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
