6.3. USING MONOCHROMATIC SOFTWARE FOR POLYCHROMATIC DATA PROCESSING 69
shows the distributions of X-ray energies contributing to 500, 30 and 10 strongest peaks. As can be clearly seen from the distributions, in the case of 30 strongest peaks, which would be considered as a sparse pattern but still easily indexable by most auto-indexing algorithms, the X-rays with the energies ranging from 13.3 to 15.25 keV are contributing. This corresponds to almost 13% of the mode energy. In the case of 10 strongest peaks, when even using monochromatic radiation auto-indexing algorithms would most likely fail because of too few peaks, the peaks are still sampled by the X-ray energy range∆E/Em>6%, which is in no way can be assumed monochromatic. This conclusion is further confirmed in Section 7, where the real pink-beam diffraction data is shown to exhibit behavior identical to Fig. 6.3b.
(a) (b)
13000 13500 14000 14500 15000 15500 k, eV
0.0 0.2 0.4 0.6 0.8
1.0 Spectrum
500 strongest peaks 30 strongest peaks 10 strongest peaks
Figure 6.3: (a) Simulated polychromatic lysozyme diffraction pattern, 30 strongest peaks are circled with the numbers pointing out the relative difference between the central X-ray energy contributing to the peak and 15.2 keV. (b) shows the spectrum used for the simulation and the distributions of X-ray energies sampling 500, 30 and 10 strongest peaks.
Despite the fact that the peaks in the pink-beam diffraction pattern are produced by the wide range of X-ray energies, it has been clearly demonstrated by Martin-Garciaet al. 2019 that monochromatic indexing algorithms can sometimes recover crystal orientation with the accuracy sufficient to successfully integrate reflection intensities and solve the crystal structure. However, using this approach it was only possible to recover low resolution structure because all high resolution data had to be discarded during the processing.
This can be explained as following: most indexing algorithms are based on finding the periodicity of the reciprocal lattice, which starts with projecting diffraction peaks onto the Ewald sphere to determine their reciprocal space coordinates. The peak detected at the angle2Θin the polychromatic diffraction pattern can be produced by the X-rays of any wavelength betweenλmin andλmax. It can therefore originate from any reciprocal lattice spot found on the interval between the limiting Ewald spheres shown in red on Fig. 6.4. When the monochromatic indexing algorithm is used to index polychromatic diffraction pattern, the inaccuracy of the determined reciprocal space position of each peak depends on the distance between the Ewald spheres: the positions of low resolution reflections are determined much more accurately. Given the sufficient number of low resolution reflections and omitting the high resolution reflections, as has been done by Martin-Garciaet al.2019, the monochromatic indexing algorithms can indeed be successful in determining crystal orientations from pink-beam diffraction data.
6.3. USING MONOCHROMATIC SOFTWARE FOR POLYCHROMATIC DATA PROCESSING 71
Figure 6.4: To-scale illustration of Ewald sphere construction in case of polychromatic beam withλ= 1Å,
∆λ= 0.15Å (15% bandwidth). Red, blue and green intervals show regions of the reciprocal space, where 2Å, 5Å and 10Å diffraction peaks can originate from.
In order to investigate the feasibility of this approach to polychromatic data processing, two lysozyme diffraction datasets were simulated using the typical undulator spectrum of 4.8% bandwidth (FWHM) with 15% low-energy tail and the Gaussian shaped spectrum with 3% bandwidth (FWHM), close to what can be achieved using multilayer monochromator (Fig. 6.5).
13000 13500 14000 14500 15000 15500 X-ray energy, eV
0.0 0.2 0.4 0.6 0.8 1.0
Intensity
Pink beam spectrum Gaussian spectrum
Figure 6.5: X-ray spectra used for simulation: pink beam spectrum (blue) with the peak at 15.2 keV, 4.8%
FWHM and 15% low-energy tail, and gaussian-shaped spectrum (yellow), centered at 15 keV with 3% FWHM.
Both datasets were generated using the same set of 100 random crystal orientations. Diffraction patterns were indexed withindexamajigprogram inCrystFEL, usingMOSFLM,asdfandDirAx monochro-matic indexing algorithms. Found unit cell parameters and crystal orientations were then compared to the ones used for simulation. If the found parameters were less than 5% different from the expected ones and the misalignment angle between the found and the target unit cell was less than 5◦, the pattern was considered to be indexed correctly. By careful optimization of the peak-finding parameters, i.e. the number of peaks and the high-resolution cut-off, and using the--retryoption inindexamajig1, it was possible to index 79% of diffraction patterns simulated using the undulator spectrum. In the case of the
1--retryoption successively rejects the weakest reflections and attempts indexing until the correct solution is found.
gaussian spectrum, no optimization was required asindexamajigyielded≥98% indexing fractions as long as at least 20 peaks were found in each diffraction pattern.
(a)
70 80 90
a= 80.51±0.31 ˚A
70 80 90
b= 80.60±0.37 ˚A
35.0 37.5 40.0
c= 38.20±0.20 ˚A
85 90 95
α= 90.02±0.14◦
85 90 95
β= 90.03±0.17◦
85 90 95
γ= 90.05±0.13◦
(b)
70 80 90
a= 79.70±2.24 ˚A
70 80 90
b= 80.48±2.27 ˚A
35.0 37.5 40.0
c= 38.01±0.89 ˚A
85 90 95
α= 90.29±1.54◦
85 90 95
β= 89.85±1.59◦
85 90 95
γ= 90.19±1.42◦
Figure 6.6: Distributions of the unit cell parameters obtained by indexing simulated polychromatic diffraction data using monochromatic indexing software in the case of (a) gaussian-shaped spectrum and (b) pink-beam spectrum. Dashed red lines show the target unit cell used for simulation:a=b= 80.5Å,c= 38.2Å,α=β =γ = 90◦.
The resulting distributions of the unit cell parameters and the misalignment angles are shown in Fig. 6.6 and 6.7 respectively. The indexing solutions demonstrate much higher discrepancy between the found and the target unit cell parameters in the case of the 4.8% BW pink-beam spectrum compared to 3% BW gaussian spectrum. Apart from the overall∼7 times larger standard deviation of each parameter, the mean values ofaandb, which should both be equal 80.5Å, differ by about 1% for pink-beam data while in the case of the gaussian spectrum the difference is only 0.1%. The root-mean-square unit cell misalignment angle in the case of pink-beam is as well about 7 times larger than in the case of the gaussian beam. As a consequence, although the majority of the simulated pink-beam diffraction patterns can be indexed with the monochromatic algorithms, the discrepancy in the found unit cell leads to inaccurate Bragg spot prediction and, therefore, incorrect integrated intensities, as it is illustrated in Fig. 6.8. While in the case of the gaussian-shaped spectrum (Fig. 6.8a) the predicted Bragg spot positions correspond
6.3. USING MONOCHROMATIC SOFTWARE FOR POLYCHROMATIC DATA PROCESSING 73 (a)
0 1 2 3 4 5
UC misalignment,◦ 0
10 20 30 40 50
Npatterns
(b)
0 1 2 3 4 5
UC misalignment,◦ 0
2 4 6 8 10
Npatterns
Figure 6.7: Distributions of misalignment angle between the simulated and found unit cells in case of (a) gaussian-shaped spectrum and (b) pink-beam spectrum. The root-mean-square deviation from zero is0.19◦in (a) and1.44◦in (b).
well with the found diffraction peaks, in the pink-beam diffraction pattern (Fig. 6.8b) the predicted and found spots are clearly misaligned. This misalignment is further quantified in Fig. 6.9, which shows how many diffraction patterns in each dataset contain a certain percentage of peaks predicted within the certain distance from the correct positions. Fig. 6.9b makes it clear that although 79% of pink-beam diffraction patterns could be indexed, far from all of them would provide valuable information. For example, given the peak size of 4-5 pixels, if the reflection intensities were integrated within 5 pixel radius around the predicted spot position, only 28% of patterns would have more than 90% of reflections correctly integrated in the case of the pink-beam as compared to 97% in the case the of gaussian beam.
(a) (b)
Figure 6.8: Lysozyme diffraction patterns simulated using (a) gaussian-shaped spectrum and (b) pink-beam spectrum in the same crystal orientation, both indexed by MOSFLM. Green circles show the peaks used for indexing, red squares show peak positions predicted using the found indexing solution.
(a)
10 20 30 40 50 60 70 80 90
% peaks 1
2 3 4 5 6
∆max,pixels
98 97 97 94 87 75 55 15 0 98 98 98 98 97 97 96 96 90 98 98 98 98 98 97 97 97 97 99 98 98 98 98 98 98 97 97 99 98 98 98 98 98 98 98 97 100 99 98 98 98 98 98 98 97
0 25 50 75 100
%patterns
(b)
10 20 30 40 50 60 70 80 90
% peaks 1
2 3 4 5 6
∆max,pixels
45 38 33 20 12 6 2 0 0 61 53 45 44 41 36 30 24 6 72 60 55 48 45 44 40 36 28 79 64 59 51 47 45 45 42 38 79 71 62 57 51 46 46 45 42 79 79 65 62 52 47 46 45 43
0 25 50 75 100
%patterns
Figure 6.9: Percentage of diffraction patterns, where % peaks are predicted within the accuracy of∆max
pixels in case of (a) gaussian-shaped spectrum and (b) pink-beam spectrum.
The analysis above leads to a conclusion that the approach of using monochromatic indexing algo-rithms for pink-beam diffraction data is rather limited: in order to obtain usable intensities from such indexing solutions one would have to either exclude the majority of patterns where the spots don’t overlap with the prediction, as it was done by Martin-Garciaet al. 2019 [115] and resulted in only 10% of the collected diffraction patterns being used, or develop a new prediction refinement algorithm as the current ones are only suitable for monochromatic diffraction. On the other hand, the method was proved to be perfectly suitable for the analysis of polychromatic diffraction data simulated using the gaussian-shaped spectrum with 3% bandwidth, which suggests it can be applied to process serial crystallographic data collected using the multilayer monochromator.