Performance of Detectors for X-ray Crystallography

Martin Stanton and Walter Phillips
Rosenstiel Basic Medical Sciences Research Center
Brandeis University
Waltham MA 02254-9110

Abstract

The performance of a detector can be characterized by its efficiency for measuring individual x-rays or for measuring Bragg peak intensities. The performance for detecting individual x-rays is well modeled by the DQE. The performance for measuring Bragg peak intensities in the presence of an x-ray background can be modeled by an expanded definition of the DQE which allows inclusion of experimental constraints, the XDCE. These constraints include the observation that by increasing the crystal-to-detector distance and using a larger detector, Bragg peaks can be better resolved and the x-ray background reduced. Calculation of the XDCE for a detector consisting of a fiberoptic taper with a phosphor x-ray convertor deposited on the large end and a CCD bonded to the small end demonstrate the need to make the detector area relatively large, possibly at the expense of a decrease in the DQE.

1. Introduction

It is not necessarily true that an area detector optimized to count individual events will be the best detector for collecting data in a real experiment. In particular, for macromolecular crystallography, it is not necessarily true that the best detector will be one optimized to count individual x-rays. Consider a typical diffraction image as shown in figure 1. Several features on this image are critical to the design of an x-ray crystallography detector: Bragg peaks (diffraction maxima) have an extended profile, any peak is likely to be surrounded by several other (possibly higher-intensity) peaks, there is considerable x-ray background, and the total x-ray diffraction pattern can cover a large angular range. An ideal detector would allow accurate integration of all Bragg peaks diffracted at one time by the crystal.

The type of x-ray detector that we are constructing converts incident x-ray photons to light photons, which are then imaged onto a sensor such as a CCD (1,2). As a result of the large solid angle and the extended profile of the Bragg peaks, a large input area is required to record a full diffraction image. Because of the relatively small size of CCDs, the image must usually be demagnified between the x-ray-to-light convertor and the sensor (3,4). The image demagnification results in the loss of light, which for a lens or fiberoptic taper goes as the square of the demagnification. In this article we discuss a method developed to resolve this trade-off between a large input area and a high-efficiency detector, and to address other detector design decisions. For the purpose of illustrating this method, we will assume a design which employs a fiberoptic taper with an x-ray-to-light converter on the large end and a CCD bonded to the small end.

2. Detector Performance Considering Individual X-rays

Detector performance is often characterized by the detective quantum efficiency (DQE) (5,6,7). The DQE is defined as the square of the signal-to-noise ratio at the output of the detector divided by the input signal-to-noise ratio. Typically, the DQE is calculated assuming that the input is a (spatial) delta function. Consider what the relative DQEs would be for two detectors, where one detector has a demagnification ratio twice that of the first. Assume that the smaller detector has the response to single x-rays shown in figure 2a. This detector would have a high output signal/noise ratio, which would indicate a high DQE. Because the efficiency (gain) of the detector falls as the demagnification squared, the signal for the larger detector would look like that shown in figure 2b. Here the detector noise is unchanged, but the signal level is 1/4 the signal level observed with the first detector. Thus, the DQE of this detector would be much lower than that of the small detector. This analysis would lead to the conclusion that the smaller detector is better.

3. Detector Performance Considering Bragg Peaks

Now consider what happens for an input consisting of peaks riding on a background. In a crystallographic experiment the x-ray diffraction pattern has two components; (coherently diffracted) Bragg peaks, and (incoherently scattered) diffuse x-ray background. For most crystallographic studies, each Bragg peak must be identified and it's signal integrated; the x-ray background is important only to the extent that it contributes to the noise. If the x-ray beam divergence is small and the crystal is well ordered, the profile of the Bragg peaks will be independent of the crystal-to-detector distance. Since x-ray background is generated at the crystal, the background flux decreases as the square of the detector-to-crystal distance. For this discussion, assume that both detectors will be used to collect data over the same solid angle. Because the large detector has twice the width of the small detector, it can be placed at twice the crystal-to-detector distance. The input x-ray pattern to each detector is shown in the left half of figure 3. The larger detector has two important advantages based on the increased crystal-to-detector distance: 1) there is less background under each peak, and 2) there is less overlap between the peaks, allowing easier identification of the peaks and more accurate integration. However, the large detector has only 1/4 the gain of the small detector, and the pixels have 4 times the area (twice the linear dimension). These factors result in the output from each detector shown on the right side of figure 3. There are several differences between the images made with the large detector relative to images made with the small detector:

4. Detector Performance Considering the Solid Angle

Increased input area can also be used to sample a larger solid angle. Considering the same two detectors, the large detector could be moved closer to the crystal such that the x-ray background and peak overlap are still small, but a larger area of the diffraction pattern is sampled ( figure 4). A useful measure of the detector efficiency should include the actual area of the detector relative to the entire area over which data must be collected. (The detector can be thought of as having a efficiency of zero for detecting x-rays not incident on the detector.) Although it is still not clear which detector is superior, this analysis suggest that the large detector has some advantages over the small detector.

5. The XDCE

To model the performance of a detector for measuring x-ray crystallographic data, and to optimize the design of the detector and the experiment, we have developed a performance measure which we call the experimental detective collection efficiency (XDCE) (8,9). The XDCE is an extension of the DQE. The DQE is equivalent to the quantum efficiency for detecting x-rays incident on the detector. The XDCE is the efficiency of the detector for measuring the integrated intensity of all Bragg peaks, regardless of whether they are included in the detector image.

To develop the formulation of experimental detective quantum efficiency, we have defined two additional terms. The relationship of these terms is shown in figure 5. The first new term, the detective collection efficiency (DCE), is defined as the DQE times ratio of the detector area to the area required to collect all the diffracted x-ray photons to a given resolution (ideal area). This term reflects the fact that we are trying to build an `area' detector, and should therefore include a measure of the area in the performance of the detector. This term assumes that all x-rays within the defined experimental area are of equal importance, independent of whether they are incident on the face of the detector. The second additional term, the experimental detective quantum efficiency (XDQE), extends the definition of the DQE to consider the quantum efficiency of the detector for measuring the experimental quantity of interest, in this case the integrated intensity of Bragg peaks in the presence of an x-ray background noise, rather than a delta function. Combining these two measures yields the XDCE.

These performance measures compare the quality of the detector to an ideal detector. The ideal detector is defined as a detector which will add no additional noise to the data, and has physical characteristics consistent with the experimental setup. As an example of the later criteria, consider a detector for a synchrotron beam line. Because of the focus and columniation of the synchrotron beam, increasing the crystal-to-detector distance will decrease the diffuse scatter per unit area, while the size of the Bragg peaks will remain nearly constant. However, because of the physical size of the experimental area it is not possible to make an arbitrarily large detector. The idea detector would have a maximum size consistent with the constraints imposed at a real beamline.

Equations describing the integrated signal and all sources of noise can be developed when the detector configuration and experimental conditions are known, allowing the XDCE to be calculated (see reference 8). The detector configuration includes factors such as the x-ray-to-light conversion efficiency, the gain of the signal chain, the sensor size and resolution, the overall point response function, and sources of electronic noise. Experimental conditions for a x-ray crystallographic experiment include factors such as the exposure time, the crystal unit cell size, the Bragg peak profile, the diffuse x-ray background, the crystal-to-detector distance, the x-ray wavelength and the desired atomic resolution for the experiment. A computer program can then optimize detector parameters, such as the demagnification ratio or the pixel size, by maximizing the XDCE for this set of experimental conditions. Experimental conditions such as the peak integration area and the crystal-to-detector distance can also be optimized using this method.

In figure 6, the XDCE is plotted as a function of the demagnification ratio for a 3x3 unit detector constructed using 2.5cm x 2.5cm CCDs (see reference 9 for a detailed description of experimental and detector characteristics). The XDCE initially increases as an increasingly larger area of the diffraction patten is be sampled, while maintaining a sufficiently low background. At some point, the full area of the diffraction image is sampled, and the x-ray background is reduced to a level comparable to the detector (electronic) background. At this point, the XDCE starts to fall as the efficiency of the detector decreases.

6. Discussion

We have outlined the need for a performance model that allows experimental considerations to be included into the design of an area detector. The XDCE allows a prediction of the performance for any detector that is required to integrate the intensity of features on an image, providing a quantitative method for determining optimal instrumental and experimental conditions. We have used the XDCE to model the performance of a CCD detector for x-ray crystallography, optimizing the detector demagnification ratio. While examining the response of the detector to individual x-rays (the DQE) predicts that any demagnification will degrade detector performance, the XDCE demonstrates that a higher performance detector can be developed by using a significant demagnification ratio (~2-5, depending on experimental conditions). In the accompanying paper, we discuss the design of a detector for x-ray crystallography (10).

This work was supported by NSF grant DIR-9102302 and NIH grant RR06017.

7. References

[1] D. O'Mara, W.C. Phillips, M. Stanton, D. Saroff, I. Naday and E.M. Westbrook, Proc. Soc. Photo-Opt. Instr. Eng. 1656 (1992) 450.

[2] E.F. Eikenberry, M.W. Tate, A.L. Belmonte, J.L. Lowrance, D. Bilderback and S.M. Gruner, IEEE Trans. Nucl. Sci. 38 (1991) 110.

[3] W. Phillips, M. Stanton, D. O'Mara, I. Naday, and E. Westbrook, Proc. Soc. Photo-Opt. Instr. Eng. 1900 (1993) in press.

[4] M.G. Strauss, E.M. Westbrook, I. Naday, T.A. Coleman, M.L. Deacon, D.J. Travis, R.M. Sweet, J.W. Pflugrath and M. Stanton, Nucl. Instr. and Meth. A297 (1990) 275.

[5] M. Stanton, W.C. Phillips, Y. Li and K Kalata, J. Appl. Cryst. 25 (1992) 638.

[6] U.W. Arndt and D.J. Gilmore, J. Appl. Cryst. 12 (1979) 1.

[7] S.M. Gruner, J.R. Milch and G.T. Reynolds, IEEE Trans. Nucl. Sci. NS-25 (1978) 562.

[8] M. Stanton, Nucl. Instr. and Meth. A325 (1993) 550.

[9] M. Stanton, W. Phillips, D. O'Mara, I. Naday and E. Westbrook, Nucl. Instr. and Meth. A325 (1993) 558.

[10] W. Phillips, M. Stanton, D. O'Mara, Li, Y., I. Naday and E. Westbrook, Proc. Soc. Photo-Opt. Instr. Eng. 2009 (1993) submitted.