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.
