Data processing requires the specification of geometrical parameters that describe the diffraction experiment. The coordinate systems used by XDS to this purpose are explained in this chapter.
Any convenient right-handed orthonormal system may be chosen with the origin at the intersection between rotation axis, direct beam and crystal. Once chosen, the coordinate system remains fixed throughout the diffraction experiment. This coordinate system serves to specify
Example of a typical experimental setup for the SIEMENS detector.
In the chosen laboratory coordinate system, the y-axis points vertically down and is defined to be collinear with the detector swing axis (for the SIEMENS). The z-axis lies within the plane spanned by the y-axis and the detector normal at swing angle 0, with +z pointing from the crystal towards the detector. The x-axis is defined to yield an orthonormal right handed laboratory coordinate system.
viewed from ABOVE:
detector
\
\
\
| \
| \
| \
| \
|
---beam->-->-O-------------- +z
|
|
|
|
|
+x
viewed from the SIDE:
swing axis
ω axis
2θ axis
rotation axis detector
|
|
| |
| |
| |
---beam->-->-O-------------- +z |
| |
| |
| |
|
|
+y
ROTATION_AXIS= 0 -1 0
if the crystal rotates counterclockwise when proceeding to the next data
frame and the crystal is viewed from above; otherwise 0 1 0.
Note, that XDS requires the frame increment
(oscillation range)
be positive.
INCIDENT_BEAM_DIRECTION= 0 0 1
The positive beam direction points along +z from the source towards
the crystal. The beam is aligned normal to the y-axis and the
detector surface at swing angle χ=0.
DIRECTION_OF_DETECTOR_X-AXIS=ED(1,1) ED(2,1) ED(3,1)=
-cosχ 0 sinχ
DIRECTION_OF_DETECTOR_Y-AXIS=ED(1,2) ED(2,2) ED(3,2)=
0.0 -1.0 0.0
ORGX=ORGX=261.45
ORGY=ORGY=268.06
DETECTOR_DISTANCE=
F=130.01
The x- and y- axes of the detector coordinate system are
specified by the two orthonormal vectors ED(:,1) and ED(:,2), respectively.
These vectors - together with their cross product
ED(:,3)=ED(:,1) X ED(:,2) - define a rotation matrix ED.
The swing axis points downwards along +y and a positive swing
angle χ (or 2θ) corresponds to a clockwise rotation of the
detector when viewed from above. In the drawing, the detector is set to
a negative swing angle χ.
The origin of the detector system, fixed at 0 0 0 in the
instrument, is specified (mm) in the laboratory coordinate system
by the vector
ORG(:) = -ORGX*QX*ED(:,1)-ORGY*QY*ED(:,2)+F*ED(:,3)
using device specific conversion factors
QX=0.19 and
QY=0.19.
Note the positive sign for the detector distance because the detector
normal ED(:,3) points away from the crystal.
A right-handed orthonormal detector coordinate system, assumed
fixed with the instrument, is used as a reference for desribing internal
details of the device. This description is invariant towards movements
of the detector. As mentioned above this detector system, i.e.
the right-handed orthonormal matrix ED, is specified by the
two input parameters
DIRECTION_OF_DETECTOR_X-AXIS= and
DIRECTION_OF_DETECTOR_Y-AXIS=.
The detector translation, namely the point vector
ORG(:) = -ORGX*QX*ED(:,1)-ORGY*QY*ED(:,2)+F*ED(:,3)
to the origin of the detector system, is specified by the
input parameters
ORGX=,
ORGY=, and
DETECTOR_DISTANCE=.
The class of detectors that can be handled by XDS consist of one or
several rectangular arrays of X-ray sensitive segments at arbitrary
orientation and translation fixed with respect to the detector
system.
The pixels of a segment are enumerated by IX,IY whereby
1≤x1≤IX≤x2 and 1≤y1≤IY≤y2. The index ranges
[x1,x2], [y1,y2] are defined by the input parameter
SEGMENT= x1 x2 y1 y2
and are chosen so that pixel numbers from other segments do not overlap.
Thus, a given pixel address IX,IY belongs to at most one segment.
The union of the index ranges from all segments is a subset of the index
set 1≤IX≤NX and 1≤IY≤NY, where NX, NY are chosen just big
enough to accommodate all pixels. Diffraction data from all segments are
stored as a linear array of NX*NY pixel values in a file that can be
displayed as a two-dimensional 'mosaic' image.
A pixel at (IX,IY) in the two-dimensional image is found at
position IX+NX*(IY-1) in the linear array, where NX is the number of
"fast" pixels, NY is the number of "slow" image pixels,
and IX= 1,..., NX, IY= 1,..., NY. Each pixel is assumed as a rectangle
of dimensions QX,QY along the
"fast" and "slow" directions, respectively.
The orientation of each segment plane is described by two orthonormal
vectors, EDS(:,1) and EDS(:,2), that specify the directions along the
"fast" and "slow" pixels with respect to the
detector system. The two vectors have to be provided as
the input parameters
DIRECTION_OF_SEGMENT_X-AXIS=EDS(1,1) EDS(2,1) EDS(3,1)
DIRECTION_OF_SEGMENT_Y-AXIS=EDS(1,2) EDS(2,2) EDS(3,2)
The third unit vector, the segment normal, is then constructed as
EDS(:,3)=EDS(:,1) X EDS(:,2)
to form a right-handed orthonormal segment system
{EDS(:,1), EDS(:,2), EDS(:,3)}.
The origin of the segment system, fixed at 0 0 0
in the segment system, is specified (mm) in the detector
coordinate system by the point vector
-ORGXS*QX*EDS(:,1)-ORGYS*QY*EDS(:,2)+FS*EDS(:,3)
where
SEGMENT_ORGX=
ORGXS,
SEGMENT_ORGY=
ORGYS, and
SEGMENT_DISTANCE=
FS.
The representation of the segment system EDS
with respect to the laboratory system ED is the
matrix product EDSL = MATMUL(ED,EDS).
Thus, a segment pixel at IX,IY has the laboratory coordinates (mm units)
x=QX*(IX-ORGXS)*EDSL(1,1)+QY*(IY-ORGYS)*EDSL(1,2)+FS*EDSL(1,3)+ORG(1)
y=QX*(IX-ORGXS)*EDSL(2,1)+QY*(IY-ORGYS)*EDSL(2,2)+FS*EDSL(2,3)+ORG(2)
z=QX*(IX-ORGXS)*EDSL(3,1)+QY*(IY-ORGYS)*EDSL(3,2)+FS*EDSL(3,3)+ORG(3)
This mapping of segment pixels IX,IY to corresponding
laboratory coordinates is assumed by XDS and the correct choice
of the involved parameter values is absolutely essential.
In the next section a procedure is described for their experimental
verification.
The parameters describing translation and orientation of a segment
with respect to the laboratory system can be refined individually by XDS
(see REFINE_SEGMENT=).
For each segment of the detector a new set of the segment parameters must be provided in XDS.INP, except for the case of a single segment for which a default set of values is used.
Example (continued from above)
NX= 512 NY= 512 QX=0.19 QY=0.19 !old SIEMENS at MPI-Heidelberg
DIRECTION_OF_DETECTOR_X-AXIS=
-1 0.0 0.0 !-cosχ 0 sinχ
DIRECTION_OF_DETECTOR_Y-AXIS= 0.0 -1.0 0.0
ORGX=261.45
ORGY=268.06
DETECTOR_DISTANCE=
130.01
SEGMENT= 1 512 1 512
SEGMENT_ORGX=
0.0,
SEGMENT_ORGY=
0.0, and
SEGMENT_DISTANCE=
0.0
DIRECTION_OF_SEGMENT_X-AXIS= 1.0 0.0 0.0
DIRECTION_OF_SEGMENT_Y-AXIS= 0.0 1.0 0.0
This detector consists of a single segment.
The segment system is identical with the detector system
which is the default for single segment devices. Specification of
all SEGMENT parameters could have been omitted from XDS.INP.
Verification of the above mapping of pixels to their location in the laboratory system is supported by using the program Pix2lab together with the XDS-Viewer graphics program. Confidence in the correctness of the parameter settings in XDS.INP can be achieved by repeating the following procedure :
WARNING: An incorrect choice of the vectors ED, EDS may well lead to incorrect signs of the anomalous intensity differences. As pointed out by Janet Smith (Purdue University, USA) for the case of a SIEMENS detector, the incorrect enantiomorph can be obtained by a negative F (instead of positive) and a detector Y-axis pointing opposite to the correct direction.
© 2009-2018, MPI for Medical Research, Heidelberg