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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:


- 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:


- scroll down a bit to the second section with the header:

# Header lines are:
# spgp_# spgp_name patterson_spgp_# patterson_spgp_name
# 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,

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 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

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

-- Ian

Ian J. Tickle, DPhil.
Director of X-ray Technology
Astex Therapeutics Ltd
436 Cambridge Science Park
Milton Road, Cambridge
Tel: +44(0)1223 226214
Fax: +44(0)1223 226201


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