Hi Clemens,
Sorry to be picky and start the 'definition game' over again, but
'Miller indices' are strictly not the numbers that index X-ray
reflections that everyone is familiar with (whether observed or not!).
Miller indices were introduced in 1839 by the British mineralogist
William Hallowes Miller (it says in WIkipedia) as a way of describing
the direction of the perpendicular to the plane faces that he observed
on mineral crystals. A condition is that no common denominator is
possible, since it defines only the direction of a vector; its
magnitude has no relevance in this context. So you can have Miller
indices (1,0,0), (1,2,0), (1,2,3) etc but you can't have (2,0,0),
(3,0,0), {2,4,0), (3,6,9) etc., or at least (1,0,0) means exactly the
same thing as (2,0,0) etc. You can multiply the MiIler index vector
by -1: this indicates the opposite face of the crystal. Imagine what
an electron density map would look like if you only collected
intensities at the Miller indices!
Miller's observation of the plane faces of mineral crystals occurred
73 years before the discovery in 1912 of X-ray diffraction by Max Laue
in Munich (he became Max von Laue in 1913 when his father was raised
to the nobility), for which Laue received the Nobel Prize in Physics
in 1914. Laue explained diffraction by means of the 'Laue equations'
which contain 3 integers corresponding exactly to the indices we are
all familiar with. I prefer to call them 'reflection indices', though
strictly I suppose we should be calling them 'Laue indices'. Almost
immediately after Laue's discovery, William Lawrence Bragg in
Cambridge devised what we now know as "Bragg's Law", wherein the
factor 'n' relates the Miller indices to the Laue indices; thus the
reflection with indices (nh,nk,nl) is the n'th order of diffraction
from the set of crystal planes with Miller indices (h,k,l). Bragg
also received the physics Nobel prize jointly with his father William
Henry Bragg in the following year, 1915, for their determination of
the crystal structures of NaCl, ZnS and diamond.
Cheers
-- Ian
On Thu, Oct 21, 2010 at 4:57 PM, Clemens Vonrhein
<vonrh...@globalphasing.com> wrote:
> Hi Herman,
>
> On Thu, Oct 21, 2010 at 05:31:51PM +0200, herman.schreu...@sanofi-aventis.com
> wrote:
>> If you process your data in a lower symmetry space group, you will have
>> more unique reflections, since reflections which are related by the
>> higher symmetry will be avaraged during scaling in a higher symmetry
>> space group (i.e. a 2fold or 3fold axis), while in lower symmetry space
>> groups they will not. So the observation to parameter ratio stays the
>> same and is only depending on resolution and solvent content.
>
> True - if you count Miller indices as observations. But if you think
> about information content than probably not (as you discuss below).
>
>> The question one has to ask of course is: are these reflections really
>> different, or are they the same only not averaged?
>
> Yes - by merging we're getting better data (better error estimate on
> the intensity due to higher multiplicity). So there isn't really
> independent information in 50% of the reflections if e.g. going from
> P21 to P1 - we've only increased the noise because the multiplicity of
> each reflection has been reduced.
>
>> In the latter case, you have more reflections, but not more
>> information. As Ed mentions, using tight "NCS" restraints would in
>> this case mimick the crystallographic symmetry.
>
> Apart from the (good) NCS argument, one could go even further:
>
> We could also just collect 36000 degree of data on a 7A Lysozyme
> crystal and refine against completely unmerged data. After all, why
> should we stop at removing only the some symmetry operators from our
> data merging ... lets get rid of all of them including th x,y,z
> operator and use unmerged data. Then we could refine Lysozyme with
> anisotropic hydrogens and no restraints against 7A data since we have
> a huge number of 'observations' ... right?
>
> But seriously: there is a difference in having reflections (H, K, L)
> and independent data (I, SIGI). Maybe we should talk more about
> (independent observations)/parameters ratio in the same way we
> look at depdencies of parameters (e.g. restraints on Bfactors etc).
>
> Cheers
>
> Clemens
>
> --
>
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