X-ray diffraction from isolated and strongly aligned gas-phase molecules with a free-electron laser
We report experimental results on x-ray diffraction of quantum-state-selected and strongly aligned ensembles of the prototypical asymmetric rotor molecule 2,5-diiodobenzonitrile using the Linac Coherent Light Source. The experiments demonstrate first steps toward a new approach to diffractive imaging of distinct structures of individual, isolated gas-phase molecules. We confirm several key ingredients of single molecule diffraction experiments: the abilities to detect and count individual scattered x-ray photons in single shot diffraction data, to deliver state-selected, e. g., structural-isomer-selected, ensembles of molecules to the x-ray interaction volume, and to strongly align the scattering molecules. Our approach, using ultrashort x-ray pulses, is suitable to study ultrafast dynamics of isolated molecules.
pacs:33.15.-e, 33.15.Dj, 33.80.-b, 37.10.-x
Present address: ]Bruker AXS GmbH, Karlsruhe, GermanyPresent address: ]ARC Centre of Excellence for Coherent X-ray Science, School of Physics, The University of Melbourne, AustraliaPresent address: ]Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, GermanyPresent address: ]Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, GermanyPresent address: ]European X-ray Free Electron Laser (XFEL) GmbH, 22761 Hamburg, GermanyPresent address: ]Stanford PULSE Institute, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USAPresent address: ]PNSensor GmbH, 81739 Munich, GermanyPresent address: ]Institute for Molecules and Materials, Radboud University Nijmegen, Heijendaalseweg 135, 6525 AJ Nijmegen, The NetherlandsPresent address: ]Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany X-ray Free-Electron Lasers (XFELs) hold the promise to determine atomically resolved structures and to trace structural dynamics of individual molecules and nanoparticles Neutze et al. (2000). Over the last decade, ground-breaking experiments were performed at the Free-Electron Laser at DESY in Hamburg (FLASH) Andruszkow et al. (2000); Chapman et al. (2006); Barty et al. (2008); Jiang et al. (2010) and the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory Emma et al. (2010); Bostedt et al. (2013); Young et al. (2010); Chapman et al. (2011); Seibert et al. (2011); Glownia et al. (2010); Boll et al. (2013). These experiments already begin to provide new insights into fundamental aspects of matter, such as hitherto unobserved structures of non-crystallizable mesoscopic objects Barty et al. (2013); Loh et al. (2012); Gorkhover et al. (2012) or the radiation damage induced by the short and very strong x-ray pulses Barty et al. (2012); Young et al. (2010). However, the path to actual determination of atomically resolved structures and dynamics of single molecules is still long Barty et al. (2013). Nevertheless, related experiments on the investigation of small-molecule structures and their dynamics utilizing molecular ensembles are within reach Filsinger et al. (2011); Barty et al. (2013).
To be able to record structural changes during ultrafast molecular processes under well-defined conditions it was proposed Filsinger et al. (2011); Barty et al. (2013) to spatially separate shapes von Helden et al. (1995), sizes Trippel et al. (2012), or individual isomers Filsinger et al. (2009a); Kierspel et al. (2014); Filsinger et al. (2008) of complex small molecules before delivery to the interaction point of an XFEL. The molecules should be one- or three-dimensionally aligned or oriented in space Stapelfeldt and Seideman (2003); Spence and Doak (2004); Spence et al. (2005); Peterson et al. (2008); Glownia et al. (2010); Pabst et al. (2010); Filsinger et al. (2011); Hensley et al. (2012); Boll et al. (2013). This controlled-delivery approach would allow for the averaging of many identical patterns, similar to recent electron diffraction experiments on aligned CFI Hensley et al. (2012) or to photoelectron imaging of 1-Ethynyl-4-fluorobenzene Boll et al. (2013). A controlled variation of the alignment direction in space allows to tomographically build up the complete three-dimensional diffraction volume of individual isomers. This ensemble- and pulse-averaging approach would allow working at appropriately low fluences to circumvent detrimental electronic damage processes that have been predicted Lorenz et al. (2012); Ziaja et al. (2012) for the very high x-ray fluences necessary to obtain classifiable single-molecule diffraction patterns. The forthcoming European XFEL facility will give the opportunity to collect patterns at a rate of 27 000 per second, which could be sufficient to collect the necessary – patterns within minutes or hours Barty et al. (2013).
Here, we record x-ray-diffraction patterns of ensembles of identical, state-selected and strongly aligned 2,5-diiodo-benzonitrile (DIBN, Fig. 1) molecules in the gas phase, demonstrating the applicability of this controlled-delivery approach. Using 2 keV (620 pm) radiation from the LCLS we succeeded to observe the two-center interference between the two iodine scattering centers, separated by approximately 700 pm, in the continuous coherent diffraction pattern. The strongly aligned samples Holmegaard et al. (2009) allow to simply average the continuous diffraction patterns from a very large number of isolated molecules Filsinger et al. (2011); Barty et al. (2013). We restricted the angular control to one-dimensional alignment of the axis containing the two iodine atoms, as this was the solely required control for this experiment. The extension to three-dimensional alignment and orientation is straightforward for the cold, state-selected samples employed Larsen et al. (2000); Nevo et al. (2009); Hansen et al. (2013). Moreover, we have previously demonstrated that for more complex molecules we could also exploit the current setup to spatially separate structural isomers and sizes Filsinger et al. (2009a); Trippel et al. (2012); Kierspel et al. (2014).
The experiment was performed at the AMO beamline at LCLS Emma et al. (2010); Bostedt et al. (2013) using the CAMP endstation Strüder et al. (2010); Foucar et al. (2012) extended by a state-of-the-art molecular beam setup Filsinger et al. (2009b).
Fig. 1 shows a scheme of the experimental arrangement. The setup contains multiple devices to simultaneously detect photons, electrons, and ions Strüder et al. (2010). A pulsed cold molecular beam is formed by expanding a few mbar of DIBN in 50 bar of helium into vacuum through an Even-Lavie valve Even et al. (2000). The molecular beam travels through an electrostatic deflector, which disperses the molecules according to their rotational quantum states, into the target region. There it is crossed by three pulsed laser beams: One laser beam consisting of 12 ns (FWHM) pulses from a Nd:YAG laser (YAG, nm, mJ, m, W/cm) is used to align the molecules. A second laser beam consists of 60 fs (FWHM) pulses from a Ti:Sapphire laser (TSL, 800 nm, J, m, W/cm) and is used to optimize the molecular beam and the alignment without LCLS. The third beam consists of the 100-fs x-ray pulses (LCLS, pm (2 keV), mJ, m, W/cm); we estimate that 35 % of the generated x-ray photons/pulse are transported to the experiment Rudek et al. (2012). All three laser beams are copropagating, overlapped using dichroic (1064 nm and 800 nm) and holey (NIR lasers and x-rays) mirrors before they intersect the sample and finally leave the setup through a gap in an on-axis pnCCD camera and another holey mirror to separate the laser beams again. Time-of-flight and velocity-map-imaging (VMI) spectrometers are installed perpendicular to the horizontal plane of the molecular and laser beams to investigate the ion- and electron-momentum distributions resulting from the Coulomb explosion due to absorption of one or a few x-ray photons.
We exploit Coulomb explosion imaging of DIBN induced either by strong field ionization using the TSL pulse or through one- or two-photon ionization by the x-ray pulse to analyze the alignment of the rotational-state-selected molecules along their I–I axis. The pertinent experimental observable is the emission direction of the recoiling ions from the Coulomb explosion, illustrated by the 2D ion images in Fig. 2.
Without the YAG pulse the images (Fig. 2 a, c) were circularly symmetric as expected for randomly aligned molecules. The circularly symmetric image obtained following ionization with the horizontally polarized LCLS beam demonstrated that the interaction of the far-off resonant radiation with the molecule was independent of the angle between the molecular axis and the x-ray polarization direction: The x-rays were a practically unbiased ideal probe of the spatial orientation of the molecules. When the YAG pulse was included the ions were strongly confined along the YAG polarization axis demonstrating tight adiabatic 1D alignment. From the corresponding 2D momentum distribution shown in Fig. 2 b and d, we extracted and for the TSL and LCLS ionization, respectively. This degree of alignment is in good agreement with previous measurements of adiabatic alignment of similar molecules Holmegaard et al. (2009) and stronger than previous aligment experiments of diatomic molecules at the LCLS Glownia et al. (2010). This demonstrated strong alignment of complex molecules, even under the constraint conditions of a temporary setup at a FEL beamline. It was made possible by the very cold molecular beam and the adiabatic alignment conditions. The demonstrated degree of alignment fulfills the requirements to observe aligned molecule diffraction Spence and Doak (2004); Filsinger et al. (2011); Hensley et al. (2012).
In subsequent experiments we recorded the x-ray diffraction data of these aligned samples on the pnCCD cameras. For these experiments the polarization of the YAG laser was rotated clockwise by . VMI data were repeatedly recorded in between diffraction experiments under the same conditions as in Fig. 2. An average value for the degree of alignment in the diffraction data of was derived, limited by the (changing) spatial overlap of the foci of the YAG and the LCLS beams. The obtained x-ray diffraction patterns are shown in Fig. S1 in the supplementary information (SI). We have analyzed diffraction data for shots with YAG and shots without (NoYAG), respectively, corresponding to 7 h (YAG) and 9 h (NoYAG) measurement time with LCLS operating at 60 Hz. This data is corrected for background and camera artifacts and individual photon hits are extracted (see SI). This results in 0.20 photons/shot which are histogrammed into the molecular diffraction pattern (Fig. S2). By subtracting the diffraction pattern of randomly-oriented molecules () from the diffraction pattern of aligned molecules (), the background is cancelled. This includes the isotropic background originating from atomic scattering of the atoms in the DIBN molecule and the helium seed gas, as well as experimental background, e. g., scattering from apertures and rest gas.
In Fig. 3 we present these diffraction-differences () for simulated (Fig. 3 a, c) and experimentally observed (Fig. 3 b, d) x-ray diffraction patterns. The data has been scaled to match the number of shots in the case. The anisotropy mainly originates in the scattering interference of the two (heavy) iodine atoms. Parts of the zeroth order scattering maximum and the first minimum (along the alignment direction ) show up most prominently on the bottom pnCCD panel. The simulated image has been calculated for a molecular beam density M of DIBN molecules of cm. The error bars correspond to the statistical errors from the subtraction (). The histograms Fig. 3 (c–d) visualize the angular anisotropy which is well beyond the statistical error in the experimentally observed image (Fig. 3 d), confirming the observation of x-ray diffraction from strongly aligned samples of DIBN.
To analyze which structural information can be derived from the x-ray diffraction of isolated DIBN molecules, the intensity in dependence of the scattering vector along the alignment direction is compared to simulated models of different iodine-iodine distances. is the scattering angle and is the angle between the beam direction and a given detector point Waasmaier and Kirfel (1995). Ab initio calculations (GAMESS-US MP2/6-311G** Schmidt et al. (1993)) predict a value of 700 pm for the iodine-iodine distance. Fig. 4 shows the experimentally obtained intensity profiles , averaged over , together with simulated profiles.
Each curve is normalized to be independent of the exact molecular beam density M of DIBN molecules, which merely changes the contrast, i. e., the depth of the minimum. Due to the relatively long wavelength (620 pm) compared to the known iodine-iodine distance (700 pm), the scattering extends to large angles and the first scattering maximum from the iodine-iodine interference is not covered by the detector in our setup. The experimentally obtained is best fitted for an iodine-iodine distance of 800 pm. Fig. 4 shows the simulated for iodine-iodine distances of 500, 700, 800, and 1000 pm. The inset of Fig. 4 depicts the calculated -values Nakamura (2010) in dependence of the iodine-iodine distance. Due to the experimental parameters, as mentioned above, the scattering features are large and vary only slightly within the recorded range of -values. We note that the structural features of small molecules, like DIBN, could be determined much more accurately with data recorded at shorter wavelength where the available range extends to cover several maxima/minima. This would be possible at wavelength of 200–100 pm, which became available at LCLS recently and will be available at upcoming facilities, e. g., the European XFEL, in the near future.
We do not observe direct signs of radiation damage in the diffraction data. While previous experiments aimed specifically at the investigation of x-ray induced damage in strongly focused x-ray beams Young et al. (2010); Erk et al. (2013), here, we have actively avoided that regime and performed the experiments using a hundred times larger cross-section of the x-ray beam. Under these moderate-fluence conditions the damage can be rationalized based on simple cross-section estimates for photoionization and elastic scattering and is detailed in the supplementary information. Since the sample is replenished for every XFEL pulse, the diffractive imaging signal is only sensitive to the dynamics of damaged molecules during the x-ray pulse ( fs). Using a simple mechanical model we estimate that most ( %) of the diffraction signal is due to (practically) intact molecules. A minor fraction of the signal is due to damaged molecules with small changes in molecular structure, which could not been resolved with the available x-ray wavelength. Damage could even be mitigated using shorter ( fs) duration pulses; see supplementary information for details. Moreover, an appropriate trade-off between pulse duration, pulse energy, and repetition rate would allow the recording of atomically resolved x-ray diffraction patterns of molecules within minutes Barty et al. (2013). At these high repetition rates one could directly observe femtosecond molecular dynamics through snapshots for many time-delays in pump-probe experiments of electronic-ground-state chemical dynamics.
In summary, we demonstrate the preparation of strongly aligned samples of polyatomic molecules at an XFEL facility. We experimentally verify that the high-frequency, far off-resonant x-rays are an ideal probe of alignment of molecular ensembles in an photoion momentum imaging approach. The employed setup and conditions are applicable for coherent diffractive imaging of single biomolecules or molecular ensembles. We show the possibility to perform spatially resolved single x-ray photon counting. Due to the weak scattering signal from small isolated molecules, averaging of many shots is necessary and possible for the observation of an analyzable diffraction signal, on top of a large background from NIR photons. We confirm that the angular structures in the single molecule diffraction patterns were preserved during averaging and that a diffraction pattern of isolated and strongly aligned DIBN molecules was successfully measured beyond experimental noise. Even with the experimentally limited range of scattering vectors , the heavy-atom distance derived from the -plot is in agreement with the computed molecular structure, demonstrating the capability to extract structural information for small molecules.
Our results provide direct evidence for the feasibility of x-ray diffractive imaging of aligned gas-phase ensembles of molecules. Analyzing radiation damage in detail shows that damage effects in the diffraction pattern could be avoided by using shorter x-ray pulses with lower fluences at higher repetition rates. This would allow to observe atomically resolved snapshots of ultrafast chemical dynamics. Combined with advanced molecular beam delivery techniques, e. g., laser desorption or helium droplet beams, considerably larger molecules could be delivered in cold beams, isomer selected, and aligned, providing a bottom-up approach toward the envisioned atomic-resolution single-molecule diffraction experiments. In contrast to ultrafast electron diffraction, pump-probe experiments with x-ray pulses will not suffer from Coulomb-repulsion broadening or pump-probe velocity mismatch and hence may permit better time resolution, i. e., in the range of 10-100 fs.
Parts of this research were carried out at the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory. LCLS is an Office of Science User Facility operated for the U. S. Department of Energy Office of Science by Stanford University. We acknowledge the Max Planck Society for funding the development and operation of the CAMP instrument within the ASG at CFEL. H. S. acknowledges support from the Carlsberg Foundation. C. C. and P. J. acknowledge support from the Swedish Research Council and the Swedish Foundation for Strategic Research. J.C.H.S. and H.N.C. acknowledge NSF STC award 1231306. A.Ru. acknowledges support from the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy. D. R. acknowledges support from the Helmholtz Gemeinschaft through the Young Investigator Program. This work has been supported by the excellence cluster “The Hamburg Center for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale” of the Deutsche Forschungsgemeinschaft.
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X-ray diffraction from isolated and strongly aligned gas-phase molecules with a free-electron laser
Appendix A Analysis of the x-ray diffraction data
Diffraction data was obtained for shots for aligned (YAG) and for shots for unaligned (NoYAG) molecules, respectively. This corresponds to 7 h (YAG) and 9 h (NoYAG) measurement time with LCLS operating at 60 Hz. Fig. 5 show example single shot data for NoYAG () (a) and YAG () (b) respectively. Different sources of experimental background and pnCCD artifacts were taken into account in order to clean the single shot data and extract single photon hits from the single shot data frames. The data contains a pnCCD-channel dependent offset and gain variation, time-wise readout-variations (common mode), and some channels (horizontally) or rows (vertically) with unusual high or low signal. The latter are marked as “bad pixel regions” and were excluded from all further steps in data analysis. The pnCCD detectors could not be fully shielded against NIR photons from the high-power YAG, resulting in severe background signal levels and saturation of the lines closest to the central gap and at the outer edges of the detector as can be seen in the single shot data in Fig. 5 b for the YAG case. We extracted hitlists of x-ray photons after carefully subtracting all backgrounds and data-acquisition-based artifacts from the pnCCD data frames. The saturated regions close to the inner edge as well as hot pixels of the pnCCDs are neglected. The resulting data of x-ray hits contain further real experimental background contained in these images, mostly due to scattering from apertures, remaining helium seedgas, and rest gas (diffused helium, residual air) in the chamber.
For a 2 keV photon the generated charge cloud created in silicon can spread over multiple pnCCD pixels. This is considered in the analysis and such events are joined to single hits. In our experiments, 64 % of the recorded 2 keV photons are single-pixel hits, 35 % are double pixel hits, and % are spread over more pixels. Only single and double pixel hits were considered as scattered x-rays in the analysis. A detailed account of this analysis will be published elsewhere Stern et al. (2013). Overall, this process results in lists of individual photons and their position on the pnCCDs, with 0.20 x-ray photons per shot that are elastically scattered from the molecular beam. These are histogrammed into diffraction patterns representing the incoherent sum of the coherent x-ray-diffraction patterns of all molecules in the interaction volume.
In Fig. 6 spectra of the completely cleaned data are shown for (a) and (b) respectively. 2 keV x-ray hits are expected at 2 600 ADU. The width of the spectrum is due to variations of the cleaned single shot frames after background subtraction, the energy spread of the pnCCD detector, and the thresholding of the event recombination process when a single photon hit is spread over more than a single pixel. The spatial distribution of all hits in the energy interval 1 500–3 200 ADU is shown in Fig. 6 c–d. A difference of the and data is only weakly visible, even at strong degrees-of-alignment in our experiment. In order to visualize the anisotropy, on top of a strong isotropic signal from the He, the atomic scattering and experimental background, the isotropic part of the data was removed by obtaining the diffraction-difference pattern. Therefore, after the NoYAG data (, i. e., isotropically distributed DIBN) was scaled to the number of shots of the YAG data (, i. e., aligned DIBN), the NoYAG diffraction data was subtracted from the YAG data. The resulting differencepattern is shown in Fig. 3 b in the main manuscript.
Appendix B Simulated x-ray-diffraction pattern
Simulated diffraction patterns were calculated using an atomistic approach in which the target molecule 2,5-diiodobenzonitrile (CHIN, DIBN) is considered to consist of single atomic scatterers at fixed positions according to the calculated molecular structure. The molecular scattering factor was derived by calculating the sum of all atomic scattering factors times their phase factors , hence Als-Nielsen and McMorrow (2001). We used atomic scattering factors from ref. Waasmaier and Kirfel, 1995 and dispersion corrections to the scattering factors from ref. Henke et al., 1993. For a given set of parameters, i. e., photon energy, number-of-photons per XFEL-pulse, number-of-shots, number density M of target molecules in our molecular beam and the given geometry of the detector, the expected number of x-ray photons scattered to a certain detector area was calculated. The experimental diffraction pattern of an ensemble will be blurred with respect to the diffraction pattern from a single molecule due to the finite (i. e., non-perfect) alignment. The diffracted intensity from a non-perfectly aligned molecular ensemble is obtained by the convolution of diffraction patterns of possible orientations distributed and an alignment angular distribution . The latter is given by Friedrich and Herschbach (1995), where is the angle between the axis given by the (linear) YAG polarisation and the molecular axis of highest polarizability, which is approximately, within a few degree, the same as the I–I axis. In order to analyse the experimentally obtained diffraction pattern, diffraction of aligned and not- aligned DIBN with a strong background from the He seed gas was simulated. Furthermore, for fitting the main structural feature, i. e., the I–I distance, to the measured diffraction data (Fig. 4 in main text), the iodine-iodine distance was varied in the range of 500–1000 pm.
Fig. 7 shows simulated data for the parameters of the experiment (563,453 shots as in the YAG case, 4.37510 incident photons/shot). The molecular beam density of DIBN is cm. The degree-of-alignment of DIBN in the YAG case was with respect to the axis of the YAG polarisation of . The first row (a, b) shows the scattering intensity (in terms of photons/pixel) for DIBN only. White rectangular frames mark the position of the top and bottom pnCCD panel in the experiment. The second row (c, d) shows the scattering intensity of DIBN (like in a, b), but with a strong background of 14 000 He atoms added per DIBN. The amount of He was chosen to match the total measured intensity (with keeping the molecular beam density M of DIBN at cm). It should be noted that in the experiment the amount of He is lower due to strong contributions from other sources of background, such as rest-gas scattering, stray light from apertures, etc., which here is effectively simulated by the He signal. Unfortunately, the experimental background without molecular beam was not measured sufficiently long, but the exact amounts of He and other background sources become obsolete when the data is analyzed in terms of the diffraction-difference, i. e., when taking the difference image of aligned and isotropically-distributed DIBN ().
In the lower row (e, f), the simulated intensities in the NoYAG and YAG case are shown for the regions covered by the pnCCD panels in the experiment. The alignment of DIBN results in a slight shift of the 0-order scattering maximum towards the left with respect to the center, but the effect is hardly visible due to the background. The difference (and hence the anisotropy in the YAG case) can be visualized best by plotting the difference of both diffraction patterns as shown in (g–i).
Appendix C Ionization and radiation damage
Ionization and two-dimensional momentum distributions of ions were measured to quantify the degree of alignment as shown in Fig. 2 in the main manuscript. When utilizing the FEL instead of the TSL, the two peaks (d) were further separated than the ones in (b), with the latter corresponding to ions from doubly and triply ionized molecules Larsen:JCP111:7774. Thus, the DIBN molecules accumulate overall higher charges when ionized by the x-rays, indicating molecular radiation damage processes and the possibility to observe these processes in ion-imaging experiments Erk et al. (2013). In time-of-flight measurements we observed iodine ions in charge states up to I; I cannot be observed because it is hidden by the much larger O signal from residual air. With a 2 keV photon creating an M-shell vacancy in iodine, one expects four or more charges per photon Kochur et al. (1994), which are quickly redistributed over the molecule Erk et al. (2013). From the observed monotonically decaying intensities of the I ion signals, we conclude that typically one and, generally, at most two photons are absorbed per molecule. Calculation of the probability of a single iodine absorbing a 2 keV photon and subsequently getting ionized during a single FEL shot is based on the photoabsorption cross section of iodine Mbarn Berger et al. (2010). The probability is given by with the number of photons impinging on the interaction area . With the FEL-pulse energy in the interaction region of 1.4 mJ, the number of 2 keV photons is . The interaction area ( m) is cm. Hence, the probability for a single iodine atom to absorb a 2 keV photon from the FEL pulse photons is and since the a DIBN molecule is made of 2 iodines, the probability for a DIBN molecule to absorb a 2 keV photon is 0.5. Thus, half of the DIBN molecules absorb a 2 keV photon during the FEL pulse and will eventually be multiply ionized by Auger relaxation and fragmented due to Coulomb explosion. The diffraction pattern of fragmenting DIBN will look different than the diffraction from intact DIBN.
Here, only nuclear damage is considered. In the following, the influence of scattering from fragmenting DIBN on the diffraction pattern is discussed in terms of an effective spatial distribution of the two main scattering centers (i. e., the two iodine atoms) “seen” by the FEL. The velocity distribution of ions upon ionization by the FEL is obtained from the measured momentum distribution shown in Fig. 2 of the main text. This distribution can be well approximated by a Gaussian with mean m/s and m/s. Since one is interested in the relative velocity of the two iodines (rather than in the velocity of in the lab frame) it is assumed that DIBN fragments into and CHIN. Momentum conservation yields a velocity distribution of the relative velocity of with respect to \ceC7H3IN^+n with m/s and m/s. The following discussion is restricted to this simple case since a complete velocity distribution of all ions has not been determined experimentally.
The velocity distribution is utilized in order to estimate an effective spatial distribution of the two iodine atoms in DIBN as seen by the FEL pulse. The pulse duration was 100 fs, estimated from the electron bunch length and pulse duration measurements Düsterer et al. (2011). In Fig. 8 the fractions of intact and ionized DIBN over the duration of the x-ray pulse is plotted. At each time the effective I–I distribution seen by the incident x-ray intensity is calculated. The effective I–I distribution depends on all times at which molecules were ionized and started to recoil with the given velocity distribution:
where and are the fractions of molecules that are ionized and intact at time , respectively. The amount of ionized molecules at depends on all times before, hence . The onset of Coulomb explosion is assumed to happen instantly upon absorption of a 2 keV photon, neglecting the finite delay due to Auger decay and subsequent charge reorganisation within the DIBN molecule. This is a worst case approximation which rather overestimates the distance of the fragmenting ions. The complete effective spatial distribution seen by the whole FEL pulse is then just the sum over all
Fig. 9 a shows a histogram of to visualize the fraction of molecules in different distance intervals as “seen” by the FEL pulse. The cumulative distribution is given in Fig. 9 b. For a FEL pulse duration of 100 fs (solid-blue) this shows that 75 % of the elastically scattered photons originate from scattering at intact molecules and that another 15 % (20 %) of the scattered photons originate from scattering of molecules corresponding to I–I distances that are less than 200 pm (330 pm) longer than the 700 pm-equilibrium distance. These changes are not significant in the current experiment, i. e., while the resolution using 620 pm wavelength radiation (and the limited range of -values) is not high enough to analyze these changes, the experimentally observed elongated I–I distance could be due to these damage effects.
If shorter x-ray pulses, e. g., of fs duration, would be used (dashed-green line in Fig. 9 b) practically no damage would be observed even for the same pulse energy. Thus, for the diffraction from ensembles of isolated “small” molecules even the dose delivered, in a single pulse, by a state-of-the art very intense x-ray laser is small enough to record diffraction patterns without radiation damage. For molecules with larger photoabsorption cross sections, the pulse energy could always be reduced to recover this regime, at correspondingly extended averaging times. With upcoming high-repetition rate light sources, for instance the European XFEL in Hamburg, Germany, this would allow the recording of atomically resolved x-ray diffraction patterns of molecules within minutes Barty et al. (2013). Moreover, at these high repetition rates one could directly observe femtosecond molecular dynamics through snapshots for many time-delays in pump- probe experiments of electronic-ground-state chemical dynamics.