Measures.
>>>>
>>>>
>>>
>>