Everyone knows that there are 230 space groups, and these belong to one 
of 32 point groups, which, in turn, belong to one of the 14 Bravais 
lattices, and 7 crystal "systems": triclinic, monoclinic, orthorhombic, 
tetragonal, hexagonal, rhombohedral and cubic.

Or are there?  If you look in ${CLIBD}/symop.lib of your nearest CCP4 
Suite install, you will find not 230 but 266 entries for "space groups", 
and 43 different kinds of point groups.  And those so-called 
"rhombohedral" systems can apparently be represented as hexagonal, so 
maybe there are only six crystal systems?

Blasphemy!  (I can almost hear the purists now)  But, the point I am 
trying to make here is that there is a disconnect between the 
traditional way that crystallography is taught (aka Chapter 1: crystal 
symmetry) and the pragmatic practice of crystallography (aka "what 
MOSFLM is doing").  It is ironic really that the first thing you must 
decide for a new crystal is its "space group" when in reality it is the 
last thing you will ever be certain about it.  Probably one of the most 
common examples of this is the P2221 and P21212 space groups.  
Technically, P2122, P2212, etc are NOT space groups!  However, given 
that orthorhombic unit cells are traditionally sorted a<b<c, simply 
giving such a unit cell with the space group P2221 is not enough 
information to be sure which axis is the "screwy" one.  Also, I'm sure 
many of you have noticed that for any trigonal/hexagonal crystal there 
is always a C222 cell that comes up in autoindexing?  This is because 
you can always index a trigonal lattice along a "diagonal" and that 
makes it look like centered orthorhombic.  But, if you try going with 
that C222 choice you find that it doesn't merge ... most of the time.

The fact of the matter is that all autoindexing algorithms give you is a 
unit cell, and that is just six numbers.  The cell dimensions generally 
allow you to EXCLUDE a great many symmetry operations, but they can 
never really INFER symmetry.  Except, of course, in the special case 
where all three angles of the reduced cell are not 90 (or 60) degrees, 
then the only possible space group is P1.  On the other hand, it is 
perfectly possible to have P1 symmetry with all three cell edges the 
same length and all angles 90 degrees.  It just isn't very "likely" (in 
the "posterior probability" sense).  This is why MOSFLM and other 
autoindexing programs pick the highest-symmetry lattice and give you a 
"space group" consistent with that lattice, even though there are plenty 
of other possibilities.  This is why you should always take the "space 
group" that comes out of autoindexing with a grain of salt. Do NOT make 
the mistake of classifying your crystals by the result of autoindexing 
alone!

Something similar is true for point groups.  A high Rsym for a given 
symmetry operator (like you will see in the output of "pointless") means 
that there is NO WAY that the given symmetry operation is part of the 
space group.  A low Rsym, however, does not mean that you have a given 
symmetry.  Could always be some kind of "twinning" or 
nearly-crystallographic NCS (NCNCS?). Twinning is relatively rare, and 
gets increasingly rare as you get into the non-merohedral stuff, but it 
is always a possibility. Yes, intensity statistics can tell you 
something is twinned, but if you have just the right mixture of twinning 
and pseudotranslation, then the twinning can go undetected.  So, in 
general, you can always have _less_ symmetry than you think, but proving 
the existence of a symmetry operation is hard.

Space groups, or narrowing down the screw vs rotation nature of various 
axes generally requires phasing and looking at a map.  The one with 
right-handed alpha helices is the correct one.  Yes, there are plenty of 
"tricks" like systematic absences, native Pattersons and the like but 
there are a lot of false positives and false negatives possible with 
each of these.  In fact, you tend to throw out more "rejects" in scaling 
than you ever have observations of systematic absences, so why trust 
those "absent" spots so much?  In fact, sometimes you need to even go 
all the way to the end of refinement to settle the space group.  It is 
possible to get "stuck" with R/Rfree too high because the crystal very 
slightly violates the symmetry you think it has.  ("NCNCS" again)

Whatever you do, don't forget to try all the possible P2122-like space 
groups if you are searching for heavy atoms or doing MR with a primitive 
orthorhombic crystal.  Far too many people have missed solving their 
structure because they didn't know to do this! Fortunately, modern 
computers tend to have 8 or so CPUs in them, and there are never more 
than 8 space groups possible on any given point group.  So, you might as 
well launch 8 parallel MR or heavy-atom site-finding jobs in different 
space groups, since it will take just as long to run 8 jobs as it will 
take to do only one.  Well, okay, some of the non-protein ones have more 
than 8 choices for a given point group, but I don't generally care about 
those.

One important pitfall to NOT get stuck in is dropping a real symmetry 
operator and replacing it with a twin operator.  This will always lower 
R/Rfree because you compress the dynamic range of your data.  Best 
"control" I can think of is to take an operator you don't think is 
twinning and "twinify" that one to see what kind of R drop you expect 
from doing the "wrong thing".  If you get the same drop for all symmetry 
operators (and if the L-test does not say "twinning") then you will have 
a hard time defending your hypothesis that the crystal was twinned.

Another unfortunate pitfall is exchanging an actual crystallographic 
operator for "NCS".  This sounds innocuous at first, because there 
really isn't a difference between these two ways to represent the same 
symmetry operation.  In fact, many frustrated crystallographers at one 
time or another have asked the obvious question: why can't we just do 
everything in P1?  The problem arises when you start thinking about 
choosing your "random" Rfree set.  For example, you could take a crystal 
that actually belongs to P212121 and "drop the symmetry" to P21 (with a 
90-degree beta angle) and then apply an "NCS" operator between the 
monomers in your new "asymmetric unit".  Problem is, if you just pick a 
"random" Rfree set in this P21 cell, then pretty much every hkl in your 
"free" set will have an "NCS symmetry mate" in the working set.  All you 
have to do is turn up the x-ray weight and voila!  You have a reasonably 
low Rfree and R-Rfree gap for what could be a completely wrong structure.

Other space groups have similar traps, and there is no good tabulation 
of all the ways a "random" Rfree set can go wrong if you are applying 
NCS or de-twinning operators.  This is because there are actually a 
large number of ways that a given "free" reflection can be "correlated" 
to one in the working set.  Thin shells don't always do the trick.  In 
fact, the mere fact that protein crystals are ~50% solvent actually 
creates "local" correlations in reciprocal space.  This is why solvent 
flattening works, but it also produces some "bias" in the Rfree.  How 
much bias?  It's actually hard to say.  I know of a few people who have 
written programs for picking free-R flags in an "unbiased" way, but as 
far as I can tell nobody is distributing such programs as "software", 
perhaps due to fear that some clever idiot will find a way to abuse it.

Personally, I try to stay away from NCS, unless I'm really sure that 
there is nothing "funny" going on in the crystal symmetry.

-James Holton
MAD Scientist

On 11/13/2012 1:55 AM, vincent Chaptal wrote:
> Dear all,
>
> I am not sure I understand point groups and relations between groups 
> and subgroups anymore, and would appreciate some guidance.
>
> I was under the impression that all point groups were related to an 
> original P1 cell, and that by applying specific lattice symmetries, 
> one could "get" higher point groups. Thus, if one knows the symmetry 
> operators, one could jump from one point group to another. Inspection 
> of the reflections can then determine the "real" point group and space 
> group.
> At least that's what I thought Mosflm was doing? Am I correct?
> P1 +(symm-opp)>C2 + (symm-opp2)>P3
> same P1 +(symm-opp3)> P2 + (symm-opp4)>P222 ....
> If that's the case, could someone point to me where to find these 
> symmetry opperators (International tables?), because it's not obvious 
> to me.
>
> Or are these relations between groups and subgroups only true for 
> certain crystals where the cell parameters are specific, and allows a 
> symmetry operator to generate a higher symmetry point group?
>
> Thank you for your help.
> vincent