Crystals Manual

Chapter 8: Fourier Routines

8.1: Scope of the Fourier section of the user guide

8.2: Input of the Fourier section limits - LIST 14

8.3: Printing the contents of LIST 14

8.4: Compute Fourier limits from the symmetry

8.5: Fourier calculations - \FOURIER

8.6: Calculation of superposition minimum functions

8.7: Processing of the peaks list - LIST 10

8.8: Printing the contents of LIST 10

8.9: Elimination of duplicated entries in LISTS 5 and 10 - \PEAKS

8.10: Slant fourier calculations - \SLANT

In this section of the user guide, the lists and commands relating to the Fourier routines are described.

      Input of the Fourier section limits                  -  \LIST 14
      Compute Fourier linits from the symmetry operators   -  \FLIMIT
      Fourier calculations                                 -  \FOURIER
      Processing of the peaks list                         -   LIST 10
      Elimination of duplicated entries in LISTS 5 and 10  -  \PEAKS
      Slant fourier calculations                           -  \SLANT

 \LIST 14
 X-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=
 Y-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=
 Z-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=
 X-PAT MINIMUM= STEP= MAXIMUM= DIVISION=
 Y-PAT MINIMUM= STEP= MAXIMUM= DIVISION=
 Z-PAT MINIMUM= STEP= MAXIMUM= DIVISION=
 ORIENTATION DOWN= ACROSS= THROUGH=
 SCALEFACTOR VALUE=

 \LIST 14
 X-AXIS 0.0 0.0 0.5 0.0
 Y-AXIS 0.0 0.0 0.9 0.0
 Z-AXIS -2 2 32 60
 ORIENTATION Z X Y
 SCALE VALUE = 10
 END

The Fourier routines will calculate a map with section edges parallel to any two of the cell axes (a, b or c). The starting and stopping points must be given for each direction (in crystal fractions). The user should choose the asymmetric unit to have one range as small as possible, and the other two approximately equal. Orientate the computation so that the sections are perpendicular to the short range direction. If the command \SPACEGROUP has been used to input the symmetry information, a LIST 14 will have been generated. This will be a valid choice, but may not be optimal.
 

      X  -  Default value
      Y
      Z

The contents of LIST 14 can be listed to the line printer by issuing the command :
 

 \FLIMIT LAUE=

 \FLIMIT
 END

This command uses the same algorithms as \SPACEGROUP to create a LIST 14. This will be a valid choice, but may not be optimal. The parameter LAUE takes a value from this table:

  Laue Group     Number      Nx  Ny  Nz         Comment
     -1                   Default value     Compute from operators
      1             1         4   4   4     Triclinic
     2/m            2         8   8   8     Monoclinic
     mmm            3         8   8   8     Orthorhombic  (Fddd 16)
     4/m            4         8   8  16     Tetragonal
     4/mmm          5         8   8  16     Tetragonal
     -3R            6         8   8   8     Rhombohedral
     -3mR           7         8   8   8     Rhombohedral
     -3             8        12  12  24     Hexagonal
     -3m1           9        12  12  24     Hexagonal
     -31m          10        12  12  24     Hexagonal
     6/m           11        12  12  24     Hexagonal
     6/mmm         12        12  12  24     Hexagonal
     m3            13        16  16  16     Cubic
     m3m           14        16  16  16     Cubic
  The values for groups 8 and 9 are OK for the order X,Y,Z, if the 2
  other orders are searched NX and NY should be 24

 \FOURIER INPUT=
 MAP TYPE= NE= PRINT= SCAN= SCALE= ORIGIN= NMAP= MONITOR=
 REFLECTIONS WEIGHT= REJECT= F000= CALC=
 LAYOUT NLINE= NCHARACTER= MARGIN= NSPACE= MIN-RHO= MAX-RHO=
 PEAKS HEIGHT= NPEAK= REJECT=
 TAPES INPUT= OUTPUT=
 END


 \FOURIER
 MAP TYPE=DIFF
 PEAK HEIGHT = 3
 END

Before a Fourier is computed, a LIST 14 must have been created or input. The routine will compute a map in any space group, the relevant symmetry being found in LIST 2 (space group information, see section 4.8).

In the ouput listing, new peaks are labelled, with the following meanings

      GOOD PEAK - The peak centre was determined by Least-Squares.
      POOR PEAK - The peak centre was determined by interpolation.
      DUBIUOS PEAK - The peak centre is only a local maximum.
      MALFORMED PEAK - The peak centre is extrapolated to be out side
                 of the asymmetric unit - usually due to very poor phasing.

      6      Default
      7      Alternative reflection list

      F-OBS     -  Default value
      F-CALC
      DIFFERENCE
      2F0-FC
      OPTIMAL
      FO-PATTERSON
      FC-PATTERSON
      EXTERNAL

      NO   -  Default value
      YES

      NO
      YES  -  Default value

      NO
      AUTOMATIC  -  Default value
      YES

      LOW
      MEDIUM  -  Default value
      HIGH

      SIM
      NO     -  Default value
      LIST-6

      NONE
      SMALL  -  Default value
      QUARTER
      HALF

      NO   -  Default value
      YES

      NPEAK = (Cell volume) / (18 * Space Group multiplicity)

                               18 is an average atomic volume.

      \OPEN MT1 filename

 Record 1: 'INFO  DOWN ACROSS SECTION'
 Record 2: 'TRAN'       9 elements of a transformation matrix
 Record 3: 'CELL'       Cell parameters, angles in radians
 Record 4: 'L14 '       List 14 information
 Record 5: 'SIZE'       number of points down, across, and number of sections
 Record 6:  number of values,  values for a section
           Record 6 is repeated for every section.
 Record n:  number of atoms, number of items per atom
 Record n+1: Items for an atom, repeated for all atoms

      NO   -  Default value
      YES

      NO   -  Default value
      YES

(Issue 7 - implementation incomplete, 1984)

(Issue 9 - implementation still incomplete, 1993 - no one seems to want it anyway!- use SHELXS if you need to).

(Issue 10 - still no change, 1996)

The Fourier routine provides a way of calculating superposition minimum functions. For each map that is produced, it is possible to specify that another map should be read in from magnetic tape at the same time (the TAPES directive). Each point of the resulting map is taken as the minimum of the newly computed map and that read off the magnetic tape. This output map may be written to a second magnetic tape, also by use of the TAPES directive.

When the input map and the calculated map are superposed, the first point calculated and the first point read off the tape are compared, the second point calculated and the second point input are compared, and so on. This implies that the first point on each map must represent the same point in real space for the output map, and that each map must contain the same number of points. The origin of each map that is to be calculated is altered by changing LIST 14 (Fourier limits - section 8.2). For example, if a 2x, 2y, 2z vector has been identified at 0.36, 0.14 and 0.28, and the 2x, 1/2-2y, 0 vector resulting from a two-fold axis has been found at 0.36, 0.36, 0, then the two LIST 14's for the superposition function might appear as :

 \LIST 14
 X-AXIS 14 4 122 400
 Y-AXIS 5 2 59 100
 Z-AXIS 12 2 66 100
 ORIENT X Y Z
 SCALE 10
 END

      and

 \LIST 14
 X-AXIS 14 4 122 400
 Y-AXIS 16 2 70 100
 Z-AXIS -2 2 52 100
 ORIENT X Y Z
 SCALE 10
 END

For the first map, the origin of real space is at 0.18, 0.07 and 0.14 in vector space. This point is moved so that it is one grid point in along each axial direction, to allow for the map scan. For the second peak, the origin in real space is at 0.18, 0.18 and 0.0. The second LIST 14 places this point one grid point in along each of the axial directions so that the real space origin of the two maps coincides. To convert the coordinates that result from the second map scan to real space coordinates, it is necessary to subtract 0.18 from x and 0.18 from y, since the coordinates are printed in Patterson space for all the maps calculated.
 

 \EDIT 10 5                \PEAKS 10 5
 \COLLECT 10 5             \REGROUP  10 5

The contents of LIST 10 can be listed with:
 

 \PEAKS INPUTLIST= OUTPUTLIST=
 SELECT REJECT= KEEP= MONI= SEQ= TYPE= REGROUP= MOVE= SYMM= TRANS=
 REFINE DISTANCE= MULTIPLIER=
 END

 \PEAKS
 SELECT REJECT=0.0001
 REFINE DISTANCE=.5
 END

This routine eliminates atoms or peaks which duplicate other entries in an atomic parameter list. When using this routine, a set of distances is calculated about each atom or peak in turn. Atoms or peaks further down the list than the current pivot are then eliminated if they have a contact distance less than a user specified maximum (the REJECT parameter). Thus, when peaks have been added to a LIST 5, the peaks corresponding to the atoms can be eliminated.
 

      5
      10  -  Default value

      5   -  Default value
      10

      LOW
      HIGH  -  Default value

      NO  -  Default value
      YES
      EXHYD

      PEAK  -  Default value
      ALL
      AVERAGE

      NO  -  Default value
      YES

      SPACEGROUP  -  Default value. The full spacegroup symmetry is used in
                                    all computations
      PATTERSON.     A centre of symmetry in introduced, and the translational
                     parts of the symmetry operators are dropped.
      NONE.          Only the identity operator is used.

      X(new) = x(atom) + mult(x(peak) - x(atom)).

 \ reject atoms or peaks with contact distances less than 0.7
 \ keep 30 entries in the output list
 \ list the atoms and peaks rejected because of both 'KEEP'
 \ and 'REJECT'
 \
 \PEAKS 10 5
 SELECT REJECT=0.7,KEEP=30,MONITOR=HIGH
 END

 \SLANT INPUT=
 MAP TYPE= MIN-RHO= SCALE= WEIGHT=
 SAVED MATRIX=
 CENTROID XO= YO= ZO=
 MATRIX R(11)= R(12)= R(13)= R(21)=  .  .  . R(33)=
 DOWN MINIMUM= NUMBER= STEP=
 ACROSS MINIMUM= NUMBER= STEP=
 SECTION MINIMUM= NUMBER= STEP=
 END

A Slant Fourier is one that is calculated through any general plane of the unit cell. For such a Fourier, the normal Beevers-Lipson expansion of the summation cannot be used, so that it will take many orders of magnitude longer than a conventional one. The algorithm adopted here is as follows :

 X    A general vector expressed in fractions of the
     unit cell edges (i.e. x/a, y/b and z/c)
 XO   The centroid of the required general fourier section,
     also expressed in crystal fractions.
 XP   The coordinates of the point 'X' when expressed
     in the coordinate system used to define the
     plane of the general section.
 'X' and 'XP' are related by the expression :  XP = R.(X-XO)
 R    'R' is the matrix that describes the transformation
     of a set of coordinates in the crystal system to
     a set of coordinates in the required plane.
 therefore :  X = S.XP + XO

     'S' is the inverse matrix of 'R'.

 The required expression in the fourier is :

     H'.X = H'.S.XP + H'XO

 H    H is a vector containing the Miller indices of
     a reflection and H' is the transpose of H.
 This may be re-expressed as :

     H'.X = H'.S.DXP + H'.(S.XPS + XO)

 DXP  'DXP' represents the increment in going from the
     first point on the section to be calculated.
 XPS  'XPS' is the coordinate of the first point on the
     section to be calculated.
     obviously :  XP = XPS + DXP.

When the Fourier is calculated, the term H'.(S.XPS + XO) is constant for each section to be calculated. The term H'.S , which may be regarded as the transformed indices, is also constant for each reflection, so that a two dimensional recurrence relation may be used to change DXP and thus Cos(2*PI*H.X - ALPHA)' over the required section for each reflection. ( ALPHA is the phase angle for the current reflection).

The input for the slant Fourier thus must include the rotation matrix R, the centroid XO, and the steps and divisions in the required plane.
 

      6      Default
      7      Alternative reflection list

      F-OBS
      F-CALC
      DIFFERENCE
      FO-PATTERSON
      FC-PATTERSON

      NO   -  Default value
      YES

      MOLAX
      TLS
      AXES

 \ the map will be a difference map
 \ we wish to compute the section 0.3 anstrom above the plane
 \ numbers less than zero will be printed as zero
 \ the molecule lies at a centre of symmetry
 \ so that the centroid in crystal fractions is 0, 0, 0
 \ the plane coordinates are in angstrom
 \ for printing the plane both across and down the page,
 \ we will start 4 angstrom from the centroid,
 \ and go 4 angstrom the other side of the centroid,
 \ making a grid 8 angstrom by 8 angstrom
 \
 MAP DIFFERENCE 0.3 0
 CENTROID 0 0 0
 MATRIX 3.4076 10.0498 6.1794
 CONT   5.0606  8.287 -9.5483
 CONT  -6.9181 11.0121 1.546
 DOWN -4 33 0.25
 ACROSS -4 33 0.25
 END