Certainly to a mathematician by training P1 is ludicrous - any sensible being wold set up an 3-D axial system with all axes of equal length and all angles 90 degrees - I can still remember my disbelief at indexing any lattice in any other way - E
On Wed, 22 Jul 2020 at 19:18, James Holton <jmhol...@lbl.gov> wrote: > 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> 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> >> 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. >>> *************************************************************** >>> 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 or jeremyk...@gmail.com >>> http://img.chem.ucl.ac.uk/www/cockcroft/homepage.htm >>> *************************************************************** >>> 6 Wellington Road >>> Horsham >>> West Sussex >>> RH12 1DD >>> +44 (0) 1403 256946 (home) >>> *************************************************************** >>> >>> >>> On Wed, 22 Jul 2020 at 14:51, Nicholas Keep <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> >>>> <0000176a9d5ebad7-dmarc-requ...@jiscmail.ac.uk> >>>> Reply-To: Eleanor Dodson <eleanor.dod...@york.ac.uk> >>>> <eleanor.dod...@york.ac.uk> >>>> To: 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> 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, >>>>> Bloomsbury >>>>> LONDON >>>>> WC1E 7HX >>>>> >>>>> Office G54a >>>>> >>>>> Dean Email; scid...@bbk.ac.uk >>>>> Dept email 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 the door >>>>> >>>>> >>>>> ------------------------------ >>>>> >>>>> To unsubscribe from the CCP4BB list, click the following link: >>>>> https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 >>>>> >>>> >>>> ------------------------------ >>>> >>>> To unsubscribe from the CCP4BB list, click the following link: >>>> https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 >>>> >>> >> ------------------------------ >> >> To unsubscribe from the CCP4BB list, click the following link: >> https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 >> > > ------------------------------ > > To unsubscribe from the CCP4BB list, click the following link: > https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 > > > > ------------------------------ > > To unsubscribe from the CCP4BB list, click the following link: > https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 > ######################################################################## To unsubscribe from the CCP4BB list, click the following link: https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a mailing list hosted by www.jiscmail.ac.uk, terms & conditions are available at https://www.jiscmail.ac.uk/policyandsecurity/
