Molecular Solvation Dynamics from Inelastic X-ray Scattering Measurements

ion of spherically symmetric hydration shells that rigidly follow a moving charge, these results indicate that chargemovement stronglymodifies the hydration structure, which evolves from a closed spherical shell to a cylindrical hydration “sleeve” with cylindrical symmetry. Finally, we discuss the strengths and weaknesses of GFID in the context of this “proof of concept” example. A. Dynamical Response Function Extraction from meV IXS Measurements The dynamic structure factor of water S(q, ω) was measured for energies to 80 meV over a q range from 0.2 to 7.2 Å−1, as described elsewhere [53]. Most of the data weremeasuredwith incident X-ray energy of 21.747 keV ( E ≈ 1.7meV for the Si(11,11,11) reflection). To improve counting statistics, data for q > 6.0 Å−1 were measured with the higher intensity Si(9,9,9) reflection with incident energy 17.794 keV ( E ≈ 3.0 meV). For large q, S(q, ω) has a broad shape, making this relatively small resolution difference insignificant. Example raw IXS spectra are shown in Fig. 6a. Corrections for sample holder scattering and different Figure 6. (a) Individual S(q, ω) scans from liquid water measured at beamline ID-28, European Synchrotron Radiation Facility. For q 2π/d, where d is the average interparticle spacing, S(q, ω) has a characteristic Brillouin line shape: a quasielastic peak centered at ω ≈ 0 and the Stokes and anti-Stokes features indicating the collective modes of the system. For q 2π/d, S(q, ω) appears as a Gaussian line shape (e.g., spectra q = 6.3 Å−1) that is centered on higher energies for increasing q. For large values of q, S(q, ω) reflects the momentum distribution of particles in a liquid [79]. For room-temperature liquid water, d = 2.8 Å. (b) χ′′(q, ω) from applying the Bose factor n(ω) to the complete measurement of S(q, ω). While χ′′(q, ω) is shown to only ω = 30 meV, it was measured to ωmax = 30 meV to assure that all features contained in χ′′ are captured in the data. Adapted from Ref. [53]. Copyright 2009 by the American Physical Society. molecular solvation dynamics from ixs 101 measurement efficiencies between analyzers were made to the raw measurements. The result is an experimental measurement of the complete dynamic structure factor of liquid water over the energy and momentum ranges relevant to molecular reorganization. As described in detail in the previous section, the function S(q, ω) is a measure of the correlation of density fluctuations in a given medium [36]. It is related to the imaginary part of the linear response function χ(q, ω) = χ′(q, ω)+ ıχ′′(q, ω) by the fluctuation-dissipation theorem, χ′′(q, ω) = −π [S(q, ω)− S(q,−ω)] (30) A few technical considerations need to be accounted for before χ can be reconstructed from measurement. IXS measurements with meV energy resolution have a low count rate due to the extreme reduction in intensity by the high-resolution monochromator. Experimentally, the compromise between measured energy range and counting statistics must be optimized to the objective of the measurement. As shown in the previous section, the detailed balance condition S(q,−ω) = e−βωS(q, ω) can be used to evaluate the energy loss from energy gain measurements. The counting statistics can be improved by measuring as little of the anti-Stokes part of each spectra as possible. For the measurement used in the following examples, the data were measured from−20 to∼80 meV to observe the full quasielastic line. At ω = 80 meV, the measured intensity is essentially at background levels. The quasielastic line is narrow in energy spread (width <1 meV) but has a high intensity. This feature is broadened by the instrumental resolution function, leading to an artificial contribution to the S(q, ω) measurements away from ω = 0. Because of this, Lorentzian fits to the elastic line are subtracted from the data to remove artifacts. Finally, we evaluate Eq. (30) experimentally by dividing the ω > 0 portion of S(q, ω) by the Bose factor n(ω) = (1− e− ω/kT )−1. Because the imaginary part of the response function is odd, we impose the condition χ′′(q,−ω) = −χ′′(q, ω) on the ω > 0 data (Fig. 6b). We calculated χ′(q, ω) from χ′′(q, ω) using KK relations, as previously described [53]. Fourier transforms require that the argument functions be defined on an infinite continuous domain. We extended the data onto a continuous interval using linear interpolation to avoid artifacts from finite, discrete data sets. The IXS data at the end points of the measurements in q and ω are essentially featureless and at background count levels. Moreover, extrapolation of the data beyond the maximum energy measured ωmax is also necessary because KK relations are defined as integrals from −∞ to ∞. Numerical truncation of the integral at ωmax causes artificial oscillations in the transformation with period 2π/ωmax to appear, influencing the characterization of any physical features. To avoid these artifacts, the data are extrapolated essentially to infinity in energy using the DHO model fit parameters described above. The form of the extrapolation affects density 102 r. h. coridan and g. c. l. wong fluctuations at much higher frequency than the temporal resolution of ourmeasurement. The best-fit DHO model parameters were consistent with those reported in other IXS experiments on water. The known acoustic phonon mode with a sound velocity of 3100± 150 ms−1 is observed at low q values as expected (see Fig. 5 for comparison) [29, 30]. S(q, ω) data are measured by scanning the energy transfer ω at a fixed momentum transfer q. For each measured spectrum, the KK relation is applied to the measured and extrapolated χ′′ to recover the real part χ′(q, ω): χ′(q, ω) = − 1 π ∫ ∞ −∞ dω′χ′′(q, ω′)P ( 1 ω − ω′ ) (31) The full complex-valued χ(q, ω) is then Fourier transformed,