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X-ray powder diffraction with a femtosecond time resolution is introduced to map ultrafast structural dynamics of polycrystalline condensed matter. Our pump-probe approach is based on photoexcitation of a powder sample with a femtosecond optical pulse and probing changes of its structure by diffracting a hard X-ray pulse generated in a laser-driven plasma source. We discuss the key aspects of this scheme including an analysis of detection sensitivity and angular resolution. Applying this technique to the prototype molecular material ammonium sulfate, up to 20 powder diffraction rings are recorded simultaneously with a time resolution of 100 fs. We describe how to derive transient charge density maps of the material from the extensive set of diffraction data in a quantitative way.
©2010 Optical Society of America
1. Introduction
X-ray diffraction is a standard method to determine the atomic structure of condensed matter. Diffracting hard X-rays with a photon energy of the order of 10 keV from crystalline materials allows for determining interatomic distances with an accuracy of a small fraction of a chemical bond length. The intensity of X-ray diffraction peaks is determined by the Fourier transform of the three-dimensional electron density. Thus, spatially resolved maps of electronic charge density can be derived from X-ray diffraction patterns. There is a variety of powerful X-ray diffraction methods such as Bragg diffraction of monochromatic X-rays from single crystals or polycrystalline materials, the latter representing the so-called Debye-Scherrer method, as well as Laue diffraction from single crystals, using X-rays in a wide spectral range.
In recent years, X-ray diffraction with a femtosecond time resolution has been implemented to observe structural dynamics in real-time, i.e., on the time scale of atomic motions [1, 2, 3, 4, 5, 6]. Such studies are based on pump-probe schemes where a femtosecond optical pulse induces a structural change in the sample which is mapped by diffracting a femtosecond hard X-ray pulse from the excited sample. The measurement of a sequence of diffraction patterns as a function of pump-probe delays allows for reconstructing the real-time changes of atomic structure and - in principle - of the charge density distribution. So far, mainly Bragg diffraction from single crystals has been applied and changes of angular position and intensity of single or a small number of Bragg peaks have been measured for a limited range of samples. A much wider class of materials, in particular molecular crystals, is accessible by powder diffraction. Up to now, the highest time resolution in powder diffraction has been in the sub-nanosecond regime where atomic motions cannot be accessed directly [7, 8, 9].
In this paper, we present, to the best of our knowledge, the first experimental implementation of femtosecond powder diffraction. We address key aspects of the method which is based on the application of a laser-driven plasma source for ultrashort hard X-ray pulses, and discuss diffraction results for a molecular prototype material, the highly polar ammonium sulfate. The analysis of such results allows for the reconstruction of transient electron density maps. The paper is organized as follows. In section 2, we briefly recall basic principles of X-ray powder diffraction and present the femtosecond powder diffraction technique, including a discussion of its sensitivity and angular resolution. The analysis of transient diffraction data and the reconstruction of transient charge density maps are addressed in section 3. A summary and a brief outlook are given in section 4.
2. X-ray powder diffraction with femtosecond time resolution
2.1. The Debye-Scherrer method
Debye and Scherrer [10] first analyzed X-ray scattering from randomly oriented crystallites. In a powder sample of crystallites, a particular set of lattice planes with the Miller indices (hkl) and the reciprocal lattice vector G(hkl) show all orientations in the full solid angle 4π. In X-ray diffraction (Fig. 1), a subset of crystallite orientations is selected via the Bragg condition for the wavevectors k of the incoming X-ray beam, k′ of the diffracted X-rays, and the reciprocal lattice vector G(hkl): G(hkl)=k′-k with |k| = |k′| for elastic scattering. This condition defines a cone of reciprocal lattice vectors G(hkl) contributing to the diffracted signal and, thus, a diffraction cone defined by k and k′ with an opening angle 2θ. Different sets of lattice planes give rise to different diffraction cones as illustrated in Fig. 1. On a planar X-ray detector, e.g., an X-ray CCD, one detects diffraction rings with a diameter determined by the angle 2θ.
The number of photons scattered per unit time into a particular ring is given by Nscatt = Pscatt/(hc/λ) where
where I 0 is the incident X-ray intensity and Pscatt is the scattered X-ray power, λ is the X-ray wavelength, re is the classical electron radius, V = ρ powder/ρ material V crystal is the effective illuminated volume with ρ powder/ρ material ≈ 0.8 for usual powders, M(hkl) is the multiplicity of the reflection, va is the unit cell volume and Fhkl is the structure factor for X-ray scattering [11]. The structure factor is proportional to the Fourier transform of the 3-dimensional density of electronic charge with respect to the reciprocal lattice vector G(hkl). The last term of Eq. (1) (in parentheses) is the Lorentz-polarization factor. For a wide range of materials, the diffraction efficiency into a single Debye-Scherrer diffraction ring has values between 10-3 and 10-5.
Compared to Bragg diffraction from single crystals, the Debye-Scherrer method offers a number of advantages [11, 12]. Large crystals of good quality are not needed. Instead, any sample can to be ground in small crystallites. The alignment of the diffraction set-up is greatly simplified due to the random orientations of crystallites in the sample. A large number of reflections/diffraction rings is measured simultaneously with the limited area of the detector being the main restricting factor. The method also offers a straightforward normalization method: single reflections can be normalized to the total diffracted signal and, in this way, the influence of fluctuations of the incident X-ray flux on the diffracted intensities can be limited.
There are, however, also drawbacks compared to Bragg diffraction from single crystals. Since the incoming X-ray photons are diffracted into all diffraction rings fulfilling the respective Bragg conditions, the signal on a single ring is smaller than for a Bragg peak from a single crystal. To exploit the full potential of powder diffraction by measuring many diffraction rings simultaneously, highly sensitive large-area detectors with high quantum efficiency and low noise are required.
2.2. Femtosecond powder diffraction
Femtosecond photoexcitation of a powder sample can induce different types of ultrafast structural modifications resulting in characteristic changes of the X-ray diffraction pattern. In the lower panels of Fig. 1, prototype reversible changes of the crystal lattice are illustrated schematically. The diffraction pattern (bottom panel) of the unperturbed crystallite (a) is shifted to new angular positions when the spatial separation of lattice planes is changed by applying external stress or by changing the temperature (b). Propagation of acoustic phonons through the crystallite (c) results in a broadening of the diffraction peaks whereas optical phonon excitations (d) are localized and modify the peak intensities via changes of the structure factor. Beyond such lattice motions, photoinduced changes of the electronic charge distribution in the material change the structure factor and, thus, the diffracted intensities as well.
Fig. 1. Upper panel: sketch of the Debye-Scherrer method. Lower panel: possible laser-induced crystal changes, adapted from [1]. The black and the red dots mimic two different kinds of atoms in a crystal. The lower plots show how the powder pattern would look like for two arbitrary reflections. (a) unperturbed crystal. (b) expansion (contraction) due, e.g., to an increase (decrease) of the temperature; the peaks shift correspondingly. (c) propagation of acoustical phonon, compression followed by expansion; the peaks get broader. (d) optical-phonon-like excitation of the crystal; only the ratio of the peak intensity is changed. This process can be very fast in low-Z materials.
In femtosecond powder diffraction, the photoinduced structural changes are monitored by hard X-ray probe pulses diffracted from the excited sample for different pump-probe delays. This method requires synchronized optical and hard X-ray pulses which, in our approach, are both derived from the output of an amplified Ti:sapphire laser system. A schematic of the experimental set-up is shown in Fig. 2a. The femtosecond laser system provides sub-50 fs pulses of up to 5 mJ energy per pulse with a repetition rate of 1 kHz. A minor part of the laser output (5 %) is reflected by a beamsplitter and serves for generating femtosecond pump pulses. Both the laser fundamental at 800 nm and frequency converted pulses in a wide wavelength range from the mid-infrared to the ultraviolet can be applied for inducing structural dynamics in the powder sample. The interaction length l in the sample is limited by elastic scattering of the pump light. Typical values are l ≤ 50 μm for a pumping wavelength of 400 nm.
Fig. 2. (a) Schematic of the experimental set-up for femtosecond powder diffraction. Explanation in the main text. (b) Sample geometry used in the femtosecond experiments. (c) Debye-Scherrer diffraction pattern from an ammonium sulfate powder sample recorded with a large-area CCD detector. The exposure time was 420 s.
2.2.1. Laser-driven plasma source of ultrashort hard X-ray pulses
The major part (95 %) of the laser output is focused onto a Cu tape target to generate probe pulses at the characteristic Cu Kα photon energy of 8.05 keV (wavelength 0.154 nm) [13, 14]. The spot diameter and the peak intensity on the target have a value of 2.8 μm (FWHM) and up to 1018 W/cm2, respectively. The 20 μm thick Cu tape is moving at a speed of several cm/s in order to provide a fresh target volume for each laser pulse. The target is placed in a vacuum chamber (pressure 10-3 mbar), together with a moving plastic tape that protects the entrance window of the chamber against deposition of Cu debris. The X-rays emitted in forward direction leave the chamber through an exit slit sealed by a second 35 μm thick Kapton tape.
The X-ray emission originates from a laser-generated plasma on the target [15]. In a single optical cycle of the very high laser field, electrons from the plasma are accelerated away from the target and then back onto the target with a net energy gain. This so-called vacuum heating (Brunel effect, [16]) accelerates electrons up to kinetic energies of several tens of keV. Such energetic electrons ionize Cu atoms in the target by generating K-shell holes. Recombination of electrons into the unoccupied K-shell is connected with the emission of characteristic Cu Kα radiation. This X-ray fluorescence is emitted into the full solid angle 4π. The time structure of the emitted X-ray burst is ultrashort because the electrons are accelerated only as long as the driving laser field is present. In parallel with the characteristic X-ray emission, Bremsstrahlung is generated by the electrons decelerating in the target. Under our focusing conditions for the driving laser pulses, the X-ray source diameter has a very small value of 10 ± 2 μm as the electrons do not have sufficient energy to migrate significantly from the interaction region. Such small source diameter facilitates imaging of the emitted X-rays by different types of X-ray optic.
Details of the most advanced version of this kilohertz plasma source have been discussed in Ref. [13]. In Table 1, some of the key parameters are summarized.
Table 1. Femtosecond X-ray plasma source
2.2.2. Set-up for femtosecond powder diffraction
A multilayer X-ray optic focuses the X-rays emitted in forward direction onto the powder sample. The optic collects X-rays in a solid angle of approximately 10-4, resulting in a X-ray flux of 106 photons per seconds on the sample. The X-ray spot size on the sample is between 30 to 200 μm (FWHM), depending on the particular optic chosen.
The sample geometry is shown in Fig. 2b. The powder sample of 1 mm diameter and 250 μm thickness is contained in a metallic sample holder. The powder is sealed by a 20 μm thick polycrystalline diamond front window (Diamond Materials GmbH, Germany) and a 6 μm thick mylar foil on the rear side. In general, heat conduction within the powder sample is rather low. Here, the high thermal conductivity of the diamond window (≈ 3000 W/(m·K)) is favorable for transferring the thermal excess energy originating from the laser excitation of the powder close to the front window to the metallic sample holder which serves as a heat sink. The high mechanical stability and the wide optical transparency ranging from the far-infrared up to the ultraviolet are other favorable properties of the diamond window.
In the following, we present data for ammonium sulfate [(NH4)2SO4, AS hereafter] at room temperature (T = 300 K). AS has an ionic orthorhombic structure with four (NH4)2SO4 entities per unit cell. Under thermal equilibrium conditions, this material undergoes a ferroelectric to paraelectric phase transition at T = 220 K [17], connected with a symmetry change from Pna21 to Pnam. A second phase transition in which the crystal symmetry is preserved, occurs at T = 420 K [18]. For the diffraction experiments, a sample was prepared by manually grinding commercially available AS powder (Alfa Aesar, purity 99.999 %) for about 10 min. The sample was ground in a heated mortar to reduce contamination with air humidity and the resulting grain size was 5 to 10 μm.
In the femtosecond experiment, the AS sample is electronically excited by a 50 fs pump pulse at 400 nm (pulse energy 75 μJ) to induce a non-equilibrium structural change in the femtosecond time domain. The penetration depth into the sample is determined by the elastic scattering length of the pump light of approximately 40 μm, rather than the optical penetration depth of 200 μm. On the 40 μm interaction length, the group velocity mismatch between the optical pump and the hard X-ray probe pulses is much less than the X-ray pulse duration of 100 fs. Sample heating by the pump light was studied with the help of a calibrated thermal camera and a maximum temperature increase of 40 K was found under our excitation conditions.
Fig. 3. Powder pattern in the measured angular range. Black solid line: measured powder pattern with 420 s exposure time. Blue dashed line: calculated powder pattern based on measurements reported in [17]. Red dotted line: synthetic spectrum used to optimize the parameters needed in the 2D-1D data reduction; note that only the exact positions are needed, not the relative intensities as discussed in the text.
The X-rays diffracted from the sample were detected with a large-area CCD detector (quantum efficiency 0.8,6Ö6 cm2 sensitive area, 2000 Ö 2000 pixels, 30 μm pixel dimension). Up to approximately 20 Debye-Scherrer rings were recorded simultaneously as a function of the time delay between pump and probe pulses. For each delay time, the X-ray signals were integrated on the CCD detector for a time interval of 300 to 600 s. The data sets shown in the following consist of up to 10 delay scans.
2.2.3. Powder diffraction with femtosecond Cu Kα pulses
Before discussing time-resolved X-ray data, we analyze diffraction patterns from the AS sample taken with the X-ray probe pulses only. In Fig. 2c, we present the Debye-Scherrer ring pattern measured with an integration time of the CCD detector of 420 s. The individual rings display spots with enhanced intensities, i.e., the ring does not have uniform intensity. This behavior points to an imperfect grinding process: crystallites with different dimensions are present in the powder, resulting in a higher intensity diffracted from crystallites with larger dimensions. In Fig. 3, the intensities integrated over each ring are plotted as a function of the diffraction angle 2θ (thick solid line). To derive this plot, we applied the data reduction method described in the appendix. One easily distinguishes approximately 10 different rings with a good signal-to-noise ratio. The absolute number of photon counts detected on the (200) ring is typically 50000 for a 420 s integration time.
It is interesting to compare the measured photon numbers with the predictions of Eq. (1). Using the diffraction angle 2θ = 22.8°, the form factor F 200 = 86, and the multiplicity M(200)= 2 of the (200) ring, the volume va = 0.496 nm3 of the AS unit cell and the experimental parameters given above, one derives N scatt = 65000 for a 420 s integration time, in good agreement with the experimental value. The relative uncertainty of the number of counts originating from photon statistics is given by For Ntot=50000, the relative uncertainty has a value of 4.47 Ö 10-3 which can be further reduced by integrating the signal over a longer time interval. The statistical uncertainty represents a fundamental limit of the dynamic range over which a photoinduced change of diffracted intensity ΔI/I 0 can be measured in a time-resolved pump-probe experiment.
Fig. 4. In (a) the different contributions to the measured line broadening are sketched. The whole X-ray probe beam can be thought of as composed of many beamlets, each with the divergence of the full beam. Each incoming beamlet (in red) along the path through the powder is diffracted at different depths. Both the divergence of the incoming beam and the finite extension of the probe beam on the sample contributes to the measured line broadening. In (b) the line broadening is modeled by taking into account the probed volume and the X-ray beam divergence. The solid black line is the line width from the unpumped sample as calculated for a sample thickness of 250 μm. The data points are the measured line widths of (200) and (212) reflections. The red dashed line is the line width as calculated by considering a sample thickness of 25 μm, to be compared with the measured line width of time-resolved changes of (200) at t = 50 fs.
Next, we discuss the angular resolution of the diffraction scheme. The measured angular width of the rings is strongly influenced by the divergence of the incoming X-ray beam and by the finite dimension of the probed powder volume. To quantify the angular resolution, Gaussian envelopes were fitted to the data of Fig. 3. For the (200) and the (212) ring, one derives an angular width of 0.57 and 0.55 degrees, respectively.
The two broadening mechanisms are sketched in Fig. 4a. The finite divergence of the incoming X-ray beam results in an angular spread of the diffracted X-rays. The integration of diffracted signal both along the interaction length and laterally along the sample leads to an additional angular broadening on the detector. Using the measured divergence of the incoming X-ray beam of 6 mrad, the focal spot diameter of 200 μm and the sample thickness of 250 μm, we calculated the solid line in Fig. 4b by adding the two contributions in quadrature. The calculated values are in good agreement with the measured width of the (200) and (212) diffraction rings.
In the pump-probe experiments, the penetration depth of the pump pulse and, thus, the sample thickness over which the structural changes occur, are a fraction of the total sample thickness only. As a result, the change of diffracted intensity is integrated over a shorter length and the angular width of the intensity change becomes smaller. The dashed line in Fig. 4b represents the angular width calculated for an effective interaction length of 25 μm (dashed line), a value close to the penetration depth of the 400 nm pump light into the AS powder sample. Again, the calculation reproduces the experimental value for the (200) ring quite well.
The analysis shows that the angular broadening originates mainly from the (lateral) X-ray spot size on the sample. A way to cope with this effect would be to put the detector further away from the sample. In turn, this implies that the detector must be as large as possible to collect diffraction rings at greater 2θ angles. Large distance also means that the signal is spread along a larger ring and, thus, the counts per CCD pixel are lower, requiring a very low noise level of the detector for accurate measurements.
3. Transient diffraction data and reconstruction of charge density maps
Results of the femtosecond diffraction experiment with the AS powder sample are presented in Figs. 5 and 6. In Fig. 5a, the diffracted intensity integrated over the respective diffraction ring is plotted as a function of the pump-probe delay and of the diffraction angle 2θ. There are pronounced changes of intensity with a clear onset around time delay zero and a fast time evolution during the first couple of picoseconds. A cut through this two-dimensional plot along the abscissa gives the time evolution for a particular diffraction angle, i.e., diffraction ring. In Fig. 5b, such a transient is shown for the (111) peak. The normalized intensity change
is plotted versus pump-probe delay. Here, I(θhkl,t) is the diffracted intensity measured for a delay time t and 〈I(θhkl,-∞)〉 is the diffracted intensity before excitation. The difference ΔI(θhkl,t) of such intensities is normalized to the intensity 〈I(θ 111,-∞)〉 of the (strongest) (111) peak. There is a sharp rise of the signal around time delay zero which is shown in more detail in the inset of Fig. 5b. From the temporal width of this rise, one infers a width of the cross-correlation of pump and probe pulses of 120 fs (solid line: calculated time-integrated cross correlation function), i.e., a duration of the X-ray pulses of the order of 100 fs. At longer time delays, the transient displays pronounced oscillations which are due to coherent lattice motions and will be discussed in detail elsewhere.
In Fig. 6a, we present the normalized intensity change measured at a delay of +50 fs as a function of the diffraction angle 2θ (cf vertical line in Fig. 5a). Pronounced intensity changes of up to several percent occur in several diffraction rings. In contrast, the angular positions of the diffraction rings remains unchanged over the full time range plotted in Fig. 5. The latter result demonstrates that the geometry of the unit cell of AS undergoes minor changes, i.e., the heavier sulfur, oxygen, and nitrogen atoms essentially stay at their original positions and volume changes of the unit cell are negligible. Thus, the intensity changes of the different diffraction rings mainly reflect a relocation of electronic charge which is induced by the femtosecond photoexcitation. This non-equilibrium change of electronic structure has not been anticipated and/or observed before and represents a new type of structural change in AS.
We now discuss how to derive transient 3-dimensional charge density maps from the time resolved diffraction data. According to equation (1), the diffracted intensity or number of photons diffracted per unit time is proportional to the square of the structure factor, i.e., N scatt ∝ C|Fhkl|2 with C = LP (θ hkl)M(hkl) containing the Lorentz polarization factor LP and the multiplicity M. To determine the charge density, one needs to derive the structure factor Fhkl = |Fhkl| exp(iϕhkl) from the data. As the diffraction signal gives only the absolute value |Fhkl|, one encounters the so-called ’phase problem’, i.e., the phase ϕhkl remains unknown. In general, recursive methods of data analysis are applied to solve this problem [11]. AS, however, represents a much simpler case. At room temperature, AS displays inversion symmetry. Therefore, ϕhkl = 0 or π, corresponding to a real-valued Fhkl. As the femtosecond excitation of the sample does not affect inversion symmetry, the value of ϕhkl remains unchanged for the excited unit cells.
Fig. 5. Time-resolved signal. (a) 2θ-delay scan. The diffracted X-ray intensity is plotted as a function of the diffraction angle 2θ and the pump-probe delay. (b) Intensity changes (time cut along the red line in plot a) normalized to the intensity of the strongest diffracted peak before excitation, 〈I(θ 111, -∞)〉. Inset: Intensity change around time delay zero. The numerical fit gives a cross-correlation width of about 120 fs.
The change of diffracted intensity is given by:
where ηis the fraction of excited unit cells in the sample and Fhklex and Fhklgr are the structure factors of the excited and the unexcited unit cells. For weak excitation as in our experiments, i.e., η)≪ 1, this expression can be simplified by keeping only terms linear in η:
Fig. 6. (a) Measured intensity changes at time t = +50 fs, corresponding to the vertical lineout of Fig. 5a (blue line). (b) Static structure factor values calculated from [17]. (c) Difference between static and transient structure factor values. Note that the magnitude of ΔF is strongly influenced by the multiplicity.
This equation directly connects the observed changes in the intensity to the change of the structure factor. It should be noted that all quantities in the denominator on the r.h.s. of the equation are known, i.e., a measurement of ΔI(θhkl,t) gives the transient structure factor Fhklex.
From Eq. (4) the change Δρ of the 3-dimensional charge density is calculated:
To calculate the absolute value of Δρ, the fraction of excited crystallites η needs to be determined in the experiment, taking into account boundary conditions set by the physical process under study. For instance, a transfer of electrons over a distance larger than the atomic radius results in a decrease of charge density at the original site and an increase at the new position. The charge integral at the new site has a value identical to the elementary charge e or a multiple of it:
where N is the number of electrons involved in the transfer process. This boundary condition allows for a calibration of η. In our experiments, η has a value of 0.06.
Equilibrium and transient charge density maps of AS are presented in Fig. 7. Fig. 7a shows the unit cell of AS. The light blue plane is parallel to the z-axis and includes the line connecting the hydrogen atoms of opposite NH4 + groups. The equilibrium charge density in this plane is plotted in Fig. 7b. This map was calculated using the atomic positions determined by neutron diffraction [17] and the known form factors of the different atoms. The changes Δρ derived from our diffraction data for a delay time of 50 fs are presented in Fig. 7c. One observes a pronounced increase of electron density in a new highly confined area in the center of the map. This area is a channel-like volume of enhanced electron density parallel to the c axis. This behavior is borne out in more detail by the transient charge density maps in Fig. 8. A detailed analysis including other planes in the unit cell shows a marked decrease of electron density on the sulfur and - to lesser extent - on the nitrogen and oxygen atoms, i.e., a migration of charge from these atoms to the channel is observed.
Fig. 7. 2D projection of the electron density in AS. (a) Unit cell of AS. The shaded plane is parallel to the z-axis, goes through (0.5 a, 0.5 b, 0.5 c) and is tilted by 60° with respect to the x-axis. (b) Electron density as calculated from [17] for this plane. (c) electron density map of the changes Δρ(x,y,z,t) = ρ(x,y,z,t) - ρ(x,y,z, - ∞) are shown at time t = +50 fs.
The results in Figs. 7 and 8 demonstrate the potential of femtosecond powder diffraction to determine ultrafast changes of the 3-dimensional charge density in a quantitative way. A detailed analysis of the physical processes underlying the behavior of AS is beyond the scope of this paper and will be presented elsewhere.
4. Summary
In summary, we have demonstrated femtosecond X-ray powder diffraction using a laser-driven plasma source for Cu Kα pulses with a kilohertz repetition rate. Studying ammonium sulfate powder as a prototype molecular material, we have measured photoinduced changes of diffracted intensity of up to 20 diffraction rings with a time resolution of 100 fs. Relative changes of intensity down to approximately 10-3 can be measured for signal integration times of the order of 2000 s. The angular resolution of the experiment is determined mainly by the X-ray spot size on the sample and has values of about 0.5 degrees. The transient diffraction data for AS display negligible changes of the lattice geometry, but pronounced modifications of the electronic charge distribution upon femtosecond excitation with 400 nm light. Quantitative charge density maps were derived from the transient diffraction data for different time delays after optical excitation, giving detailed insight into the relocation of charge within the unit cell.
Our results demonstrate the potential of femtosecond powder diffraction for studying a wide range of polycrystalline inorganic and organic materials. Even for the comparably light sulfur, oxygen, and nitrogen atoms such as in AS, changes of diffracted intensity are measured with high accuracy for signal integration times of the order of 30 minutes. The procedure for analyzing the data can be further improved by using the Rietveld method [19] to derive transient structure factors and, thus, charge densities with an even better accuracy.
Fig. 8. Δρ(x,y,z,t) for the same plane depicted in Fig. 7a is given for different time delays. This plane is parallel to the z-axis and includes the line connecting the hydrogen atoms of opposite NH+ 4 groups, marked with a dashed circle.
5. Appendix: Analysis of the ring diffraction patterns
The following step-wise procedure was applied to derive the diffracted X-ray intensity as a function of diffraction angle (one-dimensional plot, angle vs. intensity) from the two-dimensional CCD images:
- First, all the unwanted events (hot pixels, hot spots, cosmic rays) were removed or at least reduced. For this purpose a box measuring 11 pixel by 11 pixel around each pixel was considered together with the median value of the box. The final pixel value was then assigned as described in the following pseudo-code: The procedure was then repeated for all the pixels in the image. In this way, pixels or a bunch of pixels with too high a value could be effectively removed.
- The CCD pattern, as displayed in Fig. 2c is the intersection of the diffraction cones similar to those discussed in Fig. 1a (one cone for each reflection) and the detector plane. The geometry of the experiment was modeled by means of 5 parameters (the x and y position of the intersection between X-ray beam and the detector plane, the distance between sample and detector and the two tilt angles of the detector plane). It is important to note that a good parameter set is crucial to reproduce the angular resolution allowed by the experimental set-up. This, in turn, influences the possibility to connect in a clear way certain signal variations at a certain angle to one or more reflections. The better the angular resolution, the simpler and more exact is the reconstruction of the unit cell modifications. To this end, a fit was used to optimize the one-dimensional powder pattern: basically, a family of ellipses was searched that could reproduce, in dependence of the 2θ angle, the measured pattern. A calculated powder pattern was used as a goal (the red-dotted line of Fig. 3). At this point the following was done iteratively until the minimization routine (Minuit [20]) achieved a minimum:
– Given the 5 temporary fit parameters, a 2θ angle was assigned to each pixel;
– A histogram (1200 bins for 2θ in the range 0–1.2 rad) was calculated. The intensity of the pixels was used as a weighting factor.
– Background subtraction. Since the number of summed pixels roughly increases quadratically with the angle, the signal is always distributed on a baseline that grows with the angle. To find out this baseline the SNIP algorithm described in [21] was used. It mainly estimates the angle-dependent background level through a parameter: structures larger than this parameter are considered a part of the background, the rest is signal. We used a value of 50, to be compared with the line width FWHM of about 9.5 pixels; the presence of a triplet was also taken into account.
– The measured powder pattern was compared with the synthetic one; the following quantity Q was used as a measure for the goodness of the parameters:
where N = 1200 is the number of bins, P meas(θi) is the measured one-dimensional powder pattern obtained with the temporary fit parameters and P synth(θi) is the synthetic one-dimensional powder pattern shown in Fig. 3. The logarithm was used to be sensitive also on the weaker peaks; the minus sign was needed because we used a minimization routine.
– If the minimum of Q was not reached, the next best set of parameters for the new iteration was calculated otherwise the program stopped.
- A Fourier filter was applied to increase the signal-to-noise ratio.
From the data processed in this way, the charge density maps discussed in section 3 were derived.
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