It took me a long time to realize that the rhombohedral system is
actually a form of centering. Just like C, F, and I, the R lattice type
has extra translation-related points. What is unique about rhombohedral
lattices is that there are two different kinds of "extra" centers. In
C, all you have is an extra point in the middle of one face, and for F
it is all the faces. For I it is an extra point at the exact center of
the whole cell, but for R there are two extra points in the middle of
the cell.
The reason there are two is analogous to how an ellipse has two "foci"
and a circle only has one. Rhombohedral is just a distortion of a cubic
cell: grab opposing corners and pull. The relationship to hexagonal and
trigonal is clear if you look at a cube down the corner-to-corner
diagonal. It looks like a hexagon. And as it turns out if you
re-define the lattice centering operations to be "symmetry operators",
that are internal to the cell rather than relating different cells, you
can map rhombohedral onto trigonal. That's where "H" comes from. Most
any high-symmetry space group can be mapped onto another by turning
centering into a translation-only symmetry operator. However, if you
want to follow the proper definition of a unit cell, anything that
relates the whole lattice back onto itself via translation only is not a
"symmetry operator", it is a whole-cell shift.
For a lecture once, I made some movies showing how the different
centering types relate. I make movies because 3D concepts are hard to
show in 2D, and so why not provide some motion to give perspective? Some
of you may find them useful in teaching.
https://bl831.als.lbl.gov/~jamesh/powerpoint/CSHL_spacegroups_2019.pptx
I find it can be more intuitive to students to go in "reverse": start
with a cubic lattice, which has only one parameter to think about, then
introduce centering. After that rotations, and eventually arrive at P1,
which has six degrees of freedom. Strange how we consider P1 to be the
"simplest" unit cell, when to the unindoctrinated it certainly is not.
-James Holton
MAD Scientist
On 7/22/2020 9:50 AM, Ian Tickle wrote:
The original reference for the H cell is the very first edition of
Int. Tab.:
Hermann, C. (1935). Internationale Tabellen zur Bestimmung von
Kristallstrukturen. Berlin: Gebrueder Borntraeger.
Cheers
-- Ian
On Wed, 22 Jul 2020 at 17:34, Eleanor Dodson
<0000176a9d5ebad7-dmarc-requ...@jiscmail.ac.uk
<mailto:0000176a9d5ebad7-dmarc-requ...@jiscmail.ac.uk>> wrote:
Well - yes . I am a true devotee of doctored cells to match
something already in existence in a higher symmetry which has
become approximate in some new manifestation! But I hadnt realised
there were official versions of doctoring..
Eleanor
On Wed, 22 Jul 2020 at 16:29, Jeremy Karl Cockcroft
<jeremyk...@gmail.com <mailto:jeremyk...@gmail.com>> wrote:
Dear Eleanor,
What you say is absolutely spot-on! An H3 cell can be reduced
down to the smaller P3 cell as you pointed out.
However, sometimes it may be useful to use a larger unit cell.
I can't give an example for trigonal space groups or from
protein crystallography, but in a recent paper, I used a unit
cell in C-1 (i.e. a doubled P-1 triclinic cell) as this
related to the C-centred monoclinic cell as exhibited in a
higher temperature phase. I could have used P-1, but I knew
that chemists would see the relationship between the phases
more easily by using an enlarged cell. I have done this sort
of thing many times, e.g. I used F2/d for the low temperature
phase of DI (HI) many years ago instead of C2/c as this
related to the face-centred cubic form. As I am interested in
phase transitions, I tabulated a range of space-group
settings for enlarged unit cells on my site.
I am not sure that this will make the CCP4 list as I am not
subscribed to it - please feel free to echo it on there.
Best regards,
Jeremy Karl.
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Dr Jeremy Karl Cockcroft
Department of Chemistry
(University College London)
Christopher Ingold Laboratories
20 Gordon Street
London WC1H 0AJ
+44 (0) 20 7679 1004 (laboratory)
+44 (0) 7981 875 829 (cell/mobile)
j.k.cockcr...@ucl.ac.uk <mailto:j.k.cockcr...@ucl.ac.uk> or
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On Wed, 22 Jul 2020 at 14:51, Nicholas Keep
<n.k...@mail.cryst.bbk.ac.uk
<mailto:n.k...@mail.cryst.bbk.ac.uk>> wrote:
-------- Forwarded Message --------
Subject: Re: [ccp4bb] Question about P3, H3 and R3 space
groups
Date: Wed, 22 Jul 2020 14:41:27 +0100
From: Eleanor Dodson
<0000176a9d5ebad7-dmarc-requ...@jiscmail.ac.uk>
<mailto:0000176a9d5ebad7-dmarc-requ...@jiscmail.ac.uk>
Reply-To: Eleanor Dodson <eleanor.dod...@york.ac.uk>
<mailto:eleanor.dod...@york.ac.uk>
To: CCP4BB@JISCMAIL.AC.UK <mailto:CCP4BB@JISCMAIL.AC.UK>
But surely P3 symmetry ' cell ( a,a,c 90 90 120) with
origin shifts (⅓,⅔,0), +(⅔,⅓,0) can just be indexed as P3
with
cell (a/sqrt(3), a/sqrt(3), c, 90 90 120) ??
Eleanor
On Wed, 22 Jul 2020 at 14:32, Nicholas Keep
<n.k...@mail.cryst.bbk.ac.uk
<mailto:n.k...@mail.cryst.bbk.ac.uk>> wrote:
I have an answer from Jeremy Cockcroft the author of
the 'Birkbeck' Tables. Actually Jeremy has been at UCL
for a decade or so and they are hosted from their
Nick
In answer to the question regarding the use of R and H
for trigonal space groups, the letters refer to two
distinct types of lattice centring symmetry operation,
namely +(⅓,⅔,⅔), +(⅔,⅓,⅓) and +(⅓,⅔,0), +(⅔,⅓,0),
respectively. For the subset of rhombohedral space
groups, the symbol R should always be used.When the
alternative unit cell with a=b=c and α=β=γ is chosen,
this cell corresponds to a primitive Bravais lattice,
but the label P is not used as this would result in
confusion with non-rhombohedral space groups. (The
choice of symmetry operators for rhombohedral space
groups is wholly dependent of the choice of unit cell,
i.e. hexagonal versus rhombohedral, so there is no
ambiguity if the unit cell is specified.)
For the non-rhombohedral space groups, it may
occasionally be convenient to choose a larger
H-centred unit cell that is not the usual primitive P
one.(The use of larger unit cells is quite common for
systems that undergo phase transformations as it may
enable the crystallographer to keep the contents of
the unit cell the same in both phases.)Note that the
use of an H-centred lattice switches the order of the
symmetry elements in these space group symbols, e.g.
P312 becomes H321.
I am not aware of any changes to this convention,
which I believe has a long history. However, it is
possible that the letter H has been used unwittingly
for other purposes.
Jeremy Karl Cockcroft
--
NOTE NEW PHONE NUMBER JULY 2020
Prof Nicholas H. Keep
Executive Dean of School of Science
Professor of Biomolecular Science
Crystallography, Institute for Structural and Molecular Biology,
Department of Biological Sciences
Birkbeck, University of London,
Malet Street,
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LONDON
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<mailto:n.k...@mail.cryst.bbk.ac.uk>
Telephone 020-3926-3475 (Will contact me at home if working as
well as my office)
If you want to access me in person you have to come to the
crystallography entrance
and ring me or the department office from the internal phone by
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