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:[log in to unmask]] On Behalf Of Ian
Tickle
Sent: Sunday, November 22, 2009 10:57 AM
To: [log in to unmask]
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
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