Download Femtosecond powder diffraction with a laser-driven hard X

Femtosecond powder diffraction with a
laser-driven hard X-ray source
F. Zamponi, Z. Ansari, M. Woerner, T. Elsaesser
Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Str. 2A, 12489
Berlin, Germany
[email protected]
Abstract: 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.
© 2009 Optical Society of America
OCIS codes: (000.2170) Equipment and techniques; (340.7480) X-rays, soft X-rays, extreme
ultraviolet (EUV); (320.2250) Femtosecond phenomena
References and links
1. A. Rousse, C. Rischel, and J. C. Gauthier “Femtosecond X-ray crystallography,” Rev. Mod. Phys. 73, 17 (2001).
2. C. Rose-Petruck, R. Jiminez, T. Guo, A. Cavalleri, C. Siders, F. Raksi, J. A. Squier, B. C. Walker, K. R. Wilson,
and C. P. J. Barty “Picosecond-milliangstrom lattice dynamics measured by ultrafast X-ray diffraction,” Nature
398, 310 (1999).
3. K. Sokolowski-Tinten, C. Blome, J. Blums, A. Cavalleri, C. Dietrich, A. Tarasevitch, I. Uschmann, E. Förster,
M. Kammler, M. Horn-von-Hoegen, and D. von der Linde, “Femtosecond X-ray measurement of coherent lattice
vibrations near the Lindemann stability limit,” Nature 422, 287 (2003).
4. M. Bargheer, N. Zhavoronkov, Y. Gritsai, J. C. Woo, D. S. Kim, M. Woerner, and T. Elsaesser, “Coherent atomic
motions in a nanostructure studied by femtosecond X-ray diffraction,” Science 306, 1771–1773 (2004).
5. A. M. Lindenberg, J. Larsson, K. Sokolowski-Tinten, K. J. Gaffney, C. Blome, O. Synnergren, J. Sheppard, C.
Caleman, A. G. MacPhee, D. Weinstein, and others, “Atomic-Scale Visualization of Inertial Dynamics,” Science
308, 392–395 (2005).
6. M. Braun, C. v. Korff Schmising, M. Kiel, N. Zhavoronkov, J. Dreyer, M. Bargheer, T. Elsaesser, C. Root, T.
E. Schrader, P. Gilch, W. Zinth, and M. Woerner, “Ultrafast changes of molecular crystal structure induced by
dipole solvation,” Phys. Rev. Lett. 98, 248301 (2007).
7. N. C. Woolsey, J. S. Wark, and D. Riley, “Sub-nanosecond X-ray powder diffraction,” J. Appl. Cryst. 23, 441–443
(1990).
8. S. Techert, F. Schotte, and M. Wulff, “Picosecond X-ray diffraction probed transient structural changes in organic
solids,” Phys. Rev. Lett. 86, 2030 (2001).
9. S. Techert, and K. A. Zachariasse, “Structure determination of the intramolecular charge transfer state in crystalline 4-(diisopropylamino) benzonitrile from picosecond X-ray diffraction,” J. Am. Chem. Soc 126, 5593
(2004).
10. P. Debye, and P. Scherrer, “Interferenzen an regellos orientierten Teilchen im Röntgenlicht,” Phys. Z. 17, 277
(1916).
11. B. E. Warren, “X-ray Diffraction,” (Courier Dover Publications, 1990).
12. C. Blome, T. Tschentscher, J. Davaasambuu, P. Durand, and S. Techert, “Femtosecond time-resolved powder
diffraction experiments using hard X-ray free-electron lasers,” J. Sync. Rad. 11, 483–489 (2005).
13. F. Zamponi, Z. Ansari, C. v. Korff Schmising , P. Rothhardt, N. Zhavoronkov, M. Woerner, T. Elsaesser , M.
Bargheer, T. Trobitzsch-Ryll, and M. Haschke, “Femtosecond hard X-ray plasma sources with a kilohertz repetition rate,” Appl. Phys. A 96, 51–58 (2009).
14. N. Zhavoronkov, Y. Gritsai, M. Bargheer , M. Woerner, Th. Elsaesser, F. Zamponi, I. Uschmann, and E. Förster,
“Microfocus Cu K al pha source for femtosecond X-ray science,” Opt. Lett. 30, 1737 (2005).
15. P. Gibbon, and E. Förster, “Short-pulse laser-plasma interactions,” Plasma Phys. Control. Fusion 38, 769 (1996).
16. F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett. 59, 52–55 (1987).
17. E. O. Schlemper, and W. C. Hamilton, “Neutron-Diffraction Study of the Structures of Ferroelectric and Paraelectric Ammonium Sulfate,” J. Chem. Phys. 44, 4498 (1966).
18. S. Ahmed, A. M. Shamah, A. Ibrahim, and F. Hanna, “X-ray studies of the high temperature phase transition of
ammonium sulphate crystals,” Phys. Status Solidi (a) 115, K149 (1989).
19. H. Rietveld, “A profile refinement method for nuclear and magnetic structures,” J. Appl. Cryst. 2, 65 (1969).
20. R. Brun and F. Rademakers, “ROOT-An object oriented data analysis framework,” Nucl. Instrum. Methods A
389, 81–86 (1997).
21. C. G. Ryan, E. J. Clayton, W. L. Griffin, S. H. Sie, and D. R. Cousens, “SNIP, a statistics-sensitive background
treatment for the quantitative analysis of PIXE spectra in geoscience applications,” Nucl. Instrum. Methods B
34, 396–402 (1988).
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 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.
2.1.
X-ray powder diffraction with femtosecond time resolution
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 | = |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
2 re2V λ 3 M(hkl)Fhkl
1 + cos2 (2θ )
Pscatt = I0
,
(1)
4v2a
2 sin θ
where I0 is the incident X-ray intensity and Pscatt is the scatterd X-ray power, c is the velocity of
light, λ is the X-ray wavelength, re is the classical electron radius, V = ρpowder /ρmaterialVcrystal
is the effective illuminated volume with ρpowder /ρmaterial ≈ 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 3dimensional 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
Fig. 1. (Color online) 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.
(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.
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 μ m for a pumping wavelength of 400 nm.
Fig. 2. (Color online) (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 μ m (FWHM) and
up to 1018 W/cm2 , respectively. The 20 μ m 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 μ m thick scotch 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 μ m 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 Xray 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
Laser pulse energy
Repetition rate
Laser pulse duration
Laser intensity
Cu Kα photon energy
X-ray photons emitted in 4π
Source dimension
X-ray pulse duration
X-ray photons on sample
X-ray spot dimension on sample
2.2.2.
5 mJ
1 kHz
35 fs
≤ 1018 W/cm2
8.04 keV
4 × 1010 ph/s
10±2 μ m
100-200 fs
≈ 106 ph/s
30-200 μ m
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 μ m (FWHM), depending on the particular optic chosen.
The sample geometry is shown in Fig. 2b. The powder sample of 1 mm diameter and 250 μ m
thickness is contained in a metallic sample holder. The powder is sealed by a 20 μ m thick
polycrystalline diamond front window (Diamond Materials GmbH, Germany) and a 6 μ m 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 at 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 )2 SO4 , AS hereafter] at room
temperature (T = 300 K); AS has an ionic orthorhombic structure with four (NH4 )2 SO4 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 μ m.
In the femtosecond experiment, the AS sample is electronically excited by a 50 fs pump pulse
at 400 nm (pulse energy 75 μ J) 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 μ m, rather than the optical penetration depth of
200 μ m. On the 40 μ m 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.
The X-rays diffracted from the sample were detected with a large-area CCD detector (quantum efficiency 0.8, 6 × 6 cm2 sensitive area, 2000 × 2000 pixels, 30 μ m 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.
Fig. 3. (Color online) 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.
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 F200 = 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 Nscatt = 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 ΔN/Ntot = 1/ Ntot . For Ntot =50000, the relative uncertainty has a
value of 4.47 × 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/I0 can be measured in a time-resolved
pump-probe experiment.
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
Fig. 4. (Color online) 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 μ m. 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 μ m, to be compared with the measured
line width of time-resolved changes of (200) at t = 50 fs.
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 μ m and the sample thickness of 250 μ m,
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 μ m (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
ΔI(θhkl ,t)
I(θhkl ,t) − I(θhkl )−∞<t<0
=
,
I(θ111 )−∞<t<0
I(θ111 )−∞<t<0
(2)
is plotted versus pump-probe delay. Here, I(θhkl ,t) is the diffracted intensity measured for a
delay time t and I(θhkl )−∞<t<0 is the respective intensity averaged over a range of negative delay times. The difference ΔI(θhkl ,t) of such intensities is normalized to the intensity
I(θ111 )−∞<t<0 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., Nscatt ∝ C|Fhkl |2
with C = LP(θhkl )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. (Color online) 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 averaged over all the negative times, I(θ111 )−∞<t<0 . 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:
ex + (1 − η )F gr |2 − |F gr |2
|η Fhkl
ΔI(θhkl ,t)
hkl
hkl
=
C.
I(θ111 )−∞<t<0
I(θ111 )−∞<t<0
(3)
ex and F gr are the structure
where η is the fraction of excited unit cells in the sample and Fhkl
hkl
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 :
gr
ex
η ΔF = η (Fhkl
− Fhkl
)=
ΔI(θhkl ,t)
gr .
2CFhkl
(4)
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
ex .
are known, i.e., a measurement of ΔI(θhkl ,t) gives the transient structure factor Fhkl
From Eq. (4) the change Δρ of the 3-dimensional charge density is calculated:
hx ky lz
η
+ +
η Δρ (x, y, z,t) =
ΔFhkl (t) exp(iφhkl ) cos 2π
(5)
abc ∑
a
b
c
hkl
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
Fig. 6. (Color online) (a) Measured intensity changes at time t = +50 fs, corresponding to
the vertical lineout of Fig. 5a (blue line). The uncertainty of ΔI for three different angles
was estimated by taking the standard deviation of 25 delay points at negative times. (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.
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:
VCT
η Δρ (x, y, z,t) dx dy dz = N · e,
(6)
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 NH+
4 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.
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.
Fig. 7. (Color online) 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.
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. (Color online) 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 twodimensional 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:
if(PixelValue>2.0*MedianValue)
PixelValue=MedianValue
else
PixelValue=PixelValue
The procedure was then repeated for all the pixels in the image. In this way, pixels or
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 as 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:
N
Q = ∑ −Psynth (θi ) · ln(Pmeas (θi )),
i=0
where N = 1200 is the number of bins, Pmeas (θi ) is the measured one-dimensional
powder pattern obtained with the temporary fit parameters and Psynth (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.
• Once the intensity vs. angle plot was ready for each time delay the time series were put
together and I(θhkl )−∞<t<0 (see Eq. 2) was subtracted, as shown in Fig. 5(a).
• A Fourier filter was applied to increase the signal-to-noise ratio.
• A this point lineouts for every angle were taken and time resolved data for the different
reflections were obtained, as shown in Fig. 5b and Fig. 6a.
From the data processed in this way, the charge density maps discussed in section 3 were
derived.