Subsections

• Definition of the Tilt Angles

The TILT Command

The TILT command allows any non-orthogonality of the detector to the direct beam (tilt), and the direct beam centre on the detector to be determined [10]. This is achieved by least squares fitting of powder rings (or rings from other randomly orientated samples e.g. wax).

One or more powder rings are used to refine beam centre and non-orthogonality parameters. This form allows a number of choices in the manner in which this works.

This operation works on the current selected data region, and does not take into account ``masked-off'' data. You may want to use the MASK command to mask-off data prior to using the TILT command. Or, if problems are encountered during the TILT evaluation, problem data regions may be identified and masked-out prior to another try with TILT.

An initial approximate beam centre is obtained from a choice of methods. This beam centre is then used to define a search region around one powder ring. This ring should be strong and well defined. The whole of the ring should be within the search region through the defined azimuthal range.

Other rings may also be selected to be used to refine the beam centre and tilt parameters. Ideally two or more rings should be used, but one will work. Complete rings at high angle will have the largest effect.

From the starting ring coordinates a best fit circle is found, followed at an ellipse. The powder ring section centres are re-calculated for all the rings. The beam centre and tilt angles are refined to these positions. There is the option of rejecting badly fitting positions and re-fitting the data.

For a given set of ellipses, two possible beam centre / tilt angle combinations are theoretically possible and indistinghable. Only if the beam centre and/ or the tilt angles are fixed is there an unique solution. If the beam centre and the tilt are being fitted you will be asked to choose between the two solutions. (In practice owing to the uncertainty in the data the same solution may be produced for both minimisations.) If there are two solutions you may be able to choose the correct solution by other knowledge from the data e.g. scatter from the beam-stop. If you can't choose between the solutions, do not worry; the calculated two-theta angles from the data will be the same.

First the Experimental Geometry Control Form is presented. An example is shown in Figure 34.

Figure 34: The Experimental Geometry Control Form

This form is not too important, as many of the values will be determined automatically by the command. However, the pixel size should be correct if strange results are to be avoided, and DISTANCE does affect slightly the interpretation of the powder rings. Other values, such as the wavelength, are not important for the TILT command, but can be usefully set for later operations.

When the correct values have been set, click on O.K..

The TILT/ BEAM CENTRE REFINEMENT Control form now appears. This is shown in Figure 35.

Figure 35: The TILT/ BEAM CENTRE REFINEMENT Control Form

The following "buttons" are available:

ANGULAR SECTIONS: This the number of angular sections around 360 degrees of the powder rings which are used to calculate the centre of the rings. The beam centre and tilt are fitted to these positions on each ring. If the data is very noisy or shows poor powder averaging then a smaller number may be better. (No theoretical criterion exists to set an optimum value, so trail and error is recommended. Clearly, this value should not be too small.)

REJECT OUTLIERS: This option allows badly fitting "ring" positions to be rejected, and the data re-fitted. This is to make the fit procedure more robust and to allow for erroneous positions owing to contaminating Bragg peaks, etc.

REJECT LIMIT: If REJECT OUTLIERS is YES then this is the number of standard deviations away from the best fit predicted position after which ``outliers'' are rejected.

FULL INFO: YES for terminal screen output of information relating every stage of the fitting of the powder rings.

REFINE BEAM: YES to fit the beam centre position, as well as the tilt angles. NO to refine only the tilt angles (if variable). This is usually used when the beam centre has been determined from a direct beam mark.

REFINE TILT: YES to fit the detector non-orthogonality to the beam (the tilt). NO will keep the tilt angles fixed and only fit the beam centre (if variable). NO is often used when the tilt angles have been determined accurately from a high quality calibrant measurement.

Set the required parameters and click O.K. to continue.

Next the BEAM CENTRE MENU appears. This is shown in Figure 37, Page . This is used to set an initial beam centre, which will then be refined if REFINE BEAM has been set to YES.

If the beam centre has been accurately recorded, e.g. with a semi-transparent beam-stop, then fitting with a 2-D Gaussian may be the best method of setting the beam centre, and it may then be better not to refine it using the powder ring positions.

For further information on the BEAM CENTRE MENU see Section 11.4, Page .

Depending on whether or not a circle has been selected in defining the beam centre, the user may be requested to select a powder ring for the initial tilt determination.

Next the RING PROFILE SEARCH LIMIT is defined by graphical input. Relative to the defined circle this defines an annuli symmetric about the ring. This profile limit will also be used for other rings which are selected for fitting. The search is limited to ring positions within the annuli, so this should be large enough that all rings fit inside, but small enough not to include other rings.

The annuli are draw in white on the image.

Other rings to be used for the tilt refinement are now selected by clicking on them. A single ring is sufficient, but more well defined rings will clearly lead to more accurate results. When finished click within the prompt box.

Starting from the initial beam centre the centre of each sector of powder ring will be determined for the different sectors. This is then fitted with an ellipse to obtain a better estimate of the beam centre. The central positions of all sectors of all the selected rings are then found and the tilt and/or beam centre parameters are refinement to give the best fit to the experimental data positions.

Unless the beam centre is known there are theoretically two equally possible and indistinguishable solutions to the tilt angles from the shape of the ellipses. These have opposite tilt angles, but differing beam centres. FIT2D tries to find both solutions, but owing to noise it may find the same one twice. Details of both solutions are output in the terminal window and the user is asked to choose one of them. If some knowledge of the beam centre exists it may be possible to determine the true solution. If not, it is probably not important as both solutions theoretically lead to the same $2\theta$ angles when the data is integrated.

Definition of the Tilt Angles

FIT2D parametrises the ``tilt'' using two orthogonal angles so that the refinement is as stable as possible. The tilt is determined by the elliptical shape of powder rings. The tilt out of the orthogonal plane is related to the ratio of the major to minor axes, and the plane in which this tilt occurs is given by the angle of the major axis from the X-axis. To define the tilt, first an ideal detector plane is imagined. This is prefectly orthogonal to the direct beam, and passes through the intersection of the detector with the direct beam.

The ROTATION ANGLE OF TILTING PLANE (DEGREES) defines the plane in which the tilt is defined. This angle is in degrees anti-clockwise about the direct beam, from the detector X-axis, in the ``ideal detector plane''. This angle and the direct beam define a plane which in which the ANGLE OF DETECTOR TILT IN PLANE (DEGREES) is defined. This is defined as a rotation about the direct beam position anti-clockwise from the ``ideal detector plane'' to the actual detector plane. The axis of rotation is orthogonal to the tilting plane, passing through the direct beam position.

It may be noted that this system is redundant e.g. rotation angle 0.0 degrees and tilt angle 2.0 degrees is the same as rotation angle 180.0 degrees and tilt angle -2.0 degrees.