SYMLIB (CCP4: Library)

NAME

symlib - Software Library for handling symmetry operations

DESCRIPTION

From CCP4 5.0, the core handling of symmetry information is done by a set of C functions. Separate 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For Fortran programs, the original set of subroutines (held in symlib.f) has been replaced by an interface to the C library.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.

DESCRIPTION OF THE SYMMETRY LIBRARY
HOW SYMMETRY OPERATIONS ARE STORED AND APPLIED
DESCRIPTION OF THE GROUPS OF ROUTINES
FORTRAN API (arranged alphabetically)
DEFINITION OF THE CCP4 ASYMMETRIC UNIT
- a. Reciprocal space
- b. Real space

1. DESCRIPTION OF THE SYMMETRY 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).

Format of the symmetry library file

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 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.

2. HOW SYMMETRY OPERATIONS ARE STORED AND APPLIED

Storage of symmetry operations

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  )

      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 MSYMLB3 will obtain the set of symmetry matrices in this representation for a given spacegroup, whilst SYMFR2 or SYMFR3 will obtain individual matrices from the string representation mentioned above. (SYMTR4 will perform the inverse operation, converting matrices to string representation.)

Application of symmetry operations

1. Real Space Coordinates

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 library provides matrix-vector routines MATVEC4 and TRANSFRM which can be used to perform this operation.

2. Reciprocal Space

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 INVSYM will invert a 4x4 matrix stored in this representation, for the purpose of applying symmetry operations to reflection indices.

3. Axis Vectors

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.)

3. DESCRIPTION OF THE GROUPS OF ROUTINES

The routines have been broken down into groups according to function.

Group 1: Subroutines for deriving and manipulating symmetry operations

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.

Group 1a: Deriving symmetry operator matrices

MSYMLB3: Gets the symmetry operators in matrix representation for a specified spacegroup from the symmetry library. The spacegroup is specified either by the spacegroup number (which could be a non-standard CCP4 number such as 4005) or a spacegroup name. It will match to any valid s/g name but always returns the longest name.
Replaces: MSYMLB2, MSYMLB, MSYGET
SYMFR2, SYMFR3: Translates a character string containing symmetry operator(s) into matrix representation, 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.
SYMTR4: Translates symmetry matrices into character strings with the equivalent symmetry operations.
Replaces: SYMTRN, SYMTR3
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 section 2.

Internal routines:

DETERM: Calculate the determinant of 4*4 matrix.

Group 1b: Deriving information from symmetry operators

PGDEFN: Obtain/guess pointgroup and primitive set of symmetry operators from analysis of all symmetry operators.
PGMDF: Gronigen subroutine: determine the nature of the rotation and screw axes from the symmetry matrices.
PGNLAU: Determine the Laue group from pointgroup name.
PATSGP: Determine the Patterson spacegroup from true spacegroup.
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

Group 2: Subroutines for testing reflection data

a) Centric reflections

CENTRIC: Sets up symmetry elements; must be called first.
CENTR: Tests if a reflection is centric

Nb: routines 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.

b) Epsilon zones

EPSLN: Sets up tables of epsilon zones; must be called first.
EPSLON: Returns the epsilon zone of a given reflection, as well as whether the reflection is systematically absent (using a call to SYSAB).
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.

HKLEQ - used in SCALA to test if two reflections have equal indices.

Group 3: Subroutines for choosing asymmetric units

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 section 3.

a) Subroutines for choosing asymmetric units for reflection data

ASUSET: Set up symmetry for ASUPUT and ASUGET; must be called first. Calls PRTRSM.
ASUPUT: Put reflection into asymmetric unit defined in ASUSET (reverse operation of ASUGET). Calls INASU.
ASUGET: Recover original indices of a reflection in the asymmetric unit defined in ASUSET (reverse operation of ASUPUT).
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:

INASU: Function: test if reflection is in the asymmetric unit defined by ASUSET.
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.

b) Subroutines for choosing asymmetric units in real space consistent with FFT expectations; FFT grids etc.

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.
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.
SETGRD: Set up a suitable sampling grid for FFT. Calls FNSMP.

Internal routines:

FACTRZ: Function: returns TRUE if N has all prime factors < 19.
FNDSMP: Find suitable grid sample.

Group 4: Subroutines for testing coordinate data

CALC_ORIG_PS: Creates a list of equivalent origins for a given spacegroup
XSPECIALS: Finds which coordinates occupy special positions (i.e. have occupancies less than 1.0) from consideration of the symmetry operations.
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.

Group 5: Subroutines for permuting symmetry operators

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.

PRMVCI: Permutes specified column of an integer input matrix using another matrix.
PRMVCR: Equivalent to PRMVCI but operates on a real input matrix.
ROTFIX: Permutes inverse symmetry operations.

Group 6: Subroutines for generating and accessing a hash table

A set of routines used in SCALA, POSTREF and REBATCH.

CCP4_HASH_SETUP: Places a value in the internal look-up table.
CCP4_HASH_LOOKUP: Access a value stored in the table.
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 retrieved later on; the hash table is the resulting data structure.

Group 7: Miscellaneous subroutines

SETRSL: Routine to calculate set coefficients for calculation of (sin(theta)/lambda)**2, from cell dimensions and angles.
STHLSQ: Calculate (sin(theta)/lambda)**2 from h,k,l, using coefficients set by a call to SETRSL.
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)/lambda)**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.

PSTOPH: Phase angle conversion routine.

The exact function of this routine is unclear and it does not appear in any supported program.

Group 8: Multiple Spacegroups

The set of Fortran calls which mimic the original symlib.f assume you are working within a single spacegroup. All calls access the same spacegroup data structure, in analogy with the COMMON blocks of symlib.f For cases where you wish to work with multiple spacegroups (e.g. in the program REINDEX, a different set of calls is provided (the names of which generally start with "CCP4SPG_F_"). These identify the spacegroup of interest via an index "sindx" (by analogy with the "mindx" of mtzlib).

CCP4SPG_F_ASUPUT: Put reflection in asymmetric unit of spacegroup on index sindx.
CCP4SPG_F_CENTPHASE
CCP4SPG_F_EPSLON
CCP4SPG_F_EQUAL_OPS_ORDER: Compare two sets of symmetry operators to see if they are in the same order.
CCP4SPG_F_GET_LAUE: Return Laue number and name for a spacegroup onto index "sindx".
CCP4SPG_F_INASU: Test whether reflection or it's Friedel mate is in the asymmetric unit of the spacegroup on index "sindx".
CCP4SPG_F_IS_CENTRIC
CCP4SPG_F_IS_SYSABS
CCP4SPG_F_LOAD_BY_NAME: Loads a spacegroup onto index "sindx". The spacegroup is identified by the spacegroup name.
CCP4SPG_F_LOAD_BY_OPS: Loads a spacegroup onto index "sindx". The spacegroup is identified by the set of symmetry matrices.

4. FORTRAN API

The following calls are available to Fortran programs. They are arranged alphabetically.

Subroutine ASUGET(IHKL,JHKL,ISYM)

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

Subroutine ASUPHP(JHKL,LSYM,ISIGN,PHASIN,PHSOUT)

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)

Subroutine ASUPUT(IHKL,JHKL,ISYM)

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

Subroutine ASUSET(SPGNAM, NUMSPG, PGNAME, MSYM, RSYM, MSYMP, MLAUE, LPRINT)

Set up & store symmetry information for later use in ASUPUT or ASUGET. Should be used with subroutine LRSYMI to get PGNAME and subroutine LRSYMM (both in mtzlib) to get RSYM and MSYM.

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

Subroutine CALC_ORIG_PS(NAMSPG_CIF,NSYM,RSYM,NORIG,ORIG,LPAXISX,LPAXISY,LPAXISZ)

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

Integer Function CCP4_HASH_LOOKUP(NSER)

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).

Subroutine CCP4_HASH_SETUP(NSER,NFIND)

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 retrieve the pair in (at most) 3 calculations. Note that KPRI (the table size) MUST BE A PRIME in order for this method to work.

Subroutine CCP4_HASH_ZEROIT()

Initialises elements of array 'it' used in ccp4_hash_setup and ccp4_hash_lookup to zero.

Subroutine CCP4SPG_F_ASUPUT(sindx,ihkl[3],jhkl[3],isym)

Put reflection in asymmetric unit of spacegroup on index sindx.

Subroutine CCP4SPG_F_CENTPHASE(sindx,ih[3],cenphs)

Subroutine CCP4SPG_F_EPSLON(sindx,ih[3],epsi,isysab)

Integer Function CCP4SPG_F_EQUAL_OPS_ORDER(msym1,rrsym1,msym2,rrsym2)

Compare two sets of symmetry operators to see if they are in the same order. This is important for the consistent use of ISYM which encodes the operator position in the list. Returns 1 if operator lists are equal and in the same order, 0 otherwise.

Subroutine CCP4SPG_F_GET_LAUE(sindx,nlaue,launam)

Return Laue number and name for a spacegroup onto index "sindx".

Integer Function CCP4SPG_F_INASU(sindx,ihkl[3])

Test whether reflection or it's Friedel mate is in the asymmetric unit of the spacegroup on index "sindx". Return 1 if in asu, -1 if -h -k -l is in asu, 0 otherwise.

Subroutine CCP4SPG_F_IS_CENTRIC(sindx,ih[3],ic)

Subroutine CCP4SPG_F_IS_SYSABS(sindx,in[3],isysab)

Subroutine CCP4SPG_F_LOAD_BY_NAME(sindx,namspg)

Loads a spacegroup onto index "sindx". The spacegroup is identified by the spacegroup name.

Subroutine CCP4SPG_F_LOAD_BY_OPS(sindx,msym,rrsym)

Loads a spacegroup onto index "sindx". The spacegroup is identified by the set of symmetry matrices.

Subroutine CENTR(IH,IC)

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.

Logical Function CENTRC(KHKL,ICENT)

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)

Subroutine CENTRIC(NSM,RSMT,IPRINT)

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.

Subroutine DETERM(det,a)

Gets determinant of a matrix

Input A: 4*4 matrix (real)
Output DET: determinant of A.

Subroutine EPSLN(NSM,NSMP,RSMT,IPRINT)

It works out the epsilon cards using NSM and the symmetry operators stored in RSMT(4,4,NSM).

This is Randy's program description:

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

Subroutine EPSLON(IH,EPSI,ISYSAB)

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.

Logical Function FACTRZ(N)

Returns true if N has all prime factors <= 19

Subroutine FNDSMP(MINSMP, NMUL, SAMPLE, NSAMPL)

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

Logical Function HKLEQ(IH,KH)

Checks if indices are equal.

Returns true if indices ih = kh

Subroutine HANDCHANGE(lspgrp,cx,cy,cz)

Used in program phistats.

Subroutine HKLRANGE(IHRNG0,IKRNG1,IKRNG0,IKRNG1,ILRNG0,ILRNG1)

Return HKL ranges chosen in PGNLAUE

Arguments: (INTEGER) HRNG0,HRNG1,KRNG0,KRNG1,LRNG0,LRNG1

Integer Function INASU(IH, NLAUE)

Arguments:

NLAUE: - code number for this pointgroup
IH(3): - indices

Returns:

INASU = +1 if h k l chosen
INASU = -1 if -h-k-l chosen
INASU = 0 if reflection is out-of-bounds

Subroutine INVSYM(S,ST)

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)

Integer Function KROT(NS)

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.

Subroutine MSYGET(IST,LSPGRP,NSYM,ROT)

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

Subroutine MSYMLB(IST,LSPGRP,NAMSPG,NAMPG,NSYMP,NSYM,ROT)

Get symmetry operations from library file, logical name SYMINFO. Space group defined by LSPGRP - spacegroup number or NAMSPG - spacegroup name. This routine is obsolete now, and simply wraps MSYMLB3.

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

Subroutine MSYMLB2(IST,LSPGRP,NAMSPG_CIF,NAMPG,NSYMP,NSYM,ROT)

Identical to MSYMLB, except that on output NAMSPG_CIF has correct CIF format, e.g. 'P 21 21 21'. This routine is obsolete now, and simply wraps MSYMLB3.

Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RSYM)

Get symmetry operations from library file, logical name SYMINFO.

The routine first tries to identify the spacegroup by its number. This requires that the number is UNIQUE, so alternate settings are numbered n000 + Int Tab number.
If the spacegroup number is set to 0 on input, the routine will try to match the assigned NAMSPG_CIF. This uses the C function ccp4spg_load_by_ccp4_spgname which will cope with a variety of possible formats for the name, e.g. presence or absence of spaces.

On entry:

IST: stream number to read file
LSPGRP: spacegroup number, can be set to 0
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
NSYMP: number of primitive symmetry operations, only different from NSYM in non-primitive spacegroups
NSYM: total number of symmetry operations
RSYM(4,4,NSYM): Symmetry rotation/translation matrices

Subroutine PATSGP(SPGNAM, PGNAME, PATNAM, LPATSG)

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

Subroutine PGDEFN(NAMPG,NSYMP,NSYM,RSYMT,LPRINT)

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: - possibly reordered.

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.

Subroutine PGMDF(JLASS,JCENTR,JSCREW)

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.

Subroutine PGNLAU(NAMPG,NLAUE,LAUNAM)

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

Subroutine PRMVCI(PERM,JV,N,N1)

Input PERM: - 4*4 matrix (real)
Input JV: - N1*3 matrix (integer)
Output JV: - N1*3 matrix (integer)
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

Subroutine PRMVCR(PERM,AV,N,N1)

Input PERM: - 4*4 matrix (real)
Input AV: - N1*3 matrix (real)
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.

Subroutine PRTRSM(PGNAME, NSYMP, RSYMIV)

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

Subroutine PSTOPH (PSIX,PSIY,PSIZ,PHIX,PHIY,PHIZ,AVPHI)

Convert PSIX,PSIY,PSIZ (= epsx,epsy,epsz) to PHIX,PHIY,PHIZ, using AVPHI

All angles in radians

Subroutine ROTFIX

Permutes inverse symmetry operations

Matrices passed in Common block ATSYM

Subroutine SETGRD(NLAUE,SAMPLE,NXMIN,NYMIN,NZMIN,NX,NY,NZ)

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

Subroutine SETLIM(LSPGRP,XYZLIM)

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

Subroutine SETLIM_ZERO(LSPGRP,XYZLIM)

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

Subroutine SETRSL(A,B,C,ALPHA,BETA,GAMMA)

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.

Real Function STHLSQ(IH,IK,IL)

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.

Real Function STS3R4(IH,IK,IL)

Calculate (sin(theta)/lambda)**2 from h,k,l. The coefficients are set by a call to SETRSL. Works for any kind of axes.

Subroutine SYMFR2(ICOL,I1,NS,ROT)

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.

Subroutine SYMFR3(ICOL,I1,NS,ROT,EFLAG)

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.

Subroutine SYMTRN(NSM,RSM)

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!!

Subroutine SYMTR3(NSM,RSM)

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

Subroutine SYMTR4(NSYM,RSM,SYMCHS)

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

Subroutine SYSAB(IN,ISYSAB)

Input IN(3) - reflection indices

Returns ISYSAB flag. Systematic absences flagged with ISYSAB = 1 Only reflns with EPSI > 1 need be considered.

Subroutine XSPECIALS(NSYM,RSYM,XF,YF,ZF,NSPEC)

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.

5. DEFINITION OF THE CCP4 ASYMMETRIC UNIT

There is no standard defined asymmetric unit so the definitions are arbitrary and may differ between differ packages. The subroutines in 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 reciprocal space and the real space asymmetric units under the CCP4 convention.

a. Reciprocal Space Asymmetric Unit Definitions

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 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..

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..

b. Real Space Asymmetric Unit Definitions

The subroutine 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 (*)
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,

No.	Spacegroup	Upper limits on x, y, z (*)
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.

SYMLIB (CCP4: Library)

NAME

DESCRIPTION

CONTENTS

1. DESCRIPTION OF THE SYMMETRY LIBRARY

Format of the symmetry library file

2. HOW SYMMETRY OPERATIONS ARE STORED AND APPLIED

Storage of symmetry operations

Application of symmetry operations

1. Real Space Coordinates

2. Reciprocal Space

3. Axis Vectors

3. DESCRIPTION OF THE GROUPS OF ROUTINES

Group 1: Subroutines for deriving and manipulating symmetry operations

Group 1a: Deriving symmetry operator matrices

Internal routines:

Group 1b: Deriving information from symmetry operators

Group 2: Subroutines for testing reflection data

a) Centric reflections

b) Epsilon zones

Group 3: Subroutines for choosing asymmetric units

a) Subroutines for choosing asymmetric units for reflection data

Internal routines:

b) Subroutines for choosing asymmetric units in real space consistent with FFT expectations; FFT grids etc.

Internal routines:

Group 4: Subroutines for testing coordinate data

Group 5: Subroutines for permuting symmetry operators

Group 6: Subroutines for generating and accessing a hash table

Group 7: Miscellaneous subroutines

Group 8: Multiple Spacegroups

4. FORTRAN API

Subroutine ASUGET(IHKL,JHKL,ISYM)

Subroutine ASUPHP(JHKL,LSYM,ISIGN,PHASIN,PHSOUT)

Subroutine ASUPUT(IHKL,JHKL,ISYM)

Subroutine ASUSET(SPGNAM, NUMSPG, PGNAME, MSYM, RSYM, MSYMP, MLAUE, LPRINT)

Subroutine CALC_ORIG_PS(NAMSPG_CIF,NSYM,RSYM,NORIG,ORIG,LPAXISX,LPAXISY,LPAXISZ)

Integer Function CCP4_HASH_LOOKUP(NSER)

Subroutine CCP4_HASH_SETUP(NSER,NFIND)

Subroutine CCP4_HASH_ZEROIT()

Subroutine CCP4SPG_F_ASUPUT(sindx,ihkl[3],jhkl[3],isym)

Subroutine CCP4SPG_F_CENTPHASE(sindx,ih[3],cenphs)

Subroutine CCP4SPG_F_EPSLON(sindx,ih[3],epsi,isysab)

Integer Function CCP4SPG_F_EQUAL_OPS_ORDER(msym1,rrsym1,msym2,rrsym2)

Subroutine CCP4SPG_F_GET_LAUE(sindx,nlaue,launam)

Integer Function CCP4SPG_F_INASU(sindx,ihkl[3])

Subroutine CCP4SPG_F_IS_CENTRIC(sindx,ih[3],ic)

Subroutine CCP4SPG_F_IS_SYSABS(sindx,in[3],isysab)

Subroutine CCP4SPG_F_LOAD_BY_NAME(sindx,namspg)

Subroutine CCP4SPG_F_LOAD_BY_OPS(sindx,msym,rrsym)

Subroutine CENTR(IH,IC)

Logical Function CENTRC(KHKL,ICENT)

Subroutine CENTRIC(NSM,RSMT,IPRINT)

Subroutine DETERM(det,a)

Subroutine EPSLN(NSM,NSMP,RSMT,IPRINT)

Subroutine EPSLON(IH,EPSI,ISYSAB)

Logical Function FACTRZ(N)

Subroutine FNDSMP(MINSMP, NMUL, SAMPLE, NSAMPL)

Logical Function HKLEQ(IH,KH)

Subroutine HANDCHANGE(lspgrp,cx,cy,cz)

Subroutine HKLRANGE(IHRNG0,IKRNG1,IKRNG0,IKRNG1,ILRNG0,ILRNG1)

Integer Function INASU(IH, NLAUE)

Subroutine INVSYM(S,ST)

Integer Function KROT(NS)

Subroutine MSYGET(IST,LSPGRP,NSYM,ROT)

Subroutine MSYMLB(IST,LSPGRP,NAMSPG,NAMPG,NSYMP,NSYM,ROT)

Subroutine MSYMLB2(IST,LSPGRP,NAMSPG_CIF,NAMPG,NSYMP,NSYM,ROT)

Subroutine MSYMLB3(IST,LSPGRP,NAMSPG_CIF,NAMSPG_CIFS,NAMPG,NSYMP,NSYM,RSYM)

Subroutine PATSGP(SPGNAM, PGNAME, PATNAM, LPATSG)

Subroutine PGDEFN(NAMPG,NSYMP,NSYM,RSYMT,LPRINT)

Subroutine PGMDF(JLASS,JCENTR,JSCREW)

Subroutine PGNLAU(NAMPG,NLAUE,LAUNAM)

Subroutine PRMVCI(PERM,JV,N,N1)

Subroutine PRMVCR(PERM,AV,N,N1)

Subroutine PRTRSM(PGNAME, NSYMP, RSYMIV)

Subroutine PSTOPH (PSIX,PSIY,PSIZ,PHIX,PHIY,PHIZ,AVPHI)

Subroutine ROTFIX

Subroutine SETGRD(NLAUE,SAMPLE,NXMIN,NYMIN,NZMIN,NX,NY,NZ)

Subroutine SETLIM(LSPGRP,XYZLIM)

Subroutine SETLIM_ZERO(LSPGRP,XYZLIM)

Subroutine SETRSL(A,B,C,ALPHA,BETA,GAMMA)