From CCP4 5.0, the core handling of symmetry information is done by a set of C functions. Separate
CSYM library documentation describes the structures and functions used, and the API for C/C++ programs.
For Fortran programs, the original set of subroutines (held in symlib.f
) has been replaced by an CSYM library interface to the C library. From the point of view of an application programmer, this interface should be identical to the original set of subroutines. This document originates from the original Fortran library, but should be applicable to the new library.
The available Fortran calls have been arranged as much as possible into groups by function. There are often several versions of calls apparently performing the same or very similar tasks, reflecting the policy of never removing existing functionality in order to maintain compatibility with programs written using older versions of the library.
In CCP4, the primary symmetry information is held in a library data file defined by the logical name SYMINFO. In the standard configuration of CCP4 this is the file syminfo.lib which normally resides in $CLIBD. This file replaces the previous data file SYMOP (i.e. $CLIBD/symop.lib).
syminfo.lib
holds information for all the standard spacegroups in the International Tables. For each spacegroup, several alternative settings are included (e.g. "P 1 2 1", "P 1 1 2" (a.k.a. 1003) and "P 2 1 1" for spacegroup 3).
Each setting of a spacegroup is delimited by begin_spacegroup / end_spacegroup records, and contains the following items:
number = standard spacegroup number
basisop = change of basis operator
symbol ccp4 = CCP4 spacegroup number e.g. 1003
(0 if not a CCP4 group)
symbol Hall = Hall symbol
symbol xHM = extended Hermann Mauguin symbol
symbol old = CCP4 spacegroup name
(blank if not a CCP4 group)
symbol laue = Laue group symbol
symbol patt = Patterson group symbol
symbol pgrp = Point group symbol
hklasu ccp4 = reciprocal space asymmetric unit
(with respect to standard setting)
mapasu ccp4 = CCP4 real space asymmetric unit
(with respect to standard setting)
(negative ranges if not a CCP4 group)
mapasu zero = origin based real space asymmetric unit
(with respect to current setting)
mapasu nonz = non-origin based real space asymmetric unit
(with respect to current setting)
cheshire = Cheshire cell
(with respect to standard setting)
symop = list of primitive symmetry operators
cenop = list of centering operators
For example:
begin_spacegroup
number 3
basisop z,x,y
symbol ccp4 1003
symbol Hall ' P 2y (z,x,y)'
symbol xHM 'P 1 1 2'
symbol old 'P 1 1 2'
symbol laue '-P 2' '2/m'
symbol patt '-P 2' '2/m'
symbol pgrp ' P 2' '2'
hklasu ccp4 'k>=0 and (l>0 or (l=0 and h>=0))'
mapasu ccp4 0\<=x\<=1/2; 0\<=y\<1; 0\<=z\<1
mapasu zero 0\<=x\<1; 0\<=y\<=1/2; 0\<=z\<1
mapasu nonz 0\<=x\<1; 0\<=y\<=1/2; 0\<=z\<1
cheshire 0\<=x\<=1/2; 0\<=y\<=0; 0\<=z\<=1/2
symop x,y,z
symop -x,-y,z
cenop x,y,z
end_spacegroup
A call to Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RlSymmMatrx) MSYMLB3 should be used to retrieve information from the symmetry library. Note that not all the data items are compulsory for MSYMLB3, although older versions of the routine (MSYMLB2, MSYMLB, MSYGET) still need them.
In syminfo.lib the symmetry operations in each spacegroup are listed as strings, for example X,Y,Z or -Y,X-Y,Z+1/3 etc. To be useful within a program these string representations have to be converted to some mathematical representation.
Typically a symmetry operation RSym will consist of a rotation operation R and a translation operation T (basically a vector). These are applied to a vector x to obtain x':
x' = Rx + T
It is convenient to represent the rotation by a 3*3 matrix:
( R11 R12 R13 )
[R] = ( R21 R22 R23 )
( R31 R32 R33 )
and the translation by a column vector with 3 elements:
( T1 )
[T] = ( T2 )
( T3 ).
CCP4 uses 4x4 arrays to store these symmetry operations as follows:
RSym = ( R11 R12 R13 T1 )
( R21 R22 R23 T2 )
( R31 R32 R33 T3 )
( 0 0 0 1 )
or
RSym = ( [R] | [T] )
( 0 0 0 | 1 )
Essentially this is a 4x4 matrix holding 3x3 transformation matrix in the "top-left-hand corner", the 3-element column (translation) vector in the "top-right-hand corner", and then (0 0 0 1) in the bottom row.
The subroutine Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RlSymmMatrx) MSYMLB3 will obtain the set of symmetry matrices in this representation for a given spacegroup, whilst Subroutine SYMFR2(ICOL,I1,NS,ROT) SYMFR2 or Subroutine SYMFR3(ICOL,I1,NS,ROT,EFLAG) SYMFR3 will obtain individual matrices from the string representation mentioned above. (Subroutine SYMTR4(NSYM,RSM,SYMCHS) SYMTR4 will perform the inverse operation, converting matrices to string representation.)
Using the convention outlined above, post-multiplying the 4x4 matrix by a column vector as follows:
RSym . [xf]
[yf]
[zf]
[1 ]
will apply both the symmetry and the translation operations to real space coordinates with a single matrix multiplication. The CCP4 modlib MODLIB library provides matrix_vector_ops matrix-vector routines MATVEC4 and TRANSFRM which can be used to perform this operation.
The inverse of real-space symmetry matrices are applied to reflection indices by pre-multiplying them by a row vector representing hkl,
ie. h' = h R-1 or,
(h' k' l') = (h k l) R-1
Note that only the operations in the appropriate Laue group are applied to reflection indices, so there are no translational components (i.e. the vector part of the operation, [T], is zero). The subroutine Subroutine INVSYM(S,ST) INVSYM will invert a 4x4 matrix stored in this representation, for the purpose of applying symmetry operations to reflection indices.
Real space axis vectors are transformed like reciprocal space vectors, i.e.
(a' b' c') = (a b c) R-1
while reciprocal space axis vectors are transformed like real space coordinates, i.e.
(a*' b*' c*') = R (a* b* c*)
(See also the REINDEX documentation.)
The routines have been broken down into groups according to function.
This group contains routines for obtaining the lists of symmetry operators from the library, and converting between the string (eg Y,X,-Z etc) and matrix representations of symmetry operators.
- Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RlSymmMatrx) MSYMLB3
- Gets the symmetry operators in matrix representation for a specified spacegroup from the symmetry library. Will match to any valid s/g name but always returns the longest name, and will match to the spacegroup number if no name matches. Always uses the s/r PGDEFN to guess the point group and the number of primitive symmetry operations (so that any listed in the library are automatically ignored). Checks that the symmetry operations form a closed set.
Replaces: Subroutine MSYMLB2(IST,LSPGRP,NAMSPG_CIF,NAMPG,NSYMP,NSYM,ROT) MSYMLB2, Subroutine MSYMLB(IST,LSPGRP,NAMSPG,NAMPG,NSYMP,NSYM,ROT) MSYMLB, Subroutine MSYGET(IST,LSPGRP,NSYM,ROT) MSYGET
- Subroutine SYMFR2(ICOL,I1,NS,ROT) SYMFR2, Subroutine SYMFR3(ICOL,I1,NS,ROT,EFLAG) SYMFR3
- Translates a character string containing symmetry operator(s) into matrix represenation, stored in a 4*4 matrix/array.
NB: SYMFR2 will translate real space coordinate operations (e.g. x+z,z,-y
), reciprocal space operations (e.g. h,l-h,-k
) and reciprocal- and real-space axis vector operations (e.g. a*+c*,c*,-b*
and a,c-a,-b
respectively). SYMFR3 only translates real space coordinate operations.
- Subroutine SYMTR4(NSYM,RSM,SYMCHS) SYMTR4
- Translates symmetry matrices into character strings with the equivalent symmetry operations.
Replaces: Subroutine SYMTRN(NSM,RSM) SYMTRN, Subroutine SYMTR3(NSM,RSM) SYMTR3
- Subroutine INVSYM(S,ST) INVSYM
- Invert the 4*4 array holding the symmetry matrices, to get the inverse symmetry operation.
Use MSYMLB3 to obtain the set of symmetry operator matrices given the spacegroup name or number. SYMFR2/3 will generate individual symmetry operator matrices from their string representation (useful if the operators are a subset of a spacegroup). SYMTR4 performs the opposite action, and generates string representations of individual symmetry operations from the matrices.
INVSYM will generate the inverse matrix of a real space symmetry operation, to be applied to reflection indices as described in HOW SYMMETRY OPERATIONS ARE STORED AND APPLIED section 2.
- Subroutine DETERM(det,a) DETERM
- Calculate the determinant of 4*4 matrix.
- Subroutine PGDEFN(NAMPG,NSYMP,NSYM,RSYMT,LPRINT) PGDEFN
- Obtain/guess pointgroup and primitive set of symmetry operators from analysis of all symmetry operators.
- Subroutine PGMDF(JLASS,JCENTR,JSCREW) PGMDF
- Gronigen subroutine: determine the nature of the rotation and screw axes from the symmetry matrices.
- Subroutine PGNLAU(NAMPG,NLAUE,LAUNAM) PGNLAU
- Determine the Laue group from pointgroup name.
- Subroutine PATSGP(SPGNAM, PGNAME, PATNAM, LPATSG) PATSGP
- Determine the Patterson spacegroup from true spacegroup.
- Subroutine HKLRANGE(IHRNG0,IKRNG1,IKRNG0,IKRNG1,ILRNG0,ILRNG1) HKLRANGE
- Return HKL ranges chosen in PGNLAU
These routines all derive additional information from the symmetry operators or the spacegroup name. The subroutine HKLRANGE returns the information stored in the common block which it shares with PGNLAU
- Subroutine CENTRIC(NSM,RSMT,IPRINT) CENTRIC
- Sets up symmetry elements; must be called first.
- Subroutine CENTR(IH,IC) CENTR
- Tests if a reflection is centric
Nb: routines Logical Function CENTRC(KHKL,ICENT) CENTRC and CENPHS both appear to be unused.
Call CENTRIC once to set up the symmetry elements in common blocks shared with CENTR. This defines the set of centric reflections. Then for each reflection, a call to CENTR will return whether it is centric.
- Subroutine EPSLN(NSM,NSMP,RSMT,IPRINT) EPSLN
- Sets up tables of epsilon zones; must be called first.
- Subroutine EPSLON(IH,EPSI,ISYSAB) EPSLON
- Returns the epsilon zone of a given reflection, as well as whether the reflection is systematically absent (using a call to SYSAB).
- Subroutine SYSAB(IN,ISYSAB) SYSAB
- Function: determines if a reflection is systematically absent.
Call EPSLN once to generate the epsilon zones (general sets of reflections eg 00l or 0k0) and determine the multiplicity/fold. For each reflection a call to EPSLON returns the zone and if the reflection is systematically absent. SYSAB is not called directly.
Logical Function HKLEQ(IH,KH) HKLEQ - used in SCALA to test if two reflections have equal indices.
Remember that the choice of asymmetric unit is NOT UNIQUE. These routines define the set of CCP4 asymmetric units. The limits for these definitions are given in
DEFINITION OF THE CCP4 ASYMMETRIC UNIT section 3.
- Subroutine ASUSET ASUSET
- Set up symmetry for ASUPUT and ASUGET; must be called first. Calls PRTRSM.
- Subroutine ASUPUT(IHKL,JHKL,ISYM) ASUPUT
- Put reflection into asymmetric unit defined in ASUSET (reverse operation of ASUGET). Calls INASU.
- Subroutine ASUGET(IHKL,JHKL,ISYM) ASUGET
- Recover original indices of a reflection in the asymmetric unit defined in ASUSET (reverse opertaion of ASUPUT).
- Subroutine ASUPHP(JHKL,LSYM,ISIGN,PHASIN,PHSOUT) ASUPHP
- Change phase for symmetry related reflection.
Call ASUSET first to set up symmetry operations in common blocks shared with the other routines. For each reflection calls can then be made to ASUPUT (return the unique hkl indices in the asymmetric unit and symmetry number) or ASUGET (obtain real space indices given unique hkl's and symmetry number). INASU will determine whether a given reflection lies in the asymmetric unit and ASUPHP will convert the phase.
Internal routines:
- Integer Function INASU(IH, NLAUE) INASU
- Funtion: test if reflection is in the asymmetric unit defined by ASUSET.
- Subroutine PRTRSM(PGNAME, NSYMP, RSYMIV) PRTRSM
- Print reciprocal space symmetry operations.
Both these routines are called from within other routines, although they can also be called independently. ASUSET must be called before INASU can be used.
- Subroutine SETLIM(LSPGRP,XYZLIM) SETLIM
- Set the appropriate box (=asymmetric unit) for the true spacegroup (ie not the FFT spacegroup). For cubic symmetry spacegroups, this will be more than one asymmetric unit.
- Subroutine SETLIM_ZERO(LSPGRP,XYZLIM) SETLIM_ZERO
- Set the appropriate box (=asymmetric unit) for the true spacegroup (ie not the FFT spacegroup). For cubic symmetry spacegroups, this will be more than one asymmetric unit.
NB: This s/r differs from SETLIM in using asu limits derived from cctbx.
- Subroutine SETGRD(NLAUE,SAMPLE,NXMIN,NYMIN,NZMIN,NX,NY,NZ) SETGRD
- Set up a suitable sampling grid for FFT. Calls FNSMP.
- Logical Function FACTRZ(N) FACTRZ
- Function: returns TRUE if N has all prime factors < 19.
- Subroutine FNDSMP(MINSMP, NMUL, SAMPLE, NSAMPL) FNDSMP
- Find suitable grid sample.
- Subroutine CALC_ORIG_PS(NAMSPG_CIF,NSYM,RSYM,NORIG,ORIG,LPAXISX,LPAXISY,LPAXISZ) CALC_ORIG_PS
- Creates a list of equivalent origins for a given spacegroup
- Subroutine XSPECIALS(NSYM,RSYM,XF,YF,ZF,NSPEC) XSPECIALS
- Finds which coordinates occupy special positions (i.e. have occupancies less than 1.0) from consideration of the symmetry operations.
- Integer Function KROT(NS) KROT
- Function: apply symmetry operation to coordinate triplet and check if the result lies in the asymmetric unit.
Neither of the routines XSPECIALS or KROT appear to be used in supported CCP4 programs.
Three subroutines for permuting symmetry operations. They do not really belong here in symlib, but are widely used invisibly in FFT routines using symmetry operations to permute axes for easier fast fourier calculations.
- Subroutine PRMVCI(PERM,JV,N,N1) PRMVCI
- Permutes specified column of an integer input matrix using another matrix.
- Subroutine PRMVCR(PERM,AV,N,N1) PRMVCR
- Equivalent to PRMVCI but operates on a real input matrix.
- Subroutine ROTFIX ROTFIX
- Permutes inverse symmetry operations.
A set of routines used in SCALA, POSTREF and REBATCH.
- Subroutine CCP4_HASH_SETUP(NSER,NFIND) CCP4_HASH_SETUP
- Places a value in the internal look-up table.
- Integer Function CCP4_HASH_LOOKUP(NSER) CCP4_HASH_LOOKUP
- Access a value stored in the table.
- Subroutine CCP4_HASH_ZEROIT() CCP4_HASH_ZEROIT
- Initialise contents of the table to zero.
These routines are not directly related to symmetry operations. Hashing is a method of storing data value pairs in such a way that they can be be efficiently retreived later on; the hash table is the resulting data structure.
- Subroutine SETRSL(A,B,C,ALPHA,BETA,GAMMA) SETRSL
- Routine to calculate set coefficients for calculation of (sin(theta)/lambda)**2, from cell dimensions and angles.
- Real Function STHLSQ(IH,IK,IL) STHLSQ
- Calculate (sin(theta)/lambda)**2 from h,k,l, using coefficients set by a call to SETRSL.
- Real Function STS3R4(IH,IK,IL) STS3R4
- Calculate (sin(theta)/lambda)**2 from h,k,l, using coefficients set by a call to SETRSL. Duplicates STHLSQ exactly.
These three routines share the common block RECPLT. SETRSL and STHLSQ are used only in CAD, whilst STS3R4 does not appear in any supported program.
This is how the routines are used in CAD. A call to SETRSL with the cell dimensions and angles sets up coefficients in RECPLT, which are then used by the function STHLSQ to calculate the quantity "(sin(theta)/lamba)**2" for any given set of h, k, l indices. From Bragg's Law, this quantity is equal to 1/(4*d**2), that is, one-quarter of the resolution. Within CAD, multiplication by 4 yields the resolution 1/d**2.
- Subroutine PSTOPH (PSIX,PSIY,PSIZ,PHIX,PHIY,PHIZ,AVPHI) PSTOPH
- Phase angle conversion routine.
The exact function of this routine is unclear and it does not appear in any supported program.
The following calls are available to Fortran programs. They are arranged alphabetically.
Get original indices of reflection from asymmetric unit, i.e. reverse operation of ASUPUT. Symmetry defined by call to ASUSET.
On input:
- IHKL(3)
- input unique indices hkl
- ISYM
- symmetry number for output
odd numbers are for I+
even numbers are for I-
real-space symmetry operation number L = (ISYM-1)/2 + 1
On output:
- JHKL(3)
- output original indices hkl
The real-space symmetry matrices are applied in ASUPUT by premultiplying them by a row vector hkl, ie (h'k'l') = (hkl)R. So here we calculate (hkl) = (h'k'l') R**-1
Generate phase of symmetry equivalent JHKL from that of IHKL.
On input:
- JHKL(3)
- indices hkl generated in ASUPUT
- LSYM
- symmetry number for generating JHKL ( found by ASUPUT)
- ISIGN
- = 1 for I+
= -1 for I-
- PHASIN
- phase for reflection IHKL(3)
On output:
- PHSOUT
- phase for reflection JHKL(3)
Put reflection into asymmetric unit defined by call to ASUSET
On input:
- IHKL(3)
- input indices hkl
On output:
- JHKL(3)
- output indices hkl
- ISYM
- symmetry number for output
odd numbers are for I+
even numbers are for I-
real-space symmetry operation number L = (ISYM-1)/2 + 1
The real-space symmetry matrices are applied by premultiplying them by a row vector hkl, ie (h'k'l') = (hkl)R
Set up & store symmetry informtion for later use in ASUPUT or ASUGET
On input:
- SPGNAM
- space-group name (not used) ( character)
- NUMSGP
- space-group number (not used)
- PGNAME
- point-group name ( character)
- MSYM
- total number of symmetry operations
- RRSYM(4,4,MSYM)
- symmetry matrices (real-space)
- LPRINT
- printing flag. ( logical)
On output:
- PGNAME
- point-group name ( character)
- MSYMP
- number of primitive symmetry operations
- MLAUE
- Laue group number - See PGNLAU for details
Creates a list of equivalent origins for the named spacegroup.
ARGUMENTS
- (I) NAMSPG_CIF
- spacegroup name (character)
- (I) NSYM
- number of symmetry operations
- (I) RSYM(4,4,NSYM)
- symmetry ops stored as 4x4 matrices
- (O) NORIG
- number of origins.
- (O) ORIG(3,i)
- vector of alternate origin (for example : 0.5,0.0,0.5) only positive components. include vector: (0,0,0)
- (O) LPAXISX
- logical; set true if s/grp is polar along x axis
- (O) LPAXISY
- logical; set true if s/grp is polar along y axis
- (O) LPAXISZ
- logical; set true if s/grp is polar along z axis
Taken from Alexei Vagin
The function CCP4_HASH_LOOKUP returns the value NFIND (which was input when setting up the function in the subroutine CCP4_HASH_SETUP) for the large range variable NSER. Uses hashing. (see comments for CCP4_HASH_SETUP for description of hashing method).
This subroutine sets up a value for the function ccp4_hash_lookup.
When ccp4_hash_lookup(nser) is later evaluated it will return nfind
This function will allow the efficient retrieval of an identifier for a large range variable (such as a crystal number). The values of the function ccp4_hash_lookup(nser) are stored in the array it(2, kpri) where kpri is the prime number used to generate the function.
The array 'it' lives in the common block which is shared by ccp4_hash_setup and the function ccp4_hash_lookup
NOTES: A hash table is a way of storing information so that it easily be retrieved without the need for indexing or long searches. NSER is referred to as the "key", which is "hashed" (computer- science speak for "messed up") by the hashing function (in this case MOD(NSER4,KPRI) + 1) to determine where the value pair will be stored. The function LOOKUP can then search on the same basis when supplied with the key, to retreive the pair in (at most) 3 calculations. Note that KPRI (the table size) MUST BE A PRIME in order for this method to work.
Initialises elements of array 'it' used in ccp4_hash_setup and ccp4_hash_lookup to zero.
Input IH(3) - reflection indices
Returns IC
Determine whether a reflection is centric (return IC=1) or not (IC=0). If none of the zone tests is satisfied, the reflection is non-centric.
Returns value as true if reflection khkl is centric, false otherwise. It is general for all point groups - but only for the unique set of indices which conforms to the criterion of maximising the value of
(khkl(3)*256 + khkl(2))*256 + khkl(1)
as produced by e.g. subroutine turnip in protin and ulysses.
In this case the required tests are controlled by 7 flags in icent for
0KL H0L HK0 HKK HKH HHL H,-2H,L
(the last is needed in pg312)
This is Randy Read's method of defining centric reflections. It uses NSM and the symmetry operators stored in RSMT(4,4,NSM)
It decides how many centric zones there are, and flags them.
set up tests for 0kl h0l hk0 hhl hkh hkk h,-hl hk-h hk-k -h 2h l 2h -h l hkl
Zones are encoded using an idea from a program by Bricogne. If h*zone(1) + k*zone(2) + l*zone(3) is equal to 0.0, that reflection is in that zone. All that is needed is the most general conditions - a reflection is either centric or not.
Gets determinant of a matrix
- Input A
- 4*4 matrix (real)
- Output DET
- determinant of A.
It works out the epsilon cards using NSM and the symmetry operators stored in RSMT(4,4,NSM).
This is Randys program description:
- Note:
- zones defined as for centric zones, but fourth number on each line is the multiplicity corresponding to this zone. last card should always be 0 0 0 n, where n is appropriate for the lattice (1 for primitive, 2 for face- centred, etc.), so that general reflections get a value for epsilon. be very careful of the order of the zones. cards for reciprocal lattice rows should be given before those for planes, because the first test that is satisfied defines the zone.
set up tests for
h00 0k0 00l hh0 h0h 0kk h,-h0 h0-h 0k-k -h2h0 2h-h0 hhh hkl
Input IH(3) - reflection indices
Returns EPSI ( epsilon zone) , and ISYSAB flag. Systematic absences flagged with ISYSAB = 1
Find the zone a reflection falls into, and return the appropriate value for the reflection multiplicity factor. Each reflection must have a zone.
Returns true if N has all prime factors <= 19
Find suitable grid sample, approximately = SAMPLE/2 * maximum index, with required factor, and no prime factor > 19
On entry:
- MINSMP
- minimum sample, approximately 2 * maximum index
- NMUL
- required factor
- SAMPLE
- desired sample factor, ie if = 1.0 (minimum), try to get sample close to MINSMP
On exit:
- nsampl
- grid sample; if MINSMP<=0, nsampl=nmul
Checks if indices are equal.
Returns true if indices ih = kh
Return HKL ranges chosen in PGNLAUE
Arguments: (INTEGER) HRNG0,HRNG1,KRNG0,KRNG1,LRNG0,LRNG1
Arguments:
- NLAUE
- code number for this pointgroup
- IH(3)
-
Returns:
INASU = +1 if h k l chosen
INASU = -1 if -h-k-l chosen
INASU = 0 if reflection is out-of-bounds
Inverts a 4*4 matrix. Used here to get inverse symmetry operation for generating equivalent h k l, i.e.
[h'] = [h][St]
h'(j) =Sum(I=1,3)[ h(i)*St(I,J,NS)]
Arguments:
- Input S
- 4*4 matrix to be inverted
- Output ST
- 4*4 matrix (inverse of S)
Apply NS'th symmetry operation to JP to get LP, check if lies in asymmetric unit given by NAU.
Returns KROT=0 correct operation, =1 if not.
Get symmetry operations for space-group LSPGRP from library file, logical name SYMINFO.
Arguments:
- IST
- dummy parameter for backwards compatibility
- LSPGRP (input)
- Name of spacegroup
- IST (input)
- Stream of library file
- NSYM (output)
- Number of symmetry operations
- ROT(4,4,NSYM) (output)
- Rotation/translation matrices
Get symmetry operations from library file, logical name SYMINFO. Space group defined by LSPGRP - spacegroup number or NAMSPG - spacegroup name.
Arguments:
- IST
- dummy parameter for backwards compatibility
- LSPGRP (I/O)
- spacegroup number
- NAMSPG (I/O)
- spacegroup name
- NAMPG (O)
- pointgroup name
- NSYMP (O)
- number of primitive symmetry operations
- NSYM (O)
- number of symmetry operations
- ROT(4,4,NSYM)
- rotation/translation matrices
Identical to MSYMLB, except that on output NAMSPG_CIF has correct CIF format, e.g. 'P 21 21 21'
NAMSPG_CIF should be as in _symmetry.space_group_name_H-M
Another version of MSYMLB, with the following changes:
-
The routine will try first to match the assigned NAMSPG_CIF to ANY name given on the spacegroup line:
It is satisfied by the first fit it finds:
eg: 47 8 8 Pmmm PGmmm ORTHORHOMBIC 'P 2/m 2/m 2/m' 'P m m m'
You could call the subroutine with NAMSPG_CIF = 'Pmmm' or 'P 2/m 2/m 2/m' or 'P m m m'
But it will always return the LONGEST possible name. ie 'P 2/m 2/m 2/m'
-
If there is no match to the spacegroup NAME, the space group is identified by its number. This requires that the number is UNIQUE, so alternate settings are numbered n000 + Int Tab number
-
The point group name is always guessed at by the SR PGDEFN and the guess on the data line is ignored.
-
The number of primitive symmetry operators is also determined by the SR PGDEFN. The non-primitive operators are stored in the first NSYMP symmetry matrices and the guess on the data line is ignored.
-
The symmetry operators are checked to make sure they are a closed group.
In the library file, the header for each entry may contain (Order not guaranteed, but must start with)
LSPGRP NLINS and contain either NAMSPG or NAMSPG_CIF
LSPGRP NLINS NLINP NAMSPG NAMPG CRYSTAL NAMSPG_CIF
where:
- LSPGRP
- spacegroup number
- NLINS
- total number of lines of symmetry operators.
- NLINP
- number of LINES of primitive symmetry operators (not used now..)
- NAMSPG_CIF
- spacegroup name (in CIF format? presumably)
- NAMPG
- name of corresponding pointgroup (not used now..)
On entry:
- IST
- stream number to read file
- LSPGRP
- spacegroup number
- NAMSPG or NAMSPG_CIF
- any acceptable spacegroup name: this will be used to identify the spacegroup if possible
Returns:
- LSPGRP
- spacegroup number
- NAMSPG_CIF
- full spacegroup name
- NAMSPG_CIFS
- name without any spaces
- NAMPG
- pointgroup name ( obtained from pgdefn - not 100% reliable!)
- NSYMP
- number of primitive symmetry operations obtained from pgdefn- only different from NSYM in non-primitive spacegroups
- NSYM
- total number of symmetry operations
- RlSymmMatrx(4,4,NSYM)
- Symmetry Rotation/translation matrices
Determine Patterson spacegroup from true space-group
On entry:
- SPGNAM
- space-group name. Only used to determine lattice centering
- PGNAME
- point-group name
On exit:
- PATNAM
- name of Patterson spacegroup
- LPATSG
- number of Patterson spacegroup
Arguments:
- Input NSYM
- number of symmetry operators. ( integer)
- Input RSYMT
- 4*4 symmetry matrices. ( real)
- Input LPRINT
- printing flag. ( logical)
- Returns NAMPG
- name of point group. ( character)
- Returns NSYMP
- number of primitive symmetry operators. ( integer)
- Returns RSYMT
-
This subroutine chooses the primitive set of symmetry operators.
If necessary it re-orders the symmetry operators to give the primitive ones first.
This subroutine works out the point group name NAMPG. That is ; it checks rotation axes, etc etc and recognises these point groups. (It DOES NOT cope with mirror planes etc)
Gronigen MDF stuff: It now sets up the common block MDFPAR for MDF file mods and fills in the symmetry info. See subroutine for details.
Gronigen subroutine.
Use this subroutine to transfer information to and from MDFPAR.
If JLASS eq 0 then fill JLASS JCENTR JSCREW from common block.
If JLASS gt 0 then fill KLASS ICENTR ISCREW in common block.
Choose Laue group from PG name.
On entry:
- NAMPG
- point-group name ( character)
On exit:
- NLAUE
- Laue group number ( integer)
- LAUNAM
- Laue group name ( character)
This subroutine returns a laue code number used to choose the unique region of reciprocal space for each point group.
The number nlaue is the same as the one set in CAD for this purpose.
Pointgroup Laue group Limits
3 pg1 1bar hkl:l>=0 hk0:h>=0 0k0:k>=0 1,2
pg1bar
4 pg2 (b) 2/m hkl:k>=0, l>=0 hk0:h>=0 3/b,4/b....
pgm pg2/m
5 pg2 (c) 2/m hkl:k>=0, l>=0 h0l:h>=0 1003,1004
6 pg222 mmm hkl:h>=0, k>=0, l>=0 16 ...
pgmm2 pgmmm
7 pg4 4/m hkl:h>=0, l>=0 with k>=0 if h=0 and
pg4bar pg4/m k>0 if h>0
8 pg422 4/mmm hkl:h>=0, k>=0, l>=0 89..
pg4mm pg4bar2m pg4barm2 pg4/mmm
9 pg3 3bar hkl:h>=0, k>0 00l:l>0 143..
pg3bar
10 pg312 3/m hkl:h>=0, k>=0 with k\<=h for all l.
pg32 pg3m pg3m1 pg3barm1 if k = 0 l>=0
Space group numbers : 149-151-153 157 159 162 163
11 pg321 3bar1m hkl:h>=0, k>=0 with k\<=h for all l.
pg31m pg3bar1m if h = k l>=0
Space group numbers : 150-152-154
12 pg6 6/m hkl:h>=0, k>=0, l>=0 with k>=0 if h=0
pg6bar 6/m and k> 0 if h>0
13 pg622 6/mmm hkl:h>=0, k>=0, l>=0 with h>=k 177..
pg6mm pg6barm2 pg6bar2m pg 6/mmm
14 pg23 m3 hkl:h>=0, k>=0, l>=0 with l>=h, k>=h
pgm3bar
15 pg432 m3m hkl:h>=0, k>=0, l>=0 with k>=l
pg4bar3m pgm3barm
- Input PERM
-
- Input JV
-
- Output JV
-
This has been modified by permuting the Nth column by matrix PERM.
Here is the code for clarity:
C---- Permute
C
C DO 10 I = 1,3
C BV(I) = PERM(I,1)*JV(N,1) + PERM(I,2)*JV(N,2) +
C + PERM(I,3)*JV(N,3)
C 10 CONTINUE
C
C---- Copy back
C
C DO 20 I = 1,3
C JV(N,I) = NINT(BV(I))
C 20 CONTINUE
- Input PERM
-
- Input AV
-
- Output AV
- N1*3 matrix (real) This has been modified by permuting the Nth column by matrix PERM.
See PRMVCI - this routine is its real equivalent.
Print reciprocal space symmetry operations
The real-space symmetry matrices are applied by premultiplying them by a row vector hkl, ie (h'k'l') = (hkl)R
Convert PSIX,PSIY,PSIZ (= epsx,epsy,epsz) to PHIX,PHIY,PHIZ, using AVPHI
All angles in radians
Permutes inverse symmetry operations
Matrices passed in Common block ATSYM
Set up a suitable sampling grid for FFT
Input:
- NLAUE
- Laue-group for FFT/SF calculation
- SAMPLE
- default fineness of sample, ie if = 1.0 (minimum), try to get sampling as close to minimum as possible.
Typically = 1.5 to get sample at traditional 3 * maximum index
- NXMIN NYMIN NZMIN
- minimum sampling (true XYZ)
Output:
- NX,NY,NZ
- sampling intervals along X,Y,Z
The sampling intervals must satisfy the following conditions:
- approximately SAMPLE * minimum sampling
- no prime factor > 19
- special restrictions for particular space-groups
This is ALL the point groups.
PG1 PG1bar PG2 PGm PG2/m PG222 PGmm2 PGmmm
PG4 PG4bar PG4/m PG422 PG4mm PG4bar2m PG4/mmm
PG3 PG3bar PG32 PG3m PG3barm
PG6 PG6bar PG6/m PG622 PG6mm PG6bar2m PG6/mmm
PG23 PGm/3bar PG432 PG4bar3m PGm3bar m
We use:
PG1 PG1bar PG2 PG2/m PG222 PGmmm
PG4 PG4/m PG422 PG4/mmm
PG3 PG3bar PG32 PG3bar/m
PG6 PG6/m PG622 PG6/mmm
PG23 PGm/3bar PG432 PGm3barm
For grid restrictions we only need to know the laue number. Here is the table:
3 pg1 1bar hkl:l>=0 hk0:h>=0 0k0:k>=0 1,2
4 pg2 2/m hkl:k>=0, l>=0 hk0:h>=0 3/b,4/b....
5 pg2(c) 2/m hkl:k>=0, l>=0 h0l:h>=0 1003,1004
6 pg222 mmm hkl:h>=0, k>=0, l>=0 16 ...
7 pg4 4/m hkl:h>=0, l>=0 with k>=0 if h=0 and
8 pg422 4/mmm hkl:h>=0, k>=0, l>=0 89..
9 pg3 3bar hkl:h>=0, k>0 00l:l>0 143..
10 pg312 3/m hkl:h>=0, k>=0 with k\<=h for all l.
if k = 0 l>=0
Space group numbers : 149-151-153
11 pg321 3/m hkl:h>=0, k>=0 with k\<=h for all l.
if h = k l>=0
Space group numbers : 150-152-154
12 pg6 6/m hkl:h>=0, k>=0, l>=0 with k=0 if h=0
13 pg622 6/mmm
14 pg23 m3
15 pg432 m3m
Set appropriate box (asymmetric unit) for spacegroup (true spacegroup rather than FFT spacegroup) LSPGRP. For cubic symmetry spacegroups, this will be more than one asymmetric unit.
On entry:
- lspgrp
- true spacegroup (not FFT spacegroup)
On exit:
- xyzlim(2,3)
- minimum, maximum limits on x,y,z (fractions of cell); if spacegroup not recognized, returns xzylim(1,1) = -1.0
Note that the minimum limits (xyzlim(1,)) will always = 0.0
Set appropriate box (asymmetric unit) for spacegroup (true spacegroup rather than FFT spacegroup) LSPGRP. For cubic symmetry spacegroups, this will be more than one asymmetric unit.
NB This s/r differs from SETLIM in that the limits are taken from cctbx via CCP4's syminfo.lib file.
On entry:
- lspgrp
- true spacegroup (not FFT spacegroup)
On exit:
- xyzlim(2,3)
- minimum, maximum limits on x,y,z (fractions of cell); if spacegroup not recognized, returns xzylim(1,1) = -1.0
Note that the minimum limits (xyzlim(1,)) will always = 0.0
Routine to calculate coefficients for (sin(theta)/lambda)**2 from h,k,l for general axes.
First calculates the components of input axes in an orthonormal basis, then calculate components of reciprocal axes in same basis.
The input angles are in degrees.
Calculate (sin(theta)/lambda)**2 from h,k,l. The coefficients are set by a previous call to SETRSL. Works for any kind of axes.
Calculate (sin(theta)/lambda)**2 from h,k,l. The coefficients are set by a call to SETRSL. Works for any kind of axes.
Read and interpret symmetry operations
SYMFR2 recognises the following types of input:
real space symmetry operations, e.g. X+1/2,Y-X,Z
reciprocal space operations, e.g. h,l-h,-k
reciprocal axis vectors, e.g. a*+c*,c*,-b*
real space axis vectors, e.g. a,c-a,-b
The subroutine returns the appropriate 4x4 transformation matrix for each operation. The calling program must interpret the resulting matrix(ces) correctly.
On entry I1 is the first character of ICOL to look at (say after keyword 'SYMM')
NS is the number of the first symmetry operation to be read, & returns with the number of the last one read.
On exit, ROT(4,4,NS) contains the real-space symmetry matrices, in standard convention, i.e.
[x'] = [s][x]
x'(I)=Sum(J=1,3)ROT(I,J,NS)*x(J) + ROT(I,4,NS)
Input:
- ICOL
- character string containing symmetry operations
- I1
- first character in ICOL to interpret from.
Output:
- ROT(I,4,NS)
- contains the fractional translations.
Read and interpret symmetry operations.
Arguments:
- ICOL (I) CHARACTER*80
- Line containing the symmetry operations
- I1 (I) INTEGER
- First character to look at in ICOL (say after keyword 'SYM')
- NS (I/O) INTEGER
- is the number of the first symmetry operation to be read, & returns with the number of the last one read (ie you can have more than one on a line!)
- ROT (O) REAL
- Array (4,4,at_least_NS), on exit contains the real-space symmetry matrices, in standard convention, i.e.
[x'] = [s][x]
x'(I)=Sum(J=1,3)ROT(I,J,NS)*x(J) + ROT(I,4,NS)
ROT(I,4,NS) contains the fractional translations
- EFLAG (O) INTEGER
- Error flag - on exit, if 0 then OK, > 0, an error occurred.
Symmetry translation from matrix back to characters
This translates the symmetry matrices RSM(4,4,NSM) into INT TAB character strings
It gives the real and reciprocal space operations.
eg X,Y,Z H , K, L
eg -Y,X-Y, Z -H-K, H, L etc
That is more complicated than you might think!!
Symmetry translation from matrix back to characters
This translates the symmetry matrices RSM(4,4,NSM) into INT TAB character strings
It gives the real and reciprocal space operations.
eg X,Y,Z H , K, L
eg -Y,X-Y, Z -H-K, H, L etc
That is more complicated than you might think!!
Arguments :
- NSM (I) INTEGER
- Number of Symmetry operations
- RSM (I) REAL
- Array of dimension (4,4,at least NSM) containing symmetry operations on input
- SYMCHS (O) CHARACTER*(*)
- Array of dimension at least NSM containing int tab char strings on output
- IPRINT (I) INTEGER
- Print flag
=0 No printing
=1 Print the int tab strings
Symmetry translation from matrix back to characters
This translates the symmetry matrices RSM(4,4,NSM) into INT TAB character strings
It gives the real and reciprocal space operations.
eg X,Y,Z H , K, L
eg -Y,X-Y, Z -H-K, H, L etc
That is more complicated than you might think!!
Arguments :
- Nsym (I) INTEGER
- Number of Symmetry operations
- Rsm (I) REAL
- Array of dimension (4,4,at least Nsym) containing symmetry operations on input
- Symchs (O) CHARACTER*(*)
- Array of dimension at least Nsym containing int tab char strings on output
Input IN(3) - reflection indices
Returns ISYSAB flag. Systematic absences flagged with ISYSAB = 1 Only reflns with EPSI > 1 need be considered.
- Input NSYM
- number of symmetry operators. ( integer)
- Input RSYM
- 4*4*NSYM symmetry matrices. ( real)
- Input XF YF ZF
- a coordinate in fractional coordinates.
- Output NSPEC
- the multiplicity of the coordinate. eg: NSPEC = 3 for an atom on a 3fod axis.
This subroutine finds what coordinates occupy special positions i.e. have occupancies less than 1.0 from consideration of the symmetry operations.
There is no standard defined asymmetric unit so the definitions are arbitrary and may differ between differ packages. The subroutines in
a) Subroutines for choosing asymmetric units for reflection data group 3.a are used to define the CCP4 asymmetric unit, and to determine whether a reflection falls within this definition.
Below are the definitions of the a. Reciprocal Space Asymmetric Unit Definitions reciprocal space and the b. Real Space Asymmetric Unit Definitions real space asymmetric units under the CCP4 convention.
The reciprocal space asymmetric unit is defined in the subroutine ASUSET from the Laue group using calls to the s/r's PGDEFN and PGNLAU. The limits of the CCP4 asymmetric unit are (from
Subroutine PGNLAU(NAMPG,NLAUE,LAUNAM) PGNLAU):
Pointgroup | Laue group | Limits | Spacegroup Nos |
3 | pg1
pg1bar | 1bar | hkl:l>=0
hk0:h>=0
0k0:k>=0 | 1,2 |
4 | pg2 (b)
pgm pg2/m | 2/m | hkl:k>=0, l>=0
hk0:h>=0 | 3,4.... |
5 | pg2 (c) | 2/m | hkl:k>=0, l>=0
h0l:h>=0 | 1003, 1004 |
6 | pg222
pgmm2
pgmmm | mmm | hkl:h>=0, k>=0, l>=0 | 16 ... |
7 | pg4
pg4bar
pg4/m | 4/m | hkl:h>=0, l>=0 with k>=0 if h=0
and k>0 if h>0 | 75,.. |
8 | pg422 pg4mm pg4bar2m
pg4barm2 pg4/mm | 4/mmm | hkl:h>=0, k>=0, l>=0 | 89,.. |
9 | pg3
pg3bar | 3bar | hkl:h>=0, k>0 00l:l>0 | 143,.. |
10 | pg312 pg32
pg3m pg3m1 pg3barm1 | 3/m | hkl:h>=0, k>=0 with k<=h for all l.
if k=0 l>=0 | 149 151 153 157 159 162 163 |
11 | pg321 pg31m pg3bar1m | 3bar1m | hkl:h>=0, k>=0 with k<=h for all l.
if k=h l>=0 | 150 152 154 |
12 | pg6 pg6bar | 6/m | hkl:h>=0, k>=0, l>=0 with k>=0 if h=0 and k>0 if h>0 | 168.. |
13 | pg622 pg6mm pg6barm2 pg6bar2m pg6/mmm | 6/mmm | hkl:h>=0, k>=0, l>=0 with h>=k | 177.. |
14 | pg23 pgm3bar | m3 | hkl:h>=0, k>=0, l>=0 with l>=h, k>=h | 195.. |
15 | pg432 pg4bar3m pgm3barm | m3m | hkl:h>=0, k>=0, l>=0 with k\=1 | 209.. |
The subroutine
Subroutine SETLIM(LSPGRP,XYZLIM) SETLIM contains the definitions of the real space asymmetric unit. Note that not all of the spacegroups have a definition within SETLIM.
No. | Spacegroup | Upper limits on x, y, z asterisk (*) |
1 | P 1 | x < 1, y < 1, z < 1, |
2 | P -1 | x < 1, y <= 1/2, z < 1, |
3 | P 1 2 1 | x <= 1/2, y < 1, z < 1, |
4 | P 1 21 1 | x < 1, y < 1/2, z < 1, |
5 | C 1 2 1 | x <= 1/2, y < 1/2, z < 1, |
10 | P 1 2/M 1 | x <= 1/2, y <= 1/2, z < 1, |
16 | P 2 2 2 | x <= 1/2, y <= 1/2, z < 1, |
17 | P 2 2 21 | x <= 1/2, y <= 1/2, z < 1, |
18 | P 21 21 2 | x < 1, y <= 1/4, z < 1, |
19 | P 21 21 21 | x < 1, y < 1, z <= 1/4, |
20 | C 2 2 21 | x <= 1/2, y <= 1/4, z < 1, |
21 | C 2 2 2 | x <= 1/2, y <= 1/4, z < 1, |
22 | F 2 2 2 | x <= 1/4, y <= 1/4, z < 1, |
23 | I 2 2 2 | x <= 1/2, y <= 1/4, z <= 1, |
24 | I 21 21 21 | x <= 1/2, y <= 1/4, z < 1, |
47 | P 2/M 2/M 2/M | x <= 1/2, y <= 1/2, z <= 1/2, |
65 | C 2/M 2/M 2/M | x <= 1/2, y <= 1/4, z <= 1/2, |
69 | F 2/M 2/M 2/M | x <= 1/4, y <= 1/4, z <= 1/2, |
71 | I 2/M 2/M 2/M | x <= 1/2, y <= 1/4, z <= 1/2, |
75 | P 4 | x <= 1/2, y <= 1/2, z < 1, |
76 | P 41 | x < 1, y < 1, z < 1/4, |
77 | P 42 | x <= 1/2, y < 1, z < 1/2, |
78 | P 43 | x < 1, y < 1, z < 1/4, |
79 | I 4 | x <= 1/2, y <= 1/2, z <= 1/2, |
80 | I 41 | x <= 1/2, y < 1, z < 1/4, |
83 | P 4/M | x <= 1/2, y <= 1/2, z <= 1/2, |
87 | I 4/M | x <= 1/2, y <= 1/2, z <= 1/4, |
89 | P 4 2 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
90 | P 4 21 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
91 | P 41 2 2 | x < 1, y < 1, z <= 1/8, |
92 | P 41 21 2 | x < 1, y < 1, z <= 1/8, |
93 | P 42 2 2 | x <= 1/2, y < 1, z <= 1/4, |
94 | P 42 21 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
95 | P 43 2 2 | x < 1, y < 1, z <= 1/8, |
96 | P 43 21 2 | x < 1, y < 1, z <= 1/8, |
97 | I 4 2 2 | x <= 1/2, y <= 1/2, z <= 1/4, |
98 | I 41 2 2 | x <= 1/2, y < 1, z <= 1/8, |
123 | P 4/M 2/M 2/M | x <= 1/2, y <= 1/2, z <= 1/2, |
139 | I 4/M 2/M 2/M | x <= 1/2, y <= 1/2, z <= 1/4, |
143 | P 3 | x <= 2/3, y <= 2/3, z < 1, |
144 | P 31 | x < 1, y < 1, z < 1/3, |
145 | P 32 | x < 1, y < 1, z < 1/3, |
146 | H 3 | x <= 2/3, y <= 2/3, z < 1/3, |
147 | P -3 | x <= 2/3, y <= 2/3, z <= 1/2, |
148 | R -3 | x <= 2/3, y <= 2/3, z <= 1/6, |
149 | P 3 1 2 | x <= 2/3, y <= 2/3, z <= 1/2, |
150 | P 3 2 1 | x <= 2/3, y <= 2/3, z <= 1/2, |
151 | P 31 1 2 | x < 1, y < 1, z <= 1/6, |
152 | P 31 2 1 | x < 1, y < 1, z <= 1/6, |
153 | P 32 1 2 | x < 1, y < 1, z <= 1/6, |
154 | P 32 2 1 | x < 1, y < 1, z <= 1/6, |
155 | H 3 2 | x <= 2/3, y <= 2/3, z <= 1/6, |
162 | P -31 2/M | x <= 2/3, y <= 1/2, z <= 1/2, |
164 | P -3 2/M 1 | x <= 2/3, y <= 1/3, z <= 1, |
166 | R -3 2/M | x <= 2/3, y <= 2/3, z <= 1/6, |
168 | P 6 | x <= 2/3, y <= 1/2, z < 1, |
169 | P 61 | x < 1, y < 1, z < 1/6, |
170 | P 65 | x < 1, y < 1, z < 1/6, |
171 | P 62 | x < 1, y < 1, z < 1/3, |
172 | P 64 | x < 1, y < 1, z < 1/3, |
173 | P 63 | x <= 2/3, y <= 2/3, z < 1/2, |
175 | P 6/M | x <= 2/3, y <= 2/3, z <= 1/2, |
177 | P 6 2 2 | x <= 2/3, y <= 1/2, z <= 1/2, |
178 | P 61 2 2 | x < 1, y < 1, z <= 1/12, |
179 | P 65 2 2 | x < 1, y < 1, z <= 1/12, |
180 | P 62 2 2 | x < 1, y < 1, z <= 1/6, |
181 | P 64 2 2 | x < 1, y < 1, z <= 1/6, |
182 | P 63 2 2 | x <= 2/3, y <= 2/3, z <= 1/4, |
191 | P 6/M 2/M 2/M | x <= 2/3, y <= 1/3, z <= 1/2, |
195 | P 2 3 | x < 1, y < 1, z <= 1/2, |
196 | F 2 3 | x <= 1/4, y <= 1/4, z < 1, |
197 | I 2 3 | x < 1, y < 1, z <= 1/2, |
198 | P 21 3 | x <= 1/2, y <= 1/2, z < 1, |
199 | I 21 3 | x <= 1/2, y <= 1/2, z <= 1/2, |
200 | P 2/M -3 | x <= 1/2, y <= 1/2, z <= 1/2, |
202 | F 2/M -3 | x <= 1/2, y <= 1/2, z <= 1/4, |
204 | I 2/M -3 | x <= 1/2, y <= 1/2, z <= 1/2, |
207 | P 4 3 2 | x < 1, y <= 1/2, z <= 1/2, |
208 | P 42 3 2 | x <= 1/2, y < 1, z <= 1/4, |
209 | F 4 3 2 | x <= 1/2, y <= 1/2, z <= 1/2, |
210 | F 41 3 2 | x <= 1/2, y < 1, z <= 1/8, |
211 | I 4 3 2 | x <= 1/2, y <= 1/2, z <= 1/4, |
212 | P 43 3 2 | x < 1, y < 1, z <= 1/8, |
213 | P 41 3 2 | x < 1, y < 1, z <= 1/8, |
214 | I 41 3 2 | x <= 1/2, y < 1, z <= 1/8, |
221 | P 4/M -3 2/M | x <= 1/2, y <= 1/2, z <= 1/2, |
225 | F 4/M -3 2/M | x <= 1/2, y <= 1/4, z <= 1/4, |
229 | I 4/M -3 2/M | x <= 1/2, y <= 1/2, z <= 1/4, |
(*) The limits are in fractional coordinates, and the lower limits are always x=0, y=0, z=0.