Artem Evdokimov wrote:
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
While it is true that angles are defined by ratios which result in
their values being independent of the units those lengths were measured,
common sense says that a number is an insufficient description of an
angle. If I tell you I measured an angle and its value is "1.5" you
cannot perform any useful calculation with that knowledge. Yes it's
true that the confusion does not arise from a mix up of feet and meters.
I would have concluded my angle was 1.5 in either case.
The confusion arises because there are differing conventions for
describing that "unitless" angle. I could be describing my angle as
1.5 radians, 1.5 degrees, or 1.5 cycles (or 1.5 of the mysterious
"grad" on my calculator). For me to communicate my result to you
I would need to also tell you the convention I'm using, and you will
have to perform a conversion to transform my value to your favorite
convention. If it looks like a unit, and it quacks like a unit, I
think I'm free to call it a unit.
I think you will agree that if we fail to pass the convention
along with it value our space probe will crash on Mars just as hard
as if we had confused feet and meters.
The result of a Sin or Cos calculation can be treated as "unitless"
only because there is 100% agreement on how these results should be
represented. Everyone agrees that the Sin of a right angle is 1.
If I went off the deep end I could declare that the Sin of a right
angle is 12 and I could construct an entirely self-consistent description
of physics using that convention. In that case I would have to be
very careful to keep track of when I was working with traditional
Sin's and when with "crazy Tronrud Sin's". When switching between
conventions I would have to careful to use the conversion factor of
12 "crazy Tronrud Sin's"/"traditional Sin" and I'd do best if I
put a mark next to each value indicating which convention was used
for that particular value. Sounds like units to me.
Of course no one would create "crazy Tronrud Sin's" because the
pain created by the confusion of multiple conventions is not compensated
by any gain. When it comes to angles, however, that ship has sailed.
While mathematicians have very good reasons for preferring the radian
convention you are never going to convince a physicist to change from
Angstrom/cycle to Angstrom/radian when measuring wavelengths. You
will also fail to convince a crystallographer to measure fractional
coordinates in radians. We are going to have to live in a world that
has some angular quantities reported in radians and others in cycles.
That means we will have to keep track of which is being used and apply
the factor of 2 Pi radian/cycle or 1/(2 Pi) cycle/radian when switching
between.
I agree with Ian that the 8 Pi^2 factor in the conversion of
<u_x^2> to B looks suspiciously like 2 (2 Pi)^2 and it is likely
a conversion of cycle^2 to radian^2. I can even imagine that the
derivation of effect of distortions of the lattice points that lead
to these parameters would start with a description of these distortions
in cycles, but I also have enough experience with this sort of problem
to know that you can only be certain of these "units" after going
back to the root definition and tracking the algebra forward.
In my opinion the Mad Scientist is right. B and <u_x^2> represent
the same quantity reported with different units (or conventions if
you will) and the answer will be something like B in A^2 radian^2
and <u_x^2> in A^2 cycle^2. It would be much clearer it someone
figured out exactly what those units are and we started properly
stating the units of each. I'm sorry that I don't have the time
myself for this project.
Dale Tronrud
P.S. As for your distinction between the "convenience" units used to
measure angles and the "absolutely required" units of length and mass:
all units are part of the coordinate systems that we humans impose on
the universe. Length and mass are no more fundamental than angles.
Feet and meters are units chosen for our convenience and one converts
between them using an arbitrary scaling constant. In fact the whole
distinction between length and mass is simply a matter of convenience.
In the classic text on general relativity "Gravitation" by Miser,
Thorne and Wheeler they have a table in the back of "Some Useful
Numbers in Conventional and Geometrized Units" where it lists the
mass of the Sun as 147600 cm and and the distance between the Earth
and Sun as 499 sec. Those people in general relativity are great
at manipulating coordinate systems!
-----Original Message-----
From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Ian
Tickle
Sent: Sunday, November 22, 2009 10:57 AM
To: CCP4BB@JISCMAIL.AC.UK
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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