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Accurate Measurements of Dielectric and Optical Functions of Liquid
Water and Liquid Benzene in the VUV Region (1−100 eV) Using
Small-Angle Inelastic X‑ray Scattering
Hisashi Hayashi*,‡ and Nozomu Hiraoka†
‡
Department of Chemical and Biological Sciences, Faculty of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo, Tokyo
112-8681, Japan
†
National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan
S Supporting Information
*
ABSTRACT: Using a third-generation synchrotron source (the BL12XU
beamline at SPring-8), inelastic X-ray scattering (IXS) spectra of liquid water
and liquid benzene were measured at energy losses of 1−100 eV with 0.24 eV
resolution for small momentum transfers (q) of 0.23 and 0.32 au with ±0.06
au uncertainty for q. For both liquids, the IXS profiles at these values of q
converged well after we corrected for multiple scattering, and these results
confirmed the dipole approximation for q ≤ ∼0.3 au. Several dielectric and
optical functions [including the optical oscillator strength distribution
(OOS), the optical energy-loss function (OLF), the complex dielectric
function, the complex index of refraction, and the reflectance] in the vacuum
ultraviolet region were derived and tabulated from these small-angle (small q)
IXS spectra. These new data were compared with previously obtained results,
and this comparison demonstrated the strong reproducibility and accuracy of
IXS spectroscopy. For both water and benzene, there was a notable similarity
between the OOSs of the liquids and amorphous solids, and there was no evidence of plasmon excitation in the OLF. The static
structure factor [S(q)] for q ≤ ∼0.3 au was also deduced and suggests that molecular models that include electron correlation
effects can serve as a good approximation for the liquid S(q) values over the full range of q.
■
INTRODUCTION
Electronic excitation in molecular liquids by photons, electrons,
or a combination of these particles is of fundamental
importance in a variety of scientific fields, including radiation
physics,1−12 radiation chemistry,5,12,13 and radiation biology.12,14 Such excitation affects the dielectric and optical
properties of liquids (usually many-electron systems) and
characterizes their electronic responses to incident radiation.
The quantity that determines these properties is the dynamic
structure factor [S(q,E)]:11,15,16
S(q,E) =
∑ pi |⟨f| ∑ exp(iq·rj)|i⟩|2 × δ(Ef
i,f
photon wavelength, respectively. Hereafter, equations will be
expressed in atomic units (au) [i.e., we equate ℏ (the reduced
Planck’s constant), me (the electron rest mass), and e (the
elementary charge) with unity]; consequently, one atomic unit
in E and q corresponds to 27.213 eV and (1/0.5292) Å−1,
respectively.
In atomic and molecular science, it is more common to use a
quantity slightly different from S(q,E), that is, the generalized
oscillator strength distribution [df(q,E)/dE],1,11,15 which is
easily related to S(q,E) by11,15,17
df (q ,E)
2E
= 2 S(q ,E)
dE
q
− E i − E)
j
(2)
(1)
Further, df(q,E)/dE can be normalized to the number of
electrons in the system (N) using the Bethe sum rule for all
values of q:1,11,15,17
Here, the photon energy E and momentum q are transferred to
the irradiated system, where the coordinate of the jth electron
in the system is rj. Then, the system of initial states |i⟩ with
energy Ei and the corresponding probability pi is excited into its
respective final states |f⟩ with energy Ef, which are allowed by
the law of energy conservation: Ef − Ei = E. For isotropic
systems, such as liquids and gases, S(q,E) is spherically
averaged [S(q,E) = ⟨S(q,E)⟩Ω] and depends only on the
magnitude of q (i.e., q = |q|). The q value is approximately given
by q = (4π·sin θ)/λ, where 2θ and λ are the scattering angle and
© 2015 American Chemical Society
∫
df (q ,E)
dE = N
dE
(3)
Received: February 15, 2015
Revised: March 31, 2015
Published: April 2, 2015
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The Journal of Physical Chemistry B
called X-ray Raman scattering15,16,36 and is employed mainly to
study excitations of core electrons (generally having small
values of re), where the condition qre ≪ 1 is fulfilled over a wide
range of q. For valence electrons of molecular condensates (re
≈ 1), the condition qre ≪ 1 is realized for small values of q or
2θ (for q ≤ ∼0.3 au).30,35 Therefore, under the dipole
approximation, the functions for which q = 0 [e.g., S(0,E),
df(0,E)/dE, ε(0,E), and Im(−1/ε(0,E))] can be deduced from
the small-angle IXS spectra. Hereafter, the 0 is omitted when
the “optical” dielectric functions are expressed [e.g., ε(E)].
Although the previous IXS results, which were deduced
about 15 years ago, have been applied in various scientific
fields,9,11−14,37−49 their statistical uncertainties are not negligible, particularly for small values of q. This degree of
uncertainty results because the IXS intensity at small values
of q (for which the intensity is proportional to q2) is inherently
low, which hinders the accurate determination of IXS spectra,
even when second-generation synchrotron sources are used.
In this study, small-angle (small q) IXS spectra for liquid
water and liquid benzene are remeasured with a highly
improved signal-to-noise (S/N) ratio by using a thirdgeneration synchrotron source, the BL12XU beamline at
SPring-8. The dielectric and optical functions in the VUV
region are more accurately rederived and tabulated to further
extend their application. The treatment of small-angle IXS data
for molecular liquids, including correction for multiple
scattering, is also described in detail.
This equation transforms the scale of experimental df(q,E)/dE
data into absolute values.
The imaginary part of the inverse dielectric response function
(called the energy-loss function) is also directly related to
S(q,E) and is given by11,15,17−19
⎛ −1 ⎞
4π 2ne
Im⎜
S(q ,E)
⎟=
q2
⎝ ε(q ,E) ⎠
(4)
where ne is the average electron density of the system. Because
the complex dielectric function [ε(q,E) = ε1(q,E) + iε2(q,E)]
depends on several optical functions of the system (as shown
later in this paper), the dielectric and optical behaviors of
molecular liquids are therefore dominated by their S(q,E).
Furthermore, ε(q,E) describes the response of matter to the
passage of an incident charged particle.7,8,18,20,21 Consequently,
in investigations of the physical and prechemical stages of
radiation interactions and energy transport in molecular liquids,
these interrelated functions [i.e., S(q,E), df(q,E)/dE, ε(q,E),
and Im(−1/ε(q,E))] for the liquids under study must be
considered. If these functions are known over a wide range of
vacuum ultraviolet (VUV) energies, various interesting properties in the fields of radiation physics and radiation chemistry,
such as the mean excitation energy in the liquid, can be
evaluated on an absolute scale.2,6,9,22,23
Despite their importance, the above functions for molecular
liquids have not yet been fully explored above ∼7 eV, even for
chemically and biologically significant liquids, such as water and
benzene, which are investigated in this study. This lack of
exploration has resulted primarily from the difficulties of
applying the two major experimental techniques that are
currently available for detecting electronic excitations: VUV
reflectance measurements (for q = 0) and electron energy-loss
spectroscopy (EELS) (for small finite values of q). Both
techniques are extremely difficult to apply to volatile liquids
because a vacuum is required and because no practical window
materials are available for either method. Specifically, to the
best of our knowledge, there has been no EELS study
investigating molecular liquids. Furthermore, despite recent
advances in VUV technology, no UV−VUV optical study of
molecular liquids for a wide energy range has been reported
since the 1970s.24−29 Surprisingly, because of the lack of
reliable experimental data, our knowledge of radiation−liquid
interactions and electronic excitations in liquid media remains
far from complete. Therefore, there is an urgent need to
accumulate and refine the experimental results for S(q,E),
df(q,E)/dE, and Im(−1/ε(q,E)) for molecular liquids.
In our previous studies,17,30,31 we pointed out that inelastic
X-ray scattering (IXS) (for which the double-differential crosssection (∂2σ/∂Ω∂E) for isotropic samples is given by (∂2σ/
∂Ω∂E) = (∂σ/∂Ω)ThS(q,E), where (∂σ/∂Ω)Th is the Thomson
scattering cross section) is free from the difficulties inherent to
VUV and EELS measurements; in particular, no vacuum is
required. Subsequently, through IXS measurements using
second-generation synchrotron X-ray sources, df(q,E)/dE and
its related functions for several volatile liquids were obtained
over a wide range of q and E values.11,15,17,30−35
Note that if the condition qre ≪ 1 is satisfied (where re is a
measure of the electron’s spatial extent), the dipole
approximation [exp(iq·r) ≈ 1 + iq·r] is justified. In this case,
the spatial average of the matrix element in eq 1 reduces to (q2/
3)|⟨f |∑jrj|i⟩|2, which is exactly the same as the matrix element
for photoabsorption.11,15,16 This type of IXS is sometimes
■
EXPERIMENTAL METHODS
Materials. Deionized water and 99.7% benzene (Wako Pure
Chemical Industries, Ltd., Chuo, Osaka) were used as liquid
samples and underwent no additional purification. Using a
syringe, thin quartz glass tubes (Mark-tube, Hilgenberg GmbH,
Malsfeld) with an inside diameter and wall thickness of 3.2 mm
and 10 μm, respectively, were filled with the liquid samples.
Macroscopic radiation effects on the samples, including bubble
formation in the tube, were not observed in any of the IXS
measurements.
Measurements. IXS spectra of liquid water and liquid
benzene were measured at the Taiwan beamline, BL12XU at
SPring-8. The details of this beamline have been given
elsewhere.50 The background intensity from the empty glass
tube was also measured to correct the measured IXS spectra.
Monochromatic X-rays with 0.17 eV resolution were used to
irradiate the liquid samples. The incident beam sizes were 80
μm (vertical) and 120 μm (horizontal). Scattered X-rays were
collected and dispersed with a spherically bent crystal analyzer
[Si(555) with a 2 m radius of curvature] at a Bragg angle of
89.1°, and these X-rays were detected using an AMPTEK XR100CR detector. The corresponding pass energy of the analyzer
was 9.887 keV. The energy resolution, given as the full width at
half-maximum of the elastic line, was 0.24 eV. Measurements
were taken for energy-loss (E) in the range 1−100 eV. By
setting 2θ to 5° and 7° with an angular width of ±1.22°, q
values of 0.23 and 0.32 au, respectively, were realized with an
uncertainty of ±0.06 au. The accumulation times per spectrum
were 13 and 3 h for water and 4 and 1.5 h for benzene for 2θ
values of 5° and 7°, respectively. Analysis of the measured
spectra is described in the next section.
All experiments were performed in the temperature range
295−298 K (room temperature) and at ambient air pressure.
Data Treatment. First, the raw IXS data (Ir), which were
obtained at small angles, were corrected for the background
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triple scatterings are expected to have negligible impact53,54
because, compared with elastic−inelastic scatterings, these
events are presumed to be weaker and spread over a wider
range of E values.
First, the angular dependence of inelastic scattering in double
(elastic−inelastic) scattering was determined on the basis of
Monte Carlo calculations (the details of these calculations are
provided in the Supporting Information). The framework for
these calculations is similar to the framework commonly used
for corrections in Compton scattering,53,54 whereas the
probabilities of elastic scattering and inelastic scattering were
estimated from the X-ray diffraction profiles59,60 and static
structure [S(q)] values,17,30,61−63 respectively. Figure 2a shows
the q dependence of the probabilities of elastic and inelastic
scattering that were employed for the water calculations. Here,
the elastic scattering probability was estimated from the
diffraction data given by Morgan and Warren,59 and the
inelastic scattering probability was obtained from experimental
data reported by Watanabe et al.17 for 0.69 ≤ q ≤ 3.59 au and
theoretical data presented by Wang et al.61 for all other q
ranges. The fractions of total scattering (the sum of elastic and
inelastic scattering) in the attenuation coefficient (absorption
or scattering) were calculated from atomic data.64 The angular
distributions were calculated by assuming a monochromatic
primary photon beam with a wavelength of 1.25 Å (9.92 keV),
which was incident upon a cylinder with 0.5 mm in radius and
3.2 mm in thickness. The paths of 1010 photons were followed.
The inset in Figure 2b schematically describes single and
double X-ray scattering in a sample. In single (inelastic)
scattering, incident X-rays are scattered inelastically with a
scattering angle of 2θ. On the contrary, in double scattering, the
X-rays are scattered both elastically and inelastically with
scattering angles of αel and αinel, respectively. Only doubly
scattered X-rays that leave the sample within the preset 2θ
range were counted. Figure 2b shows the distribution of the
obtained αinel for 2θ = 5° ± 1.22° and 7° ± 1.22° for liquid
water. These profiles can be regarded as the overlap between
the elastic intensity distribution and the inelastic intensity
distribution in Figure 2a, with a well-known π-polarization
factor (∝cos2 2θ, ∝cos2 αel, and ∝cos2 αinel). This polarization
factor strongly suppresses the scattering intensities at αinel ≈
90°. The main inelastic components in double scattering are
the scatterings at αinel ≈ 20° and 180°, which occur because of
the strong elastic peaks at q ≈ 1 au and the relatively high
intensity in S(q) at q ≥ 4 au, respectively (Figure 2a).
Finally, the double-scattering profiles were estimated by
adding normalized IXS profiles according to the calculated
angular distributions. Figure 2c shows selected IXS data for
liquid water. Here, the original S(q,E) spectra, which were
estimated from the data in ref 17 for q ≤ 3.59 au and were
calculated as Compton scattering according to ref 65 for larger
values of q, were normalized such that their total intensities
equaled one. To prepare normalized IXS profiles for liquid
benzene, the S(q,E) spectra were estimated from ref 31 for q ≤
2.77 au and calculated as Compton scattering according to ref
66 for larger values of q. Figure 2c suggests that most of the IXS
profiles at q ≈ 1 au formed the double-scattering profile in the
measured energy range (E ≤ 100 eV), whereas the profiles at
higher values of q (e.g., 5.31 au) contributed only slightly
because of their wider spread over the entire E range. In Figure
1, the obtained double-scattering profile of liquid water is
compared with the measured scattering spectra at 2θ = 5° after
scaling. This scaling was implemented by using the ratio of the
signal from the glass tube (Ib) and for the dark counts (Id)
using the conventional method for small-angle (elastic) X-ray
scattering measurements51,52 [Ic1 = {(Ir − Id) − e−μt(Ib − Id)}/
e−(μt+μBtB)]. Here, μ and t are the linear absorption coefficient
and the thickness of the sample, respectively, and μB and tB are
the corresponding values for the tube. The (Ir − Id) and e−μt(Ib
− Id) values of water measured at 2θ = 5° are plotted in Figure
1 to provide an example of the IXS data. The vertical axis in
Figure 1. Examples of scattering data. Raw IXS data (solid line): liquid
water’s scattering spectrum, obtained at scattering angles of 5° ± 1.22°
and corrected for dark counts (Ir − Id). Cell background (broken line):
background scattering from the glass tube, obtained at the same
scattering angles and corrected for dark counts and sample absorption
[e−μt(Ib − Id)]. Multiple scattering (dash-dotted line): elastic−inelastic
double scattering profile at scattering angles of 5° ± 1.22°, calculated
for liquid water (described in more detail in the text).
Figure 1 is the average count rate, where the variation in
incident beams is corrected by the output from an ion chamber
(I0 monitor).
After background correction, the tail of the elastic scattering
line (Ie), which was found in the Ic1 spectra for 1 ≤ E ≤ 7 eV,
was fitted to a decay function, A1 exp(−(E/w1)) + A2 exp(−(E/
w2)) + A0 (using A0, A1, A2, w1, and w2 as the fitting
parameters), for subtraction from Ic1. The constant component
A0 may also include parasitic scattering from the employed
spectrometer.
The Ie-subtracted spectra (Ic2) were further corrected for
multiple scattering. Here, we explain our treatment of multiple
scattering in detail because, to the best of our knowledge, no
concrete technique for estimating multiple scattering in smallangle IXS for molecular liquids has been previously reported,
unlike the case for Compton scattering (i.e., IXS for large q
values)53−56 and IXS of simple metals.57 A challenge that arises
when calculating multiple scatterings in small-angle IXS for
molecular liquids results from the lack of a proper theoretical
model for the IXS profiles, such as the impulse approximation
for Compton scattering15,16,58 and the random-phase approximation (RPA) for metal IXS/EELS16,18,20,21 (although several
semiempirical models were recently proposed for IXS of
condensed water9,23,41,43,49). Consequently, we employed a
method that combines Monte Carlo calculations with the
experimental data, as we will describe. For simplicity, we shall
treat only double-scattering events consisting of elastic−
inelastic scatterings. Inelastic−inelastic double scatterings and
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experimental IXS intensity to the calculated sum of the singleand double-scattering intensities. The corrected spectra (Ic3)
were obtained by subtracting the double-scattering contributions from the Ic2 spectra.
■
RESULTS AND DISCUSSION
Convergence of the IXS Profiles. The IXS (Ic3) spectra
obtained in this study for liquid water and benzene are shown
in Figure 3a,b, respectively. For both liquids, the IXS intensity
Figure 3. IXS spectra at q = 0.23 and 0.32 au for (a) liquid water and
(b) liquid benzene. Insets: comparisons of normalized IXS profiles for
each liquid.
decreased with decreasing q, as has been found previously.30,35
The count rates of these IXS spectra were ∼10 times higher
than the rates reported in previous measurements,30,33,35 which
indicates a marked improvement in both the S/N ratios and the
reliability of the IXS profiles.
To examine the q dependence of the spectral shapes, peaknormalized IXS spectra of water and benzene are compared in
the insets of Figure 3a,b, respectively. The two liquids’
normalized profiles converged well within the experimental
error, and this result clearly justifies the application of the
dipole approximation and demonstrates that their profiles can
be regarded as optical profiles. Consequently, the Ic3 spectra of
the two liquids at q = 0.23 au were used to deduce the dielectric
and optical data obtained in the following sections.
Figure 2. Items for multiple scattering calculations. (a) q dependences
of liquid water’s elastic and inelastic X-ray scattering, estimated from
refs 17, 59, 61, and 64. (b) Calculated distribution of inelastic
scattering angles in double scattering (αinel) into the 2θ = 5° ± 1.22°
and 7° ± 1.22° angle regions for liquid water. Inset: schematic
description of single and double scattering. In single (inelastic)
scattering, X-rays are scattered inelastically with a scattering angle of
2θ. In double scattering, X-rays are scattered both elastically and
inelastically with scattering angles of αel and αinel, respectively. Only
doubly scattered X-rays that leave the sample within the preset 2θ
range are counted. (c) Selected normalized IXS profiles for liquid
water, which were employed to generate double scatterings.
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Table 1. Dielectric and Optical Functions of Liquid Water Obtained from Small-Angle IXS
energy
(eV)
OOS
(eV−1)
Im(−1/ε)
ε1
ε2
n
k
R
energy
(eV)
OOS
(eV−1)
Im(−1/ε)
ε1
ε2
n
k
R
1.0
2.0
4.0
6.0
7.0
7.2
7.4
7.6
7.8
8.0
8.2
8.4
8.6
8.8
9.0
9.2
9.4
9.6
9.8
10.0
10.4
10.8
11.2
11.6
12.0
12.4
12.8
13.2
13.6
14.0
14.4
14.8
15.2
15.6
16.0
16.4
16.8
17.2
17.6
18.0
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.003
0.006
0.010
0.012
0.015
0.016
0.015
0.014
0.014
0.015
0.017
0.018
0.021
0.023
0.025
0.028
0.031
0.035
0.038
0.043
0.050
0.057
0.064
0.074
0.083
0.089
0.097
0.104
0.111
0.117
0.126
0.136
0.145
0.000
0.000
0.000
0.000
0.000
0.002
0.009
0.032
0.057
0.086
0.108
0.129
0.133
0.125
0.115
0.110
0.117
0.130
0.136
0.149
0.160
0.168
0.179
0.192
0.211
0.224
0.240
0.275
0.301
0.331
0.370
0.403
0.425
0.448
0.470
0.487
0.504
0.529
0.560
0.583
1.76
1.77
1.83
1.98
2.18
2.28
2.38
2.51
2.45
2.42
2.28
2.13
2.06
2.04
2.09
2.18
2.17
2.15
2.14
2.11
2.06
2.04
2.04
2.00
1.95
1.91
1.88
1.73
1.63
1.50
1.35
1.24
1.17
1.11
1.06
1.02
0.99
0.93
0.88
0.84
0.00
0.00
0.00
0.00
0.00
0.01
0.05
0.20
0.35
0.53
0.60
0.64
0.62
0.56
0.54
0.55
0.59
0.66
0.68
0.74
0.78
0.81
0.88
0.94
1.03
1.08
1.20
1.26
1.34
1.36
1.33
1.29
1.24
1.22
1.17
1.14
1.11
1.10
1.06
1.01
1.33
1.33
1.35
1.41
1.48
1.51
1.54
1.59
1.57
1.56
1.52
1.47
1.45
1.44
1.46
1.49
1.49
1.48
1.48
1.47
1.46
1.46
1.46
1.45
1.44
1.43
1.43
1.39
1.37
1.33
1.28
1.23
1.20
1.17
1.15
1.13
1.11
1.09
1.06
1.04
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.06
0.11
0.17
0.20
0.22
0.21
0.19
0.18
0.19
0.20
0.22
0.23
0.25
0.27
0.28
0.30
0.32
0.36
0.38
0.42
0.45
0.49
0.51
0.52
0.52
0.52
0.52
0.51
0.51
0.50
0.50
0.50
0.49
0.020
0.020
0.023
0.029
0.037
0.041
0.046
0.052
0.051
0.053
0.049
0.044
0.041
0.039
0.040
0.044
0.044
0.046
0.046
0.047
0.046
0.047
0.049
0.050
0.053
0.055
0.059
0.061
0.064
0.065
0.064
0.062
0.060
0.060
0.057
0.057
0.055
0.057
0.056
0.054
18.4
18.8
19.2
19.6
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
32.0
34.0
36.0
38.0
40.0
42.0
44.0
46.0
48.0
50.0
52.0
54.0
56.0
58.0
60.0
64.0
68.0
72.0
76.0
80.0
84.0
88.0
92.0
96.0
100.0
0.155
0.166
0.178
0.184
0.188
0.202
0.206
0.204
0.198
0.192
0.182
0.174
0.174
0.166
0.168
0.167
0.160
0.150
0.136
0.130
0.125
0.117
0.110
0.104
0.100
0.092
0.086
0.084
0.077
0.071
0.066
0.064
0.047
0.045
0.032
0.032
0.022
0.019
0.016
0.013
0.610
0.637
0.669
0.678
0.681
0.694
0.677
0.641
0.596
0.555
0.505
0.467
0.448
0.414
0.406
0.377
0.340
0.301
0.258
0.235
0.215
0.192
0.172
0.157
0.144
0.127
0.115
0.108
0.096
0.086
0.075
0.068
0.047
0.043
0.029
0.028
0.018
0.015
0.012
0.009
0.81
0.77
0.74
0.73
0.73
0.72
0.73
0.74
0.76
0.78
0.79
0.81
0.81
0.83
0.83
0.82
0.82
0.83
0.85
0.85
0.85
0.86
0.87
0.87
0.88
0.88
0.89
0.89
0.89
0.89
0.90
0.91
0.91
0.92
0.93
0.93
0.94
0.94
0.95
0.95
0.97
0.93
0.86
0.82
0.79
0.69
0.62
0.55
0.48
0.44
0.40
0.37
0.35
0.33
0.32
0.29
0.25
0.22
0.20
0.18
0.16
0.15
0.13
0.12
0.11
0.10
0.09
0.09
0.08
0.07
0.06
0.06
0.04
0.04
0.02
0.02
0.02
0.01
0.01
0.01
1.02
0.99
0.97
0.96
0.95
0.93
0.92
0.91
0.91
0.91
0.92
0.92
0.92
0.93
0.92
0.92
0.92
0.92
0.93
0.92
0.93
0.93
0.93
0.94
0.94
0.94
0.94
0.94
0.95
0.95
0.95
0.95
0.96
0.96
0.96
0.97
0.97
0.97
0.97
0.98
0.48
0.47
0.45
0.43
0.41
0.37
0.34
0.30
0.26
0.24
0.22
0.20
0.19
0.18
0.17
0.16
0.14
0.12
0.11
0.10
0.09
0.08
0.07
0.07
0.06
0.05
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.00
0.053
0.052
0.049
0.046
0.044
0.037
0.032
0.026
0.021
0.018
0.015
0.012
0.011
0.010
0.009
0.008
0.007
0.006
0.004
0.004
0.003
0.003
0.002
0.002
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
IXS-derived OOS data33 (for benzene, the previous OOS was
calculated from the IXS data in ref 35). The OOS values
obtained using the Bethe sum rule (eq 3) are absolute and not
relative, with units of eV−1. Despite the differences in both the
incident X-ray energies and the spectrometers employed, the
present and previous OOS profiles for both liquids agreed fairly
well with each other, and the errors resulting from experimental
differences were within the expected range; this result confirms
the high reproducibility of the IXS method. Parts a and b of
Figure 4 also indicate that the statistical uncertainties in the
present OOS data were markedly lower than those in previous
data because of improvements in the signal intensity, which
made the OOS and related functions more reliable. The OOS
data are independent of ne in the system, unlike other dielectric
and optical functions, such as ε(E); therefore, the OOS data are
useful for examining the differences with respect to phase
changes (i.e., phase effects). The phase effects on the OOS of
the two liquids will be discussed in a later section.
Normalization of IXS Spectra and Optical Oscillator
Strength Distributions. First, the Ic3 spectra of water and
benzene were normalized using the Bethe sum rule (eq 3) to
determine the optical oscillator strength distribution (OOS,
df(0,E)/dE). N was set to 8.22 and 30.55 (exactly the same
values employed in previous studies31,33) for water and
benzene, respectively. Small estimated corrections of 0.22 (for
water) and 0.55 (for benzene) were added to the total number
of valence electrons (8 for water and 30 for benzene) to obtain
these values; these corrections were intended to account for
Pauli-excluded transitions from the K shell to the already
occupied orbitals.67 For the normalization (as in previous
studies17,31), the higher-E part of the IXS profile (E > ∼70 eV)
was fitted to the function A3E−B using A3 and B as the fitting
parameters to extrapolate the tail of the IXS profiles to infinity.
The obtained OOS data for liquid water and liquid benzene
are tabulated in the second column of Tables 1 and 2 and are
shown in Figure 4a,b, respectively, in addition to the previous
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Table 2. Dielectric and Optical Functions of Liquid Benzene Obtained from Small-Angle IXS
energy
(eV)
OOS
(eV−1)
Im(−1/ε)
ε1
ε2
n
k
R
energy
(eV)
OOS
(eV−1)
Im(−1/ε)
ε1
ε2
n
k
R
1.0
2.0
4.0
5.0
6.0
6.2
6.4
6.6
6.8
7.0
7.2
7.4
7.6
7.8
8.0
8.2
8.4
8.6
8.8
9.0
9.2
9.4
9.6
9.8
10.0
10.4
10.8
11.2
11.6
12.0
12.4
12.8
13.2
13.6
14.0
14.4
14.8
15.2
15.6
16.0
0.00
0.00
0.00
0.00
0.04
0.05
0.09
0.13
0.20
0.28
0.27
0.27
0.22
0.19
0.14
0.11
0.08
0.07
0.07
0.08
0.10
0.12
0.14
0.16
0.17
0.17
0.17
0.20
0.23
0.25
0.27
0.29
0.32
0.34
0.34
0.37
0.42
0.49
0.54
0.60
0.000
0.000
0.000
0.000
0.096
0.120
0.215
0.282
0.427
0.579
0.548
0.543
0.418
0.350
0.250
0.193
0.144
0.123
0.118
0.126
0.159
0.182
0.218
0.236
0.247
0.242
0.228
0.262
0.292
0.309
0.322
0.331
0.352
0.363
0.360
0.380
0.420
0.470
0.510
0.546
2.11
2.15
2.40
2.82
4.11
3.89
1.95
0.88
0.90
0.63
0.90
0.92
1.10
1.21
1.35
1.50
1.67
1.82
2.01
2.09
2.12
2.04
1.90
1.81
1.74
1.74
1.86
1.76
1.62
1.56
1.50
1.48
1.41
1.36
1.38
1.32
1.17
1.02
0.93
0.87
0.00
0.00
0.00
0.00
2.02
2.66
3.60
3.31
1.92
1.45
1.06
0.94
0.73
0.67
0.53
0.48
0.43
0.43
0.51
0.59
0.83
0.91
1.02
1.02
0.98
0.95
1.02
1.16
1.17
1.19
1.17
1.21
1.21
1.17
1.25
1.32
1.39
1.35
1.29
1.21
1.45
1.47
1.55
1.68
2.08
2.08
1.74
1.47
1.23
1.05
1.07
1.06
1.10
1.14
1.18
1.24
1.30
1.36
1.43
1.46
1.48
1.46
1.43
1.40
1.37
1.36
1.41
1.39
1.35
1.33
1.31
1.30
1.28
1.26
1.27
1.26
1.22
1.17
1.12
1.09
0.00
0.00
0.00
0.00
0.48
0.64
1.04
1.13
0.78
0.69
0.50
0.44
0.33
0.29
0.22
0.19
0.16
0.16
0.18
0.20
0.28
0.31
0.36
0.37
0.36
0.35
0.36
0.42
0.43
0.45
0.45
0.47
0.47
0.47
0.49
0.52
0.57
0.58
0.57
0.56
0.034
0.036
0.047
0.064
0.145
0.159
0.189
0.202
0.119
0.102
0.055
0.045
0.026
0.023
0.017
0.019
0.022
0.028
0.036
0.041
0.050
0.051
0.051
0.049
0.046
0.044
0.051
0.056
0.054
0.055
0.053
0.056
0.056
0.053
0.058
0.064
0.070
0.073
0.071
0.069
16.4
16.8
17.2
17.6
18.0
18.4
18.8
19.2
19.6
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
32.0
34.0
36.0
38.0
40.0
44.0
48.0
52.0
56.0
60.0
64.0
68.0
72.0
76.0
80.0
84.0
88.0
92.0
96.0
100.0
0.63
0.68
0.75
0.83
0.89
0.94
1.00
1.05
1.09
1.14
1.24
1.28
1.29
1.26
1.20
1.11
1.06
1.00
0.89
0.81
0.68
0.57
0.50
0.43
0.38
0.30
0.25
0.18
0.17
0.15
0.12
0.11
0.10
0.08
0.07
0.07
0.04
0.04
0.03
0.02
0.564
0.593
0.640
0.690
0.726
0.753
0.783
0.801
0.817
0.835
0.865
0.850
0.821
0.771
0.701
0.627
0.573
0.521
0.451
0.398
0.310
0.247
0.202
0.168
0.138
0.100
0.077
0.052
0.045
0.038
0.027
0.023
0.020
0.016
0.013
0.012
0.007
0.006
0.004
0.003
0.83
0.78
0.70
0.65
0.62
0.60
0.59
0.59
0.58
0.58
0.57
0.59
0.60
0.62
0.64
0.66
0.68
0.68
0.71
0.72
0.75
0.77
0.79
0.81
0.83
0.85
0.87
0.89
0.90
0.91
0.92
0.93
0.94
0.94
0.95
0.95
0.96
0.96
0.96
0.97
1.19
1.17
1.13
1.05
0.99
0.94
0.88
0.83
0.80
0.76
0.67
0.58
0.53
0.46
0.40
0.36
0.32
0.28
0.25
0.23
0.18
0.15
0.13
0.11
0.10
0.07
0.06
0.04
0.04
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.00
0.00
1.07
1.04
1.01
0.97
0.94
0.93
0.91
0.90
0.89
0.87
0.85
0.84
0.84
0.83
0.84
0.84
0.84
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
0.94
0.95
0.95
0.96
0.97
0.97
0.97
0.97
0.98
0.98
0.98
0.98
0.98
0.56
0.56
0.56
0.54
0.52
0.51
0.48
0.46
0.45
0.43
0.39
0.35
0.31
0.27
0.24
0.21
0.19
0.17
0.15
0.13
0.11
0.09
0.07
0.06
0.05
0.04
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.068
0.070
0.072
0.070
0.068
0.066
0.063
0.059
0.057
0.055
0.049
0.041
0.036
0.030
0.025
0.020
0.018
0.016
0.013
0.011
0.008
0.006
0.005
0.004
0.003
0.002
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Optical Energy-Loss Functions. Once the OOS is
obtained from the small-angle IXS spectra, the optical dynamic
structure factor, S(0,E), can be determined from eq 2, and the
optical energy-loss function (OLF, Im(−1/ε(E)) = ε2(E)/(ε1(E)2
+ ε2(E)2)) can then be deduced from eq 4. The OLF is often
employed as a cornerstone in theoretical studies investigating
the interaction between charged particles and condensed
matter.2,6−8,20,68 In particular, OLF data represent one of the
most important physical inputs for recent Monte Carlo track
structure codes69 that have aimed to simulate the full
trajectories of decelerating electrons and ions.12,13,49 Therefore,
for several condensed molecular systems, this function has been
experimentally determined using several methods24−27,29,30,33,70−72 and theoretically estimated by several
researchers.9,12,22,23,40,41,43,49,69,73 The numerator in the OLF
[ε2(E)] corresponds to single-particle transitions of an isolated
atom or molecule, whereas the denominator [ε1(E)2 + ε2(E)2]
accounts for the influence of the condensed phases (i.e., the
shielding or screening effect). This factor describes the extent
to which polarization effects in the liquids mediate the
electronic interactions by a “screened” (as opposed to a
“bare”) Coulomb potential.8,49
The OLFs of liquid water and liquid benzene are tabulated in
the third column of Tables 1 and 2 and are plotted up to E = 45
eV in Figure 5a,b, respectively. The OLFs obtained from VUV
reflectance measurements24,26,29 are also shown in Figure 5a,b
for comparison. For both liquids, the overall shapes of the
OLFs (determined by small-angle IXS and VUV reflectance
measurements) agreed with each other. However, the peak
height and valley depth of the OLFs were not in good
agreement. For example, as shown in Figure 5a, the maximum
values of the OLF of liquid water (characterized by a broad
peak at ∼21 eV) differed considerably: 0.7 (IXS) versus 1.1
(VUV). Figure 5b indicates that the valley at ∼9 eV in the OLF
of liquid benzene obtained here using IXS was markedly
shallower than that obtained by Sowers et al. using VUV,26
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Figure 4. Optical oscillator strength distributions for (a) liquid water
and (b) liquid benzene. For comparison, previous IXS results obtained
from refs 33 and 35 (“Previous IXS”) are also plotted.
Figure 5. Optical energy-loss functions for (a) liquid water and (b)
liquid benzene, which were determined from both IXS and VUV
reflectance measurements.24,26,29 In Figure 5a, some theoretically
estimated OLFs for liquid water22,23,40 are also shown.
whereas its depth was consistent with the VUV result obtained
by Williams et al.24 These differences in the OLFs significantly
affect the absolute values of ε1(E) and ε2(E), which has
historically led to the idea that the peaks found in the OLFs of
these liquids result from plasmons.2,3,5,7,24,26,29,73−75 The causes
of these differences and the validity of the plasmon
interpretation are discussed later in this paper.
Some of the theoretically estimated OLFs for liquid water are
shown in Figure 5a (many theoretical OLFs for liquid water
were compared and assessed in refs 39 and 69). Early
theoretical studies investigating the OLFs of molecular liquids
have been strongly influenced by VUV reflectance data24,26,29
because, before the development of IXS spectroscopy using
synchrotron radiation, the OLFs of volatile media were
obtained using only reflectance measurements.12,40,69 For
example, Kaplan et al.76 directly used OLF data from Heller
et al.29 in their simulations of energy degradation of fast
electrons in liquid water. Ashley22 constructed a theoretical
OLF model based on fitting calculated results to the
reflectance-derived ε2 data for liquid water.29 The intensity of
the main peak in the theoretical OLFs (∼0.86) was lower than
the corresponding VUV result (Figure 5a). However, the
theoretical peak value was higher than the value obtained from
an EELS result for amorphous ice (∼0.7).70 Consequently, the
model was considered to be reasonable within the limits of
experimental uncertainty.22 Dingfelder et al.23 also employed a
similar approach by fitting their results to reflectance-derived ε2
data.29 Reflecting the differences between these theoretical
models, their OLF curve23 differed from Ashley’s.22 However,
because the same experimental data29 were used to determine
the parameters, the intensities of the main peaks were similar.
Consequently, compared with the present IXS data, these
theoretical OLF models somewhat overestimated the 21 eV
peak intensity. Dingfelder et al. stated that the optical data at
∼20 eV were especially uncertain.23 In contrast to these earlier
OLF models, a recent theoretical OLF,40 which was based on
earlier IXS-derived ε2 data,33 was in overall agreement with the
present experimental OLF data [although it seemed to slightly
(∼5%) underestimate the 21 eV peak intensity].
As shown in Figure 4 and as already noticed by several
theorists,37,39,40,44,69,77 the IXS results were more reliable and
hence should replace the old VUV reflectance data sets24,26,29 in
theoretical OLF investigations. In particular, for liquid water,
the differences between the OLFs in the IXS and the
reflectance data sets noticeably affect both electron and proton
interaction simulations.9,39−41 For example, a recent simulation
study48 of DNA strand breaks12,78−80 induced by low-energy
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electrons showed that the OLF discrepancy led to 30%−45%
discrepancies in the single-strand break yields and 45%−80%
discrepancies in the double-strand break yields. Several recent
simulation studies9,13,39−41,43,45,49 of condensed-water-related
phenomena have already employed the IXS data set33 as input.
The present work (as highlighted in Figure 4a) firmly supports
the validity of these theoretical studies.
Complex Dielectric Functions and Related Optical
Functions. To obtain the full optical dielectric function [ε(E)
= ε1(E) + iε2(E)] from the OLF, we can use the well-known
Kramers−Kronig dispersion relation:8,11,15,16,20,22,23,74
⎛ 1 ⎞
2
Re⎜
⎟=1+ P
π
⎝ ε (E ) ⎠
∫
0
∞
⎛ −1 ⎞
E′
Im⎜
dE ′
⎟ 2
⎝ ε(E′) ⎠ E′ − E2
(5)
Here, P denotes the principal value of the integral. Both ε1(E)
and ε2(E) are readily given by ε1 = (Re(1/ε))/(Re(1/ε)2 +
Im(−1/ε)2) and ε2 = (Im(−1/ε))/(Re(1/ε)2 + Im(−1/ε)2),
respectively.74 These dielectric functions of liquid water and
liquid benzene are tabulated in the fourth and fifth columns of
Tables 1 and 2 and are also plotted up to E = 26 eV in Figures 6
and 7, respectively. In these figures, the dielectric functions
obtained from VUV reflectance measurements24,26,29 are also
shown for comparison.
With respect to the global shapes, for both liquids, ε1(E) and
ε2(E) obtained from small-angle IXS spectroscopy agreed
qualitatively with the functions obtained from the reflectance
measurements; however, there were again substantial quantitative differences. Note that, for benzene, the intensities of the
spectral structures showed a rather large discrepancy between
the two VUV results (e.g., the ε1 valley at 6.8 eV, the ε1 peak at
9.3 eV, the ε2 peak at 6.4 eV, and the ε2 shoulder at 9.6 eV),
which we attributed to the difficulty of accurately measuring
structures above 7 eV by optical methods. Therefore, it is not
surprising that the intensities of the structures in ε1 and ε2 were
also markedly different above 7 eV for water, as has already
been reported.11,33 The most significant difference between the
IXS and reflectance results was in the behavior of ε1 where the
OLF achieved a maximum; the valleys existed at ∼20 eV for
water and at 6.8 eV for benzene. In both cases, the reflectancededuced ε1 functions displayed a distinct valley with minimum
values of 0.42 in water29 (as shown in Figure 6a) and −0.6 in
benzene24 (as shown in Figure 6b). These observations
suggested that plasmons contribute to the corresponding
OLF peaks in these liquids.24,29 However, the IXS-deduced ε1
functions had a rather shallow minimum, with values of 0.72 at
∼20 eV in water and 0.90 at 6.8 eV in benzene. These larger
values do not meet the strict conditions for plasmon formation.
The plasmon hypothesis for these liquids is discussed in detail
in a later section.
The complex dielectric functions can then be converted to
various forms, such as the indices of refraction and reflectance,
for easier comparison with data from different sources. The real
(n) and imaginary (k) parts of the complex index of refraction
can be calculated as n2 = ((ε12 + ε22)1/2 + ε1)/2 and k2 = ((ε12 +
ε22)1/2 − ε1)/2, respectively, because ε1 = n2 − k2 and ε2 =
2nk.11,24,27,29,74 The reflectance (R) can be expressed by n and k
as R = ((n − 1)2 + k2)/((n + 1)2 + k2).6,18,27,29 These optical
functions (n, k, and R) for liquid water and liquid benzene are
tabulated in the sixth, seventh, and eighth columns of Tables 1
and 2 and are plotted up to E = 26 eV in Figures 8, 9, and 10,
respectively. In these figures, the n, k, and R functions obtained
from UV and VUV reflectance measurements24−27,29 are also
Figure 6. Real parts of the dielectric functions (ε1) for (a) liquid water
and (b) liquid benzene, which were determined from both IXS and
VUV reflectance measurements.24,26,29
shown for comparison. In Figure 8a,b, the calculated n values
from well-documented refractive index data in the visible and
UV regions81 are indicated by solid squares.
The small-angle IXS results for n and R for both liquids
clearly agree very well with the corresponding optical results in
the visible and UV regions (<7 eV for water and <6 eV for
benzene). In these E regions, k = ε2 ≈ 0; this relation implies
that there is no strong absorption (such regions are described
as transparent regions) and that ε1 ≈ n2 and R = (n − 1)2/(n +
1)2. Here, direct measurements of n and R encountered less
experimental difficulty, and the obtained values were
considered reliable. The condition ε2 ≈ 0 also indicates that
Im(−1/ε(E)) = ε2(E)/(ε1(E)2 + ε2(E)2) ≈ 0. During IXS data
processing, we found that the values of ε1, n, and R in the
transparent region were very sensitive to background
subtraction, multiple scattering estimation, and normalization,
including the extrapolation to E → ∞. Therefore, the good
agreement between the IXS-derived and optically measured n
and R values for the transparent region (as shown in Figures 8
and 10, respectively) confirmed both the validity of our
treatment of small-angle IXS data and the overall accuracy of
the deduced dielectric and optical functions.
Conversely, in accordance with the ε1 and ε2 results (as
shown in Figures 6 and 7), these optical functions (n, k, and R)
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Figure 8. Real parts of the complex index of refraction (n) for (a)
liquid water and (b) liquid benzene, which were determined from both
IXS and VUV reflectance measurements.24,26,29 For comparison, the n
values obtained from direct refractive measurements81 are also plotted.
Figure 7. Imaginary parts of the dielectric functions (ε2) for (a) liquid
water and (b) liquid benzene, which were determined from both IXS
and VUV reflectance measurements.24,26,29
differed significantly from one another in the VUV region (E >
∼9 eV). Here, we focus on R to discuss the discrepancy
between the VUV-optical and small-angle IXS results because R
is related to the most direct experimental data obtained from
reflectance measurements, whereas n and k are closely related
to R. As indicated in Figure 10, the VUV reflectance profiles
obtained in different studies diverged for both liquids, which
suggests the experimental difficulties and decreased reliability of
the data; furthermore, these values tended to be larger than the
values deduced from IXS. As previously suggested for
water,11,33 these discrepancies most likely resulted from the
intense absorption of vapor from the liquid surface. This
conjecture is supported by the fact that, under cooler
conditions (such as 80 K), where vapor effects are significantly
reduced, the VUV reflectance spectra of amorphous ice82 were
similar in intensity to the present IXS-deduced data for water,
particularly for E > 15 eV (as shown in Figure 10a). The
detailed phase effects are discussed in the following section.
Phase Effects of the OOS of Molecular Systems. To
examine the phase effects on the dielectric and optical functions
of water and benzene in the VUV region, the liquid OOS data
derived from small-angle IXS measurements were compared
with the values of the gas72,83 and amorphous solid,70,71 which
were calculated using the EELS results up to 30 eV, as shown in
Figure 11. In both molecular systems, the gas OOS spectra
were characterized by sharp bands followed by continua above
the ionization threshold (12.6 eV for water and 9.2 eV for
benzene), although the data shown in Figure 11 were obtained
at a resolution of ∼1 eV; therefore, the structures were
aggregates of many sharp lines.
Conversely, the liquid OOS profiles for the two systems had
no sharp features above the ionization thresholds, and these
profiles shifted entirely toward the high-E side. This result
reflects the fact that excited electrons have a special limitation
in condensed phases.8,38 The observed upward shifts imply that
the OOS spectra can be described as “harder” for liquid than
for vapor, which accounts for the greater amount of ionization
in liquids. This effect is important for track simulations in
molecular media.3,69,84,85
For both water and benzene, the OOS of amorphous solids
almost overlapped that of liquids over the entire range of E;
both the peak energy of the OOS and its magnitude were
almost identical for the two phases. In general, the (averaged)
molecular structure of amorphous solids is similar to the
structure of liquids; therefore, the optical spectra of these two
phases are expected to be rather similar. Specifically,
considering that one spectrum is from EELS/VUV measure5617
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Figure 10. Reflectances (R) for (a) liquid water and (b) liquid
benzene, which were determined from both IXS and VUV reflectance
measurements.24−27,29 For comparison, amorphous ice data82 are also
shown.
Figure 9. Imaginary parts of the complex index of refraction (k) for
(a) liquid water and (b) liquid benzene, which were determined from
both IXS and VUV reflectance measurements.24,26,29
ments of amorphous solids and the other is from small-angle
IXS measurements of liquids, the agreement cannot be
coincidental and should be viewed as evidence supporting the
accuracy of both experiments. In particular, the strong similarity
between the UV absorption spectra of liquid and solid benzene
was well-known at least 43 years ago.28 Unfortunately, likely
because of the experimental difficulties suggested in the
previous section, the expected similarities between liquids and
amorphous solids were lost for decades (e.g., see refs 22, 29,
and 85). However, Figure 11 finally eliminates this confusion.
Collective Excitations in Molecular Liquids. It has been
a matter of debate whether the strong peaks in the OLF (at ∼7
eV in benzene and ∼21 eV in water) are caused by plasmonlike collective excitation. Williams et al. first interpreted the
peak at 7 eV for benzene as a collective excitation of the π
electrons based on the ε1 profile they obtained (as shown in
Figure 6b).24 Sowers et al.26 observed rather different
experimental profiles, particularly in the VUV region (as
shown in Figures 5b, 6b, and 7b); however, their interpretation
was the same as that of Williams et al.24 This interpretation was
questioned by Inagaki,28 who noted that the liquid UV spectra
were very similar to the solid UV spectra,86 and therefore the
spectra should be interpreted on the basis of molecular exciton
theory using the Frenkel model (where the excitons are
considered to be strongly bound and relatively compact, in
accordance with the molecular excitation interpretation).4 In a
later review,2 Williams et al. commented that benzene’s 7 eV
peak was not free-electron-like, and the collective behavior
associated with this peak was relatively small. In a more recent
review,74 La Verne and Mozumder stated that although the 7
eV peak fulfilled the condition for collective excitation, it was
excitonic rather than plasmonic in character.
For the peak at ∼21 eV for water, the situation was more
controversial. It was first predicted by Platzman that collective
oscillation of electrons most likely occurs at ∼21 eV in
condensed water.87 However, Daniels did not observe such
collective oscillation in amorphous ice up to 30 eV, and the
large energy-loss structure around 20 eV was attributed to ε2
structures at ∼15 eV.70 Nevertheless, for liquid water, optical
measurements taken by Heller et al.29 (Figures 5a, 6a, 7a, 8a,
9a, and 10a) were thought to confirm the plasmon hypothesis.
In accordance with the results reported by Heller et al., some
researchers argued in favor of some delocalized collective
excitations,2−8,76 and used the plasmon concept to estimate the
OOS of liquid water. 73 Conversely, other researchers
questioned the plasmon interpretation74,75 and estimated the
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⎛ 3E ⎞
4
F ⎟ 2
Ep(q) = Ep(0) + ⎜⎜
⎟q + O(q )
5
E
(0)
⎝ p ⎠
(6)
Here, EF is the Fermi energy of NFEs and O(q4) represents the
higher terms. Although the dispersion constant [3EF/5Ep(0)]
can decrease because of (1) exchange and correlation effects in
the NFEs18,20 and (2) the contributions of interband
transitions,19,21 the q2 dispersion in the plasmon energy still
generally holds. Therefore, if plasmon excitation occurs in the
system (in the usual sense), some q dispersions for the
corresponding bands are expected (note that the plasmon
energy for liquid water, ∼21 eV, was originally based on the
RPA−NFE scheme9,87). Such plasmon dispersion was reported
not only for simple metals with NFEs16,18−21 but also for liquid
metals consisting of ions and molecules (e.g., Li−NH319,88,89),
conducting polymers [e.g., (C2H2)n90], insulating ionic crystals
(e.g., LiF, NaCl, KCl, and KBr91), and even single crystals of
large molecules (e.g., C6092). Note that the q2 dispersion
characterizing plasmon excitation indicates that the dipole
approximation does not hold at finite values of q; hence, the
corresponding IXS bands do not converge even at small values
of q. The failure of the dipole approximation is caused by the
delocalization property of NFEs; that is, for NFEs, the
condition qre ≪ 1 is not fulfilled because re is very large (re
≫ 1).
Conversely, as has been shown, the insets in Figure 3 indicate
no q dispersion for either the 7 eV peak in benzene or the 21
eV peak in water, and there were similar findings for all the
other bands in these two liquids; these results thereby justify
the application of the dipole approximation. Consequently, the
present small-angle IXS results firmly support (1) the view74
that these OLF peaks are not caused by plasmon excitation and
(2) the data treatment9,23,39−41,43 in which no plasmon
contribution is included when the OLF of water is estimated.
In addition, a recent EELS study investigating amorphous ice75
found no evidence of plasmon decay, which is in accordance
with our small-angle IXS results.
Therefore, the assignment of spectral structures to the OOS
and OLF of the two liquids should not be based on collective
electronic excitation and should instead depend on singleelectronic excitations/ionizations, which are considered to be
substantially localized at each molecular site; such assignments
have been applied in recent simulation studies.9,23,39−41,43 For
example, the OLF peaks at ∼7 eV for benzene and at ∼8.5 and
∼21 eV for water can be largely assigned to the A1g → E1u
molecular excitation,28 the X̃ 1A1 → Ã 1B1 molecular excitation,9,23,39−41,43,46 and the combination of the following
molecular excitations and ionizations: the X̃ 1A1 → B̃ 1A1
molecular excitation, the excitations to Rydberg states, and
the ionizations of the 1b 1 , 3a 1 , and 1b 2 molecular
orbitals.9,23,39−41,43
Here, it should be noted that the electronic transitions for
molecular liquids in the UV−VUV region are not fully
understood at the molecular level. For example, intriguing
questions persist regarding water absorption onset (the 8.5 eV
OLF peak), which shows a high-energy (blue) shift of ∼1 eV in
condensed phases (in contrast, the benzene 7 eV OLF peak
shows little shift). The delocalization of the (Frenkel) exciton
onto nearby hydrogen-bonded molecules in condensed water
has been proposed as a possible cause of the blue shift,38 and
both the effects of local electric fields owing to the hydrate
shell42 and the contributions of acceptor hydrogen bonds in
Figure 11. Comparisons of the OOSs in several phases for (a) water
and (b) benzene. The liquid data were obtained in the present study.
The gas72,83 and amorphous solid data70,71 were calculated from EELS
data in the literature.
OLF of liquid water without introducing the plasmon
concept.9,23,40,41,43
As shown in previous sections, there is no experimental
evidence at present to suggest plasmon excitation in molecular
liquids; at both 7 eV in benzene and ∼21 eV in water, the ε1
function did not produce a deep valley (∼0.7, as shown in
Figure 6), although the condition for plasmon excitation is ε1
≪ 1 (or ε1 ≈ 0). The strong OLF peak or resultant deep ε1
valley that was observed in previous VUV measurements24,29
likely resulted from the experimental uncertainties discussed in
the previous section.
The close convergence of the small-angle IXS profiles (as
shown in the insets in Figure 3) also provides evidence against
the perspective that plasmon excitation occurs in liquid water
and liquid benzene. As is well-known, the plasmon hypothesis
is most applicable to valence electrons in metals or to nearly
free electrons (NFEs), for which the energy bands are relatively
wide and the influence of the crystal potential can be treated as
a small perturbation. The RPA, in which electrons are assumed
to move in an average field, can fairly adequately describe the
properties of NFEs.20,49 The NFE model within the RPA
predicts the dispersion of the plasmon energy Ep(q), which is
expressed using the following equation:18−21
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water’s pentamer47 have been discussed. The accurate
experimental data provided by small-angle IXS spectroscopy
can support and encourage such detailed theoretical studies of
UV−VUV electronic excitations in molecular liquids.
Static Structure Factor at Small Values of q in
Molecular Liquids. S(q) is defined as S(q) = ∫ ∞
0 S(q,E) dE
and is very sensitive to the electron correlation.15,17,19,20,31,32,61−63,93 In early experiments,17,31 because of
low IXS intensity and low instrumental resolution, the S(q)
values for water and benzene could not be experimentally
determined at small values of q, where phase effects are
expected. The inaccessible S(q) values can now be derived
because the convergence of small-angle IXS profiles was
confirmed for q ≤ 0.32 au (as shown in Figure 3), and the
accurate OOS or OLF was obtained on an absolute scale (as
shown in Figures 4 and 5). According to eq 4, S(q) at small
values of q [S(q)q≪1] can be obtained using
S(q)q ≪ 1 ≈
q2
2
4π ne
∫0
∞
⎛ −1 ⎞
Im⎜
⎟ dE
⎝ ε( E ) ⎠
(7)
By integrating the OLFs, S(q)q≪1 values of 3.33 × q2 and 16.0 ×
q2 were obtained for liquid water and liquid benzene,
respectively. Because the contributions from core electrons
(C1s and O1s) were not considered in the present measurements, the calculated values according to Thakkar and Smith’s
formula94 were added to provide comparisons with theoretical
calculations. Any error in the estimation of the core
contributions was insignificant because the contributions were
very small (less than 1%) at small values of q.
Parts a and b of Figure 12 compare various theoretically
calculated S(q) values with the S(q)q≪1 values obtained for
liquid water and liquid benzene, respectively. The simplest way
to calculate S(q) for molecules is to add the S(q) values of the
constituent atoms. This framework is often called the
independent-atom model (IAM).17,63 Although the IAM
might appear to be a rough approximation, it can quantitatively
provide S(q) values that reproduce molecular S(q) values
reasonably well (particularly at large values of q) if the electron
correlation in atoms is included by applying the configuration
interaction (CI) method.15,17,31,63,93 The term “atomic
calculation” in Figure 12 indicates the calculated S(q) values
from atomic functions that included CI62 within the framework
of the IAM. It is evident that the calculated S(q) values deviated
from the experimental values for both liquids. The “molecular
calculation (HF)” notation in Figure 12 denotes the calculated
values from Hartree−Fock (HF) molecular wave functions [the
correlation-consistent polarized valence triple-ζ (cc-pVTZ)
basis set for water and the correlation-consistent polarized
valence double-ζ (cc-pVDZ) basis set for benzene].93
Interestingly, although the discrepancies were partially
addressed in water, the differences became large in benzene.
The “molecular calculation (CCSD)” notation in the figure
indicates the S(q) values calculated from molecular wave
functions that included CI in the molecule-based coupledcluster singles and doubles (CCSD) approach.93 It is
immediately apparent that the results of the CCSD calculations
agreed very well with the experimental S(q)q≪1 values in water,
up to an unexpectedly high value of q, such as 0.5 au. By
combining these findings with previous results,17 one can
conclude that the S(q) of water is sensitive to electron
correlation in the molecule but is not sensitive to its
Figure 12. Comparisons of experimental and theoretical S(q) values
for (a) water and (b) benzene at small values of q.
condensation over the entire q range; that is, the phase effects
on S(q) are negligible for water.
In benzene, the CCSD calculation reduced the discrepancy
considerably, but small differences remained. This tendency
was also found at larger values of q; the effects of the higher
excitation configuration and less reliable basis functions
(because of the larger number of electrons in benzene) were
suggested as possible reasons for this tendency.93 The details of
phase effects on benzene’s S(q) are not yet clear, but the fairly
small discrepancy between the experimental S(q)q≪1 values and
the CCSD S(q) values indicates that, if such effects are present,
they are relatively small.
■
CONCLUSIONS
The dielectric and optical functions in the VUV region are
indispensable for studying the interactions between matter and
photons or charged particles, including electrons. For molecular
liquids, only small-angle IXS experiments can currently provide
reliable data over a wide energy range. By applying this
technique to two important molecular liquids, water and
benzene, their dielectric and optical data were accurately
obtained, and several aspects of the phase effects on the OOS,
OLF (plasmon excitation), and S(q) were clarified.
Because of recent progress in third-generation synchrotron
X-ray sources, it is now possible to routinely apply small-angle
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IXS techniques (which are fully described in this paper) to
accurately measure the dielectric and optical functions of
common organic and inorganic solvents at room temperature.
In particular, the extension of the present work to methanol,
acetonitrile, and cyclohexane is expected to be simple, because
these liquids’ IXS data, which are necessary for the multiple
scattering correction, have already been published.31 Moreover,
low-temperature experiments to measure low boiling liquids
(such as ammonia) may not be very difficult. However, because
of the inherently weak intensity of IXS, measuring the smallangle IXS spectra for the following liquids or fluids may remain
somewhat challenging: (1) unstable liquids under X-ray
radiation, (2) low-density liquids and supercritical fluids, (3)
liquids whose measurements need thick cell windows (such as
high-pressure liquids), and (4) liquids where bubbles can easily
form (such as high-temperature liquids). Further efforts to
promote wider application of this small-angle IXS technique are
underway.
■
ASSOCIATED CONTENT
S Supporting Information
*
Details of the multiple-scattering calculations. This material is
available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*H. Hayashi. E-mail: [email protected]. Phone: +81-35981-3665.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
The authors are grateful to Prof. N. Watanabe of the Institute
of Multidisciplinary Research for Advanced Materials, Tohoku
University, for providing digital S(q) data. Both Prof. Emeritus
Y. Hatano of the Tokyo Institute of Technology and Prof. A.
Mozumder of the University of Notre Dame gave us warm
encouragement. The experiments at SPring-8 were conducted
under proposal No. 2014B4252. This study was partially
supported by JSPS KAKENHI Grant Number 26410163.
■
ABBREVIATIONS
CCSD, coupled-cluster singles and doubles; CI, configuration
interaction; EELS, electron energy-loss spectroscopy; HF,
Hartree−Fock; IAM, independent-atom model; IXS, inelastic
X-ray scattering; NFE, nearly free electron; OLF, optical
energy-loss function; OOS, optical oscillator strength distribution; RPA, random-phase approximation; S/N, signal-to-noise;
VUV, vacuum ultraviolet
■
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