Comparing images taken at 0 and 180 gives the impression of a mirror flip about an axis perpendicular to the rotation axis.  This is because Friedel mates are directly across the origin from each other and the Ewald sphere is curved.  Lets say you have a vertical rotation axis. Choose a spot at phi=0 in the "northern hemisphere" of the spindle (above the "equator") and also to the "left" of the spindle axis on the image.  Now, when you rotate the crystal 180 degrees, that spot's Friedel mate will pop up in the "southern hemisphere", but also "left" of the spindle axis.  This is because at phi=0, the Friedel mate was "right" of the spindle, still in the "southern hemisphere" but too far away from the Ewald sphere to diffract. 

Perhaps easier to understand is a similar illusion of inversion symmetry you get when you look at your crystal in the beamline microscope.  If you take a picture of the video screen, rotate the crystal 180 degrees, take another picture, and then load that second picture into photoshop and "flip" it about the rotation axis, you will find that these two images are very similar-looking.  (In fact, I use a procedure like this to find the rotation axis in my beamline's microscope).  However, the 180 degree rotation did not invert the hand of your crystal!  In actual fact, the "front" of the sample (the part facing the camera) is now the "back" after doing a 180 degree rotation.  This same front-to-back switch is also being done in the unit cells.  Anomalously-scattering atoms that used to be "first" to experience the incoming wavefront are now "last" and this is what makes the diffraction pattern sensitive to the phase lag in the scattering from these atoms.

-James Holton
MAD Scientist

On Fri, Feb 11, 2011 at 1:48 PM, Jacob Keller <[log in to unmask]> wrote:

Does the diffraction pattern change if you shoot the crystal from phi
= n or n+180 deg (a la inverse-beam geometry)? I was thinking it was
identical, but I am not sure now--is it a mirror image? Maybe
different space groups are different?

JPK

On Fri, Feb 11, 2011 at 10:35 AM, James Holton <[log in to unmask]> wrote:
>
> The indexing ambiguities do not include anomalous pair confusion because
> there is no way to rotate the lattice to make every h,k,l overlap with
> -h,-k,-l.  I.E. you can't rotate your left hand to superimpose it on your
> right.  The only way to mix those up is to change the sign of some detector
> geometry parameter (I.E. looking in a mirror).
>
> That said, anomalous differences tend to be very weak and noisy in all but
> the most exotic cases of macromolecular diffraction.  Twinning makes this
> worse because you are (to a first order approximation) averaging DANO(h,k,l)
> with DANO(k,h,-l) and the result will tend to be closer to zero than either
> one taken individually.  However, the biggest source of error in LCLS
> datasets at the moment is partiality.  Basically, you only get one shot per
> crystal, you can't rotate it appreciably in the 70 fs exposure time, the
> beam is a laser so there is essentially no divergence or dispersion, and the
> crystals are so small as to be one mosaic domain each, so there is no
> "mosaic spread".  The "3D profile" of the spots is therefore dominated by
> the finite size of the crystal itself (Sherrer broadening).  We were
> actually worried for a while that we wouldn't see any spots at all at LCLS!
>
> So, everything is a partial, and we currently don't have postrefinement
> software that can model the shape of each crystal and give us a partiality.
>  At least, not in a reasonable amount of time.  If we spent 30 s on each of
> the 3 million images, we would still be processing them for a few more
> years.  So, for the first run, it was decided to jut average out the
> partiality errors.  For example, unknown partiality means that each spot is
> measured with 100% error (at best), but if you have 700 of them, then the
> expected error of the average is ~3%.  John Spence called this a "Monte
> Carlo integration", and it turned out to be a really good idea.  We measured
> the error of the average by splitting the images into two heaps and
> comparing the merged datasets that resulted from each heap.  I proposed
> calling this "R-internal" for internal agreement, since a traditional Rmerge
> does not really apply.  However, I admit that for the PDB deposition I
> entered R-internal as "Rmerge".  Technically, R-internal is exactly what an
> Rmerge used to be: the R-factor between data from different crystals.
>
> Personally, I think "the way" to crack this "twin problem" is to scale all
> the data and look at the partial intensity histograms for each spot.  In
> situations where the "true" values of h,k,l and k,h,-l have radically
> different intensities, there will be a bimodal distribution, and that will
> allow us to re-index the ~700 images that contained a spot from one of those
> two hkls.  Which group to flip (the bright ones or the dim ones) is an
> interesting question, but probably the dim ones, since they are the least
> consistent with the average intensity.  Might need to try both.  After
> re-mergeing and re-scaling, there will be another hkl with the strongest
> bimodal distribution, and then you iterate.  That's the idea anyway.
>
> -James Holton
> MAD Scientist
>
> On 2/10/2011 6:32 AM, Jacob Keller wrote:
>>
>> Would it be true that the anomalous differences could not be measured
>> in these types of datasets, because one would not know which
>> Friedel/Bivoet reflection one is measuring in a given frame? Perhaps,
>> given anomalous signal, there would be a way to tease out which
>> orientation one was looking at from the correlations of the
>> signs/magnitudes of anomalous-scattering-induced deviations from the
>> mean intensities (derived from the whole dataset) for all of the
>> relections observed in each frame? I guess this might also detwin the
>> data?
>>
>> JPK
>>
>> On Thu, Feb 10, 2011 at 7:17 AM, Anastassis Perrakis<[log in to unmask]>
>>  wrote:
>>>>
>>>> Anyway, I thought that was a cool idea, but like so many other cool
>>>> things, it had to be cut from the Nature paper.  Admittedly, the problem
>>>> has
>>>> not actually been solved yet.  This is why we used REFMAC in TWIN mode.
>>>
>>> Is that a hint on the:
>>>
>>> a. wisdom of the editor
>>> b. wisdom of 'the third referee'
>>> c. wisdom of the dogma 'five years of eight eight lifes in 2000 words'
>>> d. All of the above
>>>
>>> ;-)
>>>
>>> A.
>>>
>>
>>
>
>

--
*******************************************
Jacob Pearson Keller
Northwestern University
Medical Scientist Training Program
cel: 773.608.9185
email: [log in to unmask]
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