X-ray diffraction at Matter in Extreme Conditions endstationZ. Xing,1, a) E. Galtier,1 H.J. Lee,1 E. Granados,1 B. Arnold,1 S. Mullane,2 C. Bolme,3 A. Fry,1 P. Hart,1 andB. Nagler11)SLAC National Accelerator Laboratory, Menlo Park CA 94025, USA2)Stanford University, Stanford CA 94305, USA3)Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA

(Dated: 8 December 2015)

Understanding dynamic response at the atomic level under extreme conditions is highly sought after goal toscience frontiers studying warm dense matter, high pressure, geoscience, astrophysics, and planetary science.Thus it is of importance to determine the high pressure phases or metastable phases of material under shockcompression. In situ X-ray diffraction technique using Linear Coherent Light Source (LCLS) free electron laserX-ray is a powerful tool to record structural behavior and microstructure evolution in dense matter. Shock-induced compression and phase transitions of material lead to changes of the lattice spacing or evolution of newX-ray diffraction patterns. In this paper, we describe a standard platform dedicated for the X-ray diffractionstudies at Matter in Extreme Conditions (MEC), which can be used to reconstruct a complete diffractionpattern from multiple detectors at different positions or orientations, optimize the detector positions in atimely manner using calibration materials, extract the lattice spacing profiles and determine texture features.This platform as well as the analysis software package, ExtremeXRD, is available to the MEC user communityfor efficient diffraction measurements and real-time data analysis.

I. INTRODUCTION

The use of X-ray diffraction (XRD) analysis to obtainbasic crystal structure is unequivocally exemplary, andattempts to leverage the strength of this approach werediscussed in the literatures as early as 19331–5. Whilehigh-pressure research has proven to be an efficient toolto improve the understanding of the fundamental phys-ical properties of common materials such as salt, metal,semiconductor and mineral, the prime contribution ofthe X-ray diffraction analysis to the field of shock com-pressed condense matter physics is in structure determi-nation. Since any microscopic theory of the solid-stateproperties of a material is structure dependent, in fact,usually structure dominated, the theoretical interpreta-tion of most observed phenomena is ambiguous withoutstructure data. Such data also give the most meaning-ful information concerning chemical bonding. In par-ticular, it has been established that materials undergoa pressure-driven transformation involving a symmetryreduction6. A particularly notable application of shock-induced polymorphic transitions is in material synthesis,exemplified by production of diamonds in shock-loadedgraphite7. X-ray Free Electron Lasers (XFELs) providethe possibility of performing lattice-level measurementsof phase transitions and deformation, and melting curveson shock timescales with unprecedented clarity and de-tail using ultra bright, monochromatic, femtosecond X-ray pulses. The crystallographic state can be deter-mined along specific adiabats with the X-ray snapshotsof the samples taken at different times8. The growth ofnanocrystalline grains can also be characterized on thenanosecond timescale after the shock compression9,10.

a)Electronic mail: zxing@slac.standford.edu

Moreover, in situ measurements are able to resolve thephase transition to warm dense matter11.

There has been a few XRD platforms and data anal-ysis tools developed at other laser facilities, synchrotronsources. Kalantar et al. has developed a platform, usinglarge area detectors, at OMEGA laser facility dedicatedfor modeling and recording the diffraction patterns asa function of the Bragg and azimuthal angle for shocklattice measurements12,13. Rygg, Eggert et al. also im-plemented a method of obtaining powder diffraction dataon dynamically compressed solids at the Jupiter (LLNL)and OMEGA laser facilities (PXRDIP)14. Prescher,Prakapenka et al. specifically designed an online analysistool (DIOPTAS) for the large amount of data collectedat XRD beamlines at synchrotrons (APS)15, which per-forms data processing and exploration of two-dimensionalX-ray diffraction area detector data. It was believed to bean improvement over the Fit2D16 which has been used forhigh pressure synchrotron XRD experiments for almost20 years.

In this paper, we present a standard platform ofXRD measurement and data analysis dedicated forthe shock physics studies carried out at the MECexperiment17, where we implement a routine to recon-struct a complete diffraction pattern from multiple de-tectors at different positions or orientations, and an im-proved approach, utilizing advanced numerical imagingprocessing tools, to perspectively transform the Debye-Scherrer rings that are generated when the diffractioncones intersect the flat photographic plate of the detec-tors, which are usually tilted with respect to the FreeElectron Laser (FEL) X-ray beam at MEC. We havealso developed an optimization strategy, based on thequasi-Newton method that approximates the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm18,19, to ex-peditiously find the orientation and position of the detec-tor that are requisite for the diffraction pattern recon-

2

struction. This prompt optimization procedure improvesthe efficiency of the real-time diffraction data analysisduring the FEL X-ray beam time. Further developmenton advanced XRD techniques are possible and promisingat the facility including Rietveld refinement algorithm,structure solution, percent crystallinity, residual stress,and orientation for texture analysis.

II. EXPERIMENTAL SETUP

The LCLS X-ray20 provides revolutionary capabilitiesfor studying the transient behavior of matter in extremeconditions. The almost perfect collimated beam, Quasi-monochromatic (∆E/E = 0.4 to 0.6%) X-ray of 5− 200fs pulse duration, high photon flux on the order of 1012

photons per pulse, make it an excellent facility for suchresearch. The particular strength of the MEC instrumentmanifests itself by the uniqueness of combining the LCLSFEL X-ray beam with high-power optical laser beams, aswell as a suite of dedicated diagnostic systems such as theXRD platform described in this paper. The MEC exper-imental chamber, which is a cylindrical vacuum vesselwith a diameter of two meters, is located roughly 460meters from the end of the electron undulator, with theX-ray beam steered in using three silicon carbide coatedoffset mirrors, which reflect photons with energies up to25 keV. The beam can be focused down to few µm2, us-ing Beryllium compound refractive lenses, centered onthe interaction point at the target sample, where the op-tical drive laser and the X-ray spatially overlap. Thisreference point in the 3-dimensional (3D) space is alsodenoted as the target chamber center (TCC) as shownin Fig. 1. The optical Nd:glass laser system consists ofa continuous-wave cavity operating at a wavelength of1054 nm, with the pulse further amplified and finallyfrequency-doubled to 527 nm. The pulse shape modula-tor has a rising time of 200 ps and intrinsic time jitter of20 ps, with pulse length possible from 2 to 200 ns. Forrelatively short pulses, the beams are intensity-limitedto approximately 1.5 J ns−1, depending on the particu-lar pulse shape.

We normally provide several Cornell SLAC pixel arraydetectors21 (CSPAD) for the diffraction measurementswith some of them having smaller active sensor sizesof 400 × 400 pixels (CSPAD 140k), and some larger of825 × 830 pixels (CSPAD 550k or CSPAD Quad). Thenominal pixel size is 110 µm with the exception thatthere are two rows of pixels in the middle of the 2 × 1tile22 that are wider by a factor of 2.5. The pixels nearthe edge of each tile usually read out more noises. Thedetectors are orientated around the primary interactionpoint in a Debye-Scherrer transmission geometry23, withan example of the setup illustrated in Fig. 1. Here fourCSPAD 140k, labeled as A, B, C and D, and one CSPAD550k, Quad, are used to record the diffractions, and theoptical laser beam path is not shown specifically as it canhave different geometrical configurations with respect to

the X-ray depending on the goal of the experiment. Allof the global coordinates that are used throughout thispaper are defined with respect to TCC, which is also de-noted as the standard LCLS coordinates. The detectorscan not be positioned exactly perpendicular to the X-raybeam but maximizing the coverage of the solid angle,due to various spatial constraints near the interactionpoint from the target orientations, optics for the drivelaser and other essential diagnostics for a typical shockdriven diffraction experiment, such as the Velocity Inter-ferometer System for Any Reflector (VISAR)24 and theThomson scattering spectrometer25. This particular ge-ometry results in a perspectively distorted image of thediffraction rings on the detectors. The analysis platformcollects the images from all the detectors, performs theperspective transform, and combines them to a completediffraction pattern with all the different detector posi-tions and orientations taken into account.

QUAD

A

BC

DTCC

y (U

p-D

own)

[mm

]

z (Beam direction) [mm]x (North-South) [mm]

0-50

-100-150

0-50

50

-100

100

0

50

100

150

200

FEL X-ray beam

FIG. 1. Standard LCLS coordinate system is shown herewith the positive x direction pointing north, positive y di-rection pointing up and positive z direction pointing down-stream of the FEL beam. Five detectors, CSPAD Quad andfour CSPAD 140k, are shown for a typical experimental setup,where some of them are titled with an angle in order to max-imize the solid angle coverage.

III. RECONSTRUCTION OF THE POWDERDIFFRACTION PATTERN

In order to reconstruct and consolidate a completediffraction pattern from multiple detectors, we need toascertain, globally, the 3D coordinates, x, y and z, thusthe corresponding Bragg (2θB) and azimuthal angles (ϕ),of all the pixels under the standard LCLS coordinate sys-tem. To obtain these coordinates, we first define a localcoordinate system of xL, yL and zL within each individ-ual detector. The xL and yL are just the 2-dimensional(2D) index of the pixel from each detector multiplied bythe pixel size, and zL is constantly zero. Then the trans-form between the two Cartesian coordinate systems can

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be performed if the detector position and orientation areknown, which are specified by three global translationcoordinates, x0, y0 and z0, of a reference point on thedetector plane and three Euler rotation angles, α, β andγ, around the reference point, respectively. Here we nor-mally choose the bottom left corner of the detector asthe reference point when looking downstream along theX-ray beam direction. The α, β and γ rotation anglesfollow the convention of intrinsic rotations, x→ y′ → x′′,with y′ as the y axis after the first rotation and x′′ as thex axis after the second rotation, or extrinsic rotationsaround x→ y → x axes. They are defined as zero whenxL and yL axes spanning the detector plane are parallelto the global x and y axes. The angle α ranges fromzero to π, when rotating the detector counter-clockwise,and to −π, clockwise if looking into negative x axis. Theangles β and γ follow the same convention of definitionwhen looking into y′ and x′′ negative axes, respectively.Equation 1 shows the explicit conversion from xL, yL andzL to x, y and z, xy

z

=

c2 s2s3 c3s2s1s2 c1c3 − c2s1s3 −c1s3 − c2c3s1−c1s2 c3s1 + c1c2s3 c1c2c3 − s1s3

×

xLyL0

+

x0y0z0

, (1)

where c stands for cosine function, s stands for sine func-tion, subscript 1, 2 and 3 represent the function argu-ments, α, β and γ, and x0, y0 and z0 are the positionof the reference point on the detector. Then 2θB can

be determined as arctan(√x2 + y2/z) for z > 0, and

π − arctan(√x2 + y2/|z|) for z < 0, so that it ranges

from 0 to π, from the forward to backward scattering re-gions. The azimuthal angle, ϕ, is defined as arctan(y/x)such that it is zero at positive x axis, and π at negative xaxis if approaching counterclockwise and −π clockwise.Special care is given to the x and y projection of thediffracted photons as the detectors are neither located atthe same z nor perpendicular to the X-ray beam. Thusthe recorded Debye-Scherrer rings are distorted to ellipsesand radially discontinuous. To correct for these distor-tions, we have to extrapolate the x and y coordinates inEq. 1 to a common z = zref plane, denoted as the virtualdetector plane, which are

xv = xzrefz,

yv = yzrefz. (2)

The diffraction patterns, reinterpreted as a function ofxv and yv, then appear as ring-shaped and continuous inradius. For the rest of this paper, we will always use xvand yv in a projection and denote them simply as x andy, providing the z position of the virtual detector plane.

To obtain the detector positions and orientations, wefirst roughly measure the six parameters, x0, y0, z0, α,

β and γ, inside the experimental chamber with relativelylarge uncertainties. These measurements provide a ini-tial “guess” for the fitter that we are going to use foroptimizing the positioning of the detectors. The metricor χ2 function of this minimization algorithm is built bysimply comparing the reconstructed diffraction peaks ofsome standard reference materials (SRM), such as LaB6

that has very narrow diffraction rings, with the expected2θB calculated from the SRM lattice parameters at acertain X-ray photon energy. While we have tried twodifferent methods to minimize this χ2 function, the firstone is just a simple grid search that scans over a givenmulti-dimensional grid consisting of the six parameters,x0, y0, z0, α, β and γ, and then locate the grid pointwhere the minimum value is achieved. Although thismethod does not require any user interactive input, itis very time consuming thus inefficient for the real-timeanalysis, depending on the given initial detector positionand the size of the grid that determines the resolution ofthe measurement. Therefore it will not be discussed anymore in this paper. Instead, we have developed a sec-ond optimization method based on the gradient descentsearch. This is a quasi-Newton method that approxi-mates the Broyden–Fletcher–Goldfarb–Shanno (BFGS)algorithm18,19, where some user interactive input isneeded to select a few reference points on the raw diffrac-tion rings within the actual detector. To be more specific,the multivariate objective or χ2 function that needs to beminimized is formulated as

fobj(x0, y0, z0, α, β, γ) =∑i

arcsin(

√x2i + y2izi

)

−2θexpectedi

2

, (3)

which is integrated over all reference points denoted by

the index i. Here 2θexpectedi stands for the expected Braggangle at the reference point, and global coordinates xi,yi and zi can be written as

xi = xiL cos(β) + yiL sin(β) sin(γ) + x0,

yi = xiL sin(α) sin(β) + yiL[cos(α) cos(γ)− cos(β) sin(α)

sin(γ)] + y0,

zi = −xiL cos(α) sin(β) + yiL[cos(γ) sin(α) + cos(α) cos(β)

sin(γ)] + z0, (4)

with xiL and yiL as the local coordinates of the ref-

erence point. Taking xiL, yiL and 2θexpectedi as con-stant coefficients, we can then analytically solve forthe search direction, thus provide both the objectivefunction, fobj(x0, y0, z0, α, β, γ), and gradient function,~∇fobj, to the BFGS algorithm for the optimal solution.This approach is greatly improved in terms of the per-formance and computation time26. In consequence, itis used as the baseline fitting routine for the XRD dataanalysis platform at MEC. Figure 2 shows a complete re-constructed diffraction pattern of the calibration sample,

4

LaB6, after applying this optimization procedure, wherethe spotty diffraction rings are caused by the large grainsize.

(a)

(b)

FIG. 2. After applying the optimization procedure, the re-constructed diffraction pattern of LaB6, integrated from fiveCSPAD detectors, are shown in (a) as the x-y projection atz = 100.54 mm with a bin size of 1.11 mm and (b) as the2θB-ϕ projection with bin size of 0.09 for 2θB and 1 for ϕ.Here the dashed green lines indicate the calculated 2θB forLaB6 at a photon energy of 8 keV and the intensity of imagestands for the photon counts.

IV. IMPROVED PERSPECTIVE TRANSFORM OF THEDIFFRACTION IMAGE

The perspective transformation is any method ofmapping three-dimensional points to a two-dimensionalplane. As most of the current methods for displayinggraphical data are based on planar two-dimensional me-dia such as a pixel detector, the use of this 3D projectionis widespread, especially in computer graphics, engineer-ing and drafting. This projection can be done basicallyin two different ways, direct mapping and reverse map-ping. Direct mapping projects the photon counts fromthe actual pixel of the detector (source) to a virtual pixelin a 2D plane (destination) for visualization as shown inFig. 3 (a), which can be implemented in a timely mannerand easily conserves the total number of photon counts.

However, there can be sampling artifacts using this ap-proach. When the pixel size of the destination imageis small, many pixels on this virtual plane have no realmapping from the source, this results in empty pixels onthe image. If the pixel size is chosen to be too large, therecould be pixels that are linked to multiple sources, thiscauses artificial over-counted pixels. To avoid these sam-pling artifacts, the mapping can be done in the reversedorder from destination to the source as shown in Fig. 3(b). That is, for each pixel (xd, yd) of the destinationimage, we compute the coordinates of the correspondingdonor pixel in the source,

dst(xd, yd)←→ src(fx(xd, yd), fy(xd, yd)), (5)

where in this case xd, yd, fx(xd, yd) and fy(xd, yd) are justthe pixel indices. When fx(xd, yd) or fy(xd, yd) are com-puted as a fractional pixel coordinate, numerical inter-polation methods, such as fitting a polynomial functionto the neighborhood of the source and taking the valueof the function at the reversely mapped point as the des-tination pixel content, are used to smooth out the gapson the projected image. If fx(xd, yd) or fy(xd, yd) falloutside of the source image, extrapolation methods areused to obtain the photon counts for these non-existingpixels. We use the same principle of reverse mapping toproject the raw detector image, as a function of x and y,to a virtual image, that can be either as a function of xand y or 2θB and ϕ.

We use the OpenCV27 package to implement this re-verse mapping, where we provide the one-to-one links be-tween all the source pixels and destination pixels and theOpenCV performs the numerical interpolations, extrap-olations and generates the projected image. However,one thing that the OpenCV does not take into accountin this transform is the conservation of the total photoncount. Thus we need to calculate the determinant of theJacobian matrix explicitly, and apply this correction tothe images that OpenCV generates28. The improvementachieved by the reverse mapping is shown in Fig. 4, wherediffraction patterns of CeO2 are generated in the 2θB andϕ representation using the two approaches. The reversemapping method gives a more uniformly distributed pro-jection as a function of 2θB and ϕ without any artificialpatterns or gaps, which helps in scrutinizing detailed tex-ture features when the sample material is under high tem-perature and pressure. Here the X-ray photon energy is9.5 keV.

V. THE 2θB OR Q-POSITION PROFILES

We can obtain the 2θB profile simply by projectingthe 2θB–ϕ image onto this axis. Here we renormalizethe raw intensity, Iraw, on the transformed image by afactor of 1/ sin(2θB) so that the pixel-to-pixel intensity

5

virtual detector plane (destination)

actual detector plane (source)

x [mm]

x [mm]

virtual detector plane (destination)

actual detector plane (source)

z [mm]

y [m

m]

z [mm]

y [m

m]

(a)

(b)

FIG. 3. Direct (a) and reverse (b) mappings between thesource and destination image are shown here. For the directmapping, the artifacts on the virtual plane are indicated bythe empty pixels that are not filled with any dashed yellowlines.

is a function of the solid angle dΩ, instead of d(2θB)dϕ,∫I(Ω)dΩ =

∫Iraw(2θB , ϕ)d(2θB)dϕ. (6)

The number of photons, Np, that is renormalized to aunit degree of ϕ as a function of 2θB can be computedby integrating I(Ω) over ϕ and dividing it by the detectoracceptance along ϕ. We can also convert the 2θB to theQ-position in reciprocal space using

Q =4π sin(θB)

λ, (7)

where the profile of Q-position is then independent onthe X-ray photon energy. The renormalized Q-positionprofiles of the calibration samples at ambient conditionsare shown in Fig. 5 for two experiments that are car-ried out at MEC, where the X-ray photon energies are8.2 keV and 8 keV, respectively. For the first experi-ment, CeO2 is used as the calibration sample, which givesthe 200, 220, 311, 222 and 400 reflections onthe recorded diffraction patterns, and for the second ex-periment, LaB6 is used as the calibration sample which

(a)

(b)

FIG. 4. The diffraction patterns of CeO2 are shown in (a) us-ing the direct mapping approach and (b) using the reversemapping approach. The green dashed lines illustrate theexpected 2θB positions for the 220, 311, 222, 400,331, 420 and 422 reflection planes at X-ray photon en-ergy of 9.5 keV. Here the intensity of image stands for thephoton counts.

generates the 100, 110, 111, 200, 210, 211,220 and 221 reflections within the acceptance of allthe detectors. Here we show the reconstructed profiles us-ing both direct and reverse mapping approach which giveconsistent results. The measured positions for all of thesereflections are consistent with the expected ones that arecalculated using the lattice parameters, a = 5.405 A forCeO2 and a = 4.15691 A for LaB6

29.

We have investigated the instrumental broadening ofthe diffraction profile by measuring the Full Width atHalf Maximum (FWHM) of various reconstructed diffrac-tion peaks in 2θB of NIST LaB6. Here the FWHM areextracted by simply fitting a Gaussian signal and linearbackground function to the data, as illustrated in Fig. 5for the 110 diffraction line, integrated over the full ac-

ceptance in ϕ, and FWHM = 2√

2 ln 2σ. It was reportedin the NIST certificate29 that the broadening caused bythe particle size of the LaB6 is notably small (±0.002

on 2θB), thus the measured FWHM is just dominatedby the overall instrumental broadening that can containcontributions from multiple sources such as the intrinsicX-ray beam profile, optimization of the detector geome-

6

SRM (LaB 6)

(a)

(b)

SRM (CeO 2)

]° [Bθ230 30.5 31

Ω/d p

N

0

2000

4000

6000

8000

10000

12000

110

FIG. 5. The renormalized Q-position profiles of the calibra-tion samples at ambient conditions are shown for (a) the firstexperiment and (b) the second experiment, using both direct(blue circle) and reverse (orange square) mapping approach.The red lines indicate the expected positions of the reflectionscalculated from the lattice parameters provided by NIST. Wealso show the 2θB distribution and fitting functions for the110 diffraction peak, where the red dotted line stands forthe signal Gaussian function and black dashed line the linearbackground function.

tries, etc. The measured FWHM, integrated over ϕ, isshown as a function of 2θB in Fig. 6, where the over-all broadening is about 0.2 in the regime from 20 to70. The level of broadening we have measured is similarto those instrument profile corrections reported in crys-tallography studies30. We also estimate the broadeningcontribution from the detector geometry uncertainties byselecting five small regions of interest along the 110diffraction line at different ϕ from approximately −180

to 100, and a similar fit is used to obtain the Gaus-sian mean and sigma of the various 2θB distributions.We fit these Gaussian mean of 2θB , alternatively, withboth a linear function and a constant, and the maximumdeviation between the two assumptions is taken as thebroadening component caused by the detector geometryvariation, which is about 18% of total measured FWHMintegrated over ϕ.

]°[Bθ220 40 60

]°FW

HM

[

0

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 6. The measured FWHM in 2θB , integrated over ϕ,is shown as a function of 2θB which is obtained from thediffraction lines of NIST LaB6.

VI. CONCLUSIONS

A platform of XRD diagnostic as well as the analysistool is implemented, which permits efficient data collec-tion during FEL X-ray beam time and real-time lattice-level analysis for the high pressure physic experimentsperformed at MEC. We have improved the techniqueof reconstructing the Debye-Scherrer diffraction rings,where a prompt optimization routine is developed toobtain, precisely, detector geometries and an integrateddiffraction pattern from multiple detectors. Advancednumerical image processing tools are also utilized to re-move sampling artifacts and improve the uniformity ofthe perspectively transformed diffraction images. Wehave already successfully used this platform to calibratethe diffraction patterns using NIST SRM materials, in-vestigate the instrumental broadening, study phase tran-sitions, lattice extensions and melting models of variousmaterials at MEC. This platform as well as the analysissoftware tool is available to the MEC user community31.

APPENDIX

The perspective transform, used in OpenCV, can bewritten as, txdtyd

t

= M

xsys1

=

M11 M12 M13

M21 M22 M23

M31 M32 M33

× xsys

1

, (8)

where M is the transform matrix, xs and ys are the sourceimage pixel index, xd and yd are the destination imagepixel index and t is a renormalization factor. Here thetransform is performed between 2D images whereas a 3Dtransform formalism is used to linearize the projection.We calculate the determinant of the Jacobian matrix for

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this transform as∣∣∣∣∂(xs, ys)

∂(xd, yd)

∣∣∣∣ = (−M11M22M33 +M11M23M32 +M12M21

M33 −M12M23M31 −M13M21M32 +M13

M22M31)2[M11M22 −M12M21 + (M12M31

−M11M32)yd + (M21M32 −M22M31)xd]−3,(9)

where

xd =M11xs +M12ys +M13

M31xs +M32ys +M33,

yd =M21xs +M22ys +M23

M31xs +M32ys +M33. (10)

To obtain the perspectively transformed image fromOpenCV, we just need to specify the mappings betweenfour pixels on the actual detected image and on the vir-tual image. This can be achieved easily by using thetransformation between the local and global coordinatesas shown in Eq. 1 and 2. We use a similar perspectivetransform function provided by the OpenCV to obtainthe projected image as a function of 2θB and ϕ. Thenagain we need to correct for the Jacobian function in thistransform which is∣∣∣∣ ∂(xd, yd)

∂(2θd, ϕd)

∣∣∣∣ = z2d∆Γsin(2θB)

cos3(2θB), (11)

where xd, yd, 2θd and ϕd are all in units of pixels, zd isthe z position of the virtual detector plane divided by thepixel size on the x and y axis, ∆ and Γ are the pixel sizeon 2θB and ϕ axis in units of radians and 2θB is the justthe diffraction angle for the corresponding virtual pixelin units of radians.

ACKNOWLEDGMENTS

We would like to thank Ray Smith (Lawrence Liv-ermore National Laboratory) and Roberto Alonso-Mori(SLAC National Accelerator Laboratory) for sharingtheir calibration data. We also thank Arianna Gleason(Stanford University) for all the discussions on the in-strumental broadening. This research was carried out atthe Linac Coherent Light Source (LCLS) at the SLACNational Accelerator Laboratory. LCLS is an Office ofScience User Facility operated for the US Departmentof Energy Office of Science by Stanford University. TheMEC instrument has additional support from the DOE

Office of Science, Office of Fusion Energy Sciences undercontract No. SF00515.

1F. C. Todd, The Physical Review 44 (1933).2B. Cullity, Elements of X-ray Diffraction (Addison-Wesley,1956).

3G. H. Stout and L. H. Jensen, X-Ray Structure Determination:A Practical Guide, 2nd Edition (Wiley-Interscience, 1990).

4V. E. Buhrke et al., A Practical Guide for the Preparation ofSpecimens for X-Ray Fluorescence and X-Ray Diffraction Anal-ysis (Wiley, 1998).

5V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Pow-der Diffraction and Structural Characterization of Materials(Kluwer Academic, 2003).

6G. E. Duvall and R. A. Graham, Rev. Mod. Phys. 49, 523 (1977).7P. S. DeCarli and J. C. Jamieson, Science 9, 1821 (1961).8M. Gorman et al., Physical Review Letters 115, 095701 (2015).9D. McGonegle, A. Higginbotham, et al., 18th APS-SCCM and24th AIRAPT Journal of Physics: Conference Series 500, 112063(2014).

10A. E. Gleason et al., Nature Communications 6, 8191 (2015).11L. B. Fletcher et al., Nature Photonics 9, 274 (2015).12D. H. Kalantar, E. Bringa, et al., Rev. Sci. Instrum. 74, 1929

(2003).13D. H. Kalantar, J. Belak, et al., Physics of Plasmas 10, 1569

(2003).14J. R. Rygg, J. H. Eggert, et al., Rev. Sci. Instrum. 83, 113904

(2012).15C. Prescher and V. Prakapenka, High Pressure Research 35, 223

(2015).16A. P. Hammersley, S. O. Svensson, et al., High Pressure Research14, 235 (1996).

17B. Nagler et al., J. Synchrotron Rad. 22, 520 (2015).18R. H. Byrd et al., SIAM Journal on Scientific and Statistical

Computing 16, 1190 (1995).19C. Zhu et al., ACM Transactions on Mathematical Software 23,

550 (1997).20P. Emma et al., Nat. Photon. 4, 641 (2010).21P. Hart et al., Proc. SPIE, X-Ray Free-Electron Lasers: Beam Di-

agnostics, Beamline Instrumentation, and Applications, 85040C8504 (2012), 10.1117/12.930924.

22CSPAD 140k contains two 2×1 tiles while CSPAD 550k containseight.

23P. Debye, Annalen der Physik 351 (6), 809 (1915).24L. M. Barker, Experimental Mechanics 12 (5), 209 (1972).25H. J. Lee et al., Phy. Rev. Lett. 102, 115001 (2009).26Generally it takes less than few seconds of CPU time to find the

optimized detection positions, given an average initial guess anda few interactively selected reference points on the diffractionrings.

27G. Bradski, Dr. Dobb’s Journal of Software Tools (2000).28See the appendix for the formalisms of the transform and calcu-

lation of the Jacobian function.29D. Black et al., National Institute of Standards and Technology,

U.S. Department of Commerce: Gaithersburg, MD 54 (2010).30S. Pratapa et al., Journal of Applied Crystallography 35, 155

(2002).31“ExtremeXRD: MEC XRD platform,” (2015), available

at: https://portal.slac.stanford.edu/sites/lcls_public/

instruments/mec/Pages/xrd.aspx.