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Crystals ManualChapter 8: Fourier Routines
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.1: Scope of the Fourier section of the user guideIn 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 Fourier calculations  \FOURIER Processing of the peaks list  LIST 10 Elimination of duplicated entries in LISTS 5 and 10  \PEAKS Slant fourier calculations  \SLANT [Top] [Index] Manuals generated on Wednesday 8 November 2006 8.2: Input of the Fourier section limits  LIST 14\LIST 14 XAXIS MINIMUM= STEP= MAXIMUM= DIVISION= YAXIS MINIMUM= STEP= MAXIMUM= DIVISION= ZAXIS MINIMUM= STEP= MAXIMUM= DIVISION= XPAT MINIMUM= STEP= MAXIMUM= DIVISION= YPAT MINIMUM= STEP= MAXIMUM= DIVISION= ZPAT MINIMUM= STEP= MAXIMUM= DIVISION= ORIENTATION DOWN= ACROSS= THROUGH= SCALEFACTOR VALUE=
\LIST 14 XAXIS 0.0 0.0 0.5 0.0 YAXIS 0.0 0.0 0.9 0.0 ZAXIS 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.
\LIST 14
XAXIS MINIMUM= STEP= MAXIMUM= DIVISION=
This directive specifies how the xaxis is to be divided. MINIMUM=
This parameter gives the initial value along the xdirection.
If it is omitted, a default value of 0.0 is assumed for MINIMUM.
STEP=
This parameter, which has a default value of 0.3, gives the
step along the xdirection.
MAXIMUM=
This parameter, which has a default value of 1.0, gives the
final value along the xdirection.
DIVISION=
If DIVISION is greater than zero, it defines the number of
divisions into which the xaxis is to be divided.
In this case, the three remaining parameters are expressed in
terms of DIVISION and give the first point ( MIN ), the
increment between successive points ( STEP ) and the final
point to be calculated ( MAX ). If the divisions of the unit
cell along the xaxis are given in this way, the user must
ensure that sufficient map is calculated for the map scan, by
adding one extra point beyond the asymmetric unit at both ends
along the xaxis. If this is not done, peaks at the edge of the
asymmetric unit may be missed by the peak search.
If DIVISION is equal to zero, which is its default value,
the Fourier routines will calculate the number of divisions
required along the xaxis. In this case, STEP is the interval
between successive points along the axis in angstrom.
If this parameter is less than 0.05, a default value of 0.3 angstrom
is used. MINIMUM And MAXIMUM define the first and last points to be
calculated and are given in fractional coordinates.
When the values of MIN and MAX are converted into unit cell
divisions, an extra point is added at each end to ensure that the
peak search functions correctly.
YAXIS MINIMUM= STEP= MAXIMUM= DIVISION=
Similar to XAXIS above.
ZAXIS MINIMUM= STEP= MAXIMUM= DIVISION=
Similar to XAXIS above.
XPAT MINIMUM= STEP= MAXIMUM= DIVISION=
This directive is similar to the XAXIS directive, but refers to the
Patterson asymmetric unit.
YPAT MINIMUM= STEP MAXIMUM= DIVISION=
Similar to XPAT above.
ZPAT MINIMUM= STEP= MAXIMUM= DIVISION=
Similar to XPAT above.
ORIENTATION DOWN= ACROSS= THROUGH=
Controls the orientation parameters for the map calculation and printing. DOWN=
X  Default value Y Z
ACROSS=
As DOWN above, but with the default value Y
indicating that the y coordinate goes across the page.
THROUGH=
As DOWN above, but with the default value Z
indicating that the z coordinate changes from section to section.
SCALEFACTOR VALUE=
VALUE=
This parameter specifies the value by which the electron density,
on the scale of /Fc/, is multiplied before it is printed.
If this parameter is omitted, a default value of 10 is assumed.
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.3: Printing the contents of LIST 14The contents of LIST 14 can be listed to the line printer
by issuing the command :
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.4: Fourier calculations  \FOURIER\FOURIER INPUT= MAP TYPE= NE= PRINT= SCAN= SCALE= ORIGIN= NMAP= MONITOR= REFLECTIONS WEIGHT= REJECT= F000= CALC= LAYOUT NLINE= NCHARACTER= MARGIN= NSPACE= MINRHO= MAXRHO= 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 LeastSquares. 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.
\FOURIER INPUT=
INPUT
Indicates which reflection list to use.
6 Default 7 Alternative reflection list
MAP TYPE= NE= PRINT= SCAN= SCALE= ORIGIN= NMAP= MONITOR=
TYPE=
FOBS  Default value FCALC DIFFERENCE 2F0FC OPTIMAL FOPATTERSON FCPATTERSON EXTERNAL
NE=
This parameter indicates which solution should be used to compute the
externally phased map, and has a default value of 1.
NE is only used in conjunction with TYPE = EXTERNAL.
PRINT=
Controls the printing pf the map.
NO  Default value YES
SCAN=
Controls automatic scanningof the map for peaks.
NO YES  Default value
SCALE=
Controls the scaling of the electron density in the map.
NO AUTOMATIC  Default value YES
If SCALE is NO, the scale factor is taken from LIST 14 for all types of Fourier maps. If SCALE is AUTOMATIC, there is automatic scaling for an external or Patterson map, while other maps take their scale factors from LIST 14. ORIGIN=
The default value for this parameter is 999, and is used when
the program calculates a scale factor (see SCALE above).
NMAP
Controls negation of the density values, with default NO.
Use YES, in which case the density values are negated,
when looking for minima. This
feature permits location of hydrogen in Neutron maps, and the location of
minima (which become maxima) generally. Set the Peak Height positive
even when searching for minina, since at the time of the search the
minima are inverted. The out put density values have the correct sign.
Use \COLLECT 10 5 rather than \PEAKS on negated maps, since PEAKS
cannot handle minima.
MONITOR=
LOW MEDIUM  Default value HIGH
REFLECTIONS WEIGHT= REJECT= F000= CALC=
WEIGHT=
SIM NO  Default value LIST6
If WEIGHT is NO , its default value, then the map is not weighted. If WEIGHT is set equal to SIM , then SIM weights are computed. This option requires both LIST 29 (atomic properties, section 4.15 and LIST 5. The occupation factors in LIST 5 are used to determine how many atoms of each type are present, and LIST 29 indicates how many should be present. See the notes under 'TYPE', above. If WEIGHT is LIST6 , then the map is weighted with the weight stored in LIST 6 (section 5.3). REJECT=
NONE SMALL  Default value QUARTER HALF
Some users like to omit reflections if Fc is smaller then a fraction of Fo. The options QUARTER and HALF are available. F000=
The default value for this parameter is zero, and specifies the value
of F(000) to be used.
CALC
NO  Default value YES
LAYOUT NLINE= NCHARACTER= MARGIN= NSPACE= MINRHO= MAXRHO=
This directive specifies how the map should be printed, if the value of the PRINT parameter on the MAP directive is YES. NLINE=
This parameter sets the number of lines per row of map, and has a default
value of 2.
NCHARACTER=
This parameter controls the number of characters for each grid
point, and has a default value of 4.
MARGIN=
This parameter, whose default value is 4, defines the number of
characters per division number down each side of the map.
NSPACE=
This parameter has a default value of 2, and defines the number of
spaces between the division number and the grid number down each
side of the map. The minimum value for NSPACE is 2.
MINRHO=
This parameter has a default value of 1000000, and points less than
MINRHO are left blank when the map is printed.
MAXRHO=
This parameter has a default value of 1000000, and points greater than
MAXRHO are left blank when the map is printed.
PEAKS HEIGHT= NPEAK= REJECT=
Controls the search for peaks when the map is searched, i.e. if the value of the SCAN parameter on the MAP directive is YES. HEIGHT=
This parameter sets the search of the map for all
peaks with an electron density greater than HEIGHT. If this
parameter is omitted, a default value of 50 is assumed
for an external or Patterson map. For all other maps, the map is scanned
for peaks greater than 1.5*SCALE, where SCALE is the map scale factor,
either taken from LIST 14 (Fourier control  section 8.2)
or computed using SCALE = YES above.
NPEAK=
This parameter, whose default value is 0, determines the number
of peaks to be retained after they have been ranked by peak height.
If NPEAK is zero or negative, the number of peaks saved is computed from
NPEAK = (Cell volume) / (18 * Space Group multiplicity) 18 is an average atomic volume.
REJECT=
This parameter, with a default value of 0.01, specifies that peaks
within a distance of REJECT angstrom of a peak already ranked on
peak height, will be rejected from the list.
TAPES INPUT= OUTPUT=
This directive is used if a map is to be read off magnetic tape,
or a computed map is to be written to a
magnetic tape. Remember that CRYSTALS will use scratch files unless given
named files. To assign a named output file, issue
\OPEN MT1 filename
The tape is unformatted. 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
Record 4 contains 6 integers, (No of points down and across the page, number of sections, and the index of these directions, 1 = x). Subsequent records contain a whole section line by line, prefixed by the total number of points in the section. INPUT=
NO  Default value YES
*** THIS FACILITY IS NOT CURRENTLY IMPLEMENTED ***
OUTPUT=
NO  Default value YES
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.5: Calculation of superposition minimum functions(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, 1/22y, 0 vector resulting from a twofold 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 XAXIS 14 4 122 400 YAXIS 5 2 59 100 ZAXIS 12 2 66 100 ORIENT X Y Z SCALE 10 END and \LIST 14 XAXIS 14 4 122 400 YAXIS 16 2 70 100 ZAXIS 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.
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.6: Processing of the peaks list  LIST 10\LIST 10
LIST 10 cannot be input bythe user. When the map scan has been completed, the resulting peaks are output to the disc as a LIST 10. Except for an external or Patterson map, the atoms already in LIST 5 are placed at the beginning of the LIST 10. A LIST 10 is usually converted to a LIST 5 by one of the following commands : \EDIT 10 5 \PEAKS 10 5 \COLLECT 10 5 \REGROUP 10 5
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.7: Printing the contents of LIST 10The contents of LIST 10 can be listed with:
[Top] [Index] Manuals generated on Wednesday 8 November 2006 8.8: Elimination of duplicated entries in LISTS 5 and 10  \PEAKS\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.
\PEAKS INPUTLIST= OUTPUTLIST=
INPUTLIST and OUTPUTLIST specify where the atoms are to be taken from, and where they will be put. INPUTLIST=
5 10  Default value
OUTPUTLIST=
5  Default value 10
SELECT REJECT= KEEP= MONI= SEQ= TYPE= REGROUP= MOVE= SYMM= TRANS=
REJECT=
REJECT is the distance above which connected atoms or peaks are assumed to
be distinct. If a contact is found which is less than REJECT
the second atom or peak of the pair in the list is eliminated, and
defaults to 0.5.
KEEP=
This parameter indicates how many entries are to be kept in the
output list. The default value of 1000000 is the maximum possible.
MONITOR=
LOW HIGH  Default value
SEQUENCE=
NO  Default value YES
TYPE=
PEAK  Default value ALL AVERAGE
REGROUP=
This parameter has two allowed values :
NO  Default value YES
MOVE=
The value of this parameter is the maximum separation for 'bonded' atoms.
The default is 2.0 A.
SYMMETRY=
This parameter controls the use of symmetry information in the calculation of
contacts, and can take three values.
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.
TRANSLATION=
This parameter controls the application of cell translations in the
calculation of contacts, and can take the values YES or NO
REFINE DISTANCE= MULTIPLIER=
Controls action of Fourier refinement. DISTANCE=
This parameter has a default value of zero, and is
the distance below which atoms and peaks are considered
to be coincident. The coordinates of an existing atom are replaced
by those of a coincident peak. Refinement takes precedence
over deletion of peaks.
MULTIPLIER=
This parameter has a default value to give automatic refinement.
It is set to 1 for a centric space group and is set to
2 for a noncentric space group. It can be set to 0.0 to preserve original
coordinates but be given new peak heights.
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 [Top] [Index] Manuals generated on Wednesday 8 November 2006 8.9: Slant fourier calculations  \SLANT\SLANT INPUT= MAP TYPE= MINRHO= 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 BeeversLipson 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.(XXO) 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 reexpressed 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.
\SLANT INPUT=
This is the command which initiates the slant fourier routines. INPUT
Indicates which reflection list to use.
6 Default 7 Alternative reflection list
MAP TYPE= MINRHO= SCALE= WEIGHT=
TYPE=
FOBS FCALC DIFFERENCE FOPATTERSON FCPATTERSON
MINRHO=
This parameter has a default value of zero, and
is the value below which all numbers on the map are replaced by
MINRHO.
SCALE=
The terms used in the Fourier are put on the same scale as Fc,
and then before the map is printed the numbers are multiplied
by SCALE . (i.e. SCALE is the map scale factor).
The default is 10.
WEIGHT=
NO  Default value YES
If WEIGHT = YES, the observed and calculated structure factors are
multiplied by the weights in LIST 6 (usually SQRT(w)). The user should
be aware that this might have a major effect on the scale if the map
density, and that SCALE may need adjusting.
SAVED MATRIX=
This directive, which excludes CENTRIOD and MATRIX, uses the matrix and centroid stored in LIST 20 by a previous GEOMETRY, MOLAX or ANISO command (see section 9.6). MATRIX=
MOLAX TLS AXES
CENTROID XO= YO= ZO=
This specifies the slant Fourier map centroid, in crystal fractions, and excludes SAVED. XO=
YO=
ZO=
The defaults value for XO,YO,ZO, the coordinates of the centroid,
are 0.0.
MATRIX R(11)= R(12)= R(13)= R(21)= . . . R(33)=
This gives the elements of the rotation matrix R, and excludes SAVED. The trnsformation generally used is from crystal fractions to orthogonal Angstroms. R(11)= R(12)= R(13=) R(21)= . . . R(33)=
There are no default values for any of these parameters.
DOWN MINIMUM= NUMBER= STEP=
This directive defines the printing of the map down the page. MINIMUM=
There is no default value for this parameter, the first point,
in Angstrom,
down the page of the plane to be calculated.
NUMBER=
There is no default value for this parameter, the number of points
of the plane to be printed down the page
STEP=
There is no default value for this parameter, the interval
in Angstrom between successive points down the page.
ACROSS MINIMUM= NUMBER= STEP=
This directive defines the printing of the map across the page. The
parameters have similar meanings to those for 'DOWN'.
SECTION MINIMUM= NUMBER= STEP=
This directive defines the printing of the map sections. The parameters have similar meanings to those for 'DOWN'. The units of MINIMUM and STEP are based on the coordinate system used to describe the plane, with the new 'x' axis going down the page and 'y' across. In general the most convenient axial system for the plane is one expressed in Angstrom, so that the initial points and the steps are all expressed in Angstrom. (The least squares best plane program prints out the centroid in crystal fractions and the rotation matrix from crystal fractions to best plane coordinates in Angstrom, which are the numbers required, and may be saved for use in SLANT by the directive 'SAVE'). \ 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
[Introduction To The System  Definitions And Conventions  The Crystals Database  Initial Data Input  Reflection Data Input  Atomic And Structural Parameters  Structure Factors And Least Squares  Fourier Routines  Analysis Of Results  Twinned Crystals  Matrix Calculations  Obsolete Commands ] 
© Copyright Chemical Crystallography Laboratory, Oxford, 2006.
Comments or queries to David Watkin 
david.watkin@chem.ox.ac.uk Telephone +44 1865 285019.
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