Dale's assertion that the exponent has units of radians comes from 
Euler's formula:

exp(i*x) = cos(x) + i*sin(x)

which does indeed require that "x" has units of radian, or whatever it 
is you feed your sin() functions.  However, not every exponential has an 
"i" in it, and the general complex-number case is:

exp(a+i*b) = exp(a)*(cos(b)+i*sin(b))

which means that "a" and "b" can have different units.  Not sure what 
the units of "a" are, but I suspect that they are probably a 
dimensionless and nameless unit.  The ratio of two real lengths.  
Remember, dimensionless does not mean unitless!  Anything that can be 
measured has some situation where the result of the measurement is "1", 
that is the "unit" of the measurement.  Sometimes units have names, 
sometimes not.

Although this mathematical discussion is getting off-thread, the 
question of the units of the coefficients in a Taylor expansion I think 
is to my original question.  This is because the B factor is the second 
term of the Taylor expansion in the exponential of:

F = F0 * exp( -A*s -B*s^2 -C*s^3 ... )

where "s" is sin(theta)/lambda, "F" is the observed structure factor and 
"F0" is the ideal structure factor (where every atom in every unit cell 
obeys the unit cell repeat exactly).  "B" is the B factor we know, and 
"A" and "C" etc. are the Taylor coefficients that Debye  (Ann. Phys. 
1915) said would probably be insignificant.  He was mostly right.

Now the coefficients of a Taylor polynomial are themselves values of the 
derivatives of the function being approximated.  Each time you take a 
derivative of "f(x)", you divide by the units (and therefore dimensions) 
of "x".  So, Pete's coefficients below: 1, -1/6, and 1/120 have 
dimension of [X]^-1, [X]^-2, [X]^-3, respectively.  Multiplying these 
coefficients by x, x^2, x^3, gives a resulting quantity that is 
dimensionless.  Or, more generally, has the same dimensions (and units) 
as f(x).  So all is right again with the world.

But now getting back to the B factor.  The radian is indeed an SI unit, 
and Dale has aptly pointed out that there is a scale factor of 2*pi 
radian/cycle in common crystallographic Fourier transforms because 
crystallographers insist on defining things in terms of "cycles" instead 
of radians.  So, the units of wavelength are Angstrom/cycle and that 
means the units of sin(theta)/lambda ( "s" ) are cycles/Angstrom and 
this makes the units of s^2 (cycles/Angstrom)^2.  The coefficient "B" 
must therefore have units of (Angstrom/cycle)^2 and converting this to 
radians requires dividing it by (2*pi radians/cycle)^2.  Then you have 
to divide by "2" because "B" is the second Taylor coefficient (second 
derivative).  So, at the present time my best guess is:

The unit of B factor is: (Angstrom/radian)^2/(8*pi^2)

The SI prefix for 0.001 is "milli" and the prefix for 0.01 is "centi".  
Does anyone know the Latin word for 0.0127? "duodeviginti scindo pi pi 
"?  centiduomilli...?

-James Holton
MAD Scientist

mb1pja wrote:
> how does the equation
>
> cos(x)= (exp(ix) + exp(-ix))/2     
>
> and the sine equivalent fit into this? Clearly exponentials are not restricted to angles ... indicating that x (and by implication angles) have no dimensions.
>
>
>
> Marc Schiltz's previously cited Taylor expansion demonstrates this even better:
>
> sin(x) = x/1! - x^3/3! + x^5/5! ..... etc to infinity
>
> If you assume for a moment that x does have a dimension, lets call it [X], then the equation is dimensionally unbalanced
>
> [?] = [X]^1 - [X]^3 + [X]^5    ...... etc
>
> and is therefore invalid. It only makes sense if x, and its sine, are dimensionless
>
> Pete
>
>
>
>
>
>
> On 23 Nov 2009, at 16:42, [log in to unmask] wrote:
>
>   
>> Dale Tronrud wrote:
>>     
>>>    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.
>>>       
>> I disagree: you can, for instance, put this number x = 1.5 (without units) into the series expansion for sin X :
>>
>> x - x^3/(3!) + x^5/(5!) - x^7/(7!) + ...
>>
>> and compute the value of sin(1.5) to any desired degree of accuracy
>> (four terms will be enough to get an accuracy of 0.0001). Note that
>> the x in the series expansion is just a real number (no dimension, no
>> unit).
>>
>>
>>
>> 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).
>>>       
>>
>> These are just symbols for dimensionless factors :
>>
>> 1 rad = 1
>> 1 degree = pi/180
>> 1 grad = pi/200
>>
>> Thus :
>>
>> 1.5 rad = 1.5
>> 1.5 degree = 0.0268
>> 1.5 grad = 0.0236
>>
>> and all these numbers (which have no units !!!) can be put into the
>> series expansions for trigonometric functions.
>>
>> In my opinion, it is actually best not to use the symbol rad. As we can
>> see from this discussion, it mostly creates confusion.
>>
>>
>>
>> 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.
>>>       
>> This is not a simple matter of agreement (or convention), it is
>> contained in the very definition of the sine function.
>>
>>
>>     
>>> 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.
>>>       
>> I challenge you to draw a right triangle on paper where the length of
>> one of the sides measures 12 times the length of the hypotenuse.
>>
>> Of course, you can say that your "crazy Tronrud Sin" is defined
>> differently, but then we are really speaking about something else. You
>> can define whatever crazy quantity you want. But the need for a function
>> which describes the ratio of the length of a side of a right triangle
>> to the length of its hypotenuse will inevitably arise at some point in
>> physics and mathematics. And the "crazy Tronrud Sin" will not do this
>> job. So the proper sine and cosine functions will eventually have to
>> be invented.
>>
>>
>>
>>
>> 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:[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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>> --
>> Marc SCHILTZ      http://lcr.epfl.ch
>>