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# Ultrafast large-amplitude relocation of electronic charge in ionic crystals

Edited by Franz Kaertner, Center for Free-Electron Laser Science, Germany, and accepted by the Editorial Board February 7, 2012 (received for review May 23, 2011)

## Abstract

The interplay of vibrational motion and electronic charge relocation in an ionic hydrogen-bonded crystal is mapped by X-ray powder diffraction with a 100 fs time resolution. Photoexcitation of the prototype material KH_{2}PO_{4} induces coherent low-frequency motions of the PO_{4} tetrahedra in the electronically excited state of the crystal while the average atomic positions remain unchanged. Time-dependent maps of electron density derived from the diffraction data demonstrate an oscillatory relocation of electronic charge with a spatial amplitude two orders of magnitude larger than the underlying vibrational lattice motions. Coherent longitudinal optical and tranverse optical phonon motions that dephase on a time scale of several picoseconds, drive the charge relocation, similar to a soft (transverse optical) mode driven phase transition between the ferro- and paraelectric phase of KH_{2}PO_{4}.

Ionic crystals are characterized by a periodic arrangement of positive and negative ions with the electronic charges essentially localized at the ionic sites. For a unit cell geometry with a finite electric dipole moment, such materials are ferroelectric and display a macroscopic electric polarization (1). Ferroelectrics have received strong attention both from the viewpoint of their basic properties (1, 2) and for device applications (3). A prototype class of ionic ferroelectrics are hydrogen-bonded potassium dihydrogen phosphate KH_{2}PO_{4} (KDP) and its isomorphs in which the electric polarization originates from a diplacement between the units and the K^{+} cations along the *c*-axis of the orthorhombic crystal structure (4⇓⇓⇓⇓–9). At a critical temperature of *T*_{C} = 123 K, the macroscopic polarization disappears and KDP transforms into a paraelectric tetragonal phase (Fig. 1*A*). The microscopic mechanisms behind this phase transition have remained controversial (cf. *Materials and Methods*).

A fundamental issue for understanding the relation between structure and (ferro)electric properties consists in the interplay of a change of ionic positions and/or hydrogen bond geometries on the one hand with relocations of electronic charge on the other (9, 10, 11). So far, most experimental and theoretical work has addressed this problem under quasi-static conditions, close to equilibrium, and/or on time scales much longer than ionic motions (12). Here, we introduce femtosecond X-ray powder diffraction (13, 14) as a real-time probe of coupled vibrational and charge motions occurring on a time scale between approximately 100 fs and a few picoseconds. As X-rays interact with the electronic charges in the KDP crystallites, a series of diffraction patterns taken with a time resolution of 100 fs allows for deriving the momentary ionic positions and the charge distributions simultaneously.

## Results and Discussion

The experiments make use of a pump–probe scheme in which ionic motions and the concomitant charge relocations are induced by electronic excitation via two-photon absorption of sub-50 fs pulses centered at a photon energy of 4.5 eV (wavelength 266 nm). The photoinduced structure changes are mapped by diffracting synchronized hard X-ray pulses (Cu Kα, *E* = 8.04 keV) of 100 fs duration (15, 16) from the excited powder sample. The pattern of diffracted X-rays consists of Debye–Scherrer diffraction rings that are recorded with a large-area CCD detector. A series of such patterns measured at different delay times after excitation allows for reconstructing the momentary nuclear positions and charge distributions. Further details of the experiment and the methods applied for data analysis are discussed in *Materials and Methods*.

Results of the femtosecond experiments are summarized in Fig. 2. Fig. 2*A, Inset* shows a diffraction pattern of the unexcited KDP powder sample as recorded with the femtosecond X-ray pulses. For an integration time as short as 7 min, one clearly distinguishes 11 different Debye–Scherrer rings. In Fig. 2*A*, the diffracted intensity integrated over the individual rings is plotted as a function of 2*θ*, *θ* being the diffraction angle. For an assignment of the different rings, the data were analyzed by calculations of the diffraction patterns based on the known tetragonal crystal structure of KDP (6) (Fig. 1*A*).

Upon photoexcitation of the powder sample, less than 5% of the crystallite volume is excited (see below) and the ring pattern displays changes of the diffracted intensities of up to 5 × 10^{-3}. Within the experimental accuracy of 0.4 °, the angular positions of the rings remain unchanged. In Fig. 2 *B*–*D*, the change of diffracted intensity integrated over the (011), (020), and (132) ring is plotted as a function of delay time between the optical pump and the X-ray probe pulses. We measured a total of ≈800 frames collected in 15 independent scans. To get a high signal-to-noise ratio we averaged the intensity changes of 46 identical and/or neighboring time delays. As a result every data point shown in Fig. 2 is averaged over an integration time of more than 5 h. The residual noise in the experimental data (cf. error bars in Fig. 2) is essentially the shot noise of detected X-ray photons. For all measured reflections the maximum time-resolved signal is at least 5 times larger than the corresponding error bars.

The data show a fast onset of Δ*I*(*hkl*)/*I*_{0}(112) on a subpicosecond time scale, superimposed by pronounced oscillatory intensity changes. We note that a sensitivity Δ*I*/*I*_{0} of much better than 1% was achieved. We also analyzed the transient X-ray intensity between the allowed reflections. For the same amount of averaging as done in Fig. 2 the intensity (integrated over the 2*θ* range of a typical allowed reflection) *I*_{outside}(*θ*,*t*) shows just noise as a function of both *θ* and *t* with a standard deviation of only .

Fig. 2*E* shows a result of the all-optical experiment, the transient absorption at a photon energy of 3 eV of a KDP crystal excited at 4.5 eV. Around delay zero, one finds a strong negative peak that is probably due to a coherent coupling of pump and probe pulses and followed by a step-like onset of absorption. The absorption increase occurs on a 100 fs time scale. The enhanced absorption decays within 200 μs.

Optical two-photon excitation with pump pulses at 4.5 eV bridges the fundamental band gap of KDP of *E*_{g} ≈ 7–8 eV (17) and generates an electronic excitation that has the character of a Frenkel exciton with a large binding energy of the order of ≈1 eV. Similar to alkali halides, such a Frenkel exciton is essentially localized (18) in a volume ( with the exciton Bohr radius *a*_{Bohr}) that covers 10 to 100 unit cells of the crystal lattice. From such numbers and the absorbed pump flux, we estimate the upper limit of 5% of the total crystallite volume that is electronically excited (see above). Electronic excitation is connected with a change of the potential energy surfaces of phonon modes that couple to the electronic transition in particular the longitudinal optical (LO) and transverse optical (TO) soft mode (10). Such impulsive change that occurs well within our time resolution of 100 fs, leads to the displacive excitation of coherent wavepacket motions along the coupling low-frequency phonon coordinates. The wavepacket motions are underdamped and decay on a time scale of a few picoseconds. The electronic excitation decays on a much longer nanosecond time scale (18), reestablishing the equilibrium charge distribution in KDP.

To confirm this excitation scheme, we performed experiments with pump pulses of 3 eV photon energy that do not allow for efficient electronic excitation but can generate coherent phonon motions in the electronic ground state via an impulsive stimulated Raman process. In such experiments, a transient change of the X-ray diffraction pattern was absent, pointing to a negligible role of ground state phonon motions for the observed structural dynamics.

The change of electronic structure upon photoexcitation and the concomitant generation of coherent lattice motions result in transient changes of the X-ray diffraction pattern. In the time-resolved diffraction data (Fig. 2), the angular positions of the diffraction rings remain unchanged up to our longest delay times of approximately 40 ps. This fact shows that the size of the unit cell and the average distances between the heavier atoms, i.e., K, P, and O, undergo minor changes upon electronic excitation.

The change of the diffracted X-ray intensity Δ*I*_{hkl}(*t*)/*I*_{0}(112) is determined by the change of the structure factor F: [1]where is the structure factor of the excited material with and *η* ≪ 1 being the structure factor and the fraction of excited unit cells. and are structure factors of the unexcited crystal. The distribution of electronic charge *ρ*(**r**,*t*) = *ηρ*_{ex}(**r**,*t*) + (1 - *η*)*ρ*_{0}(**r**) is linked to the structure factor by [2] To derive the transient charge density distribution *ρ*(**r**,*t*) from the measured intensity changes, one has to solve the so-called phasing problem, i.e., to determine the phase of the complex *F*_{hkl}(*t*). In *Materials and Methods*, we describe a method providing such phases for the specific case of KDP that lacks inversion symmetry. This analysis gives the quantity *η*Δ*ρ*(**r**,*t*) = *ρ*(**r**,*t*) - *ρ*_{0}(**r**), the change of the charge density times the fraction of excited unit cells.

Steady-state and transient charge density maps are summarized in Figs. 1 and 3. Fig. 1*B* shows the charge density *ρ*_{0}(**r**) of the unexcited KDP unit cell for the plane defined in Fig. 1*A*. Fig. 1*C* shows an isosurface of the three-dimensional equilibrium electron density for a volume containing the plane and the complete and K^{+} ions. This charge distribution was calculated from the atomic positions determined by neutron scattering (6) and the respective atomic form factors. After electronic excitation, one observes a pronounced relocation of electronic charge. In Fig. 3 *A*–*D*, the change of charge density *η*Δ*ρ*(**r**,*t*) = *ρ*(**r**,*t*) - *ρ*_{0}(**r**) is shown for delay times *t* = -1.7, 0.4, 1.1, and 2.5 ps. The electron density on the P atom displays a strong decrease, feeding a transfer of charge to the oxygen atoms of the ions. As a function of time, the charge density undergoes oscillatory relocations with a spatial amplitude of the order of 100 pm.

The signal-to-noise ratio of the transient charge density maps can be directly inferred from the difference maps *η*Δ*ρ*(**r**,*t*) at negative delay times as shown for *t* = -1.7 ps in Fig. 3*A*. A more detailed analysis gives |*η*Δ*ρ*(**r**,*t* < -1 ps)| < 1.5*e*^{-}/nm^{3} and |*η*Δ*ρ*(**r**,-1 ps < *t* < -0.2 ps)| < 4*e*^{-}/nm^{3}, which is much smaller than *η*Δ*ρ*(**r**,*t*) at positive delays.

The ultrafast charge relocations were analyzed in detail by (i) integrating *η*Δ*ρ*(**r**,*t*) over the volume *V*_{A} of a particular atom *A* giving the total charge change *η*Δ*q*_{A} and (ii) by determining its center of gravity **R**_{A} within this volume (see *Materials and Methods*). In Fig. 4*A*, we show results for an OH unit of the anion. The elongation projected on the *Y* direction (Fig. 4 *A, Inset*) and the direction perpendicular to the P-O-H-O plane is plotted as a function of delay time. Pronounced low-frequency oscillations are observed with a spatial amplitude of the order of 0.2 pm, which is characteristic for low-frequency phonon motions in the crystal lattice. It is important to note that such amplitudes are much smaller than the length scale of charge relocations in Fig. 3, which is of the order of 100 pm, i.e., of a chemical bond length.

Fig. 4*B* shows the spatially integrated charge changes *η*Δ*q* on the P atom [Fig. 4*B* (blue line)] and on an OH unit [Fig. 4*B* (red line)] as a function of delay time. The charge on the P atom displays an oscillatory decrease reaching its final value after a few picoseconds whereas the charge on the OH group increases in an oscillatory fashion. This is equivalent to a net charge flow on the length scale of the P-O chemical bond, much longer than the vibrational amplitudes in Fig. 4*A* and results in a strong change of the P-O dipole moment. The right-hand ordinate scales of Fig. 4 give the absolute values of the P-O dipole change corresponding to (Fig. 4*A*) the vibrational amplitudes and Fig. 4*B* the charge relocation. Such numbers give direct evidence that contributions due to nuclear motions to the transient electric polarization are very small compared to the effects of charge relocation.

The frequency spectrum of the oscillations in Fig. 4 *A* and *B* was determined by two different methods of spectral analysis, a linear interpolation to generate regularly spaced points as input for a fast Fourier transform and the method of ref. 19, both giving the same result. The Fourier spectra in Fig. 4*C* display frequency components of approximately 30, 55, and 85 cm^{-1}. We assign the latter two to the TO and LO phonon in electronically excited KDP [ground state frequencies: *ν*_{TO} ≈ 75 cm^{-1}, *ν*_{LO} ≈ 115 cm^{-1} (10, 11)]. The relative amplitudes of the three components are influenced by the finite time resolution of the experiment, leading to a reduced amplitude at higher frequency. For our time resolution and signal-to-noise ratio (error bars in Fig. 2), the oscillation frequency *ν*_{LO} = 85 cm^{-1} is close to the detection limit with three data points per oscillation period only.

The presence of the 55 and 85 cm^{-1} components in the Fourier spectrum demonstrates that the TO and LO phonon modes are coherently excited upon photoexcitation. The 30 cm^{-1} contribution is identical to the difference of the TO and LO frequencies and does not exist as a normal mode of KDP. There are two mechanisms from which such a component can originate. First, anharmonic coupling via higher-order terms in the vibrational potential, in the simplest case proportional to or , causes oscillations at the sum and difference frequencies. So far, there is no information on higher-order terms of vibrational potentials in the electronically excited state of KDP, making a quantitative assessment of such couplings difficult. Second, one has to consider the pecularities of transient charge density maps derived from X-ray diffraction patterns. The charge density maps reflect the symmetry of excited unit cells averaged over all possible manifestations of the structure change (14). The symmetry properties of such averaged maps can be quite different to those of local, individual unit cells. For instance, coherent excitation of any infrared active mode polarized along the *c*-axis will break the local initial inversion symmetry of 2D projections of individual unit cells but will preserve the inversion symmetry of projections of the globally averaged electron density on the *yz* and *zx* planes, as measured by X-ray diffraction. As a result, particular atoms in the averaged map split into two “copies” with 50% probability, each performing motions that are a nonlinear superposition of the normal modes of the crystal. This nonlinear relation results in sum and difference frequencies when several normal modes are excited simultaneously, and the observed frequency spectrum depends on the particular atom under investigation. The atomic elongations considered in Fig. 4 have the full symmetry of the initial KDP structure, which is different from that of TO and LO normal modes at *q* = 0 (i.e., B_{2} symmetry), the latter including complex motions of all atoms within unit cells (10). Thus, a 30 cm^{-1} component in the Fourier spectrum is expected and the spectra for different projections in Fig. 4*C* are not identical.

We now address a key finding of our experiments, the concerted motions of nuclei and electronic charge on very different length scales. The microscopic physics can be visualized with the help of the soft mode theory of ferroelectricity developed by Cochran (20). This approach treats the combination of electronic polarizations, i.e., charge transfer, and lattice polarizations, i.e., ionic displacements, within a common concept. Ionic displacements are described by normal modes of the crystal, including the so-called soft mode that displays strong changes of vibrational frequency with temperature. The soft mode in KDP is the TO phonon at 75 cm^{-1} (11). The simplest model describing ferroelectricity in KDP is schematically explained in Fig. 1*D*. The key constituents are the K^{+} cation (*q*_{K} = *Ze*_{0}) and a strongly polarizable anion consisting of a H_{2}PO_{4} ion core (*q*_{C} = *Xe*_{0}) surrounded by a spherical electron shell (*q*_{e} = *Ye*_{0}) with *X* + *Y* + *Z* = 0 (overall charge neutrality). There are three Bravais lattices, consisting of (i) the positive ions, (ii) the cores, and (iii) shells of negative ions. The forces are separated into Coulomb forces that are described by the macroscopic polarization and its corresponding Lorentz fields measured at the respective quasi-cubic environments of the cores and shells, and short range forces between ions and shells specified by two force constants *R*_{0} and *k* with *R*_{0} ≪ *k*. (We use here the notation of Cochran’s paper but the international system of units (20)). *U*_{1}, *U*_{2}, and *V*_{2} stand for deviations from the respective equilibrium positions. Our main interest lies in the relation between the difference *U*_{1} - *U*_{2}, which is a representative for all nuclear motions connected to the soft mode, and the separation of the electron shell from the anion core *V*_{2} - *U*_{2} (charge transfer) that can be derived from the following two equations describing the balance of forces and the polarization: [3][4]Expressing the result in terms of the electronic polarizability of the anion *α*_{e} = (*Ye*_{0})^{2}/(*k* + *R*_{0}) we obtain the following relation: [5]For *Z* ≈ +1, *Y* ≈ -1 and a very small polarizability per unit volume *α*_{e}/3*ε*_{0}*V* ≪ 1 the crystal is paraelectric and the term *R*_{0}/(*k* + *R*_{0}) ≪ 1 dominates in Eq. **5**. In this case, the electronic charge transfer contribution to a vibrational dipole is small compared to that connected to the anion-cation displacement. For *α*_{e}/3*ε*_{0}*V* ≈ 1, however, relation **5** completely reverses its behavior. Here, the electronic charge transfer contribution to the TO soft mode dipole is huge in comparison to that of the anion-cation displacement and may even diverge.

The key predictions of the Cochran theory are borne out by our experimental results. Electronic excitation of KDP enhances the polarizability *α*_{e}, bringing the excited system closer to the *α*_{e}/3*ε*_{0}*V* ≈ 1 limit. Coherent vibrational motions along the TO phonon (soft mode) coordinate that display a small spatial amplitude are connected with charge oscillations at the same frequency and with a two order of magnitude larger spatial amplitude, i.e., a huge electronic polarization. Our experiments give insight into the ultrafast dynamics of this mechanism under nonequilibrium conditions and provide quantitative information on the intrinsic length scale of charge relocation. In this sense, we show that the microscopic mechanisms invoked by Cochran are essential for understanding the transient behavior in electronically excited states. Such phenomena are, however, fundamentally different from the structural phase transition of KDP that occurs in the electronic ground state under thermal equilibrium conditions. In addition to elongations along normal modes, the temperature dependent entropy, e.g., in the form of an order–disorder transition, has been invoked to explain the thermal phase transition of KDP (5).

In conclusion, we have demonstrated an ultrafast relocation of charge that is induced by electronic excitation and steered by coherent low-frequency motions of the ionic crystal lattice. Our results reveal the nonequilibrium charge dynamics in ionic materials on their intrinsic length and time scales. Such insight is relevant for a large range of polar materials and their ultrafast dynamic response. In principle, low-frequency lattice motions can be controlled by interaction with tailored optical pulses. Our results suggest that such schemes allow for an ultrafast all-optical control of electric polarization in KDP and related ionic materials.

## Materials and Methods

### Phase Transitions in KDP.

Below *T*_{C} = 123 K, KDP exists in a ferroelectric orthorhombic phase in which the tetrahedral PO_{4} units are linked by O-H-O hydrogen bonds and the structure is stabilized by the ionic interaction (5). The two hydrogens per PO_{4} unit are part of two hydrogen bonds connecting the PO_{4} group with neighboring PO_{4} units in the crystal plane above or below. Within an O-H-O bond, the H atom forms a covalent bond with the neighboring O atom of the PO_{4} groups, corresponding to a asymmetric localization within the double minimum O-H-O potential. At *T*_{C} = 123 K, KDP becomes paraelectric and the crystal symmetry changes to tetragonal with space group , (No. 122, no inversion symmetry). The unit cell dimensions at *T* = 298 K are *a* = *b* = 7.4529 *Å*, *c* = 6.9751 *Å* (6). This unit cell comprises four formula units. In the tetragonal paraelectric phase existing above *T*_{C}, average hydrogen positions in the center of the O-H-O bonds have been found.

The mechanisms behind the ferro- to paraelectric phase transition have been controversial for a long time. On the one hand, the ferro- to paraelectric transition has been attributed to an order–disorder transition of proton positions in the double-minimum potential along the O-H-O hydrogen bonds, resulting in statistically oriented local dipole moments and, thus, the disappearance of the macroscopic polarization (5). This picture is challenged by experimental work giving evidence of a proton delocalization along the O-H-O bonds above *T*_{C} (7). Recent ab initio calculations suggest a prominent role of electronic charge redistributions and ionic displacements that are linked to proton ordering in the ferroelectric phase (9, 21). Lattice vibrations involved in the phase transition have been studied by steady-state infrared and Raman spectroscopies. Pronounced changes of frequency and transition dipole strength at the phase transition have been observed for a number of low-frequency modes, in particular the TO mode at 55 cm^{-1} in the paraelectric phase (10).

### Experimental Methods.

The KDP powder sample (Alfa Aesar, 99,999% purity, less than 10 ppm Si) was carefully pulverized down to a medium crystallite size of less than 1 μm and subsequently sandwiched between a 20 μm thick diamond window and a thin mylar foil. The sample thickness was 100 μm. The sample temperature was kept constant at *T* = 350 K.

The experiment was performed in an optical pump/X-ray probe scheme (13, 22). The sample was excited by a sub-50 fs pulse at 266 nm, the third harmonic of amplified pulses from a Ti:sapphire laser system (1 kHz repetition rate, 35 fs pulse duration, 800 nm central wavelength), and the resulting structure changes were mapped by diffracting a 100-fs hard X-ray pulse from a laser-driven Cu Kα (wavelength 0.154 nm) plasma source from the excited sample. The diffracted X-rays were recorded with a large-area CCD detector. In the optical excitation process, the two-photon interband absorption of KDP was exploited (band gap is ≈7–8 eV (17)). The pump intensity was about 4 × 10^{11} W/cm^{2} and, by considering the two-photon absorption coefficient *β*_{KDP} = 2.7 × 10^{-4} cm/MW (23), one obtains an absorption length of about 100 μm that is longer than the elastic scattering length of the pump light of approximately 50 μm.

In addition to the X-ray diffraction study, we performed an all-optical pump–probe experiment to measure the transient absorption of the electronically excited state (24). The sample was a (100)-cut KDP crystal of 500 μm thickness held at *T* = 300 K. The pump pulses centered at 4.5 eV had a duration and intensity (3 × 10^{11} W/cm^{2}) similar to those in the X-ray diffraction experiment and the induced absorption changes were probed by 50 fs pulses at 3 eV (400 nm).

### Analysis of X-ray Diffraction Data.

Recently, we have reconstructed the transient three-dimensional electron density *ρ*(*x*,*y*,*z*,*t*) of a crystal structure with inversion symmetry from measured femtosecond powder diffraction patterns (14). The analysis exploits the interference of X-rays diffracted from the small fraction of excited unit cells and from the majority of unexcited ones within a single crystallite to solve the phase problem. Building on that concept, we now present an extension of the method that allows for reconstructing transient charge density maps of crystals without inversion symmetry such as KDP. The present analysis is based on three fundamental assumptions. (i) The crystal structure [neutron diffraction data (6)] and, thus, the charge density map [calculated by means of atomic (ionic) form factors for Cu Kα radiation] of the initial structure are known. (ii) The fraction *η* of excited unit cells is small, i.e., *η* ≪ 1. (iii) Inversion symmetry of two-dimensional projections of *ρ*(*x*,*y*,*z*,*t*): Whereas the three-dimensional KDP structure (space group ) lacks inversion symmetry, the projections of the charge density *ρ*(*x*,*y*,*z*,*t*) onto the (*x*,*y*), (*y*,*z*), and (*z*,*x*) planes display inversion symmetry relative to an inversion center in the respective plane. We assume that this inversion symmetry of the projections is preserved after photoexcitation. This is justified by the fact that the selection rules and probability of photoexcitation are identical for the (projected) structural units and their inverted counterparts, and that the diffraction pattern reflects the statistical average of all possible manifestations of the structure change in excited unit cells (cf. Ref. (14), section II.C). The absence of any forbidden reflections in the diffraction pattern supports this assumption.

An ensemble-averaged projection onto the, e.g., *yz*-plane is defined as [6]and the projections on the three planes are given by where and are the projections for excited and unexcited unit cells.

Linearizing Eq. **1** gives the following implicit equation for the change of a complex structure factor : [7]Φ_{F0} and Φ_{ΔF} are the phases of and Δ*F*_{hkl}. In the X-ray data analysis we applied a projection method that is explained in Fig. 5. The outer and inner circle is determined by |*F*_{hkl}| and , respectively. From the time-resolved data we obtain |*F*_{hkl}|, whereas is known from the literature. The aim of the procedure is to find the best approximation for *η*Δ*F*_{hkl}, i.e., to decide which one of the small green arrows is compatible with the symmetry properties of the excited crystal structure. Similar to earlier work (25, 26) the projection method simply states exp 2*i*(Φ_{F0} - Φ_{ΔF}) = 1, i.e., the best *η*Δ*F*_{hkl} compatible with all symmetry operations is as shown in Fig. 5 (red arrow). Because of the inversion symmetry of the projections [Eq. **6**], this description gives exact amplitudes and phases of *F*(0,*k*,*l*,*t*), *F*(*h*,0,*l*,*t*), and *F*(*h*,*k*,0,*t*).

For structure factors *F*(*h* ≠ 0,*k* ≠ 0,*l* ≠ 0,*t*), the phase has been determined independently in the following way. If the charge clouds of any two different atoms within the unit cell show a negligible overlap along any of the three projection directions, a three-dimensional auxiliary function can be used that exhibits the same peaks as the two-dimensional projections [8]In KPD, however, each K atom aligns with a P atom along the c-axis. Thus, we identify the K atoms by their (transient) integrated charge observed in the and planes only, in which no alignment occurs. This information is sufficient to derive the position of the K atoms within the unit cell. Our data show a center of gravity of charge of the K atoms that remains unchanged at all pump–probe delays. Thus, we subtract the K atoms from , etc., and apply [Eq. **8**] for reconstructing . Finally, is calculated from the known *τ*^{′}(*x*,*y*,*z*,*t*) and *τ*_{K}(*x*,*y*,*z*,*t*). A Fourier back transform of gives the phase of the transient structure factors *F*(*h* ≠ 0,*k* ≠ 0,*l* ≠ 0,*t*). This procedure confirms that exp 2*i*(Φ_{F0} - Φ_{ΔF}) ≈ 1 in Eq. **7** is a very good approximation also in such cases.

Next, we describe how the transients of Fig. 4 were derived from *ρ*(*t*). We divided the volume of the unit cell into subvolumes each containing one atom (on top of the oxygen charge, we assign 50% of the charge of an H atom to the volume of the O atom) by assigning each spatial position *x*,*y*,*z* within the unit cell uniquely to the subvolume of the nearest atom defined by the nuclear positions of K, P, or O of the initial structure. As a result we get a set of polyhedrons as integration volumes *V*_{A} for calculating the transient charge change of atom *A*: [9]For moderate charge changes on each atom, i.e., , the change of the average atomic position is given by [10]This approximated relation for position changes of atoms holds if the motion of atoms in the excited unit cells is small compared to the distance between atoms: for all atoms *B* ≠ *A*. This condition is fulfilled for all vibrational motions of the lattice.

Finally, Eqs. **9** and **10** can be used to calculate two different contributions of a particular ion to the electric dipole change of the structure. In Fig. 4 the right ordinates of *A* and *B* show that the estimated contribution of the P-OH charge transfer to the P-O dipole change *η*Δ*d*_{P-O} is ≈30 times larger than that due the spatial motion of the OH-unit. This statement is based on the following equation: [11]with the elementary charge *e* and the atomic number of atom *A*, *Z*_{A}. Please note, that the ratio of the two contributions does not depend on the fraction of excited unit cells *η*. Thus, the main conclusions of the paper do not require an exact knowledge of *η*.

## Acknowledgments

This research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 247051 and from the Deutsche Forschungsgemeinschaft (Grant WO 558/13-1).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: woerner{at}mbi-berlin.de.

Author contributions: M.W. and T.E. designed research; F.Z., P.R., and J.S. performed research; F.Z., P.R., J.S., and M.W. analyzed data; and F.Z., M.W., and T.E. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. F.K. is a guest editor invited by the Editorial Board.

Freely available online through the PNAS open access option.

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