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*<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?
The dimension of an angular frequency can not be "cycles per second",
because that contradicts the definition of this quantity, which is
defined to be an angle per time. Again, there is no need to specifically
pack this information into the unit (although it can be done by
specifying rad/s as unit - but this is not strictly necessary).
Marc
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.
--
Marc SCHILTZ http://lcr.epfl.ch
