Dear Ian,

     I made a modest contribution to this discussion a long time ago, and I
will only limit myself to one point.

     I think you may be confusing "setting" and "lattice mode". A change of
setting is performed by an integer matrix with determinant 1 (a "unimodular"
matrix) whereas a change of lattice mode involves two mutually inverse
integer matrices with determinants (mutually inverse, of course) different
from 1.

     The case of R32 and H32 seems to stick out like a sore thumb because we
never use the primitive-lattice versions of the centered-lattice space
groups in the monoclinic, orthorhombic and tetragonal classes - and yet they
exist! The problem with them is that e.g. 2-fold axes are represented by
non-diagonal matrices that are somehow thought to be an eyesore, so we
sacrifice mathematical rigour (the theory of "arithmetic classes") to the
comfort of having a 2-fold axis represented by the familiar diagonal matrix
with one 1 and two -1 on it. The matrices that would reindex those primitive
lattices to the usual centered ones would have determinants 2 or 4 in one
direction, and 1/2 or 1/4 in the other. However, as we never see these
representations of "centered" space groups in a primitive lattice basis, we
are startled when we come to the trigonal class. Here, the 3-fold axis has
two distinct representations by integer matrices: one in which the three
axes undergo a circular permutation (so they have to be of equal lengths and
separated by equal angles), and the other in which one axis (z) is
invariant, and the 3-fold symmetry is represented by a 120-degree rotation
in the (x,y) plane. These two representations cannot be mapped into each
other by means of a unimodular matrix: if one reindexes one representation
into the other, the determinant is 3 in one direction and 1/3 in the other.
In this case, it is a matter of opinion which representation of a 3-fold
axis has the greatest aesthetic merit, so the two possibilities are in use,
unlike the poor non-diagonal 2-fold axis representations that no one wants
to see.

     It is a matter of convention and vocabulary whether one calls these two
modes of indexing the rhombohedral and hexagonal "lattice modes", or calls
them "settings": one thing is certain, and that is that the mathematical
phenomenon in question is of a different kind from the reindexing of P21212
into P22121 with which you draw a parallel.

     At least this is what my distant memories of space-group theory seem to
be telling me :-)) . 


     With best wishes,
     
          Gerard.

--
On Mon, Jul 30, 2012 at 04:23:02PM +0100, Ian Tickle wrote:
> Without wishing to re-ignite previous discussions on this topic
> (perhaps <FLAME> ... </FLAME> tags would be in order!), I would point
> out that this and similar confusion with other space groups has arisen
> largely from a failure of some programmers (and users!) to fully
> comprehend the important difference between a 'standard symbol' and a
> 'setting symbol' for a space group, no doubt because in many cases
> these are superficially identical, or a least very similar.  This
> point is also made in the Computational Crystallography Newsletter
> article on H3 and H32 that I referenced earlier.
> 
> The Hermann-Mauguin symbol (aka 'standard symbol') is unique to a
> space group and crucially is designed to be independent of the setting
> (orientation and/or origin).  It is used to identify a space group
> without reference to the setting, and therefore its main use is to
> provide page headings and index entries in ITC. There exist exactly
> 230 H-M standard symbols for the 230 unique 3D space groups.  The H-M
> standard symbol is the same for all settings of a particular space
> group and therefore cannot be used to define the setting: for that you
> obviously need additional information.
> 
> The standard symbol is thus of little or no relevance to practical
> crystallography: for that you must use a setting symbol.  However for
> the majority of space groups only one setting is accepted as
> 'conventional' so in those cases the standard and setting symbols are
> identical; it's only where there are multiple settings that problems
> arise.
> 
> A simple analogy might be to say that an object is called 'building'
> and that is also its standard symbol.  It describes the object without
> reference to its orientation or position and so is not relevant to the
> practical problem of defining the view of the building: for that you
> need extra symbols.  For example you might need to specify one of the
> setting symbols 'building (front elevation)', 'building (side
> elevation)' or 'building (plan)'.
> 
> So R32 is a H-M standard symbol which corresponds to the 2 alternate
> setting symbols R32:r and R32:h as described in the article.  Plainly
> you can't use the H-M symbol R32 to uniquely specify the setting since
> it is the standard symbol for both the R32:r and R32:h settings.  The
> latter are _not_ H-M symbols: they are ITC extensions of the H-M
> symbol.
> 
> For other space groups further confusion has arisen because ITC often
> uses the exact same character string for both the standard symbol and
> one of the corresponding alternate setting symbols.  An obvious
> example is P21212: this is the H-M standard symbol for SG #18 but is
> also one of the 3 ITC setting symbols for P21212, the other two being
> P22121 and P21221.  Perhaps the intention would have been clearer if
> the ITC setting symbols had all been made different from the standard
> symbol, as they are in the R32 case.  For example P21212a, P21212b and
> P21212c would have been equally valid choices for the ITC setting
> symbols but do not express a 'preferred' setting (since there isn't
> one).  Similarly the standard symbol for SG #5 (unique axis b) is C2,
> and the alternate setting symbols are A2, C2 and I2, but they could
> equally well have been (for example) C2a, C2c and C2i, which doesn't
> express a preference for any one of the alternate settings.
> 
> Either way, according to the ITC rules, the choice of 'conventional'
> setting for a space group (i.e. the recommended default choice when
> there are no other grounds such as isomorphism with a previously
> determined structure) is made by reference to the unit cell.  For R32
> the conventional cell happens to be the hexagonal one (a = b != c,
> alpha = beta = 90, gamma = 120) with symbol R32:h; for all
> orthorhombic SGs the convention is a < b < c and the setting symbol
> derives from that.
> 
> Cheers
> 
> -- Ian
> 
> On 28 July 2012 22:22, Edward A. Berry <[log in to unmask]> wrote:
> > Are all the software packages consistent in their (mis)use of these
> > symbols? Recently I scaled data (scalepack) as R3, imported to ccp4 as H3,
> > and had to make a link in $ODAT/symm from R32 to H32 (which it turned out to
> > be).
> >
> >
> >
> > Ian Tickle wrote:
> >>
> >> If we're all agreed that ITC(A) is taken as the authority on all
> >> matters of space group symbology (and I for one certainly agree that
> >> it should be), then SG symbol H32 (SG #145:
> >> http://img.chem.ucl.ac.uk/sgp/medium/145bz1.htm) has nothing to do
> >> with R32 (SG #155: http://img.chem.ucl.ac.uk/sgp/medium/155az1.htm)!
> >> According to the Hermann-Mauguin system of nomenclature H32 (more
> >> correctly written as H3_2 where the '_' indicates a subscripted screw
> >> axis) would be the hexagonal-centred (H) lattice setting of P32 (P3_2
> >> in H-M).  H32 as an alternate setting symbol for R32 is a very recent
> >> PDB invention which conflicts with the well-established H-M convention
> >> used throughout ITC.  The ITC symbols for the rhombohedral&  hexagonal
> >>
> >> axis settings of SG R32 are R32:r and R32:h respectively, i.e. obvious
> >> extensions of the H-M symbols without introducing any conflict with
> >> the existing convention, as the PDB symbol does.  The confusion has
> >> arisen from the failure to distinguish the lattice type (the first
> >> letter of the symbol) from the symbol for the basis system of the
> >> setting (the final letter after the ':').
> >>
> >> See http://cci.lbl.gov/~rwgk/my_papers/CCN_2011_01_H3_H32.pdf for an
> >> excellent explanation of all this and of the confusion that arises
> >> when programmers ignore established conventions and 're-invent the
> >> wheel' (e.g. SCALEPACK apparently swaps the meaning of the PDB symbols
> >> R32&  H32 and uses R32 for PDB H32 and vice-versa!).
> >>
> >>
> >> Cheers
> >>
> >> -- Ian
> >>
> >> On 27 July 2012 21:09, Bernhard Rupp<[log in to unmask]>  wrote:
> >>>
> >>> H32 indicates the hexagonal obverse setting (as you list) for a R
> >>> centered trigonal cell, which is 3x larger than the primitive R32 cell
> >>> indexed a=b=c, al=be=ga<>  90. Standard imho is the H32 setting, for which I
> >>> will probably get flamed.
> >>> The relation between H and R cells is depicted here:
> >>>
> >>> http://www.ruppweb.org/Garland/gallery/Ch5/pages/Biomolecular_Crystallography_Fig_5-29.htm
> >>>
> >>> This has been discussed and is explained in the ccp4 tutorials and doc
> >>> afaik, where you can find more detailed info.
> >>>
> >>> For proper format in a journal, I would suggest to adhere to the format
> >>> given in the ITC (International tables for Crystallography), I.e. Bravais
> >>> Italic, subscripted screw symbols. Note that this is not the format you put
> >>> it into most programs - their docs help.
> >>>
> >>> You can also try my old space croup decoding program to see general
> >>> positions, operators, matrices and other useful stuff.
> >>>
> >>> http://www.ruppweb.org/new_comp/spacegroup_decoder.htm
> >>>
> >>> HTH, BR
> >>>
> >>> -----Original Message-----
> >>> From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of
> >>> Theresa Hsu
> >>> Sent: Friday, July 27, 2012 12:54 PM
> >>> To: [log in to unmask]
> >>> Subject: [ccp4bb] Space group R32 and H32
> >>>
> >>> Dear all
> >>>
> >>> I have a confusion on the space group R32 and H32. For a cell parameter
> >>> of a = b not equal to c, alpha=beta, not equal to gamma, is it considered as
> >>> R32 or H32?
> >>>
> >>> I tried searching the mail list archives but it does not help a beginner
> >>> crystallographer like me.
> >>>
> >>> I also have another basic question. What is the correct way for writing
> >>> space groups? Is the Bravais lattice in italic and is there space after
> >>> that? Or it does not matter because both are used in literature?
> >>>
> >>> Thank you.
> >>
> >>
> >

-- 

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