Probing Dynamics in Colloidal Crystals with Pump-Probe Experiments at LCLS: Methodology and Analysis

applied sciences

Article Probing Dynamics in Colloidal Crystals with Pump-Probe Experiments at LCLS: Methodology and Analysis

Nastasia Mukharamova 1, Sergey Lazarev 1,2, Janne-Mieke Meijer 3,9, Matthieu Chollet 4, Andrej Singer 5, Ruslan P. Kurta 1,10, Dmitry Dzhigaev 1, Oleg Yu. Gorobtsov 1, Garth Williams 4,11, Diling Zhu 4, Yiping Feng 4, Marcin Sikorski 4,10, Sanghoon Song 4, Anatoly G. Shabalin 1,12, Tatiana Gurieva 1, Elena A. Sulyanova 6, Oleksandr M. Yefanov 7 and Ivan A. Vartanyants 1,8,*

1 Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg, Germany; [email protected] (N.M.); [email protected] (S.L.); [email protected] (R.P.K.); [email protected] (D.D.); [email protected] (O.Y.G.); [email protected] (A.G.S.); [email protected] (T.G.) 2 National Research Tomsk Polytechnic University (TPU), Pr. Lenina 30, 634050 Tomsk, Russia 3 Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterials Science, Utrecht University, Padualaan 8, 3508 TB Utrecht, The Netherlands; [email protected] 4 SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025, USA; [email protected] (M.C.); [email protected] (G.W.); [email protected] (D.Z.); [email protected] (Y.F.); [email protected] (M.S.); [email protected] (S.S.) 5 UC San Diego, 9500 Gilman Dr., La Jolla, CA 92093, USA; [email protected] 6 Shubnikov Institute of Crystallography RAS, Leninskii Pr. 59, 119333 Moscow, Russia; [email protected] 7 Center for Free-Electron Laser Science, DESY, Notkestraße 85, D-22607 Hamburg, Germany; [email protected] 8 National Research Nuclear University MEPhI, Kashirskoye Ch. 31, 115409 Moscow, Russia 9 Present address: Department of Physics, University of Konstanz, D-78457 Konstanz, Germany 10 Present address: European XFEL GmbH, Holzkoppel 4, D-22869 Schenefeld, Germany 11 Present address: NSLS-II, Brookhaven National Laboratory, Upton, NY 11973-5000, USA 12 Present address: UC San Diego, 9500 Gilman Dr., La Jolla, CA 92093, USA * Correspondence: [email protected]; Tel.: +49-040-8998-2653

Academic Editor: Kiyoshi Ueda Received: 4 April 2017; Accepted: 8 May 2017; Published: 19 May 2017

Abstract: We present results of the studies of dynamics in colloidal crystals performed by pump-probe experiments using an X-ray free-electron laser (XFEL). Colloidal crystals were pumped with an infrared laser at a wavelength of 800 nm with varying power and probed by XFEL pulses at an energy of 8 keV with a time delay up to 1000 ps. The positions of the Bragg peaks, and their radial and azimuthal widths were analyzed as a function of the time delay. The spectral analysis of the data did not reveal signiﬁcant enhancement of frequencies expected in this experiment. This allowed us to conclude that the amplitude of vibrational modes excited in colloidal crystals was less than the systematic error caused by the noise level.

Keywords: free-electron laser; pump-probe; colloidal crystal; dynamics

1. Introduction Acoustic motion in nanoscale objects induced by light is of importance for both fundamental science and applications. This interest is due to various potential applications in acousto-optical devices

Appl. Sci. 2017, 7, 519; doi:10.3390/app7050519 www.mdpi.com/journal/applsci Appl. Sci. 2017, 7, 519 2 of 13 which can be used for ultrafast manipulation and control of electromagnetic waves by hypersonic (GHz) acoustic waves [1]. Colloidal crystals, formed by the self-assembly method, have shown phononic band gaps in the GHz frequency range [2–7]. Due to the recent progress in the fabrication of high-quality colloids, this provides an opportunity to produce inexpensive phononic crystals. Dynamics in submicrometer colloidal crystals was investigated using different techniques such as Raman scattering [8], Brillouin light scattering [2], and optical pump-probe spectroscopy [4–7]. However, these methods have limited spatial resolution and are not suitable for the observation of the detailed structure of the sample. Furthermore, for some materials obtaining a sufﬁcient refractive index is a challenging task, if not impossible. At the same time, newly-developed X-ray free-electron lasers (XFELs) produce X-rays with unprecedented brightness, pulse duration and coherence, which are ideal for performing time-resolved experiments on various materials with a time resolution that outperforms that of third-generation synchrotron sources [9–11]. Moreover, the femtosecond X-ray pulses from free-electron laser sources allow an investigation of transient states of time-dependent processes [12–17]. In this respect, XFELs are especially well suited for investigating ultrafast structural dynamics of colloidal crystals. The vibrations of the colloidal crystals were recently measured at free-electron laser FLASH in Hamburg [18]. The polystyrene colloidal crystals were pumped by an infrared (IR) laser of 800 nm and probed by an X-ray pulse with a wavelength of 8 nm. This pump-probe experiment indicated the possibility of observation of the dynamics in the colloidal crystals by diffraction with a FEL. To give a reliable interpretation of the observed results, it was necessary to investigate the dynamics with a larger time interval and better statistics. Here we present the results of a time-resolved pump-probe diffraction experiment in which the ultrafast dynamics in colloidal crystals induced by an IR laser was investigated. A theoretical analysis of the vibrations of the colloidal crystals was performed, and the vibrational frequency was calculated. This analysis was applied to the experimental results.

2. Experiment The pump-probe experiments were performed at the Linac Coherent Light Source (LCLS) [9] in Stanford, USA at the X-ray Pump Probe (XPP) beamline [19]. LCLS was operated in the Self Amplified Spontaneous Emission (SASE) mode [20]. A sketch of the experiment is shown in Figure1. A double-crystal diamond (111) monochromator with the thicknesses of the monochromator crystals of 100 µm and 300 µm, enabled splitting of the primary X-ray beam into a pink (transmitted) and a monochromatic (reflected) branches and simultaneous operation of two beamlines (see Figure1)[ 21]. We employed the monochromatic regime of LCLS with the energy of a single XFEL pulse of 8 keV (1.5498 Å), energy bandwidth of 4.3 × 10−5 [22], and a pulse duration of about 50 fs as provided by electron bunch measurements at a repetition rate of 120 Hz. The X-ray beam was focused using the compound refractive lenses (CRLs) on the sample down to 50 µm full width at half maximum (FWHM). The photon flux of the beam at the sample position was about 109 ph/pulse. This allowed us to perform measurements in a non-destructive regime. In the experiment, the colloidal crystal was oriented vertically, with its surface perpendicular to the incoming XFEL pulse in the transmission diffraction geometry. Colloidal crystal films were prepared using the vertical deposition method [23] from spherical colloids consisting of two different materials, either polystyrene (PS) or silica. The crystalline films consisted of 30–40 particle layers, depending on the position on the film along the growth direction (the number of crystal layers increases at later growth times). After crystal growth, some of the films were covered with a 20–50 nm thick aluminum layer to increase interaction with the IR laser. In total, three types of samples were investigated: PS colloidal crystals, PS colloidal crystals covered by Al, and silica crystals, with different sphere diameters of 420 ± 9 nm, 376 ± 8 nm and 238 ± 7 nm, respectively. Due to the growing method, the dried colloidal crystal films used in our experiment contain cracks Appl. Sci. 2017, 7, 519 3 of 13

betweenAppl. Sci. 2017 single-crystal, 7, 519 regions of about 50 µm in size. Microscopic images of these samples are shown3 of 13 in Figure2. Appl. Sci. 2017, 7, 519 3 of 13

Figure 1. The scheme of the pump-probe experiment. X-ray free-electron laser (XFEL) pulses are FigureFigure 1.1. TheThe schemescheme ofof thethepump-probe pump-probeexperiment. experiment. X-rayX-ray free-electronfree-electron laserlaser (XFEL)(XFEL) pulsespulses areare generated by the undulator and reflected by the first diamond crystal. The second diamond crystal generatedgenerated byby thethe undulatorundulator andand reflectedreflected by by the the first first diamond diamond crystal. crystal. TheThe second second diamond diamond crystal crystal reflects the beam in the direction to the sample position. The beam is focused by the compound reflects the beam in the direction to the sample position. The beam is focused by the compound reflectsrefractive the beam lenses in (CRLs) the direction to the size to of the 50 sample µm at the position. sample position. The beam The is Cornell-SLAC focused by Pixel the compound Array refractive lenses (CRLs) to the size of 50 µm at the sample position. The Cornell-SLAC Pixel Array refractiveDetector lenses (CSPAD) (CRLs) is positioned to the size 10 of m 50downstreamµm at the from sample the colloidal position. sample. The Cornell-SLAC Pixel Array DetectorDetector (CSPAD) (CSPAD) is is positioned positioned 10 10 m m downstream downstream from from the the colloidal colloidal sample. sample.

Figure 2. Microscopic images of colloidal crystals made from (a) polystyrene (PS), (b) PS covered with Figurean2. AlMicroscopic layer, and (c) images silica samples of colloidal measured crystals at Linac made Coherent from ( aLight) polystyrene Source (LCLS). (PS); (b) PS covered with Figure 2. Microscopic images of colloidal crystals made from (a) polystyrene (PS), (b) PS covered with an Al layer; and (c) silica samples measured at Linac Coherent Light Source (LCLS). an AlSeries layer, of and X-ray (c) silicadiffraction samples imag measuredes were recordedat Linac Coherent using the Light Cornell-SLAC Source (LCLS). Pixel Array Detector (CSPAD) megapixel X-ray detector [24], positioned 10 m downstream from the sample, that is SeriescomprisedSeries ofof X-ray X-rayof 32 diffractionsilicondiffraction sensors imagesimag withes the werewere pixel recordedrecorded size of 110 using using × 110the the µm Cornell-SLAC Cornell-SLAC2 covering an area PixelPixel approximately ArrayArray DetectorDetector (CSPAD)(CSPAD)17 × 17 megapixelmegapixel cm2. This experimental X-rayX-ray detectordetector arrangement [24[24],], positionedpositioned provided 10resolution10 mm downstreamdownstream of 0.5 µm−1 fromperfrom pixel thethe in sample,sample, reciprocal thatthat isis 2 comprisedcomprisedspace. ofTheof 32 32 2.3 silicon silicon megapixels sensors sensors ofwith witheth detector the the pixel pixel were size size read of of 110 110out× ×at 110 110120µ µmHz,m 2 coveringencodedcovering at anan 14 areaarea bits approximately approximatelyper pixel. 2 2 −1 1717× × In1717 ordercmcm2.. ThisThisto prevent experimentalexperimental damage arrangementofarrangement the detector, providedprovided a 5 × 5 mm resolutionresolution Al beamstop of of 0.5 0.5 wasµ µmm placed−1 perper pixelpixelin front inin reciprocal ofreciprocal it. To reduce air scattering, a 10 m long flight tube filled with He was mounted between the sample and space.space. The The 2.3 2.3 megapixels megapixels of theof the detector detector were were read read out at out 120 at Hz, 120 encoded Hz, encoded at 14 bits at per14 pixel.bits per In orderpixel. the detector. 2 toIn prevent order to damage prevent of damage the detector, of the a 5detector,× 5 mm a Al5 × beamstop 5 mm2 Al was beamstop placed in was front placed of it. Toin reducefront of air it. A Ti:sapphire IR laser was used to excite the colloidal crystal film. The pump pulses were scattering, a 10 m long flight tube filled with He was mounted between the sample and the detector. To reducegenerated air scattering,with a wavelength a 10 m λlong = 800 flight nm (1.5497tube filled eV) and with duration He was about mounted 50 fs (FWHM).between Thethe samplesize of and the detector.Athe Ti:sapphire laser footprint IR on laser the sample was used was to100 excite µm (FWHM) the colloidal and, thus, crystal twicefilm. larger Thein comparison pump pulses to thewere generatedX-rayA Ti:sapphire withbeam. a The wavelength IRcorresponding laser λwas= 800 intensityused nm to (1.5497 ofexcite the eV) laser the and pulsescolloidal duration was crystalon about the order 50film. fs of (FWHM).The 1013 pumpW/cm The2 ”pulses The size IR of were the lasergeneratedlaser footprint pulses with on werea thewavelength propagating sample was λ =alon 100 800gµ thenmm (FWHM) XFEL(1.5497 pulses eV) and, andand thus, were duration twice synchronizedlarger about in50 to comparison fsthe (FWHM). pulses from to The the the size X-ray of beam.the laserXFEL The footprint with corresponding less thanon the 0.5 intensitysampleps jitter. was of the100 laser µm (FWHM) pulses was and, on thus, the order twice of larger 1013 W/cm in comparison2” The IR to laser the pulsesX-ray werebeam.The propagating IRThe laser corresponding energy along was thecontrolled intensity XFEL pulsesby of rotation the andlaser of were pulsesthe optical synchronized was axis on ofthe a towaveplateorder the pulsesof 10 (see13 fromW/cm Figure the2” 1), The XFEL IR withlaser lessand pulses was than werecalibrated 0.5 ps propagating jitter. by power alonsensorg theat the XFEL position pulses of the and sample. were Thesynchronized measured calibration to the pulses curve from is the presented in Figure 3. Zero degrees of waveplate angle corresponds to the minimum and 10 degrees XFELThe with IR less laser than energy 0.5 ps was jitter. controlled by rotation of the optical axis of a waveplate (see Figure1), correspond to the maximum calibrated energy and intensity of the IR laser. and wasThe calibrated IR laser energy by power was sensorcontrolled at the by position rotation of of the the sample. optical Theaxis measuredof a waveplate calibration (see Figure curve 1), is A series of pump-probe experiments were performed with the variation of the IR laser energy presented in Figure3. Zero degrees of waveplate angle corresponds to the minimum and 10 degrees and fromwas calibrated195 µJ to 691 by µJ power (3.2 × 10sensor13–1.1 ×at 10 the14 W/cm position2) for ofPS the and sample. silica samples The measured and from 195calibration µJ to 326 curveµJ is correspondpresented(3.2 × 10 in to13 –5.4Figure the × maximum 10 3.13 W/cmZero 2degrees) calibrated for PS films of waveplat energy covered and bye angle Al. intensity Measurements corresponds of the IR with to laser. the different minimum IR laser and energies 10 degrees correspondAwere series performed to of the pump-probe maximum at the new experimentscalibrated position of en the wereergy sample and performed tointensity avoid withsample of the the damage.IR variation laser. The of upper the IR limit laser of energyIR 13 14 2 fromlaserA 195 series µenergiesJ to of 691 pump-probe wasµJ (3.2set due× 10 experimentsto –1.1the damage× 10 were W/cmthreshol performedd) forof the PS with colloidal and the silica variationcrystals samples and of and thewas fromIRlower laser 195 for energyµ J to from 195 µJ to 691 µJ (3.2 × 1013–1.1 × 1014 W/cm2) for PS and silica samples and from 195 µJ to 326 µJ (3.2 × 1013–5.4 × 1013 W/cm2) for PS films covered by Al. Measurements with different IR laser energies were performed at the new position of the sample to avoid sample damage. The upper limit of IR laser energies was set due to the damage threshold of the colloidal crystals and was lower for Appl. Sci. 2017, 7, 519 4 of 13

326 µJ (3.2 × 1013–5.4 × 1013 W/cm2) for PS ﬁlms covered by Al. Measurements with different IR laser energies were performed at the new position of the sample to avoid sample damage. The upper limitAppl. of IRSci. laser2017, 7 energies, 519 was set due to the damage threshold of the colloidal crystals and was4 of 13 lower for Al-covered samples due to enhanced absorption of IR light by the Al ﬁlm. The results for the Al-coveredAppl. Sci. 2017 ,samples 7, 519 due to enhanced absorption of IR light by the Al film. The results for the highest4 of 13 highest non-destructive IR laser energies, which are marked with arrows on the calibration curve, non-destructive IR laser energies, which are marked with arrows on the calibration curve, are are presentedpresentedAl-covered in samples this work. work. due At to At enhancedenergies energies higher absorption higher than than 1of mJ, 1IR mJ, lightultrafast ultrafast by the melting Al melting film. of Thethe of the colloidalresults colloidal for crystals the crystals highest was was observedobservednon-destructive and and was was investigatedIR investigated laser energies, in in a a separate separatewhich are workwork marked [[25].25]. with arrows on the calibration curve, are presented in this work. At energies higher than 1 mJ, ultrafast melting of the colloidal crystals was observed and was investigated in a separate work [25].

FigureFigure 3. The 3. The infrared infrared (IR) (IR) laser laser energy energy calibration calibration curve. curve. TheThe maximummaximum of the the IR IR laser laser energy energy used used in

the experimentsin the experiments for three for three samples samples is marked is marked with with arrows. arrows. For For the the PS PS colloidal colloidal crystalcrystal covered with with Al AlFigure layer, 3. theThe maximum infrared (IR) IR laser laser energy energy was calibration 326 µJ, ancudrve. for The the maximumpure PS and of silica the IR samples, laser energy 691 µJ. used layer, the maximum IR laser energy was 326 µJ, and for the pure PS and silica samples, 691 µJ. in the experiments for three samples is marked with arrows. For the PS colloidal crystal covered with TheAl layer, pump-probe the maximum experiment IR laser wasenergy perf wasormed 326 µJ, with an da fortime the delay pure variationPS and silica τ from samples, −50 ps691 to µJ. +1000 ps, withThe a pump-probe 10 ps time increment. experiment This time was delay performed region withand the a timetime interval delay variation were consideredτ from in− order50 ps to +1000to ps,resolveThe with pump-probe GHz a 10 frequency ps time experiment increment. dynamics was of This perf colloidalormed time delay crystawith a regionls. time In orderdelay and to thevariation obtain time τ intervalsufficient from −50 were statistics ps to considered +1000 of theps, in ordermeasuredwith to resolvea 10 psdata, time GHz for increment. frequency each time This dynamicsdelay time a large delay of colloidalnumber region ofand crystals. diffraction the time In orderintervalpatterns to wereobtain(600 forconsidered sufficient polystyrene in statistics order and of the measured120to resolve for silica) GHz data, with frequency for and each without timedynamics delay the IRof a lasercolloidal large were number crysta measured.ls. of In diffraction order Typical to obtain patternssingle -shotsufficient (600 diffraction for statistics polystyrene patterns of the and 120 formeasured three silica) different data, with for and samples each without time are delay theshown IR a laserinlarge Figure werenumber 4. measured.Th ofe six-fold diffraction Typicalsymmetry patterns single-shot of (600the diffractionfor diffractionpolystyrene patternpatterns and is 120 for silica) with and without the IR laser were measured. Typical single-shot diffraction patterns for threedue to different the hexagonal samples close-packed are shown structure in Figure of 4the. The colloidal six-fold crystals. symmetry Two orders of the of diffraction Bragg peaks pattern can is befor seen three in different single diffraction samples are patterns shown shown in Figure in Figure 4. The 4.six-fold A family symmetry of 11.0 Bragg of the reflections,diffraction patternindicated is due to the hexagonal close-packed structure of the colloidal crystals. Two orders of Bragg peaks can indue Figure to the 4 hexagonal by squares, close-packed was used in structure further analof theysis. colloidal Indices crystals. of the structureTwo orders are of adopted Bragg peaks from canthe be seen in single diffraction patterns shown in Figure4. A family of 11.0 Bragg reflections, indicated hexagonalbe seen in singlestructure diffraction and the patternspoint symbolizes shown in theFigure four 4.th A index family equal of 11.0 to the Bragg negative reflections, sum of indicated the first in Figure4 by squares, was used in further analysis. Indices of the structure are adopted from the twoin Figure indices 4 by(11.0 squares, equals wasto 11 used20). in further analysis. Indices of the structure are adopted from the hexagonalhexagonal structure structure and and the the point point symbolizes symbolizes thethe four fourthth index index equal equal to to the the negative negative sum sum of the of thefirst first twotwo indices indices (11.0 (11.0 equals equals to to 11 1120).20).

Figure 4. Single-shot diffraction patterns (in logarithmic scale) from colloidal crystals made from (a) PS, (b) PS covered with Al layer, and (c) silica samples measured at LCLS. Squares indicate Bragg FigurepeaksFigure 4. Single-shotconsidered 4. Single-shot in diffraction the diffraction further patterns analysis. patterns (in (in logarithmic logarithmic scale) scale) from from colloidal colloidal crystals crystals made made from from (a) PS; (b) PS(a) coveredPS, (b) PS with covered Al layer, with Al and layer, (c) silicaand (c samples) silica samples measured measured at LCLS. at LCLS. Squares Squares indicate indicate Bragg Bragg peaks 3. Datapeaks Analysis considered in the further analysis. considered in the further analysis. The following strategy was implemented for the data analysis. First, for the patterns with 3. Data Analysis 3. Datasufficiently Analysis high intensity (more than 5000 counts per single diffraction pattern) of the XFEL pulse, we selectedThe following Bragg peaks strategy located was far implemented away from the for detector the data gaps analysis. (see Figure First, 4). for Patterns the patterns with a lower with signalsufficientlyThe followingwere highnot treated. intensity strategy Using (more was this implementedthan procedure, 5000 counts typi for percally the single three data diffraction to analysis. four Bragg pattern) First, peaks forof were the the XFELselected patterns pulse, for with sufficientlythewe selectedanalysis high Bragg(see intensity Figure peaks 4). (morelocated Next, than farthe away Bragg 5000 from peak counts the positions detector per single were gaps diffraction extracted (see Figure using pattern) 4). Patterns the center of thewith of XFEL massa lower of pulse, theirsignal intensity were not distributions treated. Using as this well procedure, as the relative typically distances three to between four Bragg the peaksopposite were Bragg selected peaks. for the analysis (see Figure 4). Next, the Bragg peak positions were extracted using the center of mass of their intensity distributions as well as the relative distances between the opposite Bragg peaks. Appl. Sci. 2017, 7, 519 5 of 13 we selected Bragg peaks located far away from the detector gaps (see Figure4). Patterns with a lower signal were not treated. Using this procedure, typically three to four Bragg peaks were selected for the Appl. Sci. 2017, 7, 519 5 of 13 analysisAppl. Sci. 2017 (see, 7 Figure, 519 4). Next, the Bragg peak positions were extracted using the center of mass of their5 of 13 intensity distributions as well as the relative distances between the opposite Bragg peaks. The change TheThe change change of of the the relative relative positions positions of of the the Bragg Bragg pe peaksaks as as a a function function of of the the time time delay delay allowed allowed us us to to of the relative positions of the Bragg peaks as a function of the time delay allowed us to probe the probeprobe the the lattice lattice dynamics dynamics along along specific specific crystal crystal di directions.rections. The The following following parameters parameters as as a a function function lattice dynamics along specific crystal directions. The following parameters as a function of time delay ofof time time delay delay were were analyzed analyzed in in order order to to reveal reveal th thee dynamics dynamics of of the the colloidal colloidal crystals, crystals, induced induced by by the the wereIR laser: analyzed Bragg in peak order position to reveal q(τ the), as dynamics well as FWHMs of the colloidal in the radial crystals, wq(τ) induced and azimuthal by the IRwφ( laser:τ) directions Bragg IR laser: Bragg peak position q(τ), as well as FWHMs in the radial wq(τ) and azimuthal wφ(τ) directions peakin reciprocal position qspace.(τ), as wellThe temporal as FWHMs variation in the radial of thew qmomentum(τ) and azimuthal transferwφ vector(τ) directions q(τ) is related in reciprocal to the in reciprocal space. The temporal variation of the momentum transfer vector q(τ) is related to the space.dynamics The temporalin the inter-particle variation of thespacing, momentum while transferwq(τ) corresponds vector q(τ) to is relatedthe dynamics to the dynamics of the average in the dynamics in the inter-particle spacing, while wq(τ) corresponds to the dynamics of the average inter-particleparticle size, spacing, and wφ(τ while) defineswq(τ angular) corresponds misorientation to the dynamics of coherent of the scattering average particledomains size, [26]. and wφ(τ) particle size, and wφ(τ) defines angular misorientation of coherent scattering domains [26]. defines angular misorientation of coherent scattering domains [26]. DueDue to to the the nature nature of of the the SASE SASE pr process,ocess, each each XFEL XFEL pulse pulse has has a a uniq uniqueue fine fine spatial spatial structure structure that that Due to the nature of the SASE process, each XFEL pulse has a unique fine spatial structure that waswas mappedmapped byby thethe intensityintensity distributiondistribution atat eacheach BraggBragg position.position. AA multi-spikedmulti-spiked structurestructure ofof thethe was mapped by the intensity distribution at each Bragg position. A multi-spiked structure of the BraggBragg peaks peaks varying varying from from pulse pulse to to pulse pulse was was observed observed in in our our experiment experiment (see (see Figure Figure 5). 5). From From these these Bragg peaks varying from pulse to pulse was observed in our experiment (see Figure5). From these diffractiondiffraction patterns patterns spatial spatial and and temporal temporal coherence coherence pr propertiesoperties of of hard hard XFEL XFEL could could be be determined determined by by diffraction patterns spatial and temporal coherence properties of hard XFEL could be determined by spatialspatial correlation correlation analysis analysis [27]. [27]. Due Due to to the the different different shape shape of of each each FEL FEL pulse pulse (see (see Figure Figure 5), 5), it it was was not not spatial correlation analysis [27]. Due to the different shape of each FEL pulse (see Figure5), it was not possiblepossible to to perform perform deconvolution deconvolution of of each each scattere scatteredd pulse pulse shape. shape. For For the the same same reason reason (non-Gaussian (non-Gaussian possible to perform deconvolution of each scattered pulse shape. For the same reason (non-Gaussian structurestructure of of Bragg Bragg peaks) peaks) fitting fitting of of the the peaks peaks wi withth the the two-dimensional two-dimensional Gaussian Gaussian function function was was not not structure of Bragg peaks) fitting of the peaks with the two-dimensional Gaussian function was not reliablereliable (see(see FigureFigure 6a).6a). InIn orderorder toto improveimprove BraggBragg peakpeak characterization,characterization, projectionsprojections ofof theirtheir reliable (see Figure6a). In order to improve Bragg peak characterization, projections of their intensities intensitiesintensities onon azimuthalazimuthal andand radialradial directionsdirections werewere performed.performed. TheseThese datadata werewere fittedfitted withwith thethe onone-dimensional azimuthal and radialGaussian directions function were (see performed. Figure 6b), These and data peak were broadening fitted with in the the one-dimensional radial (wq) and one-dimensional Gaussian function (see Figure 6b), and peak broadening in the radial (wq) and Gaussianazimuthal function (wφ) directions (see Figure was6b), determined. and peak broadening in the radial ( wq) and azimuthal (wφ) directions azimuthal (wφ) directions was determined. was determined.

FigureFigureFigure 5.5.5. FourFour single-shot single-shot diffraction diffraction patterns patternspatterns meas measuredmeasuredured at atat the thethe same same Bragg Bragg peakpeak position.position. TheTheThe spike-shapedspike-shaped spike-shaped peaks peaks demonstrate demonstratedemonstrate an an influence inﬂuenceinfluence of of the the the individual individual individual structure structure structure of of ofeach each each XFEL XFEL XFEL pulse pulse pulse on on onthethe the Bragg Bragg Bragg peak. peak. peak.

FigureFigureFigure 6. 6.6.( a ()a(a) The ) TheThe cross-section cross-sectioncross-section of a ofof two-dimensional aa two-dimensionaltwo-dimensional Bragg BrBr peakaggagg peak alongpeak along azimuthalalong azimuthalazimuthal direction directiondirection and (b ) and theand projection(b(b) )the the projection projection of the sameof of the the Braggsame same Bragg Bragg peak peakon peak the on on same the the same same direction direction direction . It . is.It It seenis is seen seen that that that the the the cross-section cross-section cross-section datasetdataset dataset ((aa()a) deviates)deviates deviates fromfrom from thethe the GaussianGaussian Gaussian ﬁtfit fit (red(red (red line),line), line), whereaswherea whereass thethe the projectionprojection projection pointspoints points arear aree ﬁttedfitted fitted withwith with thethe the GaussianGaussian Gaussian functionfunctionfunction withwith with higherhigher higher accuracy.accuracy. accuracy.

Appl. Sci. 2017, 7, 519 6 of 13

In order to compare dynamics of the collected data as a function of the time delay τ for different samplesAppl. Sci. 2017 and, 7 IR, 519 laser energies, the following dimensionless parameters were used: 6 of 13 D E In order to compare dynamics of∆ qthe(τ )collectedhqon( τdata)i − as qao function f f of the time delay τ for different = (1) samples and IR laser energies, the followingq dimensionlessD E parameters were used: qo f f ∆() 〈()〉 − 〈〉 = (1) 〈D〉 E ∆w(τ) hwon(τ)i − wo f f = (2) ∆w() 〈(D)〉 −E〈〉 = wo f f (2) 〈〉 where q(τ) is the distance between the opposite Bragg peaks, and w(τ) is the radial or azimuthal where q(τ) is the distance between the opposite Bragg peaks, and w(τ) is the radial or azimuthal FWHM of the Bragg peak. Subscripts “on” and “off” define measurements with and without the IR FWHM of the Bragg peak. Subscripts “on” and “off” define measurements with and without the IR laser, respectively. Brackets < ... > denote ensemble averaging of the chosen Bragg peak parameter laser, respectively. Brackets <…> denote ensemble averaging of the chosen Bragg peak parameter for for each time delay τ over all XFEL pulses. Time dependencies of the momentum transfer vector each time delay τ over all XFEL pulses. Time dependencies of the momentum transfer vector q(τ), q(τ), as well as radial wq(τ) and azimuthal wφ(τ) broadening of the Bragg peaks for three measured as well as radial wq(τ) and azimuthal wφ(τ) broadening of the Bragg peaks for three measured samples, samples, are shown in Figure7. Some points in this figure are excluded due to the unexpected drops are shown in Figure 7. Some points in this figure are excluded due to the unexpected drops of of intensity of the XFEL (such drops of intensity were observed when operation of some clystrons in intensity of the XFEL (such drops of intensity were observed when operation of some clystrons in the the accelerator complex was failing). The statistical error of these parameters was determined using accelerator complex was failing). The statistical error of these parameters was determined using the the standard approach for the error determination in the multi-parameter equation [28]. As clearly standard approach for the error determination in the multi-parameter equation [28]. As clearly seen seen in Figure7, an error of the ∆q(τ)/q values is one order of magnitude lower than for the Bragg in Figure 7, an error of the ∆q(τ)/q values is one order of magnitude lower than for the Bragg peaks peaks broadening parameters. This could be explained by the influence of the XFEL pulse shape on the broadening parameters. This could be explained by the influence of the XFEL pulse shape on the shape of the Bragg peaks with a relatively stable position of each pulse. Due to this, the Gaussian fit of shape of the Bragg peaks with a relatively stable position of each pulse. Due to this, the Gaussian fit the Bragg peaks shape was not accurate, while the influence of the XFEL pulse shape on the variation of the Bragg peaks shape was not accurate, while the influence of the XFEL pulse shape on the of the center of mass was insignificant. The error values for the silica sample are higher than for the PS variation of the center of mass was insignificant. The error values for the silica sample are higher than due to lower statistics (for the PS crystals 600 shots and for the silica crystals 120 shots were collected for the PS due to lower statistics (for the PS crystals 600 shots and for the silica crystals 120 shots were with the IR laser). collected with the IR laser).

Figure 7. Time dependence of the relative change of the distance between the two opposite Bragg Figure 7. Time dependence of the relative change of the distance between the two opposite Bragg peaks Δq(τ)/q (a–c) and Bragg peaks widths in radial Δwq(τ)/wq (d–f) and azimuthal Δwφ(τ)/wφ (g–i) peaks ∆q(τ)/q (a–c) and Bragg peaks widths in radial ∆wq(τ)/wq (d–f) and azimuthal ∆wφ(τ)/wφ (directionsg–i) directions for three for three selected selected samples. samples. Analysis Analysis of the of the data data was was performed performed for for the the IR IR pump laser intensities shownshown inin FigureFigure3 3 and and 11.0 11.0 Bragg Bragg peaks peaks indicated indicated by by a a square square in in Figure Figure4 .4.

Figure7 also clearly shows that variations of all parameters are around zero and in all cases, the signal is lower than the statistical error. To reveal characteristic frequencies excited by the IR laser, we performed the Fourier analysis of the time dependencies of the Bragg peak parameters Appl. Sci. 2017, 7, 519 7 of 13 described above. The corresponding Fourier spectra are shown in Figure8. Due to the more precise measurementsmeasurements ofof thethe momentummomentum transfertransfer vectorvector qq((ττ),), thethe averageaverage valuevalue ofof thethe FourierFourier componentscomponents ofof qq((ττ)) isis oneone orderorder ofof magnitudemagnitude lowerlower thanthan forfor thethe BraggBragg peakspeaks broadeningbroadening ww((ττ).). ThereThere waswas nono signiﬁcantsignificant enhancementenhancement ofof anyany particularparticular FourierFourier component,component, forfor allall FourierFourier spectra.spectra. We determineddetermined anan averageaverage valuevalue ofof thethe FourierFourier componentscomponents andand thethe standardstandard deviationdeviation σσ fromfrom thethe distributiondistribution ofof thesethese FourierFourier components.components. For allall FourierFourier spectraspectra shownshown inin FigureFigure8 8,, there there is is no no Fourier Fourier component component higherhigher thanthan 33σσ aboveabove thethe averageaverage FourierFourier componentcomponent value,value, whichwhich indicatesindicates thatthat thethe periodicperiodic signalsignal couldcould notnot bebe reliablyreliably detecteddetected inin thesethese conditions.conditions.

FigureFigure 8.8. FourierFourier spectraspectra fromfrom thethe distancedistance betweenbetween thethe twotwo opposite opposite Bragg Bragg peaks peaks∆ Δqq(τ(τ)/)/qq ((aa––cc)) andand Bragg peaks widths in radial Δwq(τ)/wq (d–f) and azimuthal Δwφ(τ)/wφ (g–i) directions for three Bragg peaks widths in radial ∆wq(τ)/wq (d–f) and azimuthal ∆wφ(τ)/wφ (g–i) directions for three selectedselected samples. Fourier Fourier spectra spectra calculated calculated from from th thee results results of of diffraction diffraction pattern pattern analysis analysis shown shown in inFigure Figure 7.7 Average. Average value value of ofall all Fourier Fourier components components is indicated is indicated by bythe the red red line. line. The The dashed dashed red redline linecorresponds corresponds to the to the3σ 3levelσ level above above the the average average value, value, where where σ σisis the the standard standard deviation and isis determineddetermined fromfrom thethe distributiondistribution ofof FourierFourier components.components.

4.4. Model SimulationsSimulations ToTo havehave betterbetter understandingunderstanding ofof thethe obtainedobtained results,results, wewe performedperformed simulationssimulations ofof thethe colloidalcolloidal crystalcrystal dynamicsdynamics inducedinduced byby anan externalexternal IRIR pulsepulse excitation.excitation. InIn ourour modelmodel colloidalcolloidal crystalscrystals consistconsist ofof isotropicisotropic homogeneoushomogeneous spheres.spheres. TheThe characteristiccharacteristic frequenciesfrequencies ofof vibrationvibration ofof aa colloidalcolloidal particleparticle werewere determineddetermined usingusing thethe LambLamb theorytheory describingdescribing vibrations of an isotropic elastic sphere [29]. [29]. Theoretically,Theoretically, twotwo familiesfamilies ofof LambLamb modes,modes, namelynamely spheroidalspheroidal andand torsionaltorsional ones,ones, cancan bebe derivedderived fromfrom thethe equationsequations ofof motion.motion. TheThe spheroidalspheroidal modesmodes causecause thethe changechange ofof thethe spheresphere volume,volume, andand thethe torsional modes leave the sphere volume unperturbed. During the IR pulse propagation through a torsional modes leave the sphere volume unperturbed. During the IR pulse propagation through a sphere, the changes of the sphere volume are expected and, therefore, we used spheroidal modes in sphere, the changes of the sphere volume are expected and, therefore, we used spheroidal modes in our simulations. We determined the frequencies of the first spheroidal mode for all three measured our simulations. We determined the frequencies of the ﬁrst spheroidal mode for all three measured samples using reference [30]. Sphere diameter, longitudinal cL and transverse cT sound velocities of samples using reference [30]. Sphere diameter, longitudinal cL and transverse cT sound velocities of PS PS and silica were obtained from reference [31]. These parameters, as well as the frequencies of the and silica were obtained from reference [31]. These parameters, as well as the frequencies of the ﬁrst first spheroidal mode for the colloidal crystals, are summarized in Table 1. spheroidal mode for the colloidal crystals, are summarized in Table1.

Appl. Sci. 2017, 7, 519 8 of 13

Table 1. Parameters of PS and silica nanoparticles used in our model. The corresponding frequencies of the ﬁrst spheroidal mode of colloidal particles were obtained from reference [30]. The noise level was derived from the experimental data. Relative and absolute deviation in the interparticle distance

(∆d/d0 and ∆d), which could be detected in our experiment with the probability of 60%, were calculated from the experimental data and simulations.

PS without Al PS with Al Silica Sphere diameter, nm 420 376 238 Longitudinal sound velocity cL, m/s 2350 2350 5950 Transverse sound velocity cT, m/s 1210 1210 3760 Frequency of the breathing mode, GHz 4.97 5.54 19.06 Noise level N × 10−3 1.4 1.8 1.8 −3 Relative deviation in interparticle distance ∆d/d0 × 10 0.7 0.9 0.9 Deviation in interparticle distance ∆d, nm 0.29 0.35 0.22

The values of ∆q(τ)/q were obtained from the experimental data with the smallest error in comparison to other Bragg peak parameters for all investigated samples (see Figure7a–c). Therefore, the parameter ∆q(τ)/q was considered for our simulations. Assuming an ideal close-packed crystal with the interparticle distances equal to the particles diameter, the unperturbed momentum transfer modulus of the 11.0 Bragg reﬂection can be described as q0 = 2π/d11.0 = 4π/d0, where d0 is the unperturbed particle diameter. We simulated vibrations of the colloidal spheres with the diameter d0 as a periodic sinusoidal signal: d(τ) = d0(1 − S· sin(ωτ)) (3) where S is the relative amplitude and ω is the characteristic frequency of the vibration. The dynamics of the experimentally-observed 11.0 diffraction peak can be described as:

4π 4π q(τ) = ≈ (1 + S·sin(ωτ)) (4) d(τ) d0 where we used the Taylor series expansion of the momentum transfer vector q(τ) due to the small values of the amplitude S << 1. As a result, measured changes in the Bragg peak positions and ∆q(τ)/q can be simulated as a periodic sinusoidal signal. We should also take into account systematic errors of the measurement as an additional noise. Finally, simulations of the parameter ∆q(τ)/q in our model were described using the following expression:

∆q(τ) q(τ) − q0 = = S·sin(ωτ) + N·ηnoise(τ) (5) q0 q0 where N is the amplitude of a noise and ηnoise(τ) is a random noise function with normal distribution and standard deviation value equal to one. The signal-to-noise ratio in this case is assumed to be equal to the ratio of the amplitudes S/N. Next, we defined criterion when the periodic signal is detected by the algorithm in our simulations. Due to the random behavior of the noise function ηnoise(τ), each realization of the signal ∆q(τ)/q0 (see Equation (5)) for a constant S/N is different. As an example, we demonstrate this for the experimental data of the polystyrene sample without aluminum. This sample was considered as one with the lowest noise level in our experiment. The time interval and increment of the simulated signal ∆q(τ)/q0 were chosen the same as in our pump-probe experiment. One of the typical realizations of the ∆q(τ)/q0 with the signal level S = 0.7 × 10−3 and noise level N = 1.4 × 10−3 is shown in Figure9a. From this figure it is difficult to conclude on the presence or absence of the signal. Appl. Sci. 2017, 7, 519 9 of 13 Appl. Sci. 2017, 7, 519 9 of 13

FigureFigure 9.9. ((aa)) SimulationSimulation ofof the the signal signal∆ ∆qq(τ(τ)/)/q0 and (b) FourierFourier transformtransform ofof thisthis signal.signal. TheThe averageaverage valuevalue ofof allall Fourier Fourier components components is is indicated indicated by by red red line. line. The The dashed dashed red red line line corresponds corresponds to levelto level of 3 ofσ above3σ above the the average average value, value, where whereσ is σ a is standard a standard deviation deviation and and is determined is determined from from the the distribution distribution of Fourierof Fourier components. components.

SimilarSimilar toto thethe analysisanalysis ofof thethe experimentalexperimental data,data, thethe FourierFourier spectraspectra werewere calculatedcalculated forfor thethe simulatedsimulated signalsignal with with the the noise noise (see (see Figure Figure9b). Next, 9b). theNext, behavior the behavior of the Fourier of the component Fourier component amplitude (amplitudeAω) corresponding (Aω) corresponding to the ﬁrst spheroidal to the first mode spheroidal frequency modeω = frequency 4.97 GHz wasω = studied.4.97 GHz The was criterion studied. of detectingThe criterion the signalof detecting was considered the signal as:was considered as: Aω > A + 3σ (6) > +3 (6) wherewhere A is is thethe meanmean andand σis is the the standard standard deviation deviation ofof allall FourierFourier componentscomponents inin thethe spectrum.spectrum. TheThe FourierFourier spectrumspectrum ofof thethe simulatedsimulated datadata (see(see FigureFigure9 a),9a), as as well well as as its its 3 3σσ levellevel areare shownshown inin FigureFigure9 b.9b. It It can can be be seen seen that that the the contribution contribution of the of vibrational the vibrational mode mode in the Fourierin the Fourier spectrum spectrum is higher is thanhigher the than criterion the criterion stipulated stipulated by Equation by Equation (6). Therefore, (6). Therefore, the algorithm the algorithm detected detected the presence the presence of the 4.97of the GHz 4.97 frequency GHz frequency signal. signal. WeWe calculatedcalculated thethe probabilityprobability ofof successfulsuccessful signalsignal detectiondetection asas aa functionfunction ofof signal-to-noisesignal-to-noise ratioratio SS//N inin the the range range of of ( (SS//NN) )from from 0 0to to 1.5 1.5 with with 0.01 0.01 increment. increment. During During the the simulations simulations the the amplitude amplitude of ofthe the signal signal S Swaswas changing, changing, while while the the noise noise level level NN waswas kept kept constant. The probability ofof detectingdetecting signalsignal PP((SS//N)) was was defined deﬁned as: n(S(//N)) P(S(//N))==lim detected (7)(7) n→→∞ n

where ndetected(S/N) is a number of simulations with successful detection of signal (according to the where ndetected(S/N) is a number of simulations with successful detection of signal (according to the criterioncriterion stipulatedstipulated by by Equation Equation (6)) (6)) and andn =n10,000 = 10,000 is the is the total total number number of simulations of simulations for each for Seach/N step. S/N Resultsstep. Results of these of simulations these simulations for the experimentalfor the experi datamental of the data polystyrene of the polystyrene sample without sample aluminum without arealuminum shown inare Figure shown 10 .in As Figure one can 10. seeAs fromone can Figure see 10from that, Figure according 10 that, to ouraccording criterion to stipulatedour criterion by Equationstipulated (6), bythe Equation signal is(6), 100% the signal detectable is 100% if it detectable is on the level if it ofis on the the noise level (S /ofN the = 1). noise (S/N = 1). Appl. Sci. 2017, 7, 519 10 of 13 Appl. Sci. 2017, 7, 519 10 of 13

Figure 10. The probability of successful signal detection as a function of signal-to-noise ratio (S/N). OnFigure the top10. The horizontal probability axis, theof successful change of signal the corresponding detection as interparticlea function of distance signal-to-noise for thepolystyrene ratio (S/N). sampleOn the top without horizontal Al is shown. axis, the The change red lines of the correspond corresponding to the interparticle value of S/N distance= 0.5 and forP the(S/ Npolystyrene) = 0.6. sample without Al is shown. The red lines correspond to the value of S/N = 0.5 and P(S/N) = 0.6. 5. Results and Discussion 5. Results and Discussion Simulations, described in the previous section, were performed for each sample (polystyrene sampleSimulations, without aluminum, described polystyrenein the previous sample section, with were aluminum, performed and for silica each samples) sample to(polystyrene derive the probabilitysample without function aluminum,P(S/N) ofpolystyrene detecting thesample periodic with signal aluminum, (see Figure and silica 10) in samples) each case. to Itderive is clearly the visibleprobability from function Figure 10 P that(S/N even) of detecting the value the of theperiodic signal-to-noise signal (see ratio FigureS/N 10)= 0.5in each gives case. 60% probabilityIt is clearly ofvisible detecting from theFigure signal. 10 that This even condition the value was of consideredthe signal-to-noise as a criterion ratio S for/N = the 0.5 minimum gives 60% measurable probability signal.of detecting Using the well-known signal. This relations condition between was considered the deviations as a ofcriterion the momentum for the minimum transfer vectorsmeasurable and signal. Using well-known relations between the deviations of the momentum transfer vectors and interparticle distances ∆q(τ)/q0 = −∆d(τ)/d0, as well as Equation (5), we can determine the relative interparticle distances ∆q(τ)/q0 = −∆d(τ)/d0, as well as Equation (5), we can determine the relative (and (and absolute) values of the changes of the interparticle distances ∆d(τ)/d0 (∆d(τ)) for the given values ofabsolute) the experimental values of the noise changesN. As aof result the interparticle of this analysis, distances we can ∆d obtain(τ)/d0 ( the∆d(τ minimum)) for the given values values of these of parametersthe experimental which couldnoise beN. detectedAs a result in ourof this experimental analysis, conditionswe can obtain with the the minimum probability values of 60%. of For these all measuredparameters samples which could results be of detected this analysis in our are experimental summarized conditions in Table1. with the probability of 60%. For all measured samples results of this analysis are summarized in Table 1. As it follows from Table1, the minimum relative changes of the interparticle distance ∆d/d0 that can beAs detected it follows with from the Table probability 1, the minimum of 60% for rela alltive samples changes were of about the interparticle 10−3. Such valuesdistance correspond Δd/d0 that −3 tocan relatively be detected large with amplitudes the probability of the deviations of 60% for in all interparticle samples were distances about 10 of about. Such 0.2–0.4 valuesnm. correspond Clearly, into ourrelatively experimental large amplitudes conditions of excited the deviations vibrations in interparticle were lower thandistances the noise of about level 0.2–0.4 of the nm. experiment Clearly, thatin our prevented experimental us from conditions their experimental excited vibrations determination. were lower than the noise level of the experiment that preventedFor the successful us from their measurement experimental of vibrations determination. of colloidal crystals at the FEL sources, some furtherFor improvements the successful of measurement the experiment of mayvibrations be suggested. of colloidal First, crystals the accuracy at the of FEL the sources, measurement some offurther the incoming improvements X-ray beamof theintensity experiment from may pulse be su toggested. pulse should First, bethe improved, accuracy of as the this measurement would allow measurementof the incoming of X-ray the Bragg beam peak intensity intensity from with pulse higher to pulse precision. should In be our improved, experiment as this such would an accurate allow measurementmeasurement ofof thethe BraggBragg peakpeak intensityintensity variation with higher was precision. not possible In our due experiment to the complicated such anstructure accurate ofmeasurement each XFEL pulse of the and Bragg a relatively peak intensity low accuracy variation (about was 3%)not inpossible the measurement due to the complicated of the incoming structure X-ray beamof each intensity XFEL pulse from pulseand a torelatively pulse. Second, low accuracy colloidal (about crystals 3%) may in the be coveredmeasurement by a thicker of the aluminum incoming layerX-ray which beam couldintensity lead from to a morepulse efﬁcient to pulse. energy Second transfer, colloidal from crystals an IR laser may to be the covered colloidal by crystal a thicker [32]. Thisaluminum would layer lead towhich a higher could amplitude lead to a of more vibrations efficient of energy a PS colloidal transfer crystal, from an and IR enable laser to their the detectioncolloidal incrystal IR pump [32]. andThis X-ray would probe lead experiments.to a higher amplitude of vibrations of a PS colloidal crystal, and enable their detection in IR pump and X-ray probe experiments. 6. Conclusions 6. Conclusions We performed experiments at the LCLS to investigate dynamics in the colloidal crystals. The diffractionWe performed patterns experiments from the at colloidal the LCLS crystals, to invest whichigate were dynamics pumped in by the IR colloidal laser pulses crystals. and probedThe diffraction by XFEL patterns radiation, from were the collected. colloidal The crystals, measurements which were were pumped performed by inIR alaser non-destructive pulses and regimeprobed atby the XFEL different radiation, positions were of collected. the samples. The Themeasurements pump-probe were experiments performed were in a performed non-destructive in the regime at the different positions of the samples. The pump-probe experiments were performed in the Appl. Sci. 2017, 7, 519 11 of 13 wide range of the IR laser energies, from 195 µJ up to the upper limit of the non-destructive threshold of 691 µJ. Through the analysis of the Bragg peaks, extracted from the diffraction patterns, changes in the colloidal crystals caused by the IR laser were investigated. The dynamics at different timescales was studied by the Fourier analysis of parameters associated with the momentum transfer and Bragg peak broadening in the radial and angular directions. Simulations of the colloidal crystal dynamics were performed and the probability function P(S/N) was derived. Relative and absolute deviations in the interparticle distance (∆d/d0 and ∆d), which could be detected in such an experiment with the probability of 60% (corresponding to the signal to noise level of 0.5), were calculated for all measured samples. From the experiment and simulations, we were able to conclude that, for all measured samples, the amplitudes of vibrational modes excited in colloidal crystals were less than the systematic error caused by the noise level. In future experiments with XFEL sources different scattering geometries may be foreseen. For example, performing scattering experiments on colloidal crystals in grazing incidence small-angle X-ray scattering (GISAXS) [33,34] may increase the sensitivity of the scattered radiation to changes induced by the IR pump laser.

Acknowledgments: We acknowledge fruitful discussions and support of the project by Edgar Weckert as well as help and support by Andrei V. Petukhov. We acknowledge the help of Ralf Röhlsberger in preparation of Al-covered colloidal crystals. We acknowledge careful reading of the manuscript by Tim Laarman. This work was supported by the Virtual Institute VH-VI-403 of the Helmholtz Association. The experimental work was carried out at the Linac Coherent Light Source (LCLS), a National User Facility operated by Stanford University on behalf of the U.S. Department of Energy, Office of Basic Energy Sciences. Use of the LCLS, SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences under contract number DE-AC02-76SF00515. A.S. was supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contracts No. DE- SC0001805. For G.W. portions of this research were carried out at Brookhaven National Laboratory, operated under Contract No. DE-SC0012704 from the U.S. Department of Energy (DOE) Office of Science. Author Contributions: M.C., A.S., R.P.K., O.M.Y., D.D., O.Y.G., G.W., D.Z., Y.F., M.S., S.S., and I.A.V. performed the experiments; J.-M.M. and T.G. prepared the colloidal crystal samples; I.A.V. supervised N.M. and the project; N.M., A.G.S., E.A.S., and S.L. developed the code; N.M. analyzed the data; and N.M. and S.L. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Gorishnyy, T.; Ullal, C.K.; Maldovan, M.; Fytas, G.; Thomas, E. Hypersonic phononic crystals. Phys. Rev. Lett. 2005, 94, 115501. [CrossRef][PubMed] 2. Cheng, W.; Wang, J.; Jonas, U.; Fytas, G.; Stefanou, N. Observation and tuning of hypersonic bandgaps in colloidal crystals. Nat. Mater. 2006, 5, 830. [CrossRef][PubMed] 3. Still, T.; Cheng, W.; Retsch, M.; Sainidou, R.; Wang, J.; Jonas, U.; Stefanou, N.; Fytas, G. Simultaneous occurrence of structure-directed and particle-resonance-induced phononic gaps in colloidal ﬁlms. Phys. Rev. Lett. 2008, 100, 194301. [CrossRef][PubMed] 4. Thomas, E.L.; Gorishnyy, T.; Maldovan, M. Phononics: Colloidal crystals go hypersonic. Nat. Mater. 2006, 5, 773–774. [CrossRef][PubMed] 5. Akimov, A.; Tanaka, Y.; Pevtsov, A.; Kaplan, S.; Golubev, V.; Tamura, S.; Yakovlev, D.; Bayer, M. Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials. Phys. Rev. Lett. 2008, 101, 033902. [CrossRef][PubMed] 6. Salasyuk, A.S.; Scherbakov, A.V.; Yakovlev, D.R.; Akimov, A.V.; Kaplyanskii, A.A.; Kaplan, S.F.; Grudinkin, S.A.; Nashchekin, A.V.; Pevtsov, A.B.; Golubev, V.G.; et al. Filtering of elastic waves by opal-based hypersonic crystal. Nano Lett. 2010, 10, 1319–1323. [CrossRef][PubMed] 7. Mazurenko, D.; Shan, X.; Stiefelhagen, J.; Graf, C.; Van Blaaderen, A.; Dijkhuis, J. Coherent vibrations of submicron spherical gold shells in a photonic crystal. Phys. Rev. B 2007, 75, 161102. [CrossRef] 8. Duval, E.; Boukenter, A.; Champagnon, B. Vibration eigenmodes and size of microcrystallites in glass: Observation by very-low-frequency Raman scattering. Phys. Rev. Lett. 1986, 56, 2052. [CrossRef][PubMed] Appl. Sci. 2017, 7, 519 12 of 13

9. Emma, P.; Akre, R.; Arthur, J.; Bionta, R.; Bostedt, C.; Bozek, J.; Brachmann, A.; Bucksbaum, P.; Coffee, R.; Decker, F.J.; et al. First lasing and operation of an ångstrom-wavelength free-electron laser. Nat. Photonics 2010, 4, 641–647. [CrossRef] 10. Altarelli, M.; Brinkmann, R.; Chergui, M.; Decking, W.; Dobson, B.; Dusterer, S.; Grubel, G.; Graeff, W.; Graafsma, H.; Hajdu, J.; et al. XFEL The European X-ray Free-Electron Laser; Technical Design Report, DESY Report No. 2006–097; DESY XFEL Project Group, European XFEL Project Team, Deutsches Elektronen-Synchrotron, Member of the Helmholtz Association: Hamburg, Germany, 2006; Available online: http://xfel.desy.de/technical_information/tdr/tdr/ (accessed on 17 May 2017). 11. Ishikawa, T.; Aoyagi, H.; Asaka, T.; Asano, Y.; Azumi, N.; Bizen, T.; Ego, H.; Fukami, K.; Fukui, T.; Furukawa, Y.; et al. A compact X-ray free-electron laser emitting in the sub-ångstrom region. Nat. Photonics 2012, 6, 540–544. [CrossRef] 12. McFarland, B.; Farrell, J.; Miyabe, S.; Tarantelli, F.; Aguilar, A.; Berrah, N.; Bostedt, C.; Bozek, J.; Bucksbaum, P.; Castagna, J.; et al. Ultrafast X-ray Auger probing of photoexcited molecular dynamics. Nat. Commun. 2014, 5. [CrossRef][PubMed] 13. Trigo, M.; Fuchs, M.; Chen, J.; Jiang, M.; Cammarata, M.; Fahy, S.; Fritz, D.; Gaffney, K.; Ghimire, S.; Higginbotham, A.; et al. Fourier-transform inelastic X-ray scattering from time-and momentum-dependent phonon-phonon correlations. Nat. Phys. 2013, 9, 790–794. [CrossRef] 14. Kupitz, C.; Basu, S.; Grotjohann, I.; Fromme, R.; Zatsepin, N.A.; Rendek, K.N.; Hunter, M.S.; Shoeman, R.L.; White, T.A.; Wang, D.; et al. Serial time-resolved crystallography of photosystem II using a femtosecond X-ray laser. Nature 2014, 513, 261–265. [CrossRef][PubMed] 15. Glownia, J.; Natan, A.; Cryan, J.; Hartsock, R.; Kozina, M.; Minitti, M.; Nelson, S.; Robinson, J.; Sato, T.; van Driel, T.; et al. Self-referenced coherent diffraction X-ray movie of Ångstrom-and femtosecond-scale atomic motion. Phys. Rev. Lett. 2016, 117, 153003. [CrossRef][PubMed] 16. Abbey, B.; Dilanian, R.A.; Darmanin, C.; Ryan, R.A.; Putkunz, C.T.; Martin, A.V.; Wood, D.; Streltsov, V.; Jones, M.W.; Gaffney, N.; et al. X-ray laser-induced electron dynamics observed by femtosecond diffraction from nanocrystals of Buckminsterfullerene. Sci. Adv. 2016, 2, e1601186. [CrossRef][PubMed] 17. Clark, J.; Beitra, L.; Xiong, G.; Higginbotham, A.; Fritz, D.; Lemke, H.; Zhu, D.; Chollet, M.; Williams, G.; Messerschmidt, M.; et al. Ultrafast three-dimensional imaging of lattice dynamics in individual gold nanocrystals. Science 2013, 341, 56–59. [CrossRef][PubMed] 18. Dronyak, R.; Gulden, J.; Yefanov, O.; Singer, A.; Gorniak, T.; Senkbeil, T.; Meijer, J.M.; Al-Shemmary, A.; Hallmann, J.; Mai, D.; et al. Dynamics of colloidal crystals studied by pump-probe experiments at FLASH. Phys. Rev. B 2012, 86, 064303. [CrossRef] 19. Chollet, M.; Alonso-Mori, R.; Cammarata, M.; Damiani, D.; Defever, J.; Delor, J.T.; Feng, Y.; Glownia, J.M.; Langton, J.B.; Nelson, S.; et al. The X-ray pump–probe instrument at the Linac Coherent Light Source. J. Synchrotron Radiat. 2015, 22, 503–507. [CrossRef][PubMed] 20. Saldin, E.L.; Schneidmiller, E.A.; Yurkov, M.V. The Physics of Free Electron Lasers; Springer: Berlin, Germany, 2000. 21. Stoupin, S.; Terentyev, S.; Blank, V.; Shvyd’ko, Y.V.; Goetze, K.; Assouﬁd, L.; Polyakov, S.; Kuznetsov, M.; Kornilov, N.; Katsoudas, J.; et al. All-diamond optical assemblies for a beam-multiplexing X-ray monochromator at the Linac Coherent Light Source. J. Appl. Crystallogr. 2014, 47, 1329–1336. [CrossRef] [PubMed] 22. Zhu, D.; Feng, Y.; Stoupin, S.; Terentyev, S.A.; Lemke, H.T.; Fritz, D.M.; Chollet, M.; Glownia, J.; Alonso-Mori, R.; Sikorski, M.; et al. Performance of a beam-multiplexing diamond crystal monochromator at the Linac Coherent Light Source. Rev. Sci. Instrum. 2014, 85, 063106. [CrossRef][PubMed] 23. Meijer, J.-M. Colloidal Crystals of Spheres and Cubes in Real and Reciprocal Space; Springer: Berlin, Germany, 2015; pp. 23–29. 24. Hart, P.; Boutet, S.; Carini, G.; Dubrovin, M.; Duda, B.; Fritz, D.; Haller, G.; Herbst, R.; Herrmann, S.; Kenney, C.; et al. The CSPAD megapixel X-ray camera at LCLS. In Proc. SPIE 8504, X-ray Free-Electron Lasers: Beam Diagnostics, Beamline Instrumentation, and Applications, 85040C; Moeller, S.P., Yabashi, M., Hau-Riege, S.P., Eds.; SPIE: Bellingham, WA, USA, 2012. 25. Lazarev, S.; Mukharamova, N.; Gorobtsov, O.Y.; Chollet, M.; Kurta, R.P.; Meijer, J.-M.; Williams, G.; Zhu, D.; Feng, Y.; Sikorski, M.; et al. Ultrafast melting of the colloidal crystals. 2017, in preparation. Appl. Sci. 2017, 7, 519 13 of 13

26. Sulyanova, E.A.; Shabalin, A.; Zozulya, A.V.; Meijer, J.M.; Dzhigaev, D.; Gorobtsov, O.; Kurta, R.P.; Lazarev, S.; Lorenz, U.; Singer, A.; et al. Structural evolution of colloidal crystal ﬁlms in the process of melting revealed by Bragg peak analysis. Langmuir 2015, 31, 5274–5283. [CrossRef][PubMed] 27. Gorobtsov, O.Y.; Mukharamova, N.; Lazarev, S.; Chollet, M.; Kurta, R.P.; Meijer, J.-M.; Williams, G.; Zhu, D.; Feng, Y.; Sikorski, M.; et al. Intensity interferometry on crystal diffraction patterns at hard X-ray free electron laser. 2017, in preparation. 28. Hazewinkel, M. Encyclopedia of Mathematics: An Updated and Annotated Translation of the Soviet Mathematical Encyclopedia; Springer: Berlin, Germany, 2013; Volume 2. 29. Lamb, H. On the vibrations of an elastic sphere. Proc. Lond. Math. Soc. 1881, 1, 189–212. [CrossRef] 30. Saviot, L. Vibrational Eigenmodes of an Isotropic Sphere. Available online: http://lucien.saviot.free.fr/ lamb/index.en.html (accessed on 30 March 2017). 31. Mark, J.E. Polymer Data Handbook; Oxford University Press: Oxford, UK, 2009; p. 834. 32. Tas, G; Maris, H.J. Electron diffusion in metals studied by picosecond ultrasonics. Phys. Rev. B 1994, 49, 15046. [CrossRef] 33. Huber, P.; Bunk, O.; Pietsch, U.; Textor, M.; Geue, T. Grazing Incidence Small Angle X-ray Scattering on Colloidal Crystals. J. Phys. Chem. B 2010, 114, 12473–12479. [CrossRef][PubMed] 34. Zozulya, A.; Zaluzhnyy, I; Lazarev, S.; Meijer, J.-M.; Mukharamova, N.; Shabalin, A.; Kurta, R.P.; Sprung, M.; Petukhov, A.V.; Vartanyants, I.A. Temperature-driven rearrangement of colloidal crystal domains studied in situ by GISAXS. 2017, in preparation.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).