## Coordinate Systems

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.

#### Laboratory coordinate system

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

Specification of the detector setting makes use of a right-handed orthonormal system, the detector coordinate system, imagined to be fixed in the instrument. The X-ray sensitive planar segments are specified with respect to this detector system which renders the description independent from any detector movements.

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.

#### Detector coordinate system

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 detector specification

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 :

• shield selected parts of each segment from X-ray background scatter and record test exposures each time.
• make a little drawing each time so that you do not forget the location of the obscuring object in the laboratory frame!
• put each image on screen using the XDS-Viewer program and move the mouse to the shaded part to find the pixel coordinates. (It is essential to use the XDS-Viewer program because the pixel coordinates IX,IY reported at the mouse position have the identical meaning in the XDS program.)
• it is then easy to find out which way the "fast" and "slow" pixels run in the laboratory frame.
• type a few values IX,IY,Segment-id as requested by the Pix2lab program to find the corresponding laboratory x,y,z coordinates. Do the computed "fast" and "slow" directions roughly agree? If not change parameter values to reverse direction or exchange vectors in the mapping.
• Verify segment distance. Correct sign?

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 Imprint Datenschutzhinweis.
Wolfgang.Kabsch@mpimf-heidelberg.mpg.de
page last updated: May 25, 2018