Footprint correction

If the incident slits are fixed, the fraction of the sample area illuminated by the beam increases without bound as the incident angle (theta) decreases. For high Q, you can avoid correcting for this geometrical effect and simultaneously increase the flux incident on the sample by linearly opening the slits with increasing angle such that a constant area of the sample is illuminated at each Q.

For small Q, however, the variable slit openings would then approach zero as the angle decreases and the flux incident on the sample may be too small. As a result the slits are sometimes held fixed for small Q and an ever-increasing fraction of the beam will spill over the sample surface as the incident angle approaches zero (Fig 1). For fixed or variable slits, the signal will rise sharply near zero as the detector enters the main beam.

	Fig. 1: Beam spills over the sample edges as the
	incident angle approaches zero.

Our goal with footprint correction is to compensate for the geometric effects resulting from the use of fixed slits. Assuming a rectangular sample (or a fully illuminated sample of arbitrary shape), the fraction of beam which intercepts the sample will decrease linearly with Q. Since sin(theta) ~ theta near theta=0, the portion will also decrease linearly with incident angle. To correct for this effect, you should divide by a line up until the end of the correction region, and a constant thereafter.

There are several cases:

(1) A flat region is apparent below the critical angle: Normalizes this flat region to one (e.g., by fitting the plateau) and ignore all data below the flat region.


	Fig 2: Flat region after "V".  The plateau is marked with
	red "+". Fit to intercept b (the slope m is zero).

Note that you should have some explanation for why intensity of the plateau is not unity. For example, if the slit scan was taken in air but the reflected intensity was measured in vacuum, the plateau will be above unity. Similarly, if the beam is wider than the sample, not all the beam will be reflected, and the plateau will be below unity.

(2) The data linearly increases below the critical angle: Approximate the footprint correction by fitting a line to the linear region, stopping the fit where the data curve into the critical region. The footprint correction window will tell you the Q value where the footprint correction is equal to 1. You should apply the footprint correction up to that Q value or up to the end of the fixed slit region, whichever occurs at the lowest Q.


	Fig 3: No flat region after "V". Linear portion is marked
	with red "+".  Fit to line m*Qz+b.  Note the linear scale 
	on the right hand axis.
(3) Neither a flat region nor a sloped region is evident below the critical angle: In this case calculate the footprint correction from the geometry of your sample and enter it directly (Tools for this calculation are not yet available in reflred). If your sample is not rectangular, you may not be able to calculate the footprint correction, and you will need to measure it (see below).

Since the second case is the most common, we consider it here in more detail. First zoom into the reduce graphs so that you can see both the fixed slit region simultaneous with the corresponding reflectivity. Be sure that the y-axis for the reflectivity uses a _linear_ scale so that you can locate the flat portion of the rising slope. Click Parameters... to select the footprint fit parameters.

In the footprint window, select 'Fit footprint correction' and click the 'From graph...' button. You will want to first select the lowest Q point of your data which does not include scattering from the main beam. (This is at the bottom of the "V" in the reflectivity signal assuming you have measured below about 0.005 inv Angstroms). Next choose the last point which is approximately linear. Make sure the m*Qz+b selection is active (use b if fitting a plateau, use m*Qz if you want to constrain the footprint to go through zero a Qz = 0).

Having specified the slope and intercept, or having selected the fit region from the graph, you must then select the portion of the curve over which to apply the footprint correction. You can again click 'From graph...' to select points from the graph. Usually the correction should be applied from the bottom of the "V" to the Q value corresponding to the end of the fixed slit region.

For NG1 and X-ray data, the fixed slit region will be obvious from the measured slit scan. For NG7 you will have to go back to the data selection window and examine the first file in your specular sequence to see where the fixed slit region stops. [Use the right mouse button on the specular scan you are using from the list of specular scans and select 'Edit'.] Often NG7 data is taken with continously varying slits.

Click the Apply button to fit and apply the footprint correction. Notice that the value of the footprint correction specified at the end of the application range divided through the entire Q range of your reflectivity scan. This value, listed at the bottom of the footprint dialog, should be less than one. If it is not, use the Qz value where the correction is 1.0 given on the next line as the upper end of "Correct from" range.

Your footprint correction is now complete. If you do not wish to apply it, click off the "Footprint correction" check box on the reduce screen.

Note that if your sample is not rectangular or fully illuminated, the footprint correction will no longer be linear. In that case you will have to measure the footprint correction directly (Case 3). First measure a sample of the same dimension and with a critical angle greater than the fixed slit region for your experiment. Reduce that scan (being sure to divide by a slit scan if necessary) and save it. Now select the scan you wish to correct and click 'Parameters...' to get to the footprint window. Choose the footprint scan from the 'Measured footprint correction' dropdown list. From the graph, select the region you want to correct as zero to the critical angle of the footprint scan.


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