I think that it is clear that self-vector = Harker vector.
In David Harker's original paper (Harker, D. "The Application of the Three-Dimensional Patterson Method and the Crystal Structures of Proustite, Ag3AsS3, and Pyrargyrite, Ag3SbS3" J. Chem. Physics (1936) v. 4 pp. 381-390), he explicitly derives the formulae for Harker planes (two-fold [screw] axis parallel to b) and Harker lines (mirror plane perpendicular to b, & glide plane along c) in the Patterson function, and shows the attendant dramatic speed-up in calculation of these particular sections (lines) of the Patterson. [For a Harker line, "This is a one-dimensional Fourier series and can be completely evaluated in a few hours."!!]
Harker states but does not derive that similar formulae hold for 3-, 4-, and 6-fold [screw] axes as well, and that these can be used to advantage. Also, it is obvious that the symmetry axis need not lie along b. Lastly, he states "There are other cases in which the values of some of the atomic coordinates in a crystal can be determined by evaluation of P(x,y,z) throughout a plane or along a line only." He gives a few examples, including (pseudo-)non-crystallographic symmetry, concluding with "Such examples could be multiplied almost indefinitely." Although Harker did not explicitly state the formulae for the "diagonal" symmetry arrangement discussed below, I believe that he gave enough examples to render this unnecessary: simply determine the Harker plane (line) as the locus of points satisfied by the vectors: (SymOp-of-choice)*(x,y,z) - (x,y,z).
The text in Rossmann & Arnold (Int. Tables B, 2.3.2.2) does state that all self-vectors give rise to "a concentration of vectors at characteristic locations in the Patterson." Yet, their table is indeed incomplete, as is, apparently, the output from FFT. Perhaps the code-meisters at CCP4 could fix the latter?
(
By the way:
Stout & Jensen got it right (2nd Ed., p. 285; their table 12.1 lists "some" of the Harker lines & planes; the text states "Those for any other symmetry element can be derived along the lines given above.")
Ladd & Palmer also got it right, with extensive worked examples (3rd Ed., pp. 286-305).
Bricogne also (Harker peaks only, but no mention of these peaks forming a locus [Harker lines planes]; his restriction that t(1)g is non-zero would appear to eliminate Harker planes/peaks due to simple two-fold axes, but his notation confuses me and perhaps I misinterpret; (Int. Tables B, end of section 1.3.3.1.2.10)).
)
Dave
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David W. Borhani, Ph.D.
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| Bart Hazes <[EMAIL PROTECTED]>
Sent by: [EMAIL PROTECTED] 08/04/2006 11:30 AM
|
|
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Hi Ian,
Most of the time the International Tables is the ultimate authority on
matters crystallographic because most of the time it is right. However,
I agree with you that this does not make sense. The difficulty of
plotting a section should have nothing to do with the definition of
Harker sections/vectors.
Bart
Ian Tickle wrote:
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> *** CCP4 home page http://www.ccp4.ac.uk ***
>
>
>
> All -
>
> I was just having a discussion here concerning the precise definition of
> the terms 'Harker vector' & 'Harker section'. This discussion was
> prompted by this webpage:
>
> http://www.doe-mbi.ucla.edu/~sawaya/m230d/Patterson/patterson.html
>
> - specifically the 'Table of Patterson Difference Vectors' for space
> group P43212 (right at the end). The legend states "Table of all
> possible self-vectors in space group P43212. Colored blocks appear on
> Harker sections.". The implication of this is that some self-vectors do
> not appear on Harker sections and thus are not Harker vectors. This
> conclusion is backed up by this table at the same site:
>
> http://www.doe-mbi.ucla.edu/~sawaya/tutorials/Phasing/crystal.lib
>
> - scroll down a bit to the second section with the header:
>
> #CCP4I DATA PATTERSON
> #
> # Header lines are:
> # spgp_# spgp_name patterson_spgp_# patterson_spgp_name
> #_harker_sections
> #
> # Harker sections are listed on the subsequent lines
> #
> # Info generated using EJDs harker.f jiffy
>
> with this entry for P43212:
>
> 96 P43212 123 PG422 5
> X = 0.5000 1/2 2Y+1/2 2Z+3/4
> Y = 0.5000 2X+1/2 1/2 2Z+1/4
> Z = 0.2500 X+Y+1/2 -X+Y+1/2 1/4
> Z = 0.7500 X-Y+1/2 X+Y+1/2 3/4
> Z = 0.5000 2X 2Y 1/2
>
> There will always be N-1 self-vectors where N is the number of primitive
> general equivalent positions, so there are 7 in P43212, hence 2 of the
> self-vectors are missing from the above entry, and similarly for most of
> the high-symmetry space-group entries (trigonal, tetragonal, hexagonal,
> cubic).
>
> I would contend that _all_ self-vectors are also Harker vectors without
> exception (in fact surely the terms are totally synonymous?). The 2
> Harker vectors missing from the above entry are: X-Y, X-Y, 2Z and X+Y,
> X+Y, 1/2+2Z, both of which lie on the U-V = 0 Harker section. Just
> because this isn't a section perpendicular to a principal axis, hence
> FFT is unable print it on one page (though you can of course do it via
> MAPROT) seems a totally inadequate reason for excluding it from the
> definition! The locus of the vector between 2 points related by any
> rotation/screw axis is necessarily a plane perpendicular to that axis,
> and the U-V=0 section is merely the plane perpendicular to the 2-fold in
> the Laue group 422 along (1,1,0).
>
> Then, further seeking to justify my view, I turned to International
> Tables, which I've always regarded as the ultimate authority on matters
> crystallographic. The only explicit tabulation of Harker sections I
> could find there is in Vol. B, Chap. 2.3 by Rossmann & Arnold, Table
> 2.3.2.3 on p. 240. Worryingly for my point of view, this concurs with
> the definition in the webpage, and also omits the 2-fold U-V=0 and
> similar sections in trigonal, tetragonal, hexagonal and cubic space
> groups, though it does include the (1,1,1) 3-fold section in cubic (it
> calls it a 'special Harker plane' though it's not clear what's special
> about it!). Bizarrely, in the same table, space groups with 3, 4 and
> 6-folds are defined with the unique axis parallel to the _b_ axis (I
> wonder which version of the space-group tables they were using?).
>
> Finally I recalled that FFT helpfully prints out the Harker sections
> whenever you do a Patterson, so using some data in space group P321 I
> get:
>
> Symmetry operators: 1: X, Y, Z: 2:-Y, X-Y, Z
> Harker Vector: 2 X+Y, -X+2Y, 0
> Harker Section is: Z = 0.00
>
> Harker vector Matrix:
> 1.00 1.00 0.00 0.00
> -1.00 2.00 0.00 0.00
> 0.00 0.00 0.00 0.00
> 0.00 0.00 0.00 0.00
>
> Symmetry operators: 1: X, Y, Z: 3:-X+Y, -X, Z
> Harker Vector: 3 2X-Y, X+Y, 0
> Harker Section is: Z = 0.00
>
> Harker vector Matrix:
> 2.00 -1.00 0.00 0.00
> 1.00 1.00 0.00 0.00
> 0.00 0.00 0.00 0.00
> 0.00 0.00 0.00 0.00
>
> That's it! Looking at the code, FFT specifically excludes the
> additional 3 Harker vectors that I think should be listed (on the 3
> sections U+V=0, 2U-V=0 & U-2V=0), so am I alone in the belief that all
> self-vectors are Harker vectors, or do we accept the apparent
> alternative definition that only sections that can be printed on one
> page by the FFT program can be Harker sections (with the apparent
> special exception of U+V+W=0 in cubic)?
>
> It would be interesting to know how David Harker in his original paper
> viewed his eponymous section, unfortunately I don't have easy access to
> it.
>
> -- Ian
>
> ********************************************
> Ian J. Tickle, DPhil.
> Director of X-ray Technology
> Astex Therapeutics Ltd
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